PETERSONS

MASTERAP CALCULUSAB & BC

2nd Edition

W. Michael KelleyMark Wilding,Contributing Author

An ARCO Book

ARCO is a registered trademark of Petersons, and is used herein under license by Petersons.

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2007 Petersons, a Nelnet company

Previous editions 1991, 1993, 1996, 1998, 2001, 2002, 2003, 2004, 2005

Editor: Wallie Walker Hammond; Production Editor: Bernadette Webster; Composition Manager:Linda M. Williams; Manufacturing Manager: Ray Golaszewski

ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced orused in any form or by any meansgraphic, electronic, or mechanical, including photocopying, record-ing, taping, Web distribution, or information storage and retrieval systemswithout the prior writtenpermission of the publisher.

For permission to use material from this text or product, complete the Permission Request Form athttp://www.petersons.com/permissions.

ISBN 13: 978-0-7689-2470-1ISBN 10: 0-7689-2470-7

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1 09 08 07

Second Edition

Petersons.com/publishingCheck out our Web site at www.petersons.com/publishing to see if there is any new information regarding the test andany revisions or corrections to the content of this book. Weve made sure the information in this book is accurate andup-to-date; however, the test format or content may have changed since the time of publication.

OTHER RECOMMENDED TITLESMaster AP English Language & Composition

Master AP English Literature & Composition

Master AP Chemistry

Master AP U.S. Government & Politics

Master AP U.S. History

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Before You Begin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvHow This Book Is Organized . . . . . . . . . . . . . . . . . . . . . . . . . . . xvSpecial Study Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAppendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviYoure Well on Your Way to Success . . . . . . . . . . . . . . . . . . . . . xviGive Us Your Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviMike Kelleys How to Use This Book . . . . . . . . . . . . . . . . . . . xviiQuick Reference Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxTop 10 Strategies to Raise Your Score . . . . . . . . . . . . . . . . . xxiv

PART I: AP CALCULUS AB & BC BASICS

1 All About the AP Calculus AB & BC Tests . . . . . . . . . . . . . . . 3Frequently Asked Questions About the AP Calculus Tests . . 3Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

PART II: AP CALCULUS AB & BC REVIEW

2 Calculus Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Functions and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Function Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Hands-On Activity 2.1: Transforming Functions . . . . . . . . . . 31Selected Solutions to Hands-on Activity 2.1 . . . . . . . . . . . . . . 34Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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v

Parametric Equations (BC Topic Only) . . . . . . . . . . . . . . . . . . 48Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Polar Equations (BC Topic Only) . . . . . . . . . . . . . . . . . . . . . . . . 53Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Vectors and Vector Equations (BC Topic Only) . . . . . . . . . . . 59Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Technology: Solving Equations with a GraphingCalculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Hands-On Activity 3.1: What Is a Limit? . . . . . . . . . . . . . . . . 71Selected Solutions to Hands-On Activity 3.1 . . . . . . . . . . . . . 74Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Evaluating Limits Analytically . . . . . . . . . . . . . . . . . . . . . . . . . 78Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Hands-On Activity 3.2: The Extreme Value Theorem . . . . . 88Selected Solutions to Hands-On Activity 3.2 . . . . . . . . . . . . . 90Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Hands-On Activity 3.3: The Intermediate Value Theorem . 93Solutions to Hands-On Activity 3.3 . . . . . . . . . . . . . . . . . . . . . . 95Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Limits Involving Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Special Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Technology: Evaluating Limits with a GraphingCalculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Contentsvi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Differentiating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Derivative as a Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . 119Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124The Power Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Derivatives to Memorize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136The Product Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139The Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142A Word About Respecting Variables . . . . . . . . . . . . . . . . . . . . 143Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Hands-On Activity 4.1: Approximating Derivatives . . . . . . 152Selected Solutions to Hands-On Activity 4.1 . . . . . . . . . . . . 153Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Technology: Finding Numerical Derivatives with theGraphing Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Exercise 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5 Advanced Topics in Differentiation . . . . . . . . . . . . . . . . . 165The Derivative of an Inverse Function . . . . . . . . . . . . . . . . . 165Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168Hands-On Activity 5.1: Linear Approximations . . . . . . . . . . 170Solutions to Hands-On Activity 5.1 . . . . . . . . . . . . . . . . . . . . . 171Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172LHpitals Rule (BC Topic Only) . . . . . . . . . . . . . . . . . . . . . . . 173Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Parametric Derivatives (BC Topic Only) . . . . . . . . . . . . . . . . 177

Contents vii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180Polar Derivatives (BC Topic Only) . . . . . . . . . . . . . . . . . . . . . 182Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Technology: Finding Polar and Parametric Derivativeswith Your Calculator (BC Topic Only) . . . . . . . . . . . . . . . . . . 187Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6 Applications of the Derivative . . . . . . . . . . . . . . . . . . . . . . 195Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Hands-On Activity 6.1: Rolles and Mean ValueTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203Selected Solutions to Hands-On Activity 6.1 . . . . . . . . . . . . 205Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Hands-On Activity 6.2: The First Derivative Test . . . . . . . . 210Selected Solutions to Hands-On Activity 6.2 . . . . . . . . . . . . 211Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Motion in the Plane (BC Topic Only) . . . . . . . . . . . . . . . . . . . 226Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234Technology: Modeling a Particles Movement with aGraphing Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Basic Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Contentsviii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Hands-On Activity 7.1: Approximating Area withRiemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258Selected Solutions to Hands-On Activity 7.1 . . . . . . . . . . . . 260Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264The Trapezoidal Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . 270Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277Hands-On Activity 7.2: Accumulation Functions . . . . . . . . . 281Solutions to Hands-On Activity 7.2 . . . . . . . . . . . . . . . . . . . . . 282Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285The Mean Value Theorem for Integration, AverageValue of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290U-Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299Integrating Inverse Trigonometric Functions . . . . . . . . . . . . 301Exercise 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304Technology: Evaluating Definite Integrals with YourGraphing Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305Exercise 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

8 Advanced Methods of Integration . . . . . . . . . . . . . . . . . . 315Miscellaneous Methods of Integration . . . . . . . . . . . . . . . . . . 315Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319Parts (BC Topic Only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325Powers of Trigonometric Functions (BC Topic Only) . . . . . 326Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330Partial Fractions (BC Topic Only) . . . . . . . . . . . . . . . . . . . . . . 332Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335Improper Integrals (BC Topic Only) . . . . . . . . . . . . . . . . . . . . 337Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Contents ix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Technology: Drawing Derivative and Integral Graphswith Your Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

9 Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . 353Hands-On Activity 9.1: Area Between Curves . . . . . . . . . . . 353Selected Solutions to Hands-On Activity 9.1 . . . . . . . . . . . . 355Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358The Disk and Washer Methods . . . . . . . . . . . . . . . . . . . . . . . . 362Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369The Shell Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375Finding the Volume of Regions with Known CrossSections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382Arc Length (BC Topic Only) . . . . . . . . . . . . . . . . . . . . . . . . . . . 385Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387Polar Area (BC Topic Only) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392Technology: Using Your Calculator Efficiently . . . . . . . . . . . 396Exercise 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

10 Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407Hands-On Activity 10.1: Separation of Variables . . . . . . . . 407Solutions to Hands-On Activity 10.1 . . . . . . . . . . . . . . . . . . . . 409Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411Hands-On Activity 10.2: Slope Fields . . . . . . . . . . . . . . . . . . . 413Selected Solutions to Hands-On Activity 10.2 . . . . . . . . . . . 415Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418Eulers Method (BC Topic Only) . . . . . . . . . . . . . . . . . . . . . . . . 420Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . 428Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

Contentsx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Logistic Growth (BC Topic Only) . . . . . . . . . . . . . . . . . . . . . . . 433Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435Technology: A Differential Equations CalculatorProgram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

11 Sequences and Series (BC Topics Only) . . . . . . . . . . . . . 449Introduction to Sequences and Series, Nth TermDivergence Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453Convergence Tests for Infinite Series . . . . . . . . . . . . . . . . . . . 455Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478Taylor and Maclaurin Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 480Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486Technology: Viewing and Calculating Sequences andSeries with a Graphing Calculator . . . . . . . . . . . . . . . . . . . . . 489Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492Answers and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493Summing It Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

PART III: FOUR PRACTICE TESTS

Practice Test 1: AP Calculus AB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505Section I, Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505Section I, Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512Section II, Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516Section II, Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518Answer Key and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . 519

Practice Test 2: AP Calculus AB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537Section I, Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537Section I, Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543Section II, Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547Section II, Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549Answer Key and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . 550

Practice Test 3: AP Calculus BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569Section I, Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

Contents xi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Section I, Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575Section II, Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579Section II, Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580Answer Key and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . 581

Practice Test 4: AP Calculus BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595Section I, Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595Section I, Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601Section II, Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605Section II, Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606Answer Key and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . 607

APPENDIX

College-by-College Guide to AP Credit and Placement . . . 621

Contentsxii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Acknowledgments

So many people have inspired and supported me that I balk to think of howlucky and blessed my life has been. The wisdom, love, and support of thesepeople have shaped and made me, and without them, I am nothing. Specialthanks to:

My beautiful bride Lisa, who teaches me every day that being married toyour best friend is just the greatest thing in the world.

Mom, who taught me to rise above the circumstance.

Dad, who taught me always to tip the barber.

My brother Dave (The Dawg), who taught me that life is pretty funnybut never as funny as a joke that references the human butt.

My best friends Rob Halstead, Chris Sarampote, and Matt Halnon, whotaught me what true friendship means (and why housecleaning is for theweak).

My principals, George Miller and Tommy Tucker, who have given meevery opportunity and then seven more.

My students, who make my job and my life happier than they could everimagine. My English teachers Ron Gibson, Jack Keosseian, Mary Dou-glas, and Daniel Brown, who showed me how much fun writing can be.

Sue Strickland, who inspired in me a love of teaching and who taught mymethods class over dinner at her house.

Mark Wilding, for not saying, Are you crazy?

The Finley boys: James, for his great questions, written specifically tochallenge, deceive, nauseate, and ensmarten students, and Tim, for hiscomputer prowess and ability to produce crazy, three-dimensional dia-grams in the blink of an eye.

The Big Mathematician in the Sky, who (mercifully) makes straight thepaths of us mathematicians here on Earth.

Finally, wrestling legend Koko B. Ware, because Chris thought that wouldbe pretty funny.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Before You Begin

HOW THIS BOOK IS ORGANIZEDWhether you have five months, nine weeks, or just four short weeks to preparefor the exam, Petersons Master AP Calculus AB & BC will help you develop astudy plan that caters to your individual needs and timetables. These step-by-step plans are easy to follow and are remarkably effective.

Top 10 Strategies to Raise Your Score offers you tried and truetest-taking strategies.

Part I includes the basic information about the AP Calculus test that youneed to know.

Part II provides reviews and strategies for answering the different kindsof multiple-choice and free-response questions you will encounter on theexam. You will have numerous opportunities to practice what you arelearning in the exercises that appear throughout the reviews. It is a goodidea to read the answer explanations to all of the questions, because youmay find ideas or tips that will help you better analyze the answers in thenext Practice Test.

Part III includes four additional practice tests, two for AB and two forBC. Remember to apply the test-taking system carefully, work the systemto get more correct responses, be careful of your time, and strive toanswer more questions in the time period.

The Appendix is designed to provide you with an easy reference to theAP Credit Policy Guidelines instituted for all colleges and universities.

SPECIAL STUDY FEATURESPetersons Master AP Calculus AB & BC was designed to be as user-friendly asit is complete. It includes several features to make your preparation easier.

OverviewEach chapter begins with a bulleted overview listing the topics that will becovered in the chapter. You know immediately where to look for a topic that youneed to work on.

Summing It UpEach strategy chapter ends with a point-by-point summary that captures themost important points. The summaries are a convenient way to review the

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

content of these strategy chapters. In addition, be sure to look in the page margins ofyour book for the following test-prep tools:

Bonus Information

NOTE

Notes provide interesting calculus facts, connections, and notations to improve yourunderstanding of the material. Dont overlook these little gemsthey are an importantpart of the text, highlighting important information and putting things into the propercontext.

TIP

Tips are the authors personal pointers to you, so that you can stay on track and in focusas you study the material. Typically, they contain pieces of information to help you withthe many tasks youll be required to perform on the AP calculus Test.

ALERT!

Steer clear of these common errors of judgment in mathematics.

APPENDIXPetersons College-by-College Guide to AP Credit and Placement gives you the equiva-lent classes, scores, and credit awarded at more than 400 colleges and universities. Usethis guide to find your placement status, credit, and/or exemption based on your APCalculus AB or BC score.

YOURE WELL ON YOUR WAY TO SUCCESSRemember that knowledge is power.Youwill be studying themost comprehensive guideavailable, and you will become extremely knowledgeable about the exam. We lookforward to helping you raise your score.

GIVE US YOUR FEEDBACKPetersons, a Nelnet company, publishes a full line of resources to help guide youthrough the college admission process. Petersons publications can be found at yourlocal bookstore, library, and high school guidance office, and you can access us online atwww.petersons.com.

We welcome any comments or suggestions you may have about this publication andinvite you to complete our online survey at www.petersons.com/booksurvey. Or youcan fill out the survey at the back of this book, tear it out, and mail it to us at:

Publishing DepartmentPetersons2000 Lenox DriveLawrenceville, NJ 08648

Before You Beginxvi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Your feedback will help us to provide personalized solutions for your educationaladvancement.

MIKE KELLEYS HOW TO USE THIS BOOK

The Chess DilemmaI like chess. Its a fascinating game that, although easy to grasp, is nonetheless verydifficult. Once you learn how to move the pieces around the board, you are still lightyears away from actually being any good (chess geniuses excepted). I have known howto move chess pieces since I was about nine, but I have still yet to win a single chessgame against a living person. I am just terrible at chess, even though I have alwayswanted to be good. In my holy grail-like quest to outsmart someone at this dastardlygame, I have sought wisdom from books. I can still remember my last trip to thebookstore, standing with my mouth agape before the shelf allotted to chess instructionbooks. I was hoping to find something to fortify (resuscitate) my chess game, only to findbooks with titles such as 200 Pawn andBishop Endings, 50 of the Greatest Chess GamesEver, and 35 Tactics to Capture the Queens Rook in Six Moves or Less. None of thesebooks were of any use to me at all! None of them even claimed to try to teach me what Ineeded to know.

I wanted a book to teach me how to play. I knew that a horsey moves in an L shape,but I didnt know where he (or she) should go. Often, friends gave me such advice ascontrol the middle of the board. I still am not sure what that means or how to goabout doing it. In my frustration, I bought childrens chess games for my computer,hoping to benefit from the tutorials that, surely, I had a better shot at understanding.Once I had completed the tutorials, I was confident, ready to attack the toy men andfinally take my first step toward Chess Grand Masteror whatever they call thosesmart guys. I was destroyed by the smiling toy chess piecesbeaten in under tenmoves, on the easiest level. I dont think I blinked for fifteen minutes. I have seen thatlook on my students faces before. Its a look that says, I have no idea what Im doingwrong, and I think Im more likely to sprout wings than to ever understand. Perhapsyou, too, have felt that expression creep across your face, shadowed by an oily, sickfeeling in the pit of your stomach. Perhaps you have a knack for math and are notinspired to fear and cold sweat by calculus but are just looking for practice or to tie upa few loose ends before the AP Calculus test. Either way, this book will help youmaster calculus and prepare you for the infamous test day.

Understanding Versus Mastering: Calculus ReformCalculus is a subject, like chess, that requires more than a simple understanding of itscomponent parts. Before you can truly master calculus, you need to understand howeach of its basic tenets work, what they mean, and how they interrelate. Such was notalways the case, however. In fact, you may have even heard the urban legend about theAPCalculus student who simply took the derivative of each equation and set it equal tozero and still got a three rating on the AP test! Such rumors are not true, but theyunderscore the fundamental change gripping the mathematics worlda change calledreform. In fact, for many years, calculus was treated as merely advanced arithmetic,

Before You Begin xvii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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rather than a complex and even beautiful logical system. Calculus is somuchmore thanformulas and memorizing, and when you see this, you begin to actually understand therationale behind the mathematics involved. Having these connections makes under-standing flourish and, before very long, banishes math phobias to the dark places,inhabited heretofore only by your car keys when you were in a hurry.

What Makes This Book DifferentThis book is unlike most math textbooks published today. It not only presents formulasand practice problems, but it also actually helps you understand through hands-onactivities and detailed explanations. It also puts difficult formulas into everydayEnglish, to shed some light on their meanings. In relation to my chess metaphor, mybook intends not only to tell you that bishops move diagonally, but also to offer youadvice on where to move them, how soon, and in what circumstances they are mosteffective. (Although I cannot actually give you the corresponding chess advice, since mydeep understanding of bishops ends with the knowledge that they wear pointy hats.)

Traditional math test-preparatory books state formulas, prove them, and presentexercises to practice using them. There is rarely any explanation of how to use theformulas, what they mean, and how to remember them. In short, traditional books donot offer to teach, when that is what you truly need. I have compiled in this text all ofthe strategies, insights, and advice that I have amassed as an instructor. Most of all,I have used common sense. For example, I do not think that proving a theorem alwayshelps students understand that theorem. In fact, I think that the proof can sometimescloud the matter at hand! Thus, not every theorem in the book is accompanied by aproof; instead, I have included it only when I think it would be beneficial. Also, thereare numerous activities in this book to complete that will teach you, as you progress,basic definitions, properties, and theorems. In fact, you may be able to devise themyourself! I not only want you to succeedI want you to understand.

In order to promote understanding, it is my belief that a book that intends to teachmust offer answers as well as questions. How useful are 100 practice problems if onlynumerical answers and maybe a token explanation are given? Some calculus reviewbooks even confuse calculus teachers with their lack of explanation (although I betnone of your teachers would ever admit it!). Every single question in this book has agood explanation and includes all of the important steps; this is just one of thecharacteristics that makes this book unique. No longer will you have to spend fifteenminutes deciding how the author got from one step to the next in his or her compu-tations. I know I am not the only one who, gnashing his teeth, has exclaimed, Wheredid the 4 go in the third step!? How can they just drop the 4? Did the 4 just step outfor a bite to eat and will be back later? Am I so stupid that its obvious to everyone butme?

You should also find that the problems difficulties increase as you progress. Whentopics are first introduced, they are usually of easy or medium difficulty. However, theproblems at the end of each section are harder and will help you bind your under-standing to previous topics in the book. Finally, the problems at the end of the chapterare the most challenging of all, requiring you to piece together all the important topicsand involve appropriate technology along the way.

Before You Beginxviii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Herein lies the entire premise and motivation of the book: You can prepare for the APCalculus test by understanding whats involved rather than simply practicing skillsout of context. This is not to say that you wont have to memorize any formulas or thatthe concepts themselves will be presented any less rigorously than in a textbook. Thisis not, as some might claim, soft mathematics. Instead, it is mathematics presentedin a way that ensures and promotes understanding.

The Game PiecesThe following are included to help you in your study of calculus:

Hands-on activities or guided practice to introduce and teach all of the majorelements of calculus, including calculus reform topics

Target Practice examples with detailed solutions to cement your understandingof the topics (each target practice problem is accompanied by an icon denoting itsdifficultysee belowso that you can constantly monitor your progress)

Common Errors denoted during practice problems and notes so that you canavoid them

Exercises at the end of each section to practice the skills you just learned

A technology section in each chapter with step-by-step directions for the TI-83series calculator to ensure that you are using it correctly and in accordance withCollege Board guidelines. (The TI-83, TI-83 Plus, and TI-83 Plus Silver Editioncalculators constitute the vast majority of calculators used on the AP test, sothats why we chose them.)

Additional problems at the end of each chapter to review your skills and chal-lenge you

James Diabolical Dilemma Problems at the end of each chapter are written by aformer AP Calculus student of mine, James Finley; as a former 5-er on the APtest, he has created these problems to push your understanding to the verylimitsconsider these problems concentrated, with all the pulp left in for flavor.

ConclusionThis book can be utilized for numerous reasons and towardmany ends. It is best used asa study guide to supplement your calculus textbook. As a teaching tool, it can help youto learn calculus or fill in gaps in your understanding. As a resource, it can providenumerous practice problems with full solutions. However you choose to use the book, itis my hope that it can help unlock some of themysteries of calculus for you, although itsalmost certainly going to do nothing for your chess game.

Before You Begin xix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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QUICK REFERENCE GUIDEThese pages contain just about all the formulas you need to know by heart before youcan take the AP Test. It does not contain all the theorems and techniques you need toknow. An asterisk (*) indicates a Calculus BC-only formula.

Limits

x n

c

x=lim 0

x

xx

=

01lim

sin

x

xx

=

0

10lim

cos

x

x

xe

+

=lim 1

1

DerivativesPower Rule: d

dxx nxn n( ) = 1

Product Rule: ddx

f x g x f x g x g x f x( ) ( )( ) = ( ) ( )+ ( ) ( )

Quotient Rule:ddx

f x

g x

g x f x f x g x

g x

( )( )

=

( ) ( ) ( ) ( )( )(( )2

Chain Rule: ddx

f g x f g x g x( )( )( ) = ( )( ) ( )ddx

x xsin cos( ) =

ddx

x xcos sin( ) = ddx

x xtan sec( ) = 2ddx

x xcot csc( ) = 2ddx

x x xsec sec tan( ) = ddx

x x xcsc csc cot( ) =

Before You Beginxx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ddx

xx

ln( ) = 1ddx

e ex x( ) =ddx

xa xa

logln

( ) = ( )1

ddx

a a ax x( ) = ( )lnddx

xx

arcsin( ) =

1

1 2

ddx

xx

arccos( ) =

1

1 2

ddx

xx

arctan( ) =+

1

1 2

ddx

xx

arccot( ) = +

1

1 2

ddx

xx x

arcsec( ) =

1

12

ddx

xx x

arccsc( ) =

1

12

f xf f x

( ) = ( )( )1 1

1

*Parametric derivatives: dydx

dydtdxdt

d ydx

ddt

dydx

dxdt

=

=

;

2

2

Before You Begin xxi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Derivative ApplicationsMean Value Theorem:

=

( ) ( ) ( )f c f b f ab as(t) is the position function; s(t) 5 v(t), the velocity function; v(t) 5 a(t), the accelera-tion function

Projectile position equation:

s t gt v t h( )= + +12 2 0 0 ; g 5 9.8 m/s2 or 32 ft/s2IntegrationPower Rule for Integrals: x dx x

nCn

n=

++

+ 11*cos x dx 5 sin x 1 C

*sin x dx 5 2cos x 1 C

*tan x dx 5 2ln Ucos xU 1 C

*cot x dx 5 ln Usin xU 1 C

*sec x dx 5 ln Usec x + tan xU 1 C

*csc x dx 5 2ln Ucsc x + cot xU 1 C

du

a u

ua

C2 2

= + arcsindu

a u aua

C2 2

1

+= + arctan

du

u u a a

u

aC

2 2

1

= +arcsecTrapezoidal Rule: b a

n

2(f(a) 1 2f(x1) 1 2f(x2) 1 ... 1 2f(xn 2 1) 1 f(b))

Fundamental Theorem (Part 1): f x dx F ba

b ( ) ( )= F(a), if F is the antiderivativeof f(x)

Fundamental Theorem (Part 2):

ddx a

xf t dt f x( )

( ) =

Average Value: f cb a

f x dxa

b( ) =

( )1*Integration by Parts: *udv 5 uv 2 *vdu

Before You Beginxxii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Integration ApplicationsDisk Method: r x

ab

dx( )( ) 2

Washer Method:a

bR x r x dx ( )( ) ( )( ) 2 2

Shell Method:2 d x h x dxa

b ( ) ( )*Arc Length: 1

22 2

+ ( )( ) + f x dxdt dydtabab ' ; *Polar Area: 1

22

r da

b ( )( )

Differential EquationsExponential growth/decay:

dydt

ky y Nekt= =;

*Logistic growth:dydt

ky L y yL

ce Lkt= ( ) =

+ ;

1

*Dy 5 Dx { m

*Sequences and SeriesSum of a geometric series: a

r1

Ratio Test:n

n

n

a

a+lim 1

Limit Comparison Test:n

n

n

a

blim

Taylor series for f(x) centered at x 5 c:f x f c f c x c

f c x c

f c

( ) ( ) ( )( ) ( )( )( )

= + +

+

2

2!

xx c f c x cn

n n

+ +

+( ) ( )( )( )33! !

sin! !

x xx x

= + +3 5

3 5

cos! !

xx x

= + +12 4

2 4

e xx xx

= + + + +12 3

2 3

! !

11

1 2 3 4

= + + + + +x

x x x x

Before You Begin xxiii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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TOP 10 STRATEGIES TO RAISE YOUR SCOREWhen it comes to taking anAP exam, some test-taking skills will do youmore good thanothers. There are concepts you can learn and techniques you can follow that will helpyou do your best. Heres our pick for the top 10 strategies to raise your score:1. Pace yourself. Using less time on the easier questions will give you more time for

the harder ones. Questions usually go from easiest to most difficult. Work asquickly as you can through the beginning of the test. Dont get lulled into a falsesense of security because you appear to be maintaining a good pace in the first part.

2. Educated guessing will boost your score. Although random guessing wonthelp you, anything better than random guessing will. On most questions, youshould be able to make better-than-random guesses by using common sense andthe process of elimination techniques that are developed throughout this book. Ifyou can eliminate one choice out of five, you have a 25 percent chance of guessingcorrectly. If you can knock out two choices, the odds go up to 33 percent.

3. The easy answer isnt always the best answer. Make sure you read all of thechoices before selecting your choice. Quite frequently, test makers will put anattractive, but incorrect, answer as an (A) or (B) choice. Reading all of the choicesdecreases your chance of being misled, particularly in questions where no calcu-lations are involved.

4. Use common sense. It is always important to make sure your answers makesense. On multiple-choice questions, it might be readily apparent that youvemade an error (e.g., none of the choices match your answer). However, on the freeresponse, there is no immediate feedback about the accuracy of your answer. It isimportant to inspect your work to make sure it makes sense.

5. Put down your calculator. You only get to use your calculator for certain partsof the test. On the portions of the exam where calculators are prohibited, youshould expect to deal with numbers that are fairly easy to work with.

6. Become familiar with the topics in this book. You should find that theproblemsdifficulties increase as you progress. However, the problems at the end ofeach section of this book are harder and will help you bind your understanding toprevious topics in the book. Finally, the problems at the end of each chapter arethe most challenging of all, requiring you to piece together all of the importanttopics and involve appropriate technology along the way.

7. Make sure you fill in the bubble sheet neatly. Otherwise, the machine thatscores your answers wont give you credit.

8. Show all of your work on the free-response questions. If you only show youranswer, and it happens to be incorrect, the grader has no choice but to give you nocredit for the entire question. Writing down all of your steps makes sense.

9. Know your stuff. While all of these strategies are helpful, there is no substitutefor knowledge. You may not know every bit of information on the exam, but it isimportant that you remember the information you have learned.

10. Be neat on the free-response questions. Let the grader focus on content,rather than the form. The answers are not lengthy, so do your best to be neat andorganized.

xxiv Before You Begin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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PART I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .AP CALCULUS AB & BCBASICS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 1 All About the AP CalculusAB & BC Tests

All About the APCalculus AB & BC Tests

OVERVIEW Frequently asked questions about the AP calculus tests Summing it up

Your goal and vision in any Advanced Placement class should be to take theAP test, pass it with a sufficiently high score, jump up and down like a lunaticwhen you receive your score, and attain credit for the class in the college oruniversity of your choice. All AP tests are graded on a scale from 1 to 5, with5 being the highest possible grade. Most colleges will accept a score of 3 orabove and assign credit to you for the corresponding course. Some, however,require higher scores, so its important to know the policies of the schools towhich you are applying or have been accepted. An AP course is a littledifferent from a college course. In a college course, you need only pass theclass to receive credit. In an AP course, you must score high enough on thecorresponding AP test, which is administered worldwide in the month of May.So, its essential to know that all-important AP test inside and out.

FREQUENTLY ASKED QUESTIONS ABOUT THE APCALCULUS TESTSBelow are common questions that students pose about the AP Calculus tests.For now, this test is your foe, the only thing standing in your way to glorious(and inexpensive) college credit. Spend some time understanding the enemysbattle plans so that you are prepared once you go to war.

What topics are included on the test?The list of topics changes a little bit every couple of years. The College BoardWeb site (www.collegeboard.com/ap/calculus) always has the current coursedescription.As your academic year draws to a close, use it as a checklist tomakesure you understand everything.

Whats the difference between Calculus AB and BC?The Calculus BC curriculum contains significantly more material than the ABcurriculum. Completing Calculus BC is equivalent to completing college Calcu-lus I and Calculus II courses, whereas AB covers all of college Calculus I andabout half of Calculus II. TheAB andBC curricula cover the samematerial withthe same amount of rigor; BC simply covers additional topics. However, if you

chapter1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

take the BC test, you will get both an AB and BC score (the AB score excludes all BCquestions from the test).

Of the topics on the course description, which actually appear themost on the AP test?This will vary, of course, but I asked my students to list the topics they saw the most.This is the list of topics they generated (BC topics are denotedwith an asterisk): relativeextrema (maximums and minimums), the relationships between derivatives of afunction, the difference quotient, basic integration, integral functions with variables aslimits of integration, volumes of solids with known cross-sections, motion (position,velocity, and acceleration functions), differential equations, area between curves, powerseries*, elementary series* (ex, cos x, sin x), Taylor polynomials*, radius of conver-gence*, and integration by parts*.

How is the test designed?The test is split into two sections, each of which has a calculator-active and a calculator-inactive portion. Section I has 45 multiple-choice questions and lasts 105 minutes. Ofthat time, 55 minutes are spent on 28 non-calculator questions, and 50 minutes arededicated to 17 calculator active questions. Section II has 6 free-response questions andlasts 90 minutes. Three of the free-response questions allow the use of a calculator,while 3 do not.

Should I guess on the multiple-choice questions?You lose a fraction of a point for every multiple-choice question you answer incorrectly;this penalizes random guessing. If you can eliminate even one choice in a question, theodds are in your favor if you guess. If you cannot eliminate any choices, it is best to omitthe question.

Should I have the unit circle memorized?Oh, yes. The unit circle never diesit lives to haunt your life.

Should my calculator be in degrees or radians mode?Unless specifically instructed by the question, set your calculator for radians mode.

I have heard that the AP Calculus test is written by scientists livingin the African rainforest, and that many tests are lost each year ascouriers are attacked and infected by virulent monkeys. Is thistrue?Definitely.

How many questions do I have to get right to get a 3?There is no set answer to this, as the number varies every year based on studentachievement. Unofficially, answering approximately 50 percent of the questions cor-rectly usually results in a 3.

PART I: AP Calculus AB & BC Basics4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Why is the test so hard?Many students are shocked that a 50 percent is passing on the AP test, but the examis constructed to be a super test that tests not only your knowledge but also yourability to apply your knowledge under extreme pressure and in very difficult circum-stances. No one expects you to get them all right. Youre shooting for better than 50percent, so dont panic.

How will I feel when the test is over?Hopefully, youwill still be able to function.Most students are completely exhausted anddrained at the end of the ordeal. Students who are well prepared (like those who buymybook) experience less depression than others. In general, students have a vague positivefeeling when they exit the test if they dedicated themselves to studying all year long. Ihave found that the way you feel when exiting the test is independent of how you willactually perform on the test. Feeling bad in no way implies that you will score badly.

Are there any Web sites on the Internet that could help me preparefor the AP test?There are a few good sites on the Web that are free. Among them, one stands clearlyabove the rest. It offers a newAB and BC problem each week, timed to coordinate witha year-long curriculum to help you prepare for the test. Furthermore, each problem issolved in detail the following week. Every problem ever posted is listed in an archive, soits a very valuable studying tool for practicing specific skills and reviewing for teststhroughout the year. To top it off, the site is funny, and the author is extremely talented.The site? Kelleys AP Calculus Web Page, written and shamefully advertised here byyours truly. You can log on at www.calculus-help.com. Enjoy the same hickory-smokedflavor of this book on line for free each and every week. Another good problem of theweek site is theAlvirneProblems of theWeek (www.seresc.k12.nh.us/www/alvirne.html).

How are the free-response questions graded?Each free-response question is worth up to nine points. Free response questions usuallyhave multiple parts, typically two or three, and the available points are dispersedamong them. Many points are awarded for knowing how to set up a problem; points arenot only given for correct answers. It is best to show all of the setup and steps in yoursolution in an orderly fashion to get themaximumamount of credit you can. The CollegeBoard has examples of excellent, good, and poor free response answers given by actualtest takers on theirWeb site (www.collegeboard.com/ap/calculus). In addition, they havethemost recent free-response questions with their grading rubrics.You should try theseproblems, grade yourself according to the rubrics, and see how you stack up to thenational averages.

Chapter 1: All About the AP Calculus AB & BC Tests 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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ALERT!The World Wide Web is

constantly in a state of flux.

Web sites come and go,

and their addresses

change all the time. All of

these links were active at

publication time.

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Are free-response questions the same as essays?No. Free-response questions are most similar to questions you have on a typicalclassroom quiz or test. They require you to solve a problem logically, with supportingwork shown. Theres no guessing possible like on the multiple-choice questions.Theres really no need to write an essayit just slows you down, and you need everylast second on the free responseits really quite meaty.

Should I show my work?Yes, indeedy. TheAPgraders (called readers) cannot assign you partial credit if you dontgive them the opportunity. On the other hand, if you have no idea what the problem isasking, dont write a detailed explanation of what you would do, and dont writeequations all over the paper. Pick a method and stick to itthe readers can definitelytell if you are trying to bluff your way through a problem you dont understand, so dontpull a Copperfield and try to workmagic through smoke andmirrors.Also, keep inmindthat any work erased or crossed off is not graded, even if it is completely right.

What if the problem has numerous parts and I cant get the firstpart?You should do your best to answer the first part anyway. You may not get any pointsat all, but it is still worth it. If the second part requires the correct completion of thefirst, your incorrect answer will not be penalized again. If you complete the correctsequence of steps on an incorrect solution from a previous answer, you can still receivefull credit for the subsequent parts. This means you are not doomed, so dont give up.

Is it true that a genetically engineered chicken once scored a 4 onthe AB test, but the government covered it up to avoid scandal?I am not allowed to comment on that for national security reasons. However, I can saythat free-range poultry typically score better on free-response questions.

Should I simplify my answers to lowest terms?Actually, no! The AP readers will accept an answer of 393 as readily as an answer of 13.Some free-response questions can get a littlemessy, and youre not expected tomake theanswers pretty and presentable. However, you still

need to be able to simplify for the multiple-choice questions. For example, if you reacha solution of ln13 but that is not listed among the choices, you should be able torecognize that 2ln 3 has the same value, if you apply logarithmic properties.

How accurate should my answers be?Unless specified otherwise, the answer must be correct to at least three decimal places.You may truncate (cut off) the decimal there or round the decimal there. For example, asolution of x5 4.5376219may be recorded as 4.537 (truncated) or 4.538 (rounded). If youwant to write the entire decimal, that is okay, too, but remember that time is money.

PART I: AP Calculus AB & BC Basics6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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TIPThe past free-response

questions, course outline,

and grade rubrics are all

available free of charge on

line. The College Board

Web site is updated far

more frequently than

printed material. Check

there for breaking news

and policies.

TIPCross out any of your errant

work instead of erasing it.

Erasing takes more time,

and time is money on the

AP test.

NOTE2ln 3 5 ln 321 5 ln

13

according to the log

property that states

nloga x 5 loga xn.

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Should I include units in my answer?If the problem indicates units, you need to include the appropriate units in your finalanswer. For example, if the problem involves the motion of a boat and phrases thequestion in terms of feet and minutes, velocity is in ft/min, and the acceleration will bein ft/min2.

When can I use the calculator to answer questions?Youmay use the calculator only on calculator-active questions, but you probably figuredthat out, Mr. or Ms. Smarty Pants. Occasionally, you may use a calculator to completelyanswer a question and show no work at all. You can do this only in the followingcircumstances: graphing a function, calculating a numerical derivative, calculating adefinite integral, or finding an x-intercept. In fact, your calculator is expected to havethese capabilities, and you are expected to know how to use them. Therefore, in thesefour cases, you need only show the setup of the problem and jump right to the solution.For example, you may write

*5

2(x2ex)dx 5 2508.246

without actually integrating by hand at all or showing any work. In all other circum-stances, you must show supporting work for your solutions.

How should I write an answer if I used my calculator?

As in the above example, *5

2(x2ex)dx 5 2508.246 is all you should write; the readers

understood and expected you to use your calculator. Never write from the calculatoras a justification to an answer. Also, never write calculator language in your answer.For example, a free-response answer of fnInt(x 2*e (x),x,2,5) 5 2508.246 cannot get apoint for the correct setup, though it may get points for the correct answer.

What calculators can I use on the AP test?The most current list of calculators can be found on the College Board Web site. Mostfavored among the calculators are the Texas Instruments 83 and 831 (and probably theTI 89 before too long). Its a matter of preference. Some people live and die by HPcalculators and will jump down your throat in the blink of an eye if you suggest that theTI calculators are better. Calculators like the TI-92 cannot be used because they haveQWERTY keyboards. Make sure to check the Web site to see if your calculator isacceptable.

I recently made a calculator out of tinfoil, cat food, and toenailclippings. Are you telling me I cant use it on the AP test?Sorry, but you cant. By the way, I shudder to think about how the toenail clippings wereput to use.

Chapter 1: All About the AP Calculus AB & BC Tests 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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TIPNever, never, never round a

number in a problem unless

you are giving the answer.

If you get a value of

3.5812399862 midway

through a problem, use the

entire decimal as you

complete the problem.

Rounding or truncating

during calculations almost

always results in inaccurate

final answers.

TIPBecause you can use your

calculator to find

x-intercepts, you can also

use it to solve any equation

without explaining how. See

the Technology section in

Chapter 2 for a more

detailed explanation.

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Can I have programs stored in my calculators memory?Yes. Programs are not cleared from the calculators memory before the test begins.Many of my students have stored various programs, but I dont think a single studenthas ever used a program on the test. The test writers are very careful to construct thecalculator portions of the test so that no calculator has an advantage over another. Itsreally not worth your time to load up your calculator.

If I can store programs in the calculator memory, cant I storeformulas and notes? Why do I need to memorize formulas?Technically, you can enter formulas in the calculator as programs, but the test writersalso know you can do this, so it is highly unlikely that such a practice could ever beuseful to you. Remember thatmore than half of the test is now calculator inactive! Dontbecome so calculator dependent that you cant do basic things without it.

PART I: AP Calculus AB & BC Basics8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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NOTEOnly in the four listed

circumstances can you use

the calculator to reach an

answer. For instance, most

calculators can find the

maximum or minimum

value of a function based

on the graph, but you

cannot use a calculator as

your justification on a

problem such as this.

NOTEA QWERTY keyboard, for

those not in the know, has

keys in the order of those

on a computer keyboard.

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SUMMING IT UP All AP tests are graded on a scale from 1 to 5, with 5 being the highest possible

grade. Most colleges will accept a score of 3 or above and assign credit to you forthe corresponding course (see the Appendix at the back of this book).

Completing Calculus BC is equivalent to completing college Calculus I and Cal-culus II courses. AB covers all of college Calculus I and about half of Calculus II.

The test is in two sections: Section I has 45 multiple-choice questions and lasts105 minutes; Section II has 6 free-response questions and lasts 90 minutes.

The most current list of calculators can be found on the College Board Web site.

Chapter 1: All About the AP Calculus AB & BC Tests 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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PART II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .AP CALCULUS AB & BCREVIEW

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 2 Calculus Prerequisites

CHAPTER 3 Limits and Continuity

CHAPTER 4 Differentiating

CHAPTER 5 Advanced Topics inDifferentiation

CHAPTER 6 Applications of the Derivative

CHAPTER 7 Integration

CHAPTER 8 Advanced Methods ofIntegration

CHAPTER 9 Applications of Integration

CHAPTER 10 Differential Equations

CHAPTER 11 Sequences and Series(BC Topics Only)

Calculus Prerequisites

OVERVIEW Functions and relations Function properties Inverse functions Hands-On Activity 2.1: Transforming functions Trigonometry Parametric equations (BC topic only) Polar equations (BC topic only) Vectors and vector equations (BC topic only) Technology: Solving equations with a graphing calculator Summing it up

This chapter is meant to help you review some of the mathematics that leadup to calculus. Of course, all mathematics (and your entire life, no doubt) upuntil this point has simply been a build-up to calculus, but these are the mostimportant topics. Since the focus of this book must be the actual content of theAP test, this chapter is meant only to be a review and not an in-depth courseof study. If you find yourself weak in any of these areas, make sure to reviewthem and strengthen your understanding before you undertake calculus itself.Ideally, then, you should plod through this chapter early enough to addressany of your weaknesses before its too late (read with a scary voice).

FUNCTIONS AND RELATIONSCalculus is rife with functions. It is unlikely that you can find a single page inyour textbook that isnt bursting with them, so its important that you under-stand what they are. A function is a special type of relation, much as, ingeometry, a square is a special type of rectangle. So, in that case, what is arelation? Lets begin with a simple relation called r. We will define r as follows:r(x) 5 3x 1 2. (This is read r of x equals 3 times x plus 2.) Our relation willaccept some sort of input (x) and give us something in return. In the case of r, therelation will return a number that is twomore than three times as large as yourinput. For example, if you were to apply the rule called r to the number 10, therelation would return the number 32. Mathematically, this is written r(10) 53(10)1 25 301 25 32. Thus, r(10)5 32.We say that the relation r has solutionpoint (10,32), as an input of 10 has resulted in an output of 32.

chapter2

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13

Example 1: If g(x) 5 x2 2 2x 1 9, evaluate g(4) and g(23).

Solution: Simply substitute 4 and 23 in for x, one at a time, to get the solutions:

g(4) 5 (4)22 2(4) 1 9 5 16 2 8 1 9 5 17

g(23) 5 (23)2 2 2(23) 1 9 5 9 1 6 1 9 5 24

We call g and r relations because of the way they relate numbers together. Clearly, rrelated the input 10 to the output 32 in the same way that g related 4 to 17 and 23 to24. It is conventional to express these relationships as ordered pairs, so we can saythat g created the relationships (4,17) and (23,24).

There are numerous ways to express relations. They dont always have to be writtenas equations, though most of the time they are. Sometimes, relations are definedsimply as the sets of ordered pairs that create them. Here, we have defined therelation k two ways that mean the same thing:

k: {(23,9),(2,6),(5,21),(7,12),(7,14)}

You can also express a relation as a graph of various ordered pairs that create it, inthe form (x,y). Below we have graphed the function we defined earlier, r(x) 5 3x 1 2.

Sometimes, the rule for a relation changes depending on the input of the relation.These are called piecewise-defined relations or multi-ruled relations.

PART II: AP Calculus AB & BC Review14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Example 2: Graph the piecewise-defined relation and evaluate h(22), h(21), h(0),h(2), h(2.5), and h(6).

Solution: The graph begins as the parabola x2, but once x 5 21, the new rule takesover, and the graph becomes the horizontal line y 5 2. This line stops, in turn, at x 52. The function is undefined between x 5 2 and x 5 3, but for all x . 3, the line 2x 24 gives the correct outputs.

Note: If you didnt recognize that y 5 x2 is a parabola, thats OK (for now). Later inthe chapter, we discuss how to recognize these graphs.

h(22) 5 (22)2 5 4, since 22 falls within the definition of the first of the three rules.

h(21) 5 (21)2 5 1, for the same reason.

h(0) 5 2, as 0 is between 21 and 2, the defined region for the second rule.

h(2) 5 2, as it just falls within the definition of the second rule.

h(2.5) is not defined for this functionno inputs between 2 and 3 are allowed.

h(6) 5 2(6) 2 4 5 8, since 6 . 3, the defining restriction for the last rule.

So, a relation is, in essence, some type of rule that relates a set, or collection, of inputsto a set of outputs. The set of inputs for a relation is called the domain, whereas theset of outputs is called the range. Often, it helps to look at the graph of the relation todetermine its domain and range, as the x values covered by a graph represent itsdomain, and the y values covered by a graph represent its range. Alternatively, youcan think of the domain as the numbers covered by the width of the graph, and therange as the numbers covered by the height of the graph.

Chapter 2: Calculus Prerequisites 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Example 3: Find the domain and range for the relations k and h, as already definedabove.

Solution: Finding domain and range for k is quite easy. The domain is {23,2,5,7} andthe range is {21,6,9,12,14}. The order of the numbers does not matter in these sets.The domain of h comes right from the relationwritten in interval notation, thedomain is (2`,2] (3, `). This is true because any number up to or including 2 is aninput for the relation, as is any number greater than 3. The range is easily determinedfrom the graph we created already. Look at the height or vertical span of the graph.Notice that it never dips below a height of 1, but above 1, every single number iscovered. Even though there are holes in the graph at (21,2) and