James Stewart, Daniel Clegg, Saleem Watson - Single Variable Calculus PDF

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Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Study Smarter. Ever wonder if you studied enough? WebAssign from Cengage can help. WebAssign is an online learning platform for your math, statistics, physical sciences and engineering courses. It helps you practice, focus your study time and absorb what you learn. When class comes—you’re way more confident. With WebAssign you will: Get instant feedback Know how well you and grading understand concepts Watch videos and tutorials Perform better on when you’re stuck in-class assignments Ask your instructor today how you can get access to WebAssign! cengage.com/webassign Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  REFERENCE page 1 ALGEBRA GEOMETRY Arithmetic Operations Geometric Formulas Cut here and keep for reference a c ad 1 bc asb 1 cd − ab 1 ac 1 − Formulas for area A, circumference C, and volume V: b d bd a Triangle Circle Sector of Circle a1c a c b a d ad A − 12 bh A − r 2 A − 21 r 2 − 1 − 3 − b b b c b c bc d − 12 ab sin  C − 2r s − r s in radiansd Exponents and Radicals a xm h r s x m x n − x m1n − x m2n xn ¨ r ¨ 1 b sx mdn − x m n x2n − n r x sxydn − x n y n SD x y n − xn yn Sphere Cylinder Cone x − (s x) m x 1yn − s n x x myn − s n m n V− 4 3 V − r h 2 V − 13 r 2h 3 r Î sx n x A − 4r 2 A −  rsr 2 1 h 2 s xy − s x s y n n n n − n y sy U Factoring Special Polynomials U K x 2 2 y 2 − sx 1 ydsx 2 yd K x 3 1 y 3 − sx 1 ydsx 2 2 xy 1 y 2d r x 3 2 y 3 − sx 2 ydsx 2 1 xy 1 y 2d Binomial Theorem Distance and Midpoint Formulas sx 1 yd2 − x 2 1 2xy 1 y 2  sx 2 yd2 − x 2 2 2xy 1 y 2 Distance between P1sx1, y1d and P2sx 2, y2d: sx 1 yd3 − x 3 1 3x 2 y 1 3xy 2 1 y 3 d − ssx 2 2 x1d2 1 s y2 2 y1d2 sx 2 yd3 − x 3 2 3x 2 y 1 3xy 2 2 y 3 sx 1 ydn − x n 1 nx n21y 1 nsn 2 1d n22 2 2 x y Midpoint of P1 P2: S x1 1 x 2 y1 1 y2 , D SD 2 2 n n2k k …      1 … 1 x y 1 1 nxy n21 1 y n k where SD n k − nsn 2 1d … sn 2 k 1 1d 1?2?3?…?k Lines Slope of line through P1sx1, y1d and P2sx 2, y2d: Quadratic Formula m− y2 2 y1 2b 6 sb 2 2 4ac x 2 2 x1 If ax 2 1 bx 1 c − 0, then x −. 2a Point-slope equation of line through P1sx1, y1d with slope m: Inequalities and Absolute Value y 2 y1 − msx 2 x1d If a , b and b , c, then a , c. If a , b, then a 1 c , b 1 c. Slope-intercept equation of line with slope m and y-intercept b: If a , b and c. 0, then ca , cb. y − mx 1 b If a , b and c , 0, then ca. cb. If a. 0, then Circles | |            x − a  means  x − a  or  x − 2a Equation of the circle with center sh, kd and radius r: | |            x , a  means    2a , x , a           | x |. a  means  x. a  or  x , 2a sx 2 hd2 1 s y 2 kd2 − r 2 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 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.  REFERENCE page 2 TRIGONOMETRY Angle Measurement Fundamental Identities  radians − 1808 1 1 csc  − sec  − s sin  cos   180° r 18 − rad  1 rad − sin  cos  180  ¨ tan  − cot  − r cos  sin  s − r s in radiansd 1 cot  − sin2  1 cos2  − 1 tan  Right Angle Trigonometry 1 1 tan2  − sec 2  1 1 cot 2  − csc 2  opp hyp sin  −   csc  − sins2d − 2sin  coss2d − cos  hyp opp S D hyp adj hyp opp  cos  −   sec  − tans2d − 2tan  sin 2  − cos  hyp adj ¨ 2 S D S D adj opp adj   tan  −   cot  − cos 2  − sin  tan 2  − cot  adj opp 2 2 Trigonometric Functions The Law of Sines B y r y sin  −   csc  − sin A sin B sin C r y − − a (x, y) a b c x r r cos  −   sec  − C r x c y x ¨ tan  −   cot  − x The Law of Cosines x y b a 2 − b 2 1 c 2 2 2bc cos A Graphs of Trigonometric Functions b 2 − a 2 1 c 2 2 2ac cos B y y y y=tan x c 2 − a 2 1 b 2 2 2ab cos C A y=sin x y=cos x 1 1 π 2π Addition and Subtraction Formulas 2π x π 2π x π x sinsx 1 yd − sin x cos y 1 cos x sin y _1 _1 sinsx 2 yd − sin x cos y 2 cos x sin y cossx 1 yd − cos x cos y 2 sin x sin y y y=csc x y y=sec x y y=cot x cossx 2 yd − cos x cos y 1 sin x sin y tan x 1 tan y 1 1 tansx 1 yd − 1 2 tan x tan y π 2π x π 2π x π 2π x tan x 2 tan y tansx 2 yd − _1 _1 1 1 tan x tan y Double-Angle Formulas sin 2x − 2 sin x cos x Trigonometric Functions of Important Angles cos 2x − cos 2x 2 sin 2x − 2 cos 2x 2 1 − 1 2 2 sin 2x  radians sin  cos  tan  2 tan x 08 0 0 1 0 tan 2x − 1 2 tan2x 308 y6 1y2 s3y2 s3y3 458 y4 s2y2 s2y2 1 Half-Angle Formulas 608 y3 s3y2 1y2 s3 1 2 cos 2x 1 1 cos 2x 908 y2 1 0 — sin 2x −     cos 2x − 2 2 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 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.  SINGLE VAR I A B LE CALCULUS EARLY TR AN S CE NDE NTA LS NINTH EDITION JAMES STEWART McMASTER UNIVERSITY AND UNIVERSITY OF TORONTO DANIEL CLEGG PALOMAR COLLEGE SALEEM WATSON CALIFORNIA STATE UNIVERSITY, LONG BEACH Australia Brazil Mexico Singapore United Kingdom United States Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 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.  This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. 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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  Single Variable Calculus: Early Transcendentals, © 2021, 2016 Cengage Learning, Inc. Ninth Edition Unless otherwise noted, all content is © Cengage. James Stewart, Daniel Clegg, Saleem Watson ALL RIGHTS RESERVED. No part of this work covered by the copyright herein Product Director: Mark Santee may be reproduced or distributed in any form or by any means, except as Senior Product Manager: Gary Whalen permitted by U.S. copyright law, without the prior written permission of the copyright owner. Product Assistant: Tim Rogers Executive Marketing Manager: Tom Ziolkowski Senior Learning Designer: Laura Gallus For product information and technology assistance, contact us at Cengage Customer & Sales Support, 1-800-354-9706 Digital Delivery Lead: Justin Karr or support.cengage.com. 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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.  To Lothar Redlin, our friend and colleague Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 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.  Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 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. Contents Preface x A Tribute to James Stewart xxii About the Authors xxiii Technology in the Ninth Edition xxiv To the Student xxv Diagnostic Tests xxvi A Preview of Calculus 1 1 Functions and Models 7 1.1 Four Ways to Represent a Function 8 1.2 Mathematical Models: A Catalog of Essential Functions 21 1.3 New Functions from Old Functions 36 1.4 Exponential Functions 45 1.5 Inverse Functions and Logarithms 54 Review 67 Principles of Problem Solving 70 2 Limits and Derivatives 77 2.1 The Tangent and Velocity Problems 78 2.2 The Limit of a Function 83 2.3 Calculating Limits Using the Limit Laws 94 2.4 The Precise Definition of a Limit 105 2.5 Continuity 115 2.6 Limits at Infinity; Horizontal Asymptotes 127 2.7 Derivatives and Rates of Change 140 wr i t in g pr oj ec t Early Methods for Finding Tangents 152 2.8 The Derivative as a Function 153 Review 166 Problems Plus 171 v Copyright 2021 Cengage Learning. 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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. vi CONTENTS 3 Differentiation Rules 173 3.1 Derivatives of Polynomials and Exponential Functions 174 applied pr oj ec t Building a Better Roller Coaster 184 3.2 The Product and Quotient Rules 185 3.3 Derivatives of Trigonometric Functions 191 3.4 The Chain Rule 199 applied pr oj ec t Where Should a Pilot Start Descent? 209 3.5 Implicit Differentiation 209 d is cov ery pr oj ec t Families of Implicit Curves 217 3.6 Derivatives of Logarithmic and Inverse Trigonometric Functions 217 3.7 Rates of Change in the Natural and Social Sciences 225 3.8 Exponential Growth and Decay 239 applied pr oj ec t Controlling Red Blood Cell Loss During Surgery 247 3.9 Related Rates 247 3.10 Linear Approximations and Differentials 254 d is cov ery pr oj ec t Polynomial Approximations 260 3.11 Hyperbolic Functions 261 Review 269 Problems Plus 274 4 Applications of Differentiation 279 4.1 Maximum and Minimum Values 280 applied pr oj ec t The Calculus of Rainbows 289 4.2 The Mean Value Theorem 290 4.3 What Derivatives Tell Us about the Shape of a Graph 296 4.4 Indeterminate Forms and l’Hospital’s Rule 309 wr itin g pr oj ec t The Origins of l’Hospital’s Rule 319 4.5 Summary of Curve Sketching 320 4.6 Graphing with Calculus and Technology 329 4.7 Optimization Problems 336 applied pr oj ec t The Shape of a Can 349 applied pr oj ec t Planes and Birds: Minimizing Energy 350 4.8 Newton’s Method 351 4.9 Antiderivatives 356 Review 364 Problems Plus 369 Copyright 2021 Cengage Learning. 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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  CONTENTS vii 5 Integrals 371 5.1 The Area and Distance Problems 372 5.2 The Definite Integral 384 d is cov ery pr oj ec t Area Functions 398 5.3 The Fundamental Theorem of Calculus 399 5.4 Indefinite Integrals and the Net Change Theorem 409 wr i t in g pr oj ec t Newton, Leibniz, and the Invention of Calculus 418 5.5 The Substitution Rule 419 Review 428 Problems Plus 432 6 Applications of Integration 435 6.1 Areas Between Curves 436 applied pr oj ec t The Gini Index 445 6.2 Volumes 446 6.3 Volumes by Cylindrical Shells 460 6.4 Work 467 6.5 Average Value of a Function 473 applied pr oj ec t Calculus and Baseball 476 applied pr oj ec t Where to Sit at the Movies 478 Review 478 Problems Plus 481 7 Techniques of Integration 485 7.1 Integration by Parts 486 7.2 Trigonometric Integrals 493 7.3 Trigonometric Substitution 500 7.4 Integration of Rational Functions by Partial Fractions 507 7.5 Strategy for Integration 517 7.6 Integration Using Tables and Technology 523 d is cov ery pr oj ec t Patterns in Integrals 528 7.7 Approximate Integration 529 7.8 Improper Integrals 542 Review 552 Problems Plus 556 Copyright 2021 Cengage Learning. 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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. viii CONTENTS 8 Further Applications of Integration 559 8.1 Arc Length 560 d is cov ery pr oj ec t Arc Length Contest 567 8.2 Area of a Surface of Revolution 567 d is cov ery pr oj ec t Rotating on a Slant 575 8.3 Applications to Physics and Engineering 576 d is cov ery pr oj ec t Complementary Coffee Cups 587 8.4 Applications to Economics and Biology 587 8.5 Probability 592 Review 600 Problems Plus 602 9 Differential Equations 605 9.1 Modeling with Differential Equations 606 9.2 Direction Fields and Euler’s Method 612 9.3 Separable Equations 621 applied pr oj ec t How Fast Does a Tank Drain? 630 9.4 Models for Population Growth 631 9.5 Linear Equations 641 applied pr oj ec t Which Is Faster, Going Up or Coming Down? 648 9.6 Predator-Prey Systems 649 Review 656 Problems Plus 659 10 Parametric Equations and Polar Coordinates 661 10.1 Curves Defined by Parametric Equations 662 d is cov ery pr oj ec t Running Circles Around Circles 672 10.2 Calculus with Parametric Curves 673 d is cov ery pr oj ec t Bézier Curves 684 10.3 Polar Coordinates 684 d is cov ery pr oj ec t Families of Polar Curves 694 10.4 Calculus in Polar Coordinates 694 10.5 Conic Sections 702 Copyright 2021 Cengage Learning. 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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  CONTENTS ix 10.6 Conic Sections in Polar Coordinates 711 Review 719 Problems Plus 722 11 Sequences, Series, and Power Series 723 11.1 Sequences 724 d is cov ery pr oj ec t Logistic Sequences 738 11.2 Series 738 11.3 The Integral Test and Estimates of Sums 751 11.4 The Comparison Tests 760 11.5 Alternating Series and Absolute Convergence 765 11.6 The Ratio and Root Tests 774 11.7 Strategy for Testing Series 779 11.8 Power Series 781 11.9 Representations of Functions as Power Series 787 11.10 Taylor and Maclaurin Series 795 d is cov ery pr oj ec t An Elusive Limit 810 wr i t in g pr oj ec t How Newton Discovered the Binomial Series 811 11.11 Applications of Taylor Polynomials 811 applied pr oj ec t Radiation from the Stars 820 Review 821 Problems Plus 825 Appendixes A1 A Numbers, Inequalities, and Absolute Values A2 B Coordinate Geometry and Lines A10 C Graphs of Second-Degree Equations A16 D Trigonometry A24 E Sigma Notation A36 F Proofs of Theorems A41 G The Logarithm Defined as an Integral A51 H Answers to Odd-Numbered Exercises A59 Index A121 Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 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 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. george polya The art of teaching, Mark Van Doren said, is the art of assisting discovery. In this Ninth Edition, as in all of the preceding editions, we continue the tradition of writing a book that, we hope, assists students in discovering calculus — both for its practical power and its surprising beauty. We aim to convey to the student a sense of the utility of calculus as well as to promote development of technical ability. At the same time, we strive to give some appreciation for the intrinsic beauty of the subject. Newton undoubtedly experi- enced a sense of triumph when he made his great discoveries. We want students to share some of that excitement. The emphasis is on understanding concepts. Nearly all calculus instructors agree that conceptual understanding should be the ultimate goal of calculus instruction; to imple- ment this goal we present fundamental topics graphically, numerically, algebraically, and verbally, with an emphasis on the relationships between these different representa- tions. Visualization, numerical and graphical experimentation, and verbal descriptions can greatly facilitate conceptual understanding. Moreover, conceptual understanding and technical skill can go hand in hand, each reinforcing the other. We are keenly aware that good teaching comes in different forms and that there are different approaches to teaching and learning calculus, so the exposition and exer- cises are designed to accommodate different teaching and learning styles. The features (including projects, extended exercises, principles of problem solving, and historical insights) provide a variety of enhancements to a central core of fundamental concepts and skills. Our aim is to provide instructors and their students with the tools they need to chart their own paths to discovering calculus. Alternate Versions The Stewart Calculus series includes several other calculus textbooks that might be preferable for some instructors. Most of them also come in single variable and multi- variable versions. Calculus, Ninth Edition, is similar to the present textbook except that the exponen- tial, logarithmic, and inverse trigonometric functions are covered after the chapter on integration. Essential Calculus, Second Edition, is a much briefer book (840 pages), though it contains almost all of the topics in Calculus, Ninth Edition. The relative brevity is achieved through briefer exposition of some topics and putting some features on the website. x Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 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 xi 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 under- standing even more strongly than this book. The coverage of topics is not encyclo- pedic and the material on transcendental functions and on parametric equations is woven throughout the book instead of being treated in separate chapters. Brief Applied Calculus is intended for students in business, the social sciences, and the life sciences. Biocalculus: Calculus for the Life Sciences is intended to show students in the life sciences how calculus relates to biology. Biocalculus: Calculus, Probability, and Statistics for the Life Sciences contains all the content of Biocalculus: Calculus for the Life Sciences as well as three addi- tional chapters covering probability and statistics. What’s New in the Ninth Edition? The overall structure of the text remains largely the same, but we have made many improvements that are intended to make the Ninth Edition even more usable as a teach- ing tool for instructors and as a learning tool for students. The changes are a result of conversations with our colleagues and students, suggestions from users and reviewers, insights gained from our own experiences teaching from the book, and from the copious notes that James Stewart entrusted to us about changes that he wanted us to consider for the new edition. In all the changes, both small and large, we have retained the features and tone that have contributed to the success of this book. More than 20% of the exercises are new: Basic exercises have been added, where appropriate, near the beginning of exer- cise sets. These exercises are intended to build student confidence and reinforce understanding of the fundamental concepts of a section. (See, for instance, Exer- cises 7.3.1 –  4, 9.1.1 – 5, 11.4.3 – 6.) Some new exercises include graphs intended to encourage students to understand how a graph facilitates the solution of a problem; these exercises complement subsequent exercises in which students need to supply their own graph. (See Exercises 6.2.1 –  4 and 10.4.43 –  46 as well as 53 – 54.) Some exercises have been structured in two stages, where part (a) asks for the setup and part (b) is the evaluation. This allows students to check their answer to part (a) before completing the problem. (See Exercises 6.1.1 –  4 and 6.3.3 –  4.) Some challenging and extended exercises have been added toward the end of selected exercise sets (such as Exercises 6.2.87, 9.3.56, 11.2.79 – 81, and 11.9.47). Titles have been added to selected exercises when the exercise extends a concept discussed in the section. (See, for example, Exercises 2.6.66 and 10.1.55 – 57.) Some of our favorite new exercises are 1.3.71, 3.4.99, 3.5.65, 4.5.55 – 58, 6.2.79, 6.5.18, and 10.5.69. In addition, Problem 14 in the Problems Plus following Chapter 6 is interesting and challenging. Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 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. xii PREFACE New examples have been added, and additional steps have been added to the solu- tions of some existing examples. (See, for instance, Example 2.7.5, Example 6.3.5, and Example 10.1.5.) Several sections have been restructured and new subheads added to focus the organization around key concepts. (Good illustrations of this are Sections 2.3, 11.1, and 11.2.) Many new graphs and illustrations have been added, and existing ones updated, to provide additional graphical insights into key concepts. A few new topics have been added and others expanded (within a section or in extended exercises) that were requested by reviewers. (Examples include symmet- ric difference quotients in Exercise 2.7.60 and improper integrals of more than one type in Exercises 7.8.65 – 68.) Derivatives of logarithmic functions and inverse trigonometric functions are now covered in one section (3.6) that emphasizes the concept of the derivative of an inverse function. Alternating series and absolute convergence are now covered in one section (11.5). Features Each feature is designed to complement different teaching and learning practices. Throughout the text there are historical insights, extended exercises, projects, problem- solving principles, and many opportunities to experiment with concepts by using tech- nology. We are mindful that there is rarely enough time in a semester to utilize all of these features, but their availability in the book gives the instructor the option to assign some and perhaps simply draw attention to others in order to emphasize the rich ideas of calculus and its crucial importance in the real world. n Conceptual Exercises The most important way to foster conceptual understanding is through the problems that the instructor assigns. To that end we have included 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 2.2, 2.5, and 11.2) and most exercise sets contain exercises designed to reinforce basic understanding (such as Exer- cises 2.5.3 – 10, 5.5.1 – 8, 6.1.1 – 4, 7.3.1 – 4, 9.1.1 – 5, and 11.4.3 – 6). Other exercises test conceptual understanding through graphs or tables (see Exercises 2.7.17, 2.8.36 – 38, 2.8.47 – 52, 9.1.23 – 25, and 10.1.30 – 33). Many exercises provide a graph to aid in visualization (see for instance Exer- cises 6.2.1 –  4 and 10.4.43 –  46). Another type of exercise uses verbal descriptions to gauge conceptual understanding (see Exercises 2.5.12, 2.8.66, 4.3.79 – 80, and 7.8.79). In addition, all the review sections begin with a Concept Check and a True-False Quiz. We particularly value problems that combine and compare graphical, numerical, and algebraic approaches (see Exercises 2.6.45 –  46, 3.7.29, and 9.4.4). n Graded Exercise Sets Each exercise set is carefully graded, progressing from basic conceptual exercises, to skill-development and graphical exercises, and then to more challenging exercises that Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 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   xiii often extend the concepts of the section, draw on concepts from previous sections, or involve applications or proofs. n Real-World Data Real-world data provide a tangible way to introduce, motivate, or illustrate the concepts of calculus. As a result, many of the examples and exercises deal with functions defined by such numerical data or graphs. These real-world data have been obtained by contact- ing companies and government agencies as well as researching on the Internet and in libraries. See, for instance, Figure 1 in Section 1.1 (seismograms from the Northridge earthquake), Exercise 2.8.36 (number of cosmetic surgeries), Exercise 5.1.12 (velocity of the space shuttle Endeavour), and Exercise 5.4.83 (power consumption in the New England states). n 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 accomplish- ment when completed. There are three kinds of projects in the text. Applied Projects involve applications that are designed to appeal to the imagina- tion of students. The project after Section 9.5 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). Discovery Projects anticipate results to be discussed later or encourage discovery through pattern recognition (see the project following Section 7.6, which explores pat- terns in integrals). Some projects make substantial use of 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 present-day methods with those of the founders of calculus — Fermat’s method for finding tangents, for instance, following Section 2.7. Suggested references are supplied. More projects can be found in the Instructor’s Guide. There are also extended exer- cises that can serve as smaller projects. (See Exercise 4.7.53 on the geometry of beehive cells, Exercise 6.2.87 on scaling solids of revolution, or Exercise 9.3.56 on the forma- tion of sea ice.) n Problem Solving Students usually have difficulties with problems that have no single well-defined procedure for obtaining the answer. As a student of George Polya, James Stewart experienced first-hand Polya’s delightful and penetrating insights into the process of problem solving. Accordingly, a modified version of Polya’s four-stage problem- solving strategy is presented following Chapter 1 in Principles of Problem Solving. These principles are applied, both explicitly and implicitly, throughout the book. Each of the other chapters is followed by a section called Problems Plus, which features examples of how to tackle challenging calculus problems. In selecting the Problems Plus prob- lems we have 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.” We have used these problems to great effect in our own calculus classes; it is gratifying to see how students respond to a challenge. James Stewart said, “When I put these challenging problems on assignments and tests I grade them in a different way... I reward a student significantly for ideas toward a solution and for recognizing which problem-solving principles are relevant.” Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 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. xiv PREFACE n Technology When using technology, it is particularly important to clearly understand the con- cepts that underlie the images on the screen or the results of a calculation. When properly used, graphing calculators and computers are powerful tools for discovering and understanding those concepts. This textbook can be used either with or without technology — we use two special symbols to indicate clearly when a particular type of assistance from technology is required. The icon ; indicates an exercise that definitely requires the use of graphing software or a graphing calculator to aid in sketching a graph. (That is not to say that the technology can’t be used on the other exercises as well.) The symbol means that the assistance of software or a graphing calculator is needed beyond just graphing to complete the exercise. Freely available websites such as WolframAlpha.com or Symbolab.com are often suitable. In cases where the full resources of a computer algebra system, such as Maple or Mathematica, are needed, we state this in the exercise. Of course, technology doesn’t make pencil and paper obsolete. Hand calculation and sketches are often preferable to technology for illustrating and reinforcing some concepts. Both instructors and students need to develop the ability to decide where using technology is appropriate and where more insight is gained by working out an exercise by hand. n WebAssign: www.webassign.net This Ninth Edition is available with WebAssign, a fully customizable online solution for STEM disciplines from Cengage. WebAssign includes homework, an interactive mobile eBook, videos, tutorials and Explore It interactive learning modules. Instructors can decide what type of help students can access, and when, while working on assign- ments. The patented grading engine provides unparalleled answer evaluation, giving students instant feedback, and insightful analytics highlight exactly where students are struggling. For more information, visit cengage.com/WebAssign. n Stewart Website Visit StewartCalculus.com for these additional materials: Homework Hints Solutions to the Concept Checks (from the review section of each chapter) Algebra and Analytic Geometry Review Lies My Calculator and Computer Told Me History of Mathematics, with links to recommended historical websites  dditional Topics (complete with exercise sets): Fourier Series, Rotation of Axes, A Formulas for the Remainder Theorem in Taylor Series Additional chapter on second-order differential equations, including the method of series solutions, and an appendix section reviewing complex numbers and complex exponential functions Instructor Area that includes archived problems (drill exercises that appeared in previous editions, together with their solutions) Challenge Problems (some from the Problems Plus sections from prior editions) Links, for particular topics, to outside Web resources Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-203 Copyright 2021 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 Content Diagnostic Tests The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, Func- tions, 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. 1 Functions and Models From the beginning, multiple representations of functions are stressed: verbal, numeri- cal, visual, and algebraic. A discussion of mathematical models leads to a review of the standard functions, including exponential and logarithmic functions, from these four points of view. 2 Limits and Derivatives The material on limits is motivated by a prior discussion of the tangent and veloc- ity problems. Limits are treated from descriptive, graphical, numerical, and algebraic points of view. Section 2.4, on the precise definition of a limit, is an optional section. Sections 2.7 and 2.8 deal with derivatives (including derivatives for functions defined graphically and numerically) before the differentiation rules are covered in Chapter 3. Here the examples and exercises explore the meaning of derivatives in various contexts. Higher derivatives are introduced in Section 2.8. 3 Differentiation Rules All the basic functions, including exponential, logarithmic, and inverse trigonometric functions, are differentiated here. The latter two classes of functions are now covered in one section that focuses on the derivative of an inverse function. When derivatives are computed in applied situations, students are asked to explain their meanings. Exponen- tial growth and decay are included in this chapter. 4 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 machines and the analysis of families of curves. Some substantial optimi- zation problems are provided, including an explanation of why you