CALCULUS for AP* JON ROGAWSKI University of California, Los Angeles. RAY CANNON Baylor University, TX ROGAWSKI S EARLY TRANSCENDENTALS SECOND EDITION

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1 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

2 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: ISBN-10: 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 Houndmills, Basingstoke RG21 6XS, England

3 To Julie and To the AP Teachers

4 CONTENTS ROGAWSKI S CALCULUS for AP* Early Transcendentals Chapter 1 PRECALCULUS REVIEW Real Numbers, Functions, and Graphs Linear and Quadratic Functions The Basic Classes of Functions Trigonometric Functions Inverse Functions Exponential and Logarithmic Functions Technology: Calculators and Computers 51 Chapter 2 LIMITS Limits, Rates of Change, and Tangent Lines Limits: A Numerical and Graphical Approach Basic Limit Laws Limits and Continuity Evaluating Limits Algebraically Trigonometric Limits Limits at Infinity Intermediate Value Theorem The Formal Definition of a Limit 110 AP2-1 Chapter 4 APPLICATIONS OF THE DERIVATIVE Linear Approximation and Applications Extreme Values The Mean Value Theorem and Monotonicity The Shape of a Graph L Hôpital s Rule Graph Sketching and Asymptotes Applied Optimization Newton s Method Antiderivatives 275 AP4-1 Chapter 5 THE INTEGRAL Approximating and Computing Area The Definite Integral The Fundamental Theorem of Calculus, Part I The Fundamental Theorem of Calculus, Part II Net Change as the Integral of a Rate Substitution Method Further Transcendental Functions Exponential Growth and Decay 341 AP5-1 Chapter 3 DIFFERENTIATION Definition of the Derivative The Derivative as a Function Product and Quotient Rules Rates of Change Higher Derivatives Trigonometric Functions The Chain Rule Derivatives of Inverse Functions Derivatives of General Exponential and Logarithmic Functions Implicit Differentiation Related Rates 195 AP3-1 vi Chapter 6 APPLICATIONS OF THE INTEGRAL Area Between Two Curves Setting Up Integrals: Volume, Density, Average Value Volumes of Revolution The Method of Cylindrical Shells Work and Energy 391 AP6-1 Chapter 7 TECHNIQUES OF INTEGRATION Integration by Parts Trigonometric Integrals Trigonometric Substitution Integrals Involving Hyperbolic and Inverse Hyperbolic Functions 420

5 CONTENTS CALCULUS vii 7.5 The Method of Partial Fractions Improper Integrals Probability and Integration Numerical Integration 454 AP7-1 Chapter 8 FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS Arc Length and Surface Area Fluid Pressure and Force Center of Mass Taylor Polynomials 488 AP8-1 Chapter 9 INTRODUCTION TO DIFFERENTIAL EQUATIONS Solving Differential Equations Models Involving y = k(y b) Graphical and Numerical Methods The Logistic Equation First-Order Linear Equations 528 AP9-1 Chapter 10 INFINITE SERIES Sequences Summing an Infinite Series Convergence of Series with Positive Terms Absolute and Conditional Convergence The Ratio and Root Tests Power Series Taylor Series 591 AP10-1 Chapter 11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND VECTOR FUNCTIONS Parametric Equations Arc Length and Speed Polar Coordinates Area, Arc Length, and Slope in Polar Coordinates Vectors in the Plane Dot Product and the Angle between Two Vectors Calculus of Vector-Valued Functions 660 AP11-1 Chapter 12 DIFFERENTIATION IN SEVERAL VARIABLES Functions of Two or More Variables Limits and Continuity in Several Variables Partial Derivatives Differentiability and Tangent Planes The Gradient and Directional Derivatives The Chain Rule Optimization in Several Variables 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