# Calculus 2nd Edition

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CALCULUSSECOND EDITION

Publisher: Ruth Baruth

Senior Acquisitions Editor: Terri Ward

Development Editor: Tony Palermino

Development Editor: Julie Z. Lindstrom

Associate Editor: Katrina Wilhelm

Editorial Assistant: Tyler Holzer

Market Development: Steven Rigolosi

Executive Marketing Manager: Jennifer Somerville

Media Editor: Laura Capuano

Assistant Editor: Catriona Kaplan

Senior Media Acquisitions Editor: Roland Cheyney

Photo Editor: Ted Szczepanski

Photo Researcher: Julie Tesser

Cover and Text Designer: Blake Logan

Illustrations: Network Graphics and Techsetters, Inc.

Illustration Coordinator: Bill Page

Production Coordinator: Paul W. Rohloff

Composition: Techsetters, Inc.

Printing and Binding: RR Donnelley and Sons

ISBN-13: 978-1-4292-6374-0ISBN-10: 1-4292-6374-1

2012 by W. H. Freeman and CompanyAll rights reserved

Printed in the United States of AmericaFirst printing

W. H. Freeman and Company, 41 Madison Avenue, New York, NY 10010Houndmills, Basingstoke RG21 6XS, Englandwww.whfreeman.com

W. H. FREEMAN AND COMPANYNew York

CALCULUSSECOND EDITION

JON ROGAWSKIUniversity of California, Los Angeles

To Julie

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CONTENTS CALCULUS

Chapter 1 PRECALCULUS REVIEW 1

1.1 Real Numbers, Functions, and Graphs 11.2 Linear and Quadratic Functions 131.3 The Basic Classes of Functions 211.4 Trigonometric Functions 251.5 Technology: Calculators and Computers 33

Chapter 2 LIMITS 40

2.1 Limits, Rates of Change, and Tangent Lines 402.2 Limits: A Numerical and Graphical Approach 482.3 Basic Limit Laws 582.4 Limits and Continuity 622.5 Evaluating Limits Algebraically 712.6 Trigonometric Limits 762.7 Limits at Innity 812.8 Intermediate Value Theorem 872.9 The Formal Denition of a Limit 91

Chapter 3 DIFFERENTIATION 101

3.1 Definition of the Derivative 1013.2 The Derivative as a Function 1103.3 Product and Quotient Rules 1223.4 Rates of Change 1283.5 Higher Derivatives 1383.6 Trigonometric Functions 1443.7 The Chain Rule 1483.8 Implicit Differentiation 1573.9 Related Rates 163

Chapter 4 APPLICATIONS OF THE DERIVATIVE 175

4.1 Linear Approximation and Applications 1754.2 Extreme Values 1834.3 The Mean Value Theorem and Monotonicity 1944.4 The Shape of a Graph 2014.5 Graph Sketching and Asymptotes 2084.6 Applied Optimization 2164.7 Newtons Method 2284.8 Antiderivatives 234

Chapter 5 THE INTEGRAL 244

5.1 Approximating and Computing Area 2445.2 The Denite Integral 257

5.3 The Fundamental Theorem of Calculus, Part I 2675.4 The Fundamental Theorem of Calculus, Part II 2735.5 Net Change as the Integral of a Rate 2795.6 Substitution Method 285

Chapter 6 APPLICATIONS OF THE INTEGRAL 296

6.1 Area Between Two Curves 2966.2 Setting Up Integrals: Volume, Density, Average Value 3046.3 Volumes of Revolution 3146.4 The Method of Cylindrical Shells 3236.5 Work and Energy 330

Chapter 7 EXPONENTIAL FUNCTIONS 339

7.1 Derivative of f (x) = bx and the Number e 3397.2 Inverse Functions 3477.3 Logarithms and Their Derivatives 3557.4 Exponential Growth and Decay 3647.5 Compound Interest and Present Value 3717.6 Models Involving y = k( y b) 3777.7 LHpitals Rule 3827.8 Inverse Trigonometric Functions 3907.9 Hyperbolic Functions 399

Chapter 8 TECHNIQUES OF INTEGRATION 413

8.1 Integration by Parts 4138.2 Trigonometric Integrals 4188.3 Trigonometric Substitution 4268.4 Integrals Involving Hyperbolic and Inverse Hyperbolic

Functions 4338.5 The Method of Partial Fractions 4388.6 Improper Integrals 4478.7 Probability and Integration 4598.8 Numerical Integration 465

Chapter 9 FURTHER APPLICATIONS OF THEINTEGRAL AND TAYLORPOLYNOMIALS 478

9.1 Arc Length and Surface Area 4789.2 Fluid Pressure and Force 4859.3 Center of Mass 4919.4 Taylor Polynomials 499

vi

CONTENTS C A L C U L U S vii

Chapter 10 INTRODUCTION TO DIFFERENTIALEQUATIONS 513

10.1 Solving Differential Equations 51310.2 Graphical and Numerical Methods 52210.3 The Logistic Equation 52910.4 First-Order Linear Equations 534

Chapter 11 INFINITE SERIES 543

11.1 Sequences 54311.2 Summing an Innite Series 55411.3 Convergence of Series with Positive Terms 56511.4 Absolute and Conditional Convergence 57511.5 The Ratio and Root Tests 58111.6 Power Series 58511.7 Taylor Series 597

Chapter 12 PARAMETRIC EQUATIONS, POLARCOORDINATES, AND CONICSECTIONS 613

12.1 Parametric Equations 61312.2 Arc Length and Speed 62612.3 Polar Coordinates 63212.4 Area and Arc Length in Polar Coordinates 64012.5 Conic Sections 647

Chapter 13 VECTOR GEOMETRY 663

13.1 Vectors in the Plane 66313.2 Vectors in Three Dimensions 67413.3 Dot Product and the Angle between Two Vectors 68413.4 The Cross Product 69413.5 Planes in Three-Space 70513.6 A Survey of Quadric Surfaces 71113.7 Cylindrical and Spherical Coordinates 719

Chapter 14 CALCULUS OF VECTOR-VALUEDFUNCTIONS 729

14.1 Vector-Valued Functions 72914.2 Calculus of Vector-Valued Functions 73714.3 Arc Length and Speed 74714.4 Curvature 75214.5 Motion in Three-Space 76214.6 Planetary Motion According to Kepler and Newton 771

Chapter 15 DIFFERENTIATION IN SEVERALVARIABLES 780

15.1 Functions of Two or More Variables 78015.2 Limits and Continuity in Several Variables 79215.3 Partial Derivatives 80015.4 Differentiability and Tangent Planes 81115.5 The Gradient and Directional Derivatives 81915.6 The Chain Rule 83115.7 Optimization in Several Variables 83915.8 Lagrange Multipliers: Optimizing with a Constraint 853

Chapter 16 MULTIPLE INTEGRATION 866

16.1 Integration in Two Variables 86616.2 Double Integrals over More General Regions 87816.3 Triple Integrals 89116.4 Integration in Polar, Cylindrical, and Spherical

Coordinates 90216.5 Applications of Multiple Integrals 91316.6 Change of Variables 926

Chapter 17 LINE AND SURFACE INTEGRALS 945

17.1 Vector Fields 94517.2 Line Integrals 95217.3 Conservative Vector Fields 96917.4 Parametrized Surfaces and Surface Integrals 98017.5 Surface Integrals of Vector Fields 995

Chapter 18 FUNDAMENTAL THEOREMS OFVECTOR ANALYSIS 1009

18.1 Greens Theorem 100918.2 Stokes Theorem 102118.3 Divergence Theorem 1034

APPENDICES A1A. The Language of Mathematics A1B. Properties of Real Numbers A8C. Induction and the Binomial Theorem A13D. Additional Proofs A18

ANSWERS TO ODD-NUMBERED EXERCISES A27

REFERENCES A119

PHOTO CREDITS A123

INDEX I1

ABOUT JON ROGAWSKI

As a successful teacher for more than 30 years, Jon Rogawski has listened to and learnedmuch from his own students. These valuable lessons have made an impact on his thinking,his writing, and his shaping of a calculus text.

Jon Rogawski received his undergraduate and masters degrees in mathematics si-multaneously from Yale University, and he earned his PhD in mathematics from PrincetonUniversity, where he studied under Robert Langlands. Before joining the Department ofMathematics at UCLA in 1986, where he is currently a full professor, he held teachingand visiting positions at the Institute for Advanced Study, the University of Bonn, and theUniversity of Paris at Jussieu and at Orsay.

Jons areas of interest are number theory, automorphic forms, and harmonic analysison semisimple groups. He has published numerous research articles in leading mathemat-ics journals, including the research monograph Automorphic Representations of UnitaryGroups in Three Variables (Princeton University Press). He is the recipient of a SloanFellowship and an editor of the Pacic Journal of Mathematics and the Transactions ofthe AMS.

Jon and his wife, Julie, a physician in family practice, have four children. They runa busy household and, whenever possible, enjoy family vacations in the mountains ofCalifornia. Jon is a passionate classical music lover and plays the violin and classicalguitar.

PREFACE

ABOUT CALCULUS by Jon Rogawski

On Teaching MathematicsAs a young instructor, I enjoyed teaching but I didnt appreciate how difcult it is tocommunicate mathematics effectively. Early in my teaching career, I was confronted witha student rebellion when my efforts to explain epsilon-delta proofs were not greeted withthe enthusiasm I anticipated. Experiences of this type taught me two basic principles:

1. We should try to teach students as much as possible, but not more.2. As math teachers, how we say it is as important as what we say.

The formal language of mathematics is intimidating to the uninitiated. By presentingconcepts in everyday language, which is more familiar but not less precise, we open theway for students to understand the underlying ideas and integrate them into their way ofthinking. Students are then in a better position to appreciate the need for formal denitionsand proofs and to grasp their logic.

On Writing a Calculus TextI began writing Calculus with the goal of creating a text in which exposition, graphics,and layout would work together to enhance all facets of a students calculus experience:mastery of basic skills, conceptual understanding, and an appreciation of the wide rangeof applications. I also wanted students to be aware, early in the course, of the beauty ofthe subject and the important role it will play, both in their further studies and in theirunderstanding of the wider world. I paid special attention to the following aspects of thetext:

(a) Clear, accessible exposition that anticipates and addresses student difculties.(b) Layout and gures that communicate the ow of ideas.(c) Highlighted features in the text that emphasize concepts and mathematical reason