calculus eighth edition
James Stewart M c Master University and University of Toronto
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Calculus, Eighth Edition James Stewart
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Contents Preface xi To the Student xxiii Calculators, Computers, and other graphing devices xxiv Diagnostic tests xxvi
A Preview of Calculus 1
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
Four Ways to Represent a Function 10 Mathematical Models: A Catalog of Essential Functions 23 New Functions from Old Functions 36 The Tangent and Velocity Problems 45 The Limit of a Function 50 Calculating Limits Using the Limit Laws 62 The Precise Definition of a Limit 72 Continuity 82 Review 94
Principles of Problem Solving 98
2 95
2.1
Derivatives and Rates of Change 106 Writing Project • Early Methods for Finding Tangents 117 2.2 The Derivative as a Function 117 2.3 Differentiation Formulas 130 Applied Project • Building a Better Roller Coaster 144 2.4 Derivatives of Trigonometric Functions 144 2.5 The Chain Rule 152 Applied Project • Where Should a Pilot Start Descent? 161 2.6 Implicit Differentiation 161 Laboratory Project • Families of Implicit Curves 168
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2.7 2.8 2.9
Rates of Change in the Natural and Social Sciences 169 Related Rates 181 Linear Approximations and Differentials 188 Laboratory Project • Taylor Polynomials 194
Review 195
Problems Plus 200
3
3.1
Maximum and Minimum Values 204 Applied Project • The Calculus of Rainbows 213 3.2 The Mean Value Theorem 215 3.3 How Derivatives Affect the Shape of a Graph 221 3.4 Limits at Infinity; Horizontal Asymptotes 231 3.5 Summary of Curve Sketching 244 3.6 Graphing with Calculus and Calculators 251 3.7 Optimization Problems 258 Applied Project • The Shape of a Can 270 Applied Project • Planes and Birds: Minimizing Energy 271 3.8 Newton’s Method 272 3.9 Antiderivatives 278
Review 285
Problems Plus 289
4
4.1 4.2
Areas and Distances 294 The Definite Integral 306 Discovery Project • Area Functions 319 4.3 The Fundamental Theorem of Calculus 320 4.4 Indefinite Integrals and the Net Change Theorem 330 Writing Project • Newton, Leibniz, and the Invention of Calculus 339 4.5 The Substitution Rule 340
Review 348
Problems Plus 352
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Contents v
5
5.1
Areas Between Curves 356 Applied Project • The Gini Index 364 5.2 Volumes 366 5.3 Volumes by Cylindrical Shells 377 5.4 Work 383 5.5 Average Value of a Function 389 Applied Project • Calculus and Baseball 392
Review 393
Problems Plus 395
6
6.1
Instructors may cover either Sections 6.2–6.4 or Sections 6.2*–6.4*. See the Preface.
Inverse Functions 400
6.2
Exponential Functions and Their Derivatives 408
6.2* The Natural Logarithmic Function 438
6.3
Logarithmic Functions 421
6.3* The Natural Exponential Function 447
6.4
Derivatives of Logarithmic Functions 428
6.4* General Logarithmic and Exponential Functions 455
6.5
Exponential Growth and Decay 466 Applied Project • Controlling Red Blood Cell Loss During Surgery 473
6.6
Inverse Trigonometric Functions 474 Applied Project • Where to Sit at the Movies 483
6.7
6.8
Hyperbolic Functions 484 Indeterminate Forms and l’Hospital’s Rule 491 Writing Project • The Origins of l’Hospital’s Rule 503
Review 503
Problems Plus 508
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7
7.1 7.2 7.3 7.4 7.5 7.6
Integration by Parts 512 Trigonometric Integrals 519 Trigonometric Substitution 526 Integration of Rational Functions by Partial Fractions 533 Strategy for Integration 543 Integration Using Tables and Computer Algebra Systems 548 Discovery Project • Patterns in Integrals 553 7.7 Approximate Integration 554 7.8 Improper Integrals 567
Review 577
Problems Plus 580
8
8.1
Arc Length 584 Discovery Project • Arc Length Contest 590 8.2 Area of a Surface of Revolution 591 Discovery Project • Rotating on a Slant 597 8.3 Applications to Physics and Engineering 598 Discovery Project • Complementary Coffee Cups 608 8.4 Applications to Economics and Biology 609 8.5 Probability 613
Review 621
Problems Plus 623
9
9.1 9.2 9.3
Modeling with Differential Equations 626 Direction Fields and Euler’s Method 631 Separable Equations 639 Applied Project • How Fast Does a Tank Drain? 648 Applied Project • Which Is Faster, Going Up or Coming Down? 649 9.4 Models for Population Growth 650 9.5 Linear Equations 660
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Contents vii
9.6
PredatorPrey Systems 667
Review 674
Problems Plus 677
10
10.1 Curves Defined by Parametric Equations 680 Laboratory Project • Running Circles Around Circles 688 10.2 Calculus with Parametric Curves 689 Laboratory Project • Bézier Curves 697 10.3 Polar Coordinates 698 Laboratory Project • Families of Polar Curves 708 10.4 Areas and Lengths in Polar Coordinates 709 10.5 Conic Sections 714 10.6 Conic Sections in Polar Coordinates 722
Review 729
Problems Plus 732
11
11.1 Sequences 734 Laboratory Project • Logistic Sequences 747 11.2 Series 747 11.3 The Integral Test and Estimates of Sums 759 11.4 The Comparison Tests 767 11.5 Alternating Series 772 11.6 Absolute Convergence and the Ratio and Root Tests 777 11.7 Strategy for Testing Series 784 11.8 Power Series 786 11.9 Representations of Functions as Power Series 792 11.10 Taylor and Maclaurin Series 799 Laboratory Project • An Elusive Limit 813 Writing Project • How Newton Discovered the Binomial Series 813 11.11 Applications of Taylor Polynomials 814 Applied Project • Radiation from the Stars 823
Review 824
Problems Plus 827
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12
12.1 ThreeDimensional Coordinate Systems 832 12.2 Vectors 838 12.3 The Dot Product 847 12.4 The Cross Product 854 Discovery Project • The Geometry of a Tetrahedron 863 12.5 Equations of Lines and Planes 863 Laboratory Project • Putting 3D in Perspective 873 12.6 Cylinders and Quadric Surfaces 874
7et1206un03 04/21/10 MasterID: 01462
Review 881
Problems Plus 884
13
13.1 Vector Functions and Space Curves 888 13.2 Derivatives and Integrals of Vector Functions 895 13.3 Arc Length and Curvature 901 13.4 Motion in Space: Velocity and Acceleration 910 Applied Project • Kepler’s Laws 920
Review 921
Problems Plus 924
14
14.1 Functions of Several Variables 928 14.2 Limits and Continuity 943 14.3 Partial Derivatives 951 14.4 Tangent Planes and Linear Approximations 967 Applied Project • The Speedo LZR Racer 976 14.5 The Chain Rule 977 14.6 Directional Derivatives and the Gradient Vector 986 14.7 Maximum and Minimum Values 999 Applied Project • Designing a Dumpster 1010 Discovery Project • Quadratic Approximations and Critical Points 1010
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14.8 Lagrange Multipliers 1011 Applied Project • Rocket Science 1019 Applied Project • HydroTurbine Optimization 1020
Review 1021
Problems Plus 1025
15
15.1 Double Integrals over Rectangles 1028 15.2 Double Integrals over General Regions 1041 15.3 Double Integrals in Polar Coordinates 1050 15.4 Applications of Double Integrals 1056 15.5 Surface Area 1066 15.6 Triple Integrals 1069 Discovery Project • Volumes of Hyperspheres 1080 15.7 Triple Integrals in Cylindrical Coordinates 1080 Discovery Project • The Intersection of Three Cylinders 1084 15.8 Triple Integrals in Spherical Coordinates 1085 Applied Project • Roller Derby 1092 15.9 Change of Variables in Multiple Integrals 1092
Review 1101
Problems Plus 1105
16
16.1 Vector Fields 1108 16.2 Line Integrals 1115 16.3 The Fundamental Theorem for Line Integrals 1127 16.4 Green’s Theorem 1136 16.5 Curl and Divergence 1143 16.6 Parametric Surfaces and Their Areas 1151 16.7 Surface Integrals 1162 16.8 Stokes’ Theorem 1174 Writing Project • Three Men and Two Theorems 1180
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16.9 The Divergence Theorem 1181 16.10 Summary 1187
Review 1188
Problems Plus 1191
17
17.1 17.2 17.3 17.4
SecondOrder Linear Equations 1194 Nonhomogeneous Linear Equations 1200 Applications of SecondOrder Differential Equations 1208 Series Solutions 1216
Review 1221
Numbers, Inequalities, and Absolute Values A2 Coordinate Geometry and Lines A10 Graphs of SecondDegree Equations A16 Trigonometry A24 Sigma Notation A34 Proofs of Theorems A39 Complex Numbers A48 Answers to OddNumbered Exercises A57
A B C D E F G H
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Preface A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. g e o r g e p o lya
The art of teaching, Mark Van Doren said, is the art of assisting discovery. I have tried to write a book that assists students in discovering calculus—both for its practical power and its surprising beauty. In this edition, as in the first seven editions, I aim to convey to the student a sense of the utility of calculus and develop technical competence, but I also strive to give some appreciation for the intrinsic beauty of the subject. Newton undoubtedly experienced a sense of triumph when he made his great discoveries. I want students to share some of that excitement. The emphasis is on understanding concepts. I think that nearly everybody agrees that this should be the primary goal of calculus instruction. In fact, the impetus for the current calculus reform movement came from the Tulane Conference in 1986, which formulated as their first recommendation: Focus on conceptual understanding. I have tried to implement this goal through the Rule of Three: “Topics should be presented geometrically, numerically, and algebraically.” Visualization, numerical and graphical experimentation, and other approaches have changed how we teach conceptual reasoning in fundamental ways. More recently, the Rule of Three has been expanded to become the Rule of Four by emphasizing the verbal, or descriptive, point of view as well. In writing the eighth edition my premise has been that it is possible to achieve conceptual understanding and still retain the best traditions of traditional calculus. The book contains elements of reform, but within the context of a traditional curriculum.
I have written several other calculus textbooks that might be preferable for some instructors. Most of them also come in single variable and multivariable versions. Calculus: Early Transcendentals, Eighth Edition, is similar to the present textbook except that the exponential, logarithmic, and inverse trigonometric functions are covered in the first semester. ● Essential Calculus, Second Edition, is a much briefer book (840 pages), though it contains almost all of the topics in Calculus, Eighth Edition. The relative brevity is achieved through briefer exposition of some topics and putting some features on the website. ●
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Essential Calculus: Early Transcendentals, Second Edition, resembles Essential Calculus, but the exponential, logarithmic, and inverse trigonometric functions are covered in Chapter 3. Calculus: Concepts and Contexts, Fourth Edition, emphasizes conceptual understanding even more strongly than this book. The coverage of topics is not encyclopedic and the material on transcendental functions and on parametric equations is woven throughout the book instead of being treated in separate chapters.
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Calculus: Early Vectors introduces vectors and vector functions in the first semester and integrates them throughout the book. It is suitable for students taking engineering and physics courses concurrently with calculus.
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Brief Applied Calculus is intended for students in business, the social sciences, and the life sciences.
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Biocalculus: Calculus for the Life Sciences is intended to show students in the life sciences how calculus relates to biology.
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Biocalculus: Calculus, Probability, and Statistics for the Life Sciences contains all the content of Biocalculus: Calculus for the Life Sciences as well as three additional chapters covering probability and statistics.
●
The changes have resulted from talking with my colleagues and students at the University of Toronto and from reading journals, as well as suggestions from users and reviewers. Here are some of the many improvements that I’ve incorporated into this edition: ●
●
●
●
●
The data in examples and exercises have been updated to be more timely. New examples have been added (see Examples 5.1.5, 11.2.5, and 14.3.3, for instance). And the solutions to some of the existing examples have been amplified. Three new projects have been added: The project Planes and Birds: Minimizing Energy (page 271) asks how birds can minimize power and energy by flapping their wings versus gliding. The project Controlling Red Blood Cell Loss During Surgery (page 473) describes the ANH procedure, in which blood is extracted from the patient before an operation and is replaced by saline solution. This dilutes the patient’s blood so that fewer red blood cells are lost during bleeding and the extracted blood is returned to the patient after surgery. In the project The Speedo LZR Racer (page 976) it is explained that this suit reduces drag in the water and, as a result, many swimming records were broken. Students are asked why a small decrease in drag can have a big effect on performance. I have streamlined Chapter 15 (Multiple Integrals) by combining the first two sections so that iterated integrals are treated earlier. More than 20% of the exercises in each chapter are new. Here are some of my favorites: 2.1.61, 2.2.34–36, 3.3.30, 3.3.54, 3.7.39, 3.7.67, 4.1.19–20, 4.2.67–68, 4.4.63, 5.1.51, 6.2.79, 6.7.54, 6.8.90, 8.1.39, 12.5.81, 12.6.29–30, 14.6.65–66. In addition, there are some good new Problems Plus. (See Problems 10–12 on page 201, Problem 10 on page 290, Problems 14–15 on pages 353–54, and Problem 8 on page 1026.)
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Preface xiii
Conceptual Exercises The most important way to foster conceptual understanding is through the problems that we assign. To that end I have devised various types of problems. Some exercise sets begin with requests to explain the meanings of the basic concepts of the section. (See, for instance, the first few exercises in Sections 1.5, 1.8, 11.2, 14.2, and 14.3.) Similarly, all the review sections begin with a Concept Check and a TrueFalse Quiz. Other exercises test conceptual understanding through graphs or tables (see Exercises 2.1.17, 2.2.33–36, 2.2.45–50, 9.1.11–13, 10.1.24–27, 11.10.2, 13.2.1–2, 13.3.33–39, 14.1.1–2, 14.1.32–38, 14.1.41–44, 14.3.3–10, 14.6.1–2, 14.7.3–4, 15.1.6–8, 16.1.11–18, 16.2.17–18, and 16.3.1–2). Another type of exercise uses verbal description to test conceptual understanding (see Exercises 1.8.10, 2.2.64, 3.3.57–58, and 7.8.67). I particularly value problems that combine and compare graphical, numerical, and algebraic approaches (see Exercises 2.7.25, 3.4.33–34, and 9.4.4).
Graded Exercise Sets Each exercise set is carefully graded, progressing from basic conceptual exercises and skilldevelopment problems to more challenging problems involving applications and proofs.
RealWorld Data My assistants and I spent a great deal of time looking in libraries, contacting companies and government agencies, and searching the Internet for interesting realworld data to introduce, motivate, and illustrate the concepts of calculus. As a result, many of the examples and exercises deal with functions defined by such numerical data or graphs. See, for instance, Figure 1 in Section 1.1 (seismograms from the Northridge earthquake), Exercise 2.2.33 (unemployment rates), Exercise 4.1.16 (velocity of the space shuttle Endeavour), and Figure 4 in Section 4.4 (San Francisco power consumption). Functions of two variables are illustrated by a table of values of the windchill index as a function of air temperature and wind speed (Example 14.1.2). Partial derivatives are introduced in Section 14.3 by examining a column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity. This example is pursued further in connection with linear approximations (Example 14.4.3). Directional derivatives are introduced in Section 14.6 by using a temperature contour map to estimate the rate of change of temperature at Reno in the direction of Las Vegas. Double integrals are used to estimate the average snowfall in Colorado on December 20–21, 2006 (Example 15.1.9). Vector fields are introduced in Section 16.1 by depictions of actual velocity vector fields showing San Francisco Bay wind patterns.
Projects One way of involving students and making them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplishment when completed. I have included four kinds of projects: Applied Projects involve applications that are designed to appeal to the imagination of students. The project after Section 9.3 asks whether a ball thrown upward takes longer to reach its maximum height or to fall back to its original height. (The answer might surprise you.) The project after Section 14.8 uses Lagrange multipliers to determine the masses of the three stages of a rocket so as to minimize the total mass while enabling the rocket to reach a desired Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Preface
velocity. Laboratory Projects involve technology; the one following Section 10.2 shows how to use Bézier curves to design shapes that represent letters for a laser printer. Writing Projects ask students to compare presentday methods with those of the founders of calculus—Fermat’s method for finding tangents, for instance. Suggested references are supplied. Discovery Projects anticipate results to be discussed later or encourage discovery through pattern recognition (see the one following Section 7.6). Others explore aspects of geometry: tetrahedra (after Section 12.4), hyperspheres (after Section 15.6), and intersections of three cylinders (after Section 15.7). Additional projects can be found in the Instructor’s Guide (see, for instance, Group Exercise 4.1: Position from Samples).
Problem Solving Students usually have difficulties with problems for which there is no single welldefined procedure for obtaining the answer. I think nobody has improved very much on George Polya’s fourstage problemsolving strategy and, accordingly, I have included a version of his problemsolving principles following Chapter 1. They are applied, both explicitly and implicitly, throughout the book. After the other chapters I have placed sections called Problems Plus, which feature examples of how to tackle challenging calculus problems. In selecting the varied problems for these sections I kept in mind the following advice from David Hilbert: “A mathematical problem should be difficult in order to entice us, yet not inaccessible lest it mock our efforts.” When I put these challenging problems on assignments and tests I grade them in a different way. Here I reward a student significantly for ideas toward a solution and for recognizing which problemsolving principles are relevant.
Dual Treatment of Exponential and Logarithmic Functions There are two possible ways of treating the exponential and logarithmic functions and each method has its passionate advocates. Because one often finds advocates of both approaches teaching the same course, I include full treatments of both methods. In Sections 6.2, 6.3, and 6.4 the exponential function is defined first, followed by the logarithmic function as its inverse. (Students have seen these functions introduced this way since high school.) In the alternative approach, presented in Sections 6.2*, 6.3*, and 6.4*, the logarithm is defined as an integral and the exponential function is its inverse. This latter method is, of course, less intuitive but more elegant. You can use whichever treatment you prefer. If the first approach is taken, then much of Chapter 6 can be covered before Chapters 4 and 5, if desired. To accommodate this choice of presentation there are specially identified problems involving integrals of exponential and logarithmic functions at the end of the appropriate sections of Chapters 4 and 5. This order of presentation allows a fasterpaced course to teach the transcendental functions and the definite integral in the first semester of the course. For instructors who would like to go even further in this direction I have prepared an alternate edition of this book, called Calculus: Early Transcendentals, Eighth Edition, in which the exponential and logarithmic functions are introduced in the first chapter. Their limits and derivatives are found in the second and third chapters at the same time as polynomials and the other elementary functions.
Tools for Enriching Calculus TEC is a companion to the text and is intended to enrich and complement its contents. (It is now accessible in the eBook via CourseMate and Enhanced WebAssign. Selected Visuals and Modules are available at www.stewartcalculus.com.) Developed by Harvey Keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratory Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Preface xv
approach. In sections of the book where technology is particularly appropriate, marginal icons direct students to TEC Modules that provide a laboratory environment in which they can explore the topic in different ways and at different levels. Visuals are animations of figures in text; Modules are more elaborate activities and include exercises. Instructors can choose to become involved at several different levels, ranging from simply encouraging students to use the Visuals and Modules for independent exploration, to assigning specific exercises from those included with each Module, or to creating additional exercises, labs, and projects that make use of the Visuals and Modules. TEC also includes Homework Hints for representative exercises (usually oddnumbered) in every section of the text, indicated by printing the exercise number in red. These hints are usually presented in the form of questions and try to imitate an effective teaching assistant by functioning as a silent tutor. They are constructed so as not to reveal any more of the actual solution than is minimally necessary to make further progress.
Enhanced WebAssign Technology is having an impact on the way homework is assigned to students, particularly in large classes. The use of online homework is growing and its appeal depends on ease of use, grading precision, and reliability. With the Eighth Edition we have been working with the calculus community and WebAssign to develop an online homework system. Up to 70% of the exercises in each section are assignable as online homework, including free response, multiple choice, and multipart formats. The system also includes Active Examples, in which students are guided in stepbystep tutorials through text examples, with links to the textbook and to video solutions.
Website Visit CengageBrain.com or stewartcalculus.com for these additional materials: ●
Homework Hints
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Algebra Review
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Lies My Calculator and Computer Told Me
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History of Mathematics, with links to the better historical websites
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Additional Topics (complete with exercise sets): Fourier Series, Formulas for the Remainder Term in Taylor Series, Rotation of Axes Archived Problems (drill exercises that appeared in previous editions, together with their solutions)
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Challenge Problems (some from the Problems Plus sections from prior editions)
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Links, for particular topics, to outside Web resources
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Selected Visuals and Modules from Tools for Enriching Calculus (TEC)
Diagnostic Tests
The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, Functions, and Trigonometry.
A Preview of Calculus
This is an overview of the subject and includes a list of questions to motivate the study of calculus.
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1 Functions and Limits
From the beginning, multiple representations of functions are stressed: verbal, numerical, visual, and algebraic. A discussion of mathematical models leads to a review of the standard functions from these four points of view. The material on limits is motivated by a prior discussion of the tangent and velocity problems. Limits are treated from descriptive, graphical, numerical, and algebraic points of view. Section 1.7, on the precise epsilondelta defintion of a limit, is an optional section.
2 Derivatives
The material on derivatives is covered in two sections in order to give students more time to get used to the idea of a derivative as a function. The examples and exercises explore the meanings of derivatives in various contexts. Higher derivatives are introduced in Section 2.2.
3 Applications of Differentiation
The basic facts concerning extreme values and shapes of curves are deduced from the Mean Value Theorem. Graphing with technology emphasizes the interaction between calculus and calculators and the analysis of families of curves. Some substantial optimization problems are provided, including an explanation of why you need to raise your head 42° to see the top of a rainbow.
4 Integrals
The area problem and the distance problem serve to motivate the definite integral, with sigma notation introduced as needed. (Full coverage of sigma notation is provided in Appendix E.) Emphasis is placed on explaining the meanings of integrals in various contexts and on estimating their values from graphs and tables.
5 Applications of Integration
Here I present the applications of integration—area, volume, work, average value—that can reasonably be done without specialized techniques of integration. General methods are emphasized. The goal is for students to be able to divide a quantity into small pieces, estimate with Riemann sums, and recognize the limit as an integral.
6 Inverse Functions:
As discussed more fully on page xiv, only one of the two treatments of these functions need be covered. Exponential growth and decay are covered in this chapter.
7 Techniques of Integration
All the standard methods are covered but, of course, the real challenge is to be able to recognize which technique is best used in a given situation. Accordingly, in Section 7.5, I present a strategy for integration. The use of computer algebra systems is discussed in Section 7.6.
8 Further Applications of Integration
Here are the applications of integration—arc length and surface area—for which it is useful to have available all the techniques of integration, as well as applications to biology, economics, and physics (hydrostatic force and centers of mass). I have also included a section on probability. There are more applications here than can realistically be covered in a given course. Instructors should select applications suitable for their students and for which they themselves have enthusiasm.
9 Differential Equations
Modeling is the theme that unifies this introductory treatment of differential equations. Direction fields and Euler’s method are studied before separable and linear equations are solved explicitly, so that qualitative, numerical, and analytic approaches are given equal
Exponential, Logarithmic, and Inverse Trigonometric Functions
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Preface xvii
consideration. These methods are applied to the exponential, logistic, and other models for population growth. The first four or five sections of this chapter serve as a good introduction to firstorder differential equations. An optional final section uses predatorprey models to illustrate systems of differential equations. 10 Parametric Equations and Polar Coordinates
This chapter introduces parametric and polar curves and applies the methods of calculus to them. Parametric curves are well suited to laboratory projects; the two presented here involve families of curves and Bézier curves. A brief treatment of conic sections in polar coordinates prepares the way for Kepler’s Laws in Chapter 13.
11 Infinite Sequences and Series
The convergence tests have intuitive justifications (see page 759) as well as formal proofs. Numerical estimates of sums of series are based on which test was used to prove convergence. The emphasis is on Taylor series and polynomials and their applications to physics. Error estimates include those from graphing devices.
12 Vectors and the Geometry of Space
The material on threedimensional analytic geometry and vectors is divided into two chapters. Chapter 12 deals with vectors, the dot and cross products, lines, planes, and surfaces.
13 Vector Functions
This chapter covers vectorvalued functions, their derivatives and integrals, the length and curvature of space curves, and velocity and acceleration along space curves, culminating in Kepler’s laws.
14 Partial Derivatives
Functions of two or more variables are studied from verbal, numerical, visual, and algebraic points of view. In particular, I introduce partial derivatives by looking at a specific column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity.
15 Multiple Integrals
Contour maps and the Midpoint Rule are used to estimate the average snowfall and average temperature in given regions. Double and triple integrals are used to compute probabilities, surface areas, and (in projects) volumes of hyperspheres and volumes of intersections of three cylinders. Cylindrical and spherical coordinates are introduced in the context of evaluating triple integrals.
16 Vector Calculus
Vector fields are introduced through pictures of velocity fields showing San Francisco Bay wind patterns. The similarities among the Fundamental Theorem for line integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized.
17 SecondOrder Differential Equations
Since firstorder differential equations are covered in Chapter 9, this final chapter deals with secondorder linear differential equations, their application to vibrating springs and electric circuits, and series solutions.
Calculus, Eighth Edition, is supported by a complete set of ancillaries developed under my direction. Each piece has been designed to enhance student understanding and to facilitate creative instruction. The tables on pages xxi–xxii describe each of these ancillaries.
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xviii
Preface
The preparation of this and previous editions has involved much time spent reading the reasoned (but sometimes contradictory) advice from a large number of astute reviewers. I greatly appreciate the time they spent to understand my motivation for the approach taken. I have learned something from each of them.
Eighth Edition Reviewers Jay Abramson, Arizona State University Adam Bowers, University of California San Diego Neena Chopra, The Pennsylvania State University Edward Dobson, Mississippi State University Isaac Goldbring, University of Illinois at Chicago Lea Jenkins, Clemson University Rebecca Wahl, Butler University
Technology Reviewers Maria Andersen, Muskegon Community College Eric Aurand, Eastfield College Joy Becker, University of Wisconsin–Stout Przemyslaw Bogacki, Old Dominion University Amy Elizabeth Bowman, University of Alabama in Huntsville Monica Brown, University of Missouri–St. Louis Roxanne Byrne, University of Colorado at Denver and Health Sciences Center Teri Christiansen, University of Missouri–Columbia Bobby Dale Daniel, Lamar University Jennifer Daniel, Lamar University Andras Domokos, California State University, Sacramento Timothy Flaherty, Carnegie Mellon University Lee Gibson, University of Louisville Jane Golden, Hillsborough Community College Semion Gutman, University of Oklahoma Diane Hoffoss, University of San Diego Lorraine Hughes, Mississippi State University Jay Jahangiri, Kent State University John Jernigan, Community College of Philadelphia
Brian Karasek, South Mountain Community College Jason Kozinski, University of Florida Carole Krueger, The University of Texas at Arlington Ken Kubota, University of Kentucky John Mitchell, Clark College Donald Paul, Tulsa Community College Chad Pierson, University of Minnesota, Duluth Lanita Presson, University of Alabama in Huntsville Karin Reinhold, State University of New York at Albany Thomas Riedel, University of Louisville Christopher Schroeder, Morehead State University Angela Sharp, University of Minnesota, Duluth Patricia Shaw, Mississippi State University Carl Spitznagel, John Carroll University Mohammad Tabanjeh, Virginia State University Capt. Koichi Takagi, United States Naval Academy Lorna TenEyck, Chemeketa Community College Roger Werbylo, Pima Community College David Williams, Clayton State University Zhuan Ye, Northern Illinois University
Previous Edition Reviewers B. D. Aggarwala, University of Calgary John Alberghini, Manchester Community College Michael Albert, CarnegieMellon University Daniel Anderson, University of Iowa Amy Austin, Texas A&M University Donna J. Bailey, Northeast Missouri State University Wayne Barber, Chemeketa Community College Marilyn Belkin, Villanova University Neil Berger, University of Illinois, Chicago David Berman, University of New Orleans Anthony J. Bevelacqua, University of North Dakota Richard Biggs, University of Western Ontario
Robert Blumenthal, Oglethorpe University Martina Bode, Northwestern University Barbara Bohannon, Hofstra University Jay Bourland, Colorado State University Philip L. Bowers, Florida State University Amy Elizabeth Bowman, University of Alabama in Huntsville Stephen W. Brady, Wichita State University Michael Breen, Tennessee Technological University Robert N. Bryan, University of Western Ontario David Buchthal, University of Akron Jenna Carpenter, Louisiana Tech University Jorge Cassio, MiamiDade Community College
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Preface xix
Jack Ceder, University of California, Santa Barbara Scott Chapman, Trinity University ZhenQing Chen, University of Washington—Seattle James Choike, Oklahoma State University Barbara Cortzen, DePaul University Carl Cowen, Purdue University Philip S. Crooke, Vanderbilt University Charles N. Curtis, Missouri Southern State College Daniel Cyphert, Armstrong State College Robert Dahlin M. Hilary Davies, University of Alaska Anchorage Gregory J. Davis, University of Wisconsin–Green Bay Elias Deeba, University of Houston–Downtown Daniel DiMaria, Suffolk Community College Seymour Ditor, University of Western Ontario Greg Dresden, Washington and Lee University Daniel Drucker, Wayne State University Kenn Dunn, Dalhousie University Dennis Dunninger, Michigan State University Bruce Edwards, University of Florida David Ellis, San Francisco State University John Ellison, Grove City College Martin Erickson, Truman State University Garret Etgen, University of Houston Theodore G. Faticoni, Fordham University Laurene V. Fausett, Georgia Southern University Norman Feldman, Sonoma State University Le Baron O. Ferguson, University of California—Riverside Newman Fisher, San Francisco State University José D. Flores, The University of South Dakota William Francis, Michigan Technological University James T. Franklin, Valencia Community College, East Stanley Friedlander, Bronx Community College Patrick Gallagher, Columbia University–New York Paul Garrett, University of Minnesota–Minneapolis Frederick Gass, Miami University of Ohio Bruce Gilligan, University of Regina Matthias K. Gobbert, University of Maryland, Baltimore County Gerald Goff, Oklahoma State University Stuart Goldenberg, California Polytechnic State University John A. Graham, Buckingham Browne & Nichols School Richard Grassl, University of New Mexico Michael Gregory, University of North Dakota Charles Groetsch, University of Cincinnati Paul Triantafilos Hadavas, Armstrong Atlantic State University Salim M. Haïdar, Grand Valley State University D. W. Hall, Michigan State University Robert L. Hall, University of Wisconsin–Milwaukee Howard B. Hamilton, California State University, Sacramento Darel Hardy, Colorado State University Shari Harris, John Wood Community College Gary W. Harrison, College of Charleston Melvin Hausner, New York University/Courant Institute Curtis Herink, Mercer University Russell Herman, University of North Carolina at Wilmington Allen Hesse, Rochester Community College
Randall R. Holmes, Auburn University James F. Hurley, University of Connecticut Amer Iqbal, University of Washington—Seattle Matthew A. Isom, Arizona State University Gerald Janusz, University of Illinois at UrbanaChampaign John H. Jenkins, EmbryRiddle Aeronautical University, Prescott Campus Clement Jeske, University of Wisconsin, Platteville Carl Jockusch, University of Illinois at UrbanaChampaign Jan E. H. Johansson, University of Vermont Jerry Johnson, Oklahoma State University Zsuzsanna M. Kadas, St. Michael’s College Nets Katz, Indiana University Bloomington Matt Kaufman Matthias Kawski, Arizona State University Frederick W. Keene, Pasadena City College Robert L. Kelley, University of Miami Akhtar Khan, Rochester Institute of Technology Marianne Korten, Kansas State University Virgil Kowalik, Texas A&I University Kevin Kreider, University of Akron Leonard Krop, DePaul University Mark Krusemeyer, Carleton College John C. Lawlor, University of Vermont Christopher C. Leary, State University of New York at Geneseo David Leeming, University of Victoria Sam Lesseig, Northeast Missouri State University Phil Locke, University of Maine Joyce Longman, Villanova University Joan McCarter, Arizona State University Phil McCartney, Northern Kentucky University Igor Malyshev, San Jose State University Larry Mansfield, Queens College Mary Martin, Colgate University Nathaniel F. G. Martin, University of Virginia Gerald Y. Matsumoto, American River College James McKinney, California State Polytechnic University, Pomona Tom Metzger, University of Pittsburgh Richard Millspaugh, University of North Dakota Lon H. Mitchell, Virginia Commonwealth University Michael Montaño, Riverside Community College Teri Jo Murphy, University of Oklahoma Martin Nakashima, California State Polytechnic University, Pomona Ho Kuen Ng, San Jose State University Richard Nowakowski, Dalhousie University Hussain S. Nur, California State University, Fresno Norma OrtizRobinson, Virginia Commonwealth University Wayne N. Palmer, Utica College Vincent Panico, University of the Pacific F. J. Papp, University of Michigan–Dearborn Mike Penna, Indiana University–Purdue University Indianapolis Mark Pinsky, Northwestern University Lothar Redlin, The Pennsylvania State University Joel W. Robbin, University of Wisconsin–Madison Lila Roberts, Georgia College and State University
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xx
Preface
E. Arthur Robinson, Jr., The George Washington University Richard Rockwell, Pacific Union College Rob Root, Lafayette College Richard Ruedemann, Arizona State University David Ryeburn, Simon Fraser University Richard St. Andre, Central Michigan University Ricardo Salinas, San Antonio College Robert Schmidt, South Dakota State University Eric Schreiner, Western Michigan University Mihr J. Shah, Kent State University–Trumbull Qin Sheng, Baylor University Theodore Shifrin, University of Georgia Wayne Skrapek, University of Saskatchewan Larry Small, Los Angeles Pierce College Teresa Morgan Smith, Blinn College William Smith, University of North Carolina Donald W. Solomon, University of Wisconsin–Milwaukee Edward Spitznagel, Washington University Joseph Stampfli, Indiana University Kristin Stoley, Blinn College
M. B. Tavakoli, Chaffey College Magdalena Toda, Texas Tech University Ruth Trygstad, Salt Lake Community College Paul Xavier Uhlig, St. Mary’s University, San Antonio Stan Ver Nooy, University of Oregon Andrei Verona, California State University–Los Angeles Klaus Volpert, Villanova University Russell C. Walker, Carnegie Mellon University William L. Walton, McCallie School Peiyong Wang, Wayne State University Jack Weiner, University of Guelph Alan Weinstein, University of California, Berkeley Theodore W. Wilcox, Rochester Institute of Technology Steven Willard, University of Alberta Robert Wilson, University of Wisconsin–Madison Jerome Wolbert, University of Michigan–Ann Arbor Dennis H. Wortman, University of Massachusetts, Boston Mary Wright, Southern Illinois University–Carbondale Paul M. Wright, Austin Community College Xian Wu, University of South Carolina
In addition, I would like to thank R. B. Burckel, Bruce Colletti, David Behrman, John Dersch, Gove Effinger, Bill Emerson, Dan Kalman, Quyan Khan, Alfonso GraciaSaz, Allan MacIsaac, Tami Martin, Monica Nitsche, Lamia Raffo, Norton Starr, and Jim Trefzger for their suggestions; Al Shenk and Dennis Zill for permission to use exercises from their calculus texts; COMAP for permission to use project material; George Bergman, David Bleecker, Dan Clegg, Victor Kaftal, Anthony Lam, Jamie Lawson, Ira Rosenholtz, Paul Sally, Lowell Smylie, and Larry Wallen for ideas for exercises; Dan Drucker for the roller derby project; Thomas Banchoff, Tom Farmer, Fred Gass, John Ramsay, Larry Riddle, Philip Straffin, and Klaus Volpert for ideas for projects; Dan Anderson, Dan Clegg, Jeff Cole, Dan Drucker, and Barbara Frank for solving the new exercises and suggesting ways to improve them; Marv Riedesel and Mary Johnson for accuracy in proofreading; Andy BulmanFleming, Lothar Redlin, Gina Sanders, and Saleem Watson for additional proofreading; and Jeff Cole and Dan Clegg for their careful preparation and proofreading of the answer manuscript. In addition, I thank those who have contributed to past editions: Ed Barbeau, Jordan Bell, George Bergman, Fred Brauer, Andy BulmanFleming, Bob Burton, David Cusick, Tom DiCiccio, Garret Etgen, Chris Fisher, Leon Gerber, Stuart Goldenberg, Arnold Good, Gene Hecht, Harvey Keynes, E. L. Koh, Zdislav Kovarik, Kevin Kreider, Emile LeBlanc, David Leep, Gerald Leibowitz, Larry Peterson, Mary Pugh, Lothar Redlin, Carl Riehm, John Ringland, Peter Rosenthal, Dusty Sabo, Doug Shaw, Dan Silver, Simon Smith, Norton Starr, Saleem Watson, Alan Weinstein, and Gail Wolkowicz. I also thank Kathi Townes, Stephanie Kuhns, Kristina Elliott, and Kira Abdallah of TECHarts for their production services and the following Cengage Learning staff: Cheryll Linthicum, content project manager; Stacy Green, senior content developer; Samantha Lugtu, associate content developer; Stephanie Kreuz, product assistant; Lynh Pham, media developer; Ryan Ahern, marketing manager; and Vernon Boes, art director. They have all done an outstanding job. I have been very fortunate to have worked with some of the best mathematics editors in the business over the past three decades: Ron Munro, Harry Campbell, Craig Barth, Jeremy Hayhurst, Gary Ostedt, Bob Pirtle, Richard Stratton, Liz Covello, and now Neha Taleja. All of them have contributed greatly to the success of this book. james stewart Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Instructor’s Guide by Douglas Shaw ISBN 9781305271784 Each section of the text is discussed from several viewpoints. The Instructor’s Guide contains suggested time to allot, points to stress, text discussion topics, core materials for lecture, workshop/discussion suggestions, group work exercises in a form suitable for handout, and suggested homework assignments. Complete Solutions Manual Single Variable By Daniel Anderson, Jeffery A. Cole, and Daniel Drucker ISBN 9781305276109 Multivariable By Dan Clegg and Barbara Frank ISBN 9781305276116 Includes workedout solutions to all exercises in the text. Printed Test Bank By William Steven Harmon ISBN 9781305271807 Contains textspecific multiplechoice and free response test items. Cengage Learning Testing Powered by Cognero (login.cengage.com) This flexible online system allows you to author, edit, and manage test bank content from multiple Cengage Learning solutions; create multiple test versions in an instant; and deliver tests from your LMS, your classroom, or wherever you want.
TEC TOOLS FOR ENRICHING™ CALCULUS By James Stewart, Harvey Keynes, Dan Clegg, and developer Hubert Hohn Tools for Enriching Calculus (TEC) functions as both a powerful tool for instructors and as a tutorial environment in which students can explore and review selected topics. The Flash simulation modules in TEC include instructions, written and audio explanations of the concepts, and exercises. TEC is accessible in the eBook via CourseMate and Enhanced WebAssign. Selected Visuals and Modules are available at www.stewartcalculus.com. Enhanced WebAssign® www.webassign.net Printed Access Code: ISBN 9781285858265 Instant Access Code ISBN: 9781285858258 Exclusively from Cengage Learning, Enhanced WebAssign offers an extensive online program for Stewart’s Calculus to encourage the practice that is so critical for concept mastery. The meticulously crafted pedagogy and exercises in our proven texts become even more effective in Enhanced WebAssign, supplemented by multimedia tutorial support and immediate feedback as students complete their assignments. Key features include: n T housands of homework problems that match your textbook’s endofsection exercises Opportunities for students to review prerequisite skills and content both at the start of the course and at the beginning of each section
n
Read It eBook pages, Watch It videos, Master It tutorials, and Chat About It links
n
A customizable Cengage YouBook with highlighting, notetaking, and search features, as well as links to multimedia resources
n
Personal Study Plans (based on diagnostic quizzing) that identify chapter topics that students will need to master
n
A WebAssign Answer Evaluator that recognizes and accepts equivalent mathematical responses in the same way an instructor grades
n
Stewart Website www.stewartcalculus.com Contents: Homework Hints n Algebra Review n Additional Topics n Drill exercises n Challenge Problems n Web Links n History of Mathematics n Tools for Enriching Calculus (TEC)
■ Electronic items ■ Printed items
A Show My Work feature that gives instructors the option of seeing students’ detailed solutions
n
Visualizing Calculus Animations, Lecture Videos, and more
n
(Table continues on page xxii)
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Cengage Customizable YouBook YouBook is an eBook that is both interactive and customizable. Containing all the content from Stewart’s Calculus, YouBook features a text edit tool that allows instructors to modify the textbook narrative as needed. With YouBook, instructors can quickly reorder entire sections and chapters or hide any content they don’t teach to create an eBook that perfectly matches their syllabus. Instructors can further customize the text by adding instructorcreated or YouTube video links. Additional media assets include animated figures, video clips, highlighting and notetaking features, and more. YouBook is available within Enhanced WebAssign. CourseMate CourseMate is a perfect selfstudy tool for students, and requires no set up from instructors. CourseMate brings course concepts to life with interactive learning, study, and exam preparation tools that support the printed textbook. CourseMate for Stewart’s Calculus includes an interactive eBook, Tools for Enriching Calculus, videos, quizzes, flashcards, and more. For instructors, CourseMate includes Engagement Tracker, a firstofitskind tool that monitors student engagement. CengageBrain.com To access additional course materials, please visit www.cengagebrain.com. At the CengageBrain.com home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. This will take you to the product page where these resources can be found.
Student Solutions Manual Single Variable By Daniel Anderson, Jeffery A. Cole, and Daniel Drucker ISBN 9781305271814 Multivariable By Dan Clegg and Barbara Frank ISBN 9781305271821 Provides completely workedout solutions to all oddnumbered exercises in the text, giving students a chance to
check their answer and ensure they took the correct steps to arrive at the answer. The Student Solutions Manual can be ordered or accessed online as an eBook at www.cengagebrain.com by searching the ISBN. Study Guide Single Variable By Richard St. Andre ISBN 9781305279131 Multivariable By Richard St. Andre ISBN 9781305271845 For each section of the text, the Study Guide provides students with a brief introduction, a short list of concepts to master, and summary and focus questions with explained answers. The Study Guide also contains selftests with examstyle questions. The Study Guide can be ordered or accessed online as an eBook at www.cengagebrain.com by searching the ISBN. A Companion to Calculus By Dennis Ebersole, Doris Schattschneider, Alicia Sevilla, and Kay Somers ISBN 9780495011248 Written to improve algebra and problemsolving skills of students taking a calculus course, every chapter in this companion is keyed to a calculus topic, providing conceptual background and specific algebra techniques needed to understand and solve calculus problems related to that topic. It is designed for calculus courses that integrate the review of precalculus concepts or for individual use. Order a copy of the text or access the eBook online at www.cengagebrain.com by searching the ISBN. Linear Algebra for Calculus by Konrad J. Heuvers, William P. Francis, John H. Kuisti, Deborah F. Lockhart, Daniel S. Moak, and Gene M. Ortner ISBN 9780534252489 This comprehensive book, designed to supplement the calculus course, provides an introduction to and review of the basic ideas of linear algebra. Order a copy of the text or access the eBook online at www.cengagebrain.com by searching the ISBN.
■ Electronic items ■ Printed items
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To the Student Reading a calculus textbook is different from reading a newspaper or a novel, or even a physics book. Don’t be discouraged if you have to read a passage more than once in order to understand it. You should have pencil and paper and calculator at hand to sketch a diagram or make a calculation. Some students start by trying their homework problems and read the text only if they get stuck on an exercise. I suggest that a far better plan is to read and understand a section of the text before attempting the exercises. In particular, you should look at the definitions to see the exact meanings of the terms. And before you read each example, I suggest that you cover up the solution and try solving the problem yourself. You’ll get a lot more from looking at the solution if you do so. Part of the aim of this course is to train you to think logically. Learn to write the solutions of the exercises in a connected, stepbystep fashion with explanatory sentences— not just a string of disconnected equations or formulas. The answers to the oddnumbered exercises appear at the back of the book, in Appendix H. Some exercises ask for a verbal explanation or interpretation or description. In such cases there is no single correct way of expressing the answer, so don’t worry that you haven’t found the definitive answer. In addition, there are often several different forms in which to express a numerical or algebraic answer, so if your answer differs from mine, don’t immediately assume you’re wrong. For example, if the answer given in the back of the book is s2 2 1 and you obtain 1y (1 1 s2 ), then you’re right and rationalizing the denominator will show that the answers are equivalent. The icon ; indicates an exercise that definitely requires the use of either a graphing calculator or a computer with graphing software. But that doesn’t mean that graphing devices can’t be used to check your work on the other exercises as well. The symbol CAS is reserved for problems in
which the full resources of a computer algebra system (like Maple, Mathematica, or the TI89) are required. You will also encounter the symbol , which warns you against committing an error. I have placed this symbol in the margin in situations where I have observed that a large proportion of my students tend to make the same mistake. Tools for Enriching Calculus, which is a companion to this text, is referred to by means of the symbol TEC and can be accessed in the eBook via Enhanced WebAssign and CourseMate (selected Visuals and Modules are available at www.stewartcalculus.com). It directs you to modules in which you can explore aspects of calculus for which the computer is particularly useful. You will notice that some exercise numbers are printed in red: 5. This indicates that Homework Hints are available for the exercise. These hints can be found on stewartcalculus.com as well as Enhanced WebAssign and CourseMate. The homework hints ask you questions that allow you to make progress toward a solution without actually giving you the answer. You need to pursue each hint in an active manner with pencil and paper to work out the details. If a particular hint doesn’t enable you to solve the problem, you can click to reveal the next hint. I recommend that you keep this book for reference purposes after you finish the course. Because you will likely forget some of the specific details of calculus, the book will serve as a useful reminder when you need to use calculus in subsequent courses. And, because this book contains more material than can be covered in any one course, it can also serve as a valuable resource for a working scientist or engineer. Calculus is an exciting subject, justly considered to be one of the greatest achievements of the human intellect. I hope you will discover that it is not only useful but also intrinsically beautiful. james stewart
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Calculators, Computers, and Other Graphing Devices
xxiv
© Dan Clegg
You can also use computer software such as Graphing Calculator by Pacific Tech (www.pacifict.com) to perform many of these functions, as well as apps for phones and tablets, like Quick Graph (Colombiamug) or MathStudio (Pomegranate Apps). Similar functionality is available using a web interface at WolframAlpha.com.
© Dan Clegg
© Dan Clegg
Advances in technology continue to bring a wider variety of tools for doing mathematics. Handheld calculators are becoming more powerful, as are software programs and Internet resources. In addition, many mathematical applications have been released for smartphones and tablets such as the iPad. Some exercises in this text are marked with a graphing icon ; , which indicates that the use of some technology is required. Often this means that we intend for a graphing device to be used in drawing the graph of a function or equation. You might also need technology to find the zeros of a graph or the points of intersection of two graphs. In some cases we will use a calculating device to solve an equation or evaluate a definite integral numerically. Many scientific and graphing calculators have these features built in, such as the Texas Instruments TI84 or TINspire CX. Similar calculators are made by Hewlett Packard, Casio, and Sharp.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
The CAS icon is reserved for problems in which the full resources of a computer algebra system (CAS) are required. A CAS is capable of doing mathematics (like solving equations, computing derivatives or integrals) symbolically rather than just numerically. Examples of wellestablished computer algebra systems are the computer software packages Maple and Mathematica. The WolframAlpha website uses the Mathematica engine to provide CAS functionality via the Web. Many handheld graphing calculators have CAS capabilities, such as the TI89 and TINspire CX CAS from Texas Instruments. Some tablet and smartphone apps also provide these capabilities, such as the previously mentioned MathStudio.
© Dan Clegg
© Dan Clegg
© Dan Clegg
In general, when we use the term “calculator” in this book, we mean the use of any of the resources we have mentioned.
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Diagnostic Tests Success in calculus depends to a large extent on knowledge of the mathematics that precedes calculus: algebra, analytic geometry, functions, and trigonometry. The following tests are intended to diagnose weaknesses that you might have in these areas. After taking each test you can check your answers against the given answers and, if necessary, refresh your skills by referring to the review materials that are provided.
A 1. Evaluate each expression without using a calculator. (a) s23d4 (b) 234 (c) 324
SD
22
5 23 2 (d) (e) (f) 16 23y4 5 21 3 2. Simplify each expression. Write your answer without negative exponents.
(a) s200 2 s32
s3a 3b 3 ds4ab 2 d 2 (b)
S
D
22
3x 3y2 y 3 (c) x 2 y21y2 3. Expand and simplify. sx 1 3ds4x 2 5d (a) 3sx 1 6d 1 4s2x 2 5d (b) (c) ssa 1 sb dssa 2 sb d (d) s2x 1 3d2 (e) sx 1 2d3 4. Factor each expression. (a) 4x 2 2 25 (b) 2x 2 1 5x 2 12 3 2 (c) x 2 3x 2 4x 1 12 (d) x 4 1 27x 3y2 1y2 21y2 (e) 3x 2 9x 1 6x (f) x 3 y 2 4xy 5. Simplify the rational expression. x 2 1 3x 1 2 2x 2 2 x 2 1 x13 (b) ? 2 x 2x22 x2 2 9 2x 1 1 y x 2 2 x x11 x y (c) 2 2 (d) x 24 x12 1 1 2 y x
(a)
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Diagnostic Tests
xxvii
6. Rationalize the expression and simplify. s10 s4 1 h 2 2 (a) (b) h s5 2 2 7. Rewrite by completing the square. (a) x 2 1 x 1 1 (b) 2x 2 2 12x 1 11 8. Solve the equation. (Find only the real solutions.) 2x 2x 2 1 (a) x 1 5 − 14 2 12 x (b) − x11 x (c) x 2 2 x 2 12 − 0 (d) 2x 2 1 4x 1 1 − 0


(e) x 4 2 3x 2 1 2 − 0 (f) 3 x 2 4 − 10 (g) 2xs4 2 xd21y2 2 3 s4 2 x − 0
9. Solve each inequality. Write your answer using interval notation. (a) 24 , 5 2 3x < 17 (b) x 2 , 2x 1 8 (c) xsx 2 1dsx 1 2d . 0 (d) x24 ,3 2x 2 3 (e) x2 2 1 (e) x 2 1 y 2 , 4 (f) 9x 2 1 16y 2 − 144
answers to diagnostic test b: analytic geometry 1. (a) y − 23x 1 1 (b) y − 25 (c) x −
2 (d) y − 12 x 2 6
5. (a)
y
(b)
3
2. sx 1 1d2 1 s y 2 4d2 − 52
x
_1
4. (a) 234 (b) 4x 1 3y 1 16 − 0; xintercept 24, yintercept 2 16 3 (c) s21, 24d (d) 20 (e) 3x 2 4y − 13 (f) sx 1 1d2 1 s y 1 4d2 − 100
(d)
_4
1
1 4x
0
(e)
y 2
_1
y
0
y=1 2 x 2
x
_2
y
0
(c)
2
0
3. Center s3, 25d, radius 5
y
1
x
(f ) ≈+¥=4
0
y=≈1
2
x
y 3
0
4 x
If you had difficulty with these problems, you may wish to consult 6etdtba05af 5.20.06 the review of analytic geometry in Appendixes B and C.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Diagnostic Tests
xxix
C y 1. The graph of a function f is given at the left. (a) State the value of f s21d. (b) Estimate the value of f s2d. (c) For what values of x is f sxd − 2? 1 (d) Estimate the values of x such that f sxd − 0. 0 x 1 (e) State the domain and range of f.
2. If f sxd − x 3, evaluate the difference quotient
f s2 1 hd 2 f s2d and simplify your answer. h
3. Find the domain of the function. Figure For Problem 1
3 2x 1 1 x s (a) f sxd − 2 (b) tsxd − 2 (c) hsxd − s4 2 x 1 sx 2 2 1 x 1x22 x 11
4. How are graphs of the functions obtained from the graph of f ? (a) y − 2f sxd (b) y − 2 f sxd 2 1 (c) y − f sx 2 3d 1 2 5. Without using a calculator, make a rough sketch of the graph. (a) y − x 3 (b) y − sx 1 1d3 (c) y − sx 2 2d3 1 3 2 (d) y − 4 2 x (e) y − sx (f) y − 2 sx (g) y − 22 x (h) y − 1 1 x21
H
1 2 x 2 if x < 0 6. Let f sxd − 2x 1 1 if x . 0
(a) Evaluate f s22d and f s1d.
(b) Sketch the graph of f.
7. If f sxd − x 2 1 2x 2 1 and tsxd − 2x 2 3, find each of the following functions. (a) f 8 t (b) t 8 f (c) t8t8t
answers to diagnostic test C: functions 1. (a) 22 (b) 2.8 (c) 23, 1 (d) 22.5, 0.3 (e) f23, 3g, f22, 3g
5. (a)
0
4. (a) Reflect about the xaxis (b) Stretch vertically by a factor of 2, then shift 1 unit downward (c) Shift 3 units to the right and 2 units upward
(d)
(g)
1
x
_1
(e)
2
x
(2, 3) x
0
1
x
1
x
x
0
(f)
y
0
(h)
y
y
0
1
y 1
0 _1
y
1
y 4
0
(c)
y
1
2. 12 1 6h 1 h 2 3. (a) s2`, 22d ø s22, 1d ø s1, `d (b) s2`, `d (c) s2`, 21g ø f1, 4g
(b)
y
1
x
0
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
x
xxx
Diagnostic Tests
6. (a) 23, 3 (b)
7. (a) s f 8 tdsxd − 4x 2 2 8x 1 2
y
(b) s t 8 f dsxd − 2x 2 1 4x 2 5
1 _1
0
x
(c) s t 8 t 8 tdsxd − 8x 2 21
If you had difficulty with these problems, you should look at sections 1.1–1.3 of this book. 4c3DTCax06b 10/30/08
D 1. Convert from degrees to radians. (a) 3008 (b) 2188 2. Convert from radians to degrees. (a) 5y6 (b) 2 3. Find the length of an arc of a circle with radius 12 cm if the arc subtends a central angle of 308. 4. Find the exact values. (a) tansy3d (b) sins7y6d (c) secs5y3d
5. Express the lengths a and b in the figure in terms of . 24 a 6. If sin x − 13 and sec y − 54, where x and y lie between 0 and y2, evaluate sinsx 1 yd. ¨ 7. Prove the identities. b 2 tan x (a) tan sin 1 cos − sec (b) 2 − sin 2x 1 1 tan x Figure For Problem 5
8. Find all values of x such that sin 2x − sin x and 0 < x < 2. 9. Sketch the graph of the function y − 1 1 sin 2x without using a calculator.
answers to diagnostic test D: trigonometry 1. (a) 5y3 (b) 2y10 2. (a) 1508 (b) 3608y < 114.68 3. 2 cm
1 6. 15 s4 1 6 s2 d
8. 0, y3, , 5y3, 2 y 2
9.
4. (a) s3 (b) 221 (c) 2 5. (a) 24 sin (b) 24 cos
_π
0
π
x
4c3DTDax09
If you had difficulty with these problems, you should look at Appendix D of this book. 10/30/08
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
A Preview of Calculus By the time you finish this course, you will be able to calculate the length of the curve used to design the Gateway Arch in St. Louis, determine where a pilot should start descent for a smooth landing, compute the force on a baseball bat when it strikes the ball, and measure the amount of light sensed by the human eye as the pupil changes size.
calculus is fundamentally different from the mathematics that you have studied previously: calculus is less static and more dynamic. It is concerned with change and motion; it deals with quantities that approach other quantities. For that reason it may be useful to have an overview of the subject before beginning its intensive study. Here we give a glimpse of some of the main ideas of calculus by showing how the concept of a limit arises when we attempt to solve a variety of problems.
1 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
2
a preview of calculus
The Area Problem
A¡
The origins of calculus go back at least 2500 years to the ancient Greeks, who found areas using the “method of exhaustion.” They knew how to find the area A of any polygon by dividing it into triangles as in Figure 1 and adding the areas of these triangles. It is a much more difficult problem to find the area of a curved figure. The Greek method of exhaustion was to inscribe polygons in the figure and circumscribe polygons about the figure and then let the number of sides of the polygons increase. Figure 2 illustrates this process for the special case of a circle with inscribed regular polygons.
A∞
A™
A¢
A£
A=A¡+A™+A£+A¢+A∞
FIGURE 1
A£
A¢
A∞
Aß
A¶
A¡™
FIGURE 2
Let An be the area of the inscribed polygon with n sides. As n increases, it appears that An becomes closer and closer to the area of the circle. We say that the area of the circle is the limit of the areas of the inscribed polygons, and we write
TEC In the Preview Visual, you can see how areas of inscribed and circumscribed polygons approximate the area of a circle.
y
A − lim An nl`
The Greeks themselves did not use limits explicitly. However, by indirect reasoning, Eudoxus (fifth century bc) used exhaustion to prove the familiar formula for the area of a circle: A − r 2. We will use a similar idea in Chapter 4 to find areas of regions of the type shown in Figure 3. We will approximate the desired area A by areas of rectangles (as in Figure 4), let the width of the rectangles decrease, and then calculate A as the limit of these sums of areas of rectangles. y
y
(1, 1)
y
(1, 1)
(1, 1)
(1, 1)
y=≈ A 0
1
x
0
1 4
1 2
3 4
1
x
0
1
x
0
1 n
1
x
FIGURE 3
The area problem is the central problem in the branch of calculus called integral calculus. The techniques that we will develop in Chapter 4 for finding areas will also enable us to compute the volume of a solid, the length of a curve, the force of water against a dam, the mass and center of gravity of a rod, and the work done in pumping water out of a tank.
The Tangent Problem Consider the problem of trying to find an equation of the tangent line t to a curve with equation y − f sxd at a given point P. (We will give a precise definition of a tangent line in
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a preview of calculus y
Chapter 1. For now you can think of it as a line that touches the curve at P as in Figure 5.) Since we know that the point P lies on the tangent line, we can find the equation of t if we know its slope m. The problem is that we need two points to compute the slope and we know only one point, P, on t. To get around the problem we first find an approximation to m by taking a nearby point Q on the curve and computing the slope mPQ of the secant line PQ. From Figure 6 we see that
t y=ƒ P
0
x
FIGURE 5 The tangent line at P y
1
mPQ −
m − lim mPQ Q lP
Q { x, ƒ}
ƒf(a)
P { a, f(a)}
and we say that m is the limit of mPQ as Q approaches P along the curve. Because x approaches a as Q approaches P, we could also use Equation 1 to write
xa
a
x
x
FIGURE 6 The secant line at PQ y
f sxd 2 f sad x2a
Now imagine that Q moves along the curve toward P as in Figure 7. You can see that the secant line rotates and approaches the tangent line as its limiting position. This means that the slope mPQ of the secant line becomes closer and closer to the slope m of the tangent line. We write
t
0
3
t Q P
0
FIGURE 7 Secant lines approaching the tangent line
x
2
f sxd 2 f sad x2a
m − lim
xla
Specific examples of this procedure will be given in Chapter 1. The tangent problem has given rise to the branch of calculus called differential calculus, which was not invented until more than 2000 years after integral calculus. The main ideas behind differential calculus are due to the French mathematician Pierre Fermat (1601–1665) and were developed by the English mathematicians John Wallis (1616–1703), Isaac Barrow (1630–1677), and Isaac Newton (1642–1727) and the German mathematician Gottfried Leibniz (1646–1716). The two branches of calculus and their chief problems, the area problem and the tangent problem, appear to be very different, but it turns out that there is a very close connection between them. The tangent problem and the area problem are inverse problems in a sense that will be described in Chapter 4.
Velocity When we look at the speedometer of a car and read that the car is traveling at 48 miyh, what does that information indicate to us? We know that if the velocity remains constant, then after an hour we will have traveled 48 mi. But if the velocity of the car varies, what does it mean to say that the velocity at a given instant is 48 miyh? In order to analyze this question, let’s examine the motion of a car that travels along a straight road and assume that we can measure the distance traveled by the car (in feet) at lsecond intervals as in the following chart: t − Time elapsed ssd
0
1
2
3
4
5
d − Distance sftd
0
2
9
24
42
71
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
4
a preview of calculus
As a first step toward finding the velocity after 2 seconds have elapsed, we find the average velocity during the time interval 2 < t < 4: change in position time elapsed
average velocity −
42 2 9 422
−
− 16.5 ftys Similarly, the average velocity in the time interval 2 < t < 3 is average velocity −
24 2 9 − 15 ftys 322
We have the feeling that the velocity at the instant t − 2 can’t be much different from the average velocity during a short time interval starting at t − 2. So let’s imagine that the distance traveled has been measured at 0.lsecond time intervals as in the following chart: t
2.0
2.1
2.2
2.3
2.4
2.5
d
9.00
10.02
11.16
12.45
13.96
15.80
Then we can compute, for instance, the average velocity over the time interval f2, 2.5g: average velocity −
15.80 2 9.00 − 13.6 ftys 2.5 2 2
The results of such calculations are shown in the following chart: Time interval
f2, 3g
f2, 2.5g
f2, 2.4g
f2, 2.3g
f2, 2.2g
f2, 2.1g
Average velocity sftysd
15.0
13.6
12.4
11.5
10.8
10.2
The average velocities over successively smaller intervals appear to be getting closer to a number near 10, and so we expect that the velocity at exactly t − 2 is about 10 ftys. In Chapter 2 we will define the instantaneous velocity of a moving object as the limiting value of the average velocities over smaller and smaller time intervals. In Figure 8 we show a graphical representation of the motion of the car by plotting the distance traveled as a function of time. If we write d − f std, then f std is the number of feet traveled after t seconds. The average velocity in the time interval f2, tg is
d
Q { t, f(t)}
average velocity −
which is the same as the slope of the secant line PQ in Figure 8. The velocity v when t − 2 is the limiting value of this average velocity as t approaches 2; that is,
20 10 0
change in position f std 2 f s2d − time elapsed t22
P { 2, f(2)} 1
2
FIGURE 8
3
4
5
t
v − lim
tl2
f std 2 f s2d t22
and we recognize from Equation 2 that this is the same as the slope of the tangent line to the curve at P.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
a preview of calculus
5
Thus, when we solve the tangent problem in differential calculus, we are also solving problems concerning velocities. The same techniques also enable us to solve problems involving rates of change in all of the natural and social sciences.
The Limit of a Sequence In the fifth century bc the Greek philosopher Zeno of Elea posed four problems, now known as Zeno’s paradoxes, that were intended to challenge some of the ideas concerning space and time that were held in his day. Zeno’s second paradox concerns a race between the Greek hero Achilles and a tortoise that has been given a head start. Zeno argued, as follows, that Achilles could never pass the tortoise: Suppose that Achilles starts at position a 1 and the tortoise starts at position t1. (See Figure 9.) When Achilles reaches the point a 2 − t1, the tortoise is farther ahead at position t2. When Achilles reaches a 3 − t2, the tortoise is at t3. This process continues indefinitely and so it appears that the tortoise will always be ahead! But this defies common sense.
Achilles
FIGURE 9
a¡
tortoise
a™
a£
a¢
a∞
...
t¡
t™
t£
t¢
...
One way of explaining this paradox is with the idea of a sequence. The successive positions of Achilles sa 1, a 2 , a 3 , . . .d or the successive positions of the tortoise st1, t2 , t3 , . . .d form what is known as a sequence. In general, a sequence ha nj is a set of numbers written in a definite order. For instance, the sequence
h1, 12 , 13 , 14 , 15 , . . . j can be described by giving the following formula for the nth term: an − a¢ a £
a™
0
We can visualize this sequence by plotting its terms on a number line as in Figure 10(a) or by drawing its graph as in Figure 10(b). Observe from either picture that the terms of the sequence a n − 1yn are becoming closer and closer to 0 as n increases. In fact, we can find terms as small as we please by making n large enough. We say that the limit of the sequence is 0, and we indicate this by writing
a¡ 1
(a) 1
lim
nl`
1 2 3 4 5 6 7 8
(b)
FIGURE 10
1 n
n
1 −0 n
In general, the notation lim a n − L
nl`
is used if the terms a n approach the number L as n becomes large. This means that the numbers a n can be made as close as we like to the number L by taking n sufficiently large.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
6
a preview of calculus
The concept of the limit of a sequence occurs whenever we use the decimal representation of a real number. For instance, if a 1 − 3.1 a 2 − 3.14 a 3 − 3.141 a 4 − 3.1415 a 5 − 3.14159 a 6 − 3.141592 a 7 − 3.1415926 f
then
lim a n −
nl`
The terms in this sequence are rational approximations to . Let’s return to Zeno’s paradox. The successive positions of Achilles and the tortoise form sequences ha nj and htn j, where a n , tn for all n. It can be shown that both sequences have the same limit: lim a n − p − lim tn
nl`
nl`
It is precisely at this point p that Achilles overtakes the tortoise.
The Sum of a Series Another of Zeno’s paradoxes, as passed on to us by Aristotle, is the following: “A man standing in a room cannot walk to the wall. In order to do so, he would first have to go half the distance, then half the remaining distance, and then again half of what still remains. This process can always be continued and can never be ended.” (See Figure 11.)
1 2
FIGURE 11
1 4
1 8
1 16
Of course, we know that the man can actually reach the wall, so this suggests that perhaps the total distance can be expressed as the sum of infinitely many smaller distances as follows: 3
1−
1 1 1 1 1 1 1 1 1 ∙∙∙ 1 n 1 ∙∙∙ 2 4 8 16 2
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a preview of calculus
7
Zeno was arguing that it doesn’t make sense to add infinitely many numbers together. But there are other situations in which we implicitly use infinite sums. For instance, in decimal notation, the symbol 0.3 − 0.3333 . . . means 3 3 3 3 1 1 1 1 ∙∙∙ 10 100 1000 10,000 and so, in some sense, it must be true that 3 3 3 3 1 1 1 1 1 ∙∙∙ − 10 100 1000 10,000 3 More generally, if dn denotes the nth digit in the decimal representation of a number, then 0.d1 d2 d3 d4 . . . −
d1 d2 d3 dn 1 2 1 3 1 ∙∙∙ 1 n 1 ∙∙∙ 10 10 10 10
Therefore some infinite sums, or infinite series as they are called, have a meaning. But we must define carefully what the sum of an infinite series is. Returning to the series in Equation 3, we denote by sn the sum of the first n terms of the series. Thus s1 − 12 − 0.5 s2 − 12 1 14 − 0.75 s3 − 12 1 14 1 18 − 0.875 1 s4 − 12 1 14 1 18 1 16 − 0.9375 1 1 s5 − 12 1 14 1 18 1 16 1 32 − 0.96875 1 1 1 s6 − 12 1 14 1 18 1 16 1 32 1 64 − 0.984375 1 1 1 1 s7 − 12 1 14 1 18 1 16 1 32 1 64 1 128 − 0.9921875
f 1 s10 − 12 1 14 1 ∙ ∙ ∙ 1 1024 < 0.99902344
f s16 −
1 1 1 1 1 ∙ ∙ ∙ 1 16 < 0.99998474 2 4 2
Observe that as we add more and more terms, the partial sums become closer and closer to 1. In fact, it can be shown that by taking n large enough (that is, by adding sufficiently many terms of the series), we can make the partial sum sn as close as we please to the number 1. It therefore seems reasonable to say that the sum of the infinite series is 1 and to write 1 1 1 1 1 1 1 ∙∙∙ 1 n 1 ∙∙∙ − 1 2 4 8 2 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
8
a preview of calculus
In other words, the reason the sum of the series is 1 is that lim sn − 1
nl`
In Chapter 11 we will discuss these ideas further. We will then use Newton’s idea of combining infinite series with differential and integral calculus.
Summary We have seen that the concept of a limit arises in trying to find the area of a region, the slope of a tangent to a curve, the velocity of a car, or the sum of an infinite series. In each case the common theme is the calculation of a quantity as the limit of other, easily calculated quantities. It is this basic idea of a limit that sets calculus apart from other areas of mathematics. In fact, we could define calculus as the part of mathematics that deals with limits. After Sir Isaac Newton invented his version of calculus, he used it to explain the motion of the planets around the sun. Today calculus is used in calculating the orbits of satellites and spacecraft, in predicting population sizes, in estimating how fast oil prices rise or fall, in forecasting weather, in measuring the cardiac output of the heart, in calculating life insurance premiums, and in a great variety of other areas. We will explore some of these uses of calculus in this book. In order to convey a sense of the power of the subject, we end this preview with a list of some of the questions that you will be able to answer using calculus: rays from sun
138° rays from sun
observer
42°
1. How can we explain the fact, illustrated in Figure 12, that the angle of elevation from an observer up to the highest point in a rainbow is 42°? (See page 213.)
2. How can we explain the shapes of cans on supermarket shelves? (See page 270.)
3. Where is the best place to sit in a movie theater? (See page 483.)
4. How can we design a roller coaster for a smooth ride? (See page 144.)
5. How far away from an airport should a pilot start descent? (See page 161.)
6. How can we fit curves together to design shapes to represent letters on a laser printer? (See page 697.)
7. How can we estimate the number of workers that were needed to build the Great Pyramid of Khufu in ancient Egypt? (See page 388.)
8. Where should an infielder position himself to catch a baseball thrown by an outfielder and relay it to home plate? (See page 392.)
9. Does a ball thrown upward take longer to reach its maximum height or to fall back to its original height? (See page 649.)
FIGURE 12
10. How can we explain the fact that planets and satellites move in elliptical orbits? (See page 916.) 11. How can we distribute water flow among turbines at a hydroelectric station so as to maximize the total energy production? (See page 1020.) 12. If a marble, a squash ball, a steel bar, and a lead pipe roll down a slope, which of them reaches the bottom first? (See page 1092.)
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
1 Often a graph is the best way to represent a function because it conveys so much information at a glance. Shown is a graph of the vertical ground acceleration created by the 2011 earthquake near Tohoku, Japan. The earthquake had a magnitude of 9.0 on the Richter scale and was so powerful that it moved northern Japan 8 feet closer to North America.
Functions and Limits
Pictura Collectus/Alamy
(cm/
[email protected]) 2000 1000 0
time
_1000 _2000 0
50
100
150
200
Seismological Society of America
The fundamental objects that we deal with in calculus are functions. We stress that a function can be represented in different ways: by an equation, in a table, by a graph, or in words. We look at the main types of functions that occur in calculus and describe the process of using these functions as mathematical models of realworld phenomena.
In A Preview of Calculus (page 1) we saw how the idea of a limit underlies the various branches of calculus. It is therefore appropriate to begin our study of calculus by investigating limits of functions and their properties.
9 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
10
Chapter 1 Functions and Limits
Year
Population (millions)
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080 6870
Functions arise whenever one quantity depends on another. Consider the following four situations. A. The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A − r 2. With each positive number r there is associated one value of A, and we say that A is a function of r. B. The human population of the world P depends on the time t. The table gives estimates of the world population Pstd at time t, for certain years. For instance, Ps1950d < 2,560,000,000 But for each value of the time t there is a corresponding value of P, and we say that P is a function of t. C. The cost C of mailing an envelope depends on its weight w. Although there is no simple formula that connects w and C, the post office has a rule for determining C when w is known. D. The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t, the graph provides a corresponding value of a. a
{cm/
[email protected]}
100 50
5
FIGURE 1 Vertical ground acceleration during the Northridge earthquake
10
15
20
25
30
t (seconds)
_50 Calif. Dept. of Mines and Geology
Each of these examples describes a rule whereby, given a number (r, t, w, or t), another number (A, P, C, or a) is assigned. In each case we say that the second number is a function of the first number. A function f is a rule that assigns to each element x in a set D exactly one element, called f sxd, in a set E. We usually consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function. The number f sxd is the value of f at x and is read “ f of x.” The range of f is the set of all possible values of f sxd as x varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependent variable. In Example A, for instance, r is the independent variable and A is the dependent variable.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
11
Section 1.1 Four Ways to Represent a Function
x (input)
f
ƒ (output)
FIGURE 2
Machine diagram for a function f
x
ƒ a
f(a)
f
D
It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f sxd according to the rule of the function. Thus we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs. The preprogrammed functions in a calculator are good examples of a function as a machine. For example, the square root key on your calculator computes such a function. You press the key labeled s (or s x ) and enter the input x. If x , 0, then x is not in the domain of this function; that is, x is not an acceptable input, and the calculator will indicate an error. If x > 0, then an approximation to s x will appear in the display. Thus the s x key on your calculator is not quite the same as the exact mathematical function f defined by f sxd − s x . Another way to picture a function is by an arrow diagram as in Figure 3. Each arrow connects an element of D to an element of E. The arrow indicates that f sxd is associated with x, f sad is associated with a, and so on. The most common method for visualizing a function is its graph. If f is a function with domain D, then its graph is the set of ordered pairs

hsx, f sxdd x [ Dj
E
(Notice that these are inputoutput pairs.) In other words, the graph of f consists of all points sx, yd in the coordinate plane such that y − f sxd and x is in the domain of f. The graph of a function f gives us a useful picture of the behavior or “life history” of a function. Since the ycoordinate of any point sx, yd on the graph is y − f sxd, we can read the value of f sxd from the graph as being the height of the graph above the point x (see Figure 4). The graph of f also allows us to picture the domain of f on the xaxis and its range on the yaxis as in Figure 5.
FIGURE 3
Arrow diagram for f
y
y
{ x, ƒ} range
ƒ f (2)
f (1) 0
1
2
x
x
FIGURE 4 y
0
domain
x
FIGURE 5
Example 1 The graph of a function f is shown in Figure 6. (a) Find the values of f s1d and f s5d. (b) What are the domain and range of f ?
1 0
y ƒ(x)
Solution
1
x
FIGURE 6 The notation for intervals is given in Appendix A.
(a) We see from Figure 6 that the point s1, 3d lies on the graph of f, so the value of f at 1 is f s1d − 3. (In other words, the point on the graph that lies above x − 1 is 3 units above the xaxis.) When x − 5, the graph lies about 0.7 units below the xaxis, so we estimate that f s5d < 20.7. (b) We see that f sxd is defined when 0 < x < 7, so the domain of f is the closed interval f0, 7g. Notice that f takes on all values from 22 to 4, so the range of f is

hy 22 < y < 4j − f22, 4g
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■
12
Chapter 1 Functions and Limits y
Example 2 Sketch the graph and find the domain and range of each function. (a) fsxd − 2x 2 1 (b) tsxd − x 2 Solution
y=2x1 0 1
x
1 2
FIGURE 7 y
(2, 4)
y=≈ (_1, 1)
(a) The equation of the graph is y − 2x 2 1, and we recognize this as being the equation of a line with slope 2 and yintercept 21. (Recall the slopeintercept form of the equation of a line: y − mx 1 b. See Appendix B.) This enables us to sketch a portion of the graph of f in Figure 7. The expression 2x 2 1 is defined for all real numbers, so the domain of f is the set of all real numbers, which we denote by R. The graph shows that the range is also R. (b) Since ts2d − 2 2 − 4 and ts21d − s21d2 − 1, we could plot the points s2, 4d and s21, 1d, together with a few other points on the graph, and join them to produce the graph (Figure 8). The equation of the graph is y − x 2, which represents a parabola (see Appendix C). The domain of t is R. The range of t consists of all values of tsxd, that is, all numbers of the form x 2. But x 2 > 0 for all numbers x and any positive number y is a square. So the range of t is hy y > 0j − f0, `d. This can also be seen from Figure 8. ■

1 0
1
x
Example 3 If f sxd − 2x 2 2 5x 1 1 and h ± 0, evaluate
f sa 1 hd 2 f sad . h
Solution We first evaluate f sa 1 hd by replacing x by a 1 h in the expression for f sxd:
FIGURE 8
f sa 1 hd − 2sa 1 hd2 2 5sa 1 hd 1 1 − 2sa 2 1 2ah 1 h 2 d 2 5sa 1 hd 1 1 − 2a 2 1 4ah 1 2h 2 2 5a 2 5h 1 1 The expression
Then we substitute into the given expression and simplify: f sa 1 hd 2 f sad s2a 2 1 4ah 1 2h 2 2 5a 2 5h 1 1d 2 s2a 2 2 5a 1 1d − h h
f sa 1 hd 2 f sad h in Example 3 is called a difference quotient and occurs frequently in calculus. As we will see in Chapter 2, it represents the average rate of change of f sxd between x − a and x − a 1 h.
−
2a 2 1 4ah 1 2h 2 2 5a 2 5h 1 1 2 2a 2 1 5a 2 1 h
−
4ah 1 2h 2 2 5h − 4a 1 2h 2 5 h
■
Representations of Functions There are four possible ways to represent a function: verbally ● numerically ● visually ● algebraically ●
(by a description in words) (by a table of values) (by a graph) (by an explicit formula)
If a single function can be represented in all four ways, it’s often useful to go from one representation to another to gain additional insight into the function. (In Example 2, for instance, we started with algebraic formulas and then obtained the graphs.) But certain functions are described more naturally by one method than by another. With this in mind, let’s reexamine the four situations that we considered at the beginning of this section.
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13
Section 1.1 Four Ways to Represent a Function
A. The most useful representation of the area of a circle as a function of its radius is probably the algebraic formula Asrd − r 2, though it is possible to compile a table of values or to sketch a graph (half a parabola). Because a circle has to have a positive radius, the domain is hr r . 0j − s0, `d, and the range is also s0, `d. B. We are given a description of the function in words: Pstd is the human population of the world at time t. Let’s measure t so that t − 0 corresponds to the year 1900. The table of values of world population provides a convenient representation of this function. If we plot these values, we get the graph (called a scatter plot) in Figure 9. It too is a useful representation; the graph allows us to absorb all the data at once. What about a formula? Of course, it’s impossible to devise an explicit formula that gives the exact human population Pstd at any time t. But it is possible to find an expression for a function that approximates Pstd. In fact, using methods explained in Section 1.2, we obtain the approximation

t (years since 1900)
Population (millions)
0 10 20 30 40 50 60 70 80 90 100 110
1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080 6870
Pstd < f std − s1.43653 3 10 9 d s1.01395d t Figure 10 shows that it is a reasonably good “fit.” The function f is called a mathematical model for population growth. In other words, it is a function with an explicit formula that approximates the behavior of our given function. We will see, however, that the ideas of calculus can be applied to a table of values; an explicit formula is not necessary.
P
P
5x10'
0
5x10'
20
40 60 80 Years since 1900
FIGURE 9
100
120
t
0
20
40 60 80 Years since 1900
100
120
t
FIGURE 10
A function deﬁned by a table of values is called a tabular function. w (ounces)
Cswd (dollars)
0,w 0 2x if x , 0
Using the same method as in Example 7, we see that the graph of f coincides with the line y − x to the right of the yaxis and coincides with the line y − 2x to the left of the yaxis (see Figure 16). ■
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Section 1.1 Four Ways to Represent a Function
17
y
Example 9 Find a formula for the function f graphed in Figure 17.
1
SOLUTION The line through s0, 0d and s1, 1d has slope m − 1 and yintercept b − 0, so its equation is y − x. Thus, for the part of the graph of f that joins s0, 0d to s1, 1d, we have
0
x
1
f sxd − x if 0 < x < 1
FIGURE 17
The line through s1, 1d and s2, 0d has slope m − 21, so its pointslope form is
Pointslope form of the equation of a line: y 2 y1 − msx 2 x 1 d
y 2 0 − s21dsx 2 2d or y − 2 2 x So we have
See Appendix B.
f sxd − 2 2 x if 1 , x < 2
We also see that the graph of f coincides with the xaxis for x . 2. Putting this information together, we have the following threepiece formula for f :
H
x if 0 < x < 1 f sxd − 2 2 x if 1 , x < 2 0 if x . 2
■
Example 10 In Example C at the beginning of this section we considered the cost Cswd of mailing a large envelope with weight w. In effect, this is a piecewise defined function because, from the table of values on page 13, we have
C 1.50 1.00
Cswd − 0.50
0
1
2
3
4
5
figure 18
w
0.98 1.19 1.40 1.61 ∙ ∙ ∙
if if if if
0,w 2 x 1 1 if x < 21 x2 if x . 21
21 if x < 1 7 2 2x if x . 1
45–50 Sketch the graph of the function.
 

45. f sxd − x 1 x 46. f sxd − x 1 2



 

47. tstd − 1 2 3t 48. hstd − t 1 t 1 1 49. f sxd −
H  x 1
   
if x < 1 50. tsxd − if x . 1
 x  2 1
51–56 Find an expression for the function whose graph is the given curve. 51. The line segment joining the points s1, 23d and s5, 7d 52. The line segment joining the points s25, 10d and s7, 210d 53. The bottom half of the parabola x 1 s y 2 1d2 − 0 54. The top half of the circle x 2 1 s y 2 2d 2 − 4
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22
Chapter 1 Functions and Limits
55.
56.
y
1
1 0
64. A cell phone plan has a basic charge of $35 a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost C as a function of the number x of minutes used and graph C as a function of x for 0 < x < 600.
y
0
x
1
1
x
57–61 Find a formula for the described function and state its domain. 57. A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides. 2
58. A rectangle has area 16 m . Express the perimeter of the rect angle as a function of the length of one of its sides. 59. Express the area of an equilateral triangle as a function of the length of a side. 60. A closed rectangular box with volume 8 ft3 has length twice the width. Express the height of the box as a function of the width. 61. An open rectangular box with volume 2 m3 has a square base. Express the surface area of the box as a function of the length of a side of the base. 62. A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window.
65. In a certain state the maximum speed permitted on freeways is 65 miyh and the minimum speed is 40 miyh. The fine for violating these limits is $15 for every mile per hour above the maximum speed or below the minimum speed. Express the amount of the fine F as a function of the driving speed x and graph Fsxd for 0 < x < 100. 66. An electricity company charges its customers a base rate of $10 a month, plus 6 cents per kilowatthour (kWh) for the first 1200 kWh and 7 cents per kWh for all usage over 1200 kWh. Express the monthly cost E as a function of the amount x of electricity used. Then graph the function E for 0 < x < 2000. 67. In a certain country, income tax is assessed as follows. There is no tax on income up to $10,000. Any income over $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over $20,000 is taxed at 15%. (a) Sketch the graph of the tax rate R as a function of the income I. (b) How much tax is assessed on an income of $14,000? On $26,000? (c) Sketch the graph of the total assessed tax T as a function of the income I. 68. The functions in Example 10 and Exercise 67 are called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life. 69–70 Graphs of f and t are shown. Decide whether each function is even, odd, or neither. Explain your reasoning. y
69.
70.
g
f
f x
x
63. A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the vol ume V of the box as a function of x. 20 x 12
x
x
x
x
x x
x
y
g
x
71. (a) If the point s5, 3d is on the graph of an even function, what other point must also be on the graph? (b) If the point s5, 3d is on the graph of an odd function, what other point must also be on the graph? 72. A function f has domain f25, 5g and a portion of its graph is shown. (a) Complete the graph of f if it is known that f is even. (b) Complete the graph of f if it is known that f is odd.
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Section 1.2 Mathematical Models: A Catalog of Essential Functions y
75. f sxd −
23
x 76. f sxd − x x x11
 
77. f sxd − 1 1 3x 2 2 x 4 78. f sxd − 1 1 3x 3 2 x 5 _5
0
5
x
79. If f and t are both even functions, is f 1 t even? If f and t are both odd functions, is f 1 t odd? What if f is even and t is odd? Justify your answers.
73–78 Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
80. If f and t are both even functions, is the product ft even? If f and t are both odd functions, is ft odd? What if f is even and t is odd? Justify your answers.
x x2 73. f sxd − 2 74. f sxd − 4 x 11 x 11
A mathematical model is a mathematical description (often by means of a function or an equation) of a realworld phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. Figure 1 illustrates the process of mathematical modeling. Given a realworld problem, our first task is to formulate a mathematical model by identifying and naming the independent and dependent variables and making assumptions that simplify the phenomenon enough to make it mathematically tractable. We use our knowledge of the physical situation and our mathematical skills to obtain equations that relate the variables. In situations where there is no physical law to guide us, we may need to collect data (either from a library or the Internet or by conducting our own experiments) and examine the data in the form of a table in order to discern patterns. From this numerical representation of a function we may wish to obtain a graphical representation by plotting the data. The graph might even suggest a suitable algebraic formula in some cases.
Realworld problem
Formulate
Mathematical model
Solve
Mathematical conclusions
Interpret
Realworld predictions
Test
FIGURE 1 The modeling process
The second stage is to apply the mathematics that we know (such as the calculus that will be developed throughout this book) to the mathematical model that we have formulated in order to derive mathematical conclusions. Then, in the third stage, we take those mathematical conclusions and interpret them as information about the original realworld phenomenon by way of offering explanations or making predictions. The final step is to test our predictions by checking against new real data. If the predictions don’t compare well with reality, we need to refine our model or to formulate a new model and start the cycle again. A mathematical model is never a completely accurate representation of a physical situation—it is an idealization. A good model simplifies reality enough to permit math
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24
Chapter 1 Functions and Limits
ematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say. There are many different types of functions that can be used to model relationships observed in the real world. In what follows, we discuss the behavior and graphs of these functions and give examples of situations appropriately modeled by such functions.
Linear Models The coordinate geometry of lines is reviewed in Appendix B.
When we say that y is a linear function of x, we mean that the graph of the function is a line, so we can use the slopeintercept form of the equation of a line to write a formula for the function as y − f sxd − mx 1 b where m is the slope of the line and b is the yintercept. A characteristic feature of linear functions is that they grow at a constant rate. For instance, Figure 2 shows a graph of the linear function f sxd − 3x 2 2 and a table of sample values. Notice that whenever x increases by 0.1, the value of f sxd increases by 0.3. So f sxd increases three times as fast as x. Thus the slope of the graph of y − 3x 2 2, namely 3, can be interpreted as the rate of change of y with respect to x. y
y=3x2
0
1
x
_2
figure 2
x
f sxd − 3x 2 2
1.0 1.1 1.2 1.3 1.4 1.5
1.0 1.3 1.6 1.9 2.2 2.5
Example 1 (a) As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at a height of 1 km is 10°C, express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2.5 km? SOLUTION
(a) Because we are assuming that T is a linear function of h, we can write T − mh 1 b We are given that T − 20 when h − 0, so 20 − m 0 1 b − b In other words, the yintercept is b − 20.
We are also given that T − 10 when h − 1, so 10 − m 1 1 20 The slope of the line is therefore m − 10 2 20 − 210 and the required linear function is T − 210h 1 20
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.2 Mathematical Models: A Catalog of Essential Functions T
(b) The graph is sketched in Figure 3. The slope is m − 210°Cykm, and this represents the rate of change of temperature with respect to height. (c) At a height of h − 2.5 km, the temperature is
20 10 0
25
T=_10h+20
1
h
3
T − 210s2.5d 1 20 − 25°C
■
If there is no physical law or principle to help us formulate a model, we construct an empirical model, which is based entirely on collected data. We seek a curve that “fits” the data in the sense that it captures the basic trend of the data points.
FIGURE 3
Example 2 Table 1 lists the average carbon dioxide level in the atmosphere, measured in parts per million at Mauna Loa Observatory from 1980 to 2012. Use the data in Table 1 to find a model for the carbon dioxide level. SOLUTION We use the data in Table 1 to make the scatter plot in Figure 4, where t represents time (in years) and C represents the CO2 level (in parts per million, ppm). C (ppm) 400
Table 1 Year
CO 2 level (in ppm)
1980 1982 1984 1986 1988 1990 1992 1994 1996
338.7 341.2 344.4 347.2 351.5 354.2 356.3 358.6 362.4
Year
CO 2 level (in ppm)
1998 2000 2002 2004 2006 2008 2010 2012
366.5 369.4 373.2 377.5 381.9 385.6 389.9 393.8
390 380 370 360 350 340 1980
1985
1990
1995
2000
2005
2010
t
FIGURE 4 Scatter plot for the average CO2 level
Notice that the data points appear to lie close to a straight line, so it’s natural to choose a linear model in this case. But there are many possible lines that approximate these data points, so which one should we use? One possibility is the line that passes through the first and last data points. The slope of this line is 393.8 2 338.7 55.1 − − 1.721875 < 1.722 2012 2 1980 32 We write its equation as C 2 338.7 − 1.722st 2 1980d or 1
C − 1.722t 2 3070.86
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26
Chapter 1 Functions and Limits
Equation 1 gives one possible linear model for the carbon dioxide level; it is graphed in Figure 5. C (ppm) 400 390 380 370 360 350
FIGURE 5
340
Linear model through first and last data points A computer or graphing calculator ﬁnds the regression line by the method of least squares, which is to minimize the sum of the squares of the vertical distances between the data points and the line. The details are explained in Section 14.7.
1980
1985
1990
1995
2000
2005
2010
t
Notice that our model gives values higher than most of the actual CO2 levels. A better linear model is obtained by a procedure from statistics called linear regression. If we use a graphing calculator, we enter the data from Table 1 into the data editor and choose the linear regression command. (With Maple we use the fit[leastsquare] command in the stats package; with Mathematica we use the Fit command.) The machine gives the slope and yintercept of the regression line as m − 1.71262 b − 23054.14 So our least squares model for the CO2 level is 2
C − 1.71262t 2 3054.14
In Figure 6 we graph the regression line as well as the data points. Comparing with Figure 5, we see that it gives a better fit than our previous linear model. C (ppm) 400 390 380 370 360 350
Figure 6 The regression line
340 1980
1985
1990
1995
2000
2005
2010
t
■
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Section 1.2 Mathematical Models: A Catalog of Essential Functions
27
Example 3 Use the linear model given by Equation 2 to estimate the average CO2 level for 1987 and to predict the level for the year 2020. According to this model, when will the CO2 level exceed 420 parts per million? Solution Using Equation 2 with t − 1987, we estimate that the average CO2 level in
1987 was Cs1987d − s1.71262ds1987d 2 3054.14 < 348.84 This is an example of interpolation because we have estimated a value between observed values. (In fact, the Mauna Loa Observatory reported that the average CO2 level in 1987 was 348.93 ppm, so our estimate is quite accurate.) With t − 2020, we get Cs2020d − s1.71262ds2020d 2 3054.14 < 405.35 So we predict that the average CO2 level in the year 2020 will be 405.4 ppm. This is an example of extrapolation because we have predicted a value outside the time frame of observations. Consequently, we are far less certain about the accuracy of our prediction. Using Equation 2, we see that the CO2 level exceeds 420 ppm when 1.71262t 2 3054.14 . 420 Solving this inequality, we get t.
3474.14 < 2028.55 1.71262
We therefore predict that the CO2 level will exceed 420 ppm by the year 2029. This prediction is risky because it involves a time quite remote from our observations. In fact, we see from Figure 6 that the trend has been for CO2 levels to increase rather more rapidly in recent years, so the level might exceed 420 ppm well before 2029. ■
y 2
Polynomials 0
1
x
(a) y=≈+x+1
Psxd − a n x n 1 a n21 x n21 1 ∙ ∙ ∙ 1 a 2 x 2 1 a 1 x 1 a 0 where n is a nonnegative integer and the numbers a 0 , a 1, a 2 , . . . , a n are constants called the coefficients of the polynomial. The domain of any polynomial is R − s2`, `d. If the leading coefficient a n ± 0, then the degree of the polynomial is n. For example, the function Psxd − 2x 6 2 x 4 1 25 x 3 1 s2
y 2
1
A function P is called a polynomial if
x
(b) y=_2≈+3x+1
FIGURE 7 The graphs of quadratic functions are parabolas.
is a polynomial of degree 6. A polynomial of degree 1 is of the form Psxd − mx 1 b and so it is a linear function. A polynomial of degree 2 is of the form Psxd − ax 2 1 bx 1 c and is called a quadratic function. Its graph is always a parabola obtained by shifting the parabola y − ax 2, as we will see in the next section. The parabola opens upward if a . 0 and downward if a , 0. (See Figure 7.) A polynomial of degree 3 is of the form Psxd − ax 3 1 bx 2 1 cx 1 d a ± 0
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28
Chapter 1 Functions and Limits
and is called a cubic function. Figure 8 shows the graph of a cubic function in part (a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c). We will see later why the graphs have these shapes. y
y
1
2
0
FIGURE 8
y
x
1
20
1 x
(a) y=˛x+1
x
1
(b) y=x$3≈+x
(c) y=3x%25˛+60x
Polynomials are commonly used to model various quantities that occur in the natural and social sciences. For instance, in Section 2.7 we will explain why economists often use a polynomial Psxd to represent the cost of producing x units of a commodity. In the following example we use a quadratic function to model the fall of a ball. Table 2 Time (seconds)
Height (meters)
0 1 2 3 4 5 6 7 8 9
450 445 431 408 375 332 279 216 143 61
Example 4 A ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground, and its height h above the ground is recorded at 1second intervals in Table 2. Find a model to fit the data and use the model to predict the time at which the ball hits the ground. Solution We draw a scatter plot of the data in Figure 9 and observe that a linear model is inappropriate. But it looks as if the data points might lie on a parabola, so we try a quadratic model instead. Using a graphing calculator or computer algebra system (which uses the least squares method), we obtain the following quadratic model:
3
h − 449.36 1 0.96t 2 4.90t 2
h (meters)
h
400
400
200
200
0
2
4
6
8
t (seconds)
0
2
4
6
8
FIGURE 9
FIGURE 10
Scatter plot for a falling ball
Quadratic model for a falling ball
t
In Figure 10 we plot the graph of Equation 3 together with the data points and see that the quadratic model gives a very good fit. The ball hits the ground when h − 0, so we solve the quadratic equation 24.90t 2 1 0.96t 1 449.36 − 0
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
29
Section 1.2 Mathematical Models: A Catalog of Essential Functions
The quadratic formula gives t−
20.96 6 ss0.96d2 2 4s24.90ds449.36d 2s24.90d
The positive root is t < 9.67, so we predict that the ball will hit the ground after about 9.7 seconds. ■
Power Functions A function of the form f sxd − x a, where a is a constant, is called a power function. We consider several cases. (i) a − n, where n is a positive integer The graphs of f sxd − x n for n − 1, 2, 3, 4, and 5 are shown in Figure 11. (These are polynomials with only one term.) We already know the shape of the graphs of y − x (a line through the origin with slope 1) and y − x 2 [a parabola, see Example 1.1.2(b)]. y
y=≈
y
y=x
y
1
1 0
1
x
0
y=x #
y
x
0
1
x
0
y=x%
y
1
1 1
y=x$
1 1
x
0
1
FIGURE 11 Graphs of f sxd − x n for n − 1, 2, 3, 4, 5
The general shape of the graph of f sxd − x n depends on whether n is even or odd. If n is even, then f sxd − x n is an even function and its graph is similar to the parabola y − x 2. If n is odd, then f sxd − x n is an odd function and its graph is similar to that of y − x 3. Notice from Figure 12, however, that as n increases, the graph of y − x n becomes flatter near 0 and steeper when x > 1. (If x is small, then x 2 is smaller, x 3 is even smaller, x 4 is smaller still, and so on.)
 
y
A family of functions is a collection of functions whose equations are related. Figure 12 shows two families of power functions, one with even powers and one with odd powers.
y=x ^ (_1, 1)
FIGURE 12
y
y=x $ y=≈ (1, 1)
0
y=x #
(1, 1) y=x %
0
x
x
(_1, _1)
(ii) a − 1yn, where n is a positive integer n x is a root function. For n − 2 it is the square root The function f sxd − x 1yn − s function f sxd − sx , whose domain is f0, `d and whose graph is the upper half of the
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
x
30
Chapter 1 Functions and Limits n parabola x − y 2. [See Figure 13(a).] For other even values of n, the graph of y − s x is 3 similar to that of y − sx . For n − 3 we have the cube root function f sxd − sx whose domain is R (recall that every real number has a cube root) and whose graph is shown n 3 in Figure 13(b). The graph of y − s x for n odd sn . 3d is similar to that of y − s x.
y
y (1, 1)
(1, 1)
0
FIGURE 13
y=∆ 1 0
x
1
0
x (a) ƒ=œ„
Graphs of root functions y
x
x
x (b) ƒ=Œ„
(iii) a − 21 The graph of the reciprocal function f sxd − x 21 − 1yx is shown in Figure 14. Its graph has the equation y − 1yx, or xy − 1, and is a hyperbola with the coordinate axes as its asymptotes. This function arises in physics and chemistry in connection with Boyle’s Law, which says that, when the temperature is constant, the volume V of a gas is inversely proportional to the pressure P: V−
C P
where C is a constant. Thus the graph of V as a function of P (see Figure 15) has the same general shape as the right half of Figure 14. Power functions are also used to model speciesarea relationships (Exercises 30–31), illumination as a function of distance from a light source (Exercise 29), and the period of revolution of a planet as a function of its distance from the sun (Exercise 32).
Figure 14 The reciprocal function V
Rational Functions A rational function f is a ratio of two polynomials: 0
f sxd −
P
Figure 15 Volume as a function of pressure at constant temperature
where P and Q are polynomials. The domain consists of all values of x such that Qsxd ± 0. A simple example of a rational function is the function f sxd − 1yx, whose domain is hx x ± 0j; this is the reciprocal function graphed in Figure 14. The function

y
f sxd −
FIGURE 16 ƒ=
2x 4 2 x 2 1 1 x2 2 4

is a rational function with domain hx x ± 62j. Its graph is shown in Figure 16.
20 0
Psxd Qsxd
2
2x$≈+1 ≈4
x
Algebraic Functions A function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. Here are two more examples: f sxd − sx 2 1 1 tsxd −
x 4 2 16x 2 3 1 sx 2 2ds x11 x 1 sx
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
31
Section 1.2 Mathematical Models: A Catalog of Essential Functions
When we sketch algebraic functions in Chapter 3, we will see that their graphs can assume a variety of shapes. Figure 17 illustrates some of the possibilities. y
y
y
1
1
2
1
_3
x
0
(a) ƒ=xœ„„„„ x+3
FIGURE 17
x
5
0
(b) ©=$œ„„„„„„ ≈25
x
1
(c) h(x)
[email protected]?#(x2)@
An example of an algebraic function occurs in the theory of relativity. The mass of a particle with velocity v is m − f svd −
m0 s1 2 v 2yc 2
where m 0 is the rest mass of the particle and c − 3.0 3 10 5 kmys is the speed of light in a vacuum.
Trigonometric Functions Trigonometry and the trigonometric functions are reviewed on Reference Page 2 and also in Appendix D. In calculus the convention is that radian measure is always used (except when otherwise indicated). For example, when we use the function f sxd − sin x, it is understood that sin x means the sine of the angle whose radian measure is x. Thus the graphs of the sine and cosine functions are as shown in Figure 18.
The Reference Pages are located at the back of the book.
y _ _π
π 2
y 3π 2
1 _1
0
π 2
π
_π 2π
5π 2
3π
x
π 2
1 _1
(a) ƒ=sin x
FIGURE 18
_
π 0
π 2
3π 3π 2
2π
5π 2
x
(b) ©=cos x
Notice that for both the sine and cosine functions the domain is s2`, `d and the range is the closed interval f21, 1g. Thus, for all values of x, we have 21 < sin x < 1 21 < cos x < 1 or, in terms of absolute values,
 sin x  < 1  cos x  < 1 Also, the zeros of the sine function occur at the integer multiples of ; that is, sin x − 0 when x − n n an integer
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
32
Chapter 1 Functions and Limits
An important property of the sine and cosine functions is that they are periodic functions and have period 2. This means that, for all values of x, sinsx 1 2d − sin x cossx 1 2d − cos x The periodic nature of these functions makes them suitable for modeling repetitive phenomena such as tides, vibrating springs, and sound waves. For instance, in Example 1.3.4 we will see that a reasonable model for the number of hours of daylight in Philadelphia t days after January 1 is given by the function
F
Lstd − 12 1 2.8 sin
2 st 2 80d 365
G
1 ? 1 2 2 cos x Solution This function is defined for all values of x except for those that make the denominator 0. But
Example 5 What is the domain of the function f sxd −
1 2 2 cos x − 0 &? cos x −
1 5 &? x − 1 2n or x − 1 2n 2 3 3
where n is any integer (because the cosine function has period 2). So the domain of f is the set of all real numbers except for the ones noted above. ■ y
The tangent function is related to the sine and cosine functions by the equation tan x −
1 _
3π 2
0
_π _ π 2
π 2
π
3π 2
x
sin x cos x
and its graph is shown in Figure 19. It is undefined whenever cos x − 0, that is, when x − 6y2, 63y2, . . . . Its range is s2`, `d. Notice that the tangent function has period : tansx 1 d − tan x for all x The remaining three trigonometric functions (cosecant, secant, and cotangent) are the reciprocals of the sine, cosine, and tangent functions. Their graphs are shown in Appendix D.
figure 19
y − tanxx y=tan y
1 0
Exponential Functions
y
1
(a) y=2®
figure 20
x
1 0
1
(b) y=(0.5)®
x
The exponential functions are the functions of the form f sxd − b x, where the base b is a positive constant. The graphs of y − 2 x and y − s0.5d x are shown in Figure 20. In both cases the domain is s2`, `d and the range is s0, `d. Exponential functions will be studied in detail in Chapter 6, and we will see that they are useful for modeling many natural phenomena, such as population growth (if b . 1) and radioactive decay (if b , 1d.
Logarithmic Functions The logarithmic functions f sxd − log b x, where the base b is a positive constant, are the inverse functions of the exponential functions. They will be studied in Chapter 6. Figure
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Section 1.2 Mathematical Models: A Catalog of Essential Functions y
21 shows the graphs of four logarithmic functions with various bases. In each case the domain is s0, `d, the range is s2`, `d, and the function increases slowly when x . 1.
y=log™ x y=log£ x
1 0
1
y=log∞ x
33
x
y=log¡¸ x
Example 6 Classify the following functions as one of the types of functions that we have discussed. (a) f sxd − 5 x (b) tsxd − x 5 11x (c) hsxd − (d) ustd − 1 2 t 1 5t 4 1 2 sx SOLUTION
(a) f sxd − 5 x is an exponential function. (The x is the exponent.) (b) tsxd − x 5 is a power function. (The x is the base.) We could also consider it to be a polynomial of degree 5. 11x (c) hsxd − is an algebraic function. 1 2 sx
figure 21
(d) ustd − 1 2 t 1 5t 4 is a polynomial of degree 4.
■
1. 2 Exercises 1–2 Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.
3 4. (a) y − 3x (b) y − 3 x (c) y − x 3 (d) y−s x
y
1. (a) f sxd − log 2 x (b) tsxd − sx 4
(c) hsxd −
F
2x 3 (d) ustd − 1 2 1.1t 1 2.54t 2 1 2 x2
g
(e) vstd − 5 t (f ) wsd − sin cos 2 f
y − x 2. (a) y − x (b) (c) y − x 2 s2 2 x 3 d (d) y − tan t 2 cos t
x
s sx 2 1 (f ) y− 3 11s 11s x 3
(e) y −
3–4 Match each equation with its graph. Explain your choices. (Don’t use a computer or graphing calculator.) 3. (a) y − x 2 (b) y − x 5 (c) y − x 8 y
0
f
g
G
5–6 Find the domain of the function. 5. f sxd −
cos x 1 6. tsxd − 1 2 tan x 1 2 sin x
h
x
7. (a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family. (b) Find an equation for the family of linear functions such that f s2d − 1 and sketch several members of the family. (c) Which function belongs to both families? 8. What do all members of the family of linear functions f sxd − 1 1 msx 1 3d have in common? Sketch several members of the family.
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34
Chapter 1 Functions and Limits
9. What do all members of the family of linear functions f sxd − c 2 x have in common? Sketch several members of the family. 10. Find expressions for the quadratic functions whose graphs are shown. y
(_2, 2)
f (4, 2) 0
3
x
g
y (0, 1) 0
x (1, _2.5)
11. Find an expression for a cubic function f if f s1d − 6 and f s21d − f s0d − f s2d − 0. 12. R ecent studies indicate that the average surface temperature of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T − 0.02t 1 8.50, where T is temperature in °C and t represents years since 1900. (a) What do the slope and Tintercept represent? (b) Use the equation to predict the average global surface temperature in 2100. 13. If the recommended adult dosage for a drug is D (in mg), then to determine the appropriate dosage c for a child of age a, pharmacists use the equation c − 0.0417Dsa 1 1d. Suppose the dosage for an adult is 200 mg. (a) Find the slope of the graph of c. What does it represent? (b) What is the dosage for a newborn? 14. The manager of a weekend flea market knows from past experience that if he charges x dollars for a rental space at the market, then the number y of spaces he can rent is given by the equation y − 200 2 4x. (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented can’t be negative quantities.) (b) What do the slope, the yintercept, and the xintercept of the graph represent? 15. The relationship between the Fahrenheit sFd and Celsius sCd temperature scales is given by the linear function F − 95 C 1 32. (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it represent? What is the Fintercept and what does it represent? 16. J ason leaves Detroit at 2:00 pm and drives at a constant speed west along I94. He passes Ann Arbor, 40 mi from Detroit, at 2:50 pm. (a) Express the distance traveled in terms of the time elapsed. (b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represent?
17. Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70°F and 173 chirps per minute at 80°F. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute, estimate the temperature. 18. T he manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day. (a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and what does it represent? (c) What is the yintercept of the graph and what does it represent? 19. A t the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 lbyin2. Below the surface, the water pressure increases by 4.34 lbyin2 for every 10 ft of descent. (a) Express the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure 100 lbyin2? 20. T he monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi. (a) Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship gives a suitable model. (b) Use part (a) to predict the cost of driving 1500 miles per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the Cintercept represent? (e) Why does a linear function give a suitable model in this situation? 21–22 For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices. 21. (a)
y
0
(b)
x
y
0
x
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35
Section 1.2 Mathematical Models: A Catalog of Essential Functions
22.
(a) y
(b) y
0
x
0
x
he table shows (lifetime) peptic ulcer rates (per 100 popula; 23. T tion) for various family incomes as reported by the National Health Interview Survey.
Femur length (cm)
Height (cm)
Femur length (cm)
Height (cm)
50.1 48.3 45.2 44.7
178.5 173.6 164.8 163.7
44.5 42.7 39.5 38.0
168.3 165.0 155.4 155.8
Ulcer rate (per 100 population)
Income $4,000 $6,000 $8,000 $12,000 $16,000 $20,000 $30,000 $45,000 $60,000
; 25. Anthropologists use a linear model that relates human femur (thighbone) length to height. The model allows an anthropologist to determine the height of an individual when only a partial skeleton (including the femur) is found. Here we find the model by analyzing the data on femur length and height for the eight males given in the following table. (a) Make a scatter plot of the data. (b) Find and graph the regression line that models the data. (c) An anthropologist finds a human femur of length 53 cm. How tall was the person?
14.1 13.0 13.4 12.5 12.0 12.4 10.5 9.4 8.2
; 26. When laboratory rats are exposed to asbestos fibers, some of them develop lung tumors. The table lists the results of several experiments by different scientists. (a) Find the regression line for the data. (b) Make a scatter plot and graph the regression line. Does the regression line appear to be a suitable model for the data? (c) What does the yintercept of the regression line represent?
(a) Make a scatter plot of these data and decide whether a linear model is appropriate. (b) Find and graph a linear model using the first and last data points. (c) Find and graph the least squares regression line. (d) Use the linear model in part (c) to estimate the ulcer rate for an income of $25,000. (e) According to the model, how likely is someone with an income of $80,000 to suffer from peptic ulcers? (f ) Do you think it would be reasonable to apply the model to someone with an income of $200,000?
iologists have observed that the chirping rate of crickets of ; 24. B a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures. (a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Use the linear model in part (b) to estimate the chirping rate at 100°F. Temperature (°F)
Chirping rate (chirpsymin)
Temperature (°F)
Chirping rate (chirpsymin)
50 55 60 65 70
20 46 79 91 113
75 80 85 90
140 173 198 211
Asbestos Percent of mice exposure that develop (fibersymL) lung tumors 50 400 500 900 1100
2 6 5 10 26
Asbestos Percent of mice exposure that develop (fibersymL) lung tumors 1600 1800 2000 3000
42 37 38 50
; 27. The table shows world average daily oil consumption from 1985 to 2010 measured in thousands of barrels per day. (a) Make a scatter plot and decide whether a linear model is appropriate. (b) Find and graph the regression line. (c) Use the linear model to estimate the oil consumption in 2002 and 2012. Years since 1985
Thousands of barrels of oil per day
0 5 10 15 20 25
60,083 66,533 70,099 76,784 84,077 87,302
Source: US Energy Information Administration
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
36
Chapter 1 Functions and Limits
he table shows average US retail residential prices of ; 28. T electricity from 2000 to 2012, measured in cents per kilowatt hour. (a) Make a scatter plot. Is a linear model appropriate? (b) Find and graph the regression line. (c) Use your linear model from part (b) to estimate the average retail price of electricity in 2005 and 2013.
he table shows the number N of species of reptiles and ; 31. T amphibians inhabiting Caribbean islands and the area A of the island in square miles. (a) Use a power function to model N as a function of A. (b) The Caribbean island of Dominica has area 291 mi 2. How many species of reptiles and amphibians would you expect to find on Dominica?
Years since 2000
CentsykWh
Island
0 2 4 6 8 10 12
8.24 8.44 8.95 10.40 11.26 11.54 11.58
Saba Monserrat Puerto Rico Jamaica Hispaniola Cuba
Source: US Energy Information Administration
29. Many physical quantities are connected by inverse square laws, that is, by power functions of the form f sxd − kx 22. In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light?
A 4 40 3,459 4,411 29,418 44,218
N 5 9 40 39 84 76
he table shows the mean (average) distances d of the ; 32. T planets from the sun (taking the unit of measurement to be the distance from planet Earth to the sun) and their periods T (time of revolution in years). (a) Fit a power model to the data. (b) Kepler’s Third Law of Planetary Motion states that “ The square of the period of revolution of a planet is proportional to the cube of its mean distance from the sun.” Does your model corroborate Kepler’s Third Law?
30. I t makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the speciesarea relation with a power function and, in particular, the number of species S of bats living in caves in central Mexico has been related to the surface area A of the caves by the equation S − 0.7A0.3. (a) The cave called Misión Imposible near Puebla, Mexico, has a surface area of A − 60 m2. How many species of bats would you expect to find in that cave? (b) If you discover that four species of bats live in a cave, estimate the area of the cave.
Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune
d 0.387 0.723 1.000 1.523 5.203 9.541 19.190 30.086
T 0.241 0.615 1.000 1.881 11.861 29.457 84.008 164.784
In this section we start with the basic functions we discussed in Section 1.2 and obtain new functions by shifting, stretching, and reflecting their graphs. We also show how to combine pairs of functions by the standard arithmetic operations and by composition.
Transformations of Functions By applying certain transformations to the graph of a given function we can obtain the graphs of related functions. This will give us the ability to sketch the graphs of many functions quickly by hand. It will also enable us to write equations for given graphs. Let’s first consider translations. If c is a positive number, then the graph of y − f sxd 1 c is just the graph of y − f sxd shifted upward a distance of c units (because each ycoordinate is increased by the same number c). Likewise, if tsxd − f sx 2 cd, where c . 0, then the value of t at x is the same as the value of f at x 2 c (c units to the left of x). ThereCopyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
37
Section 1.3 New Functions from Old Functions
fore the graph of y − f sx 2 cd is just the graph of y − f sxd shifted c units to the right (see Figure 1). Vertical and Horizontal Shifts Suppose c . 0. To obtain the graph of
y − f sxd 1 c, shift the graph of y − f sxd a distance c units upward y − f sxd 2 c, shift the graph of y − f sxd a distance c units downward y − f sx 2 cd, shift the graph of y − f sxd a distance c units to the right y − f sx 1 cd, shift the graph of y − f sxd a distance c units to the left y
y
y=ƒ+c
y=f(x+c)
c
y =ƒ
c 0
y=cƒ (c>1) y=f(_x)
y=f(xc)
y=ƒ y= 1c ƒ
c x
c
x
0
y=ƒc y=_ƒ
Figure 1 Translating the graph of f
Figure 2 Stretching and reflecting the graph of f
Now let’s consider the stretching and reflecting transformations. If c . 1, then the graph of y − cf sxd is the graph of y − f sxd stretched by a factor of c in the vertical direction (because each ycoordinate is multiplied by the same number c). The graph of y − 2f sxd is the graph of y − f sxd reflected about the xaxis because the point sx, yd is replaced by the point sx, 2yd. (See Figure 2 and the following chart, where the results of other stretching, shrinking, and reflecting transformations are also given.) Vertical and Horizontal Stretching and Reflecting Suppose c . 1. To obtain the
graph of
y − cf sxd, stretch the graph of y − f sxd vertically by a factor of c y − s1ycdf sxd, shrink the graph of y − f sxd vertically by a factor of c y − f scxd, shrink the graph of y − f sxd horizontally by a factor of c y − f sxycd, stretch the graph of y − f sxd horizontally by a factor of c y − 2f sxd, reflect the graph of y − f sxd about the xaxis y − f s2xd, reflect the graph of y − f sxd about the yaxis
Figure 3 illustrates these stretching transformations when applied to the cosine function with c − 2. For instance, in order to get the graph of y − 2 cos x we multiply the ycoordiCopyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
38
Chapter 1 Functions and Limits
nate of each point on the graph of y − cos x by 2. This means that the graph of y − cos x gets stretched vertically by a factor of 2. y
y=2 cos x
y
2
y=cos x
2
1
1 y= 2
0
y=cos 1 x
1
cos x x
1
y=cos 2x
2
0
x
y=cos x
Figure 3
Example 1 Given the graph of y − sx , use transformations to graph y − sx 2 2, y − sx 2 2 , y − 2sx , y − 2sx , and y − s2x .
SOLUTION The graph of the square root function y − sx , obtained from Figure 1.2.13(a), is shown in Figure 4(a). In the other parts of the figure we sketch y − sx 2 2 by shifting 2 units downward, y − sx 2 2 by shifting 2 units to the right, y − 2sx by reflecting about the xaxis, y − 2sx by stretching vertically by a factor of 2, and y − s2x by reflecting about the yaxis. y
y
y
y
y
y
1 0
x
1
x
0
0
x
2
x
0
x
0
x
0
_2
(a) y=œ„x
(b) y=œ„2 x
Figure 4
(d) y=_ œ„x
(c) y=œ„„„„ x2
(f ) y=œ„„ _x
(e) y=2 œ„x
■
Example 2 Sketch the graph of the function f sxd − x 2 1 6x 1 10. SOLUTION Completing the square, we write the equation of the graph as
y − x 2 1 6x 1 10 − sx 1 3d2 1 1 This means we obtain the desired graph by starting with the parabola y − x 2 and shifting 3 units to the left and then 1 unit upward (see Figure 5). y
y
1
(_3, 1) 0
Figure 5
(a) y=≈
x
_3
_1
0
(b) y=(x+3)@+1
x
■
Example 3 Sketch the graphs of the following functions. (a) y − sin 2x (b) y − 1 2 sin x Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
39
Section 1.3 New Functions from Old Functions
SOLUTION
(a) We obtain the graph of y − sin 2x from that of y − sin x by compressing horizontally by a factor of 2. (See Figures 6 and 7.) Thus, whereas the period of y − sin x is 2, the period of y − sin 2x is 2y2 − . y
y
y=sin x
1 0
π 2
π
y=sin 2x
1 x
0 π π 4
x
π
2
FIGURE 7
FIGURE 6
(b) To obtain the graph of y − 1 2 sin x, we again start with y − sin x. We reflect about the xaxis to get the graph of y − 2sin x and then we shift 1 unit upward to get y − 1 2 sin x. (See Figure 8.) y
y=1sin x
2 1 0
FIGURE 8
π 2
π
3π 2
x
2π
■
Example 4 Figure 9 shows graphs of the number of hours of daylight as functions of the time of the year at several latitudes. Given that Philadelphia is located at approximately 408N latitude, find a function that models the length of daylight at Philadelphia. 20 18 16 14 12
20° N 30° N 40° N 50° N
Hours 10 8
FIGURE 9 Graph of the length of daylight from March 21 through December 21 at various latitudes Source: Adapted from L. Harrison, Daylight, Twilight, Darkness and Time (New York: Silver, Burdett, 1935), 40.
6
60° N
4 2 0
Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.
SOLUTION Notice that each curve resembles a shifted and stretched sine function. By looking at the blue curve we see that, at the latitude of Philadelphia, daylight lasts about 14.8 hours on June 21 and 9.2 hours on December 21, so the amplitude of the curve (the factor by which we have to stretch the sine curve vertically) is 1 2 s14.8 2 9.2d − 2.8. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
40
Chapter 1 Functions and Limits
By what factor do we need to stretch the sine curve horizontally if we measure the time t in days? Because there are about 365 days in a year, the period of our model should be 365. But the period of y − sin t is 2, so the horizontal stretching factor is 2y365. We also notice that the curve begins its cycle on March 21, the 80th day of the year, so we have to shift the curve 80 units to the right. In addition, we shift it 12 units upward. Therefore we model the length of daylight in Philadelphia on the tth day of the year by the function
F
y
Lstd − 12 1 2.8 sin
_1
0
1
x
(a) y=≈1 y
2 st 2 80d 365
G
■
Another transformation of some interest is taking the absolute value of a function. If y − f sxd , then according to the definition of absolute value, y − f sxd when f sxd > 0 and y − 2f sxd when f sxd , 0. This tells us how to get the graph of y − f sxd from the graph of y − f sxd: the part of the graph that lies above the xaxis remains the same; the part that lies below the xaxis is reflected about the xaxis.




Example 5 Sketch the graph of the function y −  x 2 2 1 . SOLUTION We first graph the parabola y − x 2 2 1 in Figure 10(a) by shifting the
_1
0
1
(b) y= ≈1 
figure 10
x
parabola y − x 2 downward 1 unit. We see that the graph lies below the xaxis when 21 , x , 1, so we reflect that part of the graph about the xaxis to obtain the graph of y − x 2 2 1 in Figure 10(b). ■


Combinations of Functions Two functions f and t can be combined to form new functions f 1 t, f 2 t, ft, and fyt in a manner similar to the way we add, subtract, multiply, and divide real numbers. The sum and difference functions are defined by s f 1 tdsxd − f sxd 1 tsxd s f 2 tdsxd − f sxd 2 tsxd If the domain of f is A and the domain of t is B, then the domain of f 1 t is the intersection A > B because both f sxd and tsxd have to be defined. For example, the domain of f sxd − sx is A − f0, `d and the domain of tsxd − s2 2 x is B − s2`, 2g, so the domain of s f 1 tdsxd − sx 1 s2 2 x is A > B − f0, 2g. Similarly, the product and quotient functions are defined by
SD
s ftdsxd − f sxd tsxd
f f sxd sxd − t tsxd
The domain of ft is A > B. Because we can’t divide by 0, the domain of fyt is therefore hx [ A > B tsxd ± 0j. For instance, if f sxd − x 2 and tsxd − x 2 1, then the domain of the rational function s fytdsxd − x 2ysx 2 1d is hx x ± 1j, or s2`, 1d ø s1, `d. There is another way of combining two functions to obtain a new function. For example, suppose that y − f sud − su and u − tsxd − x 2 1 1. Since y is a function of u and u is, in turn, a function of x, it follows that y is ultimately a function of x. We compute this by substitution:


y − f sud − f stsxdd − f sx 2 1 1d − sx 2 1 1 The procedure is called composition because the new function is composed of the two given functions f and t. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.3 New Functions from Old Functions
In general, given any two functions f and t, we start with a number x in the domain of t and calculate tsxd. If this number tsxd is in the domain of f, then we can calculate the value of f stsxdd. Notice that the output of one function is used as the input to the next function. The result is a new function hsxd − f s tsxdd obtained by substituting t into f. It is called the composition (or composite) of f and t and is denoted by f 8 t (“ f circle t”).
x (input) g
©
41
f•g
f
Definition Given two functions f and t, the composite function f 8 t (also called the composition of f and t) is defined by s f 8 tdsxd − f stsxdd
f { ©} (output)
FIGURE 11
The f 8 t machine is composed of the t machine (first) and then the f machine.
The domain of f 8 t is the set of all x in the domain of t such that tsxd is in the domain of f. In other words, s f 8 tdsxd is defined whenever both tsxd and f s tsxdd are defined. Figure 11 shows how to picture f 8 t in terms of machines.
Example 6 If f sxd − x 2 and tsxd − x 2 3, find the composite functions f 8 t and t 8 f. SOLUTION We have
s f 8 tdsxd − f stsxdd − f sx 2 3d − sx 2 3d2 st 8 f dsxd − ts f sxdd − tsx 2 d − x 2 2 3
n
NOTE You can see from Example 6 that, in general, f 8 t ± t 8 f. Remember, the notation f 8 t means that the function t is applied first and then f is applied second. In Example 6, f 8 t is the function that first subtracts 3 and then squares; t 8 f is the function that first squares and then subtracts 3.
Example 7 If f sxd − sx and tsxd − s2 2 x , find each of the following functions and their domains. (a) f 8 t (b) t 8 f (c) f 8 f (d) t 8 t SOLUTION
(a)
4 s f 8 tdsxd − f stsxdd − f (s2 2 x ) − ss2 2 x − s 22x


The domain of f 8 t is hx 2 2 x > 0j − hx x < 2j − s2`, 2g. (b)
If 0 < a < b, then a 2 < b 2.
s t 8 f dsxd − ts f sxdd − t (sx ) − s2 2 sx
For sx to be defined we must have x > 0. For s2 2 sx to be defined we must have 2 2 sx > 0, that is, sx < 2, or x < 4. Thus we have 0 < x < 4, so the domain of t 8 f is the closed interval f0, 4g. (c)
4 s f 8 f dsxd − f s f sxdd − f (sx ) − ssx − s x
The domain of f 8 f is f0, `d. (d)
st 8 tdsxd − tstsxdd − t (s2 2 x ) − s2 2 s2 2 x
This expression is defined when both 2 2 x > 0 and 2 2 s2 2 x > 0. The first inequality means x < 2, and the second is equivalent to s2 2 x < 2, or 2 2 x < 4, or x > 22. Thus 22 < x < 2, so the domain of t 8 t is the closed interval f22, 2g. ■ Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
42
Chapter 1 Functions and Limits
It is possible to take the composition of three or more functions. For instance, the composite function f 8 t 8 h is found by first applying h, then t, and then f as follows: s f 8 t 8 hdsxd − f stshsxddd
Example 8 Find f 8 t 8 h if f sxd − xysx 1 1d, tsxd − x 10, and hsxd − x 1 3. SOLUTIOn
s f 8 t 8 hdsxd − f stshsxddd − f stsx 1 3dd − f ssx 1 3d10 d −
sx 1 3d10 sx 1 3d10 1 1
■
So far we have used composition to build complicated functions from simpler ones. But in calculus it is often useful to be able to decompose a complicated function into simpler ones, as in the following example.
Example 9 Given Fsxd − cos2sx 1 9d, find functions f , t, and h such that F − f 8 t 8 h. SOLUTION Since Fsxd − fcossx 1 9dg 2, the formula for F says: First add 9, then take
the cosine of the result, and finally square. So we let hsxd − x 1 9 tsxd − cos x f sxd − x 2 Then
s f 8 t 8 hdsxd − f stshsxddd − f stsx 1 9dd − f scossx 1 9dd
− fcossx 1 9dg 2 − Fsxd
■
1. 3 Exercises 1. S uppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the xaxis. (f ) Reflect about the yaxis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3. 2. Explain how each graph is obtained from the graph of y − f sxd. (a) y − f sxd 1 8 (b) y − f sx 1 8d (c) y − 8 f sxd (d) y − f s8xd (e) y − 2f sxd 2 1 (f ) y − 8 f s 81 xd
y
@
6
! f
3
#
$ _6
_3
%
0
3
6
x
_3
4. The graph of f is given. Draw the graphs of the following functions. (a) y − f sxd 2 3 (b) y − f sx 1 1d 1 (c) y − 2 f sxd (d) y − 2f sxd y
3. The graph of y − f sxd is given. Match each equation with its graph and give reasons for your choices. (a) y − f sx 2 4d (b) y − f sxd 1 3
2
(c) y − 13 f sxd (d) y − 2f sx 1 4d
0
1
x
(e) y − 2 f sx 1 6d Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.3 New Functions from Old Functions
5. T he graph of f is given. Use it to graph the following functions. (a) y − f s2xd (b) y − f ( 12 x) (c) y − f s2xd (d) y − 2f s2xd y
0
x
1
y=œ„„„„„„ 3x≈
1.5 0
x
3
7.
y 3
y _4
2
S D
1 21. y − x 2 2 22. y − tan x 2 4 4




5

25. The city of New Orleans is located at latitude 30°N. Use Figure 9 to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. To check the accuracy of your model, use the fact that on March 31 the sun rises at 5:51 am and sets at 6:18 pm in New Orleans. 26. A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness varies by 60.35 magnitude. Find a function that models the brightness of Delta Cephei as a function of time.
y
0
 
19. y − sin ( 21 x ) 20. y− x 22

6–7 The graph of y − s3x 2 x 2 is given. Use transformations to create a function whose graph is as shown.
6.
17. y − 2 2 sx 18. y − 3 2 2 cos x
23. y − sx 2 1 24. y − cos x
1
43
_1 0
x
_1
x
_2.5
8. (a) How is the graph of y − 2 sin x related to the graph of y − sin x? Use your answer and Figure 6 to sketch the graph of y − 2 sin x. (b) How is the graph of y − 1 1 sx related to the graph of y − sx ? Use your answer and Figure 4(a) to sketch the graph of y − 1 1 sx . 9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 9. y − 2x 2 10. y − sx 2 3d2
27. S ome of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is about 12 hours and on June 30, 2009, high tide occurred at 6:45 am. Find a function involving the cosine function that models the water depth Dstd (in meters) as a function of time t (in hours after midnight) on that day. 28. In a normal respiratory cycle the volume of air that moves into and out of the lungs is about 500 mL. The reserve and residue volumes of air that remain in the lungs occupy about 2000 mL and a single respiratory cycle for an average human takes about 4 seconds. Find a model for the total volume of air Vstd in the lungs as a function of time.
     
29. (a) How is the graph of y − f ( x ) related to the graph of f ? (b) Sketch the graph of y − sin x . (c) Sketch the graph of y − s x . 30. Use the given graph of f to sketch the graph of y − 1yf sxd. Which features of f are the most important in sketching y − 1yf sxd? Explain how they are used. y
1 11. y − x 3 1 1 12. y−12 x
1
13. y − 2 cos 3x 14. y − 2 sx 1 1
0
1
x
15. y − x 2 2 4x 1 5 16. y − 1 1 sin x Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
44
Chapter 1 Functions and Limits
31–32 Find (a) f 1 t, (b) f 2 t, (c) f t, and (d) fyt and state their domains.
(d) s t 8 f ds6d (e) s t 8 tds22d (f) s f 8 f ds4d y
31. f sxd − x 3 1 2x 2, tsxd − 3x 2 2 1
g
32. f sxd − s3 2 x , tsxd − sx 2 2 1
f
2
33–38 Find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , and (d) t 8 t and their domains.
0
33. f sxd − 3x 1 5, tsxd − x 2 1 x
x
2
34. f sxd − x 3 2 2, tsxd − 1 2 4x 54. U se the given graphs of f and t to estimate the value of f s tsxdd for x − 25, 24, 23, . . . , 5. Use these estimates to sketch a rough graph of f 8 t.
35. f sxd − sx 1 1, tsxd − 4x 2 3 36. f sxd − sin x, tsxd − x 2 1 1 37. f sxd − x 1
x11 1 , tsxd − x12 x
y
x 38. f sxd − , tsxd − sin 2x 11x
g 1 0
39–42 Find f 8 t 8 h. 39. f sxd − 3x 2 2, tsxd − sin x, hsxd − x 2

1
x
f

40. f sxd − x 2 4 , tsxd − 2 x, hsxd − sx 41. f sxd − sx 2 3 , tsxd − x 2, hsxd − x 3 1 2 42. f sxd − tan x, tsxd −
x 3 , hsxd − s x x21
55. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cmys. (a) Express the radius r of this circle as a function of the time t (in seconds). (b) If A is the area of this circle as a function of the radius, find A 8 r and interpret it.
43–48 Express the function in the form f 8 t. 43. Fsxd − s2 x 1 x 2 d 4 44. Fsxd − cos2 x 45. Fsxd −
Î
3 x x s 46. Gsxd − 3 3 11x 11s x
47. vstd − secst 2 d tanst 2 d
48. ustd −
tan t 1 1 tan t
49–51 Express the function in the form f 8 t 8 h. 8 49. Rsxd − ssx 2 1 50. Hsxd − s 21 x
 
51. Sstd − sin2scos td 52. Use the table to evaluate each expression. (a) f s ts1dd (b) ts f s1dd (c) f s f s1dd (d) ts ts1dd (e) s t 8 f ds3d (f ) s f 8 tds6d x
1
2
3
4
5
6
f sxd
3
1
4
2
2
5
tsxd
6
3
2
1
2
3
53. Use the given graphs of f and t to evaluate each expression, or explain why it is undefined. (a) f s ts2dd (b) ts f s0dd (c) s f 8 tds0d
56. A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cmys. (a) Express the radius r of the balloon as a function of the time t (in seconds). (b) If V is the volume of the balloon as a function of the radius, find V 8 r and interpret it. 57. A ship is moving at a speed of 30 kmyh parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon. (a) Express the distance s between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon; that is, find f so that s − f sdd. (b) Express d as a function of t, the time elapsed since noon; that is, find t so that d − tstd. (c) Find f 8 t. What does this function represent? 58. An airplane is flying at a speed of 350 miyh at an altitude of one mile and passes directly over a radar station at time t − 0. (a) Express the horizontal distance d (in miles) that the plane has flown as a function of t. (b) Express the distance s between the plane and the radar station as a function of d. (c) Use composition to express s as a function of t.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.4 The Tangent and Velocity Problems
59. The Heaviside function H is defined by Hstd −
H
0 if t , 0 1 if t > 0
It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t − 0 and 120 volts are applied instantaneously to the circuit. Write a formula for Vstd in terms of Hstd. (c) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t − 5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for Vstd in terms of Hstd. (Note that starting at t − 5 corresponds to a translation.) 60. The Heaviside function defined in Exercise 59 can also be used to define the ramp function y − ctHstd, which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y − tHstd. (b) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t − 0 and the voltage is gradually increased to 120 volts over a 60second time interval. Write a formula for Vstd in terms of Hstd for t < 60.
45
(c) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t − 7 seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for Vstd in terms of Hstd for t < 32.
61. Let f and t be linear functions with equations f sxd − m1 x 1 b1 and tsxd − m 2 x 1 b 2. Is f 8 t also a linear function? If so, what is the slope of its graph? 62. If you invest x dollars at 4% interest compounded annually, then the amount Asxd of the investment after one year is Asxd − 1.04x. Find A 8 A, A 8 A 8 A, and A 8 A 8 A 8 A. What do these compositions represent? Find a formula for the composition of n copies of A. 63. (a) If tsxd − 2x 1 1 and hsxd − 4x 2 1 4x 1 7, find a function f such that f 8 t − h. (Think about what operations you would have to perform on the formula for t to end up with the formula for h.) (b) If f sxd − 3x 1 5 and hsxd − 3x 2 1 3x 1 2, find a function t such that f 8 t − h. 64. If f sxd − x 1 4 and hsxd − 4x 2 1, find a function t such that t 8 f − h. 65. Suppose t is an even function and let h − f 8 t. Is h always an even function? 66. Suppose t is an odd function and let h − f 8 t. Is h always an odd function? What if f is odd? What if f is even?
In this section we see how limits arise when we attempt to find the tangent to a curve or the velocity of an object.
The Tangent Problem t
(a) P
t
Example 1 Find an equation of the tangent line to the parabola y − x 2 at the
l
point Ps1, 1d. (b)
FIGURE 1
C
The word tangent is derived from the Latin word tangens, which means “touching.” Thus a tangent to a curve is a line that touches the curve. In other words, a tangent line should have the same direction as the curve at the point of contact. How can this idea be made precise? For a circle we could simply follow Euclid and say that a tangent is a line that intersects the circle once and only once, as in Figure 1(a). For more complicated curves this definition is inadequate. Figure l(b) shows two lines l and t passing through a point P on a curve C. The line l intersects C only once, but it certainly does not look like what we think of as a tangent. The line t, on the other hand, looks like a tangent but it intersects C twice. To be specific, let’s look at the problem of trying to find a tangent line t to the parabola y − x 2 in the following example.
SOLUTION We will be able to find an equation of the tangent line t as soon as we know its slope m. The difficulty is that we know only one point, P, on t, whereas we need two points to compute the slope. But observe that we can compute an approximation to m
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
46
Chapter 1 Functions and Limits y
Q {x, ≈} y=≈
by choosing a nearby point Qsx, x 2 d on the parabola (as in Figure 2) and computing the slope mPQ of the secant line PQ. [A secant line, from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.] We choose x ± 1 so that Q ± P. Then
t
P (1, 1)
mPQ −
x
0
x2 2 1 x21
For instance, for the point Qs1.5, 2.25d we have FIGURE 2
mPQ − x
mPQ
2 1.5 1.1 1.01 1.001
3 2.5 2.1 2.01 2.001
x
mPQ
0 0.5 0.9 0.99 0.999
1 1.5 1.9 1.99 1.999
2.25 2 1 1.25 − − 2.5 1.5 2 1 0.5
The tables in the margin show the values of mPQ for several values of x close to 1. The closer Q is to P, the closer x is to 1 and, it appears from the tables, the closer mPQ is to 2. This suggests that the slope of the tangent line t should be m − 2. We say that the slope of the tangent line is the limit of the slopes of the secant lines, and we express this symbolically by writing lim mPQ − m and lim
Q lP
y
Q
xl1
x2 2 1 −2 x21
Assuming that the slope of the tangent line is indeed 2, we use the pointslope form of the equation of a line [y 2 y1 − msx 2 x 1d, see Appendix B] to write the equation of the tangent line through s1, 1d as y 2 1 − 2sx 2 1d or y − 2x 2 1 Figure 3 illustrates the limiting process that occurs in this example. As Q approaches P along the parabola, the corresponding secant lines rotate about P and approach the tangent line t. y
y
t
t
t Q
P
P
0
P
0
x
x
Q
0
x
Q approaches P from the right y
y
y
t
Q
t
P 0
Q
P
0
x
t
x
0
Q
P x
Q approaches P from the left
FIGURE 3
■
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.4 The Tangent and Velocity Problems
TEC In Visual 1.4 you can see how the process in Figure 3 works for additional functions.
t
Q
0.00 0.02 0.04 0.06 0.08 0.10
100.00 81.87 67.03 54.88 44.93 36.76
47
Many functions that occur in science are not described by explicit equations; they are defined by experimental data. The next example shows how to estimate the slope of the tangent line to the graph of such a function.
Example 2 The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The data in the table describe the charge Q remaining on the capacitor (measured in microcoulombs) at time t (measured in seconds after the flash goes off). Use the data to draw the graph of this function and estimate the slope of the tangent line at the point where t − 0.04. [Note: The slope of the tangent line represents the electric current flowing from the capacitor to the flash bulb (measured in microamperes).] SOLUTION In Figure 4 we plot the given data and use them to sketch a curve that approximates the graph of the function. Q (microcoulombs) 100 90 80 70 60 50
FIGURE 4
0
0.02
0.04
0.06
0.08
0.1
t (seconds)
Given the points Ps0.04, 67.03d and Rs0.00, 100.00d on the graph, we find that the slope of the secant line PR is mPR − R
mPR
(0.00, 100.00) (0.02, 81.87) (0.06, 54.88) (0.08, 44.93) (0.10, 36.76)
2824.25 2742.00 2607.50 2552.50 2504.50
100.00 2 67.03 − 2824.25 0.00 2 0.04
The table at the left shows the results of similar calculations for the slopes of other secant lines. From this table we would expect the slope of the tangent line at t − 0.04 to lie somewhere between 2742 and 2607.5. In fact, the average of the slopes of the two closest secant lines is 1 2 s2742
2 607.5d − 2674.75
So, by this method, we estimate the slope of the tangent line to be about 2675. Another method is to draw an approximation to the tangent line at P and measure the sides of the triangle ABC, as in Figure 5. Q (microcoulombs) 100 90 80
A P
70 60 50
FIGURE 5
0
B 0.02
C 0.04
0.06
0.08
0.1
t (seconds)
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48
Chapter 1 Functions and Limits
This gives an estimate of the slope of the tangent line as he physical meaning of the answer T in Example 2 is that the electric current flowing from the capacitor to the flash bulb after 0.04 seconds is about 2670 microamperes.
2
 AB  < 2 80.4 2 53.6 − 2670 0.06 2 0.02  BC 
■
The Velocity Problem If you watch the speedometer of a car as you travel in city traffic, you see that the speed doesn’t stay the same for very long; that is, the velocity of the car is not constant. We assume from watching the speedometer that the car has a definite velocity at each moment, but how is the “instantaneous” velocity defined? Let’s investigate the example of a falling ball.
Example 3 Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground. Find the velocity of the ball after 5 seconds. SOLUTION Through experiments carried out four centuries ago, Galileo discovered that the distance fallen by any freely falling body is proportional to the square of the time it has been falling. (This model for free fall neglects air resistance.) If the distance fallen after t seconds is denoted by sstd and measured in meters, then Galileo’s law is expressed by the equation Steve Allen / Stockbyte / Getty Images
sstd − 4.9t 2 The difficulty in finding the velocity after 5 seconds is that we are dealing with a single instant of time st − 5d, so no time interval is involved. However, we can approximate the desired quantity by computing the average velocity over the brief time interval of a tenth of a second from t − 5 to t − 5.1: average velocity −
The CN Tower in Toronto was the tallest freestanding building in the world for 32 years.
change in position time elapsed
−
ss5.1d 2 ss5d 0.1
−
4.9s5.1d2 2 4.9s5d2 − 49.49 mys 0.1
The following table shows the results of similar calculations of the average velocity over successively smaller time periods. Time interval
Average velocity smysd
5 1 8. For the function A whose graph is shown, state the following. (a) lim Asxd (b) lim2 Asxd x l23
x l2
(c) lim1 Asxd (d) lim Asxd x l2
x l21
1 1 sin x if x , 0 12. f sxd − cos x if 0 < x < sin x if x .
(e) The equations of the vertical asymptotes
; 13–14 Use the graph of the function f to state the value of each limit, if it exists. If it does not exist, explain why. (a) lim2 f sxd (b) lim1 f sxd (c) lim f sxd
y
xl0
0
_3
2
x
5
xl0
xl0
1 x2 1 x 13. f sxd − f sxd − 1yx 14. 112 sx 3 1 x 2 15–18 Sketch the graph of an example of a function f that satisfies all of the given conditions. 15. lim2 f sxd − 21, lim1 f sxd − 2, f s0d − 1
9. For the function f whose graph is shown, state the following. (a) lim f sxd (b) lim f sxd (c) lim f sxd
16. lim f sxd − 1, lim2 f sxd − 22, lim1 f sxd − 2,
lim2 f sxd (e) lim1 f sxd (d)
17. lim1 f sxd − 4, lim2 f sxd − 2, lim f sxd − 2,
x l27
x l23
xl6
xl0
xl6
(f ) The equations of the vertical asymptotes.
xl0
xl0
xl0
xl3
xl3
y
xl3
f s0d − 21, f s3d − 1 xl3
x l 22
f s3d − 3, f s22d − 1
18. lim2 f sxd − 2, lim1 f sxd − 0, lim2 f sxd − 3, _7
_3
0
6
x
xl0
xl0
xl4
lim1 f sxd − 0, f s0d − 2, f s4d − 1
xl4
19–22 Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). 10. A patient receives a 150mg injection of a drug every 4 hours. The graph shows the amount f std of the drug in the blood stream after t hours. Find lim f std and lim1 f std
tl 122
tl 12
19. lim
x l3
x 2 2 3x , x2 2 9
x − 3.1, 3.05, 3.01, 3.001, 3.0001,
2.9, 2.95, 2.99, 2.999, 2.9999
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Section 1.5 The Limit of a Function
20. lim
x l 23
x 2 2 3x , x2 2 9
x − 22.5, 22.9, 22.95, 22.99, 22.999, 22.9999,
23.5, 23.1, 23.05, 23.01, 23.001, 23.0001 sin x , x − 61, 60.5, 60.2, 60.1, 60.05, 60.01 x 1 tan x
21. lim
xl0
hl 0
40. (a) Find the vertical asymptotes of the function x2 1 1 y− 3x 2 2x 2 (b) Confirm your answer to part (a) by graphing the ; function.
h − 60.5, 60.1, 60.01, 60.001, 60.0001 23–26 Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. sin 3 1 1 p9 23. lim 24. lim l 0 tan 2 p l 21 1 1 p 15
1 1 and lim1 3 x l 1 x 21 x 21 (a) by evaluating f sxd − 1ysx 3 2 1d for values of x that approach 1 from the left and from the right, (b) by reasoning as in Example 9, and (c) from a graph of f.
41. Determine lim2 x l1
s2 1 hd5 2 32 , h
22. lim
;
3
; 42. (a) By graphing the function f sxd − stan 4xdyx and zooming in toward the point where the graph crosses the yaxis, estimate the value of lim x l 0 f sxd. (b) Check your answer in part (a) by evaluating f sxd for values of x that approach 0. 43. (a) Evaluate the function f sxd − x 2 2 s2 xy1000d for x − 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of
5t 2 1 25. lim1 x x 26. lim x l0 tl0 t
lim
xl0
2 ; 27. (a) By graphing the function f sxd − scos 2x 2 cos xdyx and zooming in toward the point where the graph crosses the yaxis, estimate the value of lim x l 0 f sxd. (b) Check your answer in part (a) by evaluating f sxd for values of x that approach 0.
; 28. (a) Estimate the value of lim
xl0
sin x sin x
by graphing the function f sxd − ssin xdyssin xd. State your answer correct to two decimal places. (b) Check your answer in part (a) by evaluating f sxd for values of x that approach 0. 29–39 Determine the infinite limit. x11 x11 29. lim1 30. lim x l5 x 2 5 x l 52 x 2 5 31. lim
x l1
33. 35.
22x sx 32. lim x l32 sx 2 3d 5 sx 2 1d2
lim
x l 221
lim
x21 x21 34. lim x l 0 x 2sx 1 2d x 2sx 1 2d
xlsy2d1
1 sec x 36. lim cot x x l2 x
x 2 2 2x 37. lim 2 x csc x 38. lim2 2 x l2 x l 2 x 2 4x 1 4 39. lim1 x l2
x 2 2 2x 2 8 x 2 2 5x 1 6
61
S
x2 2
2x 1000
D
(b) Evaluate f sxd for x − 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again.
44. (a) Evaluate hsxd − stan x 2 xdyx 3 for x − 1, 0.5, 0.1, 0.05, 0.01, and 0.005. tan x 2 x (b) Guess the value of lim . xl0 x3 (c) Evaluate hsxd for successively smaller values of x until you finally reach a value of 0 for hsxd. Are you still confident that your guess in part (b) is correct? Explain why you eventually obtained values of 0 for hsxd. (In Section 6.8 a method for evaluating this limit will be explained.) (d) Graph the function h in the viewing rectangle f21, 1g ; by f0, 1g. Then zoom in toward the point where the graph crosses the yaxis to estimate the limit of hsxd as x approaches 0. Continue to zoom in until you observe distortions in the graph of h. Compare with the results of part (c). raph the function f sxd − sinsyxd of Example 4 in the ; 45. G viewing rectangle f21, 1g by f21, 1g. Then zoom in toward the origin several times. Comment on the behavior of this function. 1 46. Consider the function f sxd − tan . x 1 1 1 , ,... (a) Show that f sxd − 0 for x − , 2 3 4 4 4 , ,... (b) Show that f sxd − 1 for x − , 5 9 1 (c) What can you conclude about lim1 tan ? xl0 x
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62
Chapter 1 Functions and Limits
se a graph to estimate the equations of all the vertical ; 47. U asymptotes of the curve y − tans2 sin xd
where m 0 is the mass of the particle at rest and c is the speed of light. What happens as v l c2?
; 49. (a) Use numerical and graphical evidence to guess the value of the limit
2 < x <
Then find the exact equations of these asymptotes.
48. I n the theory of relativity, the mass of a particle with velocity v is m0 m− s1 2 v 2yc 2
lim xl1
x3 2 1 sx 2 1
(b) How close to 1 does x have to be to ensure that the function in part (a) is within a distance 0.5 of its limit?
In Section 1.5 we used calculators and graphs to guess the values of limits, but we saw that such methods don’t always lead to the correct answer. In this section we use the following properties of limits, called the Limit Laws, to calculate limits. Limit Laws Suppose that c is a constant and the limits lim f sxd and lim tsxd
xla
xla
exist. Then 1. lim f f sxd 1 tsxdg − lim f sxd 1 lim tsxd xla
xla
xla
2. lim f f sxd 2 tsxdg − lim f sxd 2 lim tsxd xla
xla
xla
3. lim fcf sxdg − c lim f sxd xla
xla
4. lim f f sxd tsxdg − lim f sxd lim tsxd xla
xla
5. lim
lim f sxd f sxd xla − tsxd lim tsxd
xla
Sum Law Difference Law Constant Multiple Law Product Law Quotient Law
xla
xla
if lim tsxd ± 0 xla
These five laws can be stated verbally as follows: 1. The limit of a sum is the sum of the limits. 2. The limit of a difference is the difference of the limits. 3. The limit of a constant times a function is the constant times the limit of the function. 4. The limit of a product is the product of the limits. 5. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0). It is easy to believe that these properties are true. For instance, if f sxd is close to L and tsxd is close to M, it is reasonable to conclude that f sxd 1 tsxd is close to L 1 M. This gives us an intuitive basis for believing that Law 1 is true. In Section 1.7 we give a
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Section 1.6 Calculating Limits Using the Limit Laws
63
precise definition of a limit and use it to prove this law. The proofs of the remaining laws are given in Appendix F. y
f 1
0
g
1
x
Example 1 Use the Limit Laws and the graphs of f and t in Figure 1 to evaluate the following limits, if they exist. f sxd (a) lim f f sxd 1 5tsxdg (b) lim f f sxdtsxdg (c) lim x l 22 xl1 x l 2 tsxd SOLUTION
(a) From the graphs of f and t we see that lim f sxd − 1 and lim tsxd − 21
FIGURE 1
x l 22
x l 22
Therefore we have lim f f sxd 1 5tsxdg − lim f sxd 1 lim f5tsxdg (by Limit Law 1)
x l 22
x l 22
x l 22
− lim f sxd 1 5 lim tsxd (by Limit Law 3) x l 22
x l 22
− 1 1 5s21d − 24 (b) We see that lim x l 1 f sxd − 2. But lim x l 1 tsxd does not exist because the left and right limits are different: lim tsxd − 22 lim1 tsxd − 21
x l 12
xl1
So we can’t use Law 4 for the desired limit. But we can use Law 4 for the onesided limits: lim f f sxdtsxdg − lim2 f sxd lim2 tsxd − 2 s22d − 24
x l 12
x l1
x l1
lim f f sxdtsxdg − lim1 f sxd lim1 tsxd − 2 s21d − 22
x l 11
x l1
x l1
The left and right limits aren’t equal, so lim x l 1 f f sxdtsxdg does not exist. (c) The graphs show that lim f sxd < 1.4 and lim tsxd − 0
xl2
xl2
Because the limit of the denominator is 0, we can’t use Law 5. The given limit does not exist because the denominator approaches 0 while the numerator approaches a nonzero number. ■ If we use the Product Law repeatedly with tsxd − f sxd, we obtain the following law. Power Law
fx l a
g
6. lim f f sxdg n − lim f sxd n where n is a positive integer x la
In applying these six limit laws, we need to use two special limits: 7. lim c − c xla
8. lim x − a xla
These limits are obvious from an intuitive point of view (state them in words or draw graphs of y − c and y − x), but proofs based on the precise definition are requested in the exercises for Section 1.7. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
64
Chapter 1 Functions and Limits
If we now put f sxd − x in Law 6 and use Law 8, we get another useful special limit. 9. lim x n − a n where n is a positive integer xla
A similar limit holds for roots as follows. (For square roots the proof is outlined in Exercise 1.7.37.) n n 10. lim s x −s a where n is a positive integer
xla
(If n is even, we assume that a . 0.) More generally, we have the following law, which is proved in Section 1.8 as a consequence of Law 10. n n lim f sxd f sxd − s 11. lim s where n is a positive integer x la
Root Law
x la
g
f
f sxd . 0. If n is even, we assume that xlim la
Example 2 Evaluate the following limits and justify each step. x 3 1 2x 2 2 1 lim (a) lim s2x 2 2 3x 1 4d (b) x l5 x l 22 5 2 3x SOLUTION
(a)
lim s2x 2 2 3x 1 4d − lim s2x 2 d 2 lim s3xd 1 lim 4 (by Laws 2 and 1)
x l5
x l5
x l5
x l5
− 2 lim x 2 2 3 lim x 1 lim 4 (by 3)
− 2s5 2 d 2 3s5d 1 4
− 39
x l5
x l5
x l5
(by 9, 8, and 7)
(b) We start by using Law 5, but its use is fully justified only at the final stage when we see that the limits of the numerator and denominator exist and the limit of the denominator is not 0.
x 3 1 2x 2 2 1 lim − x l 22 5 2 3x
lim sx 3 1 2x 2 2 1d
x l 22
(by Law 5)
lim s5 2 3xd
x l 22
lim x 3 1 2 lim x 2 2 lim 1
−
x l 22
x l 22
x l 22
lim 5 2 3 lim x
x l 22
x l 22
s22d3 1 2s22d2 2 1 5 2 3s22d
−
−2
1 11
(by 1, 2, and 3) (by 9, 8, and 7)
■
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Section 1.6 Calculating Limits Using the Limit Laws
Newton and Limits Isaac Newton was born on Christmas Day in 1642, the year of Galileo’s death. When he entered Cambridge University in 1661 Newton didn’t know much mathematics, but he learned quickly by reading Euclid and Descartes and by attending the lectures of Isaac Barrow. Cambridge was closed because of the plague in 1665 and 1666, and Newton returned home to reflect on what he had learned. Those two years were amazingly productive for at that time he made four of his major discoveries: (1) his representation of functions as sums of infinite series, including the binomial theorem; (2) his work on differential and integral calculus; (3) his laws of motion and law of universal gravitation; and (4) his prism experiments on the nature of light and color. Because of a fear of controversy and criticism, he was reluctant to publish his discoveries and it wasn’t until 1687, at the urging of the astronomer Halley, that Newton published Principia Mathematica. In this work, the greatest scientific treatise ever written, Newton set forth his version of calculus and used it to investigate mechanics, fluid dynamics, and wave motion, and to explain the motion of planets and comets. The beginnings of calculus are found in the calculations of areas and volumes by ancient Greek scholars such as Eudoxus and Archimedes. Although aspects of the idea of a limit are implicit in their “method of exhaustion,” Eudoxus and Archimedes never explicitly formulated the concept of a limit. Likewise, mathematicians such as Cavalieri, Fermat, and Barrow, the immediate precursors of Newton in the development of calculus, did not actually use limits. It was Isaac Newton who was the first to talk explicitly about limits. He explained that the main idea behind limits is that quantities “approach nearer than by any given difference.” Newton stated that the limit was the basic concept in calculus, but it was left to later mathe maticians like Cauchy to clarify his ideas about limits.
65
NOTE If we let f sxd − 2x 2 2 3x 1 4, then f s5d − 39. In other words, we would
have gotten the correct answer in Example 2(a) by substituting 5 for x. Similarly, direct substitution provides the correct answer in part (b). The functions in Example 2 are a polynomial and a rational function, respectively, and similar use of the Limit Laws proves that direct substitution always works for such functions (see Exercises 57 and 58). We state this fact as follows. Direct Substitution Property If f is a polynomial or a rational function and a is in the domain of f, then lim f sxd − f sad x la
Functions with the Direct Substitution Property are called continuous at a and will be studied in Section 1.8. However, not all limits can be evaluated by direct substitution, as the following examples show.
Example 3 Find lim xl1
x2 2 1 . x21
SOLUTION Let f sxd − sx 2 2 1dysx 2 1d. We can’t find the limit by substituting x − 1
because f s1d isn’t defined. Nor can we apply the Quotient Law, because the limit of the denominator is 0. Instead, we need to do some preliminary algebra. We factor the numerator as a difference of squares: x2 2 1 sx 2 1dsx 1 1d − x21 x21
The numerator and denominator have a common factor of x 2 1. When we take the limit as x approaches 1, we have x ± 1 and so x 2 1 ± 0. Therefore we can cancel the common factor and then compute the limit by direct substitution as follows: lim
xl1
x2 2 1 sx 2 1dsx 1 1d − lim xl1 x21 x21 − lim sx 1 1d xl1
−111−2 The limit in this example arose in Example 1.4.1 when we were trying to find the tangent to the parabola y − x 2 at the point s1, 1d.
■
NOTE 1 In Example 3 we do not have an infinite limit even though the denominator approaches 0 as x l 1. When both numerator and denominator approach 0, the limit may be infinite or it may be some finite value. NOTE 2 In Example 3 we were able to compute the limit by replacing the given function f sxd − sx 2 2 1dysx 2 1d by a simpler function, tsxd − x 1 1, with the same limit. This is valid because f sxd − tsxd except when x − 1, and in computing a limit as x approaches 1 we don’t consider what happens when x is actually equal to 1. In general, we have the following useful fact.
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66
Chapter 1 Functions and Limits
If f sxd − tsxd when x ± a, then lim f sxd − lim tsxd, provided the limits exist. x la
xla
Example 4 Find lim tsxd where x l1
tsxd −
H
x 1 1 if x ± 1 if x − 1
SOLUTION Here t is defined at x − 1 and ts1d − , but the value of a limit as x approaches 1 does not depend on the value of the function at 1. Since tsxd − x 1 1 for x ± 1, we have
lim tsxd − lim sx 1 1d − 2
xl1
■
xl1
Note that the values of the functions in Examples 3 and 4 are identical except when x − 1 (see Figure 2) and so they have the same limit as x approaches 1. y
y
y=ƒ
3 2
2
FIGURE 2
1
1
The graphs of the functions f (from Example 3) and t (from Example 4)
0
1
2
Example 5 Evaluate lim
hl0
3
y=©
3
x
0
1
2
3
x
s3 1 hd2 2 9 . h
SOLUTION If we define
Fshd −
s3 1 hd2 2 9 h
then, as in Example 3, we can’t compute lim h l 0 Fshd by letting h − 0 since Fs0d is undefined. But if we simplify Fshd algebraically, we find that Fshd −
s9 1 6h 1 h 2 d 2 9 6h 1 h 2 hs6 1 hd − − −61h h h h
(Recall that we consider only h ± 0 when letting h approach 0.) Thus
lim
hl0
Example 6 Find lim
tl0
s3 1 hd2 2 9 − lim s6 1 hd − 6 hl0 h
■
st 2 1 9 2 3 . t2
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Section 1.6 Calculating Limits Using the Limit Laws
67
SOLUTION We can’t apply the Quotient Law immediately, since the limit of the denominator is 0. Here the preliminary algebra consists of rationalizing the numerator:
lim
tl0
st 2 1 9 2 3 st 2 1 9 2 3 st 2 1 9 1 3 − lim 2 t tl0 t2 st 2 1 9 1 3 − lim
st 2 1 9d 2 9 t 2 (st 2 1 9 1 3)
− lim
t2 t (st 2 1 9 1 3)
− lim
1 st 1 9 1 3
tl0
2
tl0
2
tl0
−
1 2 1 9d 1 3 st slim t l0
Here we use several properties of limits (5, 1, 10, 7, 9).
−
1 1 − 313 6
This calculation confirms the guess that we made in Example 1.5.2.
■
Some limits are best calculated by first finding the left and righthand limits. The following theorem is a reminder of what we discovered in Section 1.5. It says that a twosided limit exists if and only if both of the onesided limits exist and are equal.
1 Theorem lim f sxd − L if and only if lim f sxd − L − lim f sxd 2 1 xla
x la
x la
When computing onesided limits, we use the fact that the Limit Laws also hold for onesided limits.
Example 7 Show that lim  x  − 0. xl0
SOLUTION Recall that
x − The result of Example 7 looks plausible from Figure 3. y
 
H
x if x > 0 2x if x , 0
Since x − x for x . 0, we have
 
lim x − lim1 x − 0
x l 01
y= x
x l0
 
For x , 0 we have x − 2x and so
 
lim x − lim2 s2xd − 0
x l 02
0
FIGURE 3
x
Therefore, by Theorem 1,
x l0
 
lim x − 0
xl0
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■
68
Chapter 1 Functions and Limits
Example 8 Prove that lim
xl0
 x  does not exist. x
 
 
SOLUTION Using the facts that x − x when x . 0 and x − 2x when x , 0, we
have y
 x
y= x
lim
x −
lim2
x −
x l 01
1 0
x
_1
x l0
x x
lim
x − lim1 1 − 1 x l0 x
lim2
2x − lim2 s21d − 21 x l0 x
x l 01
x l0
Since the right and lefthand limits are different, it follows from Theorem 1 that lim x l 0 x yx does not exist. The graph of the function f sxd − x yx is shown in Figure 4 and supports the onesided limits that we found.
 
FIGURE 4
 
Example 9 If f sxd −
H
sx 2 4 8 2 2x
■
if x . 4 if x , 4
determine whether lim x l 4 f sxd exists. SOLUTION Since f sxd − sx 2 4 for x . 4, we have
It is shown in Example 1.7.3 that lim x l 01 sx − 0.
lim f sxd − lim1 s x 2 4 − s4 2 4 − 0
x l 41
x l4
Since f sxd − 8 2 2x for x , 4, we have
y
lim f sxd − lim2 s8 2 2xd − 8 2 2 4 − 0
x l 42
x l4
The right and lefthand limits are equal. Thus the limit exists and 0
x
4
lim f sxd − 0
xl4
FIGURE 5
The graph of f is shown in Figure 5.
■
Other notations for v x b are fxg and :x;. Example 10 The greatest integer function is defined by v x b − the largest integer The greatest integer function is somethat is less than or equal to x. (For instance, v4 b − 4, v4.8b − 4, v b − 3, vs2 b − 1, times called the floor function. 1
v22 b − 21.) Show that lim x l3 v x b does not exist.
y
SOLUTION The graph of the greatest integer function is shown in Figure 6. Since v x b − 3 for 3 < x , 4, we have
4 3
lim v x b − lim1 3 − 3
y=[ x]
2
x l 31
Since v x b − 2 for 2 < x , 3, we have
1 0
x l3
1
2
3
4
5
x
lim v x b − lim2 2 − 2
x l 32
x l3
Because these onesided limits are not equal, lim xl3 v x b does not exist by Theorem 1. ■ FIGURE 6 Greatest integer function
The next two theorems give two additional properties of limits. Their proofs can be found in Appendix F.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.6 Calculating Limits Using the Limit Laws
69
2 Theorem If f sxd < tsxd when x is near a (except possibly at a) and the limits of f and t both exist as x approaches a, then lim f sxd < lim tsxd
xla
xla
3 The Squeeze Theorem If f sxd < tsxd < hsxd when x is near a (except possibly at a) and lim f sxd − lim hsxd − L
y
xla
h
f 0
x
a
FIGURE 7
lim tsxd − L
then
g
L
xla
xla
The Squeeze Theorem, which is sometimes called the Sandwich Theorem or the Pinching Theorem, is illustrated by Figure 7. It says that if tsxd is squeezed between f sxd and hsxd near a, and if f and h have the same limit L at a, then t is forced to have the same limit L at a. 1 − 0. x SOLUTION First note that we cannot use
Example 11 Show that lim x 2 sin xl0
lim x 2 sin
xl0
1 1 − lim x 2 lim sin x l 0 x l 0 x x
because lim x l 0 sins1yxd does not exist (see Example 1.5.4). Instead we apply the Squeeze Theorem, and so we need to find a function f smaller than tsxd − x 2 sins1yxd and a function h bigger than t such that both f sxd and hsxd approach 0. To do this we use our knowledge of the sine function. Because the sine of any number lies between 21 and 1, we can write. 4
21 < sin
1 0 for all x and so, multiplying each side of the inequalities in (4) by x 2, we get y
2x 2 < x 2 sin
y=≈
1 < x2 x
as illustrated by Figure 8. We know that x
0
y=_≈
FIGURE 8 y − x 2 sins1yxd
lim x 2 − 0 and lim s2x 2 d − 0
xl0
xl0
Taking f sxd − 2x 2, tsxd − x 2 sins1yxd, and hsxd − x 2 in the Squeeze Theorem, we obtain 1 lim x 2 sin − 0 xl0 x
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
■
70
Chapter 1 Functions and Limits
1. Given that
11–32 Evaluate the limit, if it exists.
lim f sxd − 4 lim tsxd − 22 lim hsxd − 0
11. lim
find the limits that exist. If the limit does not exist, explain why. (a) lim f f sxd 1 5tsxdg (b) lim f tsxdg 3
x 2 2 6x 1 5 x 2 1 3x 12. lim 2 x l 23 x 2 x 2 12 x25
13. lim
x 2 2 5x 1 6 x 2 1 3x 14. lim 2 x l 4 x 2 x 2 12 x25
3f sxd (c) lim sf sxd (d) lim xl2 x l 2 tsxd
15. lim
xl2
xl 2
xl2
xl2
(e) lim
x l2
x l5
x l5
xl2
tsxd tsxdhsxd (f ) lim xl2 hsxd f sxd
2. The graphs of f and t are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why. (a) lim f f sxd 1 tsxdg (b) lim f f sxd 2 tsxdg x l2
xl0
f sxd (c) lim f f sxd tsxdg (d) lim x l 21 x l 3 tsxd (e) lim fx f sxdg (f ) f s21d 1 lim tsxd x l2
x l 21
y
y y=ƒ
1
t l 23
hl0
x
1
0
hl0
x12 t4 2 1 20. lim 3 3 tl1 t 2 1 x 18
21. lim
s9 1 h 2 3 s4u 1 1 2 3 22. lim ul 2 h u22
x l 22
1 1 2 x 3 s3 1 hd21 2 3 21 23. lim 24. lim x l3 x 2 3 hl0 h
y=©
25. lim
x
27. lim
1
s2 1 hd3 2 8 h
18. lim
19. lim
tl0
1
0
s25 1 hd2 2 25 h
17. lim
hl0
2
t2 2 9 2x 2 1 3x 1 1 16. lim x l 21 x 2 2 2x 2 3 2t 2 1 7t 1 3
S
1 1 s1 1 t 2 s1 2 t 26. lim 2 2 t l 0 t t t 1t
x l 16
D
4 2 sx x 2 2 4x 1 4 lim 4 2 28. x l 2 x 2 3x 2 2 4 16x 2 x
S
1
D
sx 2 1 9 2 5 30. lim xl24 x14 t
1
3–9 Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).
29. lim
3. lim s5x 3 2 3x 2 1 x 2 6d
1 1 2 2 sx 1 hd3 2 x 3 sx 1 hd2 x 31. lim 32. lim hl0 hl0 h h
tl0
x l3
4. lim sx 4 2 3xdsx 2 1 5x 1 3d xl 21
5. lim
t l 22
t4 2 2 2t 2 3t 1 2 2
6. lim su 4 1 3u 1 6 ul 22
3 7. lim s1 1 s x ds2 2 6x 2 1 x 3 d 8. lim xl8
9. lim
xl2
Î
tl2
S
t2 2 2 t 2 3t 1 5 3
2x 2 1 1 3x 2 2
10. (a) What is wrong with the following equation? x2 1 x 2 6 −x13 x22
D
; 33. (a) Estimate the value of
2
lim
x l0
x l2
is correct.
x2 1 x 2 6 − lim sx 1 3d x l2 x22
x s1 1 3x 2 1
by graphing the function f sxd − xyss1 1 3x 2 1d. (b) Make a table of values of f sxd for x close to 0 and guess the value of the limit. (c) Use the Limit Laws to prove that your guess is correct. ; 34. (a) Use a graph of f sxd −
(b) In view of part (a), explain why the equation lim
t s1 1 t
2
s3 1 x 2 s3 x
to estimate the value of lim x l 0 f sxd to two decimal places. (b) Use a table of values of f sxd to estimate the limit to four decimal places. (c) Use the Limit Laws to find the exact value of the limit.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.6 Calculating Limits Using the Limit Laws
se the Squeeze Theorem to show that ; 35. U lim x l 0 sx 2 cos 20xd − 0. Illustrate by graphing the functions f sxd − 2x 2, tsxd − x 2 cos 20x, and hsxd − x 2 on the same screen.
(b) Does lim x l 2 tsxd exist? (c) Sketch the graph of t.
50. Let f sxd −
; 36. Use the Squeeze Theorem to show that −0 lim sx 3 1 x 2 sin x l0 x Illustrate by graphing the functions f, t, and h (in the notation of the Squeeze Theorem) on the same screen. 37. If 4x 2 9 < f sxd < x 2 2 4x 1 7 for x > 0, find lim f sxd.
51. Let Bstd −
38. If 2x < tsxd < x 2 x 1 2 for all x, evaluate lim tsxd. 2
xl1
39. Prove that lim x 4 cos x l0
2 − 0. x
41–46 Find the limit, if it exists. If the limit does not exist, explain why. 2x 1 12 41. lim s2x 1 x 2 3 d 42. lim xl3 x l 26 x 1 6
x l 0.5
45. lim2 x l0

S



 

 D
S
 
D
(a) Sketch the graph of this function. (b) Find each of the following limits or explain why it does not exist. (i) lim1 sgn x (ii) lim2 sgn x
 xl0
(iii) lim sgn x (iv) lim sgn x xl0

x l0
x l0
(iv) lim1 tsxd (v) lim2 tsxd (vi) lim tsxd x l
49. Let tsxd −
x l
x l
(b) For which values of a does lim x l a tsxd not exist? (c) Sketch a graph of t. x2 1 x 2 6 . x22


xl2
x l 22
x l 22.4
(b) If n is an integer, evaluate (i) lim2 v x b (ii) lim1 v x b
(c) For what values of a does lim x l a v x b exist?
xln
54. Let f sxd − v cos x b , 2 < x < . (a) Sketch the graph of f. (b) Evaluate each limit, if it exists. (i) lim f sxd (ii) lim 2 f sxd xl0
x l sy2d
(iii) lim 1 f sxd (iv) lim f sxd x l sy2d
x l y2
(c) For what values of a does lim x l a f sxd exist?
55. If f sxd − v x b 1 v 2x b , show that lim x l 2 f sxd exists but is not equal to f s2d. 56. In the theory of relativity, the Lorentz contraction formula L − L 0 s1 2 v 2yc 2 expresses the length L of an object as a function of its velocity v with respect to an observer, where L 0 is the length of the object at rest and c is the speed of light. Find lim v l c2 L and interpret the result. Why is a lefthand limit necessary? 57. If p is a polynomial, show that lim xl a psxd − psad.
(a) Find (i) lim1 tsxd (ii) lim2 tsxd x l2
xl2
53. (a) If the symbol v b denotes the greatest integer function defined in Example 10, evaluate (i) lim 1 v x b (ii) lim v x b (iii) lim v x b
48. Let tsxd − sgnssin xd . (a) Find each of the following limits or explain why it does not exist. (i) lim1 tsxd (ii) lim2 tsxd (iii) lim tsxd x l0
x,1 x−1 1,x 2
(iv) lim2 tsxd (v) lim1 tsxd (vi) lim tsxd
47. The signum (or sign) function, denoted by sgn, is defined by
H
if t , 2
st 1 c
x 3 tsxd − 2 2 x2 x23
x l2
1 1 46. lim 2 x l 01 x x
sgn x −
4 2 12 t
x l1
2x 2 1 22 x lim 3 2 44. x l 22 2x 2 x 21x
1 1 2 x x
H
tl2
52. Let
x l0
43. lim 2
x2 1 1 if x , 1 2 sx 2 2d if x > 1
Find the value of c so that lim Bstd exists.
40. Prove that lim1 sx f1 1 sin 2s2yxdg − 0.

H
(a) Find lim x l12 f sxd and lim x l11 f sxd. (b) Does lim x l1 f sxd exist? (c) Sketch the graph of f.
xl4
4
71
x l2
58. I f r is a rational function, use Exercise 57 to show that lim x l a rsxd − rsad for every number a in the domain of r.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
72
chapter 1 Functions and Limits
59. If lim
xl1
f sxd 2 8 − 10, find lim f sxd. xl1 x21
65. Is there a number a such that lim
x l 22
f sxd 60. If lim 2 − 5, find the following limits. xl0 x f sxd lim f sxd (b) lim (a) xl0 xl0 x 61. If f sxd −
H
exists? If so, find the value of a and the value of the limit. 66. The figure shows a fixed circle C1 with equation sx 2 1d2 1 y 2 − 1 and a shrinking circle C2 with radius r and center the origin. P is the point s0, rd, Q is the upper point of intersection of the two circles, and R is the point of intersection of the line PQ and the xaxis. What happens to R as C2 shrinks, that is, as r l 0 1?
x 2 if x is rational 0 if x is irrational
y
prove that lim x l 0 f sxd − 0.
P
62. S how by means of an example that lim x l a f f sxd 1 tsxdg may exist even though neither lim x l a f sxd nor lim x l a tsxd exists.
C™
63. S how by means of an example that lim x l a f f sxd tsxdg may exist even though neither lim x l a f sxd nor lim x l a tsxd exists. 64. Evaluate lim
xl2
3x 2 1 ax 1 a 1 3 x2 1 x 2 2
Q
0
s6 2 x 2 2 . s3 2 x 2 1
R
C¡
x
The intuitive definition of a limit given in Section 1.5 is inadequate for some purposes because such phrases as “x is close to 2” and “ f sxd gets closer and closer to L” are vague. In order to be able to prove conclusively that lim
xl0
S
x3 1
cos 5x 10,000
D
− 0.0001 or lim
xl0
sin x −1 x
we must make the definition of a limit precise. To motivate the precise definition of a limit, let’s consider the function f sxd −
H
2x 2 1 if x ± 3 6 if x − 3
Intuitively, it is clear that when x is close to 3 but x ± 3, then f sxd is close to 5, and so lim x l3 f sxd − 5. To obtain more detailed information about how f sxd varies when x is close to 3, we ask the following question: How close to 3 does x have to be so that f sxd differs from 5 by less than 0.l? It is traditional to use the Greek letter (delta) in this situation.




The distance from x to 3 is x 2 3 and the distance from f sxd to 5 is f sxd 2 5 , so our problem is to find a number such that
 f sxd 2 5  , 0.1 if  x 2 3  , but x ± 3 

If x 2 3 . 0, then x ± 3, so an equivalent formulation of our problem is to find a number such that
 f sxd 2 5  , 0.1 if 0 ,  x 2 3  , Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.7 The Precise Definition of a Limit

73

Notice that if 0 , x 2 3 , s0.1dy2 − 0.05, then
 f sxd 2 5  −  s2x 2 1d 2 5  −  2x 2 6  − 2 x 2 3  , 2s0.05d − 0.1  f sxd 2 5  , 0.1 if 0 ,  x 2 3  , 0.05
that is,
Thus an answer to the problem is given by − 0.05; that is, if x is within a distance of 0.05 from 3, then f sxd will be within a distance of 0.1 from 5. If we change the number 0.l in our problem to the smaller number 0.01, then by using the same method we find that f sxd will differ from 5 by less than 0.01 provided that x differs from 3 by less than (0.01)y2 − 0.005:
 f sxd 2 5  , 0.01 if 0 ,  x 2 3  , 0.005 Similarly,
 f sxd 2 5  , 0.001 if 0 ,  x 2 3  , 0.0005 The numbers 0.1, 0.01, and 0.001 that we have considered are error tolerances that we might allow. For 5 to be the precise limit of f sxd as x approaches 3, we must not only be able to bring the difference between f sxd and 5 below each of these three numbers; we must be able to bring it below any positive number. And, by the same reasoning, we can! If we write « (the Greek letter epsilon) for an arbitrary positive number, then we find as before that 1
«
 f sxd 2 5  , « if 0 ,  x 2 3  , − 2
This is a precise way of saying that f sxd is close to 5 when x is close to 3 because (1) says that we can make the values of f sxd within an arbitrary distance « from 5 by restricting the values of x to be within a distance «y2 from 3 (but x ± 3). Note that (1) can be rewritten as follows: y
ƒ is in here
if 3 2 , x , 3 1 sx ± 3d then 5 2 « , f sxd , 5 1 «
5+∑
5
and this is illustrated in Figure 1. By taking the values of x (± 3) to lie in the interval s3 2 , 3 1 d we can make the values of f sxd lie in the interval s5 2 «, 5 1 «d. Using (1) as a model, we give a precise definition of a limit.
5∑
0
x
3
3∂
3+∂
when x is in here (x≠3)
2 Precise Definition of a Limit Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f sxd as x approaches a is L, and we write lim f sxd − L
xla
FIGURE 1
if for every number « . 0 there is a number . 0 such that




if 0 , x 2 a , then f sxd 2 L , «
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
74
Chapter 1 Functions and Limits




Since x 2 a is the distance from x to a and f sxd 2 L is the distance from f sxd to L, and since « can be arbitrarily small, the definition of a limit can be expressed in words as follows: lim x l a f sxd 5 L means that the distance between f sxd and L can be made arbitrarily small by requiring that the distance from x to a be sufficiently small (but not 0).
Alternatively, lim x l a f sxd 5 L means that the values of f sxd can be made as close as we please to L by requiring x to be close enough to a (but not equal to a).
We can also reformulate Definition 2 in terms of intervals by observing that the inequality x 2 a , is equivalent to 2 , x 2 a , , which in turn can be written as a 2 , x , a 1 . Also 0 , x 2 a is true if and only if x 2 a ± 0, that is, x ± a. Similarly, the inequality f sxd 2 L , « is equivalent to the pair of inequalities L 2 « , f sxd , L 1 «. Therefore, in terms of intervals, Definition 2 can be stated as follows:






lim x l a f sxd 5 L means that for every « . 0 (no matter how small « is) we can find . 0 such that if x lies in the open interval sa 2 , a 1 d and x ± a, then f sxd lies in the open interval sL 2 «, L 1 «d.
We interpret this statement geometrically by representing a function by an arrow diagram as in Figure 2, where f maps a subset of R onto another subset of R.
f
FIGURE 2
x
a
f(a)
ƒ
The definition of limit says that if any small interval sL 2 «, L 1 «d is given around L, then we can find an interval sa 2 , a 1 d around a such that f maps all the points in sa 2 , a 1 d (except possibly a) into the interval sL 2 «, L 1 «d. (See Figure 3.)
f x
FIGURE 3
a∂
ƒ a
a+∂
L∑
L
L+∑
Another geometric interpretation of limits can be given in terms of the graph of a function. If « . 0 is given, then we draw the horizontal lines y 5 L 1 « and y 5 L 2 « and the graph of f. (See Figure 4.) If lim x l a f sxd 5 L, then we can find a number . 0 such that if we restrict x to lie in the interval sa 2 , a 1 d and take x ± a, then the curve y 5 f sxd lies between the lines y 5 L 2 « and y 5 L 1 «. (See Figure 5.) You can see that if such a has been found, then any smaller will also work. It is important to realize that the process illustrated in Figures 4 and 5 must work for every positive number «, no matter how small it is chosen. Figure 6 shows that if a smaller « is chosen, then a smaller may be required. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
75
Section 1.7 The Precise Definition of a Limit
y=ƒ
y
y
y=L+∑
ƒ is in here
∑
L
∑
y
y=L+∑
L+∑
∑
L
∑
y=L∑
0
x
a
y=L∑
L∑
y=L∑
0
y=L+∑
a
a∂
0
x
a+∂
a∂
a
x
a+∂
when x is in here (x≠a)
FIGURE 4
FIGURE 5
FIGURE 6
Example 1 Since f sxd − x 3 2 5x 1 6 is a polynomial, we know from the Direct
Substitution Property that lim x l1 f sxd − f s1d − 13 2 5s1d 1 6 − 2. Use a graph to find a number such that if x is within of 1, then y is within 0.2 of 2, that is,




if x 2 1 , then sx 3 2 5x 1 6d 2 2 , 0.2 In other words, find a number that corresponds to « 5 0.2 in the definition of a limit for the function f sxd 5 x 3 2 5x 1 6 with a 5 1 and L 5 2. SOLUTION A graph of f is shown in Figure 7; we are interested in the region near the point s1, 2d. Notice that we can rewrite the inequality
15
 sx _3
3 _5
2.3
y=˛5x+6 (1, 2) y=1.8
FIGURE 8
1.8 , x 3 2 5x 1 6 , 2.2
So we need to determine the values of x for which the curve y 5 x 3 2 5x 1 6 lies between the horizontal lines y 5 1.8 and y 5 2.2. Therefore we graph the curves y 5 x 3 2 5x 1 6, y 5 1.8, and y 5 2.2 near the point s1, 2d in Figure 8. Then we use the cursor to estimate that the xcoordinate of the point of intersection of the line y 5 2.2 and the curve y 5 x 3 2 5x 1 6 is about 0.911. Similarly, y 5 x 3 2 5x 1 6 intersects the line y 5 1.8 when x < 1.124. So, rounding toward 1 to be safe, we can say that
y=2.2
0.8 1.7

2 5x 1 6d 2 2 , 0.2
20.2 , sx 3 2 5x 1 6d 2 2 , 0.2
as or equivalently
FIGURE 7
3
1.2
if 0.92 , x , 1.12 then 1.8 , x 3 2 5x 1 6 , 2.2 This interval s0.92, 1.12d is not symmetric about x 5 1. The distance from x 5 1 to the left endpoint is 1 2 0.92 5 0.08 and the distance to the right endpoint is 0.12. We can choose to be the smaller of these numbers, that is, 5 0.08. Then we can rewrite our inequalities in terms of distances as follows:




if x 2 1 , 0.08 then sx 3 2 5x 1 6d 2 2 , 0.2 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
76
Chapter 1 Functions and Limits
This just says that by keeping x within 0.08 of 1, we are able to keep f sxd within 0.2 of 2. Although we chose 5 0.08, any smaller positive value of would also have worked. ■
TEC In Module 1.7/3.4 you can explore the precise definition of a limit both graphically and numerically.
The graphical procedure in Example 1 gives an illustration of the definition for « 5 0.2, but it does not prove that the limit is equal to 2. A proof has to provide a for every «. In proving limit statements it may be helpful to think of the definition of limit as a challenge. First it challenges you with a number «. Then you must be able to produce a suitable . You have to be able to do this for every « . 0, not just a particular «. Imagine a contest between two people, A and B, and imagine yourself to be B. Person A stipulates that the fixed number L should be approximated by the values of f sxd to within a degree of accuracy « (say, 0.01). Person B then responds by finding a number such that if 0 , x 2 a , , then f sxd 2 L , «. Then A may become more exacting and challenge B with a smaller value of « (say, 0.0001). Again B has to respond by finding a corresponding . Usually the smaller the value of «, the smaller the corresponding value of must be. If B always wins, no matter how small A makes «, then lim x l a f sxd 5 L.




Example 2 Prove that lim s4x 2 5d − 7. x l3
SOLUTION 1. Preliminary analysis of the problem (guessing a value for ). Let « be a given
positive number. We want to find a number such that




if 0 , x 2 3 , then s4x 2 5d 2 7 , «

 
 





But s4x 2 5d 2 7 5 4x 2 12 5 4sx 2 3d 5 4 x 2 3 . Therefore we want such that




if 0 , x 2 3 , then 4 x 2 3 , « that is,


if 0 , x 2 3 , then x 2 3 ,
« 4
This suggests that we should choose 5 «y4. 2. Proof (showing that this works). Given « . 0, choose 5 «y4. If 0 , x 2 3 , , then


SD
 s4x 2 5d 2 7  −  4x 2 12  − 4 x 2 3  , 4 − 4
« 4
−«
Thus




if 0 , x 2 3 , then s4x 2 5d 2 7 , « Therefore, by the definition of a limit, lim s4x 2 5d − 7
x l3
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.7 The Precise Definition of a Limit
Cauchy and Limits After the invention of calculus in the 17th century, there followed a period of free development of the subject in the 18th century. Mathematicians like the Bernoulli brothers and Euler were eager to exploit the power of calculus and boldly explored the consequences of this new and wonderful mathematical theory without worrying too much about whether their proofs were completely correct. The 19th century, by contrast, was the Age of Rigor in mathematics. There was a movement to go back to the foundations of the subject—to provide careful definitions and rigorous proofs. At the forefront of this movement was the French mathematician AugustinLouis Cauchy (1789–1857), who started out as a military engineer before becoming a mathematics professor in Paris. Cauchy took Newton’s idea of a limit, which was kept alive in the 18th century by the French mathematician Jean d’Alembert, and made it more precise. His definition of a limit reads as follows: “When the successive values attributed to a variable approach indefinitely a fixed value so as to end by differing from it by as little as one wishes, this last is called the limit of all the others.” But when Cauchy used this definition in examples and proofs, he often employed deltaepsilon inequalities similar to the ones in this section. A typical Cauchy proof starts with: “Designate by and « two very small numbers; . . .” He used « because of the correspondence between epsilon and the French word erreur and because delta corresponds to différence. Later, the German mathematician Karl Weierstrass (1815–1897) stated the definition of a limit exactly as in our Definition 2.
77
This example is illustrated by Figure 9. y
y=4x5
7+∑ 7
7∑
0
figure 9
x
3
3∂
3+∂
■
Note that in the solution of Example 2 there were two stages—guessing and proving. We made a preliminary analysis that enabled us to guess a value for . But then in the second stage we had to go back and prove in a careful, logical fashion that we had made a correct guess. This procedure is typical of much of mathematics. Sometimes it is necessary to first make an intelligent guess about the answer to a problem and then prove that the guess is correct. The intuitive definitions of onesided limits that were given in Section 1.5 can be pre cisely reformulated as follows. 3 Definition of LeftHand Limit lim f sxd − L
x l a2
if for every number « . 0 there is a number . 0 such that


if a 2 , x , a then f sxd 2 L , «
4 Definition of RightHand Limit lim f sxd − L
x la1
if for every number « . 0 there is a number . 0 such that


if a , x , a 1 then f sxd 2 L , « Notice that Definition 3 is the same as Definition 2 except that x is restricted to lie in the left half sa 2 , ad of the interval sa 2 , a 1 d. In Definition 4, x is restricted to lie in the right half sa, a 1 d of the interval sa 2 , a 1 d.
Example 3 Use Definition 4 to prove that lim1 sx − 0. xl0
SOLUTION 1. Guessing a value for . Let « be a given positive number. Here a 5 0 and L 5 0,
so we want to find a number such that


if 0 , x , then sx 2 0 , «
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
78
Chapter 1 Functions and Limits
that is,
if 0 , x , then sx , «
or, squaring both sides of the inequality sx , «, we get if 0 , x , then x , « 2 This suggests that we should choose 5 « 2. 2. Showing that this works. Given « . 0, let 5 « 2. If 0 , x , , then sx , s 5 s« 2 5 «
 sx 2 0  , «
so
According to Definition 4, this shows that lim x l 01 sx − 0.
■
Example 4 Prove that lim x 2 5 9. xl3
SOLUTION 1. Guessing a value for . Let « . 0 be given. We have to find a number . 0
such that




if 0 , x 2 3 , then x 2 2 9 , «





 

To connect x 2 2 9 with x 2 3 we write x 2 2 9 5 sx 1 3dsx 2 3d . Then we want





if 0 , x 2 3 , then x 1 3 x 2 3 , «


Notice that if we can find a positive constant C such that x 1 3 , C, then
 x 1 3  x 2 3  , C  x 2 3  



and we can make C x 2 3 , « by taking x 2 3 , «yC, so we could choose − «yC. We can find such a number C if we restrict x to lie in some interval centered at 3. In fact, since we are interested only in values of x that are close to 3, it is reasonable to assume that x is within a distance l from 3, that is, x 2 3 , 1. Then 2 , x , 4, so 5 , x 1 3 , 7. Thus we have x 1 3 , 7, and so C 5 7 is a suitable choice for the constant. But now there are two restrictions on x 2 3 , namely






«
«
 x 2 3  , 1 and  x 2 3  , C 5 7 To make sure that both of these inequalities are satisfied, we take to be the smaller of the two numbers 1 and «y7. The notation for this is 5 minh1, «y7j. 2. Showing that this works. Given « . 0, let 5 minh1, «y7j. If 0 , x 2 3 , , then x 2 3 , 1 ? 2 , x , 4 ? x 1 3 , 7 (as in part l).






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79
Section 1.7 The Precise Definition of a Limit


We also have x 2 3 , «y7, so
x
2
 


29 − x13 x23 ,7
« −« 7
This shows that lim x l3 x 2 5 9.
■
As Example 4 shows, it is not always easy to prove that limit statements are true using the «, definition. In fact, if we had been given a more complicated function such as f sxd 5 s6x 2 2 8x 1 9dys2x 2 2 1d, a proof would require a great deal of ingenuity. Fortunately this is unnecessary because the Limit Laws stated in Section 1.6 can be proved using Definition 2, and then the limits of complicated functions can be found rigorously from the Limit Laws without resorting to the definition directly. For instance, we prove the Sum Law: If lim x l a f sxd 5 L and lim x l a tsxd 5 M both exist, then lim f f sxd 1 tsxdg − L 1 M
xla
The remaining laws are proved in the exercises and in Appendix F. Proof of the Sum Law Let « . 0 be given. We must find . 0 such that




if 0 , x 2 a , then f sxd 1 tsxd 2 sL 1 Md , « Triangle Inequality:
a 1 b < a 1 b (See Appendix A).
Using the Triangle Inequality we can write
 f sxd 1 tsxd 2 sL 1 Md  5  s f sxd 2 Ld 1 stsxd 2 Md  <  f sxd 2 L  1  tsxd 2 M  We make  f sxd 1 tsxd 2 sL 1 Md  less than « by making each of the terms  f sxd 2 L  and  tsxd 2 M  less than «y2. 5
Since «y2 . 0 and lim x l a f sxd 5 L, there exists a number 1 . 0 such that



« 2

if 0 , x 2 a , 1 then f sxd 2 L ,
Similarly, since lim x l a tsxd − M, there exists a number 2 . 0 such that



« 2

if 0 , x 2 a , 2 then tsxd 2 M , Let − minh1, 2j, the smaller of the numbers 1 and 2. Notice that






if 0 , x 2 a , then 0 , x 2 a , 1 and 0 , x 2 a , 2 «
«
 f sxd 2 L  , 2 and  tsxd 2 M  , 2
and so Therefore, by (5),
 f sxd 1 tsxd 2 sL 1 Md  <  f sxd 2 L  1  tsxd 2 M  ,
« « 1 5« 2 2
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
80
Chapter 1 Functions and Limits
To summarize,




if 0 , x 2 a , then f sxd 1 tsxd 2 sL 1 Md , « Thus, by the definition of a limit, lim f f sxd 1 tsxdg − L 1 M
■
xla
Infinite Limits Infinite limits can also be defined in a precise way. The following is a precise version of Definition 1.5.4. 6 Precise Definition of an Infinite Limit Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then lim f sxd − `
xla
means that for every positive number M there is a positive number such that


if 0 , x 2 a , then f sxd . M y
y=M
M
0
a∂
a
x
a+∂
figure 10
This says that the values of f sxd can be made arbitrarily large (larger than any given number M) by requiring x to be close enough to a (within a distance , where depends on M, but with x ± a). A geometric illustration is shown in Figure 10. Given any horizontal line y 5 M, we can find a number . 0 such that if we restrict x to lie in the interval sa 2 , a 1 d but x ± a, then the curve y 5 f sxd lies above the line y − M. You can see that if a larger M is chosen, then a smaller may be required. 1 − `. x2 SOLUTION Let M be a given positive number. We want to find a number such that
Example 5 Use Definition 6 to prove that lim
xl0


if 0 , x 2 0 , then 1yx 2 . M But
1 1 . M &? x 2 , &? sx 2 , x2 M
Î
1 1 &? x , M sM
 
 
So if we choose − 1ysM and 0 , x , − 1ysM , then 1yx 2 . M. This shows that 1yx 2 l ` as x l 0. n Similarly, the following is a precise version of Definition 1.5.5. It is illustrated by Figure 11.
y
a∂
a+∂ a
0
x
7 Definition Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then lim f sxd − 2`
y=N
N
FIGURE 11
xla
means that for every negative number N there is a positive number such that


if 0 , x 2 a , then f sxd , N
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81
Section 1.7 The Precise Definition of a Limit
1. Use the given graph of f to find a number such that




; 5. Use a graph to find a number such that
Z
if x 2 1 , then f sxd 2 1 , 0.2
if x 2
y 1.2 1 0.8
4
Z


, then tan x 2 1 , 0.2
; 6. Use a graph to find a number such that

Z

if x 2 1 , then
0
0.7
; 7. For the limit
x
1 1.1
Z
2x 2 0.4 , 0.1 x2 1 4
lim sx 3 2 3x 1 4d 5 6
xl2
2. Use the given graph of f to find a number such that



illustrate Definition 2 by finding values of that correspond to « 5 0.2 and « 5 0.1.

if 0 , x 2 3 , then f s xd 2 2 , 0.5
; 8. For the limit
y
lim
2.5
x l2
2
4x 1 1 − 4.5 3x 2 4
illustrate Definition 2 by finding values of that correspond to « 5 0.5 and « 5 0.1.
1.5
; 9. (a) Use a graph to find a number such that 0
2.6 3
x
3.8
3. Use the given graph of f sxd − sx to find a number such that




if x 2 4 , then sx 2 2 , 0.4 y
y=œ„ x
2.4 2 1.6
0
4
?
?
x
4. Use the given graph of f sxd 5 x 2 to find a number such that




if x 2 1 , then x 2 2 1 , 12 y
y=≈
1.5
if 4 , x , 4 1 then
sx 2 4
. 100
(b) What limit does part (a) suggest is true?
iven that lim x l csc2 x − `, illustrate Definition 6 by ; 10. G finding values of that correspond to (a) M − 500 and (b) M − 1000. 11. A machinist is required to manufacture a circular metal disk with area 1000 cm2. (a) What radius produces such a disk? (b) If the machinist is allowed an error tolerance of 65 cm 2 in the area of the disk, how close to the ideal radius in part (a) must the machinist control the radius? (c) In terms of the «, definition of lim x l a f sxd 5 L, what is x? What is f sxd? What is a? What is L? What value of « is given? What is the corresponding value of ? crystal growth furnace is used in research to determine how ; 12. A best to manufacture crystals used in electronic components for the space shuttle. For proper growth of the crystal, the temperature must be controlled accurately by adjusting the input power. Suppose the relationship is given by T swd − 0.1w 2 1 2.155w 1 20
1 0.5 0
x2 1 4
?
1
?
x
where T is the temperature in degrees Celsius and w is the power input in watts. (a) How much power is needed to maintain the temperature at 200°C?
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
82
chapter 1 Functions and Limits
(b) If the temperature is allowed to vary from 200°C by up to 61°C, what range of wattage is allowed for the input power? (c) In terms of the «, definition of lim x l a f sxd 5 L, what is x? What is f sxd? What is a? What is L? What value of « is given? What is the corresponding value of ?


13. (a) Find a number such that if x 2 2 , , then 4x 2 8 , «, where « 5 0.1. (b) Repeat part (a) with « 5 0.01.


14. Given that limx l 2 s5x 2 7d 5 3, illustrate Definition 2 by finding values of that correspond to « 5 0.1, « 5 0.05, and « 5 0.01. 15–18 Prove the statement using the «, definition of a limit and illustrate with a diagram like Figure 9. 15. lim s1 1 13 xd − 2 16. lim s2x 2 5d 5 3 xl3
xl4
17. lim s1 2 4xd − 13 18. lim s3x 1 5d − 21 xl23
xl22
CAS
35. (a) For the limit lim x l 1 sx 3 1 x 1 1d 5 3, use a graph to find a value of that corresponds to « 5 0.4. (b) By using a computer algebra system to solve the cubic equation x 3 1 x 1 1 5 3 1 «, find the largest possible value of that works for any given « . 0. (c) Put « 5 0.4 in your answer to part (b) and compare with your answer to part (a). 36. Prove that lim
x l2
1 1 − . x 2
37. Prove that lim sx − sa if a . 0.
F
xla

Hint: Use sx 2 sa

−
x 2 a
sx 1 sa
.
F
38. If H is the Heaviside function defined in Example 1.5.6, prove, using Definition 2, that lim t l 0 Hstd does not exist. [Hint: Use an indirect proof as follows. Suppose that the limit is L. Take « 5 12 in the definition of a limit and try to arrive at a contradiction.]
19–32 Prove the statement using the «, definition of a limit. 2 1 4x 19. lim 5 2 20. lim s3 2 45 xd − 25 x l1 x l 10 3 x 2 2 2x 2 8 9 2 4x 2 − 6 22. lim −6 21. lim xl4 x l21.5 3 1 2x x24
39. If the function f is defined by
23. lim x − a 24. lim c − c
40. By comparing Definitions 2, 3, and 4, prove Theorem 1.6.1.
xla
xla
25. lim x 2 − 0 26. lim x 3 − 0 xl0
f sxd −
41. How close to 23 do we have to take x so that 1 . 10,000 sx 1 3d4
xl0
 
0 if x is rational 1 if x is irrational
prove that lim x l 0 f sxd does not exist.
27. lim x − 0 28. lim1 s6 1 x − 0 xl0
H
8
xl 26
1 − `. sx 1 3d4
29. lim sx 2 4x 1 5d 5 1 30. lim sx 2 1 2x 2 7d 5 1
42. Prove, using Definition 6, that lim
31. lim sx 2 1d 5 3 32. lim x 3 5 8
43. Prove that lim 2
33. Verify that another possible choice of for showing that lim x l3 x 2 5 9 in Example 4 is 5 min h2, «y8j.
44. Suppose that lim x l a f sxd 5 ` and lim x l a tsxd 5 c, where c is a real number. Prove each statement. (a) lim f f sxd 1 tsxdg − `
2
xl2
xl2
2
x l 22
x l23
xl2
34. Verify, by a geometric argument, that the largest possible choice of for showing that lim x l3 x 2 − 9 is − s 9 1 « 2 3.
x l21
5 − 2`. sx 1 1d 3
xla
(b) lim f f sxd tsxdg 5 ` if c . 0 xla
(c) lim f f sxd tsxdg 5 2` if c , 0 xl a
We noticed in Section 1.6 that the limit of a function as x approaches a can often be found simply by calculating the value of the function at a. Functions with this property are called continuous at a. We will see that the mathematical definition of continuity corresponds closely with the meaning of the word continuity in everyday language. (A continuous process is one that takes place gradually, without interruption or abrupt change.) 1 Definition A function f is continuous at a number a if lim f sxd − f sad
xl a
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Section 1.8 Continuity
As illustrated in Figure 1, if f is continuous, then the points sx, f sxdd on the graph of f approach the point sa, f sadd on the graph. So there is no gap in the curve.
Notice that Definition l implicitly requires three things if f is continuous at a: 1. f sad is defined (that is, a is in the domain of f ) 2. lim f sxd exists x la
3. lim f sxd 5 f sad
y
ƒ approaches f(a).
83
x la
y=ƒ f(a)
0
x
a
As x approaches a,
figure 1
y
The definition says that f is continuous at a if f sxd approaches f sad as x approaches a. Thus a continuous function f has the property that a small change in x produces only a small change in f sxd. In fact, the change in f sxd can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not continuous at a. Physical phenomena are usually continuous. For instance, the displacement or velocity of a vehicle varies continuously with time, as does a person’s height. But discontinuities do occur in such situations as electric currents. [See Example 1.5.6, where the Heaviside function is discontinuous at 0 because lim t l 0 Hstd does not exist.] Geometrically, you can think of a function that is continuous at every number in an interval as a function whose graph has no break in it: the graph can be drawn without removing your pen from the paper.
Example 1 Figure 2 shows the graph of a function f. At which numbers is f discontinuous? Why?
0
1
figure 2
2
3
4
5
x
SOLUTION It looks as if there is a discontinuity when a − 1 because the graph has a break there. The official reason that f is discontinuous at 1 is that f s1d is not defined. The graph also has a break when a 5 3, but the reason for the discontinuity is different. Here, f s3d is defined, but lim x l3 f sxd does not exist (because the left and right limits are different). So f is discontinuous at 3. What about a 5 5? Here, f s5d is defined and lim x l5 f sxd exists (because the left and right limits are the same). But
lim f sxd ± f s5d
xl5
So f is discontinuous at 5.
n
Now let’s see how to detect discontinuities when a function is defined by a formula.
Example 2 Where are each of the following functions discontinuous? (a) f sxd 5
(c) f sxd −
H
1 if x ± 0 x2 2 x 2 2 (b) f sxd − x 2 x22 1 if x − 0
H
x 2 2 x 2 2 if x ± 2 x22 (d) f sxd − v x b 1 if x − 2
SOLUTION
(a) Notice that f s2d is not defined, so f is discontinuous at 2. Later we’ll see why f is continuous at all other numbers. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
84
Chapter 1 Functions and Limits
(b) Here f s0d 5 1 is defined but lim f sxd − lim
xl0
xl0
1 x2
does not exist. (See Example 1.5.8.) So f is discontinuous at 0. (c) Here f s2d 5 1 is defined and lim f sxd − lim
x l2
x l2
x2 2 x 2 2 sx 2 2dsx 1 1d − lim − lim sx 1 1d − 3 x l2 x l2 x22 x22
exists. But lim f sxd ± f s2d
x l2
so f is not continuous at 2. (d) The greatest integer function f sxd − v x b has discontinuities at all of the integers because lim x ln v x b does not exist if n is an integer. (See Example 1.6.10 and Exercise 1.6.53.)
n
Figure 3 shows the graphs of the functions in Example 2. In each case the graph can’t be drawn without lifting the pen from the paper because a hole or break or jump occurs in the graph. The kind of discontinuity illustrated in parts (a) and (c) is called removable because we could remove the discontinuity by redefining f at just the single number 2. [The function tsxd − x 1 1 is continuous.] The discontinuity in part (b) is called an infinite discontinuity. The discontinuities in part (d) are called jump discontinuities because the function “jumps” from one value to another. y
y
y
y
1
1
1
1
0
(a) ƒ=
1
2
x
≈x2 x2
0
1 if x≠0 (b) ƒ= ≈ 1 if x=0
0
x
(c) ƒ=
1
2
x
≈x2 if x≠2 x2 1 if x=2
0
1
2
3
x
(d) ƒ=[ x ]
FIGURE 3 Graphs of the functions in Example 2
2 Definition A function f is continuous from the right at a number a if lim f sxd − f sad
x l a1
and f is continuous from the left at a if lim f sxd − f sad
x l a2
Example 3 At each integer n, the function f sxd − v x b [see Figure 3(d)] is continuous from the right but discontinuous from the left because lim f sxd − lim1 v xb − n − f snd
x l n1
x ln
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Section 1.8 Continuity
85
lim f sxd − lim2 v x b − n 2 1 ± f snd
but
x l n2
x ln
n
3 Definition A function f is continuous on an interval if it is continuous at every number in the interval. (If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left.)
Example 4 Show that the function f sxd − 1 2 s1 2 x 2 is continuous on the interval f21, 1g. SOLUTION If 21 , a , 1, then using the Limit Laws, we have
y 1
1
0
FIGURE 4
lim f sxd − lim (1 2 s1 2 x 2 )
xla
xla
− 1 2 lim s1 2 x 2
− 1 2 s lim s1 2 x 2 d (by 11)
− 1 2 s1 2 a 2
− f sad
xla
(by Laws 2 and 7)
xla
(by 2, 7, and 9)
Thus, by Definition l, f is continuous at a if 21 , a , 1. Similar calculations show that ƒ=1œ„„„„„ 1≈
lim f sxd − 1 − f s21d and lim2 f sxd − 1 − f s1d
x l 211
1
x
x l1
so f is continuous from the right at 21 and continuous from the left at 1. Therefore, according to Definition 3, f is continuous on f21, 1g. The graph of f is sketched in Figure 4. It is the lower half of the circle x 2 1 sy 2 1d2 − 1
n
Instead of always using Definitions 1, 2, and 3 to verify the continuity of a function as we did in Example 4, it is often convenient to use the next theorem, which shows how to build up complicated continuous functions from simple ones. 4 Theorem If f and t are continuous at a and if c is a constant, then the following functions are also continuous at a: 1. f 1 t
2. f 2 t
4. ft
5.
3. cf
f if tsad ± 0 t
Proof Each of the five parts of this theorem follows from the corresponding Limit Law in Section 1.6. For instance, we give the proof of part 1. Since f and t are continuous at a, we have
lim f sxd − f sad and lim tsxd − tsad
xla
xla
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
86
Chapter 1 Functions and Limits
Therefore
lim s f 1 tdsxd − lim f f sxd 1 tsxdg
xla
xla
− lim f sxd 1 lim tsxd (by Law 1)
− f sad 1 tsad
− s f 1 tdsad
xla
xla
This shows that f 1 t is continuous at a.
n
It follows from Theorem 4 and Definition 3 that if f and t are continuous on an interval, then so are the functions f 1 t, f 2 t, cf, ft, and (if t is never 0) fyt. The following theorem was stated in Section 1.6 as the Direct Substitution Property. 5 Theorem (a) Any polynomial is continuous everywhere; that is, it is continuous on R − s2`, `d. (b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain. Proof
(a) A polynomial is a function of the form Psxd − cn x n 1 cn21 x n21 1 ∙ ∙ ∙ 1 c1 x 1 c0 where c0 , c1, . . . , cn are constants. We know that lim c0 − c0 (by Law 7)
xla
and
lim x m − a m m − 1, 2, . . . , n (by 9)
xla
This equation is precisely the statement that the function f sxd − x m is a continuous function. Thus, by part 3 of Theorem 4, the function tsxd − cx m is continuous. Since P is a sum of functions of this form and a constant function, it follows from part 1 of Theorem 4 that P is continuous. (b) A rational function is a function of the form f sxd −
Psxd Qsxd

where P and Q are polynomials. The domain of f is D − hx [ R Qsxd ± 0j. We know from part (a) that P and Q are continuous everywhere. Thus, by part 5 of Theorem 4, f is continuous at every number in D. n As an illustration of Theorem 5, observe that the volume of a sphere varies continuously with its radius because the formula Vsrd − 43 r 3 shows that V is a polynomial function of r. Likewise, if a ball is thrown vertically into the air with a velocity of 50 ftys, then the height of the ball in feet t seconds later is given by the formula h − 50t 2 16t 2. Again this is a polynomial function, so the height is a continuous function of the elapsed time, as we might expect. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.8 Continuity
87
Knowledge of which functions are continuous enables us to evaluate some limits very quickly, as the following example shows. Compare it with Example 1.6.2(b).
Example 5 Find lim
x l 22
x 3 1 2x 2 2 1 . 5 2 3x
SOLUTION The function
f sxd −
x 3 1 2x 2 2 1 5 2 3x

is rational, so by Theorem 5 it is continuous on its domain, which is h x x ± 53 j. Therefore lim
x l22
y
P(cos ¨, sin ¨) 1 0
x 3 1 2x 2 2 1 − lim f sxd − f s22d x l22 5 2 3x
¨
(1, 0)
x
FIGURE 5
−
n
It turns out that most of the familiar functions are continuous at every number in their domains. For instance, Limit Law 10 (page 64) is exactly the statement that root functions are continuous. From the appearance of the graphs of the sine and cosine functions (Figure 1.2.18), we would certainly guess that they are continuous. We know from the definitions of sin and cos that the coordinates of the point P in Figure 5 are scos , sin d. As l 0, we see that P approaches the point s1, 0d and so cos l 1 and sin l 0. Thus 6
Another way to establish the limits in (6) is to use the Squeeze Theorem with the inequality sin , (for . 0), which is proved in Section 2.4.
s22d3 1 2s22d2 2 1 1 −2 5 2 3s22d 11
lim cos − 1
lim sin − 0
l0
l0
Since cos 0 − 1 and sin 0 − 0, the equations in (6) assert that the cosine and sine functions are continuous at 0. The addition formulas for cosine and sine can then be used to deduce that these functions are continuous everywhere (see Exercises 64 and 65). It follows from part 5 of Theorem 4 that tan x −
sin x cos x
is continuous except where cos x − 0. This happens when x is an odd integer multiple of y2, so y − tan x has infinite discontinuities when x − 6y2, 63y2, 65y2, and so on (see Figure 6). y
1 3π _π
_ 2
_
π 2
0
π 2
π
3π 2
x
FIGURE 6 y − tan x Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
88
Chapter 1 Functions and Limits
7 Theorem The following types of functions are continuous at every number in their domains:
• polynomials • root functions
• rational functions • trigonometric functions
Example 6 On what intervals is each function continuous? x 2 1 2x 1 17 tsxd − (a) f sxd − x 100 2 2x 37 1 75 (b) x2 2 1 x11 x11 (c) hsxd − sx 1 2 2 x21 x 11 SOLUTION
(a) f is a polynomial, so it is continuous on s2`, `d by Theorem 5(a). (b) t is a rational function, so by Theorem 5(b), it is continuous on its domain, which is D − hx x 2 2 1 ± 0j − hx x ± 61j. Thus t is continuous on the intervals s2`, 21d, s21, 1d, and s1, `d. (c) We can write hsxd − Fsxd 1 Gsxd 2 Hsxd, where


Fsxd − sx Gsxd −
x11 x11 Hsxd − 2 x21 x 11
F is continuous on [0, `d by Theorem 7. G is a rational function, so it is continuous everywhere except when x 2 1 − 0, that is, x − 1. H is also a rational function, but its denominator is never 0, so H is continuous everywhere. Thus, by parts 1 and 2 of Theorem 4, h is continuous on the intervals [0, 1d and s1, `d. ■ sin x . 2 1 cos x SOLUTION Theorem 7 tells us that y − sin x is continuous. The function in the denominator, y − 2 1 cos x, is the sum of two continuous functions and is therefore continuous. Notice that this function is never 0 because cos x > 21 for all x and so 2 1 cos x . 0 everywhere. Thus the ratio sin x f sxd − 2 1 cos x
Example 7 Evaluate lim
x l
is continuous everywhere. Hence, by the definition of a continuous function,
lim
x l
sin x sin 0 − lim f sxd − f sd − − − 0 x l 2 1 cos x 2 1 cos 221
n
Another way of combining continuous functions f and t to get a new continuous function is to form the composite function f 8 t. This fact is a consequence of the following theorem. This theorem says that a limit symbol can be moved through a function symbol if the function is continuous and the limit exists. In other words, the order of these two symbols can be reversed.
8 Theorem If f is continuous at b and lim tsxd − b, then lim f stsxdd − f sbd. x la x la In other words, lim f stsxdd − f lim tsxd xla
S
xl a
D
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.8 Continuity
89
Intuitively, Theorem 8 is reasonable because if x is close to a, then tsxd is close to b, and since f is continuous at b, if tsxd is close to b, then fstsxdd is close to f sbd. A proof of Theorem 8 is given in Appendix F. n Let’s now apply Theorem 8 in the special case where f sxd − s x , with n being a positive integer. Then n tsxd f stsxdd − s
and
S
D
f lim tsxd − xla
tsxd s xlim la n
If we put these expressions into Theorem 8, we get n n lim tsxd lim s tsxd − s
xla
xla
and so Limit Law 11 has now been proved. (We assume that the roots exist.) 9 Theorem If t is continuous at a and f is continuous at tsad, then the composite function f 8 t given by s f 8 tds xd − f stsxdd is continuous at a. This theorem is often expressed informally by saying “a continuous function of a continuous function is a continuous function.” Proof Since t is continuous at a, we have
lim tsxd − tsad
xla
Since f is continuous at b − tsad, we can apply Theorem 8 to obtain lim f stsxdd − f stsadd
xla
which is precisely the statement that the function hsxd − f s tsxdd is continuous at a; that is, f 8 t is continuous at a.
n
Example 8 Where are the following functions continuous? (a) hsxd − sinsx 2 d (b) Fsxd −
1 sx 1 7 2 4 2
SOLUTION
(a) We have hsxd − f s tsxdd, where tsxd − x 2 and f sxd − sin x Now t is continuous on R since it is a polynomial, and f is also continuous everywhere. Thus h − f 8 t is continuous on R by Theorem 9. (b) Notice that F can be broken up as the composition of four continuous functions: F − f + t + h + k or Fsxd − f stshsksxdddd Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
90
Chapter 1 Functions and Limits
where
f sxd −
1 tsxd − x 2 4 hsxd − sx ksxd − x 2 1 7 x
We know that each of these functions is continuous on its domain (by Theorems 5 and 7), so by Theorem 9, F is continuous on its domain, which is
h x [ R  sx 2 1 7

± 4 j − hx x ± 63j − s2`, 23d ø s23, 3d ø s3, `d
n
An important property of continuous functions is expressed by the following theorem, whose proof is found in more advanced books on calculus. 10 The Intermediate Value Theorem Suppose that f is continuous on the closed interval fa, bg and let N be any number between f sad and f sbd, where f sad ± f sbd. Then there exists a number c in sa, bd such that f scd − N. The Intermediate Value Theorem states that a continuous function takes on every intermediate value between the function values f sad and f sbd. It is illustrated by Figure 7. Note that the value N can be taken on once [as in part (a)] or more than once [as in part (b)]. y
y
f(b)
f(b)
N
y=ƒ
f(a) 0
a
FIGURE 7 y f(a)
y=ƒ
N
y=N
f(b) 0
a
FIGURE 8
b
x
y=ƒ
N f(a) c b
x
0
a c¡
(a)
c™
c£
b
x
(b)
If we think of a continuous function as a function whose graph has no hole or break, then it is easy to believe that the Intermediate Value Theorem is true. In geometric terms it says that if any horizontal line y − N is given between y − f sad and y − f sbd as in Figure 8, then the graph of f can’t jump over the line. It must intersect y − N somewhere. It is important that the function f in Theorem 10 be continuous. The Intermediate Value Theorem is not true in general for discontinuous functions (see Exercise 50). One use of the Intermediate Value Theorem is in locating roots of equations as in the following example.
Example 9 Show that there is a root of the equation 4x 3 2 6x 2 1 3x 2 2 − 0 between 1 and 2. SOLUTION Let f sxd − 4x 3 2 6x 2 1 3x 2 2. We are looking for a solution of the given
equation, that is, a number c between 1 and 2 such that f scd − 0. Therefore we take a − 1, b − 2, and N − 0 in Theorem 10. We have f s1d − 4 2 6 1 3 2 2 − 21 , 0 and
f s2d − 32 2 24 1 6 2 2 − 12 . 0
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
91
Section 1.8 Continuity
Thus f s1d , 0 , f s2d; that is, N − 0 is a number between f s1d and f s2d. Now f is continuous since it is a polynomial, so the Intermediate Value Theorem says there is a number c between 1 and 2 such that f scd − 0. In other words, the equation 4x 3 2 6x 2 1 3x 2 2 − 0 has at least one root c in the interval s1, 2d. In fact, we can locate a root more precisely by using the Intermediate Value Theorem again. Since 0.2
3
f s1.2d − 20.128 , 0 and f s1.3d − 0.548 . 0 3
_1
f s1.22d − 20.007008 , 0 and f s1.23d − 0.056068 . 0
_3
_0.2
FIGURE 9
so a root lies in the interval s1.22, 1.23d.
0.2
3
1.3
1.2
a root must lie between 1.2 and 1.3. A calculator gives, by trial and error,
1.3
1.2
_0.2
FIGURE 10
n
We can use a graphing calculator or computer to illustrate the use of the Intermediate Value Theorem in Example 9. Figure 9 shows the graph of f in the viewing rectangle f21, 3g by f23, 3g and you can see that the graph crosses the xaxis between 1 and 2. Fig ure 10 shows the result of zooming in to the viewing rectangle f1.2, 1.3g by f20.2, 0.2g. In fact, the Intermediate Value Theorem plays a role in the very way these graphing devices work. A computer calculates a finite number of points on the graph and turns on the pixels that contain these calculated points. It assumes that the function is continuous and takes on all the intermediate values between two consecutive points. The computer therefore “connects the dots” by turning on the intermediate pixels.
1. Write an equation that expresses the fact that a function f is continuous at the number 4.
4. From the graph of t, state the intervals on which t is continuous.
2. If f is continuous on s2`, `d, what can you say about its graph?
y
3. (a) From the graph of f , state the numbers at which f is discontinuous and explain why. (b) For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the left, or neither.
_3
_2
0
1
2
3
x
y
5 – 8 Sketch the graph of a function f that is continuous except for the stated discontinuity. 5. Discontinuous at 2, but continuous from the right there _4
_2
0
2
4
6
x
6. Discontinuities at 21 and 4, but continuous from the left at 21 and from the right at 4 7. Removable discontinuity at 3, jump discontinuity at 5
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
92
chapter 1 Functions and Limits
8. Neither left nor right continuous at 22, continuous only from the left at 2 9. The toll T charged for driving on a certain stretch of a toll road is $5 except during rush hours (between 7 am and 10 am and between 4 pm and 7 pm) when the toll is $7. (a) Sketch a graph of T as a function of the time t, measured in hours past midnight. (b) Discuss the discontinuities of this function and their significance to someone who uses the road. 10. Explain why each function is continuous or discontinuous. (a) The temperature at a specific location as a function of time (b) The temperature at a specific time as a function of the distance due west from New York City (c) The altitude above sea level as a function of the distance due west from New York City (d) The cost of a taxi ride as a function of the distance traveled (e) The current in the circuit for the lights in a room as a function of time 11–14 Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. t 2 1 5t , a − 2 2t 1 1
a−0
2x 2 2 5x 2 3 22. f sxd − x23 6
a−3
x3 2 8 x2 2 x 2 2 24. f sxd − 2 x22 x 24
23. f sxd −
25 – 32 Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. 2x 2 2 x 2 1 x2 1 1 26. Gsxd − 2 2 x 11 2x 2 x 2 1
25. Fsxd −
3 x22 sin x s 28. hsxd − x3 2 2 x11
2
15 –16 Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.
x21 , s2`, 22d 3x 1 6
Î
11
1 32. Fsxd − sinscosssin xdd x
; 33 –34 Locate the discontinuities of the function and illustrate by graphing. 33. y −
15. f sxd − x 1 sx 2 4 , f4, `d 16. tsxd −
if x ± 3 if x − 3
23 –24 How would you “remove the discontinuity” of f ? In other words, how would you define f s2d in order to make f continuous at 2?
31. Msxd −
14. f sxd − 3x 2 5x 1 sx 1 4 , a − 2 3
1 34. y − tan sx 1 1 sin x
35 – 38 Use continuity to evaluate the limit. 35. lim x s20 2 x 2 x l2
17– 22 Explain why the function is discontinuous at the given number a. Sketch the graph of the function. 17. f sxd −
18. f sxd −
1 x12
H H
1 x12 1
a − 22
if x ± 22 if x − 22
1 2 x 2 if x , 1 19. f sxd − 1yx if x > 1
a−1
tan x 29. hsxd − coss1 2 x 2 d 30. Bsxd − s4 2 x 2
13. psvd − 2s3v 2 1 1 , a − 1 4
if x ± 1 if x − 1
cos x if x , 0 21. f sxd − 0 if x − 0 1 2 x 2 if x . 0
27. Qsxd −
11. f sxd − sx 1 2x 3 d4, a − 21 12. t std −
H H H
x2 2 x 20. f sxd − x 2 2 1 1
a − 22
36. lim sinsx 1 sin xd x l
2
37. lim x tan x 38. lim x 3ysx 2 1 x 2 2 x ly4
xl2
39 – 40 Show that f is continuous on s2`, `d. 39. f sxd −
40. f sxd − a−1
H H
1 2 x 2 if x < 1 sx 2 1 if x . 1 sin x if x , y4 cos x if x > y4
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 1.8 Continuity
41– 43 Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f .
H H H
x 2 if x , 21 if 21 < x , 1 41. f sxd − x 1yx if x > 1 x 2 1 1 if x < 1 42. f sxd − 3 2 x if 1 , x < 4 if x . 4 sx x 1 2 if x , 0 43. f sxd − 2x 2 if 0 < x < 1 2 2 x if x . 1 44. T he gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet is
Fsrd −
GMr if r , R R3 GM if r > R r2
where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r? 45. F or what value of the constant c is the function f continuous on s2`, `d? f sxd −
H
cx 2 1 2x if x , 2 x 3 2 cx if x > 2
46. Find the values of a and b that make f continuous everywhere.
f sxd −
x2 2 4 if x , 2 x22 ax 2 2 bx 1 3 if 2 < x , 3 2x 2 a 1 b if x > 3
47. Suppose f and t are continuous functions such that ts2d − 6 and lim x l2 f3 f sxd 1 f sxd tsxdg − 36. Find f s2d. 48. Let f sxd − 1yx and tsxd − 1yx 2. (a) Find s f + tds xd. (b) Is f + t continuous everywhere? Explain. 49. W hich of the following functions f has a removable discon tinuity at a? If the discontinuity is removable, find a function t that agrees with f for x ± a and is continuous at a. x4 2 1 (a) f sxd − , a − 1 x21 (b) f sxd −
x 3 2 x 2 2 2x , a − 2 x22
(c) f sxd − v sin x b , a −
93
50. S uppose that a function f is continuous on [0, 1] except at 0.25 and that f s0d − 1 and f s1d − 3. Let N − 2. Sketch two possible graphs of f, one showing that f might not satisfy the conclusion of the Intermediate Value Theorem and one showing that f might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn’t satisfy the hypothesis). 51. If f sxd − x 2 1 10 sin x, show that there is a number c such that f scd − 1000. 52. Suppose f is continuous on f1, 5g and the only solutions of the equation f sxd − 6 are x − 1 and x − 4. If f s2d − 8, explain why f s3d . 6. 53– 56 Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. 53. x 4 1 x 2 3 − 0, s1, 2d 54. 2yx − x 2 sx , s2, 3d 55. cos x − x, s0, 1d 56. sin x − x 2 2 x, s1, 2d 57 – 58 (a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root. 57. cos x − x 3 58. x 5 2 x 2 1 2x 1 3 − 0 ; 59– 60 (a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root correct to three decimal places. 1 59. x 5 2 x 2 2 4 − 0 60. sx 2 5 − x13 61– 62 Prove, without graphing, that the graph of the function has at least two xintercepts in the specified interval. 61. y − sin x 3, s1, 2d 62. y − x 2 2 3 1 1yx, s0, 2d 63. Prove that f is continuous at a if and only if lim f sa 1 hd − f sad
hl0
64. T o prove that sine is continuous, we need to show that lim x l a sin x − sin a for every real number a. By Exercise 63 an equivalent statement is that lim sinsa 1 hd − sin a
hl0
Use (6) to show that this is true. 65. Prove that cosine is a continuous function. 66. (a) Prove Theorem 4, part 3. (b) Prove Theorem 4, part 5.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
94
chapter 1 Functions and Limits
67. For what values of x is f continuous?
H
0 if x is rational f sxd − 1 if x is irrational 68. For what values of x is t continuous? tsxd −
H
0 if x is rational x if x is irrational
69. Is there a number that is exactly 1 more than its cube? 70. If a and b are positive numbers, prove that the equation a b 1 3 −0 x 3 1 2x 2 2 1 x 1x22 has at least one solution in the interval s21, 1d.
71. Show that the function f sxd −
H
x 4 sins1yxd 0
if x ± 0 if x − 0
is continuous on s2`, `d.
 
72. (a) Show that the absolute value function Fsxd − x is continuous everywhere. (b) Prove that if f is a continuous function on an interval, then so is f . (c) Is the converse of the statement in part (b) also true? In other words, if f is continuous, does it follow that f is continuous? If so, prove it. If not, find a counterexample.
 
 
73. A Tibetan monk leaves the monastery at 7:00 am and takes his usual path to the top of the mountain, arriving at 7:00 pm. The following morning, he starts at 7:00 am at the top and takes the same path back, arriving at the monastery at 7:00 pm. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.
1 Review CONCEPT CHECK
Answers to the Concept Check can be found on the back endpapers.
1. (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How can you tell whether a given curve is the graph of a function?
8. Draw, by hand, a rough sketch of the graph of each function. (a) y − sin x (b) y − cos x (c) y − tan x (d) y − 1yx (e) y − x (f) y − sx
2. Discuss four ways of representing a function. Illustrate your discussion with examples. 3. (a) What is an even function? How can you tell if a function is even by looking at its graph? Give three examples of an even function. (b) What is an odd function? How can you tell if a function is odd by looking at its graph? Give three examples of an odd function. 4. What is an increasing function? 5. What is a mathematical model? 6. Give an example of each type of function. (a) Linear function (b) Power function (c) Exponential function (d) Quadratic function (e) Polynomial of degree 5 (f ) Rational function 7. Sketch by hand, on the same axes, the graphs of the following functions. (a) f sxd − x (b) tsxd − x 2 (c) hsxd − x 3 (d) jsxd − x 4
 
9. Suppose that f has domain A and t has domain B. (a) What is the domain of f 1 t? (b) What is the domain of f t? (c) What is the domain of fyt? 10. H ow is the composite function f 8 t defined? What is its domain? 11. S uppose the graph of f is given. Write an equation for each of the graphs that are obtained from the graph of f as follows. (a) Shift 2 units upward. (b) Shift 2 units downward. (c) Shift 2 units to the right. (d) Shift 2 units to the left. (e) Reflect about the xaxis. (f) Reflect about the yaxis. (g) Stretch vertically by a factor of 2. (h) Shrink vertically by a factor of 2. (i) Stretch horizontally by a factor of 2. (j) Shrink horizontally by a factor of 2. 12. Explain what each of the following means and illustrate with a sketch. (a) lim f sxd − L (b) lim1 f sxd − L (c) lim2 f sxd − L x la
x la
x la
(d) lim f sxd − ` (e) lim f sxd − 2` x la
x la
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
chapter 1 Review
13. Describe several ways in which a limit can fail to exist. Illustrate with sketches. 14. What does it mean to say that the line x − a is a vertical asymptote of the curve y − f sxd? Draw curves to illustrate the various possibilities. 15.
State the following Limit Laws. (a) Sum Law (b) Difference Law (c) Constant Multiple Law (d) Product Law (e) Quotient Law (f ) Power Law (g) Root Law
95
16. What does the Squeeze Theorem say? 17. (a) What does it mean for f to be continuous at a? (b) What does it mean for f to be continuous on the interval s2`, `d? What can you say about the graph of such a function? 18. (a) Give examples of functions that are continuous on f21, 1g. (b) Give an example of a function that is not continuous on f0, 1g. 19. What does the Intermediate Value Theorem say?
TRUEFALSE QUIZ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f is a function, then f ss 1 td − f ssd 1 f std. 2. If f ssd − f std, then s − t.
4. If x 1 , x 2 and f is a decreasing function, then f sx 1 d . f sx 2 d. 5. A vertical line intersects the graph of a function at most once. 6. If x is any real number, then sx 2 − x.
x l4
8. lim
x l1
S
2x 8 2 x24 x24
D
− lim
x l4
2x 8 2 lim x l4 x 2 4 x24
lim sx 2 1 6x 2 7d x 2 1 6x 2 7 x l1 − lim sx 2 1 5x 2 6d x 2 1 5x 2 6 x l1
lim sx 2 3d x23 xl1 − 9. lim 2 x l 1 x 1 2x 2 4 lim sx 2 1 2x 2 4d xl1
2
10.
x 29 −x13 x23
11. lim
xl3
15. If lim x l a f sxd exists but lim x l a tsxd does not exist, then lim x l a f f sxd 1 tsxdg does not exist. 16. If lim x l 6 f f sxd tsxdg exists, then the limit must be f s6d ts6d.
3. If f is a function, then f s3xd − 3 f sxd.
7. lim
14. If neither lim x l a f sxd nor lim x l a tsxd exists, then lim x l a f f sxd 1 tsxdg does not exist.
x2 2 9 − lim sx 1 3d xl3 x23
12. If lim x l 5 f sxd − 2 and lim x l 5 tsxd − 0, then limx l 5 f f sxdytsxdg does not exist. 13. If lim x l5 f sxd − 0 and lim x l 5 tsxd − 0, then lim x l 5 f f sxdytsxdg does not exist.
17. If p is a polynomial, then lim x l b psxd − psbd. 18. If lim x l 0 f sxd − ` and lim x l 0 tsxd − `, then lim x l 0 f f sxd 2 tsxdg − 0. 19. If the line x − 1 is a vertical asymptote of y − f sxd, then f is not defined at 1. 20. If f s1d . 0 and f s3d , 0, then there exists a number c between 1 and 3 such that f scd − 0. 21. If f is continuous at 5 and f s5d − 2 and f s4d − 3, then lim x l 2 f s4x 2 2 11d − 2. 22. If f is continuous on f21, 1g and f s21d − 4 and f s1d − 3, then there exists a number r such that r , 1 and f srd − .
 
23. Let f be a function such that lim x l 0 f sxd − 6. Then there exists a positive number such that if 0 , x , , then f sxd 2 6 , 1.

 

24. If f sxd . 1 for all x and lim x l 0 f sxd exists, then lim x l 0 f sxd . 1. 25. The equation x 10 2 10x 2 1 5 − 0 has a root in the interval s0, 2d.
 
26. If f is continuous at a, so is f .
 
27. If f is continuous at a, so is f .
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
96
chapter 1 Functions and Limits
EXERCISES (c) y − 2 2 f sxd (d) y − 12 f sxd 2 1
1. Let f be the function whose graph is given.
y
y
f 1
1
x
1
0
1
x
11–16 Use transformations to sketch the graph of the function.
y − 2 sx 11. y − sx 2 2d3 12.
(a) Estimate the value of f s2d. (b) Estimate the values of x such that f sxd − 3. (c) State the domain of f. (d) State the range of f. (e) On what interval is f increasing? (f) Is f even, odd, or neither even nor odd? Explain.
1 13. y − x 2 2 2 x 1 2 14. y− x21
H
1 1 x if x , 0 15. f sxd − 2cos 2x 16. f sxd − 1 1 x 2 if x > 0
2. Determine whether each curve is the graph of a function of x. If it is, state the domain and range of the function. y
y
2
2
0
1
x
0
1
x
3. If f sxd − x 2 2 2x 1 3, evaluate the difference quotient f sa 1 hd 2 f sad h 4. Sketch a rough graph of the yield of a crop as a function of the amount of fertilizer used. 5–8 Find the domain and range of the function. Write your answer in interval notation. 5. f sxd − 2ys3x 2 1d 6. tsxd − s16 2 x 4 7. y − 1 1 sin x 8. Fstd − 3 1 cos 2t 9. Suppose that the graph of f is given. Describe how the graphs of the following functions can be obtained from the graph of f. (a) y − f sxd 1 8 (b) y − f sx 1 8d (c) y − 1 1 2 f sxd (d) y − f sx 2 2d 2 2 (e) y − 2f sxd (f ) y − 3 2 f sxd 10. The graph of f is given. Draw the graphs of the following functions. (a) y − f sx 2 8d (b) y − 2f sxd
17. Determine whether f is even, odd, or neither even nor odd. (a) f sxd − 2x 5 2 3x 2 1 2 (b) f sxd − x 3 2 x 7 (c) f sxd − cossx 2 d (d) f sxd − 1 1 sin x 18. Find an expression for the function whose graph consists of the line segment from the point s22, 2d to the point s21, 0d together with the top half of the circle with center the origin and radius 1. 19. If f sxd − sx and tsxd − sin x, find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , (d) t 8 t, and their domains. 20. Express the function Fsxd − 1ysx 1 sx as a composition of three functions. 21. Life expectancy improved dramatically in the 20th century. The table gives the life expectancy at birth (in years) of males born in the United States. Use a scatter plot to choose an appropriate type of model. Use your model to predict the life span of a male born in the year 2010. Birth year Life expectancy 1900 1910 1920 1930 1940 1950
48.3 51.1 55.2 57.4 62.5 65.6
Birth year Life expectancy 1960 1970 1980 1990 2000
66.6 67.1 70.0 71.8 73.0
22. A smallappliance manufacturer finds that it costs $9000 to produce 1000 toaster ovens a week and $12,000 to produce 1500 toaster ovens a week. (a) Express the cost as a function of the number of toaster ovens produced, assuming that it is linear. Then sketch the graph.
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chapter 1 Review
(b) What is the slope of the graph and what does it represent? (c) What is the yintercept of the graph and what does it represent?
xl2
0
if x , 0 s2x f sxd − 3 2 x if 0 < x , 3 sx 2 3d2 if x . 3
x
1
x l 23
x l0
x l2
(b) State the equations of the vertical asymptotes. (c) At what numbers is f discontinuous? Explain.
24. S ketch the graph of an example of a function f that satisfies all of the following conditions: lim1 f sxd − 22, lim2 f sxd − 1, f s0d − 21,
x l0
x l0
x l3
x l3
tsxd −
x l2
x2 2 9 lim 25. lim cossx 1 sin xd 26. x l0 x l 3 x 2 1 2x 2 3 x2 2 9 x2 2 9 28. lim1 2 x l 1 x 1 2x 2 3 x 1 2x 2 3
27. lim
2
x l 23
sh 2 1d3 1 1 t2 2 4 29. lim 30. lim 3 h l0 t l2 t 2 8 h
x l3
(b) Where is f discontinuous? (c) Sketch the graph of f.
46. Let
x l0
25–38 Find the limit.
x l0
(iv) lim2 f sxd (v) lim1 f sxd (vi) lim f sxd
lim2 f sxd − `, lim1 f sxd − 2`
x l2
(a) Evaluate each limit, if it exists. lim1 f sxd (ii) lim2 f sxd (iii) lim f sxd (i)
x l 23
(iv) lim f sxd (v) lim f sxd (vi) lim2 f sxd
x l0
H
45. Let
(a) Find each limit, or explain why it does not exist. (i) lim1 f sxd (ii) lim 1 f sxd (iii) lim f sxd
xl0
2 lim −` 43. lim sx 2 2 3xd − 22 44. xl2 x l 41 sx 2 4
1
40. Prove that lim x l 0 x 2 coss1yx 2 d − 0. 3 lim s x −0 41. lim s14 2 5xd − 4 42.
y
x l4
39. If 2x 2 1 < f sxd < x 2 for 0 , x , 3, find lim x l1 f sxd.
41–44 Prove the statement using the precise definition of a limit.
23. The graph of f is given.
x l2
97
2x 2 x 2 22x x24
if if if if
0 0 if x 2 3 , 0
x23 2x 1 3
if x > 3 if x , 3
x12 2sx 1 2d
if x 1 2 > 0 if x 1 2 , 0
x12 2x 2 2
if x > 22 if x , 22
These expressions show that we must consider three cases: x , 22 22 < x , 3 x > 3 Case I If x , 22, we have
 x 2 3  1  x 1 2  , 11 2x 1 3 2 x 2 2 , 11 22x , 10 x . 25 Case II If 22 < x , 3, the given inequality becomes 2x 1 3 1 x 1 2 , 11 5 , 11 (always true) Case iii If x > 3, the inequality becomes x 2 3 1 x 1 2 , 11 2x , 12 x,6 Combining cases I, II, and III, we see that the inequality is satisfied when 25 , x , 6. So the solution is the interval s25, 6d. ■ In the following example we first guess the answer by looking at special cases and recognizing a pattern. Then we prove our conjecture by mathematical induction. In using the Principle of Mathematical Induction, we follow three steps: Step 1 Prove that Sn is true when n − 1. Step 2 Assume that Sn is true when n − k and deduce that Sn is true when n − k 1 1. Step 3 Conclude that Sn is true for all n by the Principle of Mathematical Induction. 100 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Example 2 If f0sxd − xysx 1 1d and fn11 − f0 8 fn for n − 0, 1, 2, . . . , find a formula for fnsxd. PS Analogy: Try a similar, simpler problem
Solution We start by finding formulas for fnsxd for the special cases n − 1, 2, and 3.
S D
x f1sxd − s f0 8 f0dsxd − f0( f0sxd) − f0 x11
x x x11 x11 x − − − x 2x 1 1 2x 1 1 11 x11 x11
S
x f2sxd − s f0 8 f1 dsxd − f0( f1sxd) − f0 2x 1 1
D
x x 2x 1 1 2x 1 1 x − − − x 3x 1 1 3x 1 1 11 2x 1 1 2x 1 1
S
x f3sxd − s f0 8 f2 dsxd − f0( f2sxd) − f0 3x 1 1
D
x x 3x 1 1 3x 1 1 x − − − x 4x 1 1 4x 1 1 11 3x 1 1 3x 1 1
PS Look for a pattern
We notice a pattern: The coefficient of x in the denominator of fnsxd is n 1 1 in the three cases we have computed. So we make the guess that, in general, 1
fnsxd −
x sn 1 1dx 1 1
To prove this, we use the Principle of Mathematical Induction. We have already verified that (1) is true for n − 1. Assume that it is true for n − k, that is, fksxd −
Then
x sk 1 1dx 1 1
S
x fk11sxd − s f0 8 fk dsxd − f0( fksxd) − f0 sk 1 1dx 1 1
D
x x sk 1 1dx 1 1 sk 1 1dx 1 1 x − − − x sk 1 2dx 1 1 sk 1 2dx 1 1 11 sk 1 1dx 1 1 sk 1 1dx 1 1
This expression shows that (1) is true for n − k 1 1. Therefore, by mathematical induction, it is true for all positive integers n.
■
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In the following example we show how the problemsolving strategy of introducing something extra is sometimes useful when we evaluate limits. The idea is to change the variable—to introduce a new variable that is related to the original variable—in such a way as to make the problem simpler. Later, in Section 4.5, we will make more extensive use of this general idea. 3 1 1 cx 2 1 s , where c is a constant. xl0 x Solution As it stands, this limit looks challenging. In Section 1.6 we evaluated several limits in which both numerator and denominator approached 0. There our strategy was to perform some sort of algebraic manipulation that led to a simplifying cancellation, but here it’s not clear what kind of algebra is necessary. So we introduce a new variable t by the equation
Example 3 Evaluate lim
3 t−s 1 1 cx
We also need to express x in terms of t, so we solve this equation: t 3 − 1 1 cx x −
t3 2 1 sif c ± 0d c
Notice that x l 0 is equivalent to t l 1. This allows us to convert the given limit into one involving the variable t: 3 1 1 cx 2 1 t21 s − lim 3 t l1 st 2 1dyc xl0 x
lim
− lim t l1
cst 2 1d t3 2 1
The change of variable allowed us to replace a relatively complicated limit by a simpler one of a type that we have seen before. Factoring the denominator as a difference of cubes, we get lim t l1
cst 2 1d cst 2 1d − lim t l1 st 2 1dst 2 1 t 1 1d t3 2 1 − lim t l1
c c − t2 1 t 1 1 3
In making the change of variable we had to rule out the case c − 0. But if c − 0, the function is 0 for all nonzero x and so its limit is 0. Therefore, in all cases, the limit is cy3. n The following problems are meant to test and challenge your problemsolving skills. Some of them require a considerable amount of time to think through, so don’t be discouraged if you can’t solve them right away. If you get stuck, you might find it helpful to refer to the discussion of the principles of problem solving. Problems

   2. Solve the inequality  x 2 1  2  x 2 3  > 5. 1. Solve the equation 2x 2 1 2 x 1 5 − 3.

 

3. Sketch the graph of the function f sxd − x 2 2 4 x 1 3 .
102
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

 

4. Sketch the graph of the function tsxd − x 2 2 1 2 x 2 2 4 .
 
 
5. Draw the graph of the equation x 1 x − y 1 y . 6. Sketch the region in the plane consisting of all points sx, yd such that
x 2 y 1 x 2 y < 2 7. The notation maxha, b, . . .j means the largest of the numbers a, b, . . . . Sketch the graph of each function. (a) f sxd − maxhx, 1yxj (b) f sxd − maxhsin x, cos xj (c) f sxd − maxhx 2, 2 1 x, 2 2 xj 8. Sketch the region in the plane defined by each of the following equations or inequalities. (a) maxhx, 2yj − 1 (b) 21 < maxhx, 2yj < 1 (c) maxhx, y 2 j − 1 9. A driver sets out on a journey. For the first half of the distance she drives at the leisurely pace of 30 miyh; she drives the second half at 60 miyh. What is her average speed on this trip? 10. Is it true that f 8 s t 1 hd − f 8 t 1 f 8 h? 11. Prove that if n is a positive integer, then 7 n 2 1 is divisible by 6. 12. Prove that 1 1 3 1 5 1 ∙ ∙ ∙ 1 s2n 2 1d − n 2. 13. If f0sxd − x 2 and fn11sxd − f0s fnsxdd for n − 0, 1, 2, . . . , find a formula for fnsxd. 1 and fn11 − f0 8 fn for n − 0, 1, 2, . . . , find an expression for fnsxd 22x and use mathematical induction to prove it. (b) Graph f0 , f1, f2 , f3 on the same screen and describe the effects of repeated composition.
14. (a) If f0sxd − ;
15. Evaluate lim
x l1
3 x 21 s . sx 2 1
16. Find numbers a and b such that lim
x l0
17. Evaluate lim
xl0
y
Q
P
FIGURE FOR PROBLEM 18
 2x 2 1  2  2x 1 1  . x
18. The figure shows a point P on the parabola y − x 2 and the point Q where the perpendicular bisector of OP intersects the yaxis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, find it.
y=≈
0
sax 1 b 2 2 − 1. x
x
19. E valuate the following limits, if they exist, where v x b denotes the greatest integer function. v xb (a) lim (b) lim x v 1yx b xl0 xl0 x 20. Sketch the region in the plane defined by each of the following equations. (a) v xb 2 1 v yb 2 − 1 (b) v xb 2 2 v yb 2 − 3 2 (c) v x 1 yb − 1 (d) v xb 1 v yb − 1
103 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
21. Find all values of a such that f is continuous on R: f sxd −
H
x 1 1 if x < a x2 if x . a
22. A fixed point of a function f is a number c in its domain such that f scd − c. (The function doesn’t move c; it stays fixed.) (a) Sketch the graph of a continuous function with domain f0, 1g whose range also lies in f0, 1g. Locate a fixed point of f . (b) Try to draw the graph of a continuous function with domain f0, 1g and range in f0, 1g that does not have a fixed point. What is the obstacle? (c) Use the Intermediate Value Theorem to prove that any continuous function with domain f0, 1g and range in f0, 1g must have a fixed point. 23. If lim x l a f f sxd 1 tsxdg − 2 and lim x l a f f sxd 2 tsxdg − 1, find lim x l a f f sxd tsxdg. 24. (a) The figure shows an isosceles triangle ABC with /B − /C. The bisector of angle B intersects the side AC at the point P. Suppose that the base BC remains fixed but the altitude AM of the triangle approaches 0, so A approaches the midpoint M of BC. What happens to P during this process? Does it have a limiting position? If so, find it.


A P
B
M
C
(b) Try to sketch the path traced out by P during this process. Then find an equation of this curve and use this equation to sketch the curve.
25. (a) If we start from 0° latitude and proceed in a westerly direction, we can let Tsxd denote the temperature at the point x at any given time. Assuming that T is a continuous function of x, show that at any fixed time there are at least two diametrically opposite points on the equator that have exactly the same temperature. (b) Does the result in part (a) hold for points lying on any circle on the earth’s surface? (c) Does the result in part (a) hold for barometric pressure and for altitude above sea level?
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2
Derivatives
The maximum sustainable swimming speed S of salmon depends on the water temperature T. Exercise 58 in Section 2.1 asks you to analyze how S varies as T changes by estimating the derivative of S with respect to T. © Jody Ann / Shutterstock.com
In this chapter we begin our study of differential calculus, which is concerned with how one quantity changes in relation to another quantity. The central concept of differential calculus is the derivative, which is an outgrowth of the velocities and slopes of tangents that we considered in Chapter 1. After learning how to calculate derivatives, we use them to solve problems involving rates of change and the approximation of functions.
105 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
106
Chapter 2 Derivatives
The problem of finding the tangent line to a curve and the problem of finding the velocity of an object both involve finding the same type of limit, as we saw in Section 1.4. This special type of limit is called a derivative and we will see that it can be interpreted as a rate of change in any of the natural or social sciences or engineering.
Tangents y
Q{ x, ƒ } ƒf(a)
P { a, f(a)}
If a curve C has equation y − f sxd and we want to find the tangent line to C at the point Psa, f sadd, then we consider a nearby point Qsx, f sxdd, where x ± a, and compute the slope of the secant line PQ: mPQ −
xa
0
a
y
x
x
Then we let Q approach P along the curve C by letting x approach a. If mPQ approaches a number m, then we define the tangent t to be the line through P with slope m. (This amounts to saying that the tangent line is the limiting position of the secant line PQ as Q approaches P. See Figure 1.)
t Q
1 Definition The tangent line to the curve y − f sxd at the point Psa, f sadd is the line through P with slope
Q
P
f sxd 2 f sad x2a
m − lim
Q
xla
f sxd 2 f sad x2a
provided that this limit exists.
In our first example we confirm the guess we made in Example 1.4.1.
x
0
FIGURE 1
Example 1 Find an equation of the tangent line to the parabola y − x 2 at the point Ps1, 1d. SOLUTION Here we have a − 1 and f sxd − x 2, so the slope is
m − lim
x l1
− lim
x l1
f sxd 2 f s1d x2 2 1 − lim x l1 x 2 1 x21 sx 2 1dsx 1 1d x21
− lim sx 1 1d − 1 1 1 − 2 x l1
Pointslope form for a line through the point sx1 , y1 d with slope m:
Using the pointslope form of the equation of a line, we find that an equation of the tangent line at s1, 1d is
y 2 y1 − msx 2 x 1 d
y 2 1 − 2sx 2 1d or y − 2x 2 1
n
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107
Section 2.1 Derivatives and Rates of Change
We sometimes refer to the slope of the tangent line to a curve at a point as the slope of the curve at the point. The idea is that if we zoom in far enough toward the point, the curve looks almost like a straight line. Figure 2 illustrates this procedure for the curve y − x 2 in Example 1. The more we zoom in, the more the parabola looks like a line. In other words, the curve becomes almost indistinguishable from its tangent line.
TEC Visual 2.1 shows an animation of Figure 2. 2
1.5
(1, 1)
1.1
(1, 1)
2
0
(1, 1)
1.5
0.5
0.9
1.1
FIGURE 2 Zooming in toward the point (1, 1) on the parabola y − x 2 Q { a+h, f(a+h)}
y
t
There is another expression for the slope of a tangent line that is sometimes easier to use. If h − x 2 a, then x − a 1 h and so the slope of the secant line PQ is mPQ −
P { a, f(a)} f(a+h)f(a)
h 0
a
a+h
x
f sa 1 hd 2 f sad h
(See Figure 3 where the case h . 0 is illustrated and Q is to the right of P. If it happened that h , 0, however, Q would be to the left of P.) Notice that as x approaches a, h approaches 0 (because h − x 2 a) and so the expression for the slope of the tangent line in Definition 1 becomes
FIGURE 3
2
m − lim
hl0
f sa 1 hd 2 f sad h
Example 2 Find an equation of the tangent line to the hyperbola y − 3yx at the
point s3, 1d.
SOLUTION Let f sxd − 3yx. Then, by Equation 2, the slope of the tangent at s3, 1d is
m − lim
hl0
f s3 1 hd 2 f s3d h
3 3 2 s3 1 hd 21 31h 31h − lim − lim hl0 hl0 h h y
− lim
3 y= x
x+3y6=0
hl0
Therefore an equation of the tangent at the point s3, 1d is
(3, 1) 0
y 2 1 − 213 sx 2 3d
x
which simplifies to FIGURE 4
2h 1 1 − lim 2 −2 hl0 hs3 1 hd 31h 3
x 1 3y 2 6 − 0
The hyperbola and its tangent are shown in Figure 4.
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n
108
Chapter 2 Derivatives
position at time t=a 0
position at time t=a+h s
f(a+h)f(a)
f(a) f(a+h)
FIGURE 5 s
Velocities In Section 1.4 we investigated the motion of a ball dropped from the CN Tower and defined its velocity to be the limiting value of average velocities over shorter and shorter time periods. In general, suppose an object moves along a straight line according to an equation of motion s − f std, where s is the displacement (directed distance) of the object from the origin at time t. The function f that describes the motion is called the position function of the object. In the time interval from t − a to t − a 1 h the change in position is f sa 1 hd 2 f sad. (See Figure 5.) The average velocity over this time interval is
Q { a+h, f(a+h)} P { a, f(a)} h
0
a
mPQ=
a+h
t
f(a+h)f(a) h
average velocity
average velocity −
displacement f sa 1 hd 2 f sad − time h
which is the same as the slope of the secant line PQ in Figure 6. Now suppose we compute the average velocities over shorter and shorter time intervals fa, a 1 hg. In other words, we let h approach 0. As in the example of the falling ball, we define the velocity (or instantaneous velocity) vsad at time t − a to be the limit of these average velocities:
3
vsad − lim
hl0
FIGURE 6
f sa 1 hd 2 f sad h
This means that the velocity at time t − a is equal to the slope of the tangent line at P (compare Equations 2 and 3). Now that we know how to compute limits, let’s reconsider the problem of the falling ball.
Example 3 Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground. (a) What is the velocity of the ball after 5 seconds? (b) How fast is the ball traveling when it hits the ground? Recall from Section 1.4: The distance (in meters) fallen after t seconds is 4.9t 2.
SOLUTION We will need to find the velocity both when t − 5 and when the ball hits the ground, so it’s efficient to start by finding the velocity at a general time t. Using the equation of motion s − f std − 4.9t 2, we have v std − lim
hl0
f st 1 hd 2 f std 4.9st 1 hd2 2 4.9t 2 − lim hl0 h h
− lim
4.9st 2 1 2th 1 h 2 2 t 2 d 4.9s2th 1 h 2 d − lim hl0 h h
− lim
4.9hs2t 1 hd − lim 4.9s2t 1 hd − 9.8t hl0 h
hl0
hl0
(a) The velocity after 5 seconds is vs5d − s9.8ds5d − 49 mys. (b) Since the observation deck is 450 m above the ground, the ball will hit the ground at the time t when sstd − 450, that is, 4.9t 2 − 450 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.1 Derivatives and Rates of Change
This gives t2 −
450 and t − 4.9
Î
109
450 < 9.6 s 4.9
The velocity of the ball as it hits the ground is therefore
v
SÎ D Î 450 4.9
− 9.8
450 < 94 mys 4.9
n
Derivatives We have seen that the same type of limit arises in finding the slope of a tangent line (Equation 2) or the velocity of an object (Equation 3). In fact, limits of the form lim
h l0
f sa 1 hd 2 f sad h
arise whenever we calculate a rate of change in any of the sciences or engineering, such as a rate of reaction in chemistry or a marginal cost in economics. Since this type of limit occurs so widely, it is given a special name and notation. 4 Definition The derivative of a function f at a number a, denoted by f 9sad, is f sa 1 hd 2 f sad f 9sad − lim h l0 h if this limit exists.
f 9sad is read “ f prime of a.”
If we write x − a 1 h, then we have h − x 2 a and h approaches 0 if and only if x approaches a. Therefore an equivalent way of stating the definition of the derivative, as we saw in finding tangent lines, is
5
f 9sad − lim
xla
f sxd 2 f sad x2a
Example 4 Find the derivative of the function f sxd − x 2 2 8x 1 9 at the number a. SOLUTION From Definition 4 we have
f 9sad − lim
Definitions 4 and 5 are equivalent, so we can use either one to compute the derivative. In practice, Definition 4 often leads to simpler computations.
h l0
− lim
fsa 1 hd2 2 8sa 1 hd 1 9g 2 fa 2 2 8a 1 9g h
− lim
a 2 1 2ah 1 h 2 2 8a 2 8h 1 9 2 a 2 1 8a 2 9 h
− lim
2ah 1 h 2 2 8h − lim s2a 1 h 2 8d h l0 h
h l0
h l0
h l0
f sa 1 hd 2 f sad h
− 2a 2 8
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n
110
Chapter 2 Derivatives
We defined the tangent line to the curve y − f sxd at the point Psa, f sadd to be the line that passes through P and has slope m given by Equation 1 or 2. Since, by Definition 4, this is the same as the derivative f 9sad, we can now say the following. The tangent line to y − f sxd at sa, f sadd is the line through sa, f sadd whose slope is equal to f 9sad, the derivative of f at a.
If we use the pointslope form of the equation of a line, we can write an equation of the tangent line to the curve y − f sxd at the point sa, f sadd:
y
y=≈8x+9
y 2 f sad − f 9sadsx 2 ad
Example 5 Find an equation of the tangent line to the parabola y − x 2 2 8x 1 9 at
the point s3, 26d.
x
0
SOLUTION From Example 4 we know that the derivative of f sxd − x 2 2 8x 1 9 at
(3, _6)
the number a is f 9sad − 2a 2 8. Therefore the slope of the tangent line at s3, 26d is f 9s3d − 2s3d 2 8 − 22. Thus an equation of the tangent line, shown in Figure 7, is
y=_2x
y 2 s26d − s22dsx 2 3d or y − 22x
FIGURE 7
n
Rates of Change
y
Q { ¤, ‡} P {⁄, fl}
and the corresponding change in y is
Îy
Dy − f sx 2d 2 f sx 1d
Îx 0
⁄
The difference quotient ¤
average rate rateof ofchange change−mPQ mPQ average
Dy f sx 2d 2 f sx 1d − Dx x2 2 x1
x
instantaneous rateofofchange change− instantaneous rate slope slopeofoftangent tangentatatPP
FIGURE 8
Suppose y is a quantity that depends on another quantity x. Thus y is a function of x and we write y − f sxd. If x changes from x 1 to x 2, then the change in x (also called the increment of x) is Dx − x 2 2 x 1
is called the average rate of change of y with respect to x over the interval fx 1, x 2g and can be interpreted as the slope of the secant line PQ in Figure 8. By analogy with velocity, we consider the average rate of change over smaller and smaller intervals by letting x 2 approach x 1 and therefore letting Dx approach 0. The limit of these average rates of change is called the (instantaneous) rate of change of y with respect to x at x − x1, which (as in the case of velocity) is interpreted as the slope of the tangent to the curve y − f sxd at Psx 1, f sx 1dd:
6
instantaneous rate of change − lim
Dx l 0
Dy f sx2 d 2 f sx1d − lim x lx Dx x2 2 x1 2
1
We recognize this limit as being the derivative f 9sx 1d. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.1 Derivatives and Rates of Change
111
We know that one interpretation of the derivative f 9sad is as the slope of the tangent line to the curve y − f sxd when x − a. We now have a second interpretation: y
The derivative f 9sad is the instantaneous rate of change of y − f sxd with respect to x when x − a.
Q
P
x
FIGURE 9 The yvalues are changing rapidly at P and slowly at Q.
The connection with the first interpretation is that if we sketch the curve y − f sxd, then the instantaneous rate of change is the slope of the tangent to this curve at the point where x − a. This means that when the derivative is large (and therefore the curve is steep, as at the point P in Figure 9), the yvalues change rapidly. When the derivative is small, the curve is relatively flat (as at point Q) and the yvalues change slowly. In particular, if s − f std is the position function of a particle that moves along a straight line, then f 9sad is the rate of change of the displacement s with respect to the time t. In other words, f 9sad is the velocity of the particle at time t − a. The speed of the particle is the absolute value of the velocity, that is, f 9sad . In the next example we discuss the meaning of the derivative of a function that is defined verbally.


Example 6 A manufacturer produces bolts of a fabric with a fixed width. The cost of producing x yards of this fabric is C − f sxd dollars. (a) What is the meaning of the derivative f 9sxd? What are its units? (b) In practical terms, what does it mean to say that f 9s1000d − 9? (c) Which do you think is greater, f 9s50d or f 9s500d? What about f 9s5000d? SOLUTION
(a) The derivative f 9sxd is the instantaneous rate of change of C with respect to x; that is, f 9sxd means the rate of change of the production cost with respect to the number of yards produced. (Economists call this rate of change the marginal cost. This idea is discussed in more detail in Sections 2.7 and 3.7.) Because f 9sxd − lim
Dx l 0
Here we are assuming that the cost function is well behaved; in other words, Csxd doesn’t oscillate rapidly near x − 1000.
DC Dx
the units for f 9sxd are the same as the units for the difference quotient DCyDx. Since DC is measured in dollars and Dx in yards, it follows that the units for f 9sxd are dollars per yard. (b) The statement that f 9s1000d − 9 means that, after 1000 yards of fabric have been manufactured, the rate at which the production cost is increasing is $9yyard. (When x − 1000, C is increasing 9 times as fast as x.) Since Dx − 1 is small compared with x − 1000, we could use the approximation f 9s1000d <
DC DC − − DC Dx 1
and say that the cost of manufacturing the 1000th yard (or the 1001st) is about $9. (c) The rate at which the production cost is increasing (per yard) is probably lower when x − 500 than when x − 50 (the cost of making the 500th yard is less than the cost of the 50th yard) because of economies of scale. (The manufacturer makes more Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Chapter 2 Derivatives
efficient use of the fixed costs of production.) So f 9s50d . f 9s500d But, as production expands, the resulting largescale operation might become inefficient and there might be overtime costs. Thus it is possible that the rate of increase of costs will eventually start to rise. So it may happen that
f 9s5000d . f 9s500d
n
In the following example we estimate the rate of change of the national debt with respect to time. Here the function is defined not by a formula but by a table of values. t
Dstd
1985 1990 1995 2000 2005 2010
1945.9 3364.8 4988.7 5662.2 8170.4 14,025.2
Example 7 Let Dstd be the US national debt at time t. The table in the margin gives approximate values of this function by providing end of year estimates, in billions of dollars, from 1985 to 2010. Interpret and estimate the value of D9s2000d. SOLUTION The derivative D9s2000d means the rate of change of D with respect to t when t − 2000, that is, the rate of increase of the national debt in 2000. According to Equation 5,
Source: US Dept. of the Treasury
D9s2000d − lim
t l 2000
Dstd 2 Ds2000d t 2 2000
So we compute and tabulate values of the difference quotient (the average rates of change) as follows.
A Note on Units The units for the average rate of change D DyDt are the units for D D divided by the units for Dt, namely, billions of dollars per year. The instantaneous rate of change is the limit of the average rates of change, so it is measured in the same units: billions of dollars per year.
t
Time interval
Average rate of change −
1985 1990 1995 2005 2010
[1985, 2000] [1990, 2000] [1995, 2000] [2000, 2005] [2000, 2010]
247.75 229.74 134.70 501.64 836.30
Dstd 2 Ds2000d t 2 2000
From this table we see that D9s2000d lies somewhere between 134.70 and 501.64 billion dollars per year. [Here we are making the reasonable assumption that the debt didn’t fluctuate wildly between 1995 and 2005.] We estimate that the rate of increase of the national debt of the United States in 2000 was the average of these two numbers, namely, D9s2000d < 318 billion dollars per year Another method would be to plot the debt function and estimate the slope of the tangent line when t − 2000.
n
In Examples 3, 6, and 7 we saw three specific examples of rates of change: the velocity of an object is the rate of change of displacement with respect to time; marginal cost is the rate of change of production cost with respect to the number of items produced; the rate of change of the debt with respect to time is of interest in economics. Here is a small sample of other rates of change: In physics, the rate of change of work with respect to time is called power. Chemists who study a chemical reaction are interested in the rate of change in the concentration of a reactant with respect to time (called the rate of reaction). Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.1 Derivatives and Rates of Change
113
A biologist is interested in the rate of change of the population of a colony of bacteria with respect to time. In fact, the computation of rates of change is important in all of the natural sciences, in engineering, and even in the social sciences. Further examples will be given in Section 2.7. All these rates of change are derivatives and can therefore be interpreted as slopes of tangents. This gives added significance to the solution of the tangent problem. Whenever we solve a problem involving tangent lines, we are not just solving a problem in geometry. We are also implicitly solving a great variety of problems involving rates of change in science and engineering.
1. A curve has equation y − f sxd. (a) Write an expression for the slope of the secant line through the points Ps3, f s3dd and Qsx, f sxdd. (b) Write an expression for the slope of the tangent line at P. ; 2. Graph the curve y − sin x in the viewing rectangles f22, 2g by f22, 2g, f21, 1g by f21, 1g, and f20.5, 0.5g by f20.5, 0.5g. What do you notice about the curve as you zoom in toward the origin? 3. (a) Find the slope of the tangent line to the parabola y − 4x 2 x 2 at the point s1, 3d (i) using Definition 1 (ii) using Equation 2 (b) Find an equation of the tangent line in part (a). (c) Graph the parabola and the tangent line. As a check on ; your work, zoom in toward the point s1, 3d until the parabola and the tangent line are indistinguishable. 4. (a) Find the slope of the tangent line to the curve y − x 2 x 3 at the point s1, 0d (i) using Definition 1 (ii) using Equation 2 (b) Find an equation of the tangent line in part (a). (c) Graph the curve and the tangent line in successively ; smaller viewing rectangles centered at s1, 0d until the curve and the line appear to coincide.
;
11. (a) A particle starts by moving to the right along a horizontal line; the graph of its position function is shown in the figure. When is the particle moving to the right? Moving to the left? Standing still? (b) Draw a graph of the velocity function. s (meters) 4 2 0
6. y − x 3 2 3x 1 1, s2, 3d
7. y − sx , s1, 1d
2x 1 1 8. y − , s1, 1d x12
9. (a) Find the slope of the tangent to the curve y − 3 1 4x 2 2 2x 3 at the point where x − a. (b) Find equations of the tangent lines at the points s1, 5d and s2, 3d. (c) Graph the curve and both tangents on a common ; screen. 10. (a) Find the slope of the tangent to the curve y − 1ysx at the point where x − a.
2
4
6 t (seconds)
12. Shown are graphs of the position functions of two runners, A and B, who run a 100meter race and finish in a tie. s (meters)
5–8 Find an equation of the tangent line to the curve at the given point. 5. y − 4x 2 3x 2, s2, 24d
(b) Find equations of the tangent lines at the points s1, 1d and (4, 12 ). (c) Graph the curve and both tangents on a common screen.
80
A
40
0
B 4
8
12
t (seconds)
(a) Describe and compare how the runners run the race. (b) At what time is the distance between the runners the greatest? (c) At what time do they have the same velocity?
13. If a ball is thrown into the air with a velocity of 40 ftys, its height (in feet) after t seconds is given by y − 40t 2 16t 2. Find the velocity when t − 2.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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chapter 2 Derivatives
14. If a rock is thrown upward on the planet Mars with a velocity of 10 mys, its height (in meters) after t seconds is given by H − 10t 2 1.86t 2. (a) Find the velocity of the rock after one second. (b) Find the velocity of the rock when t − a. (c) When will the rock hit the surface? (d) With what velocity will the rock hit the surface?
19. For the function f graphed in Exercise 18: (a) Estimate the value of f 9s50d. (b) Is f 9s10d . f 9s30d? f s80d 2 f s40d (c) Is f 9s60d . ? Explain. 80 2 40
15. The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s − 1yt 2, where t is measured in seconds. Find the velocity of the particle at times t − a, t − 1, t − 2, and t − 3.
21. If an equation of the tangent line to the curve y − f sxd at the point where a − 2 is y − 4x 2 5, find f s2d and f 9s2d. 22. If the tangent line to y − f sxd at (4, 3) passes through the point (0, 2), find f s4d and f 9s4d.
16. The displacement (in feet) of a particle moving in a straight line is given by s − 12 t 2 2 6t 1 23, where t is measured in seconds. (a) Find the average velocity over each time interval: (i) f4, 8g (ii) f6, 8g (iii) f8, 10g (iv) f8, 12g (b) Find the instantaneous velocity when t − 8. (c) Draw the graph of s as a function of t and draw the secant lines whose slopes are the average velocities in part (a). Then draw the tangent line whose slope is the instantaneous velocity in part (b).
25. Sketch the graph of a function t that is continuous on its domain s25, 5d and where ts0d − 1, t9s0d − 1, t9s22d − 0, lim x l 251 tsxd − `, and lim x l52 tsxd − 3.
17. For the function t whose graph is given, arrange the following numbers in increasing order and explain your reasoning:
26. Sketch the graph of a function f where the domain is s22, 2d, f 9s0d − 22, lim xl2 f sxd − `, f is continuous at all numbers in its domain except 61, and f is odd.
t9s22d
0
t9s0d
t9s2d
y
_1
1
2
3
4
x
f s40d 2 f s10d . What does this value repre40 2 10 sent geometrically?
(d) Compute
y 800
2
29. (a) If Fsxd − 5xys1 1 x 2 d, find F9s2d and use it to find an equation of the tangent line to the curve y − 5xys1 1 x 2 d at the point s2, 2d. (b) Illustrate part (a) by graphing the curve and the tangent ; line on the same screen. 30. (a) If Gsxd − 4x 2 2 x 3, find G9sad and use it to find equations of the tangent lines to the curve y − 4x 2 2 x 3 at the points s2, 8d and s3, 9d. (b) Illustrate part (a) by graphing the curve and the tangent ; lines on the same screen. 31–36 Find f 9sad. f std − 2t 3 1 t 31. f sxd − 3x 2 2 4x 1 1 32. 2t 1 1 34. f sxd − x 22 t13 4 35. f sxd − s1 2 2x 36. f sxd − s1 2 x 33. f std −
37–42 Each limit represents the derivative of some function f at some number a. State such an f and a in each case.
400
0
24. Sketch the graph of a function t for which ts0d − ts2d − ts4d − 0, t9s1d − t9s3d − 0, t9s0d − t9s4d − 1, t9s2d − 21, lim x l52 tsxd − `, and lim x l 211 tsxd − 2`.
28. If tsxd − x 4 2 2, find t9s1d and use it to find an equation of the tangent line to the curve y − x 4 2 2 at the point s1, 21d.
y=©
0
23. Sketch the graph of a function f for which f s0d − 0, f 9s0d − 3, f 9s1d − 0, and f 9s2d − 21.
27. If f sxd − 3x 2 2 x 3, find f 9s1d and use it to find an equation of the tangent line to the curve y − 3x 2 2 x 3 at the point s1, 2d.
t9s4d
18. The graph of a function f is shown. (a) Find the average rate of change of f on the interval f20, 60g. (b) Identify an interval on which the average rate of change of f is 0. (c) Which interval gives a larger average rate of change, f40, 60g or f40, 70g?
20. Find an equation of the tangent line to the graph of y − tsxd at x − 5 if ts5d − 23 and t9s5d − 4.
20
40
60
x
37. lim
h l0
2 31h 2 8 s9 1 h 2 3 38. lim h l0 h h
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Section 2.1 Derivatives and Rates of Change
1 24 x 6 2 64 x 39. lim 40. lim x l 1y4 x 2 1 x l2 x 2 2 4
48. The number N of locations of a popular coffeehouse chain is given in the table. (The numbers of locations as of October 1 are given.)
1
41. lim
h l0
sin 2 2 coss 1 hd 1 1 42. lim l y6 2 y6 h
43–44 A particle moves along a straight line with equation of motion s − f std, where s is measured in meters and t in seconds. Find the velocity and the speed when t − 4. 43. f std − 80t 2 6t 2 44. f std − 10 1
45 t11
Year
2004
2006
2008
2010
2012
N
8569
12,440
16,680
16,858
18,066
(a) Find the average rate of growth (i) from 2006 to 2008 (ii) from 2008 to 2010 In each case, include the units. What can you conclude? (b) Estimate the instantaneous rate of growth in 2010 by taking the average of two average rates of change. What are its units? (c) Estimate the instantaneous rate of growth in 2010 by measuring the slope of a tangent.
45. A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a function of time. Is the initial rate of change of temperature greater or less than the rate of change after an hour?
46. A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F. The graph shows how the temperature of the turkey decreases and eventually approaches room temperature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.
49. The table shows world average daily oil consumption from 1985 to 2010 measured in thousands of barrels per day. (a) Compute and interpret the average rate of change from 1990 to 2005. What are the units? (b) Estimate the instantaneous rate of change in 2000 by taking the average of two average rates of change. What are its units?
T (°F) 200
Years since 1985
Thousands of barrels of oil per day
0 5 10 15 20 25
60,083 66,533 70,099 76,784 84,077 87,302
P 100
0
30
60
90
120 150
t (min)
47. Researchers measured the average blood alcohol concentration Cstd of eight men starting one hour after consumption of 30 mL of ethanol (corresponding to two alcoholic drinks). t (hours)
1.0
1.5
2.0
2.5
3.0
Cstd sgydLd
0.033
0.024
0.018
0.012
0.007
(a) Find the average rate of change of C with respect to t over each time interval: (i) f1.0, 2.0g (ii) f1.5, 2.0g (iii) f2.0, 2.5g (iv) f2.0, 3.0g In each case, include the units. (b) Estimate the instantaneous rate of change at t − 2 and interpret your result. What are the units?
Source: US Energy Information Administration
50. T he table shows values of the viral load Vstd in HIV patient 303, measured in RNA copiesymL, t days after ABT538 treatment was begun.
Source: Adapted from P. Wilkinson et al., “Pharmacokinetics of Ethanol after Oral Administration in the Fasting State,” Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 207–24.
t
4
8
11
15
22
Vstd
53
18
9.4
5.2
3.6
(a) Find the average rate of change of V with respect to t over each time interval: (i) f4, 11g (ii) f8, 11g (iii) f11, 15g (iv) f11, 22g What are the units? (b) Estimate and interpret the value of the derivative V9s11d.
Source: Adapted from D. Ho et al., “Rapid Turnover of Plasma Virions and CD4 Lymphocytes in Hiv1 Infection,” Nature 373 (1995): 123–26.
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chapter 2 Derivatives
51. The cost (in dollars) of producing x units of a certain commodity is Csxd − 5000 1 10x 1 0.05x 2. (a) Find the average rate of change of C with respect to x when the production level is changed (i) from x − 100 to x − 105 (ii) from x − 100 to x − 101 (b) Find the instantaneous rate of change of C with respect to x when x − 100. (This is called the marginal cost. Its significance will be explained in Section 2.7.)
the oxygen content of water.) The graph shows how oxygen solubility S varies as a function of the water temperature T. (a) What is the meaning of the derivative S9sT d? What are its units? (b) Estimate the value of S9s16d and interpret it. S (mg / L) 16 12
52. If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as
8 4 0
1 2 Vstd − 100,000 (1 2 60 t) 0 < t < 60
ind the rate at which the water is flowing out of the tank F (the instantaneous rate of change of V with respect to t) as a function of t. What are its units? For times t − 0, 10, 20, 30, 40, 50, and 60 min, find the flow rate and the amount of water remaining in the tank. Summarize your findings in a sentence or two. At what time is the flow rate the greatest? The least? 53. The cost of producing x ounces of gold from a new gold mine is C − f sxd dollars. (a) What is the meaning of the derivative f 9sxd? What are its units? (b) What does the statement f 9s800d − 17 mean? (c) Do you think the values of f 9sxd will increase or decrease in the short term? What about the long term? Explain.
8
16
24
32
40
T (°C)
Source: C. Kupchella et al., Environmental Science: Living Within the System of Nature, 2d ed. (Boston: Allyn and Bacon, 1989).
58. The graph shows the influence of the temperature T on the maximum sustainable swimming speed S of Coho salmon. (a) What is the meaning of the derivative S9sT d? What are its units? (b) Estimate the values of S9s15d and S9s25d and interpret them. S (cm/s) 20
54. The number of bacteria after t hours in a controlled laboratory experiment is n − f std. (a) What is the meaning of the derivative f 9s5d? What are its units? (b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, f 9s5d or f 9s10d? If the supply of nutrients is limited, would that affect your conclusion? Explain.
59–60 Determine whether f 9s0d exists.
55. Let H std be the daily cost (in dollars) to heat an office building when the outside temperature is t degrees Fahrenheit. (a) What is the meaning of H9s58d? What are its units? (b) Would you expect H9s58d to be positive or negative? Explain.
60. f sxd −
0
59. f sxd −
H H
x sin
10
1 x
0 x 2 sin 0
20
T (°C)
if x ± 0 if x − 0
1 x
if x ± 0 if x − 0
1 ; 61. (a) Graph the function f sxd − sin x 2 1000 sins1000xd in the viewing rectangle f22, 2g by f24, 4g. What slope does the graph appear to have at the origin? (b) Zoom in to the viewing window f20.4, 0.4g by f20.25, 0.25g and estimate the value of f 9s0d. Does this agree with your answer from part (a)? (c) Now zoom in to the viewing window f20.008, 0.008g by 57. The quantity of oxygen that can dissolve in water depends on f20.005, 0.005g. Do you wish to revise your estimate for the temperature of the water. (So thermal pollution influences f 9s0d?
56. The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is Q − f s pd. (a) What is the meaning of the derivative f 9s8d? What are its units? (b) Is f 9s8d positive or negative? Explain.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.2 The Derivative as a Function
writing Project
117
early methods for finding tangents The first person to formulate explicitly the ideas of limits and derivatives was Sir Isaac Newton in the 1660s. But Newton acknowledged that “If I have seen further than other men, it is because I have stood on the shoulders of giants.” Two of those giants were Pierre Fermat (1601–1665) and Newton’s mentor at Cambridge, Isaac Barrow (1630–1677). Newton was familiar with the methods that these men used to find tangent lines, and their methods played a role in Newton’s eventual formulation of calculus. The following references contain explanations of these methods. Read one or more of the references and write a report comparing the methods of either Fermat or Barrow to modern methods. In particular, use the method of Section 2.1 to find an equation of the tangent line to the curve y − x 3 1 2x at the point (1, 3) and show how either Fermat or Barrow would have solved the same problem. Although you used derivatives and they did not, point out similarities between the methods. 1. Carl Boyer and Uta Merzbach, A History of Mathematics (New York: Wiley, 1989), pp. 389, 432. 2. C. H. Edwards, The Historical Development of the Calculus (New York: SpringerVerlag, 1979), pp. 124, 132. 3. Howard Eves, An Introduction to the History of Mathematics, 6th ed. (New York: Saunders, 1990), pp. 391, 395. 4. Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford University Press, 1972), pp. 344, 346.
In the preceding section we considered the derivative of a function f at a fixed number a:
1
f 9sad − lim
hl0
f sa 1 hd 2 f sad h
Here we change our point of view and let the number a vary. If we replace a in Equation 1 by a variable x, we obtain
2
f 9sxd − lim
hl0
f sx 1 hd 2 f sxd h
Given any number x for which this limit exists, we assign to x the number f 9sxd. So we can regard f 9 as a new function, called the derivative of f and defined by Equation 2. We know that the value of f 9 at x, f 9sxd, can be interpreted geometrically as the slope of the tangent line to the graph of f at the point sx, f sxdd.
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Chapter 2 Derivatives
The function f 9 is called the derivative of f because it has been “derived” from f by the limiting operation in Equation 2. The domain of f 9 is the set hx f 9sxd existsj and may be smaller than the domain of f.

Example 1 The graph of a function f is given in Figure 1. Use it to sketch the graph of the derivative f 9.
y y=ƒ 1 0
x
1
FIGURE 1
SOLUTION We can estimate the value of the derivative at any value of x by drawing the tangent at the point sx, f sxdd and estimating its slope. For instance, for x − 5 we draw the tangent at P in Figure 2(a) and estimate its slope to be about 32, so f 9s5d < 1.5. This allows us to plot the point P9s5, 1.5d on the graph of f 9 directly beneath P. (The slope of the graph of f becomes the yvalue on the graph of f 9.) Repeating this procedure at several points, we get the graph shown in Figure 2(b). Notice that the tangents at A, B, and C are horizontal, so the derivative is 0 there and the graph of f 9 crosses the xaxis (where y − 0) at the points A9, B9, and C9, directly beneath A, B, and C. Between A and B the tangents have positive slope, so f 9sxd is positive there. (The graph is above the xaxis.) But between B and C the tangents have negative slope, so f 9sxd is negative there. y
B
1
m=0
m=0
y=ƒ
A
0
1
P
m=0
3
mÅ2
5
x
C
TEC Visual 2.2 shows an animation of Figure 2 for several functions. (a) y
y=fª(x)
1
0
FIGURE 2
(b)
Bª
Aª 1
Pª (5, 1.5)
Cª 5
x
n
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Section 2.2 The Derivative as a Function
119
Example 2 (a) If f sxd − x 3 2 x, find a formula for f 9sxd. (b) Illustrate this formula by comparing the graphs of f and f 9. 2
SOLUTION
f _2
2
(a) When using Equation 2 to compute a derivative, we must remember that the variable is h and that x is temporarily regarded as a constant during the calculation of the limit. f 9sxd − lim
hl0
_2 2
− lim
x 3 1 3x 2h 1 3xh 2 1 h 3 2 x 2 h 2 x 3 1 x h
− lim
3x 2h 1 3xh 2 1 h 3 2 h h
hl0
fª
hl0
_2
2
f sx 1 hd 2 f sxd fsx 1 hd3 2 sx 1 hdg 2 fx 3 2 xg − lim hl0 h h
− lim s3x 2 1 3xh 1 h 2 2 1d − 3x 2 2 1 hl0
(b) We use a graphing device to graph f and f 9 in Figure 3. Notice that f 9sxd − 0 when f has horizontal tangents and f 9sxd is positive when the tangents have positive slope. So these graphs serve as a check on our work in part (a). n
_2
FIGURE 3
Example 3 If f sxd − sx , find the derivative of f. State the domain of f 9. SOLUTION
f 9sxd − lim
h l0
− lim
h l0
y
− lim
h l0
1 0
1
x
(a) ƒ=œ„ x
sx 1 h 2 sx h
S
D
sx 1 h 2 sx sx 1 h 1 sx ? (Rationalize the numerator.) h sx 1 h 1 sx
− lim
h sx 1 hd 2 x − lim h l 0 h (sx 1 h 1 sx ) h (sx 1 h 1 sx )
− lim
1 1 1 − − 2sx sx 1 h 1 sx sx 1 sx
h l0
h l0
y
f sx 1 hd 2 f sxd h
We see that f 9sxd exists if x . 0, so the domain of f 9 is s0, `d. This is slightly smaller than the domain of f , which is f0, `d. n
1 0
1
1 (b) f ª (x)= x 2œ„
FIGURE 4
x
Let’s check to see that the result of Example 3 is reasonable by looking at the graphs of f and f 9 in Figure 4. When x is close to 0, sx is also close to 0, so f 9sxd − 1y(2 sx ) is very large and this corresponds to the steep tangent lines near s0, 0d in Figure 4(a) and the large values of f 9sxd just to the right of 0 in Figure 4(b). When x is large, f 9sxd is very small and this corresponds to the flatter tangent lines at the far right of the graph of f and the horizontal asymptote of the graph of f 9.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
120
Chapter 2 Derivatives
Example 4 Find f 9 if f sxd −
12x . 21x
SOLUTION
f 9sxd − lim
hl0
1 2 sx 1 hd 12x 2 2 1 sx 1 hd 21x − lim hl0 h
a c 2 ad 2 bc 1 b d − ? e bd e
− lim
s1 2 x 2 hds2 1 xd 2 s1 2 xds2 1 x 1 hd hs2 1 x 1 hds2 1 xd
− lim
s2 2 x 2 2h 2 x 2 2 xhd 2 s2 2 x 1 h 2 x 2 2 xhd hs2 1 x 1 hds2 1 xd
− lim
23h 23 3 − lim −2 h l 0 s2 1 x 1 hds2 1 xd hs2 1 x 1 hds2 1 xd s2 1 xd2
hl0
hl0
Leibniz Gottfried Wilhelm Leibniz was born in Leipzig in 1646 and studied law, theology, philosophy, and mathematics at the university there, graduating with a bachelor’s degree at age 17. After earning his doctorate in law at age 20, Leibniz entered the diplomatic service and spent most of his life traveling to the capitals of Europe on political missions. In particular, he worked to avert a French military threat against Germany and attempted to reconcile the Catholic and Protestant churches. His serious study of mathematics did not begin until 1672 while he was on a diplomatic mission in Paris. There he built a calculating machine and met scientists, like Huygens, who directed his attention to the latest developments in mathematics and science. Leibniz sought to develop a symbolic logic and system of notation that would simplify logical reasoning. In particular, the version of calculus that he published in 1684 established the notation and the rules for finding derivatives that we use today. Unfortunately, a dreadful priority dispute arose in the 1690s between the followers of Newton and those of Leibniz as to who had invented calculus first. Leibniz was even accused of plagiarism by members of the Royal Society in England. The truth is that each man invented calculus independently. Newton arrived at his version of calculus first but, because of his fear of controversy, did not publish it immediately. So Leibniz’s 1684 account of calculus was the first to be published.
f sx 1 hd 2 f sxd h
hl0
n
Other Notations If we use the traditional notation y − f sxd to indicate that the independent variable is x and the dependent variable is y, then some common alternative notations for the derivative are as follows: f 9sxd − y9 −
dy df d − − f sxd − D f sxd − Dx f sxd dx dx dx
The symbols D and dydx are called differentiation operators because they indicate the operation of differentiation, which is the process of calculating a derivative. The symbol dyydx, which was introduced by Leibniz, should not be regarded as a ratio (for the time being); it is simply a synonym for f 9sxd. Nonetheless, it is a very useful and suggestive notation, especially when used in conjunction with increment notation. Referring to Equation 2.1.6, we can rewrite the definition of derivative in Leibniz notation in the form dy Dy − lim Dx l 0 Dx dx If we want to indicate the value of a derivative dyydx in Leibniz notation at a specific number a, we use the notation dy dx
Z
or x−a
dy dx
G
x−a
which is a synonym for f 9sad. The vertical bar means “evaluate at.” 3 Definition A function f is differentiable at a if f 9sad exists. It is differentiable on an open interval sa, bd [or sa, `d or s2`, ad or s2`, `d] if it is differentiable at every number in the interval.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.2 The Derivative as a Function
121
Example 5 Where is the function f sxd −  x  differentiable?
 
SOLUTION If x . 0, then x − x and we can choose h small enough that x 1 h . 0 and hence x 1 h − x 1 h. Therefore, for x . 0, we have


f 9sxd − lim
hl0
− lim
hl0
 x 1 h  2  x  − lim h
hl0
sx 1 hd 2 x h
h − lim 1 − 1 hl0 h
and so f is differentiable for any x . 0. Similarly, for x , 0 we have x − 2x and h can be chosen small enough that x 1 h , 0 and so x 1 h − 2sx 1 hd. Therefore, for x , 0,

 

f 9sxd − lim
hl0
− lim
hl0
 x 1 h  2  x  − lim h
hl0
2sx 1 hd 2 s2xd h
2h − lim s21d − 21 hl0 h
and so f is differentiable for any x , 0. For x − 0 we have to investigate f s0 1 hd 2 f s0d h
f 9s0d − lim
hl0
y
− lim
hl0
 0 1 h  2  0  − lim  h  h
hl0
h
(if it exists)
Let’s compute the left and right limits separately: 0
and
y
h −
h l0
h
h − lim1 1 − 1 h l0 h
lim2
2h − lim2 s21d − 21 h l0 h
h l0
h l0
x _1
(b) y=f ª(x)
FIGURE 5
lim2
h
lim1
Since these limits are different, f 9s0d does not exist. Thus f is differentiable at all x except 0. A formula for f 9 is given by
1 0
h −
h l0
x
(a) y=ƒ= x 
lim1
f 9sxd −
H
1 if x . 0 21 if x , 0
and its graph is shown in Figure 5(b). The fact that f 9s0d does not exist is reflected geometrically in the fact that the curve y − x does not have a tangent line at s0, 0d. [See Figure 5(a).] n
 
Both continuity and differentiability are desirable properties for a function to have. The following theorem shows how these properties are related. 4 Theorem If f is differentiable at a, then f is continuous at a.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
122
Chapter 2 Derivatives
Proof To prove that f is continuous at a, we have to show that lim x l a f sxd − f sad. We do this by showing that the difference f sxd 2 f sad approaches 0. The given information is that f is differentiable at a, that is,
f 9sad − lim
xla
PS An important aspect of problem solving is trying to find a connection between the given and the unknown. See Step 2 (Think of a Plan) in Principles of Problem Solving on page 98.
f sxd 2 f sad x2a
exists (see Equation 2.1.5). To connect the given and the unknown, we divide and multiply f sxd 2 f sad by x 2 a (which we can do when x ± a): f sxd 2 f sad −
f sxd 2 f sad sx 2 ad x2a
Thus, using the Product Law and Equation 2.1.5, we can write lim f f sxd 2 f sadg − lim
xla
xla
− lim
xla
f sxd 2 f sad sx 2 ad x2a f sxd 2 f sad ? lim sx 2 ad xla x2a
− f 9sad ? 0 − 0 To use what we have just proved, we start with f sxd and add and subtract f sad: lim f sxd − lim f f sad 1 s f sxd 2 f saddg
xla
xla
− lim f sad 1 lim f f sxd 2 f sadg xla
xla
− f sad 1 0 − f sad Therefore f is continuous at a.
n
NOTE The converse of Theorem 4 is false; that is, there are functions that are continuous but not differentiable. For instance, the function f sxd − x is continuous at 0 because lim f sxd − lim x − 0 − f s0d
 
xl0
xl0
 
(See Example 1.6.7.) But in Example 5 we showed that f is not differentiable at 0.
How Can a Function Fail To Be Differentiable?
 
We saw that the function y − x in Example 5 is not differentiable at 0 and Figure 5(a) shows that its graph changes direction abruptly when x − 0. In general, if the graph of a function f has a “corner” or “kink” in it, then the graph of f has no tangent at this point and f is not differentiable there. [In trying to compute f 9sad, we find that the left and right limits are different.] Theorem 4 gives another way for a function not to have a derivative. It says that if f is not continuous at a, then f is not differentiable at a. So at any discontinuity (for instance, a jump discontinuity) f fails to be differentiable. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.2 The Derivative as a Function y
123
A third possibility is that the curve has a vertical tangent line when x − a; that is, f is continuous at a and
vertical tangent line


lim f 9sxd − `
xla
This means that the tangent lines become steeper and steeper as x l a. Figure 6 shows one way that this can happen; Figure 7(c) shows another. Figure 7 illustrates the three possibilities that we have discussed. 0
a
x y
y
y
FIGURE 6
0
a
0
x
a
x
0
a
x
FIGURE 7 Three ways for f not to be differentiable at a
(a) A corner
(b) A discontinuity
(c) A vertical tangent
A graphing calculator or computer provides another way of looking at differentiability. If f is differentiable at a, then when we zoom in toward the point sa, f sadd the graph straightens out and appears more and more like a line. (See Figure 8. We saw a specific example of this in Figure 2.1.2.) But no matter how much we zoom in toward a point like the ones in Figures 6 and 7(a), we can’t eliminate the sharp point or corner (see Figure 9). y
0
y
a
0
x
a
FIGURE 8
FIGURE 9
f is differentiable at a.
f is not differentiable at a.
x
Higher Derivatives If f is a differentiable function, then its derivative f 9 is also a function, so f 9 may have a derivative of its own, denoted by s f 9d9 − f 0. This new function f 0 is called the second derivative of f because it is the derivative of the derivative of f. Using Leibniz notation, we write the second derivative of y − f sxd as d dx
S D
derivative of
first derivative
dy dx
−
d 2y dx 2 second derivative
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124
Chapter 2 Derivatives
Example 6 If f sxd − x 3 2 x, find and interpret f 0sxd. SOLUTION In Example 2 we found that the first derivative is f 9sxd − 3x 2 2 1. So the
second derivative is f 99sxd − s f 9d9sxd − lim
h l0
2 f· _1.5
− lim
f3sx 1 hd2 2 1g 2 f3x 2 2 1g h
− lim
3x 2 1 6xh 1 3h 2 2 1 2 3x 2 1 1 h
h l0
fª
f 1.5
f 9sx 1 hd 2 f 9sxd h
h l0
− lim s6x 1 3hd − 6x h l0
_2
FIGURE 10 TEC In Module 2.2 you can see how changing the coefficients of a polynomial f affects the appearance of the graphs of f , f 9, and f 99.
The graphs of f , f 9, and f 0 are shown in Figure 10. We can interpret f 0sxd as the slope of the curve y − f 9sxd at the point sx, f 9sxdd. In other words, it is the rate of change of the slope of the original curve y − f sxd. Notice from Figure 10 that f 0sxd is negative when y − f 9sxd has negative slope and positive when y − f 9sxd has positive slope. So the graphs serve as a check on our calculations. n In general, we can interpret a second derivative as a rate of change of a rate of change. The most familiar example of this is acceleration, which we define as follows. If s − sstd is the position function of an object that moves in a straight line, we know that its first derivative represents the velocity v std of the object as a function of time: v std − s9std −
ds dt
The instantaneous rate of change of velocity with respect to time is called the acceleration astd of the object. Thus the acceleration function is the derivative of the velocity function and is therefore the second derivative of the position function: astd − v9std − s0std or, in Leibniz notation, a−
dv d 2s − 2 dt dt
Acceleration is the change in velocity you would feel when your car is speeding up or slowing down. The third derivative f  is the derivative of the second derivative: f − s f 0d9. So f sxd can be interpreted as the slope of the curve y − f 0sxd or as the rate of change of f 0sxd. If y − f sxd, then alternative notations for the third derivative are y − f sxd −
d dx
S D d2y dx 2
−
d 3y dx 3
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Section 2.2 The Derivative as a Function
125
We can also interpret the third derivative physically in the case where the function is the position function s − sstd of an object that moves along a straight line. Because s− ss0d9 − a9, the third derivative of the position function is the derivative of the acceleration function and is called the jerk: j−
da d 3s − 3 dt dt
Thus the jerk j is the rate of change of acceleration. It is aptly named because a large jerk means a sudden change in acceleration, which causes an abrupt movement in a vehicle. The differentiation process can be continued. The fourth derivative f + is usually denoted by f s4d. In general, the nth derivative of f is denoted by f snd and is obtained from f by differentiating n times. If y − f sxd, we write y snd − f sndsxd −
dny dx n
Example 7 If f sxd − x 3 2 x, find f sxd and f s4dsxd. SOLUTION In Example 6 we found that f 0sxd − 6x. The graph of the second derivative has equation y − 6x and so it is a straight line with slope 6. Since the derivative f sxd is the slope of f 0sxd, we have
f sxd − 6 for all values of x. So f  is a constant function and its graph is a horizontal line. Therefore, for all values of x, f s4d sxd − 0 n We have seen that one application of second and third derivatives occurs in analyzing the motion of objects using acceleration and jerk. We will investigate another application of second derivatives in Section 3.3, where we show how knowledge of f 0 gives us information about the shape of the graph of f. In Chapter 11 we will see how second and higher derivatives enable us to represent functions as sums of infinite series.
1–2 Use the given graph to estimate the value of each derivative. Then sketch the graph of f 9.
2. (a) f 9s0d (b) f 9s1d (c) f 9s2d (d) f 9s3d (e) f 9s4d (f) f 9s5d (g) f 9s6d (h) f 9s7d
1. (a) f 9s23d (b) f 9s22d (c) f 9s21d (d) f 9s0d (e) f 9s1d (f ) f 9s2d (g) f 9s3d
y
y
1 0
1 1
1
x
x
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126
chapter 2 Derivatives
3. Match the graph of each function in (a)–(d) with the graph of its derivative in I–IV. Give reasons for your choices. y
(a)
y
8.
y
(b)
0 0
0
x
y
(c)
x
x
10.
y
x
0
y
x
x
0
y
III
x
0
x
y
II
0
x
y
11.
0
I
0
x
y
(d)
0
y
9.
12. S hown is the graph of the population function Pstd for yeast cells in a laboratory culture. Use the method of Example 1 to graph the derivative P9std. What does the graph of P9 tell us about the yeast population?
y
IV
P (yeast cells) 0
x
0
x
500
4–11 Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f 9 below it. 4.
y
0
5.
y
0
x
0
5
15 t (hours)
10
13. A rechargeable battery is plugged into a charger. The graph shows Cstd, the percentage of full capacity that the battery reaches as a function of time t elapsed (in hours). (a) What is the meaning of the derivative C9std? (b) Sketch the graph of C9std. What does the graph tell you?
x C
6.
100
y
7.
y
80 Percentage of full charge
0
x
0
x
60 40 20 0
2
4
6
8
10 12
t (hours)
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Section 2.2 The Derivative as a Function
14. T he graph (from the US Department of Energy) shows how driving speed affects gas mileage. Fuel economy F is measured in miles per gallon and speed v is measured in miles per hour. (a) What is the meaning of the derivative F9svd? (b) Sketch the graph of F9svd. (c) At what speed should you drive if you want to save on gas? F 30
(mi/gal)
(b) Use symmetry to deduce the values of f 9(221 ), f 9s21d, f 9s22d, and f 9s23d. (c) Use the values from parts (a) and (b) to graph f 9. (d) Guess a formula for f 9sxd. (e) Use the definition of derivative to prove that your guess in part (d) is correct.
19–29 Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. 19. f sxd − 3x 2 8 20. f sxd − mx 1 b
20
21. f std − 2.5t 2 1 6t 22. f sxd − 4 1 8x 2 5x 2
10
1 23. f sxd − x 2 2 2x 3 24. tstd − st
0
127
10
20 30 40 50 60 70
√ (mi/ h)
15. The graph shows how the average age M of first marriage of Japanese men varied in the last half of the 20th century. Sketch the graph of the derivative function M9std. During which years was the derivative negative?
x2 2 1 25. tsxd − s9 2 x 26. f sxd − 2x 2 3 1 2 2t 27. Gstd − 28. f sxd − x 3y2 31t 29. f sxd − x 4 30. (a) Sketch the graph of f sxd − s6 2 x by starting with the graph of y − s x and using the transformations of Section 1.3. (b) Use the graph from part (a) to sketch the graph of f 9. (c) Use the definition of a derivative to find f 9sxd. What are the domains of f and f 9? (d) Use a graphing device to graph f 9 and compare with ; your sketch in part (b).
M
27 25 1960
1970
1980
1990
2000 t
16. Make a careful sketch of the graph of the sine function and below it sketch the graph of its derivative in the same manner as in Example 1. Can you guess what the derivative of the sine function is from its graph? et f sxd − x 2. ; 17. L (a) Estimate the values of f 9s0d, f 9( 12 ), f 9s1d, and f 9s2d by using a graphing device to zoom in on the graph of f. (b) Use symmetry to deduce the values of f 9(221 ), f 9s21d, and f 9s22d. (c) Use the results from parts (a) and (b) to guess a formula for f 9sxd. (d) Use the definition of derivative to prove that your guess in part (c) is correct. et f sxd − x 3. ; 18. L (a) Estimate the values of f 9s0d, f 9( 12 ), f 9s1d, f 9s2d, and f 9s3d by using a graphing device to zoom in on the graph of f.
31. (a) If f sxd − x 4 1 2x, find f 9sxd. (b) Check to see that your answer to part (a) is reasonable ; by comparing the graphs of f and f 9. 32. (a) If f sxd − x 1 1yx, find f 9sxd. (b) Check to see that your answer to part (a) is reasonable ; by comparing the graphs of f and f 9. 33. T he unemployment rate Ustd varies with time. The table gives the percentage of unemployed in the US labor force from 2003 to 2012. (a) What is the meaning of U9std? What are its units? (b) Construct a table of estimated values for U9std. t
Ustd
t
Ustd
2003 2004 2005 2006 2007
6.0 5.5 5.1 4.6 4.6
2008 2009 2010 2011 2012
5.8 9.3 9.6 8.9 8.1
Source: US Bureau of Labor Statistics
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chapter 2 Derivatives
34. T he table gives the number Nstd, measured in thousands, of minimally invasive cosmetic surgery procedures performed in the United States for various years t. t
Nstd (thousands)
2000 2002 2004 2006 2008 2010 2012
5,500 4,897 7,470 9,138 10,897 11,561 13,035
38. Suppose N is the number of people in the United States who travel by car to another state for a vacation this year when the average price of gasoline is p dollars per gallon. Do you expect dNydp to be positive or negative? Explain. 39–42 The graph of f is given. State, with reasons, the numbers at which f is not differentiable. y
39.
40.
0
_2
2
y
_2
x
2
4
2
4 x
x
Source: American Society of Plastic Surgeons
(a) What is the meaning of N9std? What are its units? (b) Construct a table of estimated values for N9std. (c) Graph N and N9. (d) How would it be possible to get more accurate values for N9std?
41.
42.
y
0
2
4
6
x
y
_2
0
35. T he table gives the height as time passes of a typical pine tree grown for lumber at a managed site. Tree age (years)
14
21
28
35
42
49
Height (feet)
41
54
64
72
78
83
Source: Arkansas Forestry Commission
If Hstd is the height of the tree after t years, construct a table of estimated values for H9 and sketch its graph. 36. W ater temperature affects the growth rate of brook trout. The table shows the amount of weight gained by brook trout after 24 days in various water temperatures. Temperature (°C)
15.5
17.7
20.0
22.4
24.4
Weight gained (g)
37.2
31.0
19.8
9.7
29.8
 
raph the function f sxd − x 1 s x . Zoom in repeatedly, ; 43. G first toward the point (21, 0) and then toward the origin. What is different about the behavior of f in the vicinity of these two points? What do you conclude about the differentiability of f ? oom in toward the points (1, 0), (0, 1), and (21, 0) on ; 44. Z the graph of the function tsxd − sx 2 2 1d2y3. What do you notice? Account for what you see in terms of the differentiability of t. 45–46 The graphs of a function f and its derivative f 9 are shown. Which is bigger, f 9s21d or f 99s1d? 45.
If Wsxd is the weight gain at temperature x, construct a table of estimated values for W9 and sketch its graph. What are the units for W9sxd?
y
0
1 x
Source: Adapted from J. Chadwick Jr., “Temperature Effects on Growth and Stress Physiology of Brook Trout: Implications for Climate Change Impacts on an Iconic ColdWater Fish.” Masters Theses. Paper 897. 2012. scholarworks.umass.edu/theses/897.
37. L et P represent the percentage of a city’s electrical power that is produced by solar panels t years after January 1, 2000. (a) What does dPydt represent in this context? (b) Interpret the statement dP dt
Z
46.
y
1
x
− 3.5 t −2
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129
Section 2.2 The Derivative as a Function
47. The figure shows the graphs of f , f 9, and f 0. Identify each curve, and explain your choices. y
a
; 51–52 Use the definition of a derivative to find f 9sxd and f 0sxd. Then graph f , f 9, and f 0 on a common screen and check to see if your answers are reasonable. 51. f sxd − 3x 2 1 2x 1 1 52. f sxd − x 3 2 3x
b
2 3 s4d ; 53. If f sxd − 2x 2 x , find f 9sxd, f 0sxd, f sxd, and f sxd. Graph f , f 9, f 0, and f  on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives?
x
c
48. The figure shows graphs of f, f 9, f 0, and f . Identify each curve, and explain your choices.
54. (a) The graph of a position function of a car is shown, where s is measured in feet and t in seconds. Use it to graph the velocity and acceleration of the car. What is the acceleration at t − 10 seconds?
a b c d
y
s
x
100 0
49. T he figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices. y
t
0
t
(b) Use the acceleration curve from part (a) to estimate the jerk at t − 10 seconds. What are the units for jerk?
55.
3 Let f sxd − s x. (a) If a ± 0, use Equation 2.1.5 to find f 9sad. (b) Show that f 9s0d does not exist. 3 (c) Show that y − s x has a vertical tangent line at s0, 0d. (Recall the shape of the graph of f . See Figure 1.2.13.)
56. ;
c
20
a b
10
(a) If tsxd − x 2y3, show that t9s0d does not exist. (b) If a ± 0, find t9sad. (c) Show that y − x 2y3 has a vertical tangent line at s0, 0d. (d) Illustrate part (c) by graphing y − x 2y3.


57. Show that the function f sxd − x 2 6 is not differentiable at 6. Find a formula for f 9 and sketch its graph. 50. The figure shows the graphs of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve, and explain your choices. y
a
0
8et0208x52 08/29/13
 
59. (a) Sketch the graph of the function f sxd − x x . (b) For what values of x is f differentiable? (c) Find a formula for f 9.
 
60. (a) Sketch the graph of the function tsxd − x 1 x . (b) For what values of x is t differentiable? (c) Find a formula for t9.
d b
58. Where is the greatest integer function f sxd − v x b not differentiable? Find a formula for f 9 and sketch its graph.
c t
61. Recall that a function f is called even if f s2xd − f sxd for all x in its domain and odd if f s2xd − 2f sxd for all such x. Prove each of the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
130
chapter 2 Derivatives
62. The lefthand and righthand derivatives of f at a are defined by f sa 1 hd 2 f sad f 92 sad − lim2 h l0 h f 91 sad − lim1
and
h l0
f sa 1 hd 2 f sad h
if these limits exist. Then f 9sad exists if and only if these onesided derivatives exist and are equal. (a) Find f 92s4d and f 91s4d for the function
f sxd −
0 52x
if x < 0 if 0 , x , 4
1 52x
if x > 4
y=c
c
slope=0
x
0
64. When you turn on a hotwater faucet, the temperature T of the water depends on how long the water has been running. (a) Sketch a possible graph of T as a function of the time t that has elapsed since the faucet was turned on. (b) Describe how the rate of change of T with respect to t varies as t increases. (c) Sketch a graph of the derivative of T. 65. Let be the tangent line to the parabola y − x 2 at the point s1, 1d. The angle of inclination of is the angle that makes with the positive direction of the xaxis. Calculate correct to the nearest degree.
(b) Sketch the graph of f. (c) Where is f discontinuous? (d) Where is f not differentiable?
y
63. Nick starts jogging and runs faster and faster for 3 mintues, then he walks for 5 minutes. He stops at an intersection for 2 minutes, runs fairly quickly for 5 minutes, then walks for 4 minutes. (a) Sketch a possible graph of the distance s Nick has covered after t minutes. (b) Sketch a graph of dsydt.
If it were always necessary to compute derivatives directly from the definition, as we did in the preceding section, such computations would be tedious and the evaluation of some limits would require ingenuity. Fortunately, several rules have been developed for finding derivatives without having to use the definition directly. These formulas greatly simplify the task of differentiation. Let’s start with the simplest of all functions, the constant function f sxd − c. The graph of this function is the horizontal line y − c, which has slope 0, so we must have f 9sxd − 0. (See Figure 1.) A formal proof, from the definition of a derivative, is also easy: f 9sxd − lim
FIGURE 1
hl0
The graph of f sxd − c is the line y − c, so f 9sxd − 0.
f sx 1 hd 2 f sxd c2c − lim − lim 0 − 0 h l 0 hl0 h h
In Leibniz notation, we write this rule as follows. Derivative of a Constant Function d scd − 0 dx
y
Power Functions
y=x
We next look at the functions f sxd − x n, where n is a positive integer. If n − 1, the graph of f sxd − x is the line y − x, which has slope 1. (See Figure 2.) So
slope=1 0
x
FIGURE 2 The graph of f sxd − x is the line y − x, so f 9sxd − 1.
1
d sxd − 1 dx
(You can also verify Equation 1 from the definition of a derivative.) We have already
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Section 2.3 Differentiation Formulas
131
investigated the cases n − 2 and n − 3. In fact, in Section 2.2 (Exercises 17 and 18) we found that d d sx 2 d − 2x sx 3 d − 3x 2 dx dx
2
For n − 4 we find the derivative of f sxd − x 4 as follows: f 9sxd − lim
f sx 1 hd 2 f sxd sx 1 hd4 2 x 4 − lim hl0 h h
− lim
x 4 1 4x 3h 1 6x 2h 2 1 4xh 3 1 h 4 2 x 4 h
− lim
4x 3h 1 6x 2h 2 1 4xh 3 1 h 4 h
hl0
hl0
hl0
− lim s4x 3 1 6x 2h 1 4xh 2 1 h 3 d − 4x 3 hl0
Thus d sx 4 d − 4x 3 dx
3
Comparing the equations in (1), (2), and (3), we see a pattern emerging. It seems to be a reasonable guess that, when n is a positive integer, sdydxdsx n d − nx n21. This turns out to be true. We prove it in two ways; the second proof uses the Binomial Theorem. The Power Rule If n is a positive integer, then d sx n d − nx n21 dx First Proof The formula
x n 2 a n − sx 2 adsx n21 1 x n22a 1 ∙ ∙ ∙ 1 xa n22 1 a n21 d can be verified simply by multiplying out the righthand side (or by summing the sec
ond factor as a geometric series). If f sxd − x n, we can use Equation 2.1.5 for f 9sad and the equation above to write f sxd 2 f sad xn 2 an − lim xla x 2 a x2a
f 9sad − lim
xla
− lim sx n21 1 x n22a 1 ∙ ∙ ∙ 1 xa n22 1 a n21 d xla
− a n21 1 a n22a 1 ∙ ∙ ∙ 1 aa n22 1 a n21 − na n21 second Proof
f 9sxd − lim
hl0
f sx 1 hd 2 f sxd sx 1 hdn 2 x n − lim hl0 h h
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132
Chapter 2 Derivatives
The Binomial Theorem is given on Reference Page 1.
In finding the derivative of x 4 we had to expand sx 1 hd4. Here we need to expand sx 1 hdn and we use the Binomial Theorem to do so:
f 9sxd − lim
F
x n 1 nx n21h 1
G
nsn 2 1d n22 2 x h 1 ∙ ∙ ∙ 1 nxh n21 1 h n 2 x n 2 h
hl0
nsn 2 1d n22 2 x h 1 ∙ ∙ ∙ 1 nxh n21 1 h n 2 h
nx n21h 1 − lim
hl0
− lim
hl0
F
nsn 2 1d n22 x h 1 ∙ ∙ ∙ 1 nxh n22 1 h n21 2
nx n21 1
G
− nx n21 because every term except the first has h as a factor and therefore approaches 0.
■
We illustrate the Power Rule using various notations in Example 1.
Example 1 (a) If f sxd − x 6, then f 9sxd − 6x 5. (c) If y − t 4, then
(b) If y − x 1000, then y9 − 1000x 999.
dy d 3 − 4t 3. (d) sr d − 3r 2 dt dr
■
New Derivatives from Old When new functions are formed from old functions by addition, subtraction, or multiplica tion by a constant, their derivatives can be calculated in terms of derivatives of the old functions. In particular, the following formula says that the derivative of a constant times a function is the constant times the derivative of the function. The Constant Multiple Rule If c is a constant and f is a differentiable function, then d d fcf sxdg − c f sxd dx dx
Geometric Interpretation of the Constant Multiple Rule y
y=2ƒ y=ƒ 0
Proof Let tsxd − cf sxd. Then x
t9sxd − lim
hl0
Multiplying by c − 2 stretches the graph vertically by a factor of 2. All the rises have been doubled but the runs stay the same. So the slopes are doubled too.
tsx 1 hd 2 tsxd cf sx 1 hd 2 cf sxd − lim hl0 h h
F
− lim c
− c lim
− cf 9sxd
hl0
hl0
f sx 1 hd 2 f sxd h
G
f sx 1 hd 2 f sxd (by Limit Law 3) h
■
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Section 2.3 Differentiation Formulas
133
Example 2 (a)
d d s3x 4 d − 3 sx 4 d − 3s4x 3 d − 12x 3 dx dx
(b)
d d d s2xd − fs21dxg − s21d sxd − 21s1d − 21 dx dx dx
■
The next rule tells us that the derivative of a sum of functions is the sum of the derivatives. The Sum Rule If f and t are both differentiable, then
Using prime notation, we can write the Sum Rule as
d d d f f sxd 1 tsxdg − f sxd 1 tsxd dx dx dx
s f 1 td9 − f 9 1 t9
Proof Let Fsxd − f sxd 1 tsxd. Then
Fsx 1 hd 2 Fsxd h
F9sxd − lim
− lim
− lim
− lim
− f 9sxd 1 t9sxd
hl0
hl0
hl0
hl0
f f sx 1 hd 1 tsx 1 hdg 2 f f sxd 1 tsxdg h
F
f sx 1 hd 2 f sxd tsx 1 hd 2 tsxd 1 h h
G
f sx 1 hd 2 f sxd tsx 1 hd 2 tsxd 1 lim (by Limit Law 1) hl 0 h h ■
The Sum Rule can be extended to the sum of any number of functions. For instance, using this theorem twice, we get s f 1 t 1 hd9 − fs f 1 td 1 hg9 − s f 1 td9 1 h9 − f 9 1 t9 1 h9 By writing f 2 t as f 1 s21dt and applying the Sum Rule and the Constant Multiple Rule, we get the following formula. The Difference Rule If f and t are both differentiable, then d d d f f sxd 2 tsxdg − f sxd 2 tsxd dx dx dx
The Constant Multiple Rule, the Sum Rule, and the Difference Rule can be combined with the Power Rule to differentiate any polynomial, as the following examples demonstrate.
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134
Chapter 2 Derivatives
Example 3 d sx 8 1 12x 5 2 4x 4 1 10x 3 2 6x 1 5d dx d d d d d d − sx 8 d 1 12 sx 5 d 2 4 sx 4 d 1 10 sx 3 d 2 6 sxd 1 s5d dx dx dx dx dx dx − 8x 7 1 12s5x 4 d 2 4s4x 3 d 1 10s3x 2 d 2 6s1d 1 0 − 8x 7 1 60x 4 2 16x 3 1 30x 2 2 6 y
■
Example 4 Find the points on the curve y − x 4 2 6x 2 1 4 where the tangent line is horizontal.
(0, 4)
SOLUTION Horizontal tangents occur where the derivative is zero. We have 0
x {œ„ 3, _5}
{_ œ„ 3, _5}
FIGURE 3 4
2
The curve y − x 2 6x 1 4 and its horizontal tangents
dy d d d − sx 4 d 2 6 sx 2 d 1 s4d dx dx dx dx − 4x 3 2 12x 1 0 − 4xsx 2 2 3d Thus dyydx − 0 if x − 0 or x 2 2 3 − 0, that is, x − 6s3 . So the given curve has horizontal tangents when x − 0, s3 , and 2s3 . The corresponding points are s0, 4d, ss3 , 25d, and s2s3 , 25d. (See Figure 3.) ■
Example 5 The equation of motion of a particle is s − 2t 3 2 5t 2 1 3t 1 4, where s is measured in centimeters and t in seconds. Find the acceleration as a function of time. What is the acceleration after 2 seconds? SOLUTION The velocity and acceleration are vstd −
ds − 6t 2 2 10t 1 3 dt
astd −
dv − 12t 2 10 dt
The acceleration after 2 s is as2d − 12s2d 2 10 − 14 cmys2.
■
Next we need a formula for the derivative of a product of two functions. By analogy with the Sum and Difference Rules, one might be tempted to guess, as Leibniz did three centuries ago, that the derivative of a product is the product of the derivatives. We can see, however, that this guess is wrong by looking at a particular example. Let f sxd − x and tsxd − x 2. Then the Power Rule gives f 9sxd − 1 and t9sxd − 2x. But s ftdsxd − x 3, so s ftd9sxd − 3x 2. Thus s ftd9 ± f 9t9. The correct formula was discovered by Leibniz (soon after his false start) and is called the Product Rule.
In prime notation: s ftd9 − ft9 1 t f 9
The Product Rule If f and t are both differentiable, then d d d f f sxdtsxdg − f sxd ftsxdg 1 tsxd f f sxdg dx dx dx
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Section 2.3 Differentiation Formulas
135
Proof Let Fsxd − f sxdtsxd. Then
F9sxd − lim
hl0
− lim
hl0
Fsx 1 hd 2 Fsxd h f sx 1 hdtsx 1 hd 2 f sxdtsxd h
In order to evaluate this limit, we would like to separate the functions f and t as in the proof of the Sum Rule. We can achieve this separation by subtracting and adding the term f sx 1 hd tsxd in the numerator:
F9sxd − lim
hl0
− lim
hl0
f sx 1 hdtsx 1 hd 2 f sx 1 hdtsxd 1 f sx 1 hd tsxd 2 f sxdtsxd h
F
f sx 1 hd
tsx 1 hd 2 tsxd f sx 1 hd 2 f sxd 1 tsxd h h
− lim f sx 1 hd lim hl0
hl0
G
tsx 1 hd 2 tsxd f sx 1 hd 2 f sxd 1 lim tsxd lim hl0 hl 0 h h
− f sxdt9sxd 1 tsxd f 9sxd Note that lim h l 0 tsxd − tsxd because tsxd is a constant with respect to the variable h.
Also, since f is differentiable at x, it is continuous at x by Theorem 2.2.4, and so lim h l 0 f sx 1 hd − f sxd. (See Exercise 1.8.63.)
■
In words, the Product Rule says that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.
Example 6 Find F9sxd if Fsxd − s6x 3 ds7x 4 d. SOLUTION By the Product Rule, we have
F9sxd − s6x 3 d
d d s7x 4 d 1 s7x 4 d s6x 3 d dx dx
− s6x 3 ds28x 3 d 1 s7x 4 ds18x 2 d
− 168x 6 1 126x 6 − 294x 6
■
Notice that we could verify the answer to Example 6 directly by first multiplying the factors: Fsxd − s6x 3 ds7x 4 d − 42x 7 ? F9sxd − 42s7x 6 d − 294x 6 But later we will meet functions, such as y − x 2 sin x, for which the Product Rule is the only possible method.
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136
Chapter 2 Derivatives
Example 7 If hsxd − xtsxd and it is known that ts3d − 5 and t9s3d − 2, find h9s3d. SOLUTION Applying the Product Rule, we get
h9sxd −
d d d fxtsxdg − x ftsxdg 1 tsxd fxg dx dx dx
− x t9sxd 1 tsxd s1d h9s3d − 3t9s3d 1 ts3d − 3 2 1 5 − 11
Therefore
In prime notation:
■
The Quotient Rule If f and t are differentiable, then
SD
f 9 t f 9 2 ft9 − t t2
d dx
F G f sxd tsxd
tsxd −
d d f f sxdg 2 f sxd ftsxdg dx dx ftsxdg 2
proof Let Fsxd − f sxdytsxd. Then
f sx 1 hd f sxd 2 Fsx 1 hd 2 Fsxd tsx 1 hd tsxd F9sxd − lim − lim h l0 hl 0 h h − lim
h l0
f sx 1 hdtsxd 2 f sxdtsx 1 hd htsx 1 hdtsxd
We can separate f and t in this expression by subtracting and adding the term f sxdtsxd in the numerator: F9sxd − lim
hl0
− lim
hl0
f sx 1 hdtsxd 2 f sxdtsxd 1 f sxdtsxd 2 f sxdtsx 1 hd htsx 1 hdtsxd tsxd
f sx 1 hd 2 f sxd tsx 1 hd 2 tsxd 2 f sxd h h tsx 1 hdtsxd
lim tsxd lim
−
hl0
hl0
f sx 1 hd 2 f sxd tsx 1 hd 2 tsxd 2 lim f sxd lim hl0 hl0 h h lim tsx 1 hd lim tsxd hl0
−
hl0
tsxd f 9sxd 2 f sxdt9sxd ftsxdg 2
Again t is continuous by Theorem 2.2.4, so lim h l 0 tsx 1 hd − tsxd.
■
In words, the Quotient Rule says that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.3 Differentiation Formulas
137
The theorems of this section show that any polynomial is differentiable on R and any rational function is differentiable on its domain. Furthermore, the Quotient Rule and the other differentiation formulas enable us to compute the derivative of any rational function, as the next example illustrates. We can use a graphing device to check that the answer to Example 8 is plausible. Figure 4 shows the graphs of the function of Example 8 and its derivative. Notice that when y grows rapidly (near 22), y9 is large. And when y grows slowly, y9 is near 0.
Example 8 Let y −
sx 3 1 6d y9 −
1.5 yª _4 y
FIGURE 4
d d sx 2 1 x 2 2d 2 sx 2 1 x 2 2d sx 3 1 6d dx dx sx 3 1 6d2
−
sx 3 1 6ds2x 1 1d 2 sx 2 1 x 2 2ds3x 2 d sx 3 1 6d2
−
s2x 4 1 x 3 1 12x 1 6d 2 s3x 4 1 3x 3 2 6x 2 d sx 3 1 6d2
−
2x 4 2 2x 3 1 6x 2 1 12x 1 6 sx 3 1 6d2
4
_1.5
x2 1 x 2 2 . Then x3 1 6
■
NOTE Don’t use the Quotient Rule every time you see a quotient. Sometimes it’s easier to first rewrite a quotient to put it in a form that is simpler for the purpose of differentiation. For instance, although it is possible to differentiate the function Fsxd −
3x 2 1 2sx x
using the Quotient Rule, it is much easier to perform the division first and write the function as Fsxd − 3x 1 2x 21y2 before differentiating.
General Power Functions The Quotient Rule can be used to extend the Power Rule to the case where the exponent is a negative integer. If n is a positive integer, then d 2n sx d − 2nx 2n21 dx
proof
d d sx 2n d − dx dx xn −
−
SD 1 xn
d d s1d 2 1 sx n d dx dx x n 0 2 1 nx n21 − n 2 sx d x 2n
2nx n21 − 2nx n2122n − 2nx 2n21 x 2n
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■
138
Chapter 2 Derivatives
Example 9 (a) If y − (b)
d dt
1 dy d 1 , then − sx 21 d − 2x 22 − 2 2 x dx dx x
SD 6 t3
−6
d 23 18 st d − 6s23dt 24 − 2 4 dt t
■
So far we know that the Power Rule holds if the exponent n is a positive or negative integer. If n − 0, then x 0 − 1, which we know has a derivative of 0. Thus the Power Rule holds for any integer n. What if the exponent is a fraction? In Example 2.2.3 we found that d 1 sx − dx 2 sx which can be written as d 1y2 sx d − 12 x21y2 dx This shows that the Power Rule is true even when n − 12. In fact, it also holds for any real number n, as we will prove in Chapter 6. (A proof for rational values of n is indicated in Exercise 2.6.48.) In the meantime we state the general version and use it in the examples and exercises. The Power Rule (General Version) If n is any real number, then d sx n d − nx n21 dx
Example 10 (a) If f sxd − x , then f 9sxd − x 21. (b) Let Then
y−
dy d 22y3 − sx d − 223 x2s2y3d21 dx dx − 223 x25y3
In Example 11, a and b are constants. It is customary in mathematics to use letters near the beginning of the alphabet to represent constants and letters near the end of the alphabet to represent variables.
1 sx 2 3
■
Example 11 Differentiate the function f std − st sa 1 btd. SOLUTION 1 Using the Product Rule, we have
f 9std − st
d d sa 1 btd 1 sa 1 btd (st ) dt dt
− st ? b 1 sa 1 btd ? 12 t 21y2 − bst 1
a 1 bt a 1 3bt − 2 st 2 st
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Section 2.3 Differentiation Formulas
139
SOLUTION 2 If we first use the laws of exponents to rewrite f std, then we can proceed directly without using the Product Rule.
f std − ast 1 btst − at 1y2 1 bt 3y2 f 9std − 12 at21y2 1 32 bt 1y2 which is equivalent to the answer given in Solution 1.
■
The differentiation rules enable us to find tangent lines without having to resort to the definition of a derivative. It also enables us to find normal lines. The normal line to a curve C at a point P is the line through P that is perpendicular to the tangent line at P. (In the study of optics, one needs to consider the angle between a light ray and the normal line to a lens.)
Example 12 Find equations of the tangent line and normal line to the curve y − sx ys1 1 x 2 d at the point s1, 12 d.
SOLUTION According to the Quotient Rule, we have
dy − dx
s1 1 x 2 d
d d ssx d 2 sx dx s1 1 x 2 d dx s1 1 x 2 d2 1
−
2 sx s2xd 2 sx s1 1 x 2 d2
−
s1 1 x 2 d 2 4x 2 1 2 3x 2 2 2 − 2 sx s1 1 x d 2 sx s1 1 x 2 d2
s1 1 x 2 d
So the slope of the tangent line at s1, 12 d is dy dx
y
y 2 12 − 2 41 sx 2 1d or y − 214 x 1 34
tangent
0
FIGURE 5
x−1
1 2 3 ? 12 1 −2 2 s1s1 1 12 d2 4
We use the pointslope form to write an equation of the tangent line at s1, 12 d:
normal 1
Z
−
2
x
The slope of the normal line at s1, 12 d is the negative reciprocal of 241, namely 4, so an equation is y 2 12 − 4sx 2 1d or y − 4x 2 72 The curve and its tangent and normal lines are graphed in Figure 5.
■
Example 13 At what points on the hyperbola xy − 12 is the tangent line parallel to the line 3x 1 y − 0? SOLUTION Since xy − 12 can be written as y − 12yx, we have
dy d 12 − 12 sx 21 d − 12s2x 22 d − 2 2 dx dx x Let the xcoordinate of one of the points in question be a. Then the slope of the tangent Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
140
Chapter 2 Derivatives y (2, 6)
line at that point is 212ya 2. This tangent line will be parallel to the line 3x 1 y − 0, or y − 23x, if it has the same slope, that is, 23. Equating slopes, we get xy=12
0
x
2
12 − 23 or a 2 − 4 or a − 62 a2
Therefore the required points are s2, 6d and s22, 26d. The hyperbola and the tangents are shown in Figure 6. ■
(_2, _6)
We summarize the differentiation formulas we have learned so far as follows.
3x+y=0
Table of Differentiation Formulas
FIGURE 6
d scd − 0 dx
d sx n d − nx n21 dx
scf d9 − cf 9
s f 1 td9 − f 91 t9
s ftd9 − ft9 1 t f 9
s f 2 td9 − f 9 2 t9
SD
f 9 t f 9 2 ft9 − t t2
2.3 Exercises 1–22 Differentiate the function.
24. Find the derivative of the function
f sxd − 1. f sxd − 2 2. 40
2
Fsxd −
3. f sxd − 5.2x 1 2.3 4. tsxd − 74 x 2 2 3x 1 12
x 4 2 5x 3 1 sx x2
7. tsxd − x 2 s1 2 2xd 8. Hsud − s3u 2 1dsu 1 2d
in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer?
9. tstd − 2t 23y4 10. Bs yd − cy26
25–44 Differentiate.
5. f std − 2t 3 2 3t 2 2 4t 6. f std − 1.4t 5 2 2.5t 2 1 6.7
5 12. y − x 5y3 2 x 2y3 r3
11. Fsrd −
25. f sxd − s5x 2 2 2dsx 3 1 3xd 26. Bsud − su 3 1 1ds2u 2 2 4u 2 1d
13. Ss pd − sp 2 p 14. y − sx s2 1 xd 3
15. Rsad − s3a 1 1d2 16. SsRd − 4 R 2 17. y −
x 2 1 4x 1 3 sx
sx 1 x 18. y− x2
s7 19. Gsqd − s1 1 q d 20. Gstd − s5t 1 t 21 2
21. u −
S
1 t
2
1 st
D
2
1 1 16t 2 22. Dstd − s4td 3
23. F ind the derivative of f sxd − s1 1 2x 2 dsx 2 x 2 d in two ways: by using the Product Rule and by performing the multiplication first. Do your answers agree?
27. Fs yd −
S
D
3 1 2 4 s y 1 5y 3 d y2 y
28. Jsvd − sv 3 2 2 vdsv24 1 v22 d 29. tsxd −
1 1 2x 6t 1 1 30. hstd − 3 2 4x 6t 2 1
31. y −
x2 1 1 1 32. y− 3 x3 2 1 t 1 2t 2 2 1
33. y −
t 3 1 3t su 1 2d 2 34. y− t 2 4t 1 3 12u
35. y −
s 2 ss s2
2
36. y −
sx 21x
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.3 Differentiation Formulas
37. f std − 39. Fsxd −
3 t cx s 38. y− 1 1 cx t23
2x 5 1 x 4 2 6x x3
40. Asvd − v 2y3s2 v 2 1 1 2 v 22 d 41. Gs yd − 43. f sxd −
B Ay 3 1 B x c x1 x
42. Fstd −
At Bt 2 1 Ct 3
ax 1 b 44. f sxd − cx 1 d
53. (a) The curve y − 1ys1 1 x 2 d is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point s21, 12 d. (b) Illustrate part (a) by graphing the curve and the tangent ; line on the same screen. 54. (a) The curve y − xys1 1 x 2 d is called a serpentine. Find an equation of the tangent line to this curve at the point s3, 0.3d. (b) Illustrate part (a) by graphing the curve and the tangent ; line on the same screen. 55–58 Find equations of the tangent line and normal line to the curve at the given point. 55. y − x 1 sx , s1, 2d
45. The general polynomial of degree n has the form Psxd − a n x n 1 a n21 x n21 1 ∙ ∙ ∙ 1 a 2 x 2 1 a 1 x 1 a 0
141
57. y −
56. y 2 − x 3, s1, 1d
3x 1 1 , s1, 2d x2 1 1
58. y −
sx , s4, 0.4d x11
where a n ± 0. Find the derivative of P. ; 46–48 Find f 9sxd. Compare the graphs of f and f 9 and use them to explain why your answer is reasonable. 46. f sxd − xysx 2 2 1d 47. f sxd − 3x 15 2 5x 3 1 3 48. f sxd − x 1
1 x
; 49. (a) Graph the function f sxd − x 4 2 3x 3 2 6x 2 1 7x 1 30
in the viewing rectangle f23, 5g by f210, 50g. (b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of f 9. (See Example 2.2.1.) (c) Calculate f 9sxd and use this expression, with a graphing device, to graph f 9. Compare with your sketch in part (b).
2 2 ; 50. (a) Graph the function tsxd − x ysx 1 1d in the viewing rectangle f24, 4g by f21, 1.5g. (b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of t9. (See Example 2.2.1.) (c) Calculate t9sxd and use this expression, with a graphing device, to graph t9. Compare with your sketch in part (b).
51–52 Find an equation of the tangent line to the curve at the given point. 2x 51. y − , s1, 1d x11 52. y − 2x 3 2 x 2 1 2, s1, 3d
59–62 Find the first and second derivatives of the function. 3 60. G srd − sr 1 s r
59. f sxd − 0.001x 5 2 0.02x 3 61. f sxd −
x2 1 62. f sxd − 1 1 2x 32x
63. T he equation of motion of a particle is s − t 3 2 3t, where s is in meters and t is in seconds. Find (a) the velocity and acceleration as functions of t, (b) the acceleration after 2 s, and (c) the acceleration when the velocity is 0. 64. The equation of motion of a particle is s − t 4 2 2t 3 1 t 2 2 t, where s is in meters and t is in seconds. (a) Find the velocity and acceleration as functions of t. (b) Find the acceleration after 1 s. (c) Graph the position, velocity, and acceleration functions ; on the same screen. 65. Biologists have proposed a cubic polynomial to model the length L of Alaskan rockfish at age A: L − 0.0155A 3 2 0.372A 2 1 3.95A 1 1.21 where L is measured in inches and A in years. Calculate dL dA
Z
A −12
and interpret your answer. 66. The number of tree species S in a given area A in the Pasoh Forest Reserve in Malaysia has been modeled by the power function SsAd − 0.882 A 0.842
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
142
chapter 2 Derivatives
where A is measured in square meters. Find S9s100d and interpret your answer. Source: Adapted from K. Kochummen et al., “Floristic Composition of Pasoh Forest Reserve, A Lowland Rain Forest in Peninsular Malaysia,” Journal of Tropical Forest Science 3 (1991):1–13.
67. Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P of the gas is inversely proportional to the volume V of the gas. (a) Suppose that the pressure of a sample of air that occupies 0.106 m 3 at 25°C is 50 kPa. Write V as a function of P. (b) Calculate dVydP when P − 50 kPa. What is the meaning of the derivative? What are its units? ; 68. Car tires need to be inflated properly because overinflation or underinflation can cause premature tread wear. The data in the table show tire life L (in thousands of miles) for a certain type of tire at various pressures P (in lbyin2).
P
26
28
31
35
38
42
45
L
50
66
78
81
74
70
59
(a) Use a calculator to model tire life with a quadratic function of the pressure. (b) Use the model to estimate dLydP when P − 30 and when P − 40. What is the meaning of the derivative? What are the units? What is the significance of the signs of the derivatives?
69. Suppose that f s5d − 1, f 9s5d − 6, ts5d − 23, and t9s5d − 2. Find the following values. (a) s ftd9s5d (b) s fytd9s5d (c) s tyf d9s5d 70. Suppose that f s4d − 2, ts4d − 5, f 9s4d − 6, and t9s4d − 23. Find h9s4d. (a) hsxd − 3 f sxd 1 8tsxd (b) hsxd − f sxdtsxd (c) hsxd −
f sxd tsxd (d) hsxd − tsxd f sxd 1 tsxd
71. If f sxd − sx tsxd, where ts4d − 8 and t9s4d − 7, find f 9s4d. 72. If hs2d − 4 and h9s2d − 23, find d dx
S DZ hsxd x
y
F
G
1 0
1
x
75. If t is a differentiable function, find an expression for the derivative of each of the following functions. x (a) y − xtsxd (b) y− tsxd tsxd (c) y− x 76. I f f is a differentiable function, find an expression for the derivative of each of the following functions. f sxd (a) y − x 2 f sxd (b) y− x2 (c) y−
x2 1 1 x f sxd (d) y− f sxd sx
77. Find the points on the curve y − 2x 3 1 3x 2 2 12x 1 1 where the tangent is horizontal. 78. For what values of x does the graph of f sxd − x 3 1 3x 2 1 x 1 3 have a horizontal tangent? 79. S how that the curve y − 6x 3 1 5x 2 3 has no tangent line with slope 4. 80. Find an equation of the tangent line to the curve y − x 4 1 1 that is parallel to the line 32 x 2 y − 15. 81. Find equations of both lines that are tangent to the curve y − x 3 2 3x 2 1 3x 2 3 and are parallel to the line 3x 2 y − 15. 82. Find equations of the tangent lines to the curve y−
x−2
73. If f and t are the functions whose graphs are shown, let usxd − f sxdtsxd and vsxd − f sxdytsxd. (a) Find u9s1d. (b) Find v9s5d.
x21 x11
that are parallel to the line x 2 2y − 2. 83. Find an equation of the normal line to the curve y − sx that is parallel to the line 2x 1 y − 1. 84. Where does the normal line to the parabola y − x 2 2 1 at the point s21, 0d intersect the parabola a second time? Illustrate with a sketch.
y
f 1 0
74. Let Psxd − FsxdGsxd and Qsxd − FsxdyGsxd, where F and G are the functions whose graphs are shown. (a) Find P9s2d. (b) Find Q9s7d.
1
g x
85. Draw a diagram to show that there are two tangent lines to the parabola y − x 2 that pass through the point s0, 24d. Find the coordinates of the points where these tangent lines intersect the parabola.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
86. (a) Find equations of both lines through the point s2, 23d that are tangent to the parabola y − x 2 1 x. (b) Show that there is no line through the point s2, 7d that is tangent to the parabola. Then draw a diagram to see why. 87. (a) Use the Product Rule twice to prove that if f , t, and h are differentiable, then s fthd9 − f 9th 1 ft9h 1 fth9. (b) Taking f − t − h in part (a), show that d f f sxdg 3 − 3f f sxdg 2 f 9sxd dx
143
Section 2.3 Differentiation Formulas
(c) Use part (b) to differentiate y − sx 4 1 3x 3 1 17x 1 82d3.
88. Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs. (a) f sxd − x n (b) f sxd − 1yx 89. F ind a seconddegree polynomial P such that Ps2d − 5, P9s2d − 3, and P99s2d − 2. 90. The equation y99 1 y9 2 2y − x 2 is called a differential equation because it involves an unknown function y and its derivatives y9 and y99. Find constants A, B, and C such that the function y − Ax 2 1 Bx 1 C satisfies this equation. (Differential equations will be studied in detail in Chapter 9.) 91. Find a cubic function y − ax 3 1 bx 2 1 cx 1 d whose graph has horizontal tangents at the points s22, 6d and s2, 0d. 92. F ind a parabola with equation y − ax 2 1 bx 1 c that has slope 4 at x − 1, slope 28 at x − 21, and passes through the point s2, 15d. 93. I n this exercise we estimate the rate at which the total personal income is rising in the RichmondPetersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,400, and the population was increasing at roughly 9200 people per year. The average annual income was $30,593 per capita, and this average was increasing at about $1400 per year (a little above the national average of about $1225 yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in the RichmondPetersburg area in 1999. Explain the meaning of each term in the Product Rule. 94. A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write q − f s pd. Then the total revenue earned with selling price p is Rs pd − pf s pd. (a) What does it mean to say that f s20d − 10,000 and f 9s20d − 2350? (b) Assuming the values in part (a), find R9s20d and interpret your answer. 95. The MichaelisMenten equation for the enzyme chymotrypsin is 0.14fSg v− 0.015 1 fSg where v is the rate of an enzymatic reaction and [S] is the concentration of a substrate S. Calculate d vyd fSg and interpret it.
96. The biomass Bstd of a fish population is the total mass of the members of the population at time t. It is the product of the number of individuals Nstd in the population and the average mass Mstd of a fish at time t. In the case of guppies, breeding occurs continually. Suppose that at time t − 4 weeks the population is 820 guppies and is growing at a rate of 50 guppies per week, while the average mass is 1.2 g and is increasing at a rate of 0.14 gyweek. At what rate is the biomass increasing when t − 4? 97. Let f sxd −
H
x 2 1 1 if x , 1 x 1 1 if x > 1
Is f differentiable at 1? Sketch the graphs of f and f 9. 98. At what numbers is the following function t differentiable?
H
2x if x < 0 tsxd − 2x 2 x 2 if 0 , x , 2 22x if x > 2 Give a formula for t9 and sketch the graphs of t and t9.

99. (a) For what values of x is the function f sxd − x 2 2 9 differentiable? Find a formula for f 9. (b) Sketch the graphs of f and f 9.

 


100. Where is the function hsxd − x 2 1 1 x 1 2 differenti able? Give a formula for h9 and sketch the graphs of h and h9. 101. For what values of a and b is the line 2x 1 y − b tangent to the parabola y − ax 2 when x − 2? 102. (a) If Fsxd − f sxd tsxd, where f and t have derivatives of all orders, show that F99 − f 99t 1 2 f 9t9 1 f t99. (b) Find similar formulas for F999 and F s4d. (c) Guess a formula for F snd. 103. Find the value of c such that the line y − 32 x 1 6 is tangent to the curve y − csx . 104. Let f sxd −
H
x2 if x < 2 mx 1 b if x . 2
Find the values of m and b that make f differentiable everywhere. 105. A n easy proof of the Quotient Rule can be given if we make the prior assumption that F9sxd exists, where F − fyt. Write f − Ft; then differentiate using the Product Rule and solve the resulting equation for F9. 106. A tangent line is drawn to the hyperbola xy − c at a point P. (a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is P. (b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where P is located on the hyperbola.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
144
Chapter 2 Derivatives
107. Evaluate lim
xl1
109. I f c . 12, how many lines through the point s0, cd are normal lines to the parabola y − x 2 ? What if c < 12 ?
x 1000 2 1 . x21
108. Draw a diagram showing two perpendicular lines that intersect on the yaxis and are both tangent to the parabola y − x 2. Where do these lines intersect?
applied Project
L¡
P
building a better roller coaster Suppose you are asked to design the first ascent and drop for a new roller coaster. By studying photographs of your favorite coasters, you decide to make the slope of the ascent 0.8 and the slope of the drop 21.6. You decide to connect these two straight stretches y − L 1sxd and y − L 2 sxd with part of a parabola y − f sxd − a x 2 1 bx 1 c, where x and f sxd are measured in feet. For the track to be smooth there can’t be abrupt changes in direction, so you want the linear segments L 1 and L 2 to be tangent to the parabola at the transition points P and Q. (See the figure.) To simplify the equations, you decide to place the origin at P.
f
Q
L™
1. (a) Suppose the horizontal distance between P and Q is 100 ft. Write equations in a, b, and c that will ensure that the track is smooth at the transition points. (b) Solve the equations in part (a) for a, b, and c to find a formula for f sxd. ; (c) Plot L 1, f , and L 2 to verify graphically that the transitions are smooth. (d) Find the difference in elevation between P and Q. 2. The solution in Problem 1 might look smooth, but it might not feel smooth because the piecewise defined function [consisting of L 1sxd for x , 0, f sxd for 0 < x < 100, and L 2sxd for x . 100] doesn’t have a continuous second derivative. So you decide to improve the design by using a quadratic function qsxd − ax 2 1 bx 1 c only on the interval 10 < x < 90 and connecting it to the linear functions by means of two cubic functions:
© Susana Ortega / Shutterstock.com
7et0301apun01 01/13/10 MasterID: 00344
110. Sketch the parabolas y − x 2 and y − x 2 2 2x 1 2. Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?
CAS
A review of the trigonometric functions is given in Appendix D.
tsxd − kx 3 1 lx 2 1 mx 1 n
0 < x , 10
hsxd − px 3 1 qx 2 1 rx 1 s
90 , x < 100
(a) Write a system of equations in 11 unknowns that ensure that the functions and their first two derivatives agree at the transition points. (b) Solve the equations in part (a) with a computer algebra system to find formulas for qsxd, tsxd, and hsxd. (c) Plot L 1, t, q, h, and L 2, and compare with the plot in Problem 1(c).
Before starting this section, you might need to review the trigonometric functions. In particular, it is important to remember that when we talk about the function f defined for all real numbers x by f sxd − sin x it is understood that sin x means the sine of the angle whose radian measure is x. A similar convention holds for the other trigonometric functions cos, tan, csc, sec, and cot. Recall from Section 1.8 that all of the trigonometric functions are continuous at every number in their domains.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
145
Section 2.4 Derivatives of Trigonometric Functions
If we sketch the graph of the function f sxd − sin x and use the interpretation of f 9sxd as the slope of the tangent to the sine curve in order to sketch the graph of f 9 (see Exercise 2.2.16), then it looks as if the graph of f 9 may be the same as the cosine curve (see Figure 1). y y=ƒ=sin x 0
TEC Visual 2.4 shows an animation of Figure 1.
π 2
π
2π
x
y y=fª(x )
0
π 2
π
x
FIGURE 1
Let’s try to confirm our guess that if f sxd − sin x, then f 9sxd − cos x. From the definition of a derivative, we have
f 9sxd − lim
hl0
We have used the addition formula for sine. See Appendix D.
− lim
hl0
− lim
hl0
− lim
hl0
1
f sx 1 hd 2 f sxd sinsx 1 hd 2 sin x − lim hl0 h h sin x cos h 1 cos x sin h 2 sin x h
F F S sin x
cos h 2 1 h
− lim sin x ? lim hl0
G S DG
sin x cos h 2 sin x cos x sin h 1 h h
hl0
D
1 cos x
sin h h
cos h 2 1 sin h 1 lim cos x ? lim hl0 hl0 h h
Two of these four limits are easy to evaluate. Since we regard x as a constant when com puting a limit as h l 0, we have lim sin x − sin x and lim cos x − cos x
hl0
hl0
The limit of ssin hdyh is not so obvious. In Example 1.5.3 we made the guess, on the basis of numerical and graphical evidence, that
2
lim
l0
sin −1
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
146
Chapter 2 Derivatives
D B
We now use a geometric argument to prove Equation 2. Assume first that lies between 0 and y2. Figure 2(a) shows a sector of a circle with center O, central angle , and radius 1. BC is drawn perpendicular to OA. By the definition of radian measure, we have arc AB − . Also BC − OB sin − sin . From the diagram we see that
  
1
E
Therefore O
  BC  ,  AB  , arc AB
sin , so
¨ C
A
(a)
Let the tangent lines at A and B intersect at E. You can see from Figure 2(b) that the circumference of a circle is smaller than the length of a circumscribed polygon, and so arc AB , AE 1 EB . Thus
   
B
    ,  AE  1  ED  −  AD  −  OA  tan
− arc AB , AE 1 EB E A
O
sin ,1
− tan (b)
FIGURE 2
(In Appendix F the inequality < tan is proved directly from the definition of the length of an arc without resorting to geometric intuition as we did here.) Therefore we have sin , cos so
cos ,
sin ,1
We know that lim l 0 1 − 1 and lim l 0 cos − 1, so by the Squeeze Theorem, we have lim
l 01
sin −1
But the function ssin dy is an even function, so its right and left limits must be equal. Hence, we have sin lim −1 l0 so we have proved Equation 2. We can deduce the value of the remaining limit in (1) as follows: We multiply numerator and denominator by cos 1 1 in order to put the function in a form in which we can use the limits we know.
lim
l0
cos 2 1 − lim l0 − lim
l0
S
2sin 2 − 2lim l0 scos 1 1d
− 2lim
l0
− 21 ?
D S
cos 2 1 cos 1 1 ? cos 1 1
− lim
l0
cos2 2 1 scos 1 1d
sin sin ? cos 1 1
D
sin sin ? lim l 0 cos 1 1
S D 0 111
− 0 (by Equation 2)
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Section 2.4 Derivatives of Trigonometric Functions
3
147
cos 2 1 −0
lim
l0
If we now put the limits (2) and (3) in (1), we get f 9sxd − lim sin x lim hl0
hl0
cos h 2 1 sin h 1 lim cos x lim hl0 hl0 h h
− ssin xd 0 1 scos xd 1 − cos x So we have proved the formula for the derivative of the sine function: d ssin xd − cos x dx
4
Example 1 Differentiate y − x 2 sin x. Figure 3 shows the graphs of the function of Example 1 and its derivative. Notice that y9 − 0 whenever y has a horizontal tangent.
SOLUTION Using the Product Rule and Formula 4, we have
dy d d − x2 ssin xd 1 sin x sx 2 d dx dx dx
5
− x 2 cos x 1 2x sin x yª
_4
y 4
Using the same methods as in the proof of Formula 4, one can prove (see Exercise 20) that d scos xd − 2sin x dx
5 _5
FIGURE 3
■
The tangent function can also be differentiated by using the definition of a derivative, but it is easier to use the Quotient Rule together with Formulas 4 and 5: d d stan xd − dx dx
S D
cos x −
sin x cos x
d d ssin xd 2 sin x scos xd dx dx cos2x
−
cos x cos x 2 sin x s2sin xd cos2x
−
cos2x 1 sin2x cos2x
−
1 − sec2x cos2x
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148
Chapter 2 Derivatives
d stan xd − sec2x dx
6
The derivatives of the remaining trigonometric functions, csc, sec, and cot, can also be found easily using the Quotient Rule (see Exercises 17–19). We collect all the differentiation formulas for trigonometric functions in the following table. Remember that they are valid only when x is measured in radians. Derivatives of Trigonometric Functions When you memorize this table, it is helpful to notice that the minus signs go with the derivatives of the “cofunctions,” that is, cosine, cosecant, and cotangent.
d ssin xd − cos x dx
d scsc xd − 2csc x cot x dx
d scos xd − 2sin x dx
d ssec xd − sec x tan x dx
d stan xd − sec2x dx
d scot xd − 2csc 2x dx
Example 2 Differentiate f sxd − of f have a horizontal tangent?
sec x . For what values of x does the graph 1 1 tan x
SOLUTION The Quotient Rule gives
s1 1 tan xd f 9sxd −
−
s1 1 tan xd sec x tan x 2 sec x sec2x s1 1 tan xd2
−
sec x stan x 1 tan2x 2 sec2xd s1 1 tan xd2
−
sec x stan x 2 1d s1 1 tan xd2
3
_3
5
_3
FIGURE 4 The horizontal tangents in Example 2
0 4 s
FIGURE 5
d d ssec xd 2 sec x s1 1 tan xd dx dx s1 1 tan xd2
In simplifying the answer we have used the identity tan2x 1 1 − sec2x. Since sec x is never 0, we see that f 9sxd − 0 when tan x − 1, and this occurs when x − n 1 y4, where n is an integer (see Figure 4). ■ Trigonometric functions are often used in modeling realworld phenomena. In particular, vibrations, waves, elastic motions, and other quantities that vary in a periodic manner can be described using trigonometric functions. In the following example we discuss an instance of simple harmonic motion.
Example 3 An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t − 0. (See Figure 5 and note that the downward direction is positive.) Its position at time t is s − f std − 4 cos t
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.4 Derivatives of Trigonometric Functions
149
Find the velocity and acceleration at time t and use them to analyze the motion of the object. SOLUTION The velocity and acceleration are
s
v−
ds d d − s4 cos td − 4 scos td − 24 sin t dt dt dt
a−
dv d d − s24 sin td − 24 ssin td − 24 cos t dt dt dt
√
a
2 0
π
_2
FIGURE 6
2π t
The object oscillates from the lowest point ss − 4 cmd to the highest point ss − 24 cmd. The period of the oscillation is 2, the period of cos t. The speed is v − 4 sin t , which is greatest when sin t − 1, that is, when cos t − 0. So the object moves fastest as it passes through its equilibrium position ss − 0d. Its speed is 0 when sin t − 0, that is, at the high and low points. The acceleration a − 24 cos t − 0 when s − 0. It has greatest magnitude at the high and low points. See the graphs in Figure 6.
 




■
Example 4 Find the 27th derivative of cos x. SOLUTION The first few derivatives of f sxd − cos x are as follows:
f 9sxd − 2sin x
PS Look for a pattern.
f 99sxd − 2cos x f999sxd − sin x f s4dsxd − cos x f s5dsxd − 2sin x We see that the successive derivatives occur in a cycle of length 4 and, in particular, f sndsxd − cos x whenever n is a multiple of 4. Therefore f s24dsxd − cos x and, differentiating three more times, we have f s27dsxd − sin x
■
Our main use for the limit in Equation 2 has been to prove the differentiation formula for the sine function. But this limit is also useful in finding certain other trigonometric limits, as the following two examples show.
Example 5 Find lim
xl0
sin 7x . 4x
SOLUTION In order to apply Equation 2, we first rewrite the function by multiplying and dividing by 7: Note that sin 7x ± 7 sin x.
sin 7x 7 − 4x 4
S D sin 7x 7x
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150
Chapter 2 Derivatives
If we let − 7x, then l 0 as x l 0, so by Equation 2 we have lim
xl0
S D
sin 7x 7 sin 7x − lim x l 0 4x 4 7x −
7 sin 7 7 lim − 1− 4 l0 4 4
■
Example 6 Calculate lim x cot x. xl0
SOLUTION Here we divide numerator and denominator by x:
lim x cot x − lim
xl0
lim cos x cos x xl0 − x l 0 sin x sin x lim xl0 x x cos 0 − (by the continuity of cosine and Equation 2) 1 − 1
− lim
1–16 Differentiate. 1. f sxd − x sin x 2. f sxd − x cos x 1 2 tan x 3. f sxd − 3 cot x 2 2 cos x 4. y − 2 sec x 2 csc x 5. y − sec tan 6. tstd − 4 sec t 1 tan t 7. y − c cos t 1 t 2 sin t
13. y −
x 10. y − sin cos 2 2 tan x sin cos x 12. y− 1 1 cos 1 2 sin x
t sin t sin t 14. y− 11t 1 1 tan t
15. f sd − cos sin 16. y − x sin x tan x 2
17. Prove that
d scsc xd − 2csc x cot x. dx
18. Prove that
d ssec xd − sec x tan x. dx
19. Prove that
d scot xd − 2csc 2x. dx
21–24 Find an equation of the tangent line to the curve at the given point. 21. y − sin x 1 cos x, s0, 1d 22. y − s1 1 xd cos x, s0, 1d 23. y − cos x 2 sin x, s, 21d
8. y − usa cos u 1 b cot ud
11. f sd −
■
20. P rove, using the definition of derivative, that if f sxd − cos x, then f 9sxd − 2sin x.
2
9. y −
xl0
x cos x sin x
24. y − x 1 tan x, s, d 25. (a) Find an equation of the tangent line to the curve y − 2x sin x at the point sy2, d. (b) Illustrate part (a) by graphing the curve and the tangent ; line on the same screen. 26. (a) Find an equation of the tangent line to the curve y − 3x 1 6 cos x at the point sy3, 1 3d. (b) Illustrate part (a) by graphing the curve and the tangent ; line on the same screen. 27. (a) If f sxd − sec x 2 x, find f 9sxd. (b) Check to see that your answer to part (a) is reasonable ; by graphing both f and f 9 for x , y2.
 
28. (a) If f sxd − sx sin x, find f 9sxd. (b) Check to see that your answer to part (a) is reasonable ; by graphing both f and f 9 for 0 < x < 2.
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Section 2.4 Derivatives of Trigonometric Functions
29. If Hsd − sin , find H9sd and H99sd. 30. If f std − sec t, find f 0sy4d. 31. (a) Use the Quotient Rule to differentiate the function tan x 2 1 f sxd − sec x
(b) Simplify the expression for f sxd by writing it in terms of sin x and cos x, and then find f 9sxd. (c) Show that your answers to parts (a) and (b) are equivalent.
32. Suppose f sy3d − 4 and f 9sy3d − 22, and let tsxd − f sxd sin x and hsxd − scos xdyf sxd. Find (a) t9sy3d (b) h9sy3d 33. For what values of x does the graph of f sxd − x 1 2 sin x have a horizontal tangent? 34. Find the points on the curve y − scos xdys2 1 sin xd at which the tangent is horizontal. 35. A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is xstd − 8 sin t, where t is in seconds and x in centimeters. (a) Find the velocity and acceleration at time t. (b) Find the position, velocity, and acceleration of the mass at time t − 2y3. In what direction is it moving at that time?
38. A n object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is F−
39–50 Find the limit. 39. lim
sin 5x sin x 40. lim xl0 sin x 3x
41. lim
tan 6t cos 2 1 42. lim l 0 sin 2t sin
43. lim
sin 3x sin 3x sin 5x 44. lim xl0 5x 3 2 4x x2
45. lim
sin 46. lim csc x sinssin xd xl0 1 tan
47. lim
cos 2 1 sinsx 2 d 48. lim 2 xl0 2 x
xl0
tl0
xl0
l0
49. lim
x l y4
0
x
x
; 36. An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s − 2 cos t 1 3 sin t, t > 0, where s is measured in centimeters and t in seconds. (Take the positive direction to be downward.) (a) Find the velocity and acceleration at time t. (b) Graph the velocity and acceleration functions. (c) When does the mass pass through the equilibrium position for the first time? (d) How far from its equilibrium position does the mass travel? (e) When is the speed the greatest? 37. A ladder 10 ft long rests against a vertical wall. Let be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to when − y3?
W sin 1 cos
where is a constant called the coefficient of friction. (a) Find the rate of change of F with respect to . (b) When is this rate of change equal to 0? (c) If W − 50 lb and − 0.6, draw the graph of F as ; a function of and use it to locate the value of for which dFyd − 0. Is the value consistent with your answer to part (b)?
l0
equilibrium position
151
1 2 tan x sinsx 2 1d 50. lim 2 x l 1 sin x 2 cos x x 1x22
51–52 Find the given derivative by finding the first few derivatives and observing the pattern that occurs. 51.
d 99 d 35 sx sin xd 99 ssin xd 52. dx dx 35
53. Find constants A and B such that the function y − A sin x 1 B cos x satisfies the differential equation y99 1 y9 2 2y − sin x. x sin ; 54. Evaluate xlim l0
1 and illustrate by graphing x
y − x sins1yxd. 55. Differentiate each trigonometric identity to obtain a new (or familiar) identity. sin x 1 (b) sec x − cos x cos x 1 1 cot x (c) sin x 1 cos x − csc x
(a) tan x −
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152
chapter 2 Derivatives
56. A semicircle with diameter PQ sits on an isosceles triangle PQR to form a region shaped like a twodimensional icecream cone, as shown in the figure. If Asd is the area of the semicircle and Bsd is the area of the triangle, find lim
l 01
57. The figure shows a circular arc of length s and a chord of length d, both subtended by a central angle . Find s lim l 01 d
Asd Bsd
d
s
¨
A(¨ ) P
B(¨)
10 cm
Q
; 58. Let f sxd − 10 cm
¨ R
x
. s1 2 cos 2x (a) Graph f . What type of discontinuity does it appear to have at 0? (b) Calculate the left and right limits of f at 0. Do these values confirm your answer to part (a)?
Suppose you are asked to differentiate the function Fsxd − sx 2 1 1
See Section 1.3 for a review of composite functions.
The differentiation formulas you learned in the previous sections of this chapter do not enable you to calculate F9sxd. Observe that F is a composite function. In fact, if we let y − f sud − su and let u − tsxd − x 2 1 1, then we can write y − Fsxd − f stsxdd, that is, F − f 8 t. We know how to differentiate both f and t, so it would be useful to have a rule that tells us how to find the derivative of F − f 8 t in terms of the derivatives of f and t. It turns out that the derivative of the composite function f 8 t is the product of the derivatives of f and t. This fact is one of the most important of the differentiation rules and is called the Chain Rule. It seems plausible if we interpret derivatives as rates of change. Regard duydx as the rate of change of u with respect to x, dyydu as the rate of change of y with respect to u, and dyydx as the rate of change of y with respect to x. If u changes twice as fast as x and y changes three times as fast as u, then it seems reasonable that y changes six times as fast as x, and so we expect that dy dy du − dx du dx The Chain Rule If t is differentiable at x and f is differentiable at tsxd, then the composite function F − f 8 t defined by Fsxd − f stsxdd is differentiable at x and F9 is given by the product F9sxd − f 9stsxdd ? t9sxd I n Leibniz notation, if y − f sud and u − tsxd are both differentiable functions, then dy dy du − dx du dx
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Section 2.5 The Chain Rule
James Gregory The first person to formulate the Chain Rule was the Scottish mathematician James Gregory (1638–1675), who also designed the first practical reflecting telescope. Gregory discovered the basic ideas of calculus at about the same time as Newton. He became the first Professor of Mathematics at the University of St. Andrews and later held the same position at the University of Edinburgh. But one year after accepting that position he died at the age of 36.
153
Comments on the Proof of the Chain Rule Let Du be the change in u correspond
ing to a change of Dx in x, that is, Du − tsx 1 Dxd 2 tsxd Then the corresponding change in y is
Dy − f su 1 Dud 2 f sud It is tempting to write
dy Dy − lim Dxl 0 Dx dx
− lim
Dy Du ? Du Dx
− lim
Dy Du ? lim Dx l 0 Du Dx
− lim
Dy Du ? lim Du Dx l 0 Dx
−
1
Dx l 0
Dx l 0
Du l 0
(Note that Du l 0 as Dx l 0 since t is continuous.)
dy du du dx
The only flaw in this reasoning is that in (1) it might happen that Du − 0 (even when Dx ± 0) and, of course, we can’t divide by 0. Nonetheless, this reasoning does at least suggest that the Chain Rule is true. A full proof of the Chain Rule is given at the end of this section. ■ The Chain Rule can be written either in the prime notation s f 8 td9sxd − f 9stsxdd ? t9sxd
2
or, if y − f sud and u − tsxd, in Leibniz notation: dy dy du − dx du dx
3
Equation 3 is easy to remember because if dyydu and duydx were quotients, then we could cancel du. Remember, however, that du has not been defined and duydx should not be thought of as an actual quotient.
Example 1 Find F9sxd if Fsxd − sx 2 1 1. SOLUTION 1 (using Equation 2): At the beginning of this section we expressed F as Fsxd − s f 8 tdsxd − f stsxdd where f sud − su and tsxd − x 2 1 1. Since
f 9sud − 12 u21y2 − we have
1 2 su
and t9sxd − 2x
F9sxd − f 9stsxdd ? t9sxd −
1 x ? 2x − 2 sx 2 1 1 sx 2 1 1
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154
Chapter 2 Derivatives
SOLUTION 2 (using Equation 3): If we let u − x 2 1 1 and y − su , then
dy du 1 1 x − s2xd − s2xd − 2 1 1 du dx 2 su 2 sx 2 1 1 sx
F9sxd −
■
When using Formula 3 we should bear in mind that dyydx refers to the derivative of y when y is considered as a function of x (called the derivative of y with respect to x), whereas dyydu refers to the derivative of y when considered as a function of u (the derivative of y with respect to u). For instance, in Example 1, y can be considered as a function of x ( y − s x 2 1 1 ) and also as a function of u ( y − su ). Note that dy x dy 1 − F9sxd − whereas − f 9sud − 2 dx du 2 su sx 1 1 NOTE In using the Chain Rule we work from the outside to the inside. Formula 2 says that we differentiate the outer function f [at the inner function tsxd] and then we multiply by the derivative of the inner function. d dx
f
stsxdd
outer function
evaluated at inner function
f9
stsxdd
derivative of outer function
evaluated at inner function
−
t9sxd
?
derivative of inner function
Example 2 Differentiate (a) y − sinsx 2 d and (b) y − sin2x. SOLUTION
7et0304note1
(a) If y − sinsx 2 d, then the outer function is the sine function and the inner function is 01/13/10 the squaring function, so the Chain Rule gives
MasterID: 01592
dy d − dx dx
sin
sx 2 d
outer function
evaluated at inner function
cos
sx 2 d
derivative of outer function
evaluated at inner function
−
2x
?
derivative of inner function
− 2x cossx 2 d (b) Note that sin2x − ssin xd2. Here the outer function is the squaring function and the 7et0304note2 inner function is the sine function. So 01/13/10
01593 dy dMasterID: − ssin xd2 − 2 dx dx inner function
See Reference Page 2 or Appendix D.
?
ssin xd
?
cos x
inner derivative derivative evaluated evaluatedderivative derivative function of outer of outer at inner at inner of inner of inner function function function function function function
The answer can be left as 2 sin x cos x or written as sin 2x (by a trigonometric identity 7et0304note3 known as the doubleangle7et0304note3 formula). ■
01/13/10 01/13/10 MasterID: MasterID: 0159401594 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.5 The Chain Rule
155
In Example 2(a) we combined the Chain Rule with the rule for differentiating the sine function. In general, if y − sin u, where u is a differentiable function of x, then, by the Chain Rule, dy dy du du − − cos u dx du dx dx d du ssin ud − cos u dx dx
Thus
In a similar fashion, all of the formulas for differentiating trigonometric functions can be combined with the Chain Rule. Let’s make explicit the special case of the Chain Rule where the outer function f is a power function. If y − ftsxdg n, then we can write y − f sud − u n where u − tsxd. By using the Chain Rule and then the Power Rule, we get dy dy du du − − nu n21 − nftsxdg n21 t9sxd dx du dx dx
4 The Power Rule Combined with the Chain Rule If n is any real number and u − tsxd is differentiable, then d du su n d − nu n21 dx dx d ftsxdg n − nftsxdg n21 t9sxd dx
Alternatively,
Notice that the derivative in Example 1 could be calculated by taking n − 12 in Rule 4.
Example 3 Differentiate y − sx 3 2 1d100. SOLUTION Taking u − tsxd − x 3 2 1 and n − 100 in (4), we have
dy d d − sx 3 2 1d100 − 100sx 3 2 1d99 sx 3 2 1d dx dx dx − 100sx 3 2 1d99 3x 2 − 300x 2sx 3 2 1d99 ■
Example 4 Find f 9sxd if f sxd − SOLUTION First rewrite f :
Thus
1 . 3 x2 1 x 1 1 s
f sxd − sx 2 1 x 1 1d21y3
f 9sxd − 213 sx 2 1 x 1 1d24y3
d sx 2 1 x 1 1d dx
− 213 sx 2 1 x 1 1d24y3s2x 1 1d
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■
156
Chapter 2 Derivatives
Example 5 Find the derivative of the function tstd −
S D t22 2t 1 1
9
SOLUTION Combining the Power Rule, Chain Rule, and Quotient Rule, we get
S D S D S D
t9std − 9
t22 2t 1 1
8
d dt
t22 2t 1 1
8
−9
s2t 1 1d 1 2 2st 2 2d 45st 2 2d8 − s2t 1 1d2 s2t 1 1d10
t22 2t 1 1
■
Example 6 Differentiate y − s2x 1 1d5sx 3 2 x 1 1d4. SOLUTION In this example we must use the Product Rule before using the Chain Rule: The graphs of the functions y and y9 in Example 6 are shown in Figure 1. Notice that y9 is large when y increases rapidly and y9 − 0 when y has a horizontal tangent. So our answer appears to be reasonable. 10
dy d d − s2x 1 1d5 sx 3 2 x 1 1d4 1 sx 3 2 x 1 1d4 s2x 1 1d5 dx dx dx
yª
_2
1
y _10
FIGURE 1
− s2x 1 1d5 4sx 3 2 x 1 1d3
d sx 3 2 x 1 1d dx
1 sx 3 2 x 1 1d4 5s2x 1 1d4
d s2x 1 1d dx
− 4s2x 1 1d5sx 3 2 x 1 1d3s3x 2 2 1d 1 5sx 3 2 x 1 1d4s2x 1 1d4 2
Noticing that each term has the common factor 2s2x 1 1d4sx 3 2 x 1 1d3, we could factor it out and write the answer as
dy − 2s2x 1 1d4sx 3 2 x 1 1d3s17x 3 1 6x 2 2 9x 1 3d dx
■
The reason for the name “Chain Rule” becomes clear when we make a longer chain by adding another link. Suppose that y − f sud, u − tsxd, and x − hstd, where f , t, and h are differentiable functions. Then, to compute the derivative of y with respect to t, we use the Chain Rule twice: dy dy dx dy du dx − − dt dx dt du dx dt
Example 7 If f sxd − sinscosstan xdd, then f 9sxd − cosscosstan xdd
d cosstan xd dx
− cosscosstan xddf2sinstan xdg
d stan xd dx
− 2cosscosstan xdd sinstan xd sec2x Notice that we used the Chain Rule twice.
■
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Section 2.5 The Chain Rule
157
Example 8 Differentiate y − ssec x 3 . SOLUTION Here the outer function is the square root function, the middle function is the secant function, and the inner function is the cubing function. So we have
dy 1 d − ssec x 3 d dx 2 ssec x 3 dx −
−
1 2 ssec x
sec x 3 tan x 3
3
d sx 3 d dx
3x 2 sec x 3 tan x 3 2 ssec x 3
■
How to Prove the Chain Rule Recall that if y − f sxd and x changes from a to a 1 Dx, we define the increment of y as Dy − f sa 1 Dxd 2 f sad According to the definition of a derivative, we have lim
Dx l 0
Dy − f 9sad Dx
So if we denote by « the difference between the difference quotient and the derivative, we obtain lim « − lim
Dx l 0
But
«−
Dx l 0
S
D
Dy 2 f 9sad − f 9sad 2 f 9sad − 0 Dx
Dy 2 f 9sad Dx
?
Dy − f 9sad Dx 1 « Dx
If we define « to be 0 when Dx − 0, then « becomes a continuous function of Dx. Thus, for a differentiable function f, we can write 5
Dy − f 9sad Dx 1 « Dx
where
« l 0 as Dx l 0
and « is a continuous function of Dx. This property of differentiable functions is what enables us to prove the Chain Rule. Proof of the Chain Rule Suppose u − tsxd is differentiable at a and y − f sud is differentiable at b − tsad. If Dx is an increment in x and Du and Dy are the corresponding increments in u and y, then we can use Equation 5 to write
6
Du − t9sad Dx 1 «1 Dx − ft9sad 1 «1 g Dx
where «1 l 0 as Dx l 0. Similarly 7
Dy − f 9sbd Du 1 «2 Du − f f 9sbd 1 «2 g Du
where «2 l 0 as Du l 0. If we now substitute the expression for Du from Equation 6 into Equation 7, we get Dy − f f 9sbd 1 «2 gft9sad 1 «1 g Dx Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
158
Chapter 2 Derivatives
Dy − f f 9sbd 1 «2 gft9sad 1 «1 g Dx
so
As Dx l 0, Equation 6 shows that Du l 0. So both «1 l 0 and «2 l 0 as Dx l 0. Therefore dy Dy − lim − lim f f 9sbd 1 «2 gft9sad 1 «1 g Dx l 0 Dx Dx l 0 dx
− f 9sbd t9sad − f 9stsadd t9sad This proves the Chain Rule.
1–6 Write the composite function in the form f s tsxdd. [Identify the inner function u − tsxd and the outer function y − f sud.] Then find the derivative dyydx.
31. y − cosssec 4xd 32. Jsd − tan 2 snd 33. y − sins1 1 x 2 34. y − ssins1 1 x 2 d
y − s2x 1 5d 1. y − s1 1 4x 2. 3
3
■
4
3. y − tan x 4. y − sinscot xd 5. y − ssin x 6. y − sin sx 7–46 Find the derivative of the function.
35. y −
S
1 2 cos 2x 1 1 cos 2x
D
4
36. y − x sin
1 x
37. y − cot 2ssin d 38. y − sin ( t 1 cos st )
Fsxd − s1 1 x 1 x 2 d 99 7. Fsxd − s5x 6 1 2x 3 d 4 8.
39. f std − tanssecscos tdd 40. tsud − fsu 2 2 1d 6 2 3ug 4
9. f sxd − s5x 1 1 10. tsxd − s2 2 sin xd 3y2
41. y − sx 1 sx 42. y − sx 1 sx 1 s x
1 1 11. Astd − 12. f sxd − 3 2 scos t 1 tan td 2 sx 2 1
43. tsxd − s2r sin rx 1 nd p 44. y − cos 4ssin3 xd
13. f sd − coss 2 d 14. tsd − cos2
45. y − cos ssinstan xd 46. y − fx 1 sx 1 sin2 xd3 g 4
3 15. hsvd − v s 1 1 v 2 16. f std − t sin t
17. f sxd − s2x 2 3d4 sx 2 1 x 1 1d5
47–50 Find y9 and y 99.
18. tsxd − sx 2 1 1d3 sx 2 1 2d6
1 y− 47. y − cosssin 3d 48. s1 1 tan xd 2
19. hstd − st 1 1d2y3 s2t 2 2 1d3 20. Fstd − s3t 2 1d4 s2t 1 1d23
S Î
21. tsud − 23. y −
u3 2 1 u3 1 1
D
8
S D S D Î
1 22. y− x1 x
x y4 1 1 24. Us yd − x11 y2 1 1
25. hsd − tans 2 sin d 26. f std − 27. y −
5
cos x s1 1 sin x
29. Hsrd −
28. Fstd −
5
t t2 1 4 t
51–54 Find an equation of the tangent line to the curve at the given point. 51. y − s3x 2 1d26, s0, 1d 52. y − s1 1 x 3 , s2, 3d 53. y − sinssin xd, s, 0d 54. y − sin 2 x cos x, sy2, 0d
2
st 1 1 3
Î
4x 49. y − s1 2 sec t 50. y− sx 1 1
sr 2 2 1d 3 1 1 sin t 30. sstd − s2r 1 1d 5 1 1 cos t
55. (a) Find an equation of the tangent line to the curve y − tansx 2y4d at the point s1, 1d. (b) Illustrate part (a) by graphing the curve and the tangent ; line on the same screen.
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Section 2.5 The Chain Rule
 
56. (a) The curve y − x ys2 2 x 2 is called a bulletnose curve. Find an equation of the tangent line to this curve at the point s1, 1d. (b) Illustrate part (a) by graphing the curve and the tangent ; line on the same screen.
66. If f is the function whose graph is shown, let hsxd − f s f sxdd and tsxd − f sx 2 d. Use the graph of f to estimate the value of each derivative. (a) h9s2d (b) t9s2d y
57. (a) If f sxd − x s2 2 x 2 , find f 9sxd. (b) Check to see that your answer to part (a) is reasonable by ; comparing the graphs of f and f 9. ; 58. The function f sxd − sinsx 1 sin 2xd, 0 < x < , arises in applications to frequency modulation (FM) synthesis. (a) Use a graph of f produced by a graphing device to make a rough sketch of the graph of f 9. (b) Calculate f 9sxd and use this expression, with a calculator, to graph f 9. Compare with your sketch in part (a).
159
y=ƒ 1 0
1
x
67. If tsxd − sf sxd , where the graph of f is shown, evaluate t9s3d. y
59. F ind all points on the graph of the function f sxd − 2 sin x 1 sin2x at which the tangent line is horizontal. 60. At what point on the curve y − s1 1 2x is the tangent line perpendicular to the line 6x 1 2y − 1? 61. If Fsxd − f stsxdd, where f s22d − 8, f 9s22d − 4, f 9s5d − 3, ts5d − 22, and t9s5d − 6, find F9s5d. 62. If hsxd − s4 1 3f sxd , where f s1d − 7 and f 9s1d − 4, find h9s1d.
x
f sxd
tsxd
f 9sxd
t9sxd
1 2 3
3 1 7
2 8 2
4 5 7
6 7 9
64. Let f and t be the functions in Exercise 63. (a) If Fsxd − f s f sxdd, find F9s2d. (b) If Gsxd − tstsxdd, find G9s3d. 65. If f and t are the functions whose graphs are shown, let usxd − f s tsxdd, vsxd − ts f sxdd, and w sxd − ts tsxdd. Find each derivative, if it exists. If it does not exist, explain why. (a) u9s1d (b) v9s1d (c) w9s1d y
g 1
x
68. Suppose f is differentiable on R and is a real number. Let Fsxd − f sx d and Gsxd − f f sxdg . Find expressions for (a) F9sxd and (b) G9sxd.
70. If t is a twice differentiable function and f sxd − x tsx 2 d, find f 99 in terms of t, t9, and t99. 71. If Fsxd − f s3f s4 f sxddd, where f s0d − 0 and f 9s0d − 2, find F9s0d.
73–74 Find the given derivative by finding the first few deriva tives and observing the pattern that occurs. 73. D103 cos 2x 74. D 35 x sin x 75. T he displacement of a particle on a vibrating string is given by the equation sstd − 10 1 14 sins10 td where s is measured in centimeters and t in seconds. Find the velocity of the particle after t seconds. 76. I f the equation of motion of a particle is given by s − A cosst 1 d, the particle is said to undergo simple harmonic motion. (a) Find the velocity of the particle at time t. (b) When is the velocity 0?
f
0
1
72. If Fsxd − f sx f sx f sxddd, where f s1d − 2, f s2d − 3, f 9s1d − 4, f 9s2d − 5, and f 9s3d − 6, find F9s1d.
(a) If hsxd − f stsxdd, find h9s1d. (b) If Hsxd − ts f sxdd, find H9s1d.
1
0
69. Let rsxd − f s tshsxddd, where hs1d − 2, ts2d − 3, h9s1d − 4, t9s2d − 5, and f 9s3d − 6. Find r9s1d.
63. A table of values for f , t, f 9, and t9 is given.
f
1
x
77. A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
160
chapter 2 Derivatives
Delta Cephei, for which the interval between times of maximum brightness is 5.4 days. The average brightness of this star is 4.0 and its brightness changes by 60.35. In view of these data, the brightness of Delta Cephei at time t, where t is mea sured in days, has been modeled by the function
S D
Bstd − 4.0 1 0.35 sin
2 t 5.4
(a) Find the rate of change of the brightness after t days. (b) Find, correct to two decimal places, the rate of increase after one day.
83. Use the Chain Rule to prove the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function. 84. U se the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule. [Hint: Write f sxdytsxd − f sxdf tsxdg 21.] 85. (a) If n is a positive integer, prove that
78. I n Example 1.3.4 we arrived at a model for the length of daylight (in hours) in Philadelphia on the tth day of the year:
F
Lstd − 12 1 2.8 sin
2 st 2 80d 365
d ssinn x cos nxd − n sinn21x cossn 1 1dx dx
G
Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 and May 21.
dv ds
87. U se the Chain Rule to show that if is measured in degrees, then
Explain the difference between the meanings of the derivatives dvydt and dvyds.
d ssin d − cos d 180 (This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: the differentiation formulas would not be as simple if we used degree measure.)
80. A ir is being pumped into a spherical weather balloon. At any time t, the volume of the balloon is Vstd and its radius is rstd. (a) What do the derivatives dVydr and dVydt represent? (b) Express dVydt in terms of drydt. CAS
CAS
(b) Find a formula for the derivative of y − cosnx cos nx that is similar to the one in part (a).
86. Suppose y − f sxd is a curve that always lies above the xaxis and never has a horizontal tangent, where f is dif ferentiable everywhere. For what value of y is the rate of change of y 5 with respect to x eighty times the rate of change of y with respect to x?
79. A particle moves along a straight line with displacement sstd, velocity vstd, and acceleration astd. Show that astd − vstd
(b) Where does the graph of f have horizontal tangents? (c) Graph f and f 9 on the same screen. Are the graphs consistent with your answer to part (b)?
81. C omputer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and so further commands may be necessary to simplify the answer. (a) Use a CAS to find the derivative in Example 5 and compare with the answer in that example. Then use the simplify command and compare again. (b) Use a CAS to find the derivative in Example 6. What happens if you use the simplify command? What happens if you use the factor command? Which form of the answer would be best for locating horizontal tangents? 82. (a) Use a CAS to differentiate the function f sxd −
Î
x4 2 x 1 1 x4 1 x 1 1
and to simplify the result.
 
88. (a) Write x − sx 2 and use the Chain Rule to show that d x − dx
 

 

(b) If f sxd − sin x , find f 9sxd and sketch the graphs of f and f 9. Where is f not differentiable? (c) If tsxd − sin x , find t9sxd and sketch the graphs of t and t9. Where is t not differentiable?
x x
 
89. If y − f sud and u − tsxd, where f and t are twice differentiable functions, show that d2y d2y 2 − dx du 2
S D du dx
2
1
dy d 2u du dx 2
90. If y − f sud and u − tsxd, where f and t possess third derivatives, find a formula for d 3 yydx 3 similar to the one given in Exercise 89.
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Section 2.6 Implicit Differentiation
applied Project
where should a pilot start descent?
y
An approach path for an aircraft landing is shown in the figure and satisfies the following conditions: y=P(x)
0
161
h
(i) The cruising altitude is h when descent starts at a horizontal distance , from touchdown at the origin. (ii) The pilot must maintain a constant horizontal speed v throughout descent. (iii) The absolute value of the vertical acceleration should not exceed a constant k (which is much less than the acceleration due to gravity).
1. F ind a cubic polynomial Psxd − ax 3 1 bx 2 1 cx 1 d that satisfies condition (i) by imposing suitable conditions on Psxd and P9sxd at the start of descent and at touchdown.
x
2. Use conditions (ii) and (iii) to show that 6h v 2 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time t and after 1 second. ; (h) Graph the position, velocity, and acceleration functions for 0 < t < 6. (i) When is the particle speeding up? When is it slowing down? f std − 0.01t 4 2 0.04t 3 1. f std − t 3 2 9t 2 1 24t 2. 9t 3. f std − sinsty2d 4. f std − 2 t 19 5. Graphs of the velocity functions of two particles are shown, where t is measured in seconds. When is each particle speeding up? When is it slowing down? Explain. (a) √ √ (b) √ √
0 0
1 1
t t
0 0 1 1
t t
6. Graphs of the position functions of two particles are shown, where t is measured in seconds. When is each particle speeding up? When is it slowing down? Explain. (a) s s (b) s s
0 0 1 1
t t
0 0 1 1
t t
7. The height (in meters) of a projectile shot vertically upward from a point 2 m above ground level with an initial velocity of 24.5 mys is h − 2 1 24.5t 2 4.9t 2 after t seconds. (a) Find the velocity after 2 s and after 4 s. (b) When does the projectile reach its maximum height? (c) What is the maximum height? (d) When does it hit the ground? (e) With what velocity does it hit the ground? 8. If a ball is thrown vertically upward with a velocity of 80 ftys, then its height after t seconds is s − 80t 2 16t 2. (a) What is the maximum height reached by the ball? (b) What is the velocity of the ball when it is 96 ft above the ground on its way up? On its way down?
9. If a rock is thrown vertically upward from the surface of Mars with velocity 15 mys, its height after t seconds is h − 15t 2 1.86t 2. (a) What is the velocity of the rock after 2 s? (b) What is the velocity of the rock when its height is 25 m on its way up? On its way down? 10. A particle moves with position function s − t 4 2 4t 3 2 20t 2 1 20t t > 0
(a) At what time does the particle have a velocity of 20 mys? (b) At what time is the acceleration 0? What is the significance of this value of t?
11. (a) A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area Asxd of a wafer changes when the side length x changes. Find A9s15d and explain its meaning in this situation. (b) Show that the rate of change of the area of a square with respect to its side length is half its perimeter. Try to explain geometrically why this is true by drawing a square whose side length x is increased by an amount Dx. How can you approximate the resulting change in area DA if Dx is small? 12. (a) Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate dVydx when x − 3 mm and explain its meaning. (b) Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube. Explain geometrically why this result is true by arguing by analogy with Exercise 11(b). 13. (a) Find the average rate of change of the area of a circle with respect to its radius r as r changes from (i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1 (b) Find the instantaneous rate of change when r − 2. (c) Show that the rate of change of the area of a circle with respect to its radius (at any r) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount Dr. How can you approximate the resulting change in area DA if Dr is small? 14. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cmys. Find the rate at which the area within the circle is increasing after (a) 1 s, (b) 3 s, and (c) 5 s. What can you conclude? 15. A spherical balloon is being inflated. Find the rate of increase of the surface area sS − 4r 2 d with respect to the radius r when r is (a) 1 ft, (b) 2 ft, and (c) 3 ft. What conclusion can you make?
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Section 2.7 Rates of Change in the Natural and Social Sciences
16. (a) The volume of a growing spherical cell is V − 43 r 3, where the radius r is measured in micrometers (1 μm − 1026 m). Find the average rate of change of V with respect to r when r changes from (i) 5 to 8 μm (ii) 5 to 6 μm (iii) 5 to 5.1 μm (b) Find the instantaneous rate of change of V with respect to r when r − 5 μm. (c) Show that the rate of change of the volume of a sphere with respect to its radius is equal to its surface area. Explain geometrically why this result is true. Argue by analogy with Exercise 13(c). 17. The mass of the part of a metal rod that lies between its left end and a point x meters to the right is 3x 2 kg. Find the linear density (see Example 2) when x is (a) 1 m, (b) 2 m, and (c) 3 m. Where is the density the highest? The lowest? 18. I f a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as 1 td 0 < t < 40 V − 5000 s1 2 40 2
Find the rate at which water is draining from the tank after (a) 5 min, (b) 10 min, (c) 20 min, and (d) 40 min. At what time is the water flowing out the fastest? The slowest? Summarize your findings. 19. The quantity of charge Q in coulombs (C) that has passed through a point in a wire up to time t (measured in seconds) is given by Qstd − t 3 2 2t 2 1 6t 1 2. Find the current when (a) t − 0.5 s and (b) t − 1 s. [See Example 3. The unit of current is an ampere (1 A − 1 Cys).] At what time is the current lowest? 20. N ewton’s Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is F−
21. The force F acting on a body with mass m and velocity v is the rate of change of momentum: F − sdydtdsmvd. If m is constant, this becomes F − ma, where a − dvydt is the acceleration. But in the theory of relativity the mass of a particle varies with v as follows: m − m 0 ys1 2 v 2yc 2 , where m 0 is the mass of the particle at rest and c is the speed of light. Show that F−
22. Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is a little more than 12 hours and on June 30, 2009, high tide occurred at 6:45 am. This helps explain the following model for the water depth D (in meters) as a function of the time t (in hours after midnight) on that day: Dstd − 7 1 5 cosf0.503st 2 6.75dg How fast was the tide rising (or falling) at the following times? (a) 3:00 am (b) 6:00 am (c) 9:00 am (d) Noon 23. Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: PV − C. (a) Find the rate of change of volume with respect to pressure. (b) A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes? Explain. (c) Prove that the isothermal compressibility (see Example 5) is given by − 1yP. 24. I f, in Example 4, one molecule of the product C is formed from one molecule of the reactant A and one molecule of the reactant B, and the initial concentrations of A and B have a common value fAg − fBg − a molesyL, then fCg − a 2ktysakt 1 1d where k is a constant. (a) Find the rate of reaction at time t. (b) Show that if x − fCg, then
GmM r2
where G is the gravitational constant and r is the distance between the bodies. (a) Find dFydr and explain its meaning. What does the minus sign indicate? (b) Suppose it is known that the earth attracts an object with a force that decreases at the rate of 2 Nykm when r − 20,000 km. How fast does this force change when r − 10,000 km?
m0a s1 2 v 2yc 2 d3y2
179
dx − ksa 2 xd2 dt ; 25. The table gives the population of the world Pstd, in millions, where t is measured in years and t − 0 corresponds to the year 1900.
t 0 10 20 30 40 50
Population (millions) 1650 1750 1860 2070 2300 2560
t 60 70 80 90 100 110
Population (millions) 3040 3710 4450 5280 6080 6870
(a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. (b) Use a graphing device to find a cubic function (a thirddegree polynomial) that models the data.
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180
chapter 2 Derivatives
(c) Use your model in part (b) to find a model for the rate of population growth. (d) Use part (c) to estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a). (e) Estimate the rate of growth in 1985.
29. Suppose that the cost (in dollars) for a company to produce x pairs of a new line of jeans is
; 26. The table shows how the average age of first marriage of Japanese women has varied since 1950.
t
Astd
t
Astd
1950 1955 1960 1965 1970 1975 1980
23.0 23.8 24.4 24.5 24.2 24.7 25.2
1985 1990 1995 2000 2005 2010
25.5 25.9 26.3 27.0 28.0 28.8
Csxd − 2000 1 3x 1 0.01x 2 1 0.0002x 3
Csqd − 84 1 0.16q 2 0.0006q 2 1 0.000003q 3
(a) Use a graphing calculator or computer to model these data with a fourthdegree polynomial. (b) Use part (a) to find a model for A9std. (c) Estimate the rate of change of marriage age for women in 1990. (d) Graph the data points and the models for A and A9.
28. T he frequency of vibrations of a vibrating violin string is given by f−
1 2L
Î
T
where L is the length of the string, T is its tension, and is its linear density. [See Chapter 11 in D. E. Hall, Musical Acoustics, 3rd ed. (Pacific Grove, CA: Brooks/Cole, 2002).] (a) Find the rate of change of the frequency with respect to (i) the length (when T and are constant), (ii) the tension (when L and are constant), and (iii) the linear density (when L and T are constant). (b) The pitch of a note (how high or low the note sounds) is determined by the frequency f . (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates,
(a) Find the marginal cost function. (b) Find C9s100d and explain its meaning. What does it predict? (c) Compare C9s100d with the cost of manufacturing the 101st pair of jeans.
30. The cost function for a certain commodity is
27. Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3000 dynesycm2, and viscosity − 0.027. (a) Find the velocity of the blood along the centerline r − 0, at radius r − 0.005 cm, and at the wall r − R − 0.01 cm. (b) Find the velocity gradient at r − 0, r − 0.005, and r − 0.01. (c) Where is the velocity the greatest? Where is the velocity changing most?
(ii) when the tension is increased by turning a tuning peg, (iii) when the linear density is increased by switching to another string.
(a) Find and interpret C9s100d. (b) Compare C9s100d with the cost of producing the 101st item.
31. If psxd is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is psxd Asxd − x
(a) Find A9sxd. Why does the company want to hire more workers if A9sxd . 0? (b) Show that A9sxd . 0 if p9sxd is greater than the average productivity. 32. I f R denotes the reaction of the body to some stimulus of strength x, the sensitivity S is defined to be the rate of change of the reaction with respect to x. A particular example is that when the brightness x of a light source is increased, the eye reacts by decreasing the area R of the pupil. The experimental formula R−
40 1 24x 0.4 1 1 4x 0.4
h as been used to model the dependence of R on x when R is measured in square millimeters and x is measured in appropriate units of brightness. (a) Find the sensitivity. (b) Illustrate part (a) by graphing both R and S as functions ; of x. Comment on the values of R and S at low levels of brightness. Is this what you would expect? 33. T he gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres, atm), and volume V (in liters) is PV − nRT, where n is the number of moles of the gas and R − 0.0821 is the gas constant. Suppose that, at a certain instant, P − 8.0 atm and is increasing at a rate of 0.10 atmymin and V − 10 L and is decreasing at a rate of 0.15 Lymin. Find the rate of change of T with respect to time at that instant if n − 10 moles. 34. Invasive species often display a wave of advance as they colonize new areas. Mathematical models based on ran
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Section 2.8 Related Rates
dom dispersal and reproduction have demonstrated that the speed with which such waves move is given by the function f srd − 2 sDr , where r is the reproductive rate of individuals and D is a parameter quantifying dispersal. Calculate the derivative of the wave speed with respect to the reproductive rate r and explain its meaning. 35. I n the study of ecosystems, predatorprey models are often used to study the interaction between species. Consider populations of tundra wolves, given by Wstd, and caribou, given by Cstd, in northern Canada. The interaction has been modeled by the equations dC dW − aC 2 bCW − 2cW 1 dCW dt dt
(a) What values of dCydt and dWydt correspond to stable populations? (b) How would the statement “The caribou go extinct” be represented mathematically?
181
(c) Suppose that a − 0.05, b − 0.001, c − 0.05, and d − 0.0001. Find all population pairs sC, W d that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct?
36. In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation
S
D
dP Pstd Pstd 2 Pstd − r0 1 2 dt Pc where r0 is the birth rate of the fish, Pc is the maximum population that the pond can sustain (called the carrying capacity), and is the percentage of the population that is harvested. (a) What value of dPydt corresponds to a stable population? (b) If the pond can sustain 10,000 fish, the birth rate is 5%, and the harvesting rate is 4%, find the stable population level. (c) What happens if is raised to 5%?
If we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. But it is much easier to measure directly the rate of increase of the volume than the rate of increase of the radius. In a related rates problem the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity (which may be more easily measured). The procedure is to find an equation that relates the two quantities and then use the Chain Rule to differentiate both sides with respect to time.
Example 1 Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3ys. How fast is the radius of the balloon increasing when the diameter is 50 cm? PS According to the Principles of Problem Solving discussed on page 98, the first step is to understand the problem. This includes reading the problem carefully, identifying the given and the unknown, and introducing suitable notation.
SOLUTION We start by identifying two things:
the given information: the rate of increase of the volume of air is 100 cm3ys
and the unknown: the rate of increase of the radius when the diameter is 50 cm
In order to express these quantities mathematically, we introduce some suggestive notation: Let V be the volume of the balloon and let r be its radius. The key thing to remember is that rates of change are derivatives. In this problem, the volume and the radius are both functions of the time t. The rate of increase of the volume with respect to time is the derivative dVydt, and the rate of increase of the radius Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
182
Chapter 2 Derivatives
is drydt. We can therefore restate the given and the unknown as follows:
PS The second stage of problem solving is to think of a plan for connecting the given and the unknown.
Given:
dV − 100 cm3ys dt
Unknown:
dr dt
when r − 25 cm
In order to connect dVydt and drydt, we first relate V and r by the formula for the volume of a sphere: V − 43 r 3 In order to use the given information, we differentiate each side of this equation with respect to t. To differentiate the right side, we need to use the Chain Rule: dV dV dr dr − − 4r 2 dt dr dt dt Now we solve for the unknown quantity: dr 1 dV − dt 4r 2 dt
Notice that, although dVydt is constant, drydt is not constant.
If we put r − 25 and dVydt − 100 in this equation, we obtain dr 1 1 − 100 − dt 4s25d2 25 wall
The radius of the balloon is increasing at the rate of 1ys25d < 0.0127 cmys.
Example 2 A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ftys, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?
10
y
x
ground
FIGURE 1
dy dt
■
=?
SOLUTION We first draw a diagram and label it as in Figure 1. Let x feet be the distance from the bottom of the ladder to the wall and y feet the distance from the top of the ladder to the ground. Note that x and y are both functions of t (time, measured in seconds). We are given that dxydt − 1 ftys and we are asked to find dyydt when x − 6 ft (see Figure 2). In this problem, the relationship between x and y is given by the Pythagorean Theorem: x 2 1 y 2 − 100
Differentiating each side with respect to t using the Chain Rule, we have y
2x x
and solving this equation for the desired rate, we obtain dx dt
FIGURE 2
dx dy 1 2y −0 dt dt
=1
dy x dx −2 dt y dt
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Section 2.8 Related Rates
183
When x − 6, the Pythagorean Theorem gives y − 8 and so, substituting these values and dxydt − 1, we have dy 6 3 − 2 s1d − 2 ftys dt 8 4 The fact that dyydt is negative means that the distance from the top of the ladder to the ground is decreasing at a rate of 34 ftys. In other words, the top of the ladder is sliding down the wall at a rate of 34 ftys. ■
Example 3 A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m 3ymin, find the rate at which the water level is rising when the water is 3 m deep. 2
r
4
SOLUTION We first sketch the cone and label it as in Figure 3. Let V, r, and h be the volume of the water, the radius of the surface, and the height of the water at time t, where t is measured in minutes. We are given that dVydt − 2 m 3ymin and we are asked to find dhydt when h is 3 m. The quantities V and h are related by the equation
V − 13 r 2h
h
FIGURE 3
but it is very useful to express V as a function of h alone. In order to eliminate r, we use the similar triangles in Figure 3 to write r 2 h − r − h 4 2 and the expression for V becomes V−
SD
1 h 3 2
2
h−
3 h 12
Now we can differentiate each side with respect to t: dV 2 dh − h dt 4 dt so
dh 4 dV − dt h 2 dt
Substituting h − 3 m and dVydt − 2 m 3ymin, we have dh 4 8 − 2 ? 2 − dt s3d 9 The water level is rising at a rate of 8ys9d < 0.28 mymin. PS Look back: What have we learned from Examples 1–3 that will help us solve future problems?
Problem Solving Strategy It is useful to recall some of the problemsolving principles from page 98 and adapt them to related rates in light of our experience in Examples 1–3: 1. 2. 3. 4.
Read the problem carefully. Draw a diagram if possible. Introduce notation. Assign symbols to all quantities that are functions of time. Express the given information and the required rate in terms of derivatives.
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■
184
Chapter 2 Derivatives
WARNING A common error is to substitute the given numerical information (for quantities that vary with time) too early. This should be done only after the differentiation. (Step 7 follows Step 6.) For instance, in Example 3 we dealt with general values of h until we finally substituted h − 3 at the last stage. (If we had put h − 3 earlier, we would have gotten dVydt − 0, which is clearly wrong.)
x
C y
z
B
A
5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution (as in Example 3). 6. Use the Chain Rule to differentiate both sides of the equation with respect to t. 7. Substitute the given information into the resulting equation and solve for the unknown rate. The following examples are further illustrations of the strategy.
Example 4 Car A is traveling west at 50 miyh and car B is traveling north at 60 miyh. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mi from the intersection? SOLUTION We draw Figure 4, where C is the intersection of the roads. At a given time t, let x be the distance from car A to C, let y be the distance from car B to C, and let z be the distance between the cars, where x, y, and z are measured in miles. We are given that dxydt − 250 miyh and dyydt − 260 miyh. (The derivatives are negative because x and y are decreasing as t increases.) We are asked to find dzydt. The equation that relates x, y, and z is given by the Pythagorean Theorem:
z2 − x 2 1 y 2
FIGURE 4
Differentiating each side with respect to t, we have 2z
dz dx dy − 2x 1 2y dt dt dt dz 1 − dt z
S
x
dx dy 1y dt dt
D
When x − 0.3 mi and y − 0.4 mi, the Pythagorean Theorem gives z − 0.5 mi, so dz 1 − f0.3s250d 1 0.4s260dg dt 0.5 − 278 miyh The cars are approaching each other at a rate of 78 miyh.
■
Example 5 A man walks along a straight path at a speed of 4 ftys. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight?
¨
SOLUTION We draw Figure 5 and let x be the distance from the man to the point on the path closest to the searchlight. We let be the angle between the beam of the searchlight and the perpendicular to the path. We are given that dxydt − 4 ftys and are asked to find dydt when x − 15. The equation that relates x and can be written from Figure 5:
FIGURE 5
x − tan x − 20 tan 20
x 20
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.8 Related Rates
185
Differentiating each side with respect to t, we get dx d − 20 sec2 dt dt d 1 dx − cos2 dt 20 dt
so
−
1 1 cos2 s4d − cos2 20 5
When x − 15, the length of the beam is 25, so cos − 45 and d 1 − dt 5
SD 4 5
2
−
16 − 0.128 125
The searchlight is rotating at a rate of 0.128 radys.
1. If V is the volume of a cube with edge length x and the cube expands as time passes, find dVydt in terms of dxydt. 2. (a) If A is the area of a circle with radius r and the circle expands as time passes, find dAydt in terms of drydt. (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 mys, how fast is the area of the spill increasing when the radius is 30 m? 3. Each side of a square is increasing at a rate of 6 cmys. At what rate is the area of the square increasing when the area of the square is 16 cm2? 4. The length of a rectangle is increasing at a rate of 8 cmys and its width is increasing at a rate of 3 cmys. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing? 5. A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3ymin. How fast is the height of the water increasing? 6. The radius of a sphere is increasing at a rate of 4 mmys. How fast is the volume increasing when the diameter is 80 mm? 7. The radius of a spherical ball is increasing at a rate of 2 cmymin. At what rate is the surface area of the ball increasing when the radius is 8 cm? 8. The area of a triangle with sides of lengths a and b and contained angle is A − 12 ab sin
(a) If a − 2 cm, b − 3 cm, and increases at a rate of 0.2 radymin, how fast is the area increasing when − y3?
■
(b) If a − 2 cm, b increases at a rate of 1.5 cmymin, and increases at a rate of 0.2 radymin, how fast is the area increasing when b − 3 cm and − y3? (c) If a increases at a rate of 2.5 cmymin, b increases at a rate of 1.5 cmymin, and increases at a rate of 0.2 radymin, how fast is the area increasing when a − 2 cm, b − 3 cm, and − y3?
9. Suppose y − s2x 1 1 , where x and y are functions of t. (a) If dxydt − 3, find dyydt when x − 4. (b) If dyydt − 5, find dxydt when x − 12.
10. Suppose 4x 2 1 9y 2 − 36, where x and y are functions of t. (a) If dyydt − 13, find dxydt when x − 2 and y − 23 s5 . (b) If dxydt − 3, find dy ydt when x − 22 and y − 23 s5 . 11. If x 2 1 y 2 1 z 2 − 9, dxydt − 5, and dyydt − 4, find dzydt when sx, y, zd − s2, 2, 1d. 12. A particle is moving along a hyperbola xy − 8. As it reaches the point s4, 2d, the ycoordinate is decreasing at a rate of 3 cmys. How fast is the xcoordinate of the point changing at that instant? 13–16 (a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem. 13. A plane flying horizontally at an altitude of 1 mi and a speed of 500 miyh passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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chapter 2 Derivatives
14. If a snowball melts so that its surface area decreases at a rate of 1 cm2ymin, find the rate at which the diameter decreases when the diameter is 10 cm. 15. A street light is mounted at the top of a 15fttall pole. A man 6 ft tall walks away from the pole with a speed of 5 ftys along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole? 16. At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 kmyh and ship B is sailing north at 25 kmyh. How fast is the distance between the ships changing at 4:00 pm? 17. T wo cars start moving from the same point. One travels south at 60 miyh and the other travels west at 25 miyh. At what rate is the distance between the cars increasing two hours later? 18. A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 mys, how fast is the length of his shadow on the building decreasing when he is 4 m from the building? 19. A man starts walking north at 4 ftys from a point P. Five minutes later a woman starts walking south at 5 ftys from a point 500 ft due east of P. At what rate are the people moving apart 15 min after the woman starts walking? 20. A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ftys. (a) At what rate is his distance from second base decreasing when he is halfway to first base? (b) At what rate is his distance from third base increasing at the same moment?
23. A t noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 kmyh and ship B is sailing north at 25 kmyh. How fast is the distance between the ships changing at 4:00 pm? 24. A particle moves along the curve y − 2 sinsxy2d. As the particle passes through the point ( 13 , 1), its xcoordinate increases at a rate of s10 cmys. How fast is the distance from the particle to the origin changing at this instant? 25. W ater is leaking out of an inverted conical tank at a rate of 10,000 cm 3ymin at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cmymin when the height of the water is 2 m, find the rate at which water is being pumped into the tank. 26. A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft 3ymin, how fast is the water level rising when the water is 6 inches deep? 27. A water trough is 10 m long and a crosssection has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 m 3ymin, how fast is the water level rising when the water is 30 cm deep? 28. A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. A crosssection is shown in the figure. If the pool is being filled at a rate of 0.8 ft 3ymin, how fast is the water level rising when the depth at the deepest point is 5 ft? 3 6 6
90 ft
21. T he altitude of a triangle is increasing at a rate of 1 cmymin while the area of the triangle is increasing at a rate of 2 cm 2ymin. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2 ?
12
16
6
29. Gravel is being dumped from a conveyor belt at a rate of 30 ft 3ymin, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?
22. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 mys, how fast is the boat approaching the dock when it is 8 m from the dock?
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.8 Related Rates
30. A kite 100 ft above the ground moves horizontally at a speed of 8 ftys. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out? 31. The sides of an equilateral triangle are increasing at a rate of 10 cmymin. At what rate is the area of the triangle increasing when the sides are 30 cm long? 32. H ow fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall? 33. The top of a ladder slides down a vertical wall at a rate of 0.15 mys. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 mys. How long is the ladder?
187
40. Brain weight B as a function of body weight W in fish has been modeled by the power function B − 0.007W 2y3, where B and W are measured in grams. A model for body weight as a function of body length L (measured in centimeters) is W − 0.12L2.53. If, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 20 cm at a constant rate, how fast was this species’ brain growing when the average length was 18 cm? 41. T wo sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2 8ymin. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60°?
34. According to the model we used to solve Example 2, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values of y?
42. Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P (see the figure). The point Q is on the floor 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ftys. How fast is cart B moving toward Q at the instant when cart A is 5 ft from Q?
35. If the minute hand of a clock has length r (in centimeters), find the rate at which it sweeps out area as a function of r.
P
; 36. A faucet is filling a hemispherical basin of diameter 60 cm with water at a rate of 2 Lymin. Find the rate at which the water is rising in the basin when it is half full. [Use the following facts: 1 L is 1000 cm3. The volume of the portion of a sphere with radius r from the bottom to a height h is V − (rh 2 2 13 h 3), as we will show in Chapter 5.] 37. B oyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV − C, where C is a constant. Suppose that at a certain instant the volume is 600 cm3, the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPaymin. At what rate is the volume decreasing at this instant? 38. When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV 1.4 − C, where C is a constant. Suppose that at a certain instant the volume is 400 cm3 and the pressure is 80 kPa and is decreasing at a rate of 10 kPaymin. At what rate is the volume increasing at this instant? 39. If two resistors with resistances R1 and R2 are connected in parallel, as in the figure, then the total resistance R, measured in ohms (V), is given by 1 1 1 − 1 R R1 R2 If R1 and R2 are increasing at rates of 0.3 Vys and 0.2 Vys, respectively, how fast is R changing when R1 − 80 V and R2 − 100 V?
R¡
R™
12 ft A
B Q
43. A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let’s assume the rocket rises vertically and its speed is 600 ftys when it has risen 3000 ft. (a) How fast is the distance from the television camera to the rocket changing at that moment? (b) If the television camera is always kept aimed at the rocket, how fast is the camera’s angle of elevation changing at that same moment? 44. A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? 45. A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is y3, this angle is decreasing at a rate of y6 radians per minute. How fast is the plane traveling at that time? 46. A Ferris wheel with a radius of 10 m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level? 47. A plane flying with a constant speed of 300 kmyh passes over a ground radar station at an altitude of 1 km and climbs
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chapter 2 Derivatives
at an angle of 308. At what rate is the distance from the plane to the radar station increasing a minute later? 48. T wo people start from the same point. One walks east at 3 miyh and the other walks northeast at 2 miyh. How fast is the distance between the people changing after 15 minutes? 49. A runner sprints around a circular track of radius 100 m at a constant speed of 7 mys. The runner’s friend is standing
50. T he minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o’clock?
We have seen that a curve lies very close to its tangent line near the point of tangency. In fact, by zooming in toward a point on the graph of a differentiable function, we noticed that the graph looks more and more like its tangent line. (See Figure 2.1.2.) This observation is the basis for a method of finding approximate values of functions. The idea is that it might be easy to calculate a value f sad of a function, but difficult (or even impossible) to compute nearby values of f . So we settle for the easily computed values of the linear function L whose graph is the tangent line of f at sa, f sadd. (See Figure 1.) In other words, we use the tangent line at sa, f sadd as an approximation to the curve y − f sxd when x is near a. An equation of this tangent line is
y
y=ƒ
{a, f(a)}
at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m?
y=L(x)
y − f sad 1 f 9sadsx 2 ad 0
x
and the approximation
FIGURE 1
1
f sxd < f sad 1 f 9sadsx 2 ad
is called the linear approximation or tangent line approximation of f at a. The linear function whose graph is this tangent line, that is,
2
Lsxd − f sad 1 f 9sadsx 2 ad
is called the linearization of f at a.
Example 1 Find the linearization of the function f sxd − sx 1 3 at a − 1 and use it to approximate the numbers s3.98 and s4.05 . Are these approximations overestimates or underestimates? SOLUTION The derivative of f sxd − sx 1 3d1y2 is
f 9sxd − 12 sx 1 3d21y2 −
1 2sx 1 3
and so we have f s1d − 2 and f 9s1d − 14. Putting these values into Equation 2, we see Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.9 Linear Approximations and Differentials
189
that the linearization is Lsxd − f s1d 1 f 9s1dsx 2 1d − 2 1 14 sx 2 1d −
7 x 1 4 4
The corresponding linear approximation (1) is sx 1 3 <
7 x 1 (when x is near 1) 4 4
In particular, we have y 7
1.05 7 s3.98 < 74 1 0.98 4 − 1.995 and s4.05 < 4 1 4 − 2.0125
x
y= 4 + 4
_3
FIGURE 2
(1, 2) 0
1
y= œ„„„„ x+3 x
The linear approximation is illustrated in Figure 2. We see that, indeed, the tangent line approximation is a good approximation to the given function when x is near l. We also see that our approximations are overestimates because the tangent line lies above the curve. Of course, a calculator could give us approximations for s3.98 and s4.05 , but the linear approximation gives an approximation over an entire interval. ■ In the following table we compare the estimates from the linear approximation in Example 1 with the true values. Notice from this table, and also from Figure 2, that the tangent line approximation gives good estimates when x is close to 1 but the accuracy of the approximation deteriorates when x is farther away from 1. Actual value
x
From Lsxd
s3.9
0.9
1.975
1.97484176. . .
s3.98
0.98
1.995
1.99499373. . .
s4
1
2
2.00000000. . .
s4.05
1.05
2.0125
2.01246117. . .
s4.1
1.1
2.025
2.02484567. . .
s5 s6
2
2.25
2.23606797. . .
3
2.5
2.44948974. . .
How good is the approximation that we obtained in Example 1? The next example shows that by using a graphing calculator or computer we can determine an interval throughout which a linear approximation provides a specified accuracy.
Example 2 For what values of x is the linear approximation sx 1 3 <
7 x 1 4 4
accurate to within 0.5? What about accuracy to within 0.1? SOLUTION Accuracy to within 0.5 means that the functions should differ by less
than 0.5:
Z
sx 1 3 2
S DZ 7 x 1 4 4
, 0.5
Equivalently, we could write sx 1 3 2 0.5 ,
7 x 1 , sx 1 3 1 0.5 4 4
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
190
Chapter 2 Derivatives 4.3 y= œ„„„„ x+3+0.5
L(x)
P
Q
y= œ„„„„ x+30.5
_4
10 _1
FIGURE 3 3
is accurate to within 0.5 when 22.6 , x , 8.6. (We have rounded 22.66 up and 8.66 down to be safe.) Similarly, from Figure 4 we see that the approximation is accurate to within 0.1 when 21.1 , x , 3.9. ■
Q
y= œ„„„„ x+3+0.1
Applications to Physics
y= œ„„„„ x+30.1
P _2
1
This says that the linear approximation should lie between the curves obtained by shifting the curve y − sx 1 3 upward and downward by an amount 0.5. Figure 3 shows the tangent line y − s7 1 xdy4 intersecting the upper curve y − sx 1 3 1 0.5 at P and Q. Zooming in and using the cursor, we estimate that the xcoordinate of P is about 22.66 and the xcoordinate of Q is about 8.66. Thus we see from the graph that the approximation 7 x sx 1 3 < 1 4 4
5
FIGURE 4
Linear approximations are often used in physics. In analyzing the consequences of an equation, a physicist sometimes needs to simplify a function by replacing it with its linear approximation. For instance, in deriving a formula for the period of a pendulum, physics textbooks obtain the expression a T − 2t sin for tangential acceleration and then replace sin by with the remark that sin is very close to if is not too large. [See, for example, Physics: Calculus, 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/ Cole, 2000), p. 431.] You can verify that the linearization of the function f sxd − sin x at a − 0 is Lsxd − x and so the linear approximation at 0 is sin x < x (see Exercise 40). So, in effect, the derivation of the formula for the period of a pendulum uses the tangent line approximation for the sine function. Another example occurs in the theory of optics, where light rays that arrive at shallow angles relative to the optical axis are called paraxial rays. In paraxial (or Gaussian) optics, both sin and cos are replaced by their linearizations. In other words, the linear approximations sin < and cos < 1 are used because is close to 0. The results of calculations made with these approximations became the basic theoretical tool used to design lenses. [See Optics, 4th ed., by Eugene Hecht (San Francisco, 2002), p. 154.] In Section 11.11 we will present several other applications of the idea of linear approximations to physics and engineering.
Differentials
If dx ± 0, we can divide both sides of Equation 3 by dx to obtain dy − f 9sxd dx We have seen similar equations before, but now the left side can genuinely be interpreted as a ratio of differentials.
The ideas behind linear approximations are sometimes formulated in the terminology and notation of differentials. If y − f sxd, where f is a differentiable function, then the differential dx is an independent variable; that is, dx can be given the value of any real number. The differential dy is then defined in terms of dx by the equation 3
dy − f 9sxd dx
So dy is a dependent variable; it depends on the values of x and dx. If dx is given a specific value and x is taken to be some specific number in the domain of f , then the numerical value of dy is determined.
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Section 2.9 Linear Approximations and Differentials y
Q
Îy dx=Îx
0
x
The geometric meaning of differentials is shown in Figure 5. Let Psx, f sxdd and Qsx 1 Dx, f sx 1 Dxdd be points on the graph of f and let dx − Dx. The corresponding change in y is Dy − f sx 1 Dxd 2 f sxd
R
P
dy
S
x+Î x
191
x
y=ƒ
FIGURE 5
The slope of the tangent line PR is the derivative f 9sxd. Thus the directed distance from S to R is f 9sxd dx − dy. Therefore dy represents the amount that the tangent line rises or falls (the change in the linearization) when x changes by an amount dx, whereas Dy represents the amount that the curve y − f sxd rises or falls when x changes by an amount dx − Dx.
Example 3 Compare the values of Dy and dy if y − f sxd − x 3 1 x 2 2 2x 1 1 and x changes (a) from 2 to 2.05 and (b) from 2 to 2.01. SOLUTION
(a) We have f s2d − 2 3 1 2 2 2 2s2d 1 1 − 9 f s2.05d − s2.05d3 1 s2.05d2 2 2s2.05d 1 1 − 9.717625 Dy − f s2.05d 2 f s2d − 0.717625 Figure 6 shows the function in Example 3 and a comparison of dy and Dy when a − 2. The viewing rectangle is f1.8, 2.5g by f6, 18g.
dy − f 9sxd dx − s3x 2 1 2x 2 2d dx
In general,
When x − 2 and dx − Dx − 0.05, this becomes dy − f3s2d2 1 2s2d 2 2g0.05 − 0.7
y=˛+≈2x+1
(b) dy
Îy
f s2.01d − s2.01d3 1 s2.01d2 2 2s2.01d 1 1 − 9.140701 Dy − f s2.01d 2 f s2d − 0.140701
(2, 9)
FIGURE 6
When dx − Dx − 0.01,
dy − f3s2d2 1 2s2d 2 2g0.01 − 0.14
■
Notice that the approximation Dy < dy becomes better as Dx becomes smaller in Example 3. Notice also that dy was easier to compute than Dy. For more complicated functions it may be impossible to compute Dy exactly. In such cases the approximation by differentials is especially useful. In the notation of differentials, the linear approximation (1) can be written as f sa 1 dxd < f sad 1 dy For instance, for the function f sxd − sx 1 3 in Example 1, we have dy − f 9sxd dx −
dx 2 sx 1 3
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192
Chapter 2 Derivatives
If a − 1 and dx − Dx − 0.05, then dy − and
0.05 − 0.0125 2 s1 1 3
s4.05 − f s1.05d < f s1d 1 dy − 2.0125
just as we found in Example 1. Our final example illustrates the use of differentials in estimating the errors that occur because of approximate measurements.
Example 4 The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of at most 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere? 4
SOLUTION If the radius of the sphere is r, then its volume is V − 3 r 3. If the error
in the measured value of r is denoted by dr − Dr, then the corresponding error in the calculated value of V is DV, which can be approximated by the differential dV − 4r 2 dr When r − 21 and dr − 0.05, this becomes dV − 4s21d2 0.05 < 277 The maximum error in the calculated volume is about 277 cm3.
■
NOTE Although the possible error in Example 4 may appear to be rather large, a better picture of the error is given by the relative error, which is computed by dividing the error by the total volume: DV dV 4r 2 dr dr < − 4 3 −3 V V r 3 r Thus the relative error in the volume is about three times the relative error in the radius. In Example 4 the relative error in the radius is approximately dryr − 0.05y21 < 0.0024 and it produces a relative error of about 0.007 in the volume. The errors could also be expressed as percentage errors of 0.24% in the radius and 0.7% in the volume.
1–4 Find the linearization Lsxd of the function at a. 1. f sxd − x 2 x 1 3, a − 22 3
2
2. f sxd − sin x, a − y6 3. f sxd − sx , a − 4 4. f sxd − 2ysx 2 2 5 , a − 3 ; 5. Find the linear approximation of the function f sxd − s1 2 x at a − 0 and use it to approximate the numbers s0.9 and s0.99 . Illustrate by graphing f and the tangent line.
; 6. Find the linear approximation of the function 3 tsxd − s 1 1 x at a − 0 and use it to approximate the 3 3 numbers s 0.95 and s 1.1 . Illustrate by graphing t and the tangent line. ; 7–10 Verify the given linear approximation at a − 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. 4 7. s 1 1 2x < 1 1 12 x 8. s1 1 xd23 < 1 2 3x
9. 1ys1 1 2xd4 < 1 2 8x 10. tan x < x
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 2.9 Linear Approximations and Differentials
11–14 Find the differential dy of each function. y − s1 2 t 4 11. (a) y − sx 2 2 3d 22 (b) 1 1 2u (b) y − 2 sin 2 1 1 3u 1 2 v2 y− 13. (a) y − tan st (b) 1 1 v2 12. (a) y −
1 14. (a) y − st 2 cos t (b) y − sin x x 15–18 (a) Find the differential dy and (b) evaluate dy for the given values of x and dx. 15. y − tan x, x − y4, dx − 20.1 16. y − cos x, x − 13, dx − 20.02 17. y − s3 1 x 2 , x − 1, dx − 20.1 18. y −
x11 , x − 2, dx − 0.05 x21
19–22 Compute Dy and dy for the given values of x and dx − Dx. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and Dy. 19. y − x 2 2 4x, x − 3, Dx − 0.5 20. y − x 2 x 3, x − 0, Dx − 20.3 21. y − sx 2 2 , x − 3, Dx − 0.8 3
22. y − x , x − 1, Dx − 0.5 23–28 Use a linear approximation (or differentials) to estimate the given number. 23. s1.999d4 24. 1y4.002 3 25. s 1001 26. s100.5
27. tan 2° 28. cos 29° 29–30 Explain, in terms of linear approximations or differentials, why the approximation is reasonable. 29. sec 0.08 < 1 30. s4.02 < 2.005 31. T he edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing (a) the volume of the cube and (b) the surface area of the cube. 32. T he radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. (a) Use differentials to estimate the maximum error in the calculated area of the disk. (b) What is the relative error? What is the percentage error?
193
33. T he circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. What is the relative error? (b) Use differentials to estimate the maximum error in the calculated volume. What is the relative error? 34. U se differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m. 35. (a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height h, inner radius r, and thickness Dr. (b) What is the error involved in using the formula from part (a)? 36. One side of a right triangle is known to be 20 cm long and the opposite angle is measured as 30°, with a possible error of 61°. (a) Use differentials to estimate the error in computing the length of the hypotenuse. (b) What is the percentage error? 37. If a current I passes through a resistor with resistance R, Ohm’s Law states that the voltage drop is V − RI. If V is constant and R is measured with a certain error, use differentials to show that the relative error in calculating I is approximately the same (in magnitude) as the relative error in R. 38. When blood flows along a blood vessel, the flux F (the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius R of the blood vessel: F − kR 4 (This is known as Poiseuille’s Law; we will show why it is true in Section 8.4.) A partially clogged artery can be expanded by an operation called angioplasty, in which a balloontipped catheter is inflated inside the artery in order to widen it and restore the normal blood flow. Show that the relative change in F is about four times the relative change in R. How will a 5% increase in the radius affect the flow of blood? 39. E stablish the following rules for working with differentials (where c denotes a constant and u and v are functions of x). (a) dc − 0 (b) dscud − c du (c) dsu 1 vd − du 1 dv (d) dsuvd − u dv 1 v du
SD
(e) d
u v
−
v du 2 u dv v2
(f) dsx n d − nx n21 dx
40. O n page 431 of Physics: Calculus, 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/Cole, 2000), in the course of deriving the formula T − 2 sLyt for the period of a pendulum of length L, the author obtains the equation a T − 2t sin for the tangential acceleration of the bob of the pendulum. He then says, “for small angles, the value of in
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
194
chapter 2 Derivatives
radians is very nearly the value of sin ; they differ by less than 2% out to about 20°.” (a) Verify the linear approximation at 0 for the sine function:
(b) Are your estimates in part (a) too large or too small? Explain. y
sin x < x ;
y=fª(x)
(b) Use a graphing device to determine the values of x for which sin x and x differ by less than 2%. Then verify Hecht’s statement by converting from radians to degrees.
41. Suppose that the only information we have about a function f is that f s1d − 5 and the graph of its derivative is as shown. (a) Use a linear approximation to estimate f s0.9d and f s1.1d.
laboratory Project
;
1 0
1
x
42. Suppose that we don’t have a formula for tsxd but we know that ts2d − 24 and t9sxd − sx 2 1 5 for all x. (a) Use a linear approximation to estimate ts1.95d and ts2.05d. (b) Are your estimates in part (a) too large or too small? Explain.
Taylor Polynomials
The tangent line approximation Lsxd is the best firstdegree (linear) approximation to f sxd near x − a because f sxd and Lsxd have the same rate of change (derivative) at a. For a better approximation than a linear one, let’s try a seconddegree (quadratic) approximation Psxd. In other words, we approximate a curve by a parabola instead of by a straight line. To make sure that the approximation is a good one, we stipulate the following: (i) Psad − f sad (P and f should have the same value at a.) (ii) P9sad − f 9sad (P and f should have the same rate of change at a.) (iii) P99sad − f 99sad (The slopes of P and f should change at the same rate at a.) 1. F ind the quadratic approximation Psxd − A 1 Bx 1 Cx 2 to the function f sxd − cos x that satisfies conditions (i), (ii), and (iii) with a − 0. Graph P, f , and the linear approximation Lsxd − 1 on a common screen. Comment on how well the functions P and L approximate f. 2. Determine the values of x for which the quadratic approximation f sxd < Psxd in Problem 1 is accurate to within 0.1. [Hint: Graph y − Psxd, y − cos x 2 0.1, and y − cos x 1 0.1 on a common screen.] 3. To approximate a function f by a quadratic function P near a number a, it is best to write P in the form Psxd − A 1 Bsx 2 ad 1 Csx 2 ad2
Show that the quadratic function that satisfies conditions (i), (ii), and (iii) is Psxd − f sad 1 f 9sadsx 2 ad 1 12 f 99sadsx 2 ad2
4. Find the quadratic approximation to f sxd − sx 1 3 near a − 1. Graph f , the quadratic approximation, and the linear approximation from Example 2.9.2 on a common screen. What do you conclude? 5. I nstead of being satisfied with a linear or quadratic approximation to f sxd near x − a, let’s try to find better approximations with higherdegree polynomials. We look for an nthdegree polynomial Tnsxd − c0 1 c1 sx 2 ad 1 c2 sx 2 ad2 1 c3 sx 2 ad3 1 ∙ ∙ ∙ 1 cn sx 2 adn
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
chapter 2 Review
such that Tn and its first n derivatives have the same values at x − a as f and its first n derivatives. By differentiating repeatedly and setting x − a, show that these conditions are satisfied if c0 − f sad, c1 − f 9sad, c2 − 12 f 99 sad, and in general ck −
f skdsad k!
where k! − 1 ? 2 ? 3 ? 4 ? ∙ ∙ ∙ ? k. The resulting polynomial Tn sxd − f sad 1 f 9sadsx 2 ad 1
195
f 99sad f sndsad sx 2 ad2 1 ∙ ∙ ∙ 1 sx 2 adn 2! n!
is called the nthdegree Taylor polynomial of f centered at a.
6. F ind the 8thdegree Taylor polynomial centered at a − 0 for the function f sxd − cos x. Graph f together with the Taylor polynomials T2 , T4 , T6 , T8 in the viewing rectangle f25, 5g by f21.4, 1.4g and comment on how well they approximate f .
2 Review CONCEPT CHECK 1. Write an expression for the slope of the tangent line to the curve y − f sxd at the point sa, f sadd. 2. Suppose an object moves along a straight line with position f std at time t. Write an expression for the instantaneous velocity of the object at time t − a. How can you interpret this velocity in terms of the graph of f ?
Answers to the Concept Check can be found on the back endpapers.
7. What are the second and third derivatives of a function f ? If f is the position function of an object, how can you interpret f 0 and f ? 8. State each differentiation rule both in symbols and in words. (a) The Power Rule (b) The Constant Multiple Rule (c) The Sum Rule (d) The Difference Rule (e) The Product Rule (f) The Quotient Rule (g) The Chain Rule
3. If y − f sxd and x changes from x 1 to x 2, write expressions for the following. (a) The average rate of change of y with respect to x over the interval fx 1, x 2 g. (b) The instantaneous rate of change of y with respect to x at x − x 1.
9. State the derivative of each function. (a) y − x n (b) y − sin x (c) y − cos x (d) y − tan x (e) y − csc x (f) y − sec x (g) y − cot x
4. Define the derivative f 9sad. Discuss two ways of interpreting this number.
10. Explain how implicit differentiation works.
5. (a) What does it mean for f to be differentiable at a? (b) What is the relation between the differentiability and continuity of a function? (c) Sketch the graph of a function that is continuous but not differentiable at a − 2. 6. Describe several ways in which a function can fail to be differentiable. Illustrate with sketches.
11. G ive several examples of how the derivative can be interpreted as a rate of change in physics, chemistry, biology, economics, or other sciences. 12. (a) Write an expression for the linearization of f at a. (b) If y − f sxd, write an expression for the differential dy. (c) If dx − Dx, draw a picture showing the geometric meanings of Dy and dy.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
196
chapter 2 Derivatives
TRUEFALSE QUIZ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f is continuous at a, then f is differentiable at a. 2. If f and t are differentiable, then d f f sxd 1 tsxdg − f 9sxd 1 t9sxd dx

 

8. If f 9srd exists, then lim x l r f sxd − f srd.
10.
d 2y − dx 2
S D dy dx
xl2
tsxd 2 ts2d − 80 x22
2
11. A n equation of the tangent line to the parabola y − x 2 at s22, 4d is y 2 4 − 2xsx 1 2d.
4. If f and t are differentiable, then d f s tsxdd − f 9s tsxdd t9sxd dx
g
12.
d f 9sxd . sf sxd − dx 2 sf sxd
5. If f is differentiable, then
d x 2 1 x − 2x 1 1 dx
9. If tsxd − x 5, then lim
3. If f and t are differentiable, then d f f sxd tsxdg − f 9sxd t9sxd dx
f
7.
d d stan2xd − ssec 2xd dx dx
13. The derivative of a polynomial is a polynomial. 14. The derivative of a rational function is a rational function.
d f 9sxd f ssx d − . 6. If f is differentiable, then dx 2 sx
15. If f sxd − sx 6 2 x 4 d 5, then f s31dsxd − 0.
EXERCISES 1. The displacement (in meters) of an object moving in a straight line is given by s − 1 1 2t 1 14 t 2, where t is measured in seconds. (a) Find the average velocity over each time period. (i) f1, 3g (ii) f1, 2g (iii) f1, 1.5g (iv) f1, 1.1g (b) Find the instantaneous velocity when t − 1.
5. The figure shows the graphs of f , f 9, and f 0. Identify each curve, and explain your choices. y
a b
2. The graph of f is shown. State, with reasons, the numbers at which f is not differentiable.
0
y
_1 0
2
4
6. Find a function f and a number a such that
x
6
lim
h l0
3–4 Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath. y 4. 3. y
0
x
0
x
c
x
s2 1 hd6 2 64 − f 9sad h
7. The total cost of repaying a student loan at an interest rate of r % per year is C − f srd. (a) What is the meaning of the derivative f 9srd? What are its units? (b) What does the statement f 9s10d − 1200 mean? (c) Is f 9srd always positive or does it change sign? 8. The total fertility rate at time t, denoted by Fstd, is an estimate of the average number of children born to each woman (assuming that current birth rates remain constant). The graph
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
chapter 2 Review
S D
of the total fertility rate in the United States shows the fluctuations from 1940 to 2010. (a) Estimate the values of F9s1950d, F9s1965d, and F9s1987d. (b) What are the meanings of these derivatives? (c) Can you suggest reasons for the values of these derivatives? y
2.0
t4 2 1 20. y − sinscos xd t4 1 1 1 21. y − tan s1 2 x 22. y− sinsx 2 sin xd 19. y −
23. xy 4 1 x 2 y − x 1 3y 24. y − secs1 1 x 2 d
3.0 2.5
1 s7 17. y − x 2 sin x 18. y− x1 2 x
baby boom
3.5
25. y −
baby bust
sec 2 26. x 2 cos y 1 sin 2y − xy 1 1 tan 2
3 27. y − s1 2 x 21 d21 28. y − 1ys x 1 sx
baby boomlet
y=F(t)
29. sinsxyd − x 2 2 y 30. y − ssin s x
1.5
sx 1 d4 31. y − cots3x 2 1 5d 32. y− 4 x 1 4
1940
1950
1960
1970
1980
1990
2000
2010 t
9. Let Pstd be the percentage of Americans under the age of 18 at time t. The table gives values of this function in census years from 1950 to 2010. t
Pstd
1950 1960 1970 1980
197
31.1 35.7 34.0 28.0
t 1990 2000 2010
Pstd 25.7 25.7 24.0
(a) What is the meaning of P9std? What are its units? (b) Construct a table of estimated values for P9std. (c) Graph P and P9. (d) How would it be possible to get more accurate values for P9std?
10–11 Find f 9sxd from first principles, that is, directly from the definition of a derivative. 10. f sxd −
42x 11. f sxd − x 3 1 5x 1 4 31x
12. (a) If f sxd − s3 2 5x , use the definition of a derivative to find f 9sxd. (b) Find the domains of f and f 9. (c) Graph f and f 9 on a common screen. Compare the ; graphs to see whether your answer to part (a) is reasonable. 13–40 Calculate y9. 1 1 13. y − sx 1 x d 14. y− 2 5 3 sx sx 2
sin mx 33. y − sx cos sx 34. y− x 35. y − tan2ssin d 36. x tan y − y 2 1 sx 2 1dsx 2 4d 5 37. y − s x tan x 38. y− sx 2 2dsx 2 3d 39. y − sinstan s1 1 x 3 d 40. y − sin2 scosssin x d 41. If f std − s4t 1 1, find f 99s2d. 42. If tsd − sin , find t99sy6d. 43. Find y99 if x 6 1 y 6 − 1. 44. Find f sndsxd if f sxd − 1ys2 2 xd. 45–46 Find the limit. sec x t3 45. lim 46. lim x l 0 1 2 sin x t l 0 tan3 2t 47–48 Find an equation of the tangent to the curve at the given point. 47. y − 4 sin2 x, sy6, 1d
48. y −
x2 2 1 , s0, 21d x2 1 1
49–50 Find equations of the tangent line and normal line to the curve at the given point. 49. y − s1 1 4 sin x , s0, 1d 50. x 2 1 4xy 1 y 2 − 13, s2, 1d
3 4
x2 2 x 1 2 tan x 15. y − 16. y− 1 1 cos x sx
51. (a) If f sxd − x s5 2 x , find f 9sxd. (b) Find equations of the tangent lines to the curve y − x s5 2 x at the points s1, 2d and s4, 4d.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
198 ; ;
chapter 2 Derivatives
61–68 Find f 9 in terms of t9.
(c) Illustrate part (b) by graphing the curve and tangent lines on the same screen. (d) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f 9.
f sxd − tsx 2 d 61. f sxd − x 2tsxd 62. 63. f sxd − f tsxdg 2 64. f sxd − x a tsx b d
52. (a) If f sxd − 4x 2 tan x, 2y2 , x , y2, find f 9 and f 0. (b) Check to see that your answers to part (a) are reason; able by comparing the graphs of f , f 9, and f 99.
65. f sxd − ts tsxdd 66. f sxd − sins tsxdd 67. f sxd − tssin xd 68. f sxd − tstan sx d
53. At what points on the curve y − sin x 1 cos x, 0 < x < 2, is the tangent line horizontal?
69–71 Find h9 in terms of f 9 and t9.
54. F ind the points on the ellipse x 2 1 2y 2 − 1 where the tangent line has slope 1.
69. hsxd −
55. F ind a parabola y − ax 2 1 bx 1 c that passes through the point s1, 4d and whose tangent lines at x − 21 and x − 5 have slopes 6 and 22, respectively.
71. hsxd − f s tssin 4xdd 72. A particle moves along a horizontal line so that its coordinate at time t is x − sb 2 1 c 2 t 2 , t > 0, where b and c are positive constants. (a) Find the velocity and acceleration functions. (b) Show that the particle always moves in the positive direction.
56. H ow many tangent lines to the curve y − xysx 1 1) pass through the point s1, 2d? At which points do these tangent lines touch the curve? 57. If f sxd − sx 2 adsx 2 bdsx 2 cd, show that f 9sxd 1 1 1 − 1 1 f sxd x2a x2b x2c 58. (a) By differentiating the doubleangle formula cos 2x − cos2x 2 sin2x obtain the doubleangle formula for the sine function. (b) By differentiating the addition formula sinsx 1 ad − sin x cos a 1 cos x sin a obtain the addition formula for the cosine function. 59. Suppose that f s1d − 2 f 9s1d − 3 f s2d − 1 f 9s2d − 2 ts1d − 3 t9s1d − 1 ts2d − 1 t9s2d − 4
73. A particle moves on a vertical line so that its coordinate at time t is y − t 3 2 12t 1 3, t > 0. (a) Find the velocity and acceleration functions. (b) When is the particle moving upward and when is it moving downward? (c) Find the distance that the particle travels in the time interval 0 < t < 3. (d) Graph the position, velocity, and acceleration functions ; for 0 < t < 3. (e) When is the particle speeding up? When is it slowing down? 74. T he volume of a right circular cone is V − 13 r 2h, where r is the radius of the base and h is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant. (b) Find the rate of change of the volume with respect to the radius if the height is constant.
(a) If Ssxd − f sxd 1 tsxd, find S9s1d. (b) If Psxd − f sxd tsxd, find P9s2d. (c) If Qsxd − f sxdytsxd, find Q9s1d. (d) If Csxd − f stsxdd, find C9s2d.
60. If f and t are the functions whose graphs are shown, let Psxd − f sxd tsxd, Qsxd − f sxdytsxd, and Csxd − f s tsxdd. Find (a) P9s2d, (b) Q9s2d, and (c) C9s2d.
75. The mass of part of a wire is x s1 1 sx d kilograms, where x is measured in meters from one end of the wire. Find the linear density of the wire when x − 4 m.
y
76. The cost, in dollars, of producing x units of a certain commodity is
g f
Csxd − 920 1 2x 2 0.02x 2 1 0.00007x 3
1 0
1
Î
f sxd tsxd f sxd 70. hsxd − f sxd 1 tsxd tsxd
x
(a) Find the marginal cost function. (b) Find C9s100d and explain its meaning. (c) Compare C9s100d with the cost of producing the 101st item.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
chapter 2 Review
77. The volume of a cube is increasing at a rate of 10 cm3ymin. How fast is the surface area increasing when the length of an edge is 30 cm? 78. A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of 2 cm3ys, how fast is the water level rising when the water is 5 cm deep? 79. A balloon is rising at a constant speed of 5 ftys. A boy is cycling along a straight road at a speed of 15 ftys. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later? 80. A waterskier skis over the ramp shown in the figure at a speed of 30 ftys. How fast is she rising as she leaves the ramp?
3 83. (a) Find the linearization of f sxd − s 1 1 3x at a − 0. State the corresponding linear approximation and use 3 it to give an approximate value for s 1.03 . (b) Determine the values of x for which the linear approxi; mation given in part (a) is accurate to within 0.1.
84. Evaluate dy if y − x 3 2 2x 2 1 1, x − 2, and dx − 0.2. 85. A window has the shape of a square surmounted by a semicircle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window. 86–88 Express the limit as a derivative and evaluate. 86. lim
x l1
88. lim
4 x 17 2 1 16 1 h 2 2 s 87. lim hl0 x21 h
l y3
cos 2 0.5 2 y3
4 ft 15 ft
89. Evaluate lim
xl0
81. The angle of elevation of the sun is decreasing at a rate of 0.25 radyh. How fast is the shadow cast by a 400fttall building increasing when the angle of elevation of the sun is y6? ; 82. (a) Find the linear approximation to f sxd − s25 2 x 2 near 3. (b) Illustrate part (a) by graphing f and the linear approximation. (c) For what values of x is the linear approximation accurate to within 0.1?
199
s1 1 tan x 2 s1 1 sin x . x3
90. Suppose f is a differentiable function such that f s tsxdd − x and f 9sxd − 1 1 f f sxdg 2. Show that t9sxd − 1ys1 1 x 2 d. 91. Find f 9sxd if it is known that d f f s2xdg − x 2 dx 92. Show that the length of the portion of any tangent line to the astroid x 2y3 1 y 2y3 − a 2y3 cut off by the coordinate axes is constant.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Problems Plus
Before you look at the example, cover up the solution and try it yourself first. Example How many lines are tangent to both of the parabolas y − 21 2 x 2 and y − 1 1 x 2 ? Find the coordinates of the points at which these tangents touch the parabolas. SOLUTION To gain insight into this problem, it is essential to draw a diagram. So we sketch the parabolas y − 1 1 x 2 (which is the standard parabola y − x 2 shifted 1 unit upward) and y − 21 2 x 2 (which is obtained by reflecting the first parabola about the xaxis). If we try to draw a line tangent to both parabolas, we soon discover that there are only two possibilities, as illustrated in Figure 1. Let P be a point at which one of these tangents touches the upper parabola and let a be its xcoordinate. (The choice of notation for the unknown is important. Of course we could have used b or c or x 0 or x1 instead of a. However, it’s not advisable to use x in place of a because that x could be confused with the variable x in the equation of the parabola.) Then, since P lies on the parabola y − 1 1 x 2, its ycoordinate must be 1 1 a 2. Because of the symmetry shown in Figure 1, the coordinates of the point Q where the tangent touches the lower parabola must be s2a, 2s1 1 a 2 dd. To use the given information that the line is a tangent, we equate the slope of the line PQ to the slope of the tangent line at P. We have
y
P 1
x
Q
_1
FIGURE 1
mPQ −
1 1 a 2 2 s21 2 a 2 d 1 1 a2 − a 2 s2ad a
If f sxd − 1 1 x 2, then the slope of the tangent line at P is f 9sad − 2a. Thus the condition that we need to use is that 1 1 a2 − 2a a Solving this equation, we get 1 1 a 2 − 2a 2, so a 2 − 1 and a − 61. Therefore the points are (1, 2) and s21, 22d. By symmetry, the two remaining points are s21, 2d and s1, 22d. Problems
1. Find points P and Q on the parabola y − 1 2 x 2 so that the triangle ABC formed by the xaxis and the tangent lines at P and Q is an equilateral triangle. (See the figure.)
y
Find the point where the curves y − x 3 2 3x 1 4 and y − 3sx 2 2 xd are tangent to each ; 2. other, that is, have a common tangent line. Illustrate by sketching both curves and the common tangent.
A
P B
3. Show that the tangent lines to the parabola y − ax 2 1 bx 1 c at any two points with xcoordinates p and q must intersect at a point whose xcoordinate is halfway between p and q.
Q 0
■
C
x
4. Show that d dx
FIGURE FOR PROBLEM 1 5. If f sxd − lim tlx
S
cos2 x sin2 x 1 1 1 cot x 1 1 tan x
D
− 2cos 2x
sec t 2 sec x , find the value of f 9sy4d. t2x
6. Find the values of the constants a and b such that lim
xl0
3 ax 1 b 2 2 5 s − x 12
200 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
7. Prove that
dn ssin4 x 1 cos4 xd − 4n21 coss4x 1 ny2d. dx n
8. If f is differentiable at a, where a . 0, evaluate the following limit in terms of f 9sad: lim
y
xla
y=≈
1
10. Find all values of c such that the parabolas y − 4x 2 and x − c 1 2y 2 intersect each other at right angles.
1
11. How many lines are tangent to both of the circles x 2 1 y 2 − 4 and x 2 1 sy 2 3d 2 − 1? At what points do these tangent lines touch the circles?
x
x 46 1 x 45 1 2 , calculate f s46ds3d. Express your answer using factorial notation: 11x n! − 1 ? 2 ? 3 ? ∙ ∙ ∙ ? sn 2 1d ? n. 12. If f sxd −
FIGURE FOR PROBLEM 9 y
13. The figure shows a rotating wheel with radius 40 cm and a connecting rod AP with length 1.2 m. The pin P slides back and forth along the xaxis as the wheel rotates counter clockwise at a rate of 360 revolutions per minute. (a) Find the angular velocity of the connecting rod, dydt, in radians per second, when − y3. (b) Express the distance x − OP in terms of . (c) Find an expression for the velocity of the pin P in terms of .
A å
¨
sx 2 sa
9. The figure shows a circle with radius 1 inscribed in the parabola y − x 2. Find the center of the circle.
0
O
f sxd 2 f sad

P (x, 0) x

14. T angent lines T1 and T2 are drawn at two points P1 and P2 on the parabola y − x 2 and they intersect at a point P. Another tangent line T is drawn at a point between P1 and P2 ; it intersects T1 at Q1 and T2 at Q2. Show that
FIGURE FOR PROBLEM 13
 PQ  1  PQ  − 1  PP   PP 
y
yT
0
yN
P
xN N
2
2
15. L et T and N be the tangent and normal lines to the ellipse x 2y9 1 y 2y4 − 1 at any point P on the ellipse in the first quadrant. Let x T and yT be the x and yintercepts of T and x N and yN be the intercepts of N. As P moves along the ellipse in the first quadrant (but not on the axes), what values can x T , yT , x N, and yN take on? First try to guess the answers just by looking at the figure. Then use calculus to solve the problem and see how good your intuition is.
T
2
1
1
3
FIGURE FOR PROBLEM 15
xT
x
16. Evaluate lim
xl0
sins3 1 xd2 2 sin 9 . x
17. (a) Use the identity for tansx 2 yd (see Equation 14b in Appendix D) to show that if two lines L 1 and L 2 intersect at an angle , then tan −
m 2 2 m1 1 1 m1 m 2
where m1 and m 2 are the slopes of L 1 and L 2, respectively. (b) The angle between the curves C1 and C2 at a point of intersection P is defined to be the angle between the tangent lines to C1 and C2 at P (if these tangent lines exist). Use part (a) to find, correct to the nearest degree, the angle between each pair of curves at each point of intersection. (i) y − x 2 and y − sx 2 2d2 (ii) x 2 2 y 2 − 3 and x 2 2 4x 1 y 2 1 3 − 0
201 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
y
0
y=›
∫ P(⁄, ›)
å
18. Let Psx 1, y1d be a point on the parabola y 2 − 4px with focus Fs p, 0d. Let be the angle between the parabola and the line segment FP, and let be the angle between the horizontal line y − y1 and the parabola as in the figure. Prove that − . (Thus, by a principle of geometrical optics, light from a source placed at F will be reflected along a line parallel to the xaxis. This explains why paraboloids, the surfaces obtained by rotating parabolas about their axes, are used as the shape of some automobile headlights and mirrors for telescopes.)
x
F(p, 0) ¥=4px
19. Suppose that we replace the parabolic mirror of Problem 18 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, C is a semicircle with center O. A ray of light coming in toward the mirror parallel to the axis along the line PQ will be reflected to the point R on the axis so that /PQO − /OQR (the angle of incidence is equal to the angle of reflection). What happens to the point R as P is taken closer and closer to the axis?
FIGURE FOR PROBLEM 18
Q ¨ A
R
P
¨
20. If f and t are differentiable functions with f s0d − ts0d − 0 and t9s0d ± 0, show that lim
O
xl0
C
21. Evaluate lim
xl0
f sxd f 9s0d − tsxd t9s0d
sinsa 1 2xd 2 2 sinsa 1 xd 1 sin a . x2
22. G iven an ellipse x 2ya 2 1 y 2yb 2 − 1, where a ± b, find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals.
FIGURE FOR PROBLEM 19
23. Find the two points on the curve y − x 4 2 2x 2 2 x that have a common tangent line. 24. S uppose that three points on the parabola y − x 2 have the property that their normal lines intersect at a common point. Show that the sum of their xcoordinates is 0. 25. A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius r are drawn using all lattice points as centers. Find the smallest value of r such that any line with slope 25 intersects some of these circles. 26. A cone of radius r centimeters and height h centimeters is lowered point first at a rate of 1 cmys into a tall cylinder of radius R centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely submerged? 27. A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surface area of a cone is rl, where r is the radius and l is the slant height.) If we pour the liquid into the container at a rate of 2 cm3ymin, then the height of the liquid decreases at a rate of 0.3 cmymin when the height is 10 cm. If our goal is to keep the liquid at a constant height of 10 cm, at what rate should we pour the liquid into the container? CAS
28. (a) The cubic function f sxd − xsx 2 2dsx 2 6d has three distinct zeros: 0, 2, and 6. Graph f and its tangent lines at the average of each pair of zeros. What do you notice? (b) Suppose the cubic function f sxd − sx 2 adsx 2 bdsx 2 cd has three distinct zeros: a, b, and c. Prove, with the help of a computer algebra system, that a tangent line drawn at the average of the zeros a and b intersects the graph of f at the third zero.
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3
Applications of Differentiation
When flying, some small birds—like the finch pictured here—alternate between flapping their wings and keeping them folded while gliding. In the project on page 271, we will investigate how frequently a bird should flap its wings in order to minimize the energy required. © Targn Pleiades / Shutterstock.com
We have already investigated some of the applications of derivatives, but now that we know the differentiation rules we are in a better position to pursue the applications of differentiation in greater depth. Here we learn how derivatives affect the shape of a graph of a function and, in particular, how they help us locate maximum and minimum values of functions. Many practical problems require us to minimize a cost or maximize an area or somehow find the best possible outcome of a situation. In particular, we will be able to investigate the optimal shape of a can and to explain the location of rainbows in the sky.
203 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
204
Chapter 3 Applications of Differentiation
Some of the most important applications of differential calculus are optimization problems, in which we are required to find the optimal (best) way of doing something. Here are examples of such problems that we will solve in this chapter:
• • • •
hat is the shape of a can that minimizes manufacturing costs? W What is the maximum acceleration of a space shuttle? (This is an important question to the astronauts who have to withstand the effects of acceleration.) What is the radius of a contracted windpipe that expels air most rapidly during a cough? At what angle should blood vessels branch so as to minimize the energy expended by the heart in pumping blood?
These problems can be reduced to finding the maximum or minimum values of a function. Let’s first explain exactly what we mean by maximum and minimum values. We see that the highest point on the graph of the function f shown in Figure 1 is the point s3, 5d. In other words, the largest value of f is f s3d − 5. Likewise, the smallest value is f s6d − 2. We say that f s3d − 5 is the absolute maximum of f and f s6d − 2 is the absolute minimum. In general, we use the following definition.
y 4 2 0
4
2
x
6
1 Definition Let c be a number in the domain D of a function f. Then f scd is the • absolute maximum value of f on D if f scd > f sxd for all x in D.
Figure 1
•
y
f(d) f(a) a
0
b
c
d
e
x
Figure 2
Abs min f sad, abs max f sdd, loc min f scd, f sed, loc max f sbd, f sdd
6 4
loc min
2 0
Figure 3
loc max
loc and abs min
I
J
K
4
8
12
An absolute maximum or minimum is sometimes called a global maximum or minimum. The maximum and minimum values of f are called extreme values of f . Figure 2 shows the graph of a function f with absolute maximum at d and absolute minimum at a. Note that sd, f sddd is the highest point on the graph and sa, f sadd is the lowest point. In Figure 2, if we consider only values of x near b [for instance, if we restrict our attention to the interval sa, cd], then f sbd is the largest of those values of f sxd and is called a local maximum value of f. Likewise, f scd is called a local minimum value of f because f scd < f sxd for x near c [in the interval sb, dd, for instance]. The function f also has a local minimum at e. In general, we have the following definition. 2 Definition The number f scd is a
• •
y
x
absolute minimum value of f on D if f scd < f sxd for all x in D.
local maximum value of f if f scd > f sxd when x is near c. local minimum value of f if f scd < f sxd when x is near c.
In Definition 2 (and elsewhere), if we say that something is true near c, we mean that it is true on some open interval containing c. For instance, in Figure 3 we see that f s4d − 5 is a local minimum because it’s the smallest value of f on the interval I. It’s not the absolute minimum because f sxd takes smaller values when x is near 12 (in the interval K, for instance). In fact f s12d − 3 is both a local minimum and the absolute minimum. Similarly, f s8d − 7 is a local maximum, but not the absolute maximum because f takes larger values near 1.
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Section 3.1 Maximum and Minimum Values
205
Example 1 The function f sxd − cos x takes on its (local and absolute) maximum value of 1 infinitely many times, since cos 2n − 1 for any integer n and 21 < cos x < 1 for all x. (See Figure 4.) Likewise, coss2n 1 1d − 21 is its minimum value, where n is any integer. y
Local and absolute maximum
0
FIGURE 4 y − cosx y
y=≈
0
π
2π
3π
Local and absolute minimum
x
n
Example 2 If f sxd − x 2, then f sxd > f s0d because x 2 > 0 for all x. Therefore
x
FIGURE 5 Mimimum value 0, no maximum
y
f s0d − 0 is the absolute (and local) minimum value of f. This corresponds to the fact that the origin is the lowest point on the parabola y − x 2. (See Figure 5.) However, there is no highest point on the parabola and so this function has no maximum value. n
Example 3 From the graph of the function f sxd − x 3, shown in Figure 6, we see that this function has neither an absolute maximum value nor an absolute minimum value. In fact, it has no local extreme values either. n
Example 4 The graph of the function
y=˛
f sxd − 3x 4 2 16x 3 1 18x 2 21 < x < 4 0
x
FIGURE 6 No mimimum, no maximum
is shown in Figure 7. You can see that f s1d − 5 is a local maximum, whereas the absolute maximum is f s21d − 37. (This absolute maximum is not a local maximum because it occurs at an endpoint.) Also, f s0d − 0 is a local minimum and f s3d − 227 is both a local and an absolute minimum. Note that f has neither a local nor an absolute maximum at x − 4. y (_1, 37)
y=3x$16˛+18≈
(1, 5) _1
FIGURE 7
1
2
3
4
5
x
(3, _27)
n
We have seen that some functions have extreme values, whereas others do not. The following theorem gives conditions under which a function is guaranteed to possess extreme values.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
206
Chapter 3 Applications of Differentiation
3 The Extreme Value Theorem If f is continuous on a closed interval fa, bg, then f attains an absolute maximum value f scd and an absolute minimum value f sdd at some numbers c and d in fa, bg.
The Extreme Value Theorem is illustrated in Figure 8. Note that a function can attain an extreme value more than once. Although the Extreme Value Theorem is intuitively very plausible, it is difficult to prove and so we omit the proof. y
y
y
FIGURE 8 Functions continuous on a closed interval always attain extreme values.
0
a
c
d b
0
x
a
c
d=b
x
0
a c¡
d
c™ b
x
Figures 9 and 10 show that a function need not possess extreme values if either hypothesis (continuity or closed interval) is omitted from the Extreme Value Theorem. y
y
3
1 0
1 2
x
FIGURE 9 This function has minimum value f(2)=0, but no maximum value.
y
{c, f(c)}
{d, f (d )} 0
c
FIGURE 11
d
x
0
2
x
FIGURE 10
This continuous function g has no maximum or minimum.
The function f whose graph is shown in Figure 9 is defined on the closed interval [0, 2] but has no maximum value. (Notice that the range of f is [0, 3). The function takes on values arbitrarily close to 3, but never actually attains the value 3.) This does not contradict the Extreme Value Theorem because f is not continuous. [Nonetheless, a discontinuous function could have maximum and minimum values. See Exercise 13(b).] The function t shown in Figure 10 is continuous on the open interval (0, 2) but has neither a maximum nor a minimum value. [The range of t is s1, `d. The function takes on arbitrarily large values.] This does not contradict the Extreme Value Theorem because the interval (0, 2) is not closed. The Extreme Value Theorem says that a continuous function on a closed interval has a maximum value and a minimum value, but it does not tell us how to find these extreme values. Notice in Figure 8 that the absolute maximum and minimum values that are between a and b occur at local maximum or minimum values, so we start by looking for local extreme values. Figure 11 shows the graph of a function f with a local maximum at c and a local minimum at d. It appears that at the maximum and minimum points the tangent lines are horizontal and therefore each has slope 0. We know that the derivative is the slope of the
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Section 3.1 Maximum and Minimum Values
207
tangent line, so it appears that f 9scd − 0 and f 9sdd − 0. The following theorem says that this is always true for differentiable functions. Fermat Fermat’s Theorem is named after Pierre Fermat (1601–1665), a French lawyer who took up mathematics as a hobby. Despite his amateur status, Fermat was one of the two inventors of analytic geometry (Descartes was the other). His methods for finding tangents to curves and maximum and minimum values (before the invention of limits and derivatives) made him a forerunner of Newton in the creation of differential calculus.
4 Fermat’s Theorem If f has a local maximum or minimum at c, and if f 9scd exists, then f 9scd − 0.
Proof Suppose, for the sake of definiteness, that f has a local maximum at c. Then, according to Definition 2, f scd > f sxd if x is sufficiently close to c. This implies that if h is sufficiently close to 0, with h being positive or negative, then
f scd > f sc 1 hd and therefore f sc 1 hd 2 f scd < 0
5
We can divide both sides of an inequality by a positive number. Thus, if h . 0 and h is sufficiently small, we have f sc 1 hd 2 f scd 0 h , 0 h So, taking the lefthand limit, we have f 9scd − lim
hl0
f sc 1 hd 2 f scd f sc 1 hd 2 f scd − lim2 >0 h l 0 h h
We have shown that f 9scd > 0 and also that f 9scd < 0. Since both of these inequalities must be true, the only possibility is that f 9scd − 0. We have proved Fermat’s Theorem for the case of a local maximum. The case of a local minimum can be proved in a similar manner, or we could use Exercise 70 to n deduce it from the case we have just proved (see Exercise 71). The following examples caution us against reading too much into Fermat’s Theorem: We can’t expect to locate extreme values simply by setting f 9sxd − 0 and solving for x. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
208
Chapter 3 Applications of Differentiation y
Example 5 If f sxd − x 3, then f 9sxd − 3x 2, so f 9s0d − 0. But f has no maximum
or minimum at 0, as you can see from its graph in Figure 12. (Or observe that x 3 . 0 for x . 0 but x 3 , 0 for x , 0.) The fact that f 9s0d − 0 simply means that the curve y − x 3 has a horizontal tangent at s0, 0d. Instead of having a maximum or minimum at s0, 0d, the curve crosses its horizontal tangent there. n
y=˛ 0
x
 
Example 6 The function f sxd − x has its (local and absolute) minimum value at 0, but that value can’t be found by setting f 9sxd − 0 because, as was shown in Example 2.2.5, f 9s0d does not exist. (See Figure 13.)
Figure 12
If f sxd − x 3, then f 9s0d − 0, but f has no maximum or minimum.
warning Examples 5 and 6 show that we must be careful when using Fermat’s Theorem. Example 5 demonstrates that even when f 9scd − 0 there need not be a maximum or minimum at c. (In other words, the converse of Fermat’s Theorem is false in general.) Furthermore, there may be an extreme value even when f 9scd does not exist (as in Example 6). Fermat’s Theorem does suggest that we should at least start looking for extreme values of f at the numbers c where f 9scd − 0 or where f 9scd does not exist. Such numbers are given a special name.
y
y= x 0
n
x
Figure 13
6 Definition A critical number of a function f is a number c in the domain of f such that either f 9scd − 0 or f 9scd does not exist.
 
If f sxd − x , then f s0d − 0 is a minimum value, but f 9s0d does not exist.
Example 7 Find the critical numbers of f sxd − x 3y5s4 2 xd. SOLUTION The Product Rule gives Figure 14 shows a graph of the function f in Example 7. It supports our answer because there is a horizontal tangent when x − 1.5 fwhere f 9sxd − 0g and a vertical tangent when x − 0 fwhere f 9sxd is undefinedg. 3.5
_0.5
5
f 9sxd − x 3y5s21d 1 s4 2 xd(53 x22y5) − 2x 3y5 1
−
3s4 2 xd 5x 2 y5
25x 1 3s4 2 xd 12 2 8x − 5x 2y5 5x 2y5
[The same result could be obtained by first writing f sxd − 4x 3y5 2 x 8y5.] Therefore f 9sxd − 0 if 12 2 8x − 0, that is, x − 32, and f 9sxd does not exist when x − 0. Thus the critical numbers are 32 and 0. n In terms of critical numbers, Fermat’s Theorem can be rephrased as follows (compare Definition 6 with Theorem 4):
_2
Figure 14
7 If f has a local maximum or minimum at c, then c is a critical number of f.
To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local [in which case it occurs at a critical number by (7)] or it occurs at an endpoint of the interval, as we see from the examples in Figure 8. Thus the following threestep procedure always works.
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Section 3.1 Maximum and Minimum Values
209
The Closed Interval Method To find the absolute maximum and minimum values of a continuous function f on a closed interval fa, bg: 1. Find the values of f at the critical numbers of f in sa, bd. 2. Find the values of f at the endpoints of the interval. 3. The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.
Example 8 Find the absolute maximum and minimum values of the function f sxd − x 3 2 3x 2 1 1 212 < x < 4
f
g
SOLUTION Since f is continuous on 2 12 , 4 , we can use the Closed Interval Method:
f sxd − x 3 2 3x 2 1 1 f 9sxd − 3x 2 2 6x − 3xsx 2 2d Since f 9sxd exists for all x, the only critical numbers of f occur when f 9sxd − 0, that is, x − 0 or x − 2. Notice that each of these critical numbers lies in the interval s212 , 4d. The values of f at these critical numbers are
y 20
y=˛3≈+1 (4, 17)
15 10 5 _1 0 _5
f s0d − 1 f s2d − 23 The values of f at the endpoints of the interval are
1
f s212 d − 18 f s4d − 17
2 (2, _3)
3
x
4
FIGURE 15
Comparing these four numbers, we see that the absolute maximum value is f s4d − 17 and the absolute minimum value is f s2d − 23. Note that in this example the absolute maximum occurs at an endpoint, whereas the absolute minimum occurs at a critical number. The graph of f is sketched in Figure 15. n
If you have a graphing calculator or a computer with graphing software, it is possible to estimate maximum and minimum values very easily. But, as the next example shows, calculus is needed to find the exact values.
Example 9 (a) Use a graphing device to estimate the absolute minimum and maximum values of the function f sxd − x 2 2 sin x, 0 < x < 2. (b) Use calculus to find the exact minimum and maximum values.
8
0 _1
FIGURE 16
2π
SOLUTION (a) Figure 16 shows a graph of f in the viewing rectangle f0, 2g by f21, 8g. By moving the cursor close to the maximum point, we see that the ycoordinates don’t change very much in the vicinity of the maximum. The absolute maximum value is about 6.97 and it occurs when x < 5.2. Similarly, by moving the cursor close to the minimum point, we see that the absolute minimum value is about 20.68 and it occurs when x < 1.0. It is possible to get more accurate estimates by zooming in toward the
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210
Chapter 3 Applications of Differentiation
maximum and minimum points (or using a builtin maximum or minimum feature), but instead let’s use calculus. (b) The function f sxd − x 2 2 sin x is continuous on f0, 2g. Since f 9sxd − 1 2 2 cos x, we have f 9sxd − 0 when cos x − 12 and this occurs when x − y3 or 5y3. The values of f at these critical numbers are and
f sy3d − f s5y3d −
2 2 sin − 2 s3 < 20.684853 3 3 3 5 5 5 2 2 sin − 1 s3 < 6.968039 3 3 3
The values of f at the endpoints are f s0d − 0 and f s2d − 2 < 6.28 Comparing these four numbers and using the Closed Interval Method, we see that the absolute minimum value is f sy3d − y3 2 s3 and the absolute maximum value is f s5y3d − 5y3 1 s3 . The values from part (a) serve as a check on our work. n
Example 10 The Hubble Space Telescope was deployed on April 24, 1990, by the space shuttle Discovery. A model for the velocity of the shuttle during this mission, from liftoff at t − 0 until the solid rocket boosters were jettisoned at t − 126 seconds, is given by vstd − 0.001302t 3 2 0.09029t 2 1 23.61t 2 3.083
NASA
(in feet per second). Using this model, estimate the absolute maximum and minimum values of the acceleration of the shuttle between liftoff and the jettisoning of the boosters. SOLUTION We are asked for the extreme values not of the given velocity function, but rather of the acceleration function. So we first need to differentiate to find the acceleration:
astd − v9std −
d s0.001302t 3 2 0.09029t 2 1 23.61t 2 3.083d dt
− 0.003906t 2 2 0.18058t 1 23.61 We now apply the Closed Interval Method to the continuous function a on the interval 0 < t < 126. Its derivative is a9std − 0.007812t 2 0.18058 The only critical number occurs when a9std − 0: t1 −
0.18058 < 23.12 0.007812
Evaluating astd at the critical number and at the endpoints, we have as0d − 23.61 ast1 d < 21.52 as126d < 62.87 So the maximum acceleration is about 62.87 ftys2 and the minimum acceleration is about 21.52 ftys2.
n
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
211
Section 3.1 Maximum and Minimum Values
1. Explain the difference between an absolute minimum and a local minimum.
2. Suppose f is a continuous function defined on a closed interval fa, bg. (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for f ? (b) What steps would you take to find those maximum and minimum values?
12. (a) Sketch the graph of a function on [21, 2] that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on [21, 2] that has a local maximum but no absolute maximum.
3–4 For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum. 3.
4.
y
0 a b
c d
r
s x
y
0
a
b
c d
r
s x
5–6 Use the graph to state the absolute and local maximum and minimum values of the function. 6. y
5. y
0
14. (a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. (b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers. 15–28 Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) 15. f sxd − 12 s3x 2 1d, x < 3 16. f sxd − 2 2 13 x, x > 22 17. f sxd − 1yx, x > 1
19. f sxd − sin x, 0 < x , y2 20. f sxd − sin x, 0 , x < y2
y=ƒ 1
13. (a) Sketch the graph of a function on [21, 2] that has an absolute maximum but no absolute minimum. (b) Sketch the graph of a function on [21, 2] that is discontinuous but has both an absolute maximum and an absolute minimum.
18. f sxd − 1yx, 1 , x , 3
y=©
1
(c) Sketch the graph of a function that has a local maximum at 2 and is not continuous at 2.
21. f sxd − sin x, 2y2 < x < y2
1
x
0
1
x
22. f std − cos t, 23y2 < t < 3y2 23. f sxd − 1 1 sx 1 1d 2, 22 < x , 5
7–10 Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. 7. Absolute maximum at 5, absolute minimum at 2, local maximum at 3, local minima at 2 and 4 8. Absolute maximum at 4, absolute minimum at 5, local maximum at 2, local minimum at 3 9. Absolute minimum at 3, absolute maximum at 4, local maximum at 2 10. Absolute maximum at 2, absolute minimum at 5, 4 is a critical number but there is no local maximum or minimum there. 11. (a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2. (b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2.
 
24. f sxd − x
25. f sxd − 1 2 sx 26. f sxd − 1 2 x 3 27. f sxd −
28. f sxd −
H H
x2 if 21 < x < 0 2 2 3x if 0 , x < 1 2x 1 1 if 0 < x , 1 4 2 2x if 1 < x < 3
29–42 Find the critical numbers of the function. f sxd − x 3 1 6x 2 2 15x 29. f sxd − 4 1 13 x 2 12 x 2 30. 31. f sxd − 2x 3 2 3x 2 2 36x 32. f sxd − 2x 3 1 x 2 1 2x

33. tstd − t 4 1 t 3 1 t 2 1 1 34. tstd − 3t 2 4
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

212
Chapter 3 Applications of Differentiation
35. tsyd −
y21 p21 36. hs pd − 2 y2 2 y 1 1 p 14
60. f sxd − x 4 2 3x 3 1 3x 2 2 x, 0 < x < 2 61. f sxd − x sx 2 x 2
3 tsxd − s 4 2 x2 37. hstd − t 3y4 2 2 t 1y4 38.
62. f sxd − x 2 2 cos x, 22 < x < 0
39. Fsxd − x 4y5sx 2 4d 2 40. tsd − 4 2 tan tsxd − s1 2 x 2 41. f sd − 2 cos 1 sin2 42.
63. Between 0°C and 30°C, the volume V (in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula
; 43–44 A formula for the derivative of a function f is given. How many critical numbers does f have? 43. f 9sxd − 1 1
V − 999.87 2 0.06426T 1 0.0085043T 2 2 0.0000679T 3
210 sin x x 2 2 6x 1 10
Find the temperature at which water has its maximum density.
100 cos 2 x 21 10 1 x 2
44. f 9sxd −
45–56 Find the absolute maximum and absolute minimum values of f on the given interval.
where is a positive constant called the coefficient of friction and where 0 < < y2. Show that F is minimized when tan − .
45. f sxd − 12 1 4x 2 x 2, f0, 5g 46. f sxd − 5 1 54x 2 2x 3, f0, 4g 47. f sxd − 2x 3 2 3x 2 2 12x 1 1, f22, 3g
65. The water level, measured in feet above mean sea level, of Lake Lanier in Georgia, USA, during 2012 can be modeled by the function
48. f sxd − x 2 6x 1 5, f23, 5g 3
2
49. f sxd − 3x 4 2 4x 3 2 12x 2 1 1, f22, 3g
Lstd − 0.01441t 3 2 0.4177t 2 1 2.703t 1 1060.1
50. f std − st 2 2 4d 3, f22, 3g 1 , f0.2, 4g x x 52. f sxd − 2 , f0, 3g x 2x11
where t is measured in months since January 1, 2012. Estimate when the water level was highest during 2012.
51. f sxd − x 1
3 53. f std − t 2 s t , f21, 4g
54. f std −
st , f0, 2g 1 1 t2
55. f std − 2 cos t 1 sin 2t, f0, y2g 56. f std − t 1 cot sty2d, fy4, 7y4g
; 66. On May 7, 1992, the space shuttle Endeavour was launched on mission STS49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. (a) Use a graphing calculator or computer to find the cubic polynomial that best models the velocity of the shuttle for the time interval t [ f0, 125g. Then graph this polynomial. (b) Find a model for the acceleration of the shuttle and use it to estimate the maximum and minimum values of the acceleration during the first 125 seconds.
57. If a and b are positive numbers, find the maximum value of f sxd − x as1 2 xd b, 0 < x < 1. ; 58. Use a graph to estimate the critical numbers of f sxd − 1 1 5x 2 x 3 correct to one decimal place.


; 59–62 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. 59. f sxd − x 5 2 x 3 1 2, 21 < x < 1
64. An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is W F− sin 1 cos
Event
Time (s)
Velocity (ftys)
Launch Begin roll maneuver End roll maneuver Throttle to 89% Throttle to 67% Throttle to 104% Maximum dynamic pressure Solid rocket booster separation
0 10 15 20 32 59 62 125
0 185 319 447 742 1325 1445 4151
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Applied Project The Calculus of Rainbows
67. When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the airstream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about twothirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity v of the airstream is related to the radius r of the trachea by the equation
where k is a constant and r0 is the normal radius of the trachea. The restriction on r is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than 12 r0 is prevented (otherwise the person would suffocate). (a) Determine the value of r in the interval 12 r0 , r0 at which v has an absolute maximum. How does this compare with experimental evidence? (b) What is the absolute maximum value of v on the interval? (c) Sketch the graph of v on the interval f0, r0 g.
f
68. Show that 5 is a critical number of the function tsxd − 2 1 sx 2 5d 3 but t does not have a local extreme value at 5. 69. Prove that the function f sxd − x 101 1 x 51 1 x 1 1 has neither a local maximum nor a local minimum. 70. If f has a local minimum value at c, show that the function tsxd − 2f sxd has a local maximum value at c.
vsrd − ksr0 2 rdr 2 12 r0 < r < r0
213
g
71. Prove Fermat’s Theorem for the case in which f has a local minimum at c. 72. A cubic function is a polynomial of degree 3; that is, it has the form f sxd − ax 3 1 bx 2 1 cx 1 d, where a ± 0. (a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities. (b) How many local extreme values can a cubic function have?
The calculus of rainbows
APPLIED Project
Rainbows are created when raindrops scatter sunlight. They have fascinated mankind since ancient times and have inspired attempts at scientific explanation since the time of Aristotle. In this project we use the ideas of Descartes and Newton to explain the shape, location, and colors of rainbows. å A from sun
∫
∫
O
B ∫
D(å )
∫
å to observer
C
Formation of the primary rainbow
1. The figure shows a ray of sunlight entering a spherical raindrop at A. Some of the light is reflected, but the line AB shows the path of the part that enters the drop. Notice that the light is refracted toward the normal line AO and in fact Snell’s Law says that sin − k sin , where is the angle of incidence, is the angle of refraction, and k < 43 is the index of refraction for water. At B some of the light passes through the drop and is refracted into the air, but the line BC shows the part that is reflected. (The angle of incidence equals the angle of reflection.) When the ray reaches C, part of it is reflected, but for the time being we are more interested in the part that leaves the raindrop at C. (Notice that it is refracted away from the normal line.) The angle of deviation Dsd is the amount of clockwise rotation that the ray has undergone during this threestage process. Thus Dsd − s 2 d 1 s 2 2d 1 s 2 d − 1 2 2 4 Show that the minimum value of the deviation is Dsd < 1388 and occurs when < 59.48.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
214
Chapter 3 Applications of Differentiation
The significance of the minimum deviation is that when < 59.48 we have D9sd < 0, so DDyD < 0. This means that many rays with < 59.48 become deviated by approximately the same amount. It is the concentration of rays coming from near the direction of minimum deviation that creates the brightness of the primary rainbow. The figure at the left shows that the angle of elevation from the observer up to the highest point on the rainbow is 180 8 2 1388 − 428. (This angle is called the rainbow angle.)
rays from sun
138° rays from sun
42°
observer
C
laboratory Project ∫ D ∫
∫
å
to observer from sun
å
∫
∫
∫
A
Formation of the secondary rainbow
2. Problem 1 explains the location of the primary rainbow, but how do we explain the colors? Sunlight comprises a range of wavelengths, from the red range through orange, yellow, green, blue, indigo, and violet. As Newton discovered in his prism experiments of 1666, the index of refraction is different for each color. (The effect is called dispersion.) For red light the refractive index is k < 1.3318, whereas for violet light it is k < 1.3435. By repeating the calculation of Problem 1 for these values of k, show that the rainbow angle is about 42.38 for the red bow and 40.68 for the violet bow. So the rainbow really consists of seven bows corresponding to the seven colors. Theindividual Paradox
3. PerhapsFigure you have seen a data fainter secondary rainbow above the primary bow. That Bimportant. 1 displays from an observational study that clearly depicts thisresults trend. from the part of a ray that enters a raindrop and is refracted at A, reflected twice (at B and C), andthe refracted it leaves thecorresponds drop at D (see theinitial figureexpectationt. at the left). This time the devia 1. Draw causal as diagram that to the tion angle Dsd is the total amount of counterclockwise rotation that the ray undergoes in 2. Suppose. this fourstage process. Show that B 3. Suppose. Dsd − 2 2 6 1 2 and Dsd has a minimum value when cos −
Î
k2 2 1 8
Taking k − 43, show that the minimum deviation is about 1298 and so the rainbow angle for the secondary rainbow is about 518, as shown in the figure at the left. 4. Show that the colors in the secondary rainbow appear in the opposite order from those in the primary rainbow.
© Pichugin Dmitry / Shutterstock.com
42° 51°
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
215
Section 3.2 The Mean Value Theorem
We will see that many of the results of this chapter depend on one central fact, which is called the Mean Value Theorem. But to arrive at the Mean Value Theorem we first need the following result. Rolle’s Theorem Let f be a function that satisfies the following three hypotheses: 1. f is continuous on the closed interval fa, bg. 2. f is differentiable on the open interval sa, bd. 3. f sad − f sbd Then there is a number c in sa, bd such that f 9scd − 0.
Rolle Rolle’s Theorem was first published in 1691 by the French mathematician Michel Rolle (1652–1719) in a book entitled Méthode pour resoudre les Egalitez. He was a vocal critic of the methods of his day and attacked calculus as being a “collection of ingenious fallacies.” Later, however, he became convinced of the essential correctness of the methods of calculus.
y
0
Before giving the proof let’s take a look at the graphs of some typical functions that satisfy the three hypotheses. Figure 1 shows the graphs of four such functions. In each case it appears that there is at least one point sc, f scdd on the graph where the tangent is horizontal and therefore f 9scd − 0. Thus Rolle’s Theorem is plausible. y
a
c¡
c™ b
(a)
FIGURE 1
x
0
y
y
a
c
b
x
(b)
0
a
c¡
c™
b
x
0
a
(c)
c
b
x
(d)
Proof There are three cases: PS Take cases
CASE I f sxd − k, a constant Then f 9sxd − 0, so the number c can be taken to be any number in sa, bd. CASE II f sxd . f sad for some x in sa, bd [as in Figure 1(b) or (c)] y the Extreme Value Theorem (which we can apply by hypothesis 1), f has a maxiB
mum value somewhere in fa, bg. Since f sad − f sbd, it must attain this maximum value at a number c in the open interval sa, bd. Then f has a local maximum at c and, by hypothesis 2, f is differentiable at c. Therefore f 9scd − 0 by Fermat’s Theorem. CASE III f sxd , f sad for some x in sa, bd [as in Figure 1(c) or (d)] By the Extreme Value Theorem, f has a minimum value in fa, bg and, since f sad − f sbd, it attains this minimum value at a number c in sa, bd. Again f 9scd − 0 by Fermat’s Theorem. n
Example 1 Let’s apply Rolle’s Theorem to the position function s − f std of a moving object. If the object is in the same place at two different instants t − a and t − b, then f sad − f sbd. Rolle’s Theorem says that there is some instant of time t − c between a and b when f 9scd − 0; that is, the velocity is 0. (In particular, you can see that this is true when a ball is thrown directly upward.) n Example 2 Prove that the equation x 3 1 x 2 1 − 0 has exactly one real root. SOLUTION First we use the Intermediate Value Theorem (1.8.10) to show that a root exists. Let f sxd − x 3 1 x 2 1. Then f s0d − 21 , 0 and f s1d − 1 . 0. Since f is a Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
216
Chapter 3 Applications of Differentiation
Figure 2 shows a graph of the function f sxd − x 3 1 x 2 1 discussed in Example 2. Rolle’s Theorem shows that, no matter how much we enlarge the viewing rectangle, we can never find a second xintercept. 3
_2
polynomial, it is continuous, so the Intermediate Value Theorem states that there is a number c between 0 and 1 such that f scd − 0. Thus the given equation has a root. To show that the equation has no other real root, we use Rolle’s Theorem and argue by contradiction. Suppose that it had two roots a and b. Then f sad − 0 − f sbd and, since f is a polynomial, it is differentiable on sa, bd and continuous on fa, bg. Thus, by Rolle’s Theorem, there is a number c between a and b such that f 9scd − 0. But f 9sxd − 3x 2 1 1 > 1 for all x
2
(since x 2 > 0) so f 9sxd can never be 0. This gives a contradiction. Therefore the equation can’t have two real roots. n Our main use of Rolle’s Theorem is in proving the following important theorem, which was first stated by another French mathematician, JosephLouis Lagrange.
_3
FIGURE 2
The Mean Value Theorem Let f be a function that satisfies the following hypotheses: 1. f is continuous on the closed interval fa, bg.
The Mean Value Theorem is an example of what is called an existence theorem. Like the Intermediate Value Theorem, the Extreme Value Theorem, and Rolle’s Theorem, it guarantees that there exists a number with a certain property, but it doesn’t tell us how to find the number.
2. f is differentiable on the open interval sa, bd. Then there is a number c in sa, bd such that 1
f 9scd −
or, equivalently, 2
f sbd 2 f sad b2a
f sbd 2 f sad − f 9scdsb 2 ad
Before proving this theorem, we can see that it is reasonable by interpreting it geomet rically. Figures 3 and 4 show the points Asa, f sadd and Bsb, f sbdd on the graphs of two differentiable functions. The slope of the secant line AB is 3
mAB −
f sbd 2 f sad b2a
which is the same expression as on the right side of Equation 1. Since f 9scd is the slope of the tangent line at the point sc, f scdd, the Mean Value Theorem, in the form given by Equa tion 1, says that there is at least one point Psc, f scdd on the graph where the slope of the tangent line is the same as the slope of the secant line AB. In other words, there is a point P where the tangent line is parallel to the secant line AB. (Imagine a line far away that stays parallel to AB while moving toward AB until it touches the graph for the first time.) y
y
P { c, f(c)}
P¡
B
P™
A
A{ a, f(a)} B { b, f(b)} 0
FIGURE 3
a
c
b
x
0
a
c¡
c™
b
x
FIGURE 4
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 3.2 The Mean Value Theorem
y
h(x)
A ƒ
0
a
y=ƒ
B x
f(a)+
b
f(b)f(a) (xa) ba
x
Proof We apply Rolle’s Theorem to a new function h defined as the difference between f and the function whose graph is the secant line AB. Using Equation 3 and the pointslope equation of a line, we see that the equation of the line AB can be written as f sbd 2 f sad y 2 f sad − sx 2 ad b2a
y − f sad 1
or as
f sbd 2 f sad sx 2 ad b2a
So, as shown in Figure 5,
figure 5
4 Lagrange and the Mean Value Theorem The Mean Value Theorem was first formulated by JosephLouis Lagrange (1736–1813), born in Italy of a French father and an Italian mother. He was a child prodigy and became a professor in Turin at the tender age of 19. Lagrange made great contributions to number theory, theory of functions, theory of equations, and analytical and celestial mechanics. In particular, he applied calculus to the analysis of the stability of the solar system. At the invitation of Frederick the Great, he succeeded Euler at the Berlin Academy and, when Frederick died, Lagrange accepted King Louis XVI’s invitation to Paris, where he was given apartments in the Louvre and became a professor at the Ecole Polytechnique. Despite all the trappings of luxury and fame, he was a kind and quiet man, living only for science.
217
hsxd − f sxd 2 f sad 2
f sbd 2 f sad sx 2 ad b2a
First we must verify that h satisfies the three hypotheses of Rolle’s Theorem. 1. The function h is continuous on fa, bg because it is the sum of f and a firstdegree polynomial, both of which are continuous. 2. The function h is differentiable on sa, bd because both f and the firstdegree polynomial are differentiable. In fact, we can compute h9 directly from Equation 4: h9sxd − f 9sxd 2
f sbd 2 f sad b2a
(Note that f sad and f f sbd 2 f sadgysb 2 ad are constants.) 3.
hsad − f sad 2 f sad 2
f sbd 2 f sad sa 2 ad − 0 b2a
hsbd − f sbd 2 f sad 2
f sbd 2 f sad sb 2 ad b2a
− f sbd 2 f sad 2 f f sbd 2 f sadg − 0
Therefore hsad − hsbd. Since h satisfies the hypotheses of Rolle’s Theorem, that theorem says there is a number c in sa, bd such that h9scd − 0. Therefore 0 − h9scd − f 9scd 2 and so
f 9scd −
f sbd 2 f sad b2a
f sbd 2 f sad b2a
n
Example 3 To illustrate the Mean Value Theorem with a specific function, let’s consider f sxd − x 3 2 x, a − 0, b − 2. Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on f0, 2g and differentiable on s0, 2d. Therefore, by the Mean Value Theorem, there is a number c in s0, 2d such that f s2d 2 f s0d − f 9scds2 2 0d Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
218
Chapter 3 Applications of Differentiation y
y=˛ x B
Now f s2d − 6, f s0d − 0, and f 9sxd − 3x 2 2 1, so this equation becomes 6 − s3c 2 2 1d2 − 6c 2 2 2
O c
FIGURE 6
2
x
which gives c 2 − 43, that is, c − 62ys3 . But c must lie in s0, 2d, so c − 2ys3 . Figure 6 illustrates this calculation: The tangent line at this value of c is parallel to the secant line OB. n
Example 4 If an object moves in a straight line with position function s − f std, then the average velocity between t − a and t − b is f sbd 2 f sad b2a and the velocity at t − c is f 9scd. Thus the Mean Value Theorem (in the form of Equation 1) tells us that at some time t − c between a and b the instantaneous velocity f 9scd is equal to that average velocity. For instance, if a car traveled 180 km in 2 hours, then the speedometer must have read 90 kmyh at least once. In general, the Mean Value Theorem can be interpreted as saying that there is a number at which the instantaneous rate of change is equal to the average rate of change over an interval. n The main significance of the Mean Value Theorem is that it enables us to obtain information about a function from information about its derivative. The next example provides an instance of this principle.
Example 5 Suppose that f s0d − 23 and f 9sxd < 5 for all values of x. How large can f s2d possibly be?
SOLUTION We are given that f is differentiable (and therefore continuous) everywhere. In particular, we can apply the Mean Value Theorem on the interval f0, 2g. There exists a number c such that
f s2d 2 f s0d − f 9scds2 2 0d so
f s2d − f s0d 1 2f 9scd − 23 1 2f 9scd
We are given that f 9sxd < 5 for all x, so in particular we know that f 9scd < 5. Multiplying both sides of this inequality by 2, we have 2f 9scd < 10, so f s2d − 23 1 2f 9scd < 23 1 10 − 7 The largest possible value for f s2d is 7.
n
The Mean Value Theorem can be used to establish some of the basic facts of differential calculus. One of these basic facts is the following theorem. Others will be found in the following sections. 5 Theorem If f 9sxd − 0 for all x in an interval sa, bd, then f is constant on sa, bd. Proof Let x 1 and x 2 be any two numbers in sa, bd with x 1 , x 2. Since f is differentiable on sa, bd, it must be differentiable on sx 1, x 2 d and continuous on fx 1, x 2 g. By Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 3.2 The Mean Value Theorem
219
applying the Mean Value Theorem to f on the interval fx 1, x 2 g, we get a number c such that x 1 , c , x 2 and f sx 2 d 2 f sx 1d − f 9scdsx 2 2 x 1d
6
Since f 9sxd − 0 for all x, we have f 9scd − 0, and so Equation 6 becomes f sx 2 d 2 f sx 1 d − 0 or f sx 2 d − f sx 1 d Therefore f has the same value at any two numbers x 1 and x 2 in sa, bd. This means that f is constant on sa, bd. n Corollary 7 says that if two functions have the same derivatives on an interval, then their graphs must be vertical translations of each other there. In other words, the graphs have the same shape, but could be shifted up or down.
7 Corollary If f 9sxd − t9sxd for all x in an interval sa, bd, then f 2 t is constant on sa, bd; that is, f sxd − tsxd 1 c where c is a constant. Proof Let Fsxd − f sxd 2 tsxd. Then
F9sxd − f 9sxd 2 t9sxd − 0 for all x in sa, bd. Thus, by Theorem 5, F is constant; that is, f 2 t is constant.
n
NOTE Care must be taken in applying Theorem 5. Let f sxd −
H
x 1 if x . 0 − x 21 if x , 0
 

The domain of f is D − hx x ± 0j and f 9sxd − 0 for all x in D. But f is obviously not a constant function. This does not contradict Theorem 5 because D is not an interval. Notice that f is constant on the interval s0, `d and also on the interval s2`, 0d.
1. The graph of a function f is shown. Verify that f satisfies the hypotheses of Rolle’s Theorem on the interval f0, 8g. Then estimate the value(s) of c that satisfy the conclusion of Rolle’s Theorem on that interval. y
3. The graph of a function t is shown. y y=©
y=ƒ 1 0
1 0
1
x
2. Draw the graph of a function defined on f0, 8g such that f s0d − f s8d − 3 and the function does not satisfy the conclusion of Rolle’s Theorem on f0, 8g.
1
x
(a) Verify that t satisfies the hypotheses of the Mean Value Theorem on the interval f0, 8g. (b) Estimate the value(s) of c that satisfy the conclusion of the Mean Value Theorem on the interval f0, 8g. (c) Estimate the value(s) of c that satisfy the conclusion of the Mean Value Theorem on the interval f2, 6g.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
220
Chapter 3 Applications of Differentiation
4. Draw the graph of a function that is continuous on f0, 8g where f s0d − 1 and f s8d − 4 and that does not satisfy the conclusion of the Mean Value Theorem on f0, 8g. 5–8 Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem. 5. f sxd − 2 x 2 2 4 x 1 5, f21, 3g 6. f sxd − x 3 2 2x 2 2 4x 1 2, f22, 2g
g
8. f sxd − x 1 1y x, 12 , 2
9. Let f sxd − 1 2 x 2y3. Show that f s21d − f s1d but there is no number c in s21, 1d such that f 9scd − 0. Why does this not contradict Rolle’s Theorem? 10. Let f sxd − tan x. Show that f s0d − f sd but there is no number c in s0, d such that f 9scd − 0. Why does this not contradict Rolle’s Theorem? 11–14 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all num bers c that satisfy the conclusion of the Mean Value Theorem. 11. f sxd − 2x 2 2 3x 1 1, f0, 2g 12. f sxd − x 3 2 3x 1 2, f22, 2g 3 13. f sxd − s x , f0, 1g
24. (a) Suppose that f is differentiable on R and has two roots. Show that f 9 has at least one root. (b) Suppose f is twice differentiable on R and has three roots. Show that f 0 has at least one real root. (c) Can you generalize parts (a) and (b)? 25. If f s1d − 10 and f 9sxd > 2 for 1 < x < 4, how small can f s4d possibly be?
7. f sxd − sins xy2d, fy2, 3y2g
f
23. (a) Show that a polynomial of degree 3 has at most three real roots. (b) Show that a polynomial of degree n has at most n real roots.
14. f sxd − 1yx, f1, 3g
; 15–16 Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at sc, f scdd. Are the secant line and the tangent line parallel?
26. Suppose that 3 < f 9sxd < 5 for all values of x. Show that 18 < f s8d 2 f s2d < 30. 27. Does there exist a function f such that f s0d − 21, f s2d − 4, and f 9sxd < 2 for all x? 28. Suppose that f and t are continuous on fa, bg and differentiable on sa, bd. Suppose also that f sad − tsad and f 9sxd , t9sxd for a , x , b. Prove that f sbd , tsbd. [Hint: Apply the Mean Value Theorem to the function h − f 2 t.] 29. Show that sin x , x if 0 , x , 2. 30. Suppose f is an odd function and is differentiable everywhere. Prove that for every positive number b, there exists a number c in s2b, bd such that f 9scd − f sbdyb. 31. Use the Mean Value Theorem to prove the inequality
 sin a 2 sin b  <  a 2 b  for all a and b 32. If f 9sxd − c (c a constant) for all x, use Corollary 7 to show that f sxd − cx 1 d for some constant d. 33. Let f sxd − 1yx and
15. f sxd − sx , f0, 4g 16. f sxd − x 3 2 2x, f22, 2g 17. Let f sxd − s x 2 3d22. Show that there is no value of c in s1, 4d such that f s4d 2 f s1d − f 9scds4 2 1d. Why does this not contradict the Mean Value Theorem?


18. Let f sxd − 2 2 2 x 2 1 . Show that there is no value of c such that f s3d 2 f s0d − f 9scds3 2 0d. Why does this not contradict the Mean Value Theorem? 19–20 Show that the equation has exactly one real root. 19. 2 x 1 cos x − 0 20. 2x 2 1 2 sin x − 0 21. Show that the equation x 3 2 15x 1 c − 0 has at most one root in the interval f22, 2g. 22. Show that the equation x 4 1 4x 1 c − 0 has at most two real roots.
tsxd −
1 x
1 11 x
if x . 0 if x , 0
Show that f 9sxd − t9sxd for all x in their domains. Can we conclude from Corollary 7 that f 2 t is constant? 34. At 2:00 pm a car’s speedometer reads 30 miyh. At 2:10 pm it reads 50 miyh. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120 miyh2. 35. Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider f std − tstd 2 hstd, where t and h are the position functions of the two runners.] 36. A number a is called a fixed point of a function f if f sad − a. Prove that if f 9sxd ± 1 for all real numbers x, then f has at most one fixed point.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 3.3 How Derivatives Affect the Shape of a Graph
y
221
Many of the applications of calculus depend on our ability to deduce facts about a function f from information concerning its derivatives. Because f 9sxd represents the slope of the curve y − f sxd at the point sx, f sxdd, it tells us the direction in which the curve proceeds at each point. So it is reasonable to expect that information about f 9sxd will provide us with information about f sxd.
D B
What Does f 9 Say About f ? C
A
x
0
FIGURE 1
To see how the derivative of f can tell us where a function is increasing or decreasing, look at Figure 1. (Increasing functions and decreasing functions were defined in Section 1.1.) Between A and B and between C and D, the tangent lines have positive slope and so f 9sxd . 0. Between B and C, the tangent lines have negative slope and so f 9sxd , 0. Thus it appears that f increases when f 9sxd is positive and decreases when f 9sxd is negative. To prove that this is always the case, we use the Mean Value Theorem. Increasing/Decreasing Test
Let’s abbreviate the name of this test to the I/D Test.
(a) If f 9sxd . 0 on an interval, then f is increasing on that interval. (b) If f 9sxd , 0 on an interval, then f is decreasing on that interval. Proof
(a) Let x 1 and x 2 be any two numbers in the interval with x1 , x2. According to the definition of an increasing function (page 19), we have to show that f sx1 d , f sx2 d. Because we are given that f 9sxd . 0, we know that f is differentiable on fx1, x2 g. So, by the Mean Value Theorem, there is a number c between x1 and x2 such that 1
f sx 2 d 2 f sx 1 d − f 9scdsx 2 2 x 1 d
Now f 9scd . 0 by assumption and x 2 2 x 1 . 0 because x 1 , x 2. Thus the right side of Equation 1 is positive, and so f sx 2 d 2 f sx 1 d . 0 or f sx 1 d , f sx 2 d This shows that f is increasing. Part (b) is proved similarly.
n
Example 1 Find where the function f sxd − 3x 4 2 4x 3 2 12x 2 1 5 is increasing and where it is decreasing. SOLUTION We start by differentiating f :
f 9sxd − 12x 3 2 12x 2 2 24x − 12xsx 2 2dsx 1 1d
_1
0
2
x
To use the IyD Test we have to know where f 9sxd . 0 and where f 9sxd , 0. To solve these inequalities we first find where f 9sxd − 0, namely, at x − 0, 2, and 21. These are the critical numbers of f , and they divide the domain into four intervals (see the number line at the left). Within each interval, f 9sxd must be always positive or always negative. (See Examples 3 and 4 in Appendix A.) We can determine which is the case for each interval from the signs of the three factors of f 9sxd, namely, 12x, x 2 2, and x 1 1, as shown in the following chart. A plus sign indicates that the given expression is positive, and a minus sign indicates that it is negative. The last col
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
222
Chapter 3 Applications of Differentiation
umn of the chart gives the conclusion based on the IyD Test. For instance, f 9sxd , 0 for 0 , x , 2, so f is decreasing on (0, 2). (It would also be true to say that f is decreasing on the closed interval f0, 2g.)
20
Interval _2
3
x22
x11
f 9sxd
f
2
2
2
2
decreasing on (2`, 21)
21 , x , 0
2
2
1
1
increasing on (21, 0)
0,x,2 x.2
1 1
2 1
1 1
2 1
decreasing on (0, 2) increasing on (2, `)
x , 21
_30
FIGURE 2
12x
The graph of f shown in Figure 2 confirms the information in the chart.
n
Local Extreme Values Recall from Section 3.1 that if f has a local maximum or minimum at c, then c must be a critical number of f (by Fermat’s Theorem), but not every critical number gives rise to a maximum or a minimum. We therefore need a test that will tell us whether or not f has a local maximum or minimum at a critical number. You can see from Figure 2 that f s0d − 5 is a local maximum value of f because f increases on s21, 0d and decreases on s0, 2d. Or, in terms of derivatives, f 9sxd . 0 for 21 , x , 0 and f 9sxd , 0 for 0 , x , 2. In other words, the sign of f 9sxd changes from positive to negative at 0. This observation is the basis of the following test. The First Derivative Test Suppose that c is a critical number of a continuous function f. (a) If f 9 changes from positive to negative at c, then f has a local maximum at c. (b) If f 9 changes from negative to positive at c, then f has a local minimum at c. (c) If f 9 is positive to the left and right of c, or negative to the left and right of c, then f has no local maximum or minimum at c. The First Derivative Test is a consequence of the IyD Test. In part (a), for instance, since the sign of f 9sxd changes from positive to negative at c, f is increasing to the left of c and decreasing to the right of c. It follows that f has a local maximum at c. It is easy to remember the First Derivative Test by visualizing diagrams such as those in Figure 3. y
y
fª(x)>0
y
fª(x) 0 for all x, we have f 0sxd , 0 for x , 0 and for 0 , x , 6 and f 0sxd . 0 for x . 6. So f is concave downward on s2`, 0d and s0, 6d and concave upward on s6, `d, and the only inflection point is s6, 0d. The graph is sketched in Figure 12. Note that the curve has vertical tangents at s0, 0d and s6, 0d because f 9sxd l ` as x l 0 and as x l 6.
Try reproducing the graph in Figure 12 with a graphing calculator or computer. Some machines produce the complete graph, some produce only the portion to the right of the yaxis, and some produce only the portion between x − 0 and x − 6. For an explanation and cure, see Example 7 in “Graphing Calculators and Computers” at www.stewartcalculus.com. An equivalent expression that gives the correct graph is 62x y − sx 2 d1y3 ? 6 2 x 1y3 62x

42x

y 4
(4, 2%?# )
3
2 0


1
1–2 Use the given graph of f to find the following. (a) The open intervals on which f is increasing. (b) The open intervals on which f is decreasing. (c) The open intervals on which f is concave upward. (d) The open intervals on which f is concave downward. (e) The coordinates of the points of inflection.
1
x
0
1
3. Suppose you are given a formula for a function f. (a) How do you determine where f is increasing or decreasing?
3
4
7 x
5
n
(b) How do you determine where the graph of f is concave upward or concave downward? (c) How do you locate inflection points?
4. (a) State the First Derivative Test. (b) State the Second Derivative Test. Under what circum stances is it inconclusive? What do you do if it fails?
1
1 0
2. y
y
2
y=x @ ?#(6x)!?#
FIGURE 12
1.

x
5–6 The graph of the derivative f 9 of a function f is shown. (a) On what intervals is f increasing or decreasing? (b) At what values of x does f have a local maximum or minimum? 5. y y y=fª(x)
0
2
4
y=fª(x)
6
7et0403x05–06 09/10/09
x
0
2
4
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
6
x
228
Chapter 3 Applications of Differentiation
6. y
18. (a) Find the critical numbers of f sxd − x 4sx 2 1d3. (b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers? (c) What does the First Derivative Test tell you?
y=fª(x) 0
2
4
6
8
x
19. Suppose f 0 is continuous on s2`, `d. (a) If f 9s2d − 0 and f 0s2d − 25, what can you say about f ? (b) If f 9s6d − 0 and f 0s6d − 0, what can you say about f ?
7. In each part state the xcoordinates of the inflection points of f. Give reasons for your answers. (a) The curve is the graph of f. (b) The curve is the graph of f 9. (c) The curve is the graph of f 0. y
0
2
4
6
x
8
8. The graph of7et0403x07 the first derivative f 9 of a function f is shown. (a) On what09/10/09 intervals is f increasing? Explain. (b) At what values of x does f have a local maximum or MasterID: minimum? Explain. 00490 (c) On what intervals is f concave upward or concave down ward? Explain. (d) What are the xcoordinates of the inflection points of f ? Why? y
y=fª(x) 0
2
4
6
8
x
9–14 (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. 9. f sxd − x 3 2 3x 2 2 9x 1 4 10. f sxd − 2x 3 2 9x 2 1 12x 2 3
x 11. f sxd − x 4 2 2x 2 1 3 12. f sxd − 2 x 11 13. f sxd − sin x 1 cos x, 0 < x < 2 14. f sxd − cos2 x 2 2 sin x, 0 < x < 2 15–17 Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? x2 15. f sxd − 1 1 3x 2 2 2x 3 16. f sxd − x21 4 17. f sxd − sx 2 s x
20–27 Sketch the graph of a function that satisfies all of the given conditions. 20. (a) f 9sxd , 0 and f 0sxd , 0 for all x (b) f 9sxd . 0 and f 0sxd . 0 for all x 21. (a) f 9sxd . 0 and f 0sxd , 0 for all x (b) f 9sxd , 0 and f 0sxd . 0 for all x 22. Vertical asymptote x − 0, f 9sxd . 0 if x , 22, f 9sxd , 0 if x . 22 sx ± 0d, f 0sxd , 0 if x , 0, f 0sxd . 0 if x . 0 23.
f 9s0d − f 9s2d − f 9s4d − 0, f 9sxd . 0 if x , 0 or 2 , x , 4, f 9sxd , 0 if 0 , x , 2 or x . 4, f 0sxd . 0 if 1 , x , 3, f 0sxd , 0 if x , 1 or x . 3
24. f 9sxd . 0 for all x ± 1, vertical asymptote x − 1, f 99sxd . 0 if x , 1 or x . 3, f 99sxd , 0 if 1 , x , 3 25.
f 9s5d − 0, f 9sxd , 0 when x , 5, f 9sxd . 0 when x . 5, f 99s2d − 0, f 99s8d − 0, f 99sxd , 0 when x , 2 or x . 8, f 0sxd . 0 for 2 , x , 8
26.
f 9s0d − f 9s4d − 0, f 9sxd − 1 if x , 21, f 9sxd . 0 if 0 , x , 2, f 9sxd , 0 if 21 , x , 0 or 2 , x , 4 or x . 4, lim2 f 9sxd − `, lim1 f 9sxd − 2`, x l2
x l2
f 99sxd . 0 if 21 , x , 2 or 2 , x , 4, f 0sxd , 0 if x . 4 27.
f s0d − f 9s0d − f 9s2d − f 9s4d − f 9s6d − 0, f 9sxd . 0 if 0 , x , 2 or 4 , x , 6, f 9sxd , 0 if 2 , x , 4 or x . 6, f 0sxd . 0 if 0 , x , 1 or 3 , x , 5, f 0sxd , 0 if 1 , x , 3 or x . 5, f s2xd − f sxd
28. Suppose f s3d − 2, f 9s3d − 12, and f 9sxd . 0 and f 0sxd , 0 for all x. (a) Sketch a possible graph for f. (b) How many solutions does the equation f sxd − 0 have? Why? (c) Is it possible that f 9s2d − 13? Why? 29. Suppose f is a continuous function where f sxd . 0 for all x, f s0d − 4, f 9s xd . 0 if x , 0 or x . 2, f 9s xd , 0 if 0 , x , 2, f 99s21d − f 99s1d − 0, f 99s xd . 0 if x , 21 or x . 1, f 99s xd , 0 if 21 , x , 1. (a) Can f have an absolute maximum? If so, sketch a possible graph of f. If not, explain why.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 3.3 How Derivatives Affect the Shape of a Graph
(b) Can f have an absolute minimum? If so, sketch a possible graph of f. If not, explain why. (c) Sketch a possible graph for f that does not achieve an absolute minimum.
229
33–44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)–(c) to sketch the graph. Check your work with a graphing device if you have one.
30. T he graph of a function y − f sxd is shown. At which point(s) are the following true?
33. f sxd − x 3 2 12x 1 2
2
dy d y (a) and are both positive. dx dx 2 dy d 2y are both negative. (b) and dx dx 2 dy d 2y (c) is negative but is positive. dx dx 2 y C A
D
34. f sxd − 36x 1 3x 2 2 2x 3 tsxd − 200 1 8x 3 1 x 4 35. f sxd − 12 x 4 2 4x 2 1 3 36. 37. hsxd − sx 1 1d5 2 5x 2 2 38. hsxd − 5x 3 2 3x 5 39. Fsxd − x s6 2 x 40. G sxd − 5x 2y3 2 2x 5y3 41. Csxd − x 1y3sx 1 4d 42. f sxd − 2 sx 2 4x 2 E
43. f sd − 2 cos 1 cos 2 , 0 < < 2 44. Ssxd − x 2 sin x, 0 < x < 4
B
45. S uppose the derivative of a function f is f 9sxd − sx 1 1d2 sx 2 3d5 sx 2 6d 4. On what interval is f increasing?
x
0
31–32 The graph of the derivative f 9 of a continuous function f is shown. (a) On what intervals is f increasing? Decreasing? (b) At what values of x does f have a local maximum? Local minimum? (c) On what intervals is f concave upward? Concave downward? (d) State the xcoordinate(s) of the point(s) of inflection. (e) Assuming that f s0d − 0, sketch a graph of f. 31. y y=fª(x)
46. U se the methods of this section to sketch the curve y − x 3 2 3a 2x 1 2a 3, where a is a positive constant. What do the members of this family of curves have in common? How do they differ from each other? ; 47–48 (a) Use a graph of f to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of x at which f increases most rapidly. Then find the exact value. x11 47. f sxd − sx 2 1 1 48. f sxd − x 1 2 cos x, 0 < x < 2
2 0
2
4
6
8 x
_2
32.
; 49–50 (a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of f 0 to give better estimates. 49. f sxd − sin 2x 1 sin 4x, 0 < x <
y
50. f sxd − sx 2 1d2 sx 1 1d3 y=fª(x) CAS
2 0 _2
2
4
6
8 x
51–52 Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph f 0. 51. f sxd − 52. f sxd −
x4 1 x3 1 1 sx 2 1 x 1 1 sx 1 1d3sx 2 1 5d sx 3 1 1dsx 2 1 4d
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230
Chapter 3 Applications of Differentiation
53. A graph of a population of yeast cells in a new laboratory culture as a function of time is shown. (a) Describe how the rate of population increase varies. (b) When is this rate highest? (c) On what intervals is the population function concave upward or downward? (d) Estimate the coordinates of the inflection point.
f sxd − ax 3 1 bx 2 1 cx 1 d that has a local maximum value of 3 at x − 22 and a local minimum value of 0 at x − 1. 60. Show that the curve y−
700 600 500 400
11x 1 1 x2
h as three points of inflection and they all lie on one straight line.
Number of yeast cells 300 200 100
0
59. Find a cubic function
2
4
6
8
10 12 14 16 18
Time (in hours)
54. I n an episode of The Simpsons television show, Homer reads from a newspaper and announces “Here’s good news! According to this eyecatching article, SAT scores are declining at a slower rate.” Interpret Homer’s statement in terms of a function and its first and second derivatives. 55. T he president announces that the national deficit is increasing, but at a decreasing rate. Interpret this statement in terms of a function and its first and second derivatives. 56. L et f std be the temperature at time t where you live and suppose that at time t − 3 you feel uncomfortably hot. How do you feel about the given data in each case? (a) f 9s3d − 2, f 0s3d − 4 (b) f 9s3d − 2, f 0s3d − 24 (c) f 9s3d − 22, f 0s3d − 4 (d) f 9s3d − 22, f 0s3d − 24 57. Let Kstd be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, Ks8d 2 Ks7d or Ks3d 2 Ks2d? Is the graph of K concave upward or concave downward? Why? 58. C offee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity. What is the significance of the inflection point?
61. (a) If the function f sxd − x 3 1 ax 2 1 bx has the local minimum value 292 s3 at x − 1ys3 , what are the values of a and b? (b) Which of the tangent lines to the curve in part (a) has the smallest slope? 62. F or what values of a and b is s2, 2.5d an inflection point of the curve x 2 y 1 ax 1 by − 0? What additional inflection points does the curve have? 63. S how that the inflection points of the curve y − x sin x lie on the curve y 2sx 2 1 4d − 4x 2. 64–66 Assume that all of the functions are twice differentiable and the second derivatives are never 0. 64. (a) If f and t are concave upward on I, show that f 1 t is concave upward on I. (b) If f is positive and concave upward on I, show that the function tsxd − f f sxdg 2 is concave upward on I. 65. (a) If f and t are positive, increasing, concave upward functions on I, show that the product function f t is concave upward on I. (b) Show that part (a) remains true if f and t are both decreasing. (c) Suppose f is increasing and t is decreasing. Show, by giving three examples, that f t may be concave upward, concave downward, or linear. Why doesn’t the argument in parts (a) and (b) work in this case? 66. S uppose f and t are both concave upward on s2`, `d. Under what condition on f will the composite function hsxd − f s tsxdd be concave upward? 67. Show that tan x . x for 0 , x , y2. [Hint: Show that f sxd − tan x 2 x is increasing on s0, y2d.] 68. Prove that, for all x . 1, 2 sx . 3 2
1 x
69. S how that a cubic function (a thirddegree polynomial) always has exactly one point of inflection. If its graph has three xintercepts x 1, x 2, and x 3, show that the xcoordinate of the inflection point is sx 1 1 x 2 1 x 3 dy3. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
231
Section 3.4 Limits at Infinity; Horizontal Asymptotes
or what values of c does the polynomial ; 70. F Psxd − x 4 1 cx 3 1 x 2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as c decreases?
76. For what values of c is the function f sxd − cx 1
71. P rove that if sc, f scdd is a point of inflection of the graph of f and f 0 exists in an open interval that contains c, then f 0scd − 0. [Hint: Apply the First Derivative Test and Fermat’s Theorem to the function t − f 9.]
increasing on s2`, `d?
77. T he three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions f, t, and h whose values at 0 are all 0 and, for x ± 0, 1 1 f sxd − x 4 sin tsxd − x 4 2 1 sin x x
72. S how that if f sxd − x 4, then f 0s0d − 0, but s0, 0d is not an inflection point of the graph of f .
S
 
73. S how that the function tsxd − x x has an inflection point at s0, 0d but t0s0d does not exist.
hsxd − x 4 22 1 sin
74. Suppose that f 09 is continuous and f 9scd − f 0scd − 0, but f scd . 0. Does f have a local maximum or minimum at c? Does f have a point of inflection at c? 75. Suppose f is differentiable on an interval I and f 9sxd . 0 for all numbers x in I except for a single number c. Prove that f is increasing on the entire interval I.
1 x 13 2
S D
D
1 x
(a) Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0. (b) Show that f has neither a local maximum nor a local min imum at 0, t has a local minimum, and h has a local maximum.
In Sections 1.5 and 1.7 we investigated infinite limits and vertical asymptotes. There we let x approach a number and the result was that the values of y became arbitrarily large (positive or negative). In this section we let x become arbitrarily large (positive or negative) and see what happens to y. We will find it very useful to consider this socalled end behavior when sketching graphs. Let’s begin by investigating the behavior of the function f defined by f sxd − x 0 61 62 63 64 65 610 650 6100 61000
f sxd 21 0 0.600000 0.800000 0.882353 0.923077 0.980198 0.999200 0.999800 0.999998
x2 2 1 x2 1 1
as x becomes large. The table at the left gives values of this function correct to six decimal places, and the graph of f has been drawn by a computer in Figure 1. y
0
y=1
1
y=
≈1 ≈+1
x
FIGURE 1
As x grows larger and larger you can see that the values of f sxd get closer and closer to 1. (The graph of f approaches the horizontal line y − 1 as we look to the right.) In fact, it seems that we can make the values of f sxd as close as we like to 1 by taking x sufficiently large. This situation is expressed symbolically by writing lim
xl`
x2 2 1 −1 x2 1 1
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
232
Chapter 3 Applications of Differentiation
In general, we use the notation lim f sxd − L
xl`
to indicate that the values of f sxd approach L as x becomes larger and larger. 1 Intuitive Definition of a Limit at Infinity Let f be a function defined on some interval sa, `d. Then lim f sxd − L xl`
means that the values of f sxd can be made arbitrarily close to L by requiring x to be sufficiently large. Another notation for lim x l ` f sxd − L is f sxd l L as x l ` The symbol ` does not represent a number. Nonetheless, the expression lim f sxd − L x l` is often read as “the limit of f sxd, as x approaches infinity, is L” or
“the limit of f sxd, as x becomes infinite, is L”
or
“the limit of f sxd, as x increases without bound, is L”
The meaning of such phrases is given by Definition 1. A more precise definition, similar to the «, definition of Section 1.7, is given at the end of this section. Geometric illustrations of Definition 1 are shown in Figure 2. Notice that there are many ways for the graph of f to approach the line y − L (which is called a horizontal asymptote) as we look to the far right of each graph. y
y
y=L
y
y=ƒ
y=ƒ 0
FIGURE 2
xl`
y=ƒ
y=L x
Examples illustrating lim f sxd − L
y=L
0
x
0
x
Referring back to Figure 1, we see that for numerically large negative values of x, the values of f sxd are close to 1. By letting x decrease through negative values without bound, we can make f sxd as close to 1 as we like. This is expressed by writing lim
x l2`
x2 2 1 −1 x2 1 1
The general definition is as follows. 2 Definition Let f be a function defined on some interval s2`, ad. Then lim f sxd − L
x l 2`
means that the values of f sxd can be made arbitrarily close to L by requiring x to be sufficiently large negative.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 3.4 Limits at Infinity; Horizontal Asymptotes y
233
Again, the symbol 2` does not represent a number, but the expression lim f sxd − L x l 2` is often read as
y=ƒ
“the limit of f sxd, as x approaches negative infinity, is L” y=L 0
x
Definition 2 is illustrated in Figure 3. Notice that the graph approaches the line y − L as we look to the far left of each graph.
y
3 Definition The line y − L is called a horizontal asymptote of the curve y − f sxd if either lim f sxd − L or lim f sxd − L
y=ƒ
y=L
x l`
0
x l 2`
x
FIGURE 3
Examples illustrating lim f sxd − L
For instance, the curve illustrated in Figure 1 has the line y − 1 as a horizontal asymp tote because
x l 2`
lim
xl`
x2 2 1 −1 x2 1 1
The curve y − f sxd sketched in Figure 4 has both y − 21 and y − 2 as horizontal asymptotes because lim f sxd − 21 xl`
and
lim f sxd − 2
x l2`
y 2
y=2
0
y=_1
y=ƒ _1
x
FIGURE 4 y
Example 1 Find the infinite limits, limits at infinity, and asymptotes for the function f whose graph is shown in Figure 5. SOLUTION We see that the values of f sxd become large as x l 21 from both sides, so
2 0
2
lim f sxd − `
x l21
x
Notice that f sxd becomes large negative as x approaches 2 from the left, but large posi tive as x approaches 2 from the right. So FIGURE 5
lim f sxd − 2` and lim1 f sxd − `
x l 22
x l2
Thus both of the lines x − 21 and x − 2 are vertical asymptotes. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
234
Chapter 3 Applications of Differentiation
As x becomes large, it appears that f sxd approaches 4. But as x decreases through negative values, f sxd approaches 2. So lim f sxd − 4 and lim f sxd − 2
xl`
x l2`
This means that both y − 4 and y − 2 are horizontal asymptotes.
Example 2 Find lim
xl`
n
1 1 and lim . x l2` x x
SOLUTION Observe that when x is large, 1yx is small. For instance,
1 1 1 − 0.01 − 0.0001 − 0.000001 100 10,000 1,000,000
y
y=∆
0
FIGURE 6 lim
xl`
1 1 − 0, lim −0 x l2` x x
In fact, by taking x large enough, we can make 1yx as close to 0 as we please. There fore, according to Definition 1, we have lim
x
xl`
1 −0 x
Similar reasoning shows that when x is large negative, 1yx is small negative, so we also have 1 lim −0 x l2` x It follows that the line y − 0 (the xaxis) is a horizontal asymptote of the curve y − 1yx. (This is an equilateral hyperbola; see Figure 6.)
n
Most of the Limit Laws that were given in Section 1.6 also hold for limits at infinity. It can be proved that the Limit Laws listed in Section 1.6 (with the exception of Laws 9 and 10) are also valid if “x l a” is replaced by “x l `” or “x l 2`.” In particular, if we combine Laws 6 and 11 with the results of Example 2, we obtain the following important rule for calculating limits. 4 Theorem If r . 0 is a rational number, then lim
xl`
1 −0 xr
If r . 0 is a rational number such that x r is defined for all x, then lim
x l2`
1 −0 xr
Example 3 Evaluate lim
x l`
3x 2 2 x 2 2 5x 2 1 4x 1 1
and indicate which properties of limits are used at each stage. SOLUTION As x becomes large, both numerator and denominator become large, so it isn’t obvious what happens to their ratio. We need to do some preliminary algebra. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 3.4 Limits at Infinity; Horizontal Asymptotes
235
To evaluate the limit at infinity of any rational function, we first divide both the numerator and denominator by the highest power of x that occurs in the denominator. (We may assume that x ± 0, since we are interested only in large values of x.) In this case the highest power of x in the denominator is x 2, so we have 3x 2 2 x 2 2 1 2 32 2 2 3x 2 x 2 2 x2 x x lim − lim − lim x l ` 5x 2 1 4x 1 1 x l ` 5x 2 1 4x 1 1 x l` 4 1 51 1 2 2 x x x 2
S S
D D
1 2 2 2 x x − 4 1 lim 5 1 1 2 xl` x x lim 3 2
xl`
1 2 2 lim x l` x l` x x l` − 1 lim 5 1 4 lim 1 lim x l` x l` x x l`
y
lim 3 2 lim
y=0.6 0
FIGURE 7 2
y−
3x 2 x 2 2 5x 2 1 4x 1 1
1
(by Limit Law 5)
x
−
32020 51010
−
3 5
1 x2 1 x2
(by 1, 2, and 3)
(by 7 and Theorem 4)
A similar calculation shows that the limit as x l 2` is also 35. Figure 7 illustrates the results of these calculations by showing how the graph of the given rational function approaches the horizontal asymptote y − 35 − 0.6. n
Example 4 Find the horizontal and vertical asymptotes of the graph of the function f sxd −
s2x 2 1 1 3x 2 5
SOLUTION Dividing both numerator and denominator by x and using the properties of limits, we have
Î
s2x 2 1 1 x s2x 2 1 1 lim − lim − lim x l ` 3x 2 5 xl` xl` 3x 2 5 x lim
−
xl`
lim
xl`
2x 2 1 1 x2 (since sx 2 − x for x . 0) 3x 2 5 x
Î Î S D 21
32
5 x
1 x2
1 x2 s2 1 0 s2 − − − 325?0 3 1 lim 3 2 5 lim xl` xl` x lim 2 1 lim
xl`
xl`
Therefore the line y − s2 y3 is a horizontal asymptote of the graph of f. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
236
Chapter 3 Applications of Differentiation
In computing the limit as x l 2`, we must remember that for x , 0, we have sx 2 − x − 2x. So when we divide the numerator by x, for x , 0 we get
 
Î
s2x 2 1 1 s2x 2 1 1 − −2 x 2 sx 2
Î
2x 2 1 1 −2 x2
21
1 x2
Therefore
s2x 1 1 lim − lim x l2` x l2` 3x 2 5 2
21
1 x2
5 32 x
Î
2 −
1 s2 x2 −2 1 3 3 2 5 lim x l2` x 2 1 lim
x l2`
Thus the line y − 2s2 y3 is also a horizontal asymptote. A vertical asymptote is likely to occur when the denominator, 3x 2 5, is 0, that is, when x − 53. If x is close to 53 and x . 53, then the denominator is close to 0 and 3x 2 5 is positive. The numerator s2x 2 1 1 is always positive, so f sxd is positive. Therefore
y
œ„2
y= 3
lim
x
x l s5y3d1
œ„2
y=_ 3
x=
s2x 2 1 1 −` 3x 2 5
(Notice that the numerator does not approach 0 as x l 5y3). If x is close to 53 but x , 53, then 3x 2 5 , 0 and so f sxd is large negative. Thus
5 3
lim
FIGURE 8 y−
Î
2
x l s5y3d2
s2 x 2 1 1
s2x 2 1 1 − 2` 3x 2 5
The vertical asymptote is x − 53. All three asymptotes are shown in Figure 8.
3x 2 5
n
Example 5 Compute lim (sx 2 1 1 2 x). x l`
SOLUTION Because both sx 2 1 1 and x are large when x is large, it’s difficult to see
We can think of the given function as having a denominator of 1.
what happens to their difference, so we use algebra to rewrite the function. We first multiply numerator and denominator by the conjugate radical: lim (sx 2 1 1 2 x) − lim (sx 2 1 1 2 x)
x l`
x l`
− lim
y
x l`
y=œ„„„„„x ≈+1
FIGURE 9
1
sx 2 1 1d 2 x 2 1 − lim x l ` sx 2 1 1 1 x sx 2 1 1 1 x
Notice that the denominator of this last expression (sx 2 1 1 1 x) becomes large as x l ` (it’s bigger than x). So
1 0
sx 2 1 1 1 x sx 2 1 1 1 x
x
lim (sx 2 1 1 2 x) − lim
x l`
Figure 9 illustrates this result.
x l`
1 −0 sx 2 1 1 1 x n
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 3.4 Limits at Infinity; Horizontal Asymptotes
237
1 x
Example 6 Evaluate lim sin . xl`
PS The problemsolving strategy for Example 6 is introducing something extra (see page 98). Here, the something extra, the auxiliary aid, is the new variable t.
SOLUTION If we let t − 1yx, then t l 01 as x l `. Therefore
lim sin xl`
1 − lim1 sin t − 0 tl0 x
(See Exercise 73.)
n
Example 7 Evaluate lim sin x. xl`
SOLUTION As x increases, the values of sin x oscillate between 1 and 21 infinitely often and so they don’t approach any definite number. Thus lim x l` sin x does not exist. n
Infinite Limits at Infinity The notation lim f sxd − `
x l`
is used to indicate that the values of f sxd become large as x becomes large. Similar mean ings are attached to the following symbols: lim f sxd − ` lim f sxd − 2` lim f sxd − 2`
x l 2`
x l`
x l 2`
Example 8 Find lim x 3 and lim x 3. xl`
x l2`
SOLUTION When x becomes large, x 3 also becomes large. For instance, y
10 3 − 1000 100 3 − 1,000,000 1000 3 − 1,000,000,000
y=˛ 0
x
In fact, we can make x 3 as big as we like by requiring x to be large enough. Therefore we can write lim x 3 − ` xl`
Similarly, when x is large negative, so is x 3. Thus lim x 3 − 2`
FIGURE 10
x l2`
lim x 3 − `, lim x 3 − 2`
xl`
x l2`
These limit statements can also be seen from the graph of y − x 3 in Figure 10.
n
Example 9 Find lim sx 2 2 xd. x l`
SOLUTION It would be wrong to write
lim sx 2 2 xd − lim x 2 2 lim x − ` 2 `
x l`
x l`
x l`
The Limit Laws can’t be applied to infinite limits because ` is not a number (` 2 ` can’t be defined). However, we can write lim sx 2 2 xd − lim xsx 2 1d − `
x l`
x l`
because both x and x 2 1 become arbitrarily large and so their product does too. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
n
238
Chapter 3 Applications of Differentiation
Example 10 Find lim
xl`
x2 1 x . 32x
SOLUTION As in Example 3, we divide the numerator and denominator by the highest power of x in the denominator, which is just x:
lim
x l`
x2 1 x x11 − lim − 2` x l` 3 32x 21 x
because x 1 1 l ` and 3yx 2 1 l 0 2 1 − 21 as x l `.
n
The next example shows that by using infinite limits at infinity, together with inter cepts, we can get a rough idea of the graph of a polynomial without computing derivatives.
Example 11 Sketch the graph of y − sx 2 2d4sx 1 1d3sx 2 1d by finding its inter
cepts and its limits as x l ` and as x l 2`.
SOLUTION The yintercept is f s0d − s22d4s1d3s21d − 216 and the xintercepts are
found by setting y − 0: x − 2, 21, 1. Notice that since sx 2 2d4 is never negative, the function doesn’t change sign at 2; thus the graph doesn’t cross the xaxis at 2. The graph crosses the axis at 21 and 1. When x is large positive, all three factors are large, so
y
0
_1
1
2
x
lim sx 2 2d4sx 1 1d3sx 2 1d − `
xl`
When x is large negative, the first factor is large positive and the second and third fac tors are both large negative, so
_16
lim sx 2 2d4sx 1 1d3sx 2 1d − `
x l2`
FIGURE 11
y − sx 2 2d4 sx 1 1d3 sx 2 1d
Combining this information, we give a rough sketch of the graph in Figure 11.
n
Precise Definitions Definition 1 can be stated precisely as follows. 5 Precise Definition of a Limit at Infinity Let f be a function defined on some interval sa, `d. Then lim f sxd − L xl`
means that for every « . 0 there is a corresponding number N such that


if x . N then f sxd 2 L , «
In words, this says that the values of f sxd can be made arbitrarily close to L (within a distance «, where « is any positive number) by requiring x to be sufficiently large (larger than N, where N depends on «). Graphically it says that by keeping x large enough (larger than some number N) we can make the graph of f lie between the given hori zontal lines y − L 2 « and y − L 1 « as in Figure 12. This must be true no matter how small we choose «. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
239
Section 3.4 Limits at Infinity; Horizontal Asymptotes y
y=ƒ
y=L+∑ ∑ L ∑ y=L∑
ƒ is in here
0
FIGURE 12
x
N
lim f sxd − L
when x is in here
xl`
Figure 13 shows that if a smaller value of « is chosen, then a larger value of N may be required.
L
FIGURE 13 lim f sxd − L
y=ƒ
y=L+∑ y=L∑
0
N
x
xl`
Similarly, a precise version of Definition 2 is given by Definition 6, which is illus trated in Figure 14. 6 Definition Let f be a function defined on some interval s2`, ad. Then lim f sxd − L
x l 2`
means that for every « . 0 there is a corresponding number N such that


if x , N then f sxd 2 L , «
y
y=ƒ
y=L+∑ L y=L∑
FIGURE 14
0
N
x
lim f sxd − L
x l2`
In Example 3 we calculated that lim
xl`
3x 2 2 x 2 2 3 − 2 5x 1 4x 1 1 5
In the next example we use a graphing device to relate this statement to Definition 5 with L − 35 − 0.6 and « − 0.1. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
240
Chapter 3 Applications of Differentiation
TEC In Module 1.7y3.4 you can explore the precise definition of a limit both graphically and numerically.
Example 12 Use a graph to find a number N such that if x . N then
Z
3x 2 2 x 2 2 2 0.6 5x 2 1 4x 1 1
Z
, 0.1
SOLUTION We rewrite the given inequality as
3x 2 2 x 2 2 , 0.7 5x 2 1 4x 1 1
0.5 , 1
We need to determine the values of x for which the given curve lies between the hori zontal lines y − 0.5 and y − 0.7. So we graph the curve and these lines in Figure 15. Then we use the cursor to estimate that the curve crosses the line y − 0.5 when x < 6.7. To the right of this number it seems that the curve stays between the lines y − 0.5 and y − 0.7. Rounding up to be safe, we can say that
y=0.7 y=0.5 y=
3≈x2 5≈+4x+1
if x . 7 then
15
0
FIGURE 15
Z
3x 2 2 x 2 2 2 0.6 5x 2 1 4x 1 1
Z
, 0.1
In other words, for « − 0.1 we can choose N − 7 (or any larger number) in Definition 5. 1 − 0. x
Example 13 Use Definition 5 to prove that lim
xl`
SOLUTION Given « . 0, we want to find N such that
if x . N then
n
Z
1 20 x
Z
,«
In computing the limit we may assume that x . 0. Then 1yx , « &? x . 1y«. Let’s choose N − 1y«. So if x . N −
Z
1 1 then 20 « x
Z
−
1 ,« x
Therefore, by Definition 5, lim
xl`
1 −0 x
Figure 16 illustrates the proof by showing some values of « and the corresponding values of N. y
y
∑=1 0
FIGURE 16
N=1
x
∑=0.2
0
y
N=5
x
∑=0.1
0
N=10
x
n
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
241
Section 3.4 Limits at Infinity; Horizontal Asymptotes y M
Finally we note that an infinite limit at infinity can be defined as follows. The geomet ric illustration is given in Figure 17.
y=M
0
x
N
FIGURE 17
7 Definition of an Infinite Limit at Infinity Let f be a function defined on some interval sa, `d. Then lim f sxd − ` xl`
means that for every positive number M there is a corresponding positive number N such that if x . N then f sxd . M
lim f sxd − `
xl`
Similar definitions apply when the symbol ` is replaced by 2`. (See Exercise 74.)
y
1. Explain in your own words the meaning of each of the following. (a) lim f sxd − 5 (b) lim f sxd − 3 xl`
x l 2`
1
2. (a) Can the graph of y − f sxd intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs. (b) How many horizontal asymptotes can the graph of y − f sxd have? Sketch graphs to illustrate the possibilities. 3. For the function f whose graph is given, state the following. (a) lim f sxd (b) lim f sxd
; 5. Guess the value of the limit
(c) lim f sxd (d) lim f sxd
x l`
x l`
x l 2`
x l1
lim
x l3
y
; 6. (a) Use a graph of 1 x
4. For the function t whose graph is given, state the following. (a) lim tsxd (b) lim tsxd (e) lim1 tsxd x l2
xl2
(f ) The equations of the asymptotes
2 x
x
to estimate the value of lim x l ` f sxd correct to two decimal places. (b) Use a table of values of f sxd to estimate the limit to four decimal places.
7–8 Evaluate the limit and justify each step by indicating the appropriate properties of limits.
x l 2`
(c) lim tsxd (d) lim2 tsxd xl0
S D
f sxd − 1 2
x l`
x2 2x
by evaluating the function f sxd − x 2y2 x for x − 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of f to support your guess.
(e) The equations of the asymptotes
1
x
1
7. lim
xl`
Î
2x 2 2 7 9x 3 1 8x 2 4 8. lim 2 xl` 5x 1 x 2 3 3 2 5x 1 x 3
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
242
chapter 3 Applications of Differentiation
9–32 Find the limit or show that it does not exist. 2
9. lim
xl`
3x 2 2 12x 10. lim xl` x3 2 x 1 1 2x 1 1
x22 4x 3 1 6x 2 2 2 11. lim 2 12. lim x l 2` x 1 1 x l 2` 2x 3 2 4x 1 5
35–40 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. 35. y −
5 1 4x 2x 2 1 1 36. y− 2 x13 3x 1 2x 2 1
13. lim
t 2 t st st 1 t 2 14. lim tl ` 2t 3y2 1 3t 2 5 2t 2 t 2
37. y −
2x 2 1 x 2 1 1 1 x4 38. y− 2 2 x 1x22 x 2 x4
15. lim
s2x 2 1 1d2 x2 16. lim 2 2 x l ` sx 4 1 1 sx 2 1d sx 1 xd
39. y −
x3 2 x x29 40. y− 2 x 2 6x 1 5 s4x 1 3x 1 2
17. lim
s1 1 4x 6 s1 1 4x 6 18. lim 3 x l 2` 22x 2 2 x3
19. lim
x 1 3x 2 sx 1 3x 2 20. lim x l ` 4x 2 1 4x 2 1
tl`
xl`
xl`
xl`
; 41. Estimate the horizontal asymptote of the function f sxd −
x l`
22. lim (s4x 2 1 3x 1 2 x ) x l2`
23. lim (sx 1 ax 2 sx 1 bx 2
x l`
3x 3 1 500x 2 x 1 500x 2 1 100x 1 2000 3
by graphing f for 210 < x < 10. Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?
21. lim (s9x 2 1 x 2 3x)
2
2
; 42. (a) Graph the function
)
f sxd −
24. lim cos x x l`
How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits
x 4 2 3x 2 1 x 25. lim 3 26. lim sx 2 1 1 xl` x 2 x 1 2 x l` 1 1 x6 27. lim sx 2 1 2x 7 d 28. lim 4 x l 2` x l 2` x 1 1
29. lim s x 2 sx d 30. lim sx 2 2 x 4 d
xl`
xl`
31. lim x sin xl`
1 1 32. lim sx sin xl` x x
; 33. (a) Estimate the value of
s2x 2 1 1 3x 2 5
lim
x l`
s2x 2 1 1 s2x 2 1 1 and lim x l 2` 3x 2 5 3x 2 5
(b) By calculating values of f sxd, give numerical estimates of the limits in part (a). (c) Calculate the exact values of the limits in part (a). Did you get the same value or different values for these two limits? [In view of your answer to part (a), you might have to check your calculation for the second limit.]
43. Let P and Q be polynomials. Find
lim (sx 2 1 x 1 1 1 x)
x l 2`
lim
by graphing the function f sxd − sx 1 x 1 1 1 x.
xl`
2
(b) Use a table of values of f sxd to guess the value of the limit. (c) Prove that your guess is correct.
; 34. (a) Use a graph of f sxd − s3x 2 1 8x 1 6 2 s3x 2 1 3x 1 1
to estimate the value of lim x l ` f sxd to one decimal place. (b) Use a table of values of f sxd to estimate the limit to four decimal places. (c) Find the exact value of the limit.
Psxd Qsxd
if the degree of P is (a) less than the degree of Q and (b) greater than the degree of Q. 44. M ake a rough sketch of the curve y − x n (n an integer) for the following five cases: (i) n − 0 (ii) n . 0, n odd (iii) n . 0, n even (iv) n , 0, n odd (v) n , 0, n even Then use these sketches to find the following limits. (a) lim1 x n (b) lim2 x n x l0
x l0
(c) lim x n (d) lim x n x l`
x l 2`
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 3.4 Limits at Infinity; Horizontal Asymptotes
45. F ind a formula for a function f that satisfies the following conditions: lim f sxd − 0, lim f sxd − 2`, f s2d − 0,
x l 6`
x l0
243
60. ts0d − 0, t0sxd , 0 for x ± 0, lim x l2` tsxd − `, lim x l ` tsxd − 2`, lim x l 02 t9sxd − 2`, lim x l 01 t9sxd − `
lim2 f sxd − `, lim1 f sxd − 2`
x l3
x l3
46. F ind a formula for a function that has vertical asymptotes x − 1 and x − 3 and horizontal asymptote y − 1.
sin x . x (b) Graph f sxd − ssin xdyx. How many times does the graph cross the asymptote?
61. (a) Use the Squeeze Theorem to evaluate lim ;
xl`
47. A function f is a ratio of quadratic functions and has a vertical asymptote x − 4 and just one xintercept, x − 1. ; 62. By the end behavior of a function we mean the behavior of It is known that f has a removable discontinuity at x − 21 its values as x l ` and as x l 2`. and lim x l21 f sxd − 2. Evaluate (a) Describe and compare the end behavior of the functions (a) f s0d (b) lim f sxd xl`
Psxd − 3x 5 2 5x 3 1 2x Qsxd − 3x 5
48–51 Find the horizontal asymptotes of the curve and use them, together with concavity and intervals of increase and decrease, to sketch the curve. 48. y − 50. y −
1 1 2x 2 12x 49. y− 1 1 x2 11x x sx 2 1 1
x 51. y− 2 x 11
52–56 Find the limits as x l ` and as x l 2`. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 11. 52. y − 2x 3 2 x 4 53. y − x 4 2 x6 54. y − x sx 1 2d sx 2 1d 3
2
55. y − s3 2 xds1 1 xd 2s1 2 xd 4
by graphing both functions in the viewing rectangles f22, 2g by f22, 2g and f210, 10g by f210,000, 10,000g. (b) Two functions are said to have the same end behavior if their ratio approaches 1 as x l `. Show that P and Q have the same end behavior.
63. Find lim x l ` f sxd if 4x 2 1 4x 2 1 3x , f sxd , x x2 for all x . 5. 64. (a) A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 Lymin. Show that the concentration of salt after t minutes (in grams per liter) is
56. y − x 2sx 2 2 1d 2sx 1 2d
57–60 Sketch the graph of a function that satisfies all of the given conditions. 57. f 9s2d − 0, f s2d − 21, f s0d − 0, f 9sxd , 0 if 0 , x , 2, f 9sxd . 0 if x . 2, f 0sxd , 0 if 0 < x , 1 or if x . 4, f 0sxd . 0 if 1 , x , 4, lim x l ` f sxd − 1, f s2xd − f sxd for all x
Cstd −
(b) What happens to the concentration as t l `?
; 65. Use a graph to find a number N such that
Z
if x . N then
f 0sxd . 0 if x . 4, lim x l ` f sxd − 0, f s2xd − 2f sxd for all x 59. f s1d − f 9s1d − 0, lim x l21 f sxd − `, lim x l22 f sxd − 2`, lim x l 0 f sxd − 2`, lim x l2` f sxd − `, lim x l ` f sxd − 0, f 0sxd . 0 for x . 2, f 0sxd , 0 for x , 0 and for
0,x,2
Z
3x2 1 1 2 1.5 , 0.05 2x 2 1 x 1 1
; 66. For the limit lim
xl`
58. f 9s2d − 0, f 9s0d − 1, f 9sxd . 0 if 0 , x , 2, f 9sxd , 0 if x . 2, f 0sxd , 0 if 0 , x , 4,
30t 200 1 t
1 2 3x sx 2 1 1
− 23
illustrate Definition 5 by finding values of N that correspond to « − 0.1 and « − 0.05. ; 67. For the limit lim
x l2`
1 2 3x sx 2 1 1
−3
illustrate Definition 6 by finding values of N that correspond to « − 0.1 and « − 0.05.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
244
chapter 3 Applications of Differentiation
; 68. For the limit lim
xl`
71. Use Definition 6 to prove that lim
3x sx 2 3
x l2`
−`
illustrate Definition 7 by finding a value of N that corresponds to M − 100. 69. (a) How large do we have to take x so that 1yx 2 , 0.0001? (b) Taking r − 2 in Theorem 4, we have the statement
(b) Taking r − 12 in Theorem 4, we have the statement lim
xl`
lim f sxd − lim1 f s1ytd tl0
xl`
and
f s1ytd lim f sxd − t lim l 02
xl2`
1 sx
lim x sin
x l 01
lim f sxd − 2`
x l2`
−0
y=8˛21≈+18x+2
_2
4 _10
FIGURE 1 8
y=8˛21≈+18x+2 6
FIGURE 2
2
1 x
74. Formulate a precise definition of
Then use your definition to prove that
Prove this directly using Definition 5.
30
0
xl`
73. (a) Prove that
Prove this directly using Definition 5.
70. (a) How large do we have to take x so that 1ysx , 0.0001?
72. Prove, using Definition 7, that lim x 3 − `.
if these limits exist. (b) Use part (a) and Exercise 61 to find
1 −0 x2
lim
xl`
1 − 0. x
lim s1 1 x 3 d − 2`
x l2`
So far we have been concerned with some particular aspects of curve sketching: domain, range, symmetry, limits, continuity, and vertical asymptotes in Chapter 1; derivatives and tangents in Chapter 2; and extreme values, intervals of increase and decrease, concavity, points of inflection, and horizontal asymptotes in this chapter. It is now time to put all of this information together to sketch graphs that reveal the important features of functions. You might ask: Why don’t we just use a graphing calculator or computer to graph a curve? Why do we need to use calculus? It’s true that current technology is capable of producing very accurate graphs. But even the best graphing devices have to be used intelligently. It is easy to arrive at a misleading graph, or to miss important details of a curve, when relying solely on technology. (See “Graphing Calculators and Computers” at www.stewartcalculus.com, especially Examples 1, 3, 4, and 5. See also Section 3.6.) The use of calculus enables us to discover the most interesting aspects of graphs and in many cases to calculate maximum and minimum points and inflection points exactly instead of approximately. For instance, Figure 1 shows the graph of f sxd − 8x 3 2 21x 2 1 18x 1 2. At first glance it seems reasonable: it has the same shape as cubic curves like y − x 3, and it appears to have no maximum or minimum point. But if you compute the derivative, you will see that there is a maximum when x − 0.75 and a minimum when x − 1. Indeed, if we zoom in to this portion of the graph, we see that behavior exhibited in Figure 2. Without calculus, we could easily have overlooked it. In the next section we will graph functions by using the interaction between calculus and graphing devices. In this section we draw graphs by first considering the following information. We don’t assume that you have a graphing device, but if you do have one you should use it as a check on your work.
Guidelines for Sketching a Curve The following checklist is intended as a guide to sketching a curve y − f sxd by hand. Not every item is relevant to every function. (For instance, a given curve might not have an
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
245
Section 3.5 Summary of Curve Sketching
y
0
x
(a) Even function: reflectional symmetry y
0
x
(b) Odd function: rotational symmetry
FIGURE 3
asymptote or possess symmetry.) But the guidelines provide all the information you need to make a sketch that displays the most important aspects of the function. A. Domain It’s often useful to start by determining the domain D of f , that is, the set of values of x for which f sxd is defined. B. Intercepts The yintercept is f s0d and this tells us where the curve intersects the yaxis. To find the xintercepts, we set y − 0 and solve for x. (You can omit this step if the equation is difficult to solve.) C. Symmetry (i ) If f s2xd − f sxd for all x in D, that is, the equation of the curve is unchanged when x is replaced by 2x, then f is an even function and the curve is symmetric about the yaxis. This means that our work is cut in half. If we know what the curve looks like for x > 0, then we need only reflect about the yaxis to obtain the complete curve [see Figure 3(a)]. Here are some examples: y − x 2, y − x 4, y − x , and y − cos x. (ii) If f s2xd − 2f sxd for all x in D, then f is an odd function and the curve is symmetric about the origin. Again we can obtain the complete curve if we know what it looks like for x > 0. [Rotate 180° about the origin; see Figure 3(b).] Some simple examples of odd functions are y − x, y − x 3, y − x 5, and y − sin x. (iii) If f sx 1 pd − f sxd for all x in D, where p is a positive constant, then f is called a periodic function and the smallest such number p is called the period. For instance, y − sin x has period 2 and y − tan x has period . If we know what the graph looks like in an interval of length p, then we can use translation to sketch the entire graph (see Figure 4).
 
y
FIGURE 4 Periodic function: translational symmetry
ap
0
period p
a
a+p
a+2p
x
D. Asymptotes (i) Horizontal Asymptotes. Recall from Section 3.4 that if either lim x l ` f sxd − L or lim x l2 ` f sxd − L, then the line y − L is a horizontal asymptote of the curve y − f sxd. If it turns out that lim x l ` f sxd − ` (or 2`), then we do not have an asymptote to the right, but this fact is still useful information for sketching the curve. (ii) Vertical Asymptotes. Recall from Section 1.5 that the line x − a is a vertical asymptote if at least one of the following statements is true: 1
lim f sxd − `
x l a1
lim f sxd − 2`
x l a1
lim f sxd − `
x l a2
lim f sxd − 2`
x l a2
(For rational functions you can locate the vertical asymptotes by equating the denominator to 0 after canceling any common factors. But for other functions this method does not apply.) Furthermore, in sketching the curve it is very useful to know exactly which of the statements in (1) is true. If f sad is not defined but a is an endpoint of the domain of f, then you should compute lim x l a2 f sxd or lim x l a1 f sxd, whether or not this limit is infinite. (iii) Slant Asymptotes. These are discussed at the end of this section. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
246
Chapter 3 Applications of Differentiation
E. Intervals of Increase or Decrease Use the I/D Test. Compute f 9sxd and find the intervals on which f 9sxd is positive ( f is increasing) and the intervals on which f 9sxd is negative ( f is decreasing). F. Local Maximum and Minimum Values Find the critical numbers of f [the numbers c where f 9scd − 0 or f 9scd does not exist]. Then use the First Derivative Test. If f 9 changes from positive to negative at a critical number c, then f scd is a local maximum. If f 9 changes from negative to positive at c, then f scd is a local minimum. Although it is usually preferable to use the First Derivative Test, you can use the Second Derivative Test if f 9scd − 0 and f 0scd ± 0. Then f 0scd . 0 implies that f scd is a local minimum, whereas f 0scd , 0 implies that f scd is a local maximum. G. Concavity and Points of Inflection Compute f 0sxd and use the Concavity Test. The curve is concave upward where f 0sxd . 0 and concave downward where f 0sxd , 0. Inflection points occur where the direction of concavity changes. H. Sketch the Curve Using the information in items A–G, draw the graph. Sketch the asymptotes as dashed lines. Plot the intercepts, maximum and minimum points, and inflection points. Then make the curve pass through these points, rising and falling according to E, with concavity according to G, and approaching the asymptotes. If additional accuracy is desired near any point, you can compute the value of the derivative there. The tangent indicates the direction in which the curve proceeds.
Example 1 Use the guidelines to sketch the curve y − A. The domain is hx
x
2
2 1 ± 0j − hx
2x 2 . x 21 2
 x ± 61j − s2`, 21d ø s21, 1d ø s1, `d
B. The x and yintercepts are both 0. C. Since f s2xd − f sxd, the function f is even. The curve is symmetric about the yaxis. D. y
lim
x l6`
Therefore the line y − 2 is a horizontal asymptote. Since the denominator is 0 when x − 61, we compute the following limits: lim1
x l1
y=2
lim 1
x l 21
0
x=_1
x
x=1
FIGURE 5 Preliminary sketch We have shown the curve approaching its horizontal asymptote from above in Figure 5. This is confirmed by the intervals of increase and decrease.
2x 2 2 − lim −2 2 x l6` 1 2 1yx 2 x 21
2x 2 −` x2 2 1 2x 2 − 2` x 21 2
lim2
2x 2 − 2` x 21
lim 2
2x 2 −` x 21
x l1
x l 21
2
2
Therefore the lines x − 1 and x − 21 are vertical asymptotes. This information about limits and asymptotes enables us to draw the preliminary sketch in Figure 5, showing the parts of the curve near the asymptotes. E.
f 9sxd −
sx 2 2 1ds4xd 2 2x 2 ? 2x 24x − 2 2 2 sx 2 1d sx 2 1d2
Since f 9sxd . 0 when x , 0 sx ± 21d and f 9sxd , 0 when x . 0 sx ± 1d, f is increasing on s2`, 21d and s21, 0d and decreasing on s0, 1d and s1, `d. F. The only critical number is x − 0. Since f 9 changes from positive to negative at 0, f s0d − 0 is a local maximum by the First Derivative Test.
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Section 3.5 Summary of Curve Sketching y
G.
y=2
f 0sxd −
 
f 0sxd . 0 &? x 2 2 1 . 0 &? x . 1
x
 
and f 0sxd , 0 &? x , 1. Thus the curve is concave upward on the intervals s2`, 21d and s1, `d and concave downward on s21, 1d. It has no point of inflection since 1 and 21 are not in the domain of f. H. Using the information in E–G, we finish the sketch in Figure 6. n
x=1
FIGURE 6 Finished sketch of y −
sx 2 2 1d2 s24d 1 4x ? 2sx 2 2 1d2x 12x 2 1 4 − 2 2 4 sx 2 1d sx 2 1d3
Since 12x 2 1 4 . 0 for all x, we have
0
x=_1
247
2x 2 x 21 2
Example 2 Sketch the graph of f sxd −


x2 . sx 1 1
A. Domain − hx x 1 1 . 0j − hx x . 21j − s21, `d B. The x and yintercepts are both 0. C. Symmetry: None D. Since x2 lim −` x l ` sx 1 1 there is no horizontal asymptote. Since sx 1 1 l 0 as x l 211 and f sxd is always positive, we have x2 lim 1 −` x l 21 sx 1 1 E.
and so the line x − 21 is a vertical asymptote. f 9sxd −
3x 2 1 4x xs3x 1 4d sx 1 1 s2xd 2 x 2 ? 1y( 2sx 1 1 ) − − x11 2sx 1 1d3y2 2sx 1 1d3y2
e see that f 9sxd − 0 when x − 0 (notice that 243 is not in the domain of f ), so the W only critical number is 0. Since f 9sxd , 0 when 21 , x , 0 and f 9sxd . 0 when x . 0, f is decreasing on s21, 0d and increasing on s0, `d. F. Since f 9s0d − 0 and f 9 changes from negative to positive at 0, f s0d − 0 is a local (and absolute) minimum by the First Derivative Test.
y
G. y= x=_1
0
FIGURE 7
≈ œ„„„„ x+1 x
f 0sxd −
2sx 1 1d3y2s6x 1 4d 2 s3x 2 1 4xd3sx 1 1d1y2 3x 2 1 8x 1 8 − 3 4sx 1 1d 4sx 1 1d5y2
Note that the denominator is always positive. The numerator is the quadratic 3x 2 1 8x 1 8, which is always positive because its discriminant is b 2 2 4ac − 232, which is negative, and the coefficient of x 2 is positive. Thus f 0sxd . 0 for all x in the domain of f, which means that f is concave upward on s21, `d and there is no point of inflection. H. The curve is sketched in Figure 7. n
Example 3 Sketch the graph of f sxd −
cos x . 2 1 sin x
A. The domain is R. B. The yintercept is f s0d − 12. The xintercepts occur when cos x − 0, that is, x − sy2d 1 n, where n is an integer. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
248
Chapter 3 Applications of Differentiation
C. f is neither even nor odd, but f sx 1 2d − f sxd for all x and so f is periodic and has period 2. Thus, in what follows, we need to consider only 0 < x < 2 and then extend the curve by translation in part H. D. Asymptotes: None E.
f 9sxd −
s2 1 sin xds2sin xd 2 cos x scos xd 2 sin x 1 1 −2 2 s2 1 sin xd s2 1 sin xd 2
The denominator is always positive, so f 9sxd . 0 when 2 sin x 1 1 , 0 &? sin x , 221 &? 7y6 , x , 11y6. So f is increasing on s7y6, 11y6d and decreasing on s0, 7y6d and s11y6, 2d. F. From part E and the First Derivative Test, we see that the local minimum value is f s7y6d − 21ys3 and the local maximum value is f s11y6d − 1ys3 . G. If we use the Quotient Rule again and simplify, we get f 0sxd − 2
2 cos x s1 2 sin xd s2 1 sin xd 3
Because s2 1 sin xd 3 . 0 and 1 2 sin x > 0 for all x, we know that f 0sxd . 0 when cos x , 0, that is, y2 , x , 3y2. So f is concave upward on sy2, 3y2d and concave downward on s0, y2d and s3y2, 2d. The inflection points are sy2, 0d and s3y2, 0d. H. The graph of the function restricted to 0 < x < 2 is shown in Figure 8. Then we extend it, using periodicity, to the complete graph in Figure 9. y
”
1 2
11π 1 6 , œ„ 3
π
π 2
y
’
3π 2
1 2
2π x
_π
π
2π
3π
x
1  ’ ” 7π 6 , œ„3
FIGURE 8
n
FIGURE 9
Slant Asymptotes y
Some curves have asymptotes that are oblique, that is, neither horizontal nor vertical. If y=ƒ
lim f f sxd 2 smx 1 bdg − 0
xl`
ƒ(mx+b) y=mx+b 0
x
FIGURE 10
where m ± 0, then the line y − mx 1 b is called a slant asymptote because the vertical distance between the curve y − f sxd and the line y − mx 1 b approaches 0, as in Figure 10. (A similar situation exists if we let x l 2`.) For rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the slant asymptote can be found by long division as in the following example.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
249
Section 3.5 Summary of Curve Sketching
Example 4 Sketch the graph of f sxd −
x3 . x 11 2
A. The domain is R − s2`, `d. B. The x and yintercepts are both 0. C. Since f s2xd − 2f sxd, f is odd and its graph is symmetric about the origin. D. Since x 2 1 1 is never 0, there is no vertical asymptote. Since f sxd l ` as x l ` and f sxd l 2` as x l 2`, there is no horizontal asymptote. But long division gives x3 x f sxd − 2 −x2 2 x 11 x 11
This equation suggests that y − x is a candidate for a slant asymptote. In fact,
f sxd 2 x − 2
x −2 x2 1 1
1 x 11
1 x2
l 0 as x l 6`
So the line y − x is a slant asymptote.
E.
f 9sxd −
sx 2 1 1ds3x 2 d 2 x 3 ? 2x x 2sx 2 1 3d − 2 2 sx 1 1d sx 2 1 1d2
Since f 9sxd . 0 for all x (except 0), f is increasing on s2`, `d. F. Although f 9s0d − 0, f 9 does not change sign at 0, so there is no local maximum or minimum. G.
y
y=
˛ ≈+1
f 0sxd −
sx 2 1 1d2 s4x 3 1 6xd 2 sx 4 1 3x 2 d ? 2sx 2 1 1d2x 2xs3 2 x 2 d − 2 4 sx 1 1d sx 2 1 1d3
Since f 0sxd − 0 when x − 0 or x − 6s3 , we set up the following chart:
Interval ”œ„3, 0
”_œ„3, _
3œ„ 3 ’ 4
y=x
FIGURE 11
3œ„ 3 ’ 4
x
inflection points
x , 2s3 2s3 , x , 0 0 , x , s3
x . s3
x
3 2 x2
sx 2 1 1d3
f 99sxd
f
2
2
1
1
CU on (2`, 2s3 )
2
1
1
2
1
1
1
1
1
2
1
2
CD on (2s3 , 0) CU on ( 0, s3 )
CD on (s3 , `)
The points of inflection are (2s3 , 243 s3 ), s0, 0d, and (s3 , 34 s3 ). H. The graph of f is sketched in Figure 11.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
n
250
chapter 3 Applications of Differentiation
1–40 Use the guidelines of this section to sketch the curve. 3
2
2
3
y − 2 1 3x 2 x 1. y − x 1 3x 2.
3. y − x 4 2 4x 4. y − x 4 2 8x 2 1 8
where m 0 is the rest mass of the particle, m is the mass when the particle moves with speed v relative to the observer, and c is the speed of light. Sketch the graph of m as a function of v. 42. In the theory of relativity, the energy of a particle is
5. y − xsx 2 4d3 6. y − x 5 2 5x 7. y −
1 5 5x
2
8 3 3x
1 16x 8. y − s4 2 x d
2 5
2
9. y −
x x 1 5x 10. y− x21 25 2 x 2
11. y −
x 2 x2 1 1 12. y−11 1 2 2 2 3x 1 x 2 x x
13. y −
x 1 14. y− 2 x2 2 4 x 24
15. y −
x2 sx 2 1d2 16. y− 2 x 13 x 11
17. y −
x21 x 18. y− 3 2 x x 21
19. y −
x3 x3 20. y− x 11 x22
2
E − sm 02 c 4 1 h 2 c 2y2 where m 0 is the rest mass of the particle, is its wave length, and h is Planck’s constant. Sketch the graph of E as a function of . What does the graph say about the energy? 43. The figure shows a beam of length L embedded in concrete walls. If a constant load W is distributed evenly along its length, the beam takes the shape of the deflection curve y−2
W 4 WL 3 WL 2 2 x 1 x 2 x 24EI 12EI 24EI
where E and I are positive constants. (E is Young’s modulus of elasticity and I is the moment of inertia of a crosssection of the beam.) Sketch the graph of the deflection curve. y
W
3
0
3 21. y − sx 2 3dsx 22. y − sx 2 4ds x
L
23. y − sx 2 1 x 2 2 24. y − sx 2 1 x 2 x 25. y −
27. y −
x sx 1 1 2
26. y − x s2 2 x 2
x s1 2 x 2 28. y− x sx 2 2 1
29. y − x 2 3x 1y3 30. y − x 5y3 2 5x 2y3 3 3 31. y − s y−s x 2 2 1 32. x3 1 1
33. y − sin3 x 34. y − x 1 cos x 35. y − x tan x, 2y2 , x , y2 36. y − 2x 2 tan x, 2y2 , x , y2
44. Coulomb’s Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles with charge 1 located at positions 0 and 2 on a coordinate line and a particle with charge 21 at a position x between them. It follows from Coulomb’s Law that the net force acting on the middle particle is k k Fsxd − 2 2 1 0,x,2 x sx 2 2d2 where k is a positive constant. Sketch the graph of the net force function. What does the graph say about the force?
37. y − sin x 1 s3 cos x, 22 < x < 2
+1
_1
+1
38. y − csc x 2 2 sin x, 0 , x ,
0
x
2
39. y −
sin x sin x 40. y− 1 1 cos x 2 1 cos x
41. In the theory of relativity, the mass of a particle is m−
m0 s1 2 yc v2
2
x
7et0405x60 45–48 Find an equation of the slant asymptote. Do not sketch the curve. 09/11/09 45. y −
x 2 1 1MasterID: x11
00518
46. y −
4x 3 2 10x 2 2 11x 1 1 x 2 2 3x
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
251
Section 3.6 Graphing with Calculus and Calculators
47. y −
57. Show that the lines y − sbyadx and y − 2sbyadx are slant asymptotes of the hyperbola
2x 3 2 5x 2 1 3x 26x 4 1 2x 3 1 3 48. y− 2 x 2x22 2x 3 2 x
x2 y2 2 2 −1 2 a b
49–54 Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.
58. Let f sxd − sx 3 1 1dyx. Show that
x2 1 1 5x 2 2x 2 50. y− 49. y − x21 x22 51. y −
x3 1 4 x3 52. y− 2 x sx 1 1d2
53. y −
sx 1 1d3 2x 3 1 x 2 1 1 54. y− 2 x 11 sx 2 1d2
lim f f sxd 2 x 2 g − 0
x l 6`
This shows that the graph of f approaches the graph of y − x 2, and we say that the curve y − f sxd is asymptotic to the parabola y − x 2. Use this fact to help sketch the graph of f. 59. Discuss the asymptotic behavior of f sxd −
x4 1 1 x
55. Show that the curve y − s4x 2 1 9 has two slant asymptotes: y − 2x and y − 22x. Use this fact to help sketch the curve.
in the same manner as in Exercise 58. Then use your results to help sketch the graph of f.
56. Show that the curve y − sx 2 1 4x has two slant asymptotes: y − x 1 2 and y − 2x 2 2. Use this fact to help sketch the curve.
60. Use the asymptotic behavior of f sxd − cos x 1 1yx 2 to sketch its graph without going through the curvesketching procedure of this section.
You may want to read “Graphing Calculators and Computers” at www.stewartcalculus.com if you haven’t already. In particular, it explains how to avoid some of the pitfalls of graphing devices by choosing appropriate viewing rectangles.
The method we used to sketch curves in the preceding section was a culmination of much of our study of differential calculus. The graph was the final object that we produced. In this section our point of view is completely different. Here we start with a graph produced by a graphing calculator or computer and then we refine it. We use calculus to make sure that we reveal all the important aspects of the curve. And with the use of graphing devices we can tackle curves that would be far too complicated to consider without technology. The theme is the interaction between calculus and calculators.
Example 1 Graph the polynomial f sxd − 2x 6 1 3x 5 1 3x 3 2 2x 2. Use the graphs of f 9 and f 0 to estimate all maximum and minimum points and intervals of concavity. SOLUTION If we specify a domain but not a range, many graphing devices will deduce a suitable range from the values computed. Figure 1 shows the plot from one such device if we specify that 25 < x < 5. Although this viewing rectangle is useful for showing that the asymptotic behavior (or end behavior) is the same as for y − 2x 6, it is obviously hiding some finer detail. So we change to the viewing rectangle f23, 2g by f250, 100g shown in Figure 2. 100
41,000 y=ƒ y=ƒ _5
FIGURE 1
_1000
_3 5
2 _50
FIGURE 2
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252
Chapter 3 Applications of Differentiation 20
From Figure 2 it appears that there is an absolute minimum value of about 215.33 when x < 21.62 (by using the cursor) and f is decreasing on s2`, 21.62d and increasing on s21.62, `d. Also there appears to be a horizontal tangent at the origin and inflection points when x − 0 and when x is somewhere between 22 and 21. Now let’s try to confirm these impressions using calculus. We differentiate and get
y=fª(x)
_3
2
f 9sxd − 12x 5 1 15x 4 1 9x 2 2 4x
_5
f 0sxd − 60x 4 1 60x 3 1 18x 2 4
FIGURE 3 1 y=ƒ _1
1
_1
FIGURE 4 10 _3
2 y=f·(x)
_30
FIGURE 5
When we graph f 9 in Figure 3 we see that f 9sxd changes from negative to positive when x < 21.62; this confirms (by the First Derivative Test) the minimum value that we found earlier. But, perhaps to our surprise, we also notice that f 9sxd changes from positive to negative when x − 0 and from negative to positive when x < 0.35. This means that f has a local maximum at 0 and a local minimum when x < 0.35, but these were hidden in Figure 2. Indeed, if we now zoom in toward the origin in Figure 4, we see what we missed before: a local maximum value of 0 when x − 0 and a local minimum value of about 20.1 when x < 0.35. What about concavity and inflection points? From Figures 2 and 4 there appear to be inflection points when x is a little to the left of 21 and when x is a little to the right of 0. But it’s difficult to determine inflection points from the graph of f, so we graph the second derivative f 0 in Figure 5. We see that f 0 changes from positive to negative when x < 21.23 and from negative to positive when x < 0.19. So, correct to two decimal places, f is concave upward on s2`, 21.23d and s0.19, `d and concave downward on s21.23, 0.19d. The inflection points are s21.23, 210.18d and s0.19, 20.05d. We have discovered that no single graph reveals all the important features of this polynomial. But Figures 2 and 4, when taken together, do provide an accurate picture. n
Example 2 Draw the graph of the function f sxd −
x 2 1 7x 1 3 x2
in a viewing rectangle that contains all the important features of the function. Estimate the maximum and minimum values and the intervals of concavity. Then use calculus to find these quantities exactly. SOLUTION Figure 6, produced by a computer with automatic scaling, is a disaster. Some graphing calculators use f210, 10g by f210, 10g as the default viewing rectangle, so let’s try it. We get the graph shown in Figure 7; it’s a major improvement. 3 10!*
10 y=ƒ _10
y=ƒ _5
FIGURE 6
5
10
_10
FIGURE 7
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Section 3.6 Graphing with Calculus and Calculators
253
The yaxis appears to be a vertical asymptote and indeed it is because lim
xl0
10 y=ƒ y=1 _20
20
Figure 7 also allows us to estimate the xintercepts: about 20.5 and 26.5. The exact values are obtained by using the quadratic formula to solve the equation x 2 1 7x 1 3 − 0; we get x − (27 6 s37 )y2. To get a better look at horizontal asymptotes, we change to the viewing rectangle f220, 20g by f25, 10g in Figure 8. It appears that y − 1 is the horizontal asymptote and this is easily confirmed:
_5
FIGURE 8
lim
x l 6`
2 _3
0
x 2 1 7x 1 3 −` x2
S
x 2 1 7x 1 3 7 3 − lim 1 1 1 2 2 x l 6` x x x
D
−1
To estimate the minimum value we zoom in to the viewing rectangle f23, 0g by f24, 2g in Figure 9. The cursor indicates that the absolute minimum value is about 23.1 when x < 20.9, and we see that the function decreases on s2`, 20.9d and s0, `d and increases on s20.9, 0d. The exact values are obtained by differentiating:
y=ƒ
f 9sxd − 2 _4
7 6 7x 1 6 2 2 3 − 2 x x x3
This shows that f 9sxd . 0 when 267 , x , 0 and f 9sxd , 0 when x , 267 and when 37 x . 0. The exact minimum value is f (2 67 ) − 2 12 < 23.08. Figure 9 also shows that an inflection point occurs somewhere between x − 21 and x − 22. We could estimate it much more accurately using the graph of the second derivative, but in this case it’s just as easy to find exact values. Since
FIGURE 9
f 0sxd −
14 18 2s7x 1 9d 3 1 4 − x x x4
we see that f 0sxd . 0 when x . 297 sx ± 0d. So f is concave upward on s297 , 0d and s0, `d and concave downward on s2`, 297 d. The inflection point is s297 , 271 27 d. The analysis using the first two derivatives shows that Figure 8 displays all the major aspects of the curve. n
Example 3 Graph the function f sxd − 10
_10
y=ƒ
10
_10
FIGURE 10
x 2sx 1 1d3 . sx 2 2d2sx 2 4d4
SOLUTION Drawing on our experience with a rational function in Example 2, let’s start by graphing f in the viewing rectangle f210, 10g by f210, 10g. From Figure 10 we have the feeling that we are going to have to zoom in to see some finer detail and also zoom out to see the larger picture. But, as a guide to intelligent zooming, let’s first take a close look at the expression for f sxd. Because of the factors sx 2 2d2 and sx 2 4d4 in the denominator, we expect x − 2 and x − 4 to be the vertical asymptotes. Indeed
lim
x l2
x 2sx 1 1d3 x 2sx 1 1d3 − ` and lim −` x l 4 sx 2 2d2sx 2 4d4 sx 2 2d2sx 2 4d4
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254
Chapter 3 Applications of Differentiation
To find the horizontal asymptotes, we divide numerator and denominator by x 6: x 2 sx 1 1d3 ? x 2sx 1 1d3 x3 x3 − 2 4 − 2 sx 2 2d sx 2 4d sx 2 2d sx 2 4d4 ? x2 x4
y
_1
1
2
3
4
S D S DS D 12
2 x
2
12
4 x
4
This shows that f sxd l 0 as x l 6`, so the xaxis is a horizontal asymptote. It is also very useful to consider the behavior of the graph near the xintercepts using an analysis like that in Example 3.4.11. Since x 2 is positive, f sxd does not change sign at 0 and so its graph doesn’t cross the xaxis at 0. But, because of the factor sx 1 1d3, the graph does cross the xaxis at 21 and has a horizontal tangent there. Putting all this information together, but without using derivatives, we see that the curve has to look something like the one in Figure 11. Now that we know what to look for, we zoom in (several times) to produce the graphs in Figures 12 and 13 and zoom out (several times) to get Figure 14.
x
FIGURE 11
0.05
0.0001
500 y=ƒ
y=ƒ _100
3
1 1 11 x x
1
_1.5
0.5
y=ƒ _0.05
_0.0001
FIGURE 13
FIGURE 12
_1
_10
10
FIGURE 14
We can read from these graphs that the absolute minimum is about 20.02 and occurs when x < 220. There is also a local maximum 0 if x , 0
then the root of the equation f sxd − 0 is x − 0. Explain why Newton’s method fails to find the root no matter which initial approximation x 1 ± 0 is used. Illustrate your explanation with a sketch. 33. (a) Use Newton’s method to find the critical numbers of the function f sxd − x 6 2 x 4 1 3x 3 2 2x
277
correct to six decimal places.
48xs1 1 xd60 2 s1 1 xd60 1 1 − 0 Use Newton’s method to solve this equation. 40. The figure shows the sun located at the origin and the earth at the point s1, 0d. (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU < 1.496 3 10 8 km.) There are five locations L 1, L 2, L 3, L 4, and L 5 in this plane of rotation of the earth about the sun where a satellite remains motionless with respect to the earth because the forces acting on the satellite (including the gravitational attractions of the earth and the sun) balance each other. These locations are called libration points. (A solar research satellite has been placed
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278
Chapter 3 Applications of Differentiation
at one of these libration points.) If m1 is the mass of the sun, m 2 is the mass of the earth, and r − m 2ysm1 1 m 2 d, it turns out that the xcoordinate of L 1 is the unique root of the fifthdegree equation
Using the value r < 3.04042 3 10 26, find the locations of the libration points (a) L 1 and (b) L 2. y
psxd − x 5 2 s2 1 rdx 4 1 s1 1 2rdx 3 2 s1 2 rdx 2
L¢
sun
1 2s1 2 rdx 1 r 2 1 − 0
earth
L∞
and the xcoordinate of L 2 is the root of the equation
L¡
L™
x
L£
psxd 2 2rx 2 − 0
A physicist who knows the velocity of a particle might wish to know its position at a given time. An engineer who can measure the variable rate at which water is leaking from a tank wants to know the amount leaked over a certain time period. A biologist who knows the rate at which a bacteria population is increasing might want to deduce what the size of the population will be at some future time. In each case, the problem is to find a function F whose derivative is a known function f. If such a function F exists, it is called an antiderivative of f. Definition A function F is called an antiderivative of f on an interval I if F9sxd − f sxd for all x in I. For instance, let f sxd − x 2. It isn’t difficult to discover an antiderivative of f if we keep the Power Rule in mind. In fact, if Fsxd − 13 x 3, then F9sxd − x 2 − f sxd. But the function Gsxd − 13 x 3 1 100 also satisfies G9sxd − x 2. Therefore both F and G are antiderivatives of f. Indeed, any function of the form Hsxd − 13 x 3 1 C, where C is a constant, is an antiderivative of f. The question arises: are there any others? To answer this question, recall that in Section 3.2 we used the Mean Value Theorem to prove that if two functions have identical derivatives on an interval, then they must differ by a constant (Corollary 3.2.7). Thus if F and G are any two antiderivatives of f , then y
˛
y= 3 +3 ˛ y= 3 +2 ˛ y= 3 +1
0
x
y= ˛ 3
˛ y= 3 1 ˛
y= 3 2
FIGURE 1 Members of the family of antiderivatives of f sxd − x 2
F9sxd − f sxd − G9sxd so Gsxd 2 Fsxd − C, where C is a constant. We can write this as Gsxd − Fsxd 1 C, so we have the following result. 1 Theorem If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is Fsxd 1 C where C is an arbitrary constant. Going back to the function f sxd − x 2, we see that the general antiderivative of f is 1 C. By assigning specific values to the constant C, we obtain a family of functions whose graphs are vertical translates of one another (see Figure 1). This makes sense because each curve must have the same slope at any given value of x. 1 3 3x
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Section 3.9 Antiderivatives
279
Example 1 Find the most general antiderivative of each of the following functions. (a) f sxd − sin x (b) f sxd − x n, n > 0 (c) f sxd − x 23 SOLUTION (a) If Fsxd − 2cos x , then F9sxd − sin x, so an antiderivative of sin x is 2cos x. By Theorem 1, the most general antiderivative is Gsxd − 2cos x 1 C. (b) We use the Power Rule to discover an antiderivative of x n:
d dx
S D x n11 n11
−
sn 1 1dx n − xn n11
Therefore the general antiderivative of f sxd − x n is Fsxd −
x n11 1C n11
This is valid for n > 0 because then f sxd − x n is defined on an interval. (c) If we put n − 23 in the antiderivative formula from part (b), we get the particular antiderivative Fsxd − x 22ys22d. But notice that f sxd − x 23 is not defined at x − 0. Thus Theorem 1 tells us only that the general antiderivative of f is x 22ys22d 1 C on any interval that does not contain 0. So the general antiderivative of f sxd − 1yx 3 is 1 1 C1 if x . 0 2x 2 F sxd − 1 2 2 1 C2 if x , 0 2x 2
■
As in Example 1, every differentiation formula, when read from right to left, gives rise to an antidifferentiation formula. In Table 2 we list some particular antiderivatives. Each formula in the table is true because the derivative of the function in the right column appears in the left column. In particular, the first formula says that the antiderivative of a constant times a function is the constant times the antiderivative of the function. The second formula says that the antiderivative of a sum is the sum of the antiderivatives. (We use the notation F9− f , G9 − t.) able of 2 T Antidifferentiation Formulas To obtain the most general antiderivative from the particular ones in Table 2, we have to add a constant (or constants), as in Example 1.
Function
Particular antiderivative
Function
Particular antiderivative
c f sxd
cFsxd
cos x
sin x
f sxd 1 tsxd
Fsxd 1 Gsxd
sin x
2cos x
x n sn ± 21d
x n11 n11
sec2x
tan x
sec x tan x
sec x
Example 2 Find all functions t such that t9sxd − 4 sin x 1
2x 5 2 sx x
SOLUTION We first rewrite the given function as follows:
t9sxd − 4 sin x 1
2x 5 1 sx 2 − 4 sin x 1 2x 4 2 x x sx
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280
Chapter 3 Applications of Differentiation
Thus we want to find an antiderivative of t9sxd − 4 sin x 1 2x 4 2 x21y2 Using the formulas in Table 2 together with Theorem 1, we obtain We often use a capital letter F to represent an antiderivative of a function f. If we begin with derivative notation, f 9, an antiderivative is f, of course.
tsxd − 4s2cos xd 1 2
x5 x1y2 2 1 1C 5 2
− 24 cos x 1 25 x 5 2 2sx 1 C
n
In applications of calculus it is very common to have a situation as in Example 2, where it is required to find a function, given knowledge about its derivatives. An equation that involves the derivatives of a function is called a differential equation. These will be studied in some detail in Chapter 9, but for the present we can solve some elementary differential equations. The general solution of a differential equation involves an arbitrary constant (or constants) as in Example 2. However, there may be some extra conditions given that will determine the constants and therefore uniquely specify the solution.
Example 3 Find f if f 9sxd − xsx and f s1d − 2. SOLUTION The general antiderivative of
f 9sxd − xsx − x 3y2 f sxd −
is
x 5y2 5 2
1 C − 25 x 5y2 1 C
To determine C we use the fact that f s1d − 2: f s1d − 25 1 C − 2 Solving for C, we get C − 2 2 25 − 85, so the particular solution is f sxd −
2x 5y2 1 8 5
n
Example 4 Find f if f 0sxd − 12x 2 1 6x 2 4, f s0d − 4, and f s1d − 1. SOLUTION The general antiderivative of f 0sxd − 12x 2 1 6x 2 4 is
f 9sxd − 12
x3 x2 16 2 4x 1 C − 4x 3 1 3x 2 2 4x 1 C 3 2
Using the antidifferentiation rules once more, we find that f sxd − 4
x4 x3 x2 13 24 1 Cx 1 D − x 4 1 x 3 2 2x 2 1 Cx 1 D 4 3 2
To determine C and D we use the given conditions that f s0d − 4 and f s1d − 1. Since f s0d − 0 1 D − 4, we have D − 4. Since f s1d − 1 1 1 2 2 1 C 1 4 − 1 we have C − 23. Therefore the required function is
f sxd − x 4 1 x 3 2 2x 2 2 3x 1 4
n
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Section 3.9 Antiderivatives
281
If we are given the graph of a function f, it seems reasonable that we should be able to sketch the graph of an antiderivative F. Suppose, for instance, that we are given that Fs0d − 1. Then we have a place to start, the point s0,1d, and the direction in which we move our pencil is given at each stage by the derivative F9sxd − f sxd. In the next example we use the principles of this chapter to show how to graph F even when we don’t have a formula for f. This would be the case, for instance, when f sxd is determined by experimental data.
Example 5 The graph of a function f is given in Figure 2. Make a rough sketch of an antiderivative F, given that Fs0d − 2.
y
y=ƒ 0
1
2
3
4
x
FIGURE 2 y
y=F(x)
2
Rectilinear Motion
1 0
SOLUTION We are guided by the fact that the slope of y − Fsxd is f sxd. We start at the point s0, 2d and draw F as an initially decreasing function since f sxd is negative when 0 , x , 1. Notice that f s1d − f s3d − 0, so F has horizontal tangents when x − 1 and x − 3. For 1 , x , 3, f sxd is positive and so F is increasing. We see that F has a local minimum when x − 1 and a local maximum when x − 3. For x . 3, f sxd is negative and so F is decreasing on s3, `d. Since f sxd l 0 as x l `, the graph of F becomes flatter as x l `. Also notice that F0sxd − f 9sxd changes from positive to negative at x − 2 and from negative to positive at x − 4, so F has inflection points when x − 2 and x − 4. We use this information to sketch the graph of the antiderivative in Figure 3. n
1
FIGURE 3
x
Antidifferentiation is particularly useful in analyzing the motion of an object moving in a straight line. Recall that if the object has position function s − f std, then the velocity function is vstd − s9std. This means that the position function is an antiderivative of the velocity function. Likewise, the acceleration function is astd − v9std, so the velocity function is an antiderivative of the acceleration. If the acceleration and the initial values ss0d and vs0d are known, then the position function can be found by antidifferentiating twice.
Example 6 A particle moves in a straight line and has acceleration given by astd − 6t 1 4. Its initial velocity is vs0d − 26 cmys and its initial displacement is ss0d − 9 cm. Find its position function sstd. SOLUTION Since v9std − astd − 6t 1 4, antidifferentiation gives vstd − 6
t2 1 4t 1 C − 3t 2 1 4t 1 C 2
Note that vs0d − C. But we are given that vs0d − 26, so C − 26 and vstd − 3t 2 1 4t 2 6
Since vstd − s9std, s is the antiderivative of v: sstd − 3
t3 t2 14 2 6t 1 D − t 3 1 2t 2 2 6t 1 D 3 2
This gives ss0d − D. We are given that ss0d − 9, so D − 9 and the required position function is sstd − t 3 1 2t 2 2 6t 1 9 n An object near the surface of the earth is subject to a gravitational force that produces a downward acceleration denoted by t. For motion close to the ground we may assume that t is constant, its value being about 9.8 mys2 (or 32 ftys2). Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
282
Chapter 3 Applications of Differentiation
Example 7 A ball is thrown upward with a speed of 48 ftys from the edge of a cliff 432 ft above the ground. Find its height above the ground t seconds later. When does it reach its maximum height? When does it hit the ground? SOLUTION The motion is vertical and we choose the positive direction to be upward. At time t the distance above the ground is sstd and the velocity vstd is decreasing. Therefore the acceleration must be negative and we have
astd −
dv − 232 dt
Taking antiderivatives, we have vstd − 232t 1 C
To determine C we use the given information that vs0d − 48. This gives 48 − 0 1 C, so vstd − 232t 1 48
The maximum height is reached when vstd − 0, that is, after 1.5 seconds. Since s9std − vstd, we antidifferentiate again and obtain Figure 4 shows the position function of the ball in Example 7. The graph corroborates the conclusions we reached: The ball reaches its maximum height after 1.5 seconds and hits the ground after about 6.9 seconds. 500
sstd − 216t 2 1 48t 1 D Using the fact that ss0d − 432, we have 432 − 0 1 D and so sstd − 216t 2 1 48t 1 432 The expression for sstd is valid until the ball hits the ground. This happens when sstd − 0, that is, when 216t 2 1 48t 1 432 − 0 or, equivalently,
t 2 2 3t 2 27 − 0
Using the quadratic formula to solve this equation, we get t− 8
0
FIGURE 4
3 6 3s13 2
We reject the solution with the minus sign since it gives a negative value for t. Therefore the ball hits the ground after 3(1 1 s13 )y2 < 6.9 seconds. n
1–20 Find the most general antiderivative of the function. (Check your answer by differentiation.) 1. f sxd − 4x 1 7 2. f sxd − x 2 2 3x 1 2 3. f sxd − 2x 3 2 23 x 2 1 5x 4. f sxd − 6x 5 2 8x 4 2 9x 2 5. f sxd − xs12 x 1 8d 6. f sxd − sx 2 5d 2 7. f sxd − 7x 2y5 1 8x 24y5 8. f sxd − x 3.4 2 2x s221 9. f sxd − s2 10. f sxd − 2 3 3 11. f sxd − 3sx 2 2 s x 12. f sxd − s x 2 1 x sx
13. f sxd − 15. tstd −
10 5 2 4x 3 1 2x 6 tsxd − 9 14. x x6 1 1 t 1 t2 st
16. f std − 3 cos t 2 4 sin t
17. hsd − 2 sin 2 sec 2 18. tsv d − 5 1 3 sec 2 v 19. f std − 8st 2 sec t tan t 20. f sxd − 1 1 2 sin x 1 3ysx
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 3.9 Antiderivatives
; 21–22 Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F.
283
47. The graph of a function is shown in the figure. Make a rough sketch of an antiderivative F, given that Fs0d − 1. y
21. f sxd − 5x 4 2 2x 5, Fs0d − 4 22. f sxd − x 1 2 sin x, Fs0d − 26
y=ƒ
0
x
1
23–42 Find f. 48. The graph of the velocity function of a particle is shown in the figure. Sketch the graph of a position function.
23. f 0sxd − 20x 3 2 12x 2 1 6x 24. f 0sxd − x 6 2 4x 4 1 x 1 1
√
3 25. f 0sxd − 4 2 s x 26. f 0sxd − x 2y3 1 x 22y3
27. f std − 12 1 sin t 28. f std − st 2 2 cos t
0
29. f 9sxd − 1 1 3sx , f s4d − 25
t
30. f 9sxd − 5x 4 2 3x 2 1 4, f s21d − 2 31. f 9sxd − sx s6 1 5xd, f s1d − 10
49. The graph of f 9 is shown in the figure. Sketch the graph of f if f is continuous on f0, 3g and f s0d − 21.
32. f 9std − t 1 1yt , t . 0, f s1d − 6 3
y
33. f 9std − sec t ssec t 1 tan td, 2y2 , t , y2, f sy4d − 21
2
34. f 9sxd − sx 1 1dysx , f s1d − 5 35. f 0sxd − 22 1 12x 2 12x 2, f s0d − 4, 3
36. f 0sxd − 8x 1 5,
f s1d − 0,
f 9s0d − 12
0 _1
f 9s1d − 8
37. f 0sd − sin 1 cos , f s0d − 3, f 9s0d − 4 38. f 0std − 4 2 6yt 4, f s1d − 6, f 9s2d − 9, t . 0 39. f 0sxd − 4 1 6x 1 24x , f s0d − 3, f s1d − 10 40. f 0sxd − 20x 3 1 12x 2 1 4, f s0d − 8, f s1d − 5 3 41. f 0std − s t 2 cos t, f s0d − 2, f s1d − 2
42. f sxd − cos x, f s0d − 1, f 9s0d − 2, f 0s0d − 3
51. f sxd −
44. Find a function f such that f 9sxd − x 3 and the line x 1 y − 0 is tangent to the graph of f .
f x
c
53. vstd − sin t 2 cos t, ss0d − 0
a x
b
sin x , 22 < x < 2 1 1 x2
53–58 A particle is moving with the given data. Find the position of the particle.
46. y b
x
52. f sxd − sx 4 2 2 x 2 1 2 2 2, 23 < x < 3
45–46 The graph of a function f is shown. Which graph is an antiderivative of f and why? f
2
; 51–52 Draw a graph of f and use it to make a rough sketch of the antiderivative that passes through the origin.
43. Given that the graph of f passes through the point (2, 5) and that the slope of its tangent line at sx, f sxdd is 3 2 4x, find f s1d.
a
1
; 50. (a) Use a graphing device to graph f sxd − 2x 2 3 sx . (b) Starting with the graph in part (a), sketch a rough graph of the antiderivative F that satisfies Fs0d − 1. (c) Use the rules of this section to find an expression for Fsxd. (d) Graph F using the expression in part (c). Compare with your sketch in part (b).
2
45. y
y=fª(x)
1
54. vstd − t 2 2 3 st , ss4d − 8 55. astd − 2t 1 1, ss0d − 3, v s0d − 22
c
56. astd − 3 cos t 2 2 sin t, ss0d − 0, v s0d − 4
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
284
chapter 3 Applications of Differentiation
57. astd − 10 sin t 1 3 cos t, ss0d − 0, ss2d − 12 58. astd − t 2 2 4t 1 6, ss0d − 0, ss1d − 20 59. A stone is dropped from the upper observation deck (the Space Deck) of the CN Tower, 450 m above the ground. (a) Find the distance of the stone above ground level at time t. (b) How long does it take the stone to reach the ground? (c) With what velocity does it strike the ground? (d) If the stone is thrown downward with a speed of 5 mys, how long does it take to reach the ground? 60. Show that for motion in a straight line with constant acceleration a, initial velocity v 0, and initial displacement s 0, the dis placement after time t is
66. The linear density of a rod of length 1 m is given by sxd − 1ysx , in grams per centimeter, where x is measured in centimeters from one end of the rod. Find the mass of the rod. 67. Since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 mys and its downward acceleration is a−
H
9 2 0.9t 0
if 0 < t < 10 if t . 10
If the raindrop is initially 500 m above the ground, how long does it take to fall?
s − 12 at 2 1 v 0 t 1 s 0
68. A car is traveling at 50 miyh when the brakes are fully applied, producing a constant deceleration of 22 ftys2. What is the distance traveled before the car comes to a stop?
61. An object is projected upward with initial velocity v 0 meters per second from a point s0 meters above the ground. Show that
69. What constant acceleration is required to increase the speed of a car from 30 miyh to 50 miyh in 5 seconds?
fvstdg 2 − v02 2 19.6fsstd 2 s0 g 62. Two balls are thrown upward from the edge of the cliff in Example 7. The first is thrown with a speed of 48 ftys and the other is thrown a second later with a speed of 24 ftys. Do the balls ever pass each other? 63. A stone was dropped off a cliff and hit the ground with a speed of 120 ftys. What is the height of the cliff? 64. If a diver of mass m stands at the end of a diving board with length L and linear density , then the board takes on the shape of a curve y − f sxd, where EI y 0 − mtsL 2 xd 1 12 tsL 2 xd2
E and I are positive constants that depend on the material of the board and t s, 0d is the acceleration due to gravity. (a) Find an expression for the shape of the curve. (b) Use f sLd to estimate the distance below the horizontal at the end of the board. y
0
x
65. A company estimates that the marginal cost (in dollars per item) of producing x items is 1.92 2 0.002x. If the cost of producing one item is $562, find the cost of producing 100 items.
70. A car braked with a constant deceleration of 16 ftys2, producing skid marks measuring 200 ft before coming to a stop. How fast was the car traveling when the brakes were first applied? 71. A car is traveling at 100 kmyh when the driver sees an accident 80 m ahead and slams on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup? 72. A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is astd − 60t, at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (downward) velocity slows linearly to 218 ftys in 5 seconds. The rocket then “floats” to the ground at that rate. (a) Determine the position function s and the velocity function v (for all times t). Sketch the graphs of s and v. (b) At what time does the rocket reach its maximum height, and what is that height? (c) At what time does the rocket land? 73. A highspeed bullet train accelerates and decelerates at the rate of 4 ftys2. Its maximum cruising speed is 90 miyh. (a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes? (b) Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions? (c) Find the minimum time that the train takes to travel between two consecutive stations that are 45 miles apart. (d) The trip from one station to the next takes 37.5 minutes. How far apart are the stations?
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
CHAPTER 3 Review
285
3 Review CONCEPT CHECK
Answers to the Concept Check can be found on the back endpapers.
1. Explain the difference between an absolute maximum and a local maximum. Illustrate with a sketch.
8. Explain the meaning of each of the following statements. (a) lim f sxd − L (b) lim f sxd − L (c) lim f sxd − `
2. What does the Extreme Value Theorem say?
3. (a) State Fermat’s Theorem. (b) Define a critical number of f.
xl`
xl`
9. If you have a graphing calculator or computer, why do you need calculus to graph a function?
4. Explain how the Closed Interval Method works.
10. (a) Given an initial approximation x1 to a root of the equation f sxd − 0, explain geometrically, with a diagram, how the second approximation x 2 in Newton’s method is obtained. (b) Write an expression for x 2 in terms of x1, f sx 1 d, and f 9sx 1d. (c) Write an expression for x n11 in terms of x n , f sx n d, and f 9sx n d. (d) Under what circumstances is Newton’s method likely to fail or to work very slowly?
5. (a) State Rolle’s Theorem. (b) State the Mean Value Theorem and give a geometric interpretation. 6. (a) State the Increasing/Decreasing Test. (b) What does it mean to say that f is concave upward on an interval I? (c) State the Concavity Test. (d) What are inflection points? How do you find them? 7. (a) State the First Derivative Test. (b) State the Second Derivative Test. (c) What are the relative advantages and disadvantages of these tests?
x l 2`
(d) The curve y − f sxd has the horizontal asymptote y − L.
11. (a) What is an antiderivative of a function f ? (b) Suppose F1 and F2 are both antiderivatives of f on an interval I. How are F1 and F2 related?
TRUEFALSE QUIZ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f 9scd − 0, then f has a local maximum or minimum at c. 2. If f has an absolute minimum value at c, then f 9scd − 0. 3. If f is continuous on sa, bd, then f attains an absolute maximum value f scd and an absolute minimum value f sd d at some numbers c and d in sa, bd.
10. T here exists a function f such that f sxd , 0, f 9sxd , 0, and f 0 sxd . 0 for all x. 11. If f and t are increasing on an interval I, then f 1 t is increasing on I. 12. I f f and t are increasing on an interval I, then f 2 t is increasing on I. 13. I f f and t are increasing on an interval I, then f t is increasing on I.
4. If f is differentiable and f s21d − f s1d, then there is a number c such that c , 1 and f 9scd − 0.
14. If f and t are positive increasing functions on an interval I, then f t is increasing on I.
5. If f 9sxd , 0 for 1 , x , 6, then f is decreasing on (1, 6).
15. If f is increasing and f sxd . 0 on I, then tsxd − 1yf sxd is decreasing on I.
 
6. If f 0s2d − 0, then s2, f s2dd is an inflection point of the curve y − f sxd. 7. If f 9sxd − t9sxd for 0 , x , 1, then f sxd − tsxd for 0 , x , 1. 8. There exists a function f such that f s1d − 22, f s3d − 0, and f 9sxd . 1 for all x. 9. There exists a function f such that f sxd . 0, f 9sxd , 0, and f 0 sxd . 0 for all x.
16. If f is even, then f 9 is even. 17. If f is periodic, then f 9 is periodic. 18. The most general antiderivative of f sxd − x 22 is Fsxd − 2
1 1C x
19. If f 9sxd exists and is nonzero for all x, then f s1d ± f s0d.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
286
Chapter 3 Applications of Differentiation
EXERCISES 1–6 Find the local and absolute extreme values of the function on the given interval. 1. f sxd − x 3 2 9x 2 1 24 x 2 2, f0, 5g
(b) For what values of x does f have a local maximum or minimum? (c) Sketch the graph of f 0. (d) Sketch a possible graph of f.
2. f sxd − x s1 2 x , f21, 1g 3. f sxd −
y
3x 2 4 , f22, 2g x2 1 1
y=f ª(x)
_2 _1
4. f sxd − sx 1 x 1 1 , f22, 1g 2
0
1
2
3
4
5
6
7
x
5. f sxd − x 1 2 cos x, f2, g 6. f sxd − sin x 1 cos 2 x, f0, g
17–28 Use the guidelines of Section 3.5 to sketch the curve. 17. y − 2 2 2x 2 x 3
7–12 Find the limit.
18. y − 22 x 3 2 3x 2 1 12 x 1 5
3x 4 1 x 2 5 7. lim x l ` 6x 4 2 2x 2 1 1 8. lim
tl`
9. lim
t3 2 t 1 2 s2t 2 1dst 2 1 t 1 1d
x l 2`
s4x 2 1 1 10. lim sx 2 1 x 3 d x l 2` 3x 2 1
sin 4 x 11. lim ss4x 2 1 3x 2 2xd 12. lim xl` x l ` sx 13–15 Sketch the graph of a function that satisfies the given conditions.
x 19. y − 3x 4 2 4x 3 1 2 20. y− 1 2 x2 21. y −
1 1 1 22. y− 2 2 xsx 2 3d2 x sx 2 2d 2
23. y −
sx 2 1d 3 24. y − s1 2 x 1 s1 1 x x2
25. y − x s2 1 x 26. y − x 2y3sx 2 3d 2 27. y − sin 2 x 2 2 cos x 28. y − 4x 2 tan x, 2y2 , x , y2
13. f s0d − 0, f 9s22d − f 9s1d − f 9s9d − 0, lim f sxd − 0,
xl`
lim f sxd − 2`,
x l6
f 9sxd , 0 on s2`, 22d, s1, 6d, and s9, `d, f 9sxd . 0 on s22, 1d and s6, 9d, f 0sxd . 0 on s2`, 0d and s12, `d, f 0sxd , 0 on s0, 6d and s6, 12d 14. f s0d − 0, f is continuous and even, f 9sxd − 2x if 0 , x , 1, f 9sxd − 21 if 1 , x , 3, f 9sxd − 1 if x . 3
; 29–32 Produce graphs of f that reveal all the important aspects of the curve. Use graphs of f 9 and f 0 to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. In Exercise 29 use calculus to find these quantities exactly. 29. f sxd −
x2 2 1 x3
30. f sxd −
x3 1 1 x6 1 1
31. f sxd − 3x 6 2 5x 5 1 x 4 2 5x 3 2 2x 2 1 2
15. f is odd, f 9sxd , 0 for 0 , x , 2, f 9sxd . 0 for x . 2, f 0sxd . 0 for 0 , x , 3, f 0sxd , 0 for x . 3, lim f sxd − 22 xl`
16. The figure shows the graph of the derivative f 9of a function f. (a) On what intervals is f increasing or decreasing?
32. f sxd − x 2 1 6.5 sin x, 25 < x < 5 33. Show that the equation 3x 1 2 cos x 1 5 − 0 has exactly one real root. 34. Suppose that f is continuous on f0, 4g, f s0d − 1, and 2 < f 9sxd < 5 for all x in s0, 4d. Show that 9 < f s4d < 21.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
chapter 3 Review
35. By applying the Mean Value Theorem to the function f sxd − x 1y5 on the interval f32, 33g, show that
; 48. A manufacturer determines that the cost of making x units of a commodity is Csxd − 1800 1 25x 2 0.2x 2 1 0.001x 3
5 2,s 33 , 2.0125
36. For what values of the constants a and b is s1, 3d a point of inflection of the curve y − ax 3 1 bx 2 ? 37. Let tsxd − f sx 2 d, where f is twice differentiable for all x, f 9sxd . 0 for all x ± 0, and f is concave downward on s2`, 0d and concave upward on s0, `d. (a) At what numbers does t have an extreme value? (b) Discuss the concavity of t. 38. Find two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible. 39. Show that the shortest distance from the point sx 1, y1 d to the straight line Ax 1 By 1 C − 0 is
 Ax
1

1 By1 1 C
sA2 1 B 2
40. Find the point on the hyperbola x y − 8 that is closest to the point s3, 0d. 41. Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r. 42. Find the volume of the largest circular cone that can be inscribed in a sphere of radius r.

 

43. In D ABC, D lies on AB, CD AB, AD − BD − 4 cm, and CD − 5 cm. Where should a point P be chosen on CD so that the sum PA 1 PB 1 PC is a minimum?


      44. Solve Exercise 43 when  CD  − 2 cm.
and the demand function is psxd − 48.2 2 0.03x. (a) Graph the cost and revenue functions and use the graphs to estimate the production level for maximum profit. (b) Use calculus to find the production level for maximum profit. (c) Estimate the production level that minimizes the average cost.
49. Use Newton’s method to find the root of the equation x 5 2 x 4 1 3x 2 2 3x 2 2 − 0 in the interval f1, 2g correct to six decimal places. 50. Use Newton’s method to find all solutions of the equation sin x − x 2 2 3x 1 1 correct to six decimal places. 51. Use Newton’s method to find the absolute maximum value of the function f std − cos t 1 t 2 t 2 correct to eight decimal places. 52. Use the guidelines in Section 3.5 to sketch the curve y − x sin x, 0 < x < 2. Use Newton’s method when necessary. 53–54 Find the most general antiderivative of the function. 53. f sxd − 4 sx 2 6x 2 1 3 54. tsxd − cos x 1 2 sec 2 x 55–58 Find f. 55. f 9std − 2t 2 3 sin t, f s0d − 5
45. The velocity of a wave of length L in deep water is v−K
Î
287
L C 1 C L
where K and C are known positive constants. What is the length of the wave that gives the minimum velocity?
56. f 9sud −
u 2 1 su , f s1d − 3 u
57. f 0sxd − 1 2 6x 1 48x 2, f s0d − 1, f 9s0d − 2 58. f 0sxd − 5x 3 1 6x 2 1 2, f s0d − 3, f s1d − 22
46. A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?
59–60 A particle is moving according to the given data. Find the position of the particle.
47. A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at $12, average attendance at a game has been 11,000. A market survey indicates that for each dollar the ticket price is lowered, average attendance will increase by 1000. How should the owners of the team set the ticket price to maximize their revenue from ticket sales?
60. astd − sin t 1 3 cos t, ss0d − 0, v s0d − 2
59. vstd − 2t 2 sin t, ss0d − 3
; 61. Use a graphing device to draw a graph of the function f sxd − x 2 sinsx 2 d, 0 < x < , and use that graph to sketch the antiderivative F of f that satisfies the initial condition Fs0d − 0.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
288
chapter 3 Applications of Differentiation
tal, as shown in the figure. Show that the range of the projectile, measured up the slope, is given by
; 62. Investigate the family of curves given by f sxd − x 4 1 x 3 1 cx 2
Rsd −
In particular you should determine the transitional value of c at which the number of critical numbers changes and the transitional value at which the number of inflection points changes. Illustrate the various possible shapes with graphs.
63. A canister is dropped from a helicopter 500 m above the ground. Its parachute does not open, but the canister has been designed to withstand an impact velocity of 100 mys. Will it burst?
(b) Determine so that R is a maximum. (c) Suppose the plane is at an angle below the horizontal. Determine the range R in this case, and determine the angle at which the projectile should be fired to maximize R. y
64. In an automobile race along a straight road, car A passed car B twice. Prove that at some time during the race their accelerations were equal. State the assumptions that you make.
¨
65. A rectangular beam will be cut from a cylindrical log of radius 10 inches. (a) Show that the beam of maximal crosssectional area is a square. (b) Four rectangular planks will be cut from the four sections of the log that remain after cutting the square beam. Determine the dimensions of the planks that will have maximal crosssectional area.
depth
¨ h
(a) Suppose the projectile is fired from the base of a plane that is inclined at an angle , . 0, from the horizon
d 40
66. If a projectile is fired with an initial velocity v at an angle of inclination from the horizontal, then its trajectory, neglecting air resistance, is the parabola
x
7et04rx80
(c) Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log.
t x 2 0 , , 2v 2 cos 2 2
R
67. A light is to be placed atop a pole of height h feet to illuminate a09/09/09 busy traffic circle, which has a radius of 40 ft. MasterID: 00594 The intensity of illumination I at any point P on the circle is directly proportional to the cosine of the angle (see the figure) and inversely proportional to the square of the distance d from the source. (a) How tall should the light pole be to maximize I? (b) Suppose that the light pole is h feet tall and that a woman is walking away from the base of the pole at the rate of 4 ftys. At what rate is the intensity of the light at the point on her back 4 ft above the ground decreasing when she reaches the outer edge of the traffic circle?
10
y − stan dx 2
å
0
width
2v 2 cos sins 2 d t cos2
CAS
68. If f sxd −
P
cos 2 x
, 2 < x < , use the graphs of f, sx 1 x 1 1 f 9, and f 0 to estimate the xcoordinates of the maximum and minimum points and inflection points of f. 2
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Problems Plus
One of the most important principles of problem solving is analogy (see page 98). If you are having trouble getting started on a problem, it is sometimes helpful to start by solving a similar, but simpler, problem. The following example illustrates the principle. Cover up the solution and try solving it yourself first. Example If x, y, and z are positive numbers, prove that sx 2 1 1dsy 2 1 1dsz 2 1 1d >8 xyz SOLUTION It may be difficult to get started on this problem. (Some students have tackled it by multiplying out the numerator, but that just creates a mess.) Let’s try to think of a similar, simpler problem. When several variables are involved, it’s often helpful to think of an analogous problem with fewer variables. In the present case we can reduce the number of variables from three to one and prove the analogous inequality
1
x2 1 1 >2 x
for x . 0
In fact, if we are able to prove (1), then the desired inequality follows because sx 2 1 1ds y 2 1 1dsz 2 1 1d − xyz
S DS DS D x2 1 1 x
y2 1 1 y
z2 1 1 z
>2?2?2−8
The key to proving (1) is to recognize that it is a disguised version of a minimum problem. If we let f sxd −
x2 1 1 1 − x 1 x . 0 x x
then f 9sxd − 1 2 s1yx 2 d, so f 9sxd − 0 when x − 1. Also, f 9sxd , 0 for 0 , x , 1 and f 9sxd . 0 for x . 1. Therefore the absolute minimum value of f is f s1d − 2. This means that x2 1 1 >2 x PS Look Back What have we learned from the solution to this example? ● To solve a problem involving several variables, it might help to solve a similar problem with just one variable. ● When trying to prove an inequality, it might help to think of it as a maximum or minimum problem.
for all positive values of x
and, as previously mentioned, the given inequality follows by multiplication. The inequality in (1) could also be proved without calculus. In fact, if x . 0, we have x2 1 1 > 2 &? x
x 2 1 1 > 2x &? x 2 2 2x 1 1 > 0
&? sx 2 1d2 > 0 Because the last inequality is obviously true, the first one is true too.
■
289 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.


1. Show that sin x 2 cos x < s2 for all x.
Problems
 
2. S how that x y s4 2 x ds4 2 y 2 d < 16 for all numbers x and y such that x < 2 and y < 2. 2
2
2
 
3. S how that the inflection points of the curve y − ssin xdyx lie on the curve y 2 sx 4 1 4d − 4. 4. Find the point on the parabola y − 1 2 x 2 at which the tangent line cuts from the first quadrant the triangle with the smallest area. 5. Find the highest and lowest points on the curve x 2 1 x y 1 y 2 − 12. 6. Water is flowing at a constant rate into a spherical tank. Let Vstd be the volume of water in the tank and Hstd be the height of the water in the tank at time t. (a) What are the meanings of V9std and H9std? Are these derivatives positive, negative, or zero? (b) Is V 0std positive, negative, or zero? Explain. (c) Let t1, t 2, and t 3 be the times when the tank is onequarter full, half full, and threequarters full, respectively. Are the values H 0st1d, H 0st 2 d, and H 0st 3 d positive, negative, or zero? Why? 7. Find the absolute maximum value of the function f sxd − y
1 1 1 11 x 11 x22
 


8. Find a function f such that f 9s21d − 12 , f 9s0d − 0, and f 0sxd . 0 for all x, or prove that such a function cannot exist.
Q
9. I f Psa, a 2 d is any point on the parabola y − x 2, except for the origin, let Q be the point where the normal line at P intersects the parabola again (see the figure). (a) Show that the ycoordinate of Q is smallest when a − 1ys2 . (b) Show that the line segment PQ has the shortest possible length when a − 1ys2 .
P
10. An isosceles triangle is circumscribed about the unit circle so that the equal sides meet at the point s0, ad on the yaxis (see the figure). Find the value of a that minimizes the lengths of the equal sides. (You may be surprised that the result does not give an equilateral triangle.).
x
0
FIGURE for PROBLEM 9
y
y=≈
B
11. T he line y − mx 1 b intersects the parabola y − x 2 in points A and B. (See the figure.) Find the point P on the arc AOB of the parabola that maximizes the area of the triangle PAB.
A y=mx+b O
P
FIGURE for PROBLEM 11
x
 
12. Sketch the graph of a function f such that f 9sxd , 0 for all x, f 0sxd . 0 for x . 1, f 0sxd , 0 for x , 1, and lim x l6` f f sxd 1 xg − 0.
 
13. Determine the values of the number a for which the function f has no critical number: f sxd − sa 2 1 a 2 6d cos 2x 1 sa 2 2dx 1 cos 1
290 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
14. Sketch the region in the plane consisting of all points sx, yd such that


2xy < x 2 y < x 2 1 y 2
  

15. Let ABC be a triangle with /BAC − 1208 and AB AC − 1. (a) Express the length of the angle bisector AD in terms of x − AB . (b) Find the largest possible value of AD .


C

16. (a) Let ABC be a triangle with right angle A and hypotenuse a − BC . (See the figure.) If the inscribed circle touches the hypotenuse at D, show that D
 CD  − 12 s BC  1  AC  2  AB d
A
 

B
FIGURE for PROBLEM 16
(b) If − 12 /C, express the radius r of the inscribed circle in terms of a and . (c) If a is fixed and varies, find the maximum value of r.
17. A triangle with sides a, b, and c varies with time t, but its area never changes. Let be the angle opposite the side of length a and suppose always remains acute. (a) Express dydt in terms of b, c, , dbydt, and dcydt. (b) Express daydt in terms of the quantities in part (a). 18. ABCD is a square piece of paper with sides of length 1 m. A quartercircle is drawn from B to D with center A. The piece of paper is folded along EF, with E on AB and F on AD, so that A falls on the quartercircle. Determine the maximum and minimum areas that the triangle AEF can have. 19. The speeds of sound c1 in an upper layer and c2 in a lower layer of rock and the thickness h of the upper layer can be determined by seismic exploration if the speed of sound in the lower layer is greater than the speed in the upper layer. A dynamite charge is detonated at a point P and the transmitted signals are recorded at a point Q, which is a distance D from P. The first signal to arrive at Q travels along the surface and takes T1 seconds. The next signal travels from P to a point R, from R to S in the lower layer, and then to Q, taking T2 seconds. The third signal is reflected off the lower layer at the midpoint O of RS and takes T3 seconds to reach Q. (See the figure.) P
Q
D speed of sound=c¡
h
¨
¨ R
O
S
speed of sound=c™
(a) Express T1, T2, and T3 in terms of D, h, c1, c2, and . (b) Show that T2 is a minimum when sin − c1yc2. (c) Suppose that D − 1 km, T1 − 0.26 s, T2 − 0.32 s, and T3 − 0.34 s. Find c1, c2, and h.
N ote: Geophysicists use this technique when studying the structure of the earth’s crust, whether searching for oil or examining fault lines.
20. For what values of c is there a straight line that intersects the curve y − x 4 1 cx 3 1 12x 2 2 5x 1 2
in four distinct points?
291 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
d B
E
x
C r
F
D
FIGURE for PROBLEM 21
21. O ne of the problems posed by the Marquis de l’Hospital in his calculus textbook Analyse des Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C by a rope of length r. At another point B on the ceiling, at a distance d from C (where d . r), a rope of length , is attached and passed through the pulley at F and connected to a weight W. The weight is released and comes to rest at its equilibrium position D. (See the figure.) As l’Hospital argued, this happens when the distance ED is maximized. Show that when the system reaches equilibrium, the value of x is


r (r 1 sr 2 1 8d 2 ) 4d
Notice that this expression is independent of both W and ,.
22. Given a sphere with radius r, find the height of a pyramid of minimum volume whose base is a square and whose base and triangular faces are all tangent to the sphere. What if the base of the pyramid is a regular ngon? (A regular ngon is a polygon with n equal sides and angles.) (Use the fact that the volume of a pyramid is 13 Ah, where A is the area of the base.) 23. Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for the snowball to melt completely? 24. A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the first one. This process is continued until n chambers, including the sphere, are formed. (The figure shows the case n − 4.) Use mathematical induction to prove that the maximum height of any bubble tower with n chambers is 1 1 sn .
292 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
4
Integrals
The photo shows Lake Lanier, which is a reservoir in Georgia, USA. In Exercise 63 in Section 4.4 you will estimate the amount of water that flowed into Lake Lanier during a certain time period. JRC, Inc. / Alamy
In chapter 2 we used the tangent and velocity problems to introduce the derivative, which is the central idea in differential calculus. In much the same way, this chapter starts with the area and distance problems and uses them to formulate the idea of a definite integral, which is the basic concept of integral calculus. We will see in Chapters 5 and 8 how to use the integral to solve problems concerning volumes, lengths of curves, population predictions, cardiac output, forces on a dam, work, consumer surplus, and baseball, among many others. There is a connection between integral calculus and differential calculus. The Fundamental Theorem of Calculus relates the integral to the derivative, and we will see in this chapter that it greatly simplifies the solution of many problems.
293 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
294
Chapter 4 Integrals
Now is a good time to read (or reread) A Preview of Calculus (see page 1). It discusses the unifying ideas of calculus and helps put in perspective where we have been and where we are going. y
y=ƒ x=a
S
x=b
a
0
x
b
In this section we discover that in trying to find the area under a curve or the distance traveled by a car, we end up with the same special type of limit.
The Area Problem We begin by attempting to solve the area problem: find the area of the region S that lies under the curve y − f sxd from a to b. This means that S, illustrated in Figure 1, is bounded by the graph of a continuous function f [where f sxd > 0], the vertical lines x − a and x − b, and the xaxis. In trying to solve the area problem we have to ask ourselves: what is the meaning of the word area? This question is easy to answer for regions with straight sides. For a rectangle, the area is defined as the product of the length and the width. The area of a triangle is half the base times the height. The area of a polygon is found by dividing it into triangles (as in Figure 2) and adding the areas of the triangles.
FIGURE 1 S − hsx, yd a < x < b, 0 < y < f sxdj

A™
w
h
A¢
A¡
b
l
FIGURE 2
A£
A= 21 bh
A=lw
A=A¡+A™+A£+A¢
However, it isn’t so easy to find the area of a region with curved sides. We all have an intuitive idea of what the area of a region is. But part of the area problem is to make this intuitive idea precise by giving an exact definition of area. Recall that in defining a tangent we first approximated the slope of the tangent line by slopes of secant lines and then we took the limit of these approximations. We pursue a similar idea for areas. We first approximate the region S by rectangles and then we take the limit of the areas of these rectangles as we increase the number of rectangles. The following example illustrates the procedure. y
Example 1 Use rectangles to estimate the area under the parabola y − x 2 from 0 to 1
(1, 1)
(the parabolic region S illustrated in Figure 3). SOLUTION We first notice that the area of S must be somewhere between 0 and 1 because S is contained in a square with side length 1, but we can certainly do better than that. Suppose we divide S into four strips S1, S2, S3, and S4 by drawing the vertical lines x − 14, x − 12, and x − 34 as in Figure 4(a).
y=≈ S 0
1
x
y
y
(1, 1)
(1, 1)
y=≈
FIGURE 3
S¡ 0
FIGURE 4
S¢
S™ 1 4
S£ 1 2
(a)
3 4
1
x
0
1 4
1 2
3 4
1
x
(b)
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295
Section 4.1 Areas and Distances
We can approximate each strip by a rectangle that has the same base as the strip and whose height is the same as the right edge of the strip [see Figure 4(b)]. In other words, the heights of these rectangles are the values of the function f sxd − x 2 at the right end
f g f 41 , 12 g, f 12 , 34 g, and f 34 , 1g.
points of the subintervals 0, 14 ,
Each rectangle has width 14 and the heights are ( 14 ) , ( 12 ) , ( 34 ) , and 12. If we let R 4 be the sum of the areas of these approximating rectangles, we get 2
2
2
R4 − 14 ? ( 14 ) 1 14 ? ( 12 ) 1 14 ? ( 34 ) 1 14 ? 12 − 15 32 − 0.46875 2
2
2
From Figure 4(b) we see that the area A of S is less than R 4, so A , 0.46875 y
Instead of using the rectangles in Figure 4(b) we could use the smaller rectangles in Figure 5 whose heights are the values of f at the left endpoints of the subintervals. (The leftmost rectangle has collapsed because its height is 0.) The sum of the areas of these approximating rectangles is
(1, 1)
y=≈
7 L 4 − 14 ? 0 2 1 14 ? ( 14 ) 1 14 ? ( 12 ) 1 14 ? ( 34 ) − 32 − 0.21875 2
2
2
We see that the area of S is larger than L 4, so we have lower and upper estimates for A: 0
1 4
1 2
3 4
1
x
0.21875 , A , 0.46875 We can repeat this procedure with a larger number of strips. Figure 6 shows what happens when we divide the region S into eight strips of equal width.
FIGURE 5
y
y (1, 1)
y=≈
0
FIGURE 6 Approximating S with eight rectangles
1 8
1
(a) Using left endpoints
(1, 1)
x
0
1 8
1
x
(b) Using right endpoints
By computing the sum of the areas of the smaller rectangles sL 8 d and the sum of the areas of the larger rectangles sR 8 d, we obtain better lower and upper estimates for A: 0.2734375 , A , 0.3984375 n
Ln
Rn
10 20 30 50 100 1000
0.2850000 0.3087500 0.3168519 0.3234000 0.3283500 0.3328335
0.3850000 0.3587500 0.3501852 0.3434000 0.3383500 0.3338335
So one possible answer to the question is to say that the true area of S lies somewhere between 0.2734375 and 0.3984375. We could obtain better estimates by increasing the number of strips. The table at the left shows the results of similar calculations (with a computer) using n rectangles whose heights are found with left endpoints sL n d or right endpoints sR n d. In particular, we see by using 50 strips that the area lies between 0.3234 and 0.3434. With 1000 strips we narrow it down even more: A lies between 0.3328335 and 0.3338335. A good estimate is obtained by averaging these numbers: A < 0.3333335. n
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
296
Chapter 4 Integrals
From the values in the table in Example 1, it looks as if R n is approaching increases. We confirm this in the next example.
1 3
as n
Example 2 For the region S in Example 1, show that the sum of the areas of the upper approximating rectangles approaches 13, that is, lim R n − 13
nl`
SOLUTION R n is the sum of the areas of the n rectangles in Figure 7. Each rectangle has width 1yn and the heights are the values of the function f sxd − x 2 at the points 1yn, 2yn, 3yn, . . . , nyn; that is, the heights are s1ynd2, s2ynd2, s3ynd2, . . . , snynd2. Thus
y (1, 1)
y=≈
Rn −
0
1
1 n
x
FIGURE 7
1 n
SD SD SD 1 n
2
1
1 n
2
2 n
1
1 n
3 n
2
1 ∙∙∙ 1
−
1 1 2 s1 1 2 2 1 3 2 1 ∙ ∙ ∙ 1 n 2 d n n2
−
1 2 s1 1 2 2 1 3 2 1 ∙ ∙ ∙ 1 n 2 d n3
1 n
SD n n
2
Here we need the formula for the sum of the squares of the first n positive integers:
1
12 1 2 2 1 3 2 1 ∙ ∙ ∙ 1 n 2 −
nsn 1 1ds2n 1 1d 6
Perhaps you have seen this formula before. It is proved in Example 5 in Appendix E. Putting Formula 1 into our expression for R n, we get Rn − Here we are computing the limit of the sequence hR n j. Sequences and their limits were discussed in A Preview of Calculus and will be studied in detail in Section 11.1. The idea is very similar to a limit at infinity (Section 3.4) except that in writing lim n l ` we restrict n to be a positive integer. In particular, we know that 1 lim − 0 nl ` n
Thus we have
When we write lim n l ` Rn − 13 we mean that we can make Rn as close to 13 as we like by taking n sufficiently large.
1 nsn 1 1ds2n 1 1d sn 1 1ds2n 1 1d − 3 n 6 6n 2
lim R n − lim
nl `
nl `
sn 1 1ds2n 1 1d 6n 2
− lim
1 6
− lim
1 6
nl`
nl`
−
S DS D S DS D n11 n
11
1 n
2n 1 1 n
21
1 n
1 1 12− 6 3
n
It can be shown that the lower approximating sums also approach 13, that is, lim L n − 13
nl`
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297
Section 4.1 Areas and Distances
From Figures 8 and 9 it appears that, as n increases, both L n and R n become better and better approximations to the area of S. Therefore we define the area A to be the limit of the sums of the areas of the approximating rectangles, that is, TEC In Visual 4.1 you can create pictures like those in Figures 8 and 9 for other values of n.
A − lim R n − lim L n − 13 nl`
nl`
y
y
n=10 R¡¸=0.385
0
y
n=50 R∞¸=0.3434
n=30 R£¸Å0.3502
1
x
0
1
x
0
1
x
1
x
FIGURE 8 Right endpoints produce upper sums because f sxd − x 2 is increasing. y
y
n=10 L¡¸=0.285
0
y
n=50 L∞¸=0.3234
n=30 L£¸Å0.3169
1
x
0
1
x
0
FIGURE 9 Left endpoints produce lower sums because f sxd − x 2 is increasing.
Let’s apply the idea of Examples 1 and 2 to the more general region S of Figure 1. We start by subdividing S into n strips S1, S2 , . . . , Sn of equal width as in Figure 10. y
y=ƒ
S¡
FIGURE 10
0
a
S™ ⁄
S£ x2
Si ‹
. . . xi1
Sn xi
. . . xn1
b
x
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298
Chapter 4 Integrals
The width of the interval fa, bg is b 2 a, so the width of each of the n strips is Dx −
b2a n
These strips divide the interval fa, bg into n subintervals fx 0 , x 1 g, fx 1, x 2 g, fx 2 , x 3 g, . . . ,
fx n21, x n g
where x 0 − a and x n − b. The right endpoints of the subintervals are x 1 − a 1 Dx, x 2 − a 1 2 Dx, x 3 − a 1 3 Dx, ∙ ∙ ∙ Let’s approximate the ith strip Si by a rectangle with width Dx and height f sx i d, which is the value of f at the right endpoint (see Figure 11). Then the area of the ith rectangle is f sx i d Dx. What we think of intuitively as the area of S is approximated by the sum of the areas of these rectangles, which is R n − f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dx y
Îx
f(xi)
0
FIGURE 11
a
⁄
x2
‹
xi1
b
xi
x
Figure 12 shows this approximation for n − 2, 4, 8, and 12. Notice that this approximation appears to become better and better as the number of strips increases, that is, as n l `. Therefore we define the area A of the region S in the following way. y
y
0
a
⁄
(a) n=2
b x
0
y
a
⁄
x2
(b) n=4
‹
b
x
0
y
b
a
(c) n=8
x
0
b
a
x
(d) n=12
FIGURE 12
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299
Section 4.1 Areas and Distances
2 Definition The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: A − lim R n − lim f f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dxg nl`
nl`
It can be proved that the limit in Definition 2 always exists, since we are assuming that f is continuous. It can also be shown that we get the same value if we use left endpoints:
3
A − lim L n − lim f f sx 0 d Dx 1 f sx 1 d Dx 1 ∙ ∙ ∙ 1 f sx n21 d Dxg nl`
nl`
In fact, instead of using left endpoints or right endpoints, we could take the height of the ith rectangle to be the value of f at any number x*i in the ith subinterval fx i21, x i g. We call the numbers x1*, x2*, . . . , x n* the sample points. Figure 13 shows approximating rectangles when the sample points are not chosen to be endpoints. So a more general expression for the area of S is
A − lim f f sx1* d Dx 1 f sx2* d Dx 1 ∙ ∙ ∙ 1 f sx*n d Dxg
4
nl`
y
Îx
f(x *) i
0
FIGURE 13
a
⁄
x*¡
x2 x™*
‹ x£*
xi1
xi
b
xn1
x *i
x
x n*
Note It can be shown that an equivalent definition of area is the following: A is the unique number that is smaller than all the upper sums and bigger than all the lower sums. We saw in Examples 1 and 2, for instance, that the area s A − 13 d is trapped between all the left approximating sums L n and all the right approximating sums Rn. The function in those examples, f sxd − x 2, happens to be increasing on f0, 1g and so the lower sums arise from left endpoints and the upper sums from right endpoints. (See Figures 8 and 9.) In general, we form lower (and upper) sums by choosing the sample points x*i so that f sx*i d is the minimum (and maximum) value of f on the ith subinterval. (See Figure 14 and Exercises 7–8.) y
FIGURE 14 Lower sums (short rectangles) and upper sums (tall rectangles)
0
a
b
x
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300
Chapter 4 Integrals
This tells us to end with i=n. This tells us to add. This tells us to start with i=m.
We often use sigma notation to write sums with many terms more compactly. For instance, n
µ
n
o f sx i d Dx − f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dx i−1
f(xi) Îx
i=m
So the expressions for area in Equations 2, 3, and 4 can be written as follows: n
o f sx i d Dx n l ` i−1
If you need practice with sigma notation, look at the examples and try some of the exercises in Appendix E.
A − lim
n
o f sx i21 d Dx n l ` i−1
A − lim
n
o f sxi*d Dx
A − lim
n l ` i−1
We can also rewrite Formula 1 in the following way: n
o i2 − i−1
nsn 1 1ds2n 1 1d 6
Example 3 Let A be the area of the region that lies under the graph of f sxd − cos x between x − 0 and x − b, where 0 < b < y2. (a) Using right endpoints, find an expression for A as a limit. Do not evaluate the limit. (b) Estimate the area for the case b − y2 by taking the sample points to be midpoints and using four subintervals. SOLUTION
(a) Since a − 0, the width of a subinterval is Dx −
b20 b − n n
So x 1 − byn, x 2 − 2byn, x 3 − 3byn, x i − ibyn, and x n − nbyn. The sum of the areas of the approximating rectangles is R n − f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dx − scos x 1 d Dx 1 scos x 2 d Dx 1 ∙ ∙ ∙ 1 scos x n d Dx
S D S D
− cos
b 2b 1 cos n n
b n
S D
b nb 1 ∙ ∙ ∙ 1 cos n n
b n
According to Definition 2, the area is A − lim R n − lim nl`
nl`
b n
S
cos
b 2b 3b nb 1 cos 1 cos 1 ∙ ∙ ∙ 1 cos n n n n
D
Using sigma notation we could write b nl` n
A − lim
n
o cos i−1
ib n
It is very difficult to evaluate this limit directly by hand, but with the aid of a computer algebra system it isn’t hard (see Exercise 31). In Section 4.3 we will be able to find A more easily using a different method. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
301
Section 4.1 Areas and Distances
(b) With n − 4 and b − y2 we have Dx − sy2dy4 − y8, so the subintervals are f0, y8g, fy8, y4g, fy4, 3y8g, and f3y8, y2g. The midpoints of these subinter vals are 3 5 7 x1* − x2* − x3* − x4* − 16 16 16 16
y
y=cos x
1
0
π 8
π 4
3π 8
π 2
x
and the sum of the areas of the four approximating rectangles (see Figure 15) is 4
FIGURE 15
M4 −
o f sxi*d Dx
i−1
− f sy16d Dx 1 f s3y16d Dx 1 f s5y16d Dx 1 f s7y16d Dx
S D S D S D S D S D
− cos
−
8
16
cos
3 1 cos 8 16
5 1 cos 8 16
7 1 cos 8 16
3 5 7 1 cos 1 cos 1 cos 16 16 16 16
8
< 1.006
So an estimate for the area is A < 1.006
n
The Distance Problem Now let’s consider the distance problem: find the distance traveled by an object during a certain time period if the velocity of the object is known at all times. (In a sense this is the inverse problem of the velocity problem that we discussed in Section 1.4.) If the velocity remains constant, then the distance problem is easy to solve by means of the formula distance − velocity 3 time But if the velocity varies, it’s not so easy to find the distance traveled. We investigate the problem in the following example.
Example 4 Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30second time interval. We take speedometer readings every five seconds and record them in the following table: Time (s)
0
5
10
15
20
25
30
Velocity (miyh)
17
21
24
29
32
31
28
In order to have the time and the velocity in consistent units, let’s convert the velocity readings to feet per second (1 miyh − 5280y3600 ftys): Time (s)
0
5
10
15
20
25
30
Velocity (ftys)
25
31
35
43
47
45
41
During the first five seconds the velocity doesn’t change very much, so we can estimate the distance traveled during that time by assuming that the velocity is constant. If we Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
302
Chapter 4 Integrals
take the velocity during that time interval to be the initial velocity (25 ftys), then we obtain the approximate distance traveled during the first five seconds: 25 ftys 3 5 s − 125 ft Similarly, during the second time interval the velocity is approximately constant and we take it to be the velocity when t − 5 s. So our estimate for the distance traveled from t − 5 s to t − 10 s is 31 ftys 3 5 s − 155 ft If we add similar estimates for the other time intervals, we obtain an estimate for the total distance traveled: s25 3 5d 1 s31 3 5d 1 s35 3 5d 1 s43 3 5d 1 s47 3 5d 1 s45 3 5d − 1130 ft We could just as well have used the velocity at the end of each time period instead of the velocity at the beginning as our assumed constant velocity. Then our estimate becomes s31 3 5d 1 s35 3 5d 1 s43 3 5d 1 s47 3 5d 1 s45 3 5d 1 s41 3 5d − 1210 ft If we had wanted a more accurate estimate, we could have taken velocity readings every two seconds, or even every second. n √ 40
20
0
10
FIGURE 16
20
30
t
Perhaps the calculations in Example 4 remind you of the sums we used earlier to estimate areas. The similarity is explained when we sketch a graph of the velocity function of the car in Figure 16 and draw rectangles whose heights are the initial velocities for each time interval. The area of the first rectangle is 25 3 5 − 125, which is also our estimate for the distance traveled in the first five seconds. In fact, the area of each rectangle can be interpreted as a distance because the height represents velocity and the width represents time. The sum of the areas of the rectangles in Figure 16 is L 6 − 1130, which is our initial estimate for the total distance traveled. In general, suppose an object moves with velocity v − f std, where a < t < b and f std > 0 (so the object always moves in the positive direction). We take velocity readings at times t0 s− ad, t1, t2 , . . . , tn s− bd so that the velocity is approximately constant on each subinterval. If these times are equally spaced, then the time between consecutive readings is Dt − sb 2 adyn. During the first time interval the velocity is approximately f st0 d and so the distance traveled is approximately f st0 d Dt. Similarly, the distance traveled during the second time interval is about f st1 d Dt and the total distance traveled during the time interval fa, bg is approximately f st0 d Dt 1 f st1 d Dt 1 ∙ ∙ ∙ 1 f stn21 d Dt −
n
o f sti21 d Dt i−1
If we use the velocity at right endpoints instead of left endpoints, our estimate for the total distance becomes f st1 d Dt 1 f st2 d Dt 1 ∙ ∙ ∙ 1 f stn d Dt −
n
o f sti d Dt i−1
The more frequently we measure the velocity, the more accurate our estimates become,
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 4.1 Areas and Distances
303
so it seems plausible that the exact distance d traveled is the limit of such expressions: n
n
lim o f sti d Dt o f sti21 d Dt − nl` nl` i−1 i−1
5
d − lim
We will see in Section 4.4 that this is indeed true. Because Equation 5 has the same form as our expressions for area in Equations 2 and 3, it follows that the distance traveled is equal to the area under the graph of the velocity function. In Chapter 5 we will see that other quantities of interest in the natural and social sciences—such as the work done by a variable force or the cardiac output of the heart—can also be interpreted as the area under a curve. So when we compute areas in this chapter, bear in mind that they can be interpreted in a variety of practical ways.
1. (a) By reading values from the given graph of f, use five rect angles to find a lower estimate and an upper estimate for the area under the given graph of f from x − 0 to x − 10. In each case sketch the rectangles that you use. (b) Find new estimates using ten rectangles in each case.
4. (a) Estimate the area under the graph of f sxd − sin x from x − 0 to x − y2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.
y 4
y=ƒ
2 0
3. (a) Estimate the area under the graph of f sxd − 1yx from x − 1 to x − 2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.
8
4
x
2. (a) Use six rectangles to find estimates of each type for the area under the given graph of f from x − 0 to x − 12. (i) L 6 (sample points are left endpoints) (ii) R 6 (sample points are right endpoints) (iii) M6 (sample points are midpoints) (b) Is L 6 an underestimate or overestimate of the true area? (c) Is R 6 an underestimate or overestimate of the true area? (d) Which of the numbers L 6, R 6, or M6 gives the best estimate? Explain.
5. (a) Estimate the area under the graph of f sxd − 1 1 x 2 from x − 21 to x − 2 using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles. (b) Repeat part (a) using left endpoints. (c) Repeat part (a) using midpoints. (d) From your sketches in parts (a)–(c), which appears to be the best estimate? ; 6. (a) Graph the function f sxd − 1ys1 1 x 2 d
y 8
y=ƒ
4
(b) Estimate the area under the graph of f using four approximating rectangles and taking the sample points to be (i) right endpoints and (ii) midpoints. In each case sketch the curve and the rectangles. (c) Improve your estimates in part (b) by using eight rectangles.
7. Evaluate the upper and lower sums for f sxd − 2 1 sin x, 0 < x < , with n − 2, 4, and 8. Illustrate with diagrams like Figure 14.
4
0
22 < x < 2
8
12 x
8. Evaluate the upper and lower sums for f sxd − 1 1 x 2, 21 < x < 1, with n − 3 and 4. Illustrate with diagrams like Figure 14.
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304
Chapter 4 Integrals
9–10 With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of n, using looping. (On a TI use the Is. command or a ForEndFor loop, on a Casio use Isz, on an HP or in BASIC use a FORNEXT loop.) Compute the sum of the areas of approximating rect angles using equal subintervals and right endpoints for n − 10, 30, 50, and 100. Then guess the value of the exact area. 9. The region under y − x 4 from 0 to 1 10. The region under y − cos x from 0 to y2
time period using the velocities at the beginning of the time intervals. (b) Give another estimate using the velocities at the end of the time periods. (c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.
15. Oil leaked from a tank at a rate of rstd liters per hour. The rate decreased as time passed and values of the rate at twohour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out. t (h)
CAS
CAS
11. Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if x*i is a left or right endpoint. (For instance, in Maple use leftbox, rightbox, leftsum, and rightsum.) (a) If f sxd − 1ysx 2 1 1d, 0 < x < 1, find the left and right sums for n − 10, 30, and 50. (b) Illustrate by graphing the rectangles in part (a). (c) Show that the exact area under f lies between 0.780 and 0.791. 12. (a) If f sxd − xysx 1 2d, 1 < x < 4, use the commands discussed in Exercise 11 to find the left and right sums for n − 10, 30, and 50. (b) Illustrate by graphing the rectangles in part (a). (c) Show that the exact area under f lies between 1.603 and 1.624. 13. The speed of a runner increased steadily during the first three seconds of a race. Her speed at halfsecond intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds. t (s)
0
0.5
1.0
1.5
2.0
2.5
3.0
v (ftys)
0
6.2
10.8
14.9
18.1
19.4
20.2
14. The table shows speedometer readings at 10second intervals during a 1minute period for a car racing at the Daytona International Speedway in Florida. (a) Estimate the distance the race car traveled during this
Time (s) 0 10 20 30 40 50 60
Velocity (miyh) 182.9 168.0 106.6 99.8 124.5 176.1 175.6
rstd (Lyh)
0
2
4
6
8
10
8.7
7.6
6.8
6.2
5.7
5.3
16. W hen we estimate distances from velocity data, it is sometimes necessary to use times t0 , t1, t2 , t3 , . . . that are not equally spaced. We can still estimate distances using the time periods Dt i − t i 2 t i21. For example, on May 7, 1992, the space shuttle Endeavour was launched on mission STS49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table, provided by NASA, gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. Use these data to estimate the height above the earth’s surface of the Endeavour, 62 seconds after liftoff. Time ssd Velocity sftysd
Event Launch
0
0
Begin roll maneuver
10
185
End roll maneuver
15
319
Throttle to 89%
20
447
Throttle to 67%
32
742
Throttle to 104%
59
1325
Maximum dynamic pressure
62
1445
Solid rocket booster separation
125
4151
17. T he velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied. √ (ft /s) 60 40 20 0
2
4
t 6 (seconds)
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305
Section 4.1 Areas and Distances
18. The velocity graph of a car accelerating from rest to a speed of 120 kmyh over a period of 30 seconds is shown. Estimate the distance traveled during this period. √ (km / h)
Source: A. Gumel et al., “Modelling Strategies for Controlling SARS Outbreaks,” Proceedings of the Royal Society of London: Series B 271 (2004): 2223–32.
80
21–23 Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.
40 0
Singapore between March 1 and May 24, 2003, using both left endpoints and right endpoints. (b) How would you interpret the number of SARS deaths as an area under a curve?
t 30 (seconds)
20
10
19. I n someone infected with measles, the virus level N (measured in number of infected cells per mL of blood plasma) reaches a peak density at about t − 12 days (when a rash appears) and then decreases fairly rapidly as a result of immune response. The area under the graph of Nstd from t − 0 to t − 12 (as shown in the figure) is equal to the total amount of infection needed to develop symptoms (measured in density of infected cells 3 time). The function N has been modeled by the function
21. f sxd −
2x , 1 < x < 3 x2 1 1
22. f sxd − x 2 1 s1 1 2x , 4 < x < 7 23. f sxd − ssin x , 0 < x <
24–25 Determine a region whose area is equal to the given limit. Do not evaluate the limit. n
o n l ` i−1
24. lim
3 n
Î
11
3i n
n
o n l ` i−1
25. lim
i tan 4n 4n
f std − 2tst 2 21dst 1 1d Use this model with six subintervals and their midpoints to estimate the total amount of infection needed to develop symptoms of measles. N 1000
N=f(t)
0
12
13 1 2 3 1 3 3 1 ∙ ∙ ∙ 1 n 3 −
21
t (days)
Source: J. M. Heffernan et al., “An InHost Model of Acute Infection: Measles as a Case Study,” Theoretical Population Biology 73 (2006): 134– 47.
20. The table shows the number of people per day who died from SARS in Singapore at twoweek intervals beginning on March 1, 2003.
26. (a) Use Definition 2 to find an expression for the area under the curve y − x 3 from 0 to 1 as a limit. (b) The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a).
Date
Deaths per day
Date
Deaths per day
March 1 March 15 March 29 April 12
0.0079 0.0638 0.1944 0.4435
April 26 May 10 May 24
0.5620 0.4630 0.2897
(a) By using an argument similar to that in Example 4, estimate the number of people who died of SARS in
F
nsn 1 1d 2
G
2
27. Let A be the area under the graph of an increasing continuous function f from a to b, and let L n and Rn be the approximations to A with n subintervals using left and right endpoints, respectively. (a) How are A, L n, and Rn related? (b) Show that b2a f f sbd 2 f sadg n Then draw a diagram to illustrate this equation by showing that the n rectangles representing R n 2 L n can be reassembled to form a single rectangle whose area is the right side of the equation. (c) Deduce that Rn 2 L n −
Rn 2 A ,
b2a f f sbd 2 f sadg n
28. If A is the area under the curve y − sin x from 0 to y2, use Exercise 27 to find a value of n such that Rn 2 A , 0.0001.
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306 CAS
CAS
CAS
Chapter 4 Integrals
29. (a) Express the area under the curve y − x 5 from 0 to 2 as a limit. (b) Use a computer algebra system to find the sum in your expression from part (a). (c) Evaluate the limit in part (a). 30. (a) Express the area under the curve y − x 4 1 5x 2 1 x from 2 to 7 as a limit. (b) Use a computer algebra system to evaluate the sum in part (a). (c) Use a computer algebra system to find the exact area by evaluating the limit of the expression in part (b). 31. Find the exact area under the cosine curve y − cos x from x − 0 to x − b, where 0 < b < y2. (Use a computer
algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if b − y2? 32. (a) Let A n be the area of a polygon with n equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle 2yn, show that A n − 12 nr 2 sin
S D 2 n
(b) Show that lim n l ` A n − r 2. [Hint: Use Equation 2.4.2 on page 145.]
We saw in Section 4.1 that a limit of the form n
1
f f sx1*d Dx 1 f sx2*d Dx 1 ∙ ∙ ∙ 1 f sx n*d Dxg o f sx*i d Dx − nlim l` n l ` i−1 lim
arises when we compute an area. We also saw that it arises when we try to find the dis tance traveled by an object. It turns out that this same type of limit occurs in a wide vari ety of situations even when f is not necessarily a positive function. In Chapters 5 and 8 we will see that limits of the form (1) also arise in finding lengths of curves, volumes of solids, centers of mass, force due to water pressure, and work, as well as other quantities. We therefore give this type of limit a special name and notation. 2 Definition of a Definite Integral If f is a function defined for a < x < b, we divide the interval fa, bg into n subintervals of equal width Dx − sb 2 adyn. We let x 0 s− ad, x 1, x 2 , . . . , x n s− bd be the endpoints of these subintervals and we let x1*, x2*, . . . , x n* be any sample points in these subintervals, so x*i lies in the ith subinterval fx i21, x i g. Then the definite integral of f from a to b is
y
n
o f sx*i d Dx n l ` i−1
b
f sxd dx − lim
a
p rovided that this limit exists and gives the same value for all possible choices of sample points. If it does exist, we say that f is integrable on fa, bg. The precise meaning of the limit that defines the integral is as follows: For every number « . 0 there is an integer N such that
Zy
b
a
f sxd dx 2
n
o f sx*i d Dx i−1
Z
,«
for every integer n . N and for every choice of x*i in fx i21, x i g.
Note 1 The symbol y was introduced by Leibniz and is called an integral sign. It is an elongated S and was chosen because an integral is a limit of sums. In the notation
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307
Section 4.2 The Definite Integral
ya f sxd dx, f sxd is called the integrand and a and b are called the limits of integration; b
a is the lower limit and b is the upper limit. For now, the symbol dx has no meaning by b itself; ya f sxd dx is all one symbol. The dx simply indicates that the independent variable is x. The procedure of calculating an integral is called integration. Note 2 The definite integral ya f sxd dx is a number; it does not depend on x. In fact, b
we could use any letter in place of x without changing the value of the integral:
y
b
a
f sxd dx − y f std dt − y f srd dr b
a
Note 3 The sum
Bernhard Riemann received his Ph.D. under the direction of the legendary Gauss at the University of Göttingen and remained there to teach. Gauss, who was not in the habit of praising other mathematicians, spoke of Riemann’s “creative, active, truly mathematical mind and gloriously fertile originality.” The definition (2) of an integral that we use is due to Riemann. He also made major contributions to the theory of functions of a complex variable, mathematical physics, number theory, and the foundations of geometry. Riemann’s broad concept of space and geometry turned out to be the right setting, 50 years later, for Einstein’s general relativity theory. Riemann’s health was poor throughout his life, and he died of tuberculosis at the age of 39.
n
b
x
FIGURE 3
o f sx*i d Dx is an approximation
to the net area.
0 a
FIGURE 4 b
y
Îx
0
a
x *i
y=ƒ
0
x
b
a
b
x
FIGURE 2 b If f sxd > 0, the integral ya f sxd dx is the area under the curve y − f sxd from a to b.
If f takes on both positive and negative values, as in Figure 3, then the Riemann sum is the sum of the areas of the rectangles that lie above the xaxis and the negatives of the areas of the rectangles that lie below the xaxis (the areas of the blue rectangles minus the areas of the gold rectangles). When we take the limit of such Riemann sums, we get the situation illustrated in Figure 4. A definite integral can be interpreted as a net area, that is, a difference of areas: b y f sxd dx − A 1 2 A 2 where A 1 is the area of the region above the xaxis and below the graph of f , and A 2 is the area of the region below the xaxis and above the graph of f .
y=ƒ
a
y
a
y
y
that occurs in Definition 2 is called a Riemann sum after the German mathematician Bernhard Riemann (1826 –1866). So Definition 2 says that the definite integral of an integrable function can be approximated to within any desired degree of accuracy by a Riemann sum. We know that if f happens to be positive, then the Riemann sum can be interpreted as a sum of areas of approximating rectangles (see Figure 1). By comparing Definition 2 b with the definition of area in Section 4.1, we see that the definite integral ya f sxd dx can be interpreted as the area under the curve y − f sxd from a to b. (See Figure 2.)
FIGURE 1 If f sxd > 0, the Riemann sum o f sx*i d Dx is the sum of areas of rectangles.
y=ƒ
0 a
a
o f sx*i d Dx i−1
Riemann
y
b
f sxd dx is the net area.
b x
Note 4 Although we have defined ya f sxd dx by dividing fa, bg into subintervals of b
equal width, there are situations in which it is advantageous to work with subintervals of unequal width. For instance, in Exercise 4.1.16, NASA provided velocity data at times that were not equally spaced, but we were still able to estimate the distance traveled. And there are methods for numerical integration that take advantage of unequal subintervals.
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308
Chapter 4 Integrals
If the subinterval widths are Dx 1, Dx 2 , . . . , Dx n , we have to ensure that all these widths approach 0 in the limiting process. This happens if the largest width, max Dx i , approaches 0. So in this case the definition of a definite integral becomes
y
b
a
f sxd dx −
n
o f sx*i d Dx i
lim
max Dx i l 0 i−1
Note 5 We have defined the definite integral for an integrable function, but not all functions are integrable (see Exercises 71–72). The following theorem shows that the most commonly occurring functions are in fact integrable. The theorem is proved in more advanced courses.
3 Theorem If f is continuous on fa, bg, or if f has only a finite number of jump discontinuities, then f is integrable on fa, bg; that is, the definite integral yab f sxd dx exists. If f is integrable on fa, bg, then the limit in Definition 2 exists and gives the same value no matter how we choose the sample points x*i . To simplify the calculation of the integral we often take the sample points to be right endpoints. Then x*i − x i and the definition of an integral simplifies as follows.
4 Theorem If f is integrable on fa, bg, then
y
b
a
where
Dx −
f sxd dx − lim
n
o f sx id Dx
nl` i−1
b2a and x i − a 1 i Dx n
Example 1 Express n
o sx 3i 1 x i sin x i d Dx n l ` i−1 lim
as an integral on the interval f0, g. SOLUTION Comparing the given limit with the limit in Theorem 4, we see that they will be identical if we choose f sxd − x 3 1 x sin x. We are given that a − 0 and b − . Therefore, by Theorem 4, we have n
o sx 3i 1 x i sin x i d Dx − y0 sx 3 1 x sin xd dx n l ` i−1 lim
n
Later, when we apply the definite integral to physical situations, it will be important to recognize limits of sums as integrals, as we did in Example 1. When Leibniz chose the notation for an integral, he chose the ingredients as reminders of the limiting process. In
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Section 4.2 The Definite Integral
309
general, when we write n
o f sx *i d Dx − ya f sxd dx n l ` i−1 b
lim
we replace lim o by y, x*i by x, and Dx by dx.
Evaluating Integrals When we use a limit to evaluate a definite integral, we need to know how to work with sums. The following three equations give formulas for sums of powers of positive integers. Equation 5 may be familiar to you from a course in algebra. Equations 6 and 7 were discussed in Section 4.1 and are proved in Appendix E. n
nsn 1 1d 2
oi− i−1
5
n
nsn 1 1ds2n 1 1d 6
o i2 − i−1
6
F
n
o
7
i−1
i3 −
nsn 1 1d 2
G
2
The remaining formulas are simple rules for working with sigma notation: Formulas 8 –11 are proved by writing out each side in expanded form. The left side of Equation 9 is ca 1 1 ca 2 1 ∙ ∙ ∙ 1 ca n
n
o c − nc
8
i−1 n
o
9
i−1
The right side is csa 1 1 a 2 1 ∙ ∙ ∙ 1 a n d
These are equal by the distributive property. The other formulas are discussed in Appendix E.
n
o
10
i−1 n
o
11
i−1
n
ca i − c
sa i 1 bi d − sa i 2 bi d −
o ai
i−1
n
o
n
ai 1
i−1 n
o
i−1
o bi
i−1 n
ai 2
o bi
i−1
Example 2 (a) Evaluate the Riemann sum for f sxd − x 3 2 6x, taking the sample points to be right endpoints and a − 0, b − 3, and n − 6. (b) Evaluate y sx 3 2 6xd dx. 3
0
SOLUTION
(a) With n − 6 the interval width is Dx −
b2a 320 1 − − n 6 2
and the right endpoints are x 1 − 0.5, x 2 − 1.0, x 3 − 1.5, x 4 − 2.0, x 5 − 2.5, and
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310
Chapter 4 Integrals
x 6 − 3.0. So the Riemann sum is 6
R6 −
o f sx i d Dx i−1
− f s0.5d Dx 1 f s1.0d Dx 1 f s1.5d Dx 1 f s2.0d Dx 1 f s2.5d Dx 1 f s3.0d Dx y
− 12 s22.875 2 5 2 5.625 2 4 1 0.625 1 9d
5
y=˛6x
− 23.9375
0
x
3
FIGURE 5
Notice that f is not a positive function and so the Riemann sum does not represent a sum of areas of rectangles. But it does represent the sum of the areas of the blue rectangles (above the xaxis) minus the sum of the areas of the gold rectangles (below the xaxis) in Figure 5. (b) With n subintervals we have Dx −
b2a 3 − n n
So x 0 − 0, x 1 − 3yn, x 2 − 6yn, x 3 − 9yn, and, in general, x i − 3iyn. Since we are using right endpoints, we can use Theorem 4:
y
3
0
In the sum, n is a constant (unlike i), so we can move 3yn in front of the o sign.
− lim
3 n
− lim
3 n
nl`
nl`
− lim
nl`
− lim
nl`
y 5
y=˛6x
− lim
nl`
A¡ 0
A™
3
x
FIGURE 6
y
3
0
sx 3 2 6xd dx − A1 2 A2 − 26.75
n
SD o FS D S DG oF G n
3i n
o f sx i d Dx − nlim of n l ` i−1 l ` i−1
sx 3 2 6xd dx − lim
−
F
n
i−1
n
3i n
26
(Equation 9 with c − 3yn)
27 3 18 i 3 i 2 n n
i−1
81 n4
3
3i n
3 n
n
o
i3 2
i−1
54 n2
n
G
oi
i−1
(Equations 11 and 9)
H F G J F S D S DG 81 n4
nsn 1 1d 2
81 4
11
1 n
2
2
54 nsn 1 1d (Equations 7 and 5) n2 2
2
2 27 1 1
1 n
81 27 2 27 − 2 − 26.75 4 4
This integral can’t be interpreted as an area because f takes on both positive and negative values. But it can be interpreted as the difference of areas A 1 2 A 2, where A 1 and A 2 are shown in Figure 6.
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Section 4.2 The Definite Integral
311
Figure 7 illustrates the calculation by showing the positive and negative terms in the right Riemann sum R n for n − 40. The values in the table show the Riemann sums approaching the exact value of the integral, 26.75, as n l `. y
n 5
40 100 500 1000 5000
y=˛6x
0
x
3
FIGURE 7 R40 < 26.3998
Rn
26.3998 26.6130 26.7229 26.7365 26.7473
n
A much simpler method for evaluating the integral in Example 2 will be given in Example 4.4.3.
Example 3 (a) Set up an expression for y25 x 4 dx as a limit of sums. (b) Use a computer algebra system to evaluate the expression.
Because f sxd − x 4 is positive, the integral in Example 3 represents the area shown in Figure 8.
SOLUTION
y
(a) Here we have f sxd − x 4, a − 2, b − 5, and Dx −
y=x$ 300
b2a 3 − n n
x0 − 2, x 1 − 2 1 3yn, x 2 − 2 1 6yn, x 3 − 2 1 9yn, and xi − 2 1
0
2
5
x
3i n
From Theorem 4, we get
y
FIGURE 8
5
2
n
o n l ` i −1
x 4 dx − lim − lim
nl`
3 n
n
of n l ` i −1
f sx i d Dx − lim n
o i−1
S D 21
3i n
S D 21
3i n
3 n
4
(b) If we ask a computer algebra system to evaluate the sum and simplify, we obtain n
o i−1
S D 21
3i n
4
−
2062n 4 1 3045n 3 1 1170n 2 2 27 10n 3
Now we ask the computer algebra system to evaluate the limit:
y
5
2
x 4 dx − lim
nl`
−
3 n
n
o i−1
S D 21
3i n
4
− lim
nl`
3s2062n 4 1 3045n 3 1 1170n 2 2 27d 10n 4
3s2062d 3093 − − 618.6 10 5
We will learn a much easier method for the evaluation of integrals in the next section. n Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
312
Chapter 4 Integrals
y
Example 4 Evaluate the following integrals by interpreting each in terms of areas.
(a) y s1 2 x 2 dx (b) y sx 2 1d dx
y= œ„„„„„ 1≈ or ≈+¥=1
1
0
1
0
SOLUTION
(a) Since f sxd − s1 2 x 2 > 0, we can interpret this integral as the area under the curve y − s1 2 x 2 from 0 to 1. But, since y 2 − 1 2 x 2, we get x 2 1 y 2 − 1, which shows that the graph of f is the quartercircle with radius 1 in Figure 9. Therefore
x
1
3
0
y s1 2 x 1
FIGURE 9 y
(3, 2)
y
A¡ 1
3
dx − 14 s1d2 −
4
(In Section 7.3 we will be able to prove that the area of a circle of radius r is r 2.) (b) The graph of y − x 2 1 is the line with slope 1 shown in Figure 10. We compute the integral as the difference of the areas of the two triangles:
y=x1
0 A™
2
0
x
_1
FIGURE 10
TEC Module 4.2 / 7.7 shows how the Midpoint Rule estimates improve as n increases.
3
0
sx 2 1d dx − A 1 2 A 2 − 12 s2 ∙ 2d 2 12 s1 ∙ 1d − 1.5
n
The Midpoint Rule We often choose the sample point x*i to be the right endpoint of the ith subinterval because it is convenient for computing the limit. But if the purpose is to find an approximation to an integral, it is usually better to choose x*i to be the midpoint of the interval, which we denote by x i . Any Riemann sum is an approximation to an integral, but if we use midpoints we get the following approximation. Midpoint Rule
y
b
a
f sxd dx <
where
Dx −
n
o f sx i d Dx − Dx f f sx 1 d 1 ∙ ∙ ∙ 1 f sx n dg
i−1
b2a n
x i − 12 sx i21 1 x i d − midpoint of fx i21, x i g
and
Example 5 Use the Midpoint Rule with n − 5 to approximate y
2
1
1 dx. x
SOLUTION The endpoints of the five subintervals are 1, 1.2, 1.4, 1.6, 1.8, and 2.0, so the midpoints are 1.1, 1.3, 1.5, 1.7, and 1.9. The width of the subintervals is Dx − s2 2 1dy5 − 15, so the Midpoint Rule gives
y
2
1
1 dx < Dx f f s1.1d 1 f s1.3d 1 f s1.5d 1 f s1.7d 1 f s1.9dg x −
1 5
S
1 1 1 1 1 1 1 1 1 1.1 1.3 1.5 1.7 1.9
D
< 0.691908 Since f sxd − 1yx . 0 for 1 < x < 2, the integral represents an area, and the approxi Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 4.2 The Definite Integral y
mation given by the Midpoint Rule is the sum of the areas of the rectangles shown in Figure 11. n
1 y= x
0
1
313
2
x
At the moment we don’t know how accurate the approximation in Example 5 is, but in Section 7.7 we will learn a method for estimating the error involved in using the Midpoint Rule. At that time we will discuss other methods for approximating definite integrals. If we apply the Midpoint Rule to the integral in Example 2, we get the picture in Fig ure 12. The approximation M40 < 26.7563 is much closer to the true value 26.75 than the right endpoint approximation, R 40 < 26.3998, shown in Figure 7.
FIGURE 11
y
TEC In Visual 4.2 you can compare left, right, and midpoint approximations to the integral in Example 2 for different values of n.
5
y=˛6x
0
3
x
FIGURE 12 M40 < 26.7563
Properties of the Definite Integral b
When we defined the definite integral ya f sxd dx, we implicitly assumed that a , b. But the definition as a limit of Riemann sums makes sense even if a . b. Notice that if we reverse a and b, then Dx changes from sb 2 adyn to sa 2 bdyn. Therefore
y
a
b
f sxd dx − 2y f sxd dx b
a
If a − b, then Dx − 0 and so
y
a
a
f sxd dx − 0
We now develop some basic properties of integrals that will help us to evaluate integrals in a simple manner. We assume that f and t are continuous functions. Properties of the Integral 1. y c dx − csb 2 ad, where c is any constant b
a
2. y f f sxd 1 tsxdg dx − y f sxd dx 1 y tsxd dx b
b
a
a
b
a
3. y cf sxd dx − c y f sxd dx, where c is any constant b
a
b
a
4. y f f sxd 2 tsxdg dx − y f sxd dx 2 y tsxd dx b
a
b
a
b
a
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
314
Chapter 4 Integrals
Property 1 says that the integral of a constant function f sxd − c is the constant times the length of the interval. If c . 0 and a , b, this is to be expected because csb 2 ad is the area of the shaded rectangle in Figure 13. y
y=c
c
area=c(ba)
y
FIGURE 13 b
a
a
b
f+g
y
f
b
a
n
o f f sx i d 1 tsx i dg Dx n l ` i−1
f f sxd 1 tsxdg dx − lim
Fo n
0
b x
a
− lim
nl`
b
a
f f sxd 1 tsxdg dx −
y
b
a
i−1
n
f sxd dx 1
G
o tsx i d Dx i−1
f sx i d Dx 1
n
FIGURE 14
y
x
Property 2 says that the integral of a sum is the sum of the integrals. For positive functions it says that the area under f 1 t is the area under f plus the area under t. Figure 14 helps us understand why this is true: in view of how graphical addition works, the corresponding vertical line segments have equal height. In general, Property 2 follows from Theorem 4 and the fact that the limit of a sum is the sum of the limits:
y
g
0
c dx − csb 2 ad
n
o f sx i d Dx 1 nlim o tsx i d Dx n l ` i−1 l ` i−1
− lim
y tsxd dx
− y f sxd dx 1 y tsxd dx
b
b
a
Property 3 seems intuitively reasonable because we know that multiplying a function by a positive number c stretches or shrinks its graph vertically by a factor of c. So it stretches or shrinks each approximating rectangle by a factor c and therefore it has the effect of multiplying the area by c.
b
a
a
Property 3 can be proved in a similar manner and says that the integral of a constant times a function is the constant times the integral of the function. In other words, a constant (but only a constant) can be taken in front of an integral sign. Property 4 is proved by writing f 2 t − f 1 s2td and using Properties 2 and 3 with c − 21.
Example 6 Use the properties of integrals to evaluate y s4 1 3x 2 d dx. 1
0
SOLUTION Using Properties 2 and 3 of integrals, we have
y
1
0
s4 1 3x 2 d dx − y 4 dx 1 y 3x 2 dx − y 4 dx 1 3 y x 2 dx 1
1
0
1
0
1
0
0
We know from Property 1 that
y
1
0
4 dx − 4s1 2 0d − 4
and we found in Example 4.1.2 that y x 2 dx − 13. So 1
0
y
1
0
s4 1 3x 2 d dx − y 4 dx 1 3 y x 2 dx 1
0
1
0
− 4 1 3 ∙ 13 − 5
n
The next property tells us how to combine integrals of the same function over adjacent intervals. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 4.2 The Definite Integral y
y
5. y=ƒ
0
a
c
b
c
a
315
f sxd dx 1 y f sxd dx − y f sxd dx b
b
c
a
This is not easy to prove in general, but for the case where f sxd > 0 and a , c , b Property 5 can be seen from the geometric interpretation in Figure 15: the area under y − f sxd from a to c plus the area from c to b is equal to the total area from a to b.
x
Example 7 If it is known that y010 f sxd dx − 17 and y08 f sxd dx − 12, find y810 f sxd dx.
FIGURE 15
SOLUTION By Property 5, we have
y
8
0
so
y
10
8
f sxd dx 1 y f sxd dx − y f sxd dx 10
10
8
0
f sxd dx − y f sxd dx 2 y f sxd dx − 17 2 12 − 5 10
8
0
0
n
Properties 1–5 are true whether a , b, a − b, or a . b. The following properties, in which we compare sizes of functions and sizes of integrals, are true only if a < b. Comparison Properties of the Integral 6. If f sxd > 0 for a < x < b, then y f sxd dx > 0. b
a
7. If f sxd > tsxd for a < x < b, then y f sxd dx > y tsxd dx. b
b
a
a
8. If m < f sxd < M for a < x < b, then msb 2 ad < y f sxd dx < Msb 2 ad b
a
y
b
M
y=ƒ m 0
If f sxd > 0, then ya f sxd dx represents the area under the graph of f, so the geometric interpretation of Property 6 is simply that areas are positive. (It also follows directly from the definition because all the quantities involved are positive.) Property 7 says that a bigger function has a bigger integral. It follows from Properties 6 and 4 because f 2 t > 0. Property 8 is illustrated by Figure 16 for the case where f sxd > 0. If f is continuous, we could take m and M to be the absolute minimum and maximum values of f on the interval fa, bg. In this case Property 8 says that the area under the graph of f is greater than the area of the rectangle with height m and less than the area of the rectangle with height M.
a
FIGURE 16
b
x
Proof of Property 8 Since m < f sxd < M, Property 7 gives
y
b
a
m dx < y f sxd dx < y M dx b
a
b
a
Using Property 1 to evaluate the integrals on the left and right sides, we obtain
msb 2 ad < y f sxd dx < Msb 2 ad b
a
n
Property 8 is useful when all we want is a rough estimate of the size of an integral without going to the bother of using the Midpoint Rule. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
316
Chapter 4 Integrals
Example 8 Use Property 8 to estimate y sx dx. 4
1
y
y=œ„x
y=2
2
SOLUTION Since f sxd − sx is an increasing function, its absolute minimum on f1, 4g is m − f s1d − 1 and its absolute maximum on f1, 4g is M − f s4d − s4 − 2. Thus Property 8 gives
1s4 2 1d < y sx dx < 2s4 2 1d 4
1
1
y=1
0
1
4
FIGURE 17
3 < y sx dx < 6 4
or x
The result of Example 8 is illustrated in Figure 17. The area under y − sx from 1 to 4 is greater than the area of the lower rectangle and less than the area of the large rectangle.
1. Evaluate the Riemann sum for f sxd − x 2 1, 26 < x < 4, with five subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents. 2. If
n
1
4 6. The graph of t is shown. Estimate y22 tsxd dx with six sub intervals using (a) right endpoints, (b) left endpoints, and (c) midpoints.
y
f sxd − cos x 0 < x < 3y4
1
evaluate the Riemann sum with n − 6, taking the sample points to be left endpoints. (Give your answer correct to six decimal places.) What does the Riemann sum represent? Illustrate with a diagram.
x
1
3. If f sxd − x 2 2 4, 0 < x < 3, find the Riemann sum with n − 6, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram. 4. (a) Find the Riemann sum for f sxd − 1yx, 1 < x < 2, with four terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.) Explain what the Riemann sum represents with the aid of a sketch. (b) Repeat part (a) with midpoints as the sample points. 5. The graph of a function f is given. Estimate y010 f sxd dx using five subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. y
1 0
1
x
7. A table of values of an increasing function f is shown. Use 30 the table to find lower and upper estimates for y10 f sxd dx. x
10
14
18
22
26
30
f sxd
212
26
22
1
3
8
8. The table gives the values of a function obtained from an 9 experiment. Use them to estimate y3 f sxd dx using three equal subintervals with (a) right endpoints, (b) left end points, and (c) midpoints. If the function is known to be an increasing function, can you say whether your estimates are less than or greater than the exact value of the integral? x f sxd
3
4
5
23.4 22.1 20.6
6
7
8
9
0.3
0.9
1.4
1.8
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
317
Section 4.2 The Definite Integral
26. (a) Find an approximation to the integral y04 sx 2 2 3xd dx using a Riemann sum with right endpoints and n − 8. (b) Draw a diagram like Figure 3 to illustrate the approximation in part (a). (c) Use Theorem 4 to evaluate y04 sx 2 2 3xd dx. (d) Interpret the integral in part (c) as a difference of areas and illustrate with a diagram like Figure 4.
9–12 Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 9.
y
8
11.
y
2
0
sin sx dx, n − 4 10. y sx 3 1 1 dx, n − 5 1
0
x dx, n − 5 12. x sin2 x dx, n − 4 y 0 x11
0
27. Prove that y x dx − b
CAS
13. If you have a CAS that evaluates midpoint approximations and graphs the corresponding rectangles (use RiemannSum or middlesum and middlebox commands in Maple), check the answer to Exercise 11 and illustrate with a graph. Then repeat with n − 10 and n − 20. 14. With a programmable calculator or computer (see the instructions for Exercise 4.1.9), compute the left and right Riemann sums for the function f sxd − xysx 1 1d on the interval f0, 2g with n − 100. Explain why these estimates show that 2 x 0.8946 , y dx , 0.9081 0 x 1 1 15. Use a calculator or computer to make a table of values of right Riemann sums R n for the integral y0 sin x dx with n − 5, 10, 50, and 100. What value do these numbers appear to be approaching? 16. Use a calculator or computer to make a table of values of left and right Riemann sums L n and R n for the integral y02 s1 1 x 4 dx with n − 5, 10, 50, and 100. Between what two numbers must the value of the integral lie? Can you 2 make a similar statement for the integral y21 s1 1 x 4 dx? Explain.
a
28. Prove that y x 2 dx − b
a
18. lim
o x i s1 1 x i3
n l ` i−1
b3 2 a3 . 3
29–30 Express the integral as a limit of Riemann sums. Do not evaluate the limit. 1 3 5 29. y s4 1 x 2 dx 30. y2 x 2 1 x dx 1
S
CAS
D
31–32 Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit. 31.
y
0
sin 5x dx 32. y x 6 dx 10
2
33. The graph of f is shown. Evaluate each integral by inter preting it in terms of areas. (a) y f sxd dx (b) y f sxd dx 2
5
0
0
(c) y f sxd dx (d) y f sxd dx 7
9
5
0
y
17–20 Express the limit as a definite integral on the given interval. n sin x i 17. lim o Dx, f0, g nl` i−1 1 1 x i n
b2 2 a2 . 2
y=ƒ
2 0
2
4
6
8
x
Dx, f2, 5g
n
o f5sx*i d3 2 4 x*i g Dx, f2, 7] n l ` i−1
19. lim
n
20. lim
o
n l ` i−1
x *i Dx, f1, 3g sx *i d2 1 4
34. The graph of t consists of two straight lines and a semicircle. Use it to evaluate each integral. (a) y tsxd dx (b) y tsxd dx (c) y tsxd dx 6
2
21–25 Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. 21.
y
5
23.
y
0
25.
y
1
2
s4 2 2xd dx 22. y sx 2 2 4x 1 2 d dx
2
y=©
2
22
0
sx 2 1 x d dx 24. y s2x 2 x 3 d dx
0
y 4
4
1
7
2
0
sx 3 2 3x 2 d dx
0
0
4
7 x
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318
Chapter 4 Integrals
35–40 Evaluate the integral by interpreting it in terms of areas. 35.
y
37.
y (1 1 s9 2 x ) dx 38. y ( x 2 s25 2 x ) dx
39.
y  12 x  dx 40. y  2x 2 1  dx
2
21
s1 2 xd dx 36. y ( 13 x 2 2) dx 9
0
0
5
2
23
52. If Fsxd − y2x f std dt, where f is the function whose graph is given, which of the following values is largest? (A) Fs0d (B) Fs1d (C) Fs2d (D) Fs3d (E) Fs4d
2
y
25
3
1
24
0
y=f(t) 0
41. Evaluate y s1 1 x 4 dx. 1
1
2
3
t
4
1
42. Given that y sin4 x dx − 38 , what is y sin4 d? 0
0
43. In Example 4.1.2 we showed that y0 x 2 dx − 13. Use this fact 1
53. Each of the regions A, B, and C bounded by the graph of f and the xaxis has area 3. Find the value of
y
and the properties of integrals to evaluate y0 s5 2 6x 2 d dx. 1
44. Use the properties of integrals and the result of Example 3 to evaluate y25 s1 1 3x 4 d dx.
47. Write as a single integral in the form yab f sxd dx:
y
2
22
f sxd dx 1 y f sxd dx 2 y 5
21
2
22
f sxd dx
48. If y f sxd dx − 7.3 and y f sxd dx − 5.9, find y f sxd dx. 8 2
4 2
8 4
49. If y09 f sxd dx − 37 and y09 tsxd dx − 16, find
y
9
0
50. Find y f sxd dx if 5 0
_4
y
_2
A
0
C
2
x
54. Suppose f has absolute minimum value m and absolute maximum value M. Between what two values must y02 f sxd dx lie? Which property of integrals allows you to make your conclusion? 55–58 Use the properties of integrals to verify the inequality without evaluating the integrals. 55. 56.
f2 f sxd 1 3tsxdg dx
f f sxd 1 2x 1 5g dx
B
45. Use the results of Exercises 27 and 28 and the properties of integrals to evaluate y14 s2x 2 2 3x 1 1d dx. 46. Use the result of Exercise 27 and the fact that y2 y0 cos x dx − 1 (from Exercise 4.1.31), together with the properties of integrals, to evaluate y0y2 s2 cos x 2 5xd dx.
2
24
y
4
0
y
1
0
sx 2 2 4x 1 4d dx > 0 s1 1 x 2 dx < y s1 1 x dx 1
0
57. 2 < y s1 1 x 2 dx < 2 s2 1
21
H
3 f sxd − x
58.
for x , 3 for x > 3
51. For the function f whose graph is shown, list the following quantities in increasing order, from smallest to largest, and explain your reasoning.
y3 s3 < y sin x dx < y6 12 12
59–64 Use Property 8 of integrals to estimate the value of the integral. 3 1 x 3 dx 60. y0 x 1 4 dx
59.
y
1
y f sxd dx (E) f 9s1d (D)
61.
y
y3
y
63.
y
1
2
65–66 Use properties of integrals, together with Exercises 27 and 28, to prove the inequality.
(A) y f sxd dx (B) y f sxd dx (C) y f sxd dx 8 0
3 0
8 3
8 4
0
5
x
65.
0
y4
3
1
2
0
s1 1 x 4 dx 64. y sx 2 2 sin xd dx 2
21
y
tan x dx 62. y sx 3 2 3x 1 3d dx
sx 4 1 1 dx >
26 y2 2 66. x sin x dx < y 0 3 8
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
discovery project Area Functions
67. W hich of the integrals y12 sx dx, y12 s1yx dx, and y12 ssx dx has the largest value? Why? 0.5 0.5 68. W hich of the integrals y0 cossx 2 d dx, y0 cos sx dx is larger? Why?
69. Prove Property 3 of integrals.
uy
b
a

u
f sxd dx < y

a
 f sxd  dx


b
[Hint: 2 f sxd < f sxd < f sxd .] (b) Use the result of part (a) to show that
uy
2
0
u
72. Let f s0d − 0 and f sxd − 1yx if 0 , x < 1. Show that f is not integrable on f0, 1g. [Hint: Show that the first term in the Riemann sum, f sx1* d Dx, can be made arbitrarily large.] 73–74 Express the limit as a definite integral. n
70. (a) If f is continuous on fa, bg, show that
f sxd sin 2x dx < y
2
0
 f sxd  dx
o n l ` i−1
73. lim 74. lim
nl`
1 n
i4 [Hint: Consider f sxd − x 4.] n5 n
o i−1
1 1 1 siynd2
2
75. Find y1 x 22 dx. Hint: Choose x *i to be the geometric mean of x i21 and x i (that is, x *i − sx i21 x i ) and use the identity 1 1 1 − 2 msm 1 1d m m11
71. Let f sxd − 0 if x is any rational number and f sxd − 1 if x is any irrational number. Show that f is not integrable on f0, 1g.
discovery Project
319
area functions 1. (a) Draw the line y − 2t 1 1 and use geometry to find the area under this line, above the taxis, and between the vertical lines t − 1 and t − 3. (b) If x . 1, let Asxd be the area of the region that lies under the line y − 2t 1 1 between t − 1 and t − x. Sketch this region and use geometry to find an expression for Asxd. (c) Differentiate the area function Asxd. What do you notice? 2. (a) If x > 21, let Asxd − y s1 1 t 2 d dt x
21
Asxd represents the area of a region. Sketch that region. (b) Use the result of Exercise 4.2.28 to find an expression for Asxd. (c) Find A9sxd. What do you notice? (d) If x > 21 and h is a small positive number, then Asx 1 hd 2 Asxd represents the area of a region. Describe and sketch the region. (e) Draw a rectangle that approximates the region in part (d). By comparing the areas of these two regions, show that Asx 1 hd 2 Asxd < 1 1 x2 h (f) Use part (e) to give an intuitive explanation for the result of part (c).
3. (a) Draw the graph of the function f sxd − cossx 2 d in the viewing rectangle f0, 2g ; by f21.25, 1.25g. (b) If we define a new function t by tsxd − y cos st 2 d dt x
0
then tsxd is the area under the graph of f from 0 to x [until f sxd becomes negative, at which point tsxd becomes a difference of areas]. Use part (a) to determine the value
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
320
Chapter 4 Integrals
of x at which tsxd starts to decrease. [Unlike the integral in Problem 2, it is impossible to evaluate the integral defining t to obtain an explicit expression for tsxd.] (c) Use the integration command on your calculator or computer to estimate ts0.2d, ts0.4d, ts0.6d, . . . , ts1.8d, ts2d. Then use these values to sketch a graph of t. (d) Use your graph of t from part (c) to sketch the graph of t9 using the interpretation of t9sxd as the slope of a tangent line. How does the graph of t9 compare with the graph of f ? 4. Suppose f is a continuous function on the interval fa, bg and we define a new function t by the equation x tsxd − y f std dt a
Based on your results in Problems 1–3, conjecture an expression for t9sxd.
y
The Fundamental Theorem of Calculus is appropriately named because it establishes a connection between the two branches of calculus: differential calculus and integral calculus. Differential calculus arose from the tangent problem, whereas integral calculus arose from a seemingly unrelated problem, the area problem. Newton’s mentor at Cambridge, Isaac Barrow (1630 –1677), discovered that these two problems are actually closely related. In fact, he realized that differentiation and integration are inverse processes. The Fundamental Theorem of Calculus gives the precise inverse relationship between the derivative and the integral. It was Newton and Leibniz who exploited this relationship and used it to develop calculus into a systematic mathematical method. In particular, they saw that the Fundamental Theorem enabled them to compute areas and integrals very easily without having to compute them as limits of sums as we did in Sections 4.1 and 4.2. The first part of the Fundamental Theorem deals with functions defined by an equa tion of the form
y=f(t )
1
area=©
0
a
x
b
t
FIGURE 1 y 2
x
a
where f is a continuous function on fa, bg and x varies between a and b. Observe that t depends only on x, which appears as the variable upper limit in the integral. If x is a fixed number, then the integral yax f std dt is a definite number. If we then let x vary, the number yax f std dt also varies and defines a function of x denoted by tsxd. If f happens to be a positive function, then tsxd can be interpreted as the area under the graph of f from a to x, where x can vary from a to b. (Think of t as the “area so far” function; see Figure 1.)
y=f(t)
1 0
tsxd − y f std dt
1
2
4
t
Example 1 If f is the function whose graph is shown in Figure 2 and tsxd − y0x f std dt, find the values of ts0d, ts1d, ts2d, ts3d, ts4d, and ts5d. Then sketch a rough graph of t. SOLUTION First we notice that ts0d − y0 f std dt − 0. From Figure 3 we see that ts1d is 0
the area of a triangle: FIGURE 2
ts1d − y f std dt − 12 s1 2d − 1 1
0
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
321
Section 4.3 The Fundamental Theorem of Calculus
To find ts2d we add to ts1d the area of a rectangle: ts2d − y f std dt − y f std dt 1 y f std dt − 1 1 s1 ? 2d − 3 2
1
0
2
0
1
We estimate that the area under f from 2 to 3 is about 1.3, so ts3d − ts2d 1 y f std dt < 3 1 1.3 − 4.3 3
2
y 2
y 2
y 2
y 2
y 2
1
1
1
1
1
0
t
1
0
1
g(1)=1
2
t
0
g(2)=3
1
2
3
t
0
1
2
4
t
0
1
2
4
t
g(3)Å4.3 g(4)Å3
g(5)Å1.7
FIGURE 3 y
For t . 3, f std is negative and so we start subtracting areas:
4
g
3
ts4d − ts3d 1 y f std dt < 4.3 1 s21.3d − 3.0 4
3
2
ts5d − ts4d 1 y f std dt < 3 1 s21.3d − 1.7 5
4
1 0
1
2
4
3
We use these values to sketch the graph of t in Figure 4. Notice that, because f std is positive for t , 3, we keep adding area for t , 3 and so t is increasing up to x − 3, where it attains a maximum value. For x . 3, t decreases because f std is negative. n
5 x
FIGURE 4 tsxd − y f std dt x
If we take f std − t and a − 0, then, using Exercise 4.2.27, we have
0
tsxd − y t dt − x
0
Notice that t9sxd − x, that is, t9 − f . In other words, if t is defined as the integral of f by Equation 1, then t turns out to be an antiderivative of f , at least in this case. And if we sketch the derivative of the function t shown in Figure 4 by estimating slopes of tangents, we get a graph like that of f in Figure 2. So we suspect that t9 − f in Example 1 too. To see why this might be generally true we consider any continuous function f with x f sxd > 0. Then tsxd − ya f std dt can be interpreted as the area under the graph of f from a to x, as in Figure 1. In order to compute t9sxd from the definition of a derivative we first observe that, for h . 0, tsx 1 hd 2 tsxd is obtained by subtracting areas, so it is the area under the graph of f from x to x 1 h (the blue area in Figure 5). For small h you can see from the figure that this area is approximately equal to the area of the rectangle with height f sxd and width h:
y
h ƒ 0
a
FIGURE 5
x
x2 2
x+h
b
tsx 1 hd 2 tsxd < hf sxd
t
so
tsx 1 hd 2 tsxd < f sxd h
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
322
Chapter 4 Integrals
Intuitively, we therefore expect that t9sxd − lim
hl0
tsx 1 hd 2 tsxd − f sxd h
The fact that this is true, even when f is not necessarily positive, is the first part of the Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus, Part 1 If f is continuous on fa, bg, then the function t defined by
We abbreviate the name of this theorem as FTC1. In words, it says that the derivative of a definite integral with respect to its upper limit is the integrand evaluated at the upper limit.
tsxd − y f std dt a < x < b x
a
is continuous on fa, bg and differentiable on sa, bd, and t9sxd − f sxd. Proof If x and x 1 h are in sa, bd, then
tsx 1 hd 2 tsxd − y
y
x1h
a
Sy
−
−y
x
a
x
a
x1h
x
f std dt 2 y f std dt f std dt 1 y
x1h
x
D
f std dt 2 y f std dt (by Property 5) x
a
f std dt
and so, for h ± 0, y=ƒ
2
m
0
x u
y
x1h
x
f std dt
For now let’s assume that h . 0. Since f is continuous on fx, x 1 hg, the Extreme Value Theorem says that there are numbers u and v in fx, x 1 hg such that f sud − m and f svd − M, where m and M are the absolute minimum and maximum values of f on fx, x 1 hg. (See Figure 6.) By Property 8 of integrals, we have
M
√=x+h
tsx 1 hd 2 tsxd 1 − h h
x
FIGURE 6
mh < y
x1h
f sudh < y
x1h
x
that is,
x
f std dt < Mh f std dt < f svdh
Since h . 0, we can divide this inequality by h: f sud <
1 h
y
x1h
x
f std dt < f svd
Now we use Equation 2 to replace the middle part of this inequality: 3 TEC Module 4.3 provides visual evidence for FTC1.
f sud <
tsx 1 hd 2 tsxd < f svd h
Inequality 3 can be proved in a similar manner for the case where h , 0. (See Exercise 69.)
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 4.3 The Fundamental Theorem of Calculus
323
Now we let h l 0. Then u l x and v l x, since u and v lie between x and x 1 h. Therefore lim f sud − lim f sud − f sxd and lim f svd − lim f svd − f sxd
hl0
ulx
hl0
vlx
because f is continuous at x. We conclude, from (3) and the Squeeze Theorem, that 4
t9sxd − lim
hl0
tsx 1 hd 2 tsxd − f sxd h
If x − a or b, then Equation 4 can be interpreted as a onesided limit. Then Theo rem 2.2.4 (modified for onesided limits) shows that t is continuous on fa, bg.
n
Using Leibniz notation for derivatives, we can write FTC1 as 5
d dx
y
x
a
f std dt − f sxd
when f is continuous. Roughly speaking, Equation 5 says that if we first integrate f and then differentiate the result, we get back to the original function f.
Example 2 Find the derivative of the function tsxd − y s1 1 t 2 dt. x
0
SOLUTION Since f std − s1 1 t 2 is continuous, Part 1 of the Fundamental Theorem
of Calculus gives
y 1
f
0
t9sxd − s1 1 x 2
n
Example 3 Although a formula of the form tsxd − yax f std dt may seem like a strange
S
way of defining a function, books on physics, chemistry, and statistics are full of such functions. For instance, the Fresnel function
x
1
Ssxd − y sinst 2y2d dt x
0
is named after the French physicist Augustin Fresnel (1788 –1827), who is famous for his works in optics. This function first appeared in Fresnel’s theory of the diffraction of light waves, but more recently it has been applied to the design of highways. Part 1 of the Fundamental Theorem tells us how to differentiate the Fresnel function:
FIGURE 7 f sxd − sinsx 2y2d Ssxd − y sinst 2y2d dt x
0
S9sxd − sinsx 2y2d
y 0.5 1
FIGURE 8 The Fresnel function
Ssxd − y sinst 2y2d dt x
0
x
This means that we can apply all the methods of differential calculus to analyze S (see Exercise 63). Figure 7 shows the graphs of f sxd − sinsx 2y2d and the Fresnel function Ssxd − y0x f std dt. A computer was used to graph S by computing the value of this integral for many values of x. It does indeed look as if Ssxd is the area under the graph of f from 0 to x [until x < 1.4 when Ssxd becomes a difference of areas]. Figure 8 shows a larger part of the graph of S. If we now start with the graph of S in Figure 7 and think about what its derivative should look like, it seems reasonable that S9sxd − f sxd. [For instance, S is increasing when f sxd . 0 and decreasing when f sxd , 0.] So this gives a visual confirmation of Part 1 of the Fundamental Theorem of Calculus. n
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324
Chapter 4 Integrals
d x4 y sec t dt. dx 1 SOLUTION Here we have to be careful to use the Chain Rule in conjunction with FTC1. Let u − x 4. Then
Example 4 Find
d dx
sec t dt −
d dx
−
d du
− sec u
− secsx 4 d ? 4x 3
y
x4
1
y
u
1
sec t dt
Fy
u
1
G
sec t dt
du dx
du dx
(by the Chain Rule)
(by FTC1)
n
In Section 4.2 we computed integrals from the definition as a limit of Riemann sums and we saw that this procedure is sometimes long and difficult. The second part of the Fundamental Theorem of Calculus, which follows easily from the first part, provides us with a much simpler method for the evaluation of integrals. The Fundamental Theorem of Calculus, Part 2 If f is continuous on fa, bg, then
y
We abbreviate this theorem as FTC2.
b
a
f sxd dx − Fsbd 2 Fsad
where F is any antiderivative of f, that is, a function F such that F9 − f. Proof Let tsxd − ya f std dt. We know from Part 1 that t9sxd − f sxd; that is, t is an x
antiderivative of f. If F is any other antiderivative of f on fa, bg, then we know from Corollary 3.2.7 that F and t differ by a constant: 6
Fsxd − tsxd 1 C
for a , x , b. But both F and t are continuous on fa, bg and so, by taking limits of both sides of Equation 6 (as x l a1 and x l b2), we see that it also holds when x − a and x − b. So Fsxd − tsxd 1 C for all x in fa, bg. If we put x − a in the formula for tsxd, we get tsad − y f std dt − 0 a
a
So, using Equation 6 with x − b and x − a, we have Fsbd 2 Fsad − f tsbd 1 Cg 2 f tsad 1 Cg
− tsbd 2 tsad − tsbd − y f std dt b
a
n
Part 2 of the Fundamental Theorem states that if we know an antiderivative F of f, then we can evaluate yab f sxd dx simply by subtracting the values of F at the endpoints of the interval fa, bg. It’s very surprising that yab f sxd dx, which was defined by a complicated procedure involving all of the values of f sxd for a < x < b, can be found by knowing the values of Fsxd at only two points, a and b. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 4.3 The Fundamental Theorem of Calculus
325
Although the theorem may be surprising at first glance, it becomes plausible if we interpret it in physical terms. If vstd is the velocity of an object and sstd is its position at time t, then vstd − s9std, so s is an antiderivative of v. In Section 4.1 we considered an object that always moves in the positive direction and made the guess that the area under the velocity curve is equal to the distance traveled. In symbols:
y
b
a
vstd dt − ssbd 2 ssad
That is exactly what FTC2 says in this context.
Example 5 Evaluate the integral y x 3 dx. 1
22
SOLUTION The function f sxd − x 3 is continuous on f22, 1g and we know from Sec
tion 3.9 that an antiderivative is Fsxd − 14 x 4, so Part 2 of the Fundamental Theorem gives
y
1
22
x 3 dx − Fs1d 2 Fs22d − 14 s1d4 2 14 s22d4 − 2 15 4
Notice that FTC2 says we can use any antiderivative F of f. So we may as well use the simplest one, namely Fsxd − 14 x 4, instead of 14 x 4 1 7 or 14 x 4 1 C. n We often use the notation
g
b
Fsxd a − Fsbd 2 Fsad So the equation of FTC2 can be written as
y
b
a
g
b
f sxd dx − Fsxd a where F9 − f
Other common notations are Fsxd

b a
and fFsxdg ba .
Example 6 Find the area under the parabola y − x 2 from 0 to 1. SOLUTION An antiderivative of f sxd − x 2 is Fsxd − 13 x 3. The required area A is found
using Part 2 of the Fundamental Theorem: In applying the Fundamental Theorem we use a particular antiderivative F of f . It is not necessary to use the most general antiderivative.
A − y x 2 dx − 1
0
x3 3
G
1
− 0
13 03 1 2 − 3 3 3
n
If you compare the calculation in Example 6 with the one in Example 4.1.2, you will see that the Fundamental Theorem gives a much shorter method. y 1
Example 7 Find the area under the cosine curve from 0 to b, where 0 < b < y2. y=cos x
SOLUTION Since an antiderivative of f sxd − cos x is Fsxd − sin x, we have
0
FIGURE 9
π 2
x
g
A − y cos x dx − sin x 0 − sin b 2 sin 0 − sin b b
area=1
0
b
In particular, taking b − y2, we have proved that the area under the cosine curve from 0 to y2 is sinsy2d − 1. (See Figure 9.) n When the French mathematician Gilles de Roberval first found the area under the sine and cosine curves in 1635, this was a very challenging problem that required a great deal of ingenuity. If we didn’t have the benefit of the Fundamental Theorem, we would
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326
Chapter 4 Integrals
have to compute a difficult limit of sums using obscure trigonometric identities (or a computer algebra system as in Exercise 4.1.31). It was even more difficult for Roberval because the apparatus of limits had not been invented in 1635. But in the 1660s and 1670s, when the Fundamental Theorem was discovered by Barrow and exploited by Newton and Leibniz, such problems became very easy, as you can see from Example 7.
Example 8 What is wrong with the following calculation?
y
3
21
1 x21 dx − x2 21
G
3
21
−2
4 1 21−2 3 3
SOLUTION To start, we notice that this calculation must be wrong because the answer is negative but f sxd − 1yx 2 > 0 and Property 6 of integrals says that yab f sxd dx > 0 when f > 0. The Fundamental Theorem of Calculus applies to continuous functions. It can’t be applied here because f sxd − 1yx 2 is not continuous on f21, 3g. In fact, f has an infinite discontinuity at x − 0, so
y
3
21
1 dx does not exist. x2
n
Differentiation and Integration as Inverse Processes We end this section by bringing together the two parts of the Fundamental Theorem. The Fundamental Theorem of Calculus Suppose f is continuous on fa, bg. 1. If tsxd − ya f std dt, then t9sxd − f sxd. x
2. ya f sxd dx − Fsbd 2 Fsad, where F is any antiderivative of f , that is, F9− f. b
We noted that Part 1 can be rewritten as d dx
y
x
a
f std dt − f sxd
which says that if f is integrated and then the result is differentiated, we arrive back at the original function f. Since F9sxd − f sxd, Part 2 can be rewritten as
y
b
a
F9sxd dx − Fsbd 2 Fsad
This version says that if we take a function F, first differentiate it, and then integrate the result, we arrive back at the original function F, but in the form Fsbd 2 Fsad. Taken together, the two parts of the Fundamental Theorem of Calculus say that differentiation and integration are inverse processes. Each undoes what the other does. The Fundamental Theorem of Calculus is unquestionably the most important theo rem in calculus and, indeed, it ranks as one of the great accomplishments of the human mind. Before it was discovered, from the time of Eudoxus and Archimedes to the time of Galileo and Fermat, problems of finding areas, volumes, and lengths of curves were so difficult that only a genius could meet the challenge. But now, armed with the systematic method that Newton and Leibniz fashioned out of the Fundamental Theorem, we will see in the chapters to come that these challenging problems are accessible to all of us.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
327
Section 4.3 The Fundamental Theorem of Calculus
1. Explain exactly what is meant by the statement that “differentiation and integration are inverse processes.” x
2. Let tsxd − y0 f std dt, where f is the function whose graph is shown. (a) Evaluate tsxd for x − 0, 1, 2, 3, 4, 5, and 6. (b) Estimate ts7d. (c) Where does t have a maximum value? Where does it have a minimum value? (d) Sketch a rough graph of t. y
f
1 0
4
1
t
6
y
f 1
7. tsxd −
y
9. tssd −
y
x
t
4. Let tsxd − y0x f std dt, where f is the function whose graph is shown. (a) Evaluate ts0d and ts6d. (b) Estimate tsxd for x − 1, 2, 3, 4, and 5. (c) On what interval is t increasing? (d) Where does t have a maximum value? (e) Sketch a rough graph of t. (f) Use the graph in part (e) to sketch the graph of t9sxd. Compare with the graph of f. y 2
f 2
5
t
8. tsxd − y cosst 2 d dt x
st 1 t 3 dt
0
1
st 2 t 2 d8 dt 10. hsud − y
u
s
0
5
x
st dt t11
F
G
Hint: y s1 1 sec t dt − 2y s1 1 sec t dt 0
x
0
x
12. Rs yd − y t 3 sin t dt 2
y
13. hsxd − y
1yx
15. y − y
3x12
17. y − y
y4
1
5
x
0
7–18 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
2
1
6. tsxd − y s2 1 sin td dt
x
1
0
x
0
5. tsxd − y t 2 dt
11. Fsxd − y s1 1 sec t dt
3. Let tsxd − y0 f std dt, where f is the function whose graph is shown. (a) Evaluate ts0d, ts1d, ts2d, ts3d, and ts6d. (b) On what interval is t increasing? (c) Where does t have a maximum value? (d) Sketch a rough graph of t.
0
5–6 Sketch the area represented by tsxd. Then find t9sxd in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating.
sx
sin 4t dt 14. hsxd − y sx
1
z2 dz z 11 4
x4 t dt 16. y − y cos2 d 0 1 1 t3
tan d 18. y − ysin x s1 1 t 2 dt 1
19–38 Evaluate the integral. 19.
y
21.
y ( 45 t
23.
y
9
25.
y
27.
y
1
29.
y
4
31.
y
y2
3
1
sx 2 1 2x 2 4d dx 20. y x 100 dx 1
21
2
0
1
2 34 t 2 1 25 t ) dt 22. y (1 2 8v 3 1 16v7) d v 1
0
sx dx 24. y x 22y3 dx 8
1
sin d 26. y dx 5
y6
0
3
25
su 1 2dsu 2 3d du 28. y s4 2 td st dt 4
0
2 1 x2
1
y6
sx
dx 30. y s3u 2 2dsu 1 1d du 2
21
csc t cot t dt 32. y csc 2 d y3
y4
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328 33.
Chapter 4 Integrals
y
1
0
35.
y
2
37.
y
38.
y
2
54. tsxd − y
4 2 s 1 1 s1 1 rd3 dr 34. y1 s 2 ds 5
v 1 3v v4
1
0
6
y
55. hsxd − y
Î
22
x2
tan x
1 dt s2 1 t 4
cos t dt. Find an equation of the tangent line t to the curve y − Fsxd at the point with xcoordinate . 57. Let Fsxd − y
sin x if 0 < x , y2 f sxd dx where f sxd − cos x if y2 < x < f sxd dx where f sxd −
t sin t dt
cosst 2 d dt 56. tsxd − y
x3
sx
18
3 d v 36. dz 1 z
H H
112x
122x
x
x
58. If f sxd − y0 s1 2 t 2 d cos 2 t dt, on what interval is f increasing?
2 if 22 < x < 0 4 2 x 2 if 0 , x < 2
59. On what interval is the curve ; 39–42 Sketch the region enclosed by the given curves and calculate its area.
y−y
39. y − sx , y − 0, x − 4
x 60. Let Fsxd − y1 f std dt, where f is the function whose graph is shown. Where is F concave downward?
41. y − 4 2 x 2, y − 0
y
42. y − 2 x 2 x 2, y − 0 ; 43–46 Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. 3 43. y − s x , 0 < x < 27
44. y − x 24, 1 < x < 6
45. y − sin x, 0 < x <
46. y − sec 2 x, 0 < x < y3
_1
62. If f sxd − y0 t 0sy6d.
sin x
x 3 dx 48. y cos x dx
2
2
21
y6
; 49–52 What is wrong with the equation?
y
1
50.
y
2
51.
y
y
49.
52.
x 24 dx −
22
G G
1
y3
−2
22
4 2 dx − 2 2 x3 x
21
0
x23 23
g
0
g
CAS
y3
F
3x
2x
Sisxd − y
−0
u2 2 1 du u2 1 1
2x
0
2x
x
0
G
Hint: y f sud du − y f sud du 1 y f sud du 3x
sins t 2y2d dt − 0.2
64. The sine integral function
− 23
53–56 Find the derivative of the function. 53. tsxd − y
x
0
3x
0
t
y s1 1 t 2 dt and ts yd − y3 f sxd dx, find
y
3 − 2 21
1
63. The Fresnel function S was defined in Example 3 and graphed in Figures 7 and 8. (a) At what values of x does this function have local maxi mum values? (b) On what intervals is the function concave upward? (c) Use a graph to solve the following equation correct to two decimal places:
2
sec tan d − sec
sec 2 x dx − tan x
CAS
3 8
0
61. If f s1d − 12, f 9 is continuous, and y14 f 9sxd dx − 17, what is the value of f s4d?
47–48 Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.
y
t2 dt t 1t12 2
concave downward?
40. y − x 3, y − 0, x − 1
47.
x
0
sin t dt t
is important in electrical engineering. [The integrand f std − ssin tdyt is not defined when t − 0, but we know that its limit is 1 when t l 0. So we define f s0d − 1 and this makes f a continuous function everywhere.] (a) Draw the graph of Si. (b) At what values of x does this function have local maxi mum values? (c) Find the coordinates of the first inflection point to the right of the origin.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Section 4.3 The Fundamental Theorem of Calculus
(d) Does this function have horizontal asymptotes? (e) Solve the following equation correct to one decimal place:
y
x
0
73. Show that 0 0. (b) Show that 1 < y01 s1 1 x 3 dx < 1.25. 72. (a) Show that cossx 2d > cos x for 0 < x < 1.
x,0 0 0 on f3, 4g. Thus, from Equation 3, the distance traveled is
y  vstd  dt − y 4
To integrate the absolute value of vstd, we use Property 5 of integrals from Section 4.2 to split the integral into two parts, one where vstd < 0 and one where vstd > 0.
1
3
1
f2vstdg dt 1 y vstd dt 4
3
− y s2t 2 1 t 1 6d dt 1 y st 2 2 t 2 6d dt 3
4
1
3
F
G F
t3 t2 − 2 1 1 6t 3 2
−
3
1
1
G
4
t3 t2 2 2 6t 3 2
61 < 10.17 m 6
3
Example 7 Figure 4 shows the power consumption in the city of San Francisco for a day in September (P is measured in megawatts; t is measured in hours starting at midnight). Estimate the energy used on that day. P 800 600 400 200
FIGURE 4
0
3
6
9
12
15
18
21
t
Pacific Gas & Electric
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n
336
Chapter 4 Integrals
SOLUTION Power is the rate of change of energy: Pstd − E9std. So, by the Net Change
Theorem,
y
24
0
Pstd dt − y E9std dt − Es24d 2 Es0d 24
0
is the total amount of energy used on that day. We approximate the value of the integral using the Midpoint Rule with 12 subintervals and Dt − 2:
y
24
0
Pstd dt < fPs1d 1 Ps3d 1 Ps5d 1 ∙ ∙ ∙ 1 Ps21d 1 Ps23dg Dt < s440 1 400 1 420 1 620 1 790 1 840 1 850 1 840 1 810 1 690 1 670 1 550ds2d − 15,840
The energy used was approximately 15,840 megawatthours. A note on units
How did we know what units to use for energy in Example 7? The integral y024 Pstd dt is defined as the limit of sums of terms of the form Psti*d Dt. Now Psti*d is measured in megawatts and Dt is measured in hours, so their product is measured in megawatthours. The same is true of the limit. In general, the unit of measurement for yab f sxd dx is the product of the unit for f sxd and the unit for x.
1–4 Verify by differentiation that the formula is correct.
1 1 sx 1 x
11.
y
12.
y
3. y tan2 x dx − tan x 2 x 1 C
13.
y s2 1 tan
2 4. y xsa 1 bx dx − s3bx 2 2adsa 1 bxd3y2 1 C 15b2
15.
y
s1 1 x 2 dx − 2 1C 1. y 2 2 x x s1 1 x 1
2.
y cos x dx − 12 x 1 14 sin 2x 1 C 2
5. y sx 1.3 1 7x 2.5 d dx
17.
4 x 5 dx 6. y s
7. y (5 1
u2 1 1 1 2
1 u2
dx
D
du
d d 14. y sec t ssec t 1 tan td dt
1 2 sin3 t sin 2x dt 16. y sin x dx sin2 t
y (cos x 1 12 x) dx 18. y s1 2 x d dx 2 2
19–42 Evaluate the integral. 1
) dx
3 3 4x
8. y (u 6 2 2u 5 2 u 3 1 27 ) du 9.
S
sx
; 17–18 Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen.
5–16 Find the general indefinite integral.
2 2 3x
n
y su 1 4ds2u 1 1d du 10. y st st 1 3t 1 2d dt 2
19.
y
21.
y ( 12 t
22.
y
3
0
22 3
0
sx 2 2 3d dx 20. y s4x 3 2 3x 2 1 2xd dx 2
22
1
4
1 14 t 3 2 t) dt
s1 1 6w 2 2 10w 4 d dw
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Section 4.4 Indefinite Integrals and the Net Change Theorem
y
2
25.
y
27.
y
4
23.
29.
0
0
s2x 2 3ds4x 2 1 1d dx 24. y ts1 2 td 2 dt 1
y
21
1
S yS yS
s4 sin 2 3 cos d d 26. y 2
1
S D yÎ 4 1 6u su
1
D D
1 4 2 3 x2 x
2 1 du 28. 22 2 1 p
1
y
4
33.
y
y4
34.
y
y3
35.
y
8
37.
y ssx
39.
y  x 2 3  dx 40. y  2x 2 1  dx
41.
3y2 y21 ( x 2 2  x ) dx 42. y0  sin x  dx
1 1
0
D
y4
0
1 1 cos2 d cos2
21t 3 2 t s
4
5
dt 36. y su (u 2 s3 u ) du 64
0
5 1s x 4 d dx 38. y s1 1 x 2 d3 dx
1
0
5
2
2
0
2
a graph to estimate the xintercepts of the curve ; 43. Use y − 1 2 2x 2 5x 4. Then use this information to estimate the area of the region that lies under the curve and above the xaxis. 4 6 ; 44. Repeat Exercise 43 for the curve y − 2x 1 3x 2 2x .
45. The area of the region that lies to the right of the yaxis and to the left of the parabola x − 2y 2 y 2 (the shaded region in the figure) is given by the integral y02 s2y 2 y 2 d dy. (Turn your head clockwise and think of the region as lying below the curve x − 2y 2 y 2 from y − 0 to y − 2.) Find the area of the region. y 2
0
1
x
47. I f w9std is the rate of growth of a child in pounds per year, what does y510 w9std dt represent? 48. The current in a wire is defined as the derivative of the charge: Istd − Q9std. (See Example 2.7.3.) What does yab Istd dt represent? 49. If oil leaks from a tank at a rate of rstd gallons per minute at time t, what does y120 0 rstd dt represent?
sin 1 sin tan2 d sec2
0
y=$œ„ x
dp
st s1 1 td dt 32. y sec tan d
31.
0
y=1
2
5 8 2 3 dx 30. 2s w dw 3 1 x sw
4
1
dx
337
50. A honeybee population starts with 100 bees and increases at a rate of n9std bees per week. What does 100 1 y15 0 n9std dt represent? 51. In Section 3.7 we defined the marginal revenue function R9sxd as the derivative of the revenue function Rsxd, where 5000 x is the number of units sold. What does y1000 R9sxd dx represent? 52. I f f sxd is the slope of a trail at a distance of x miles from the start of the trail, what does y35 f sxd dx represent? 53. If x is measured in meters and f sxd is measured in newtons, what are the units for y0100 f sxd dx? 54. If the units for x are feet and the units for asxd are pounds per foot, what are the units for daydx? What units does y28 asxd dx have? 55–56 The velocity function (in meters per second) is given for a particle moving along a line. Find (a) the displacement and (b) the distance traveled by the particle during the given time interval. 55. vstd − 3t 2 5, 0 < t < 3 56. vstd − t 2 2 2t 2 3, 2 < t < 4 57–58 The acceleration function (in mys2 ) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time t and (b) the distance traveled during the given time interval.
x=2y¥
57. astd − t 1 4, v s0d − 5, 0 < t < 10 0 1
x
46. The boundaries of the shaded region in the figure are the 4 yaxis, the line y − 1, and the curve y − s x . Find the area of this region by writing x as a function of y and integrating with respect to y (as in Exercise 45).
58. astd − 2t 1 3, v s0d − 24, 0 < t < 3 59. The linear density of a rod of length 4 m is given by sxd − 9 1 2 sx measured in kilograms per meter, where x is measured in meters from one end of the rod. Find the total mass of the rod.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
338
Chapter 4 Integrals
60. Water flows from the bottom of a storage tank at a rate of rstd − 200 2 4t liters per minute, where 0 < t < 50. Find the amount of water that flows from the tank during the first 10 minutes.
64. Water flows into and out of a storage tank. A graph of the rate of change rstd of the volume of water in the tank, in liters per day, is shown. If the amount of water in the tank at time t − 0 is 25,000 L, use the Midpoint Rule to estimate the amount of water in the tank four days later.
61. The velocity of a car was read from its speedometer at 10second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. t (s)
v smiyhd
t (s)
v smiyhd
0 10 20 30 40 50
0 38 52 58 55 51
60 70 80 90 100
56 53 50 47 45
1
2
3
4
5
6
rstd
2
10
24
36
46
54
60
(a) Give upper and lower estimates for the total quantity Qs6d of erupted materials after six seconds. (b) Use the Midpoint Rule to estimate Qs6d.
63. Lake Lanier in Georgia, USA, is a reservoir created by Buford Dam on the Chattahoochee River. The table shows the rate of inflow of water, in cubic feet per second, as measured every morning at 7:30 am by the US Army Corps of Engineers. Use the Midpoint Rule to estimate the amount of water that flowed into Lake Lanier from July 18th, 2013, at 7:30 am to July 26th at 7:30 am.
July 18 July 19 July 20 July 21 July 22 July 23 July 24 July 25 July 26
5275 6401 2554 4249 3016 3821 2462 2628 3003
2
3
4 t
_1000
0
Inflow rate sft 3ysd
1
0
t
Day
2000 1000
62. Suppose that a volcano is erupting and readings of the rate rstd at which solid materials are spewed into the atmosphere are given in the table. The time t is measured in seconds and the units for rstd are tonnes (metric tons) per second.
r
65. The graph of the acceleration astd of a car measured in ftys2 is shown. Use the Midpoint Rule to estimate the increase in the velocity of the car during the sixsecond time interval. a 12 8 4 0
2
4
6 t (seconds)
66. Shown is the graph of traffic on an Internet service provider’s T1 data line from midnight to 8:00 am. D is the data throughput, measured in megabits per second. Use the Midpoint Rule to estimate the total amount of data transmitted during that time period. D 0.8
0.4
0
2
4
6
8 t (hours)
67. The following graph shows the power consumption in the province of Ontario, Canada, for December 9, 2004 (P is measured in megawatts; t is measured in hours starting at midnight). Using the fact that power is the rate of change of energy, estimate the energy used on that day.
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339
Writing project Newton, Leibniz, and the Invention of Calculus P 22,000
The following exercises are intended only for those who have already covered Chapter 6.
20,000
69–73 Evaluate the integral.
18,000
69.
y ssin x 1 sinh xd dx
70.
y
71.
y
72.
y
2
1
sx 2 1d3 dx x2
73.
y
1ys3
t2 2 1 dt t4 2 1
16,000 0
3
6
9
12
15
18
21
t
Independent Electricity Market Operator
May 7, 1992, the space shuttle Endeavour was launched ; 68. On on mission STS49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. (a) Use a graphing calculator or computer to model these data by a thirddegree polynomial. (b) Use the model in part (a) to estimate the height reached by the Endeavour, 125 seconds after liftoff. Event Launch Begin roll maneuver End roll maneuver Throttle to 89% Throttle to 67% Throttle to 104% Maximum dynamic pressure Solid rocket booster separation
writing Project
Time (s) 0 10 15 20 32 59 62 125
Velocity sftysd 0 185 319 447 742 1325 1445 4151
10
210
S
2e x dx sinh x 1 cosh x x2 1 1 1
0
1 x2 1 1
D
dx
74. The area labeled B is three times the area labeled A. Express b in terms of a. y
A 0
y
y=´
y=´
B a
x
0
b
x
Newton, leibniz, and the invention of calculus We sometimes read that the inventors of calculus were Sir Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646 –1716). But we know that the basic ideas behind integration were investigated 2500 years ago by ancient Greeks such as Eudoxus and Archimedes, and methods for finding tangents were pioneered by Pierre Fermat (1601–1665), Isaac Barrow (1630 –1677), and others. Barrow––who taught at Cambridge and was a major influence on Newton––was the first to understand the inverse relationship between differentiation and integration. What Newton and Leibniz did was to use this relationship, in the form of the Fundamental Theorem of Calculus, in order to develop calculus into a systematic mathematical discipline. It is in this sense that Newton and Leibniz are credited with the invention of calculus. Read about the contributions of these men in one or more of the given references and write a report on one of the following three topics. You can include biographical details, but the main
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340
Chapter 4 Integrals
focus of your report should be a description, in some detail, of their methods and notations. In particular, you should consult one of the sourcebooks, which give excerpts from the original publications of Newton and Leibniz, translated from Latin to English. l
The Role of Newton in the Development of Calculus
l
The Role of Leibniz in the Development of Calculus
l
The Controversy between the Followers of Newton and Leibniz over Priority in the Invention of Calculus
References
1. C arl Boyer and Uta Merzbach, A History of Mathematics (New York: Wiley, 1987), Chapter 19. 2. Carl Boyer, The History of the Calculus and Its Conceptual Development (New York: Dover, 1959), Chapter V. . H. Edwards, The Historical Development of the Calculus (New York: SpringerVerlag, 3. C 1979), Chapters 8 and 9. 4. Howard Eves, An Introduction to the History of Mathematics, 6th ed. (New York: Saunders, 1990), Chapter 11. 5. C. C. Gillispie, ed., Dictionary of Scientific Biography (New York: Scribner’s, 1974). See the article on Leibniz by Joseph Hofmann in Volume VIII and the article on Newton by I. B. Cohen in Volume X. 6. Victor Katz, A History of Mathematics: An Introduction (New York: HarperCollins, 1993), Chapter 12. 7. Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford University Press, 1972), Chapter 17. Sourcebooks
1. John Fauvel and Jeremy Gray, eds., The History of Mathematics: A Reader (London: MacMillan Press, 1987), Chapters 12 and 13. 2. D. E. Smith, ed., A Sourcebook in Mathematics (New York: Dover, 1959), Chapter V. 3. D. J. Struik, ed., A Sourcebook in Mathematics, 1200 –1800 (Princeton, NJ: Princeton University Press, 1969), Chapter V.
Because of the Fundamental Theorem, it’s important to be able to find antiderivatives. But our antidifferentiation formulas don’t tell us how to evaluate integrals such as 1 PS
Differentials were defined in Section 2.9. If u − f sxd, then du − f 9sxd dx
y 2xs1 1 x
2
dx
To find this integral we use the problemsolving strategy of introducing something extra. Here the “something extra” is a new variable; we change from the variable x to a new variable u. Suppose that we let u be the quantity under the root sign in (1), u − 1 1 x 2. Then the differential of u is du − 2x dx. Notice that if the dx in the notation for an inte
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Section 4.5 The Substitution Rule
341
gral were to be interpreted as a differential, then the differential 2x dx would occur in (1) and so, formally, without justifying our calculation, we could write
y 2xs1 1 x
2
2
dx − y s1 1 x 2 2x dx − y su du − 23 u 3y2 1 C − 23 s1 1 x 2 d3y2 1 C
But now we can check that we have the correct answer by using the Chain Rule to differentiate the final function of Equation 2: d dx
f 23 s1 1 x d
2 3y2
g
1 C − 23 32 s1 1 x 2 d1y2 2x − 2xs1 1 x 2
In general, this method works whenever we have an integral that we can write in the form y f stsxdd t9sxd dx. Observe that if F9− f , then
y F9stsxdd t9sxd dx − Fs tsxdd 1 C
3
because, by the Chain Rule, d fFs tsxddg − F9stsxdd t9sxd dx If we make the “change of variable” or “substitution” u − tsxd, then from Equation 3 we have
y F9stsxdd t9sxd dx − Fs tsxdd 1 C − Fsud 1 C − y F9sud du or, writing F9 − f , we get
y f stsxdd t9sxd dx − y f sud du Thus we have proved the following rule. 4 The Substitution Rule If u − tsxd is a differentiable function whose range is an interval I and f is continuous on I, then
y f stsxdd t9sxd dx − y f sud du Notice that the Substitution Rule for integration was proved using the Chain Rule for differentiation. Notice also that if u − tsxd, then du − t9sxd dx, so a way to remember the Substitution Rule is to think of dx and du in (4) as differentials. Thus the Substitution Rule says: it is permissible to operate with dx and du after integral signs as if they were differentials.
Example 1 Find y x 3 cossx 4 1 2d dx. SOLUTION We make the substitution u − x 4 1 2 because its differential is
du − 4x 3 dx, which, apart from the constant factor 4, occurs in the integral. Thus, using
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
342
Chapter 4 Integrals
x 3 dx −
1 4
du and the Substitution Rule, we have
yx
3
cossx 4 1 2d dx − y cos u
1 4
du − 14 y cos u du
− 14 sin u 1 C − 14 sinsx 4 1 2d 1 C Check the answer by differentiating it.
Notice that at the final stage we had to return to the original variable x.
n
The idea behind the Substitution Rule is to replace a relatively complicated integral by a simpler integral. This is accomplished by changing from the original variable x to a new variable u that is a function of x. Thus in Example 1 we replaced the integral y x 3 cossx 4 1 2d dx by the simpler integral 14 y cos u du. The main challenge in using the Substitution Rule is to think of an appropriate substitution. You should try to choose u to be some function in the integrand whose differential also occurs (except for a constant factor). This was the case in Example 1. If that is not possible, try choosing u to be some complicated part of the integrand (perhaps the inner function in a composite function). Finding the right substitution is a bit of an art. It’s not unusual to guess wrong; if your first guess doesn’t work, try another substitution.
Example 2 Evaluate y s2x 1 1 dx.
1 2
SOLUTION 1 Let u − 2x 1 1. Then du − 2 dx, so dx −
Rule gives
y s2x 1 1 dx − y su −
du. Thus the Substitution
12 du − 12 y u 1y2 du
1 u 3y2 1 C − 13 u 3y2 1 C 2 3y2
− 13 s2x 1 1d3y2 1 C SOLUTION 2 Another possible substitution is u − s2x 1 1. Then
du −
dx so dx − s2x 1 1 du − u du s2x 1 1
(Or observe that u 2 − 2x 1 1, so 2u du − 2 dx.) Therefore
y s2x 1 1 dx − y u u du − y u
−
Example 3 Find y
2
du
u3 1 C − 13 s2x 1 1d3y2 1 C 3
n
x dx. s1 2 4x 2
SOLUTION Let u − 1 2 4x 2. Then du − 28x dx, so x dx − 218 du and
y
x 1 dx − 218 y du − 218 y u 21y2 du 2 s1 2 4x su − 218 (2su ) 1 C − 214 s1 2 4x 2 1 C
n
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343
Section 4.5 The Substitution Rule 1 f _1
1 ©= ƒ dx
Example 4 Evaluate y cos 5x dx.
_1
FIGURE 1 f sxd −
SOLUTION If we let u − 5x, then du − 5 dx, so dx −
x
s1 2 4x 2
tsxd − y f sxd dx −
The answer to Example 3 could be checked by differentiation, but instead let’s check it with a graph. In Figure 1 we have used a computer to graph both the integrand f sxd − xys1 2 4x 2 and its indefinite integral tsxd − 214 s1 2 4x 2 (we take the case C − 0). Notice that tsxd decreases when f sxd is negative, increases when f sxd is positive, and has its minimum value when f sxd − 0. So it seems reasonable, from the graphical evidence, that t is an antiderivative of f.
214 s1
2 4x
2
1 5
du. Therefore
y cos 5x dx − 15 y cos u du − 15 sin u 1 C − 15 sin 5x 1 C
n
Note With some experience, you might be able to evaluate integrals like those in Examples 1– 4 without going to the trouble of making an explicit substitution. By recognizing the pattern in Equation 3, where the integrand on the left side is the product of the derivative of an outer function and the derivative of the inner function, we could work Example 1 as follows:
yx
3
cossx 4 1 2d dx − y cossx 4 1 2d x 3 dx − 14 y cossx 4 1 2d s4x 3 d dx −
1 4
y cossx
4
1 2d
d sx 4 1 2d dx − 14 sinsx 4 1 2d 1 C dx
Similarly, the solution to Example 4 could be written like this: d ssin 5xd dx − 15 sin 5x 1 C dx
y cos 5x dx − 15 y 5 cos 5x dx − 15 y
The following example, however, is more complicated and so an explicit substitution is advisable.
Example 5 Find y s1 1 x 2 x 5 dx.
SOLUTION An appropriate substitution becomes more obvious if we factor x 5 as x 4 ? x.
Let u − 1 1 x 2. Then du − 2x dx, so x dx − 12 du. Also x 2 − u 2 1, so x 4 − su 2 1d2:
y s1 1 x
2
x 5 dx − y s1 1 x 2 x 4 x dx − y su su 2 1d2 −
1 2
y su
5y2
1 2
du − 12 y su su 2 2 2u 1 1d du
2 2u 3y2 1 u 1y2 d du
− 12 ( 27 u 7y2 2 2 25 u 5y2 1 23 u 3y2) 1 C − 17 s1 1 x 2 d7y2 2 25 s1 1 x 2 d5y2 1 13 s1 1 x 2 d3y2 1 C
n
Deﬁnite Integrals When evaluating a definite integral by substitution, two methods are possible. One method is to evaluate the indefinite integral first and then use the Fundamental Theorem.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
344
Chapter 4 Integrals
For instance, using the result of Example 2, we have
y
4
0
g
s2x 1 1 dx − y s2x 1 1 dx
g
4 0
4
− 13 s2x 1 1d3y2 0 − 13 s9d3y2 2 13 s1d3y2 − 13 s27 2 1d − 26 3 Another method, which is usually preferable, is to change the limits of integration when the variable is changed. 5 The Substitution Rule for Definite Integrals If t9 is continuous on fa, bg and f is continuous on the range of u − tsxd, then
This rule says that when using a substitution in a definite integral, we must put everything in terms of the new variable u, not only x and dx but also the limits of integration. The new limits of integration are the values of u that correspond to x − a and x − b.
y
b
a
f stsxdd t9sxd dx − y
tsbd
tsad
f sud du
Proof Let F be an antiderivative of f. Then, by (3), Fs tsxdd is an antiderivative of
f stsxdd t9sxd, so by Part 2 of the Fundamental Theorem, we have
y
b
a
g
b
f stsxdd t9sxd dx − Fstsxdd a − Fstsbdd 2 Fstsadd
But, applying FTC2 a second time, we also have
y
g
tsbd
f sud du − Fsud
tsad
tsbd tsad
− Fstsbdd 2 Fstsadd
n
Example 6 Evaluate y s2x 1 1 dx using (5). 4
0
SOLUTION Using the substitution from Solution 1 of Example 2, we have u − 2x 1 1 and dx − 12 du. To find the new limits of integration we note that
when x − 0, u − 2s0d 1 1 − 1 and when x − 4, u − 2s4d 1 1 − 9 Therefore
The integral given in Example 7 is an abbreviation for 2 1 y1 s3 2 5xd2 dx
y
4
0
s2x 1 1 dx − y
9 1
1
2
su du
g
9
− 12 23 u 3y2
− 13 s9 3y2 2 13y2 d − 26 3
1
Observe that when using (5) we do not return to the variable x after integrating. We simply evaluate the expression in u between the appropriate values of u.
Example 7 Evaluate y
2
1
n
dx . s3 2 5xd2
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Section 4.5 The Substitution Rule
345
SOLUTION Let u − 3 2 5x. Then du − 25 dx, so dx − 215 du. When x − 1, u − 22
and when x − 2, u − 27. Thus
y
dx 1 2 − 2 s3 2 5xd 5
2
1
1 −2 5
1 5
−
S
y
27
22
du u2
F G 1 2 u
2
27
1 − 5u 22
1 1 1 7 2
D
−
G
27
22
1 14
n
Symmetry The next theorem uses the Substitution Rule for Definite Integrals (5) to simplify the calculation of integrals of functions that possess symmetry properties. 6 Integrals of Symmetric Functions Suppose f is continuous on f2a, ag. (a) If f is even f f s2xd − f sxdg, then y2a f sxd dx − 2 y0 f sxd dx. a
a
(b) If f is odd f f s2xd − 2f sxdg, then y2a f sxd dx − 0. a
Proof We split the integral in two:
7
y
a
2a
f sxd dx − y f sxd dx 1 y f sxd dx − 2y 0
a
0
2a
2a
0
f sxd dx 1 y f sxd dx a
0
In the first integral on the far right side we make the substitution u − 2x. Then du − 2dx and when x − 2a, u − a. Therefore 2y
2a
0
y
a
(a) ƒ even, j ƒ dx=2 j ƒ dx a
x
y
a
2a
f sxd dx − y f s2ud du 1 y f sxd dx a
a
0
0
(a) If f is even, then f s2ud − f sud so Equation 8 gives
y
a
_a
0
a
2a
f sxd dx − y f sud du 1 y f sxd dx − 2 y f sxd dx a
a
0
a
0
0
(b) If f is odd, then f s2ud − 2f sud and so Equation 8 gives
y
0
a
(b) ƒ odd, j ƒ dx=0 a
_a
FIGURE 2
a
0
and so Equation 7 becomes
0
_a
a
0
8 _a
f sxd dx − 2y f s2ud s2dud − y f s2ud du
x
y
a
2a
f sxd dx − 2y f sud du 1 y f sxd dx − 0 a
0
a
0
n
Theorem 6 is illustrated by Figure 2. For the case where f is positive and even, part (a) says that the area under y − f sxd from 2a to a is twice the area from 0 to a because of symmetry. Recall that an integral yab f sxd dx can be expressed as the area above the xaxis and below y − f sxd minus the area below the axis and above the curve. Thus part (b) says the integral is 0 because the areas cancel.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
346
Chapter 4 Integrals
Example 8 Since f sxd − x 6 1 1 satisfies f s2xd − f sxd, it is even and so
y
2
22
sx 6 1 1d dx − 2 y sx 6 1 1d dx 2
0
f
g
− 2 17 x 7 1 x
2 0
284 − 2(128 7 1 2) − 7
n
Example 9 Since f sxd − stan xdys1 1 x 2 1 x 4 d satisfies f s2xd − 2f sxd, it is odd and so
y
1–6 Evaluate the integral by making the given substitution. 1. y cos 2x dx, u − 2x 2. y xs2x 1 3d dx, u − 2x 1 3 2
3.
yx
2
4
2
sx 3 1 1 dx, u − x 3 1 1
4. y sin 2 cos d, u − sin 5. y
tan x dx − 0 1 1 x2 1 x4
1
21
x3 dx , u − x 4 2 5 sx 2 5d 2 4
z2
dt dz 24. y 2 cos t s1 1 tan t s1 1 z
23.
y
25.
y scot x csc x dx 26. y tan x dx
27.
y sec x tan x dx 28. y x s2 1 x dx
29.
y xs2x 1 5d
3
3
2
3
7. y x s1 2 x 2 dx 8. y x 2 sinsx 3d dx 9. y s1 2 2xd9 dx 10. y sin t s1 1 cost dt
y sin x cos x dx 34. y sin x cos x dx
y
1
y cos s1 1 5td dt 16. y sx dx
39.
y
y6
17.
y sec tan d 18. y sin x sinscos xd dx
41.
y
y4
19.
y sx
43.
y
13
45.
y
a
y sec 3t tan 3t dt 14. y y s4 2 y d
15.
2
3 2y3
dy
sin sx
3
1 1dsx 3 1 3xd 4 dx 20. y x sx 1 2 dx
s3ax 1 bx 3
cossyxd dx 22. y x 2 dx
2 1d3 dx 32. y tan2 sec 2 d 4
35–51 Evaluate the definite integral.
37.
13.
2
3
1
2
a 1 bx 2
dx 30. y x 3sx 2 1 1 dx
33.
y
y sins2y3d d 12. y sec 2 d
y
8
y xsx
35.
11.
21.
2
31. 7–30 Evaluate the indefinite integral.
2
sec 2 x
2
; 31–34 Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C − 0).
6. y s2t 1 1 dt, u − 2t 1 1
2
n
0
0
coss ty2d dt 36. y s3t 2 1d50 dt 1
0
3 1 1 7x dx 38. s y x cossx 2 d dx
s
0
2y3 sin t dt 40. yy3 csc 2 ( 12 t) dt cos2 t
0
0
0
sx 3 1 x 4 tan xd dx 42. y cos x sinssin xd dx y2
2y4
0
dx ss1 1 2xd 3
2
44.
y
a
0
x sa 2 2 x 2 dx
x sx 2 1 a 2 dx sa . 0d 46. y
y3
2y3
x 4 sin x dx
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
347
Section 4.5 The Substitution Rule 4 x x sx 2 1 dx 48. y0 s1 1 2x dx
47.
y
2
49.
y
1
50.
y
Ty2
51.
y (1 1 sx )
1
1y2
0 1
0
62. If f is continuous on R, prove that
y
cossx 22d dx x3
b
a
f sx 1 cd dx − y
b1c
a1c
f sxd dx
For the case where f sxd > 0, draw a diagram to interpret this equation geometrically as an equality of areas.
sins2 tyT 2 d dt
63. If a and b are positive numbers, show that
y
dx
x a s1 2 xd b dx − y x b s1 2 xd a dx
1
1
0
4
52. Verify that f sxd − sin sx is an odd function and use that fact to show that 3
3 0 < y sin s x dx < 1
0
64. If f is continuous on f0, g, use the substitution u − 2 x to show that
y
0
3
22
; 53–54 Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area.
2
y
y
y2
f scos xd dx − y
y2
0
66. Use Exercise 65 to evaluate y0
y2
54. y − 2 sin x 2 sin 2x, 0 < x < 2 55. Evaluate y22 sx 1 3ds4 2 x 2 dx by writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.
56. Evaluate y01 x s1 2 x 4 dx by making a substitution and interpreting the resulting integral in terms of an area.
0
f ssin xd dx
65. If f is continuous, prove that 0
53. y − s2x 1 1, 0 < x < 1
x f ssin xd dx −
f ssin xd dx
cos 2 x dx and y0
y2
sin 2 x dx.
The following exercises are intended only for those who have already covered Chapter 6. 67–84 Evaluate the integral. 67.
y
dx 68. y e25r dr 5 2 3x
69.
y
sln xd2 dx dx 70. y ax 1 b sa ± 0d x
71.
ye
73.
y
sarctan xd2 x dx 74. y x 2 1 4 dx x2 1 1
75.
y
11x sinsln xd dx 76. y x dx 1 1 x2
77.
59. If f is continuous and y f sxd dx − 10, find y f s2xd dx.
y
sin 2x sin x dx 78. y 1 1 cos2x dx 1 1 cos2x
79.
y cot x dx 80. y 1 1 x dx
60. If f is continuous and y f sxd dx − 4, find y x f sx 2 d dx.
81.
y
e4
83.
y
1
57. Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 Lys. This explains, in part, why the function f std − 12 sins2 ty5d has often been used to model the rate of air flow into the lungs. Use this model to find the volume of inhaled air in the lungs at time t. 58. A model for the basal metabolism rate, in kcalyh, of a young man is Rstd − 85 2 0.18 coss ty12d, where t is the time in hours measured from 5:00 am. What is this man’s total basal metabolism, y024 Rstd dt, over a 24hour time period? 4
2
0
0
9
0
61. If f is continuous on R, prove that
y
b
a
f s2xd dx − y
2a
2b
f sxd dx
For the case where f sxd > 0 and 0 , a , b, draw a dia gram to interpret this equation geometrically as an equality of areas.
s1 1 e x dx 72. y e cos t sin t dt
x
4
3
0
x
e
0
dx 1 2 82. y0 xe2x dx x sln x ez 1 1 2 2 dz 84. y0 sx 2 1de sx21d dx ez 1 z
85. Use Exercise 64 to evaluate the integral
y
0
x sin x dx 1 1 cos2x
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
348
Chapter 4 Integrals
4 Review CONCEPT CHECK
Answers to the Concept Check can be found on the back endpapers.
1. (a) Write an expression for a Riemann sum of a function f on an interval fa, bg. Explain the meaning of the notation that you use. (b) If f sxd > 0, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram. (c) If f sxd takes on both positive and negative values, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram.
5. (a) State the Net Change Theorem. (b) If rstd is the rate at which water flows into a reservoir, what does ytt12 rstd dt represent?
2. (a) Write the definition of the definite integral of a continuous function from a to b. (b) What is the geometric interpretation of yab f sxd dx if f sxd > 0? (c) What is the geometric interpretation of yab f sxd dx if f sxd takes on both positive and negative values? Illustrate with a diagram.
120 (b) What is the meaning of y60 vstd dt?
(c) What is the meaning of y
6. Suppose a particle moves back and forth along a straight line with velocity vstd, measured in feet per second, and accelera tion astd. 120 (a) What is the meaning of y60 vstd dt?

120 60

astd dt?
7. (a) Explain the meaning of the indefinite integral y f sxd dx. (b) What is the connection between the definite integral yab f sxd dx and the indefinite integral y f sxd dx?
3. State the Midpoint Rule.
8. Explain exactly what is meant by the statement that “differen tiation and integration are inverse processes.”
4. State both parts of the Fundamental Theorem of Calculus.
9. State the Substitution Rule. In practice, how do you use it?
truefalse quiz Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
7. If f and t are continuous and f sxd > tsxd for a < x < b, then
1. If f and t are continuous on fa, bg, then
8. If f and t are differentiable and f sxd > tsxd for a , x , b, then f 9sxd > t9sxd for a , x , b.
y
b
a
f f sxd 1 tsxdg dx − y f sxd dx 1 y tsxd dx b
b
a
y
a
f f sxd tsxdg dx −
Sy
b
a
DSy
f sxd dx
b
a
D
tsxd dx
b
a
5f sxd dx − 5 y f sxd dx b
y
a
b
a
10.
y
1
S
x 5 2 6x 9 1
21 5
25
sin x s1 1 x 4 d2
D
dx − 0
5
0
11. All continuous functions have derivatives.
13.
y
2
sin x 3 sin x 2 sin x dx − y dx 1 y3 dx x x x
14. If y01 f sxd dx − 0, then f sxd − 0 for 0 < x < 1. 15. If f is continuous on fa, bg, then
x f sxd dx − x y f sxd dx
sf sxd dx −
b
a
sax 2 1 bx 1 cd dx − 2 y sax 2 1 cd dx
b
d dx
a
5. If f is continuous on fa, bg and f sxd > 0, then
y
y
a
4. If f is continuous on fa, bg, then b
9.
f sxd dx > y tsxd dx
12. All continuous functions have antiderivatives.
3. If f is continuous on fa, bg, then
y
b
a
a
2. If f and t are continuous on fa, bg, then b
y
Îy
b
a
f sxd dx
6. If f 9 is continuous on f1, 3g, then y f 9svd dv − f s3d 2 f s1d. 3
1
Sy
b
a
D
f sxd dx − f sxd
2 16. y0 sx 2 x 3 d dx represents the area under the curve y − x 2 x 3 from 0 to 2.
17.
y
1
22
1 3 4 dx − 2 x 8
18. If f has a discontinuity at 0, then y
1
21
f sxd dx does not exist.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
CHAPTER 4 Review
349
exercises 1. Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be (a) left endpoints and (b) midpoints. In each case draw a diagram and explain what the Riemann sum represents.
7. The figure shows the graphs of f, f 9, and y0x f std dt. Identify each graph, and explain your choices. y
b
y
c y=ƒ
2 0
2
x 6
a
x
8. Evaluate: (a) y
y2
0
2. (a) Evaluate the Riemann sum for
with four subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents. (b) Use the definition of a definite integral (with right end points) to calculate the value of the integral
y
2
x x cos 2 3