ROGAWSKI S CALCULUS for AP* EARLY TRANSCENDENTALS JON ROGAWSKI University of California, Los Angeles RAY CANNON Baylor University, TX W. H. FREEMAN AND COMPANY New York SECOND EDITION
Director, BFW High School: Craig Bleyer Executive Editor: Ann Heath Publisher: Ruth Baruth Senior Acquisitions Editor: Terri Ward Development Editor: Tony Palermino Development Editor: Julie Z. Lindstrom, Andrew Sylvester Associate Editor: Katrina Wilhelm Assistant Editor: Dora Figueiredo Editorial Assistant: Tyler Holzer Market Development: Steven Rigolosi Executive Marketing Manager: Cindi Weiss Media Editor: Laura Capuano Assistant Media 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 Library of Congress Control Number ISBN-13: 978-1-4292-5074-0 ISBN-10: 1-4292-5074-7 2012 by W. H. Freeman and Company All rights reserved Printed in the United States of America First printing W. H. Freeman and Company, 41 Madison Avenue, New York, NY 10010 Houndmills, Basingstoke RG21 6XS, England www.whfreeman.com
To Julie and To the AP Teachers
CONTENTS ROGAWSKI S CALCULUS for AP* Early Transcendentals Chapter 1 PRECALCULUS REVIEW 1 1.1 Real Numbers, Functions, and Graphs 1 1.2 Linear and Quadratic Functions 13 1.3 The Basic Classes of Functions 21 1.4 Trigonometric Functions 25 1.5 Inverse Functions 33 1.6 Exponential and Logarithmic Functions 43 1.7 Technology: Calculators and Computers 51 Chapter 2 LIMITS 59 2.1 Limits, Rates of Change, and Tangent Lines 59 2.2 Limits: A Numerical and Graphical Approach 67 2.3 Basic Limit Laws 77 2.4 Limits and Continuity 81 2.5 Evaluating Limits Algebraically 90 2.6 Trigonometric Limits 95 2.7 Limits at Infinity 100 2.8 Intermediate Value Theorem 106 2.9 The Formal Definition of a Limit 110 AP2-1 Chapter 4 APPLICATIONS OF THE DERIVATIVE 207 4.1 Linear Approximation and Applications 207 4.2 Extreme Values 215 4.3 The Mean Value Theorem and Monotonicity 226 4.4 The Shape of a Graph 234 4.5 L Hôpital s Rule 241 4.6 Graph Sketching and Asymptotes 248 4.7 Applied Optimization 257 4.8 Newton s Method 269 4.9 Antiderivatives 275 AP4-1 Chapter 5 THE INTEGRAL 286 5.1 Approximating and Computing Area 286 5.2 The Definite Integral 299 5.3 The Fundamental Theorem of Calculus, Part I 309 5.4 The Fundamental Theorem of Calculus, Part II 316 5.5 Net Change as the Integral of a Rate 322 5.6 Substitution Method 328 5.7 Further Transcendental Functions 336 5.8 Exponential Growth and Decay 341 AP5-1 Chapter 3 DIFFERENTIATION 120 3.1 Definition of the Derivative 120 3.2 The Derivative as a Function 129 3.3 Product and Quotient Rules 143 3.4 Rates of Change 150 3.5 Higher Derivatives 159 3.6 Trigonometric Functions 165 3.7 The Chain Rule 169 3.8 Derivatives of Inverse Functions 178 3.9 Derivatives of General Exponential and Logarithmic Functions 182 3.10 Implicit Differentiation 188 3.11 Related Rates 195 AP3-1 vi Chapter 6 APPLICATIONS OF THE INTEGRAL 357 6.1 Area Between Two Curves 357 6.2 Setting Up Integrals: Volume, Density, Average Value 365 6.3 Volumes of Revolution 375 6.4 The Method of Cylindrical Shells 384 6.5 Work and Energy 391 AP6-1 Chapter 7 TECHNIQUES OF INTEGRATION 400 7.1 Integration by Parts 400 7.2 Trigonometric Integrals 405 7.3 Trigonometric Substitution 413 7.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions 420
CONTENTS CALCULUS vii 7.5 The Method of Partial Fractions 426 7.6 Improper Integrals 436 7.7 Probability and Integration 448 7.8 Numerical Integration 454 AP7-1 Chapter 8 FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS 467 8.1 Arc Length and Surface Area 467 8.2 Fluid Pressure and Force 474 8.3 Center of Mass 480 8.4 Taylor Polynomials 488 AP8-1 Chapter 9 INTRODUCTION TO DIFFERENTIAL EQUATIONS 502 9.1 Solving Differential Equations 502 9.2 Models Involving y = k(y b) 511 9.3 Graphical and Numerical Methods 516 9.4 The Logistic Equation 524 9.5 First-Order Linear Equations 528 AP9-1 Chapter 10 INFINITE SERIES 537 10.1 Sequences 537 10.2 Summing an Infinite Series 548 10.3 Convergence of Series with Positive Terms 559 10.4 Absolute and Conditional Convergence 569 10.5 The Ratio and Root Tests 575 10.6 Power Series 579 10.7 Taylor Series 591 AP10-1 Chapter 11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS 607 11.1 Parametric Equations 607 11.2 Arc Length and Speed 620 11.3 Polar Coordinates 626 11.4 Area, Arc Length, and Slope in Polar Coordinates 634 11.5 Vectors in the Plane 641 11.6 Dot Product and the Angle between Two Vectors 653 11.7 Calculus of Vector-Valued Functions 660 AP11-1 Chapter 12 DIFFERENTIATION IN SEVERAL VARIABLES 672 12.1 Functions of Two or More Variables 672 12.2 Limits and Continuity in Several Variables 684 12.3 Partial Derivatives 692 12.4 Differentiability and Tangent Planes 703 12.5 The Gradient and Directional Derivatives 711 12.6 The Chain Rule 723 12.7 Optimization in Several Variables 731 12.8 Lagrange Multipliers: Optimizing with a Constraint 745 APPENDICES A1 A. The Language of Mathematics A1 B. Properties of Real Numbers A8 C. Induction and the Binomial Theorem A13 D. Additional Proofs A18 ANSWERS TO ODD-NUMBERED EXERCISES ANSWERS TO THE ODD-NUMBERED PREPARING FOR THE AP EXAM QUESTIONS REFERENCES PHOTO CREDITS INDEX A27 A104 A113 A116 I1