James Stewart Calculus PDF - Early Transcendentals 8th Edition

 CALCULUS Early TranscEndEnTals EighTh EdiTion...

 CALCULUS Early TranscEndEnTals EighTh EdiTion JamEs sTEwarT M C Master University and University of toronto Australia Brazil Mexico Singapore United Kingdom United States 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.  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. 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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Calculus: Early Transcendentals, Eighth Edition © 2016, 2012 Cengage Learning James Stewart WCN: 02-200-203 Product Manager: Neha Taleja ALL RIGHTS RESERVED. No part of this work covered by the copyright Senior Content Developer: Stacy Green herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to Associate Content Developer: Samantha Lugtu photocopying, recording, scanning, digitizing, taping, Web distribution, Product Assistant: Stephanie Kreuz information networks, or information storage and retrieval systems, except Media Developer: Lynh Pham as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. 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Windows is a registered trademark of the Microsoft Corporation and used herein under license. Macintosh is a registered trademark of Apple Computer, Inc. k12T14 Used herein under license. Maple is a registered trademark of Waterloo Maple, Inc. Mathematica is a registered trademark of Wolfram Research, Inc. Tools for Enriching Calculus is a trademark used herein under license. Printed in the United States of America Print Number: 01 Print Year: 2014 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.  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 Four Ways to Represent a Function 10 1.2 Mathematical Models: A Catalog of Essential Functions 23 1.3 New Functions from Old Functions 36 1.4 Exponential Functions 45 1.5 Inverse Functions and Logarithms 55 Review 68 Principles of Problem Solving 71 2 2.1 The Tangent and Velocity Problems 78 2.2 The Limit of a Function 83 2.3 Calculating Limits Using the Limit Laws 95 2.4 The Precise Definition of a Limit 104 2.5 Continuity 114 2.6 Limits at Infinity; Horizontal Asymptotes 126 2.7 Derivatives and Rates of Change 140 Writing Project Early Methods for Finding Tangents 152 2.8 The Derivative as a Function 152 Review 165 Problems Plus 169 iii 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. iv Contents 3 3.1 Derivatives of Polynomials and Exponential Functions 172 Applied Project Building a Better Roller Coaster 182 3.2 The Product and Quotient Rules 183 3.3 Derivatives of Trigonometric Functions 190 3.4 The Chain Rule 197 Applied Project Where Should a Pilot Start Descent? 208 3.5 Implicit Differentiation 208 Laboratory Project Families of Implicit Curves 217 3.6 Derivatives of Logarithmic Functions 218 3.7 Rates of Change in the Natural and Social Sciences 224 3.8 Exponential Growth and Decay 237 Applied Project Controlling Red Blood Cell Loss During Surgery 244 3.9 Related Rates 245 3.10 Linear Approximations and Differentials 251 Laboratory Project Taylor Polynomials 258 3.11 Hyperbolic Functions 259 Review 266 Problems Plus 270 4 4.1 Maximum and Minimum Values 276 Applied Project The Calculus of Rainbows 285 4.2 The Mean Value Theorem 287 4.3 How Derivatives Affect the Shape of a Graph 293 4.4 Indeterminate Forms and l’Hospital’s Rule 304 Writing Project The Origins of l’Hospital’s Rule 314 4.5 Summary of Curve Sketching 315 4.6 Graphing with Calculus and Calculators 323 4.7 Optimization Problems 330 Applied Project The Shape of a Can 343 Applied Project Planes and Birds: Minimizing Energy 344 4.8 Newton’s Method 345 4.9 Antiderivatives 350 Review 358 Problems Plus 363 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.  Contents v 5 5.1 Areas and Distances 366 5.2 The Definite Integral 378 Discovery Project Area Functions 391 5.3 The Fundamental Theorem of Calculus 392 5.4 Indefinite Integrals and the Net Change Theorem 402 Writing Project Newton, Leibniz, and the Invention of Calculus 411 5.5 The Substitution Rule 412 Review 421 Problems Plus 425 6 6.1 Areas Between Curves 428 Applied Project The Gini Index 436 6.2 Volumes 438 6.3 Volumes by Cylindrical Shells 449 6.4 Work 455 6.5 Average Value of a Function 461 Applied Project Calculus and Baseball 464 Applied Project Where to Sit at the Movies 465 Review 466 Problems Plus 468 7 7.1 Integration by Parts 472 7.2 Trigonometric Integrals 479 7.3 Trigonometric Substitution 486 7.4 Integration of Rational Functions by Partial Fractions 493 7.5 Strategy for Integration 503 7.6 Integration Using Tables and Computer Algebra Systems 508 Discovery Project Patterns in Integrals 513 7.7 Approximate Integration 514 7.8 Improper Integrals 527 Review 537 Problems Plus 540 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. vi Contents 8 8.1 Arc Length 544 Discovery Project Arc Length Contest 550 8.2 Area of a Surface of Revolution 551 Discovery Project Rotating on a Slant 557 8.3 Applications to Physics and Engineering 558 Discovery Project Complementary Coffee Cups 568 8.4 Applications to Economics and Biology 569 8.5 Probability 573 Review 581 Problems Plus 583 9 9.1 Modeling with Differential Equations 586 9.2 Direction Fields and Euler’s Method 591 9.3 Separable Equations 599 Applied Project How Fast Does a Tank Drain? 608 Applied Project Which Is Faster, Going Up or Coming Down? 609 9.4 Models for Population Growth 610 9.5 Linear Equations 620 9.6 Predator-Prey Systems 627 Review 634 Problems Plus 637 10 10.1 Curves Defined by Parametric Equations 640 Laboratory Project Running Circles Around Circles 648 10.2 Calculus with Parametric Curves 649 Laboratory Project Bézier Curves 657 10.3 Polar Coordinates 658 Laboratory Project Families of Polar Curves 668 10.4 Areas and Lengths in Polar Coordinates 669 Copyright 2016 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 10.5 Conic Sections 674 10.6 Conic Sections in Polar Coordinates 682 Review 689 Problems Plus 692 11 11.1 Sequences 694 Laboratory Project Logistic Sequences 707 11.2 Series 707 11.3 The Integral Test and Estimates of Sums 719 11.4 The Comparison Tests 727 11.5 Alternating Series 732 11.6 Absolute Convergence and the Ratio and Root Tests 737 11.7 Strategy for Testing Series 744 11.8 Power Series 746 11.9 Representations of Functions as Power Series 752 11.10 Taylor and Maclaurin Series 759 Laboratory Project An Elusive Limit 773 Writing Project How Newton Discovered the Binomial Series 773 11.11 Applications of Taylor Polynomials 774 Applied Project Radiation from the Stars 783 Review 784 Problems Plus 787 12 12.1 Three-Dimensional Coordinate Systems 792 12.2 Vectors 798 12.3 The Dot Product 807 12.4 The Cross Product 814 Discovery Project The Geometry of a Tetrahedron 823 12.5 Equations of Lines and Planes 823 Laboratory Project Putting 3D in Perspective 833 12.6 Cylinders and Quadric Surfaces 834 Review 841 Problems Plus 844 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. viii Contents 13 13.1 Vector Functions and Space Curves 848 13.2 Derivatives and Integrals of Vector Functions 855 13.3 Arc Length and Curvature 861 13.4 Motion in Space: Velocity and Acceleration 870 Applied Project Kepler’s Laws 880 Review 881 Problems Plus 884 14 14.1 Functions of Several Variables 888 14.2 Limits and Continuity 903 14.3 Partial Derivatives 911 14.4 Tangent Planes and Linear Approximations 927 Applied Project The Speedo LZR Racer 936 14.5 The Chain Rule 937 14.6 Directional Derivatives and the Gradient Vector 946 14.7 Maximum and Minimum Values 959 Applied Project Designing a Dumpster 970 Discovery Project Quadratic Approximations and Critical Points 970 14.8 Lagrange Multipliers 971 Applied Project Rocket Science 979 Applied Project Hydro-Turbine Optimization 980 Review 981 Problems Plus 985 15 15.1 Double Integrals over Rectangles 988 15.2 Double Integrals over General Regions 1001 15.3 Double Integrals in Polar Coordinates 1010 15.4 Applications of Double Integrals 1016 15.5 Surface Area 1026 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.  Contents ix 15.6 Triple Integrals 1029 Discovery Project Volumes of Hyperspheres 1040 15.7 Triple Integrals in Cylindrical Coordinates 1040 Discovery Project The Intersection of Three Cylinders 1044 15.8 Triple Integrals in Spherical Coordinates 1045 Applied Project Roller Derby 1052 15.9 Change of Variables in Multiple Integrals 1052 Review 1061 Problems Plus 1065 16 16.1 Vector Fields 1068 16.2 Line Integrals 1075 16.3 The Fundamental Theorem for Line Integrals 1087 16.4 Green’s Theorem 1096 16.5 Curl and Divergence 1103 16.6 Parametric Surfaces and Their Areas 1111 16.7 Surface Integrals 1122 16.8 Stokes’ Theorem 1134 Writing Project Three Men and Two Theorems 1140 16.9 The Divergence Theorem 1141 16.10 Summary 1147 Review 1148 Problems Plus 1151 17 17.1 Second-Order Linear Equations 1154 17.2 Nonhomogeneous Linear Equations 1160 17.3 Applications of Second-Order Differential Equations 1168 17.4 Series Solutions 1176 Review 1181 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 Contents 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 A34 F Proofs of Theorems A39 G The Logarithm Defined as an Integral A50 H Complex Numbers A57 I Answers to Odd-Numbered Exercises A65 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 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 cur- rent calculus reform movement came from the Tulane Conference in 1986, which for- mulated as their first recommendation: Focus on conceptual understanding. I have tried to implement this goal through the Rule of Three: “Topics should be pre- sented geometrically, numerically, and algebraically.” Visualization, numerical and graphical experimentation, and other approaches have changed how we teach concep- tual 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 con- ceptual 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 instruc- tors. Most of them also come in single variable and multivariable versions. Calculus, Eighth Edition, is similar to the present textbook except that the exponen- tial, logarithmic, and inverse trigonometric functions are covered in the second 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. Essential Calculus: Early Transcendentals, Second Edition, resembles Essential Calculus, but the exponential, logarithmic, and inverse trigonometric functions are covered in Chapter 3. xi 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. xii Preface 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. Calculus: Early Vectors introduces vectors and vector functions in the first semester and integrates them throughout the book. It is suitable for students taking engineer- ing and physics courses concurrently with calculus. 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. The changes have resulted from talking with my colleagues and students at the Univer- sity of Toronto and from reading journals, as well as suggestions from users and review- ers. 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 6.1.5, 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 Controlling Red Blood Cell Loss During Surgery (page 244) 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 bleed- ing and the extracted blood is returned to the patient after surgery. The project Planes and Birds: Minimizing Energy (page 344) asks how birds can minimize power and energy by flapping their wings versus gliding. In the project The Speedo LZR Racer (page 936) 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 sec- tions so that iterated integrals are treated earlier. More than 20% of the exercises in each chapter are new. Here are some of my favorites: 2.7.61, 2.8.36–38, 3.1.79–80, 3.11.54, 4.1.69, 4.3.34, 4.3.66, 4.4.80, 4.7.39, 4.7.67, 5.1.19–20, 5.2.67–68, 5.4.70, 6.1.51, 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 12–14 on page 272, Problem 13 on page 363, Problems 16–17 on page 426, and Problem 8 on page 986.) 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 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 irst few exercises in Sections 2.2, 2.5, 11.2, 14.2, and 14.3.) Similarly, all the review sections begin with a Concept Check and a True-False Quiz. Other exercises test conceptual understanding through graphs or tables (see Exercises 2.7.17, 2.8.35–38, 2.8.47–52, 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 2.5.10, 2.8.66, 4.3.69–70, and 7.8.67). I particularly value problems that combine and compare graphical, numerical, and algebraic approaches (see Exercises 2.6.45–46, 3.7.27, and 9.4.4). graded Exercise sets Each exercise set is carefully graded, progressing from basic conceptual exercises and skill-development problems to more challenging problems involving applications and proofs. real-world 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 real-world data to introduce, motivate, and illustrate the concepts of calculus. As a result, many of the examples and exercises deal with functions deined by such numerical data or graphs. See, for instance, Figure 1 in Section 1.1 (seismograms from the Northridge earthquake), Exercise 2.8.35 (unemployment rates), Exercise 5.1.16 (velocity of the space shuttle Endeavour), and Figure 4 in Section 5.4 (San Francisco power consumption). Functions of two variables are illustrated by a table of values of the wind-chill 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 ields are introduced in Section 16.1 by depictions of actual velocity vector ields 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 accomplish- ment 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. xiv 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. Writ- ing Projects ask students to compare present-day methods with those of the founders of calculus—Fermat’s method for inding tangents, for instance. Suggested references are supplied. Discovery Projects anticipate results to be discussed later or encourage dis- covery 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 5.1: Position from Samples). Problem solving Students usually have dificulties with problems for which there is no single well-deined procedure for obtaining the answer. I think nobody has improved very much on George Polya’s four-stage problem-solving strategy and, accordingly, I have included a version of his problem-solving 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 dificult 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 signii- cantly for ideas toward a solution and for recognizing which problem-solving principles are relevant. Technology The availability of technology makes it not less important but more important to clearly understand the concepts that underlie the images on the screen. But, when properly used, graphing calculators and computers are powerful tools for discovering and understand- ing those concepts. This textbook can be used either with or without technology and I use two special symbols to indicate clearly when a particular type of machine is required. The icon ; indicates an exercise that deinitely requires the use of such technology, but that is not to say that it can’t be used 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 TI-89) are required. But 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 the hand or the machine is appropriate. 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 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 anima- tions of igures in text; Modules are more elaborate activities and include exercises. Instructors can choose to become involved at several different levels, ranging from sim- ply encouraging students to use the Visuals and Modules for independent exploration, to assigning speciic exercises from those included with each Module, or to creating additional exercises, labs, and projects that make use of the Visuals and Modules. 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 TEC also includes Homework Hints for representative exercises (usually odd-num- bered) 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, particu- larly 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 multi-part formats. The system also includes Active Examples, in which students are guided in step-by- step 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 Algebra Review Lies My Calculator and Computer Told Me History of Mathematics, with links to the better historical websites 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) Challenge Problems (some from the Problems Plus sections from prior editions) Links, for particular topics, to outside Web resources Selected Visuals and Modules from Tools for Enriching Calculus (TEC) 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 velocity 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 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. xvi Preface 2.7 and 2.8 deal with derivatives (especially with functions defined graphically and numerically) before the differentiation rules are covered in Chapter 3. Here the exam- ples and exercises explore the meanings of derivatives in various contexts. Higher deriva- tives are introduced in Section 2.8. 3 diferentiation rules All the basic functions, including exponential, logarithmic, and inverse trigonometric functions, are differentiated here. When derivatives are computed in applied situations, students are asked to explain their meanings. Exponential growth and decay are now covered in this chapter. 4 applications of diferentiation 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 optimi- zation problems are provided, including an explanation of why you need to raise your head 42° to see the top of a rainbow. 5 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. 6 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. 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 Here are the applications of integration—arc length and surface area—for which it is of integration useful to have available all the techniques of integration, as well as applications to biol- ogy, 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 diferential 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 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 first-order differential equations. An optional final section uses predator- prey models to illustrate systems of differential equations. 10 Parametric Equations This chapter introduces parametric and polar curves and applies the methods of calculus and Polar coordinates 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. 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 11 ininite sequences and series The convergence tests have intuitive justifications (see page 719) 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 The material on three-dimensional analytic geometry and vectors is divided into two geometry of space chapters. Chapter 12 deals with vectors, the dot and cross products, lines, planes, and surfaces. 13 vector functions This chapter covers vector-valued functions, their derivatives and integrals, the length and curvature of space curves, and velocity and acceleration along space curves, culmi- nating in Kepler’s laws. 14 Partial derivatives Functions of two or more variables are studied from verbal, numerical, visual, and alge- braic 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 second-order Since first-order differential equations are covered in Chapter 9, this final chapter deals diferential Equations with second-order linear differential equations, their application to vibrating springs and electric circuits, and series solutions. Calculus, Early Transcendentals, 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. 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 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. xviii Preface 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 Brian karasek, South Mountain Community College Eric Aurand, Eastield College Jason kozinski, University of Florida Joy Becker, University of Wisconsin–Stout Carole krueger, The University of Texas at Arlington Przemyslaw Bogacki, Old Dominion University ken kubota, University of Kentucky Amy Elizabeth Bowman, University of Alabama John Mitchell, Clark College in Huntsville Donald Paul, Tulsa Community College Monica Brown, University of Missouri–St. Louis Chad Pierson, University of Minnesota, Duluth Roxanne Byrne, University of Colorado at Denver and Lanita Presson, University of Alabama in Huntsville Health Sciences Center karin Reinhold, State University of New York at Albany Teri Christiansen, University of Missouri–Columbia Thomas Riedel, University of Louisville Bobby Dale Daniel, Lamar University Christopher Schroeder, Morehead State University Jennifer Daniel, Lamar University Angela Sharp, University of Minnesota, Duluth Andras Domokos, California State University, Sacramento Patricia Shaw, Mississippi State University Timothy Flaherty, Carnegie Mellon University Carl Spitznagel, John Carroll University Lee Gibson, University of Louisville Mohammad Tabanjeh, Virginia State University Jane Golden, Hillsborough Community College Capt. koichi Takagi, United States Naval Academy Semion Gutman, University of Oklahoma Lorna TenEyck, Chemeketa Community College Diane Hoffoss, University of San Diego Roger Werbylo, Pima Community College Lorraine Hughes, Mississippi State University David Williams, Clayton State University Jay Jahangiri, Kent State University Zhuan Ye, Northern Illinois University John Jernigan, Community College of Philadelphia Previous Edition reviewers B. D. Aggarwala, University of Calgary David Buchthal, University of Akron John Alberghini, Manchester Community College Jenna Carpenter, Louisiana Tech University Michael Albert, Carnegie-Mellon University Jorge Cassio, Miami-Dade Community College Daniel Anderson, University of Iowa Jack Ceder, University of California, Santa Barbara Amy Austin, Texas A&M University Scott Chapman, Trinity University Donna J. Bailey, Northeast Missouri State University Zhen-Qing Chen, University of Washington—Seattle Wayne Barber, Chemeketa Community College James Choike, Oklahoma State University Marilyn Belkin, Villanova University Barbara Cortzen, DePaul University Neil Berger, University of Illinois, Chicago Carl Cowen, Purdue University David Berman, University of New Orleans Philip S. Crooke, Vanderbilt University Anthony J. Bevelacqua, University of North Dakota Charles N. Curtis, Missouri Southern State College Richard Biggs, University of Western Ontario Daniel Cyphert, Armstrong State College Robert Blumenthal, Oglethorpe University Robert Dahlin Martina Bode, Northwestern University M. Hilary Davies, University of Alaska Anchorage Barbara Bohannon, Hofstra University Gregory J. Davis, University of Wisconsin–Green Bay Jay Bourland, Colorado State University Elias Deeba, University of Houston–Downtown Philip L. Bowers, Florida State University Daniel DiMaria, Suffolk Community College Amy Elizabeth Bowman, University of Alabama in Huntsville Seymour Ditor, University of Western Ontario Stephen W. Brady, Wichita State University Greg Dresden, Washington and Lee University Michael Breen, Tennessee Technological University Daniel Drucker, Wayne State University Robert N. Bryan, University of Western Ontario kenn Dunn, Dalhousie University 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 xix Dennis Dunninger, Michigan State University Marianne korten, Kansas State University Bruce Edwards, University of Florida Virgil kowalik, Texas A&I University David Ellis, San Francisco State University kevin kreider, University of Akron John Ellison, Grove City College Leonard krop, DePaul University Martin Erickson, Truman State University Mark krusemeyer, Carleton College Garret Etgen, University of Houston John C. Lawlor, University of Vermont Theodore G. Faticoni, Fordham University Christopher C. Leary, State University of New York at Geneseo Laurene V. Fausett, Georgia Southern University David Leeming, University of Victoria Norman Feldman, Sonoma State University Sam Lesseig, Northeast Missouri State University Le Baron O. Ferguson, University of California—Riverside Phil Locke, University of Maine Newman Fisher, San Francisco State University Joyce Longman, Villanova University José D. Flores, The University of South Dakota Joan McCarter, Arizona State University William Francis, Michigan Technological University Phil McCartney, Northern Kentucky University James T. Franklin, Valencia Community College, East Igor Malyshev, San Jose State University Stanley Friedlander, Bronx Community College Larry Mansield, Queens College Patrick Gallagher, Columbia University–New York Mary Martin, Colgate University Paul Garrett, University of Minnesota–Minneapolis Nathaniel F. G. Martin, University of Virginia Frederick Gass, Miami University of Ohio Gerald Y. Matsumoto, American River College Bruce Gilligan, University of Regina James Mckinney, California State Polytechnic University, Pomona Matthias k. Gobbert, University of Maryland, Baltimore County Tom Metzger, University of Pittsburgh Gerald Goff, Oklahoma State University Richard Millspaugh, University of North Dakota Stuart Goldenberg, California Polytechnic State University Lon H. Mitchell, Virginia Commonwealth University John A. Graham, Buckingham Browne & Nichols School Michael Montaño, Riverside Community College Richard Grassl, University of New Mexico Teri Jo Murphy, University of Oklahoma Michael Gregory, University of North Dakota Martin Nakashima, California State Polytechnic University, Charles Groetsch, University of Cincinnati Pomona Paul Triantailos Hadavas, Armstrong Atlantic State University Ho kuen Ng, San Jose State University Salim M. Haïdar, Grand Valley State University Richard Nowakowski, Dalhousie University D. W. Hall, Michigan State University Hussain S. Nur, California State University, Fresno Robert L. Hall, University of Wisconsin–Milwaukee Norma Ortiz-Robinson, Virginia Commonwealth University Howard B. Hamilton, California State University, Sacramento Wayne N. Palmer, Utica College Darel Hardy, Colorado State University Vincent Panico, University of the Paciic Shari Harris, John Wood Community College F. J. Papp, University of Michigan–Dearborn Gary W. Harrison, College of Charleston Mike Penna, Indiana University–Purdue University Indianapolis Melvin Hausner, New York University/Courant Institute Mark Pinsky, Northwestern University Curtis Herink, Mercer University Lothar Redlin, The Pennsylvania State University Russell Herman, University of North Carolina at Wilmington Joel W. Robbin, University of Wisconsin–Madison Allen Hesse, Rochester Community College Lila Roberts, Georgia College and State University Randall R. Holmes, Auburn University E. Arthur Robinson, Jr., The George Washington University James F. Hurley, University of Connecticut Richard Rockwell, Paciic Union College Amer Iqbal, University of Washington—Seattle Rob Root, Lafayette College Matthew A. Isom, Arizona State University