Petersons Master AP Calculus

PETERSON’S MASTER AP CALCULUS AB & BC 2nd Edition W Michael Kelley Mark Wilding, Contributing Author An ARCO Book ARCO is a registered trademark of Peterson’s, and is used herein under license by Peterson’s About Peterson’s, a Nelnet company Peterson’s (www.petersons.com) is a leading provider of education information and advice, with books and online resources focusing on education search, test preparation, and financial aid Its Web site offers searchable databases and interactive tools for contacting educational institutions, online practice tests and instruction, and planning tools for securing financial aid Peterson’s serves 110 million education consumers annually For more information, contact Peterson’s, 2000 Lenox Drive, Lawrenceville, NJ 08648; 800-338-3282; or find us on the World Wide Web at: www.petersons.com/about ® 2007 Peterson’s, 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 or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, or information storage and retrieval systems—without the prior written permission of the publisher For permission to use material from this text or product, complete the Permission Request Form at http://www.petersons.com/permissions ISBN 13: 978-0-7689-2470-1 ISBN 10: 0-7689-2470-7 Printed in the United States of America 10 Second Edition 09 08 07 Petersons.com/publishing Check out our Web site at www.petersons.com/publishing to see if there is any new information regarding the test and any revisions or corrections to the content of this book We’ve made sure the information in this book is accurate and up-to-date; however, the test format or content may have changed since the time of publication OTHER RECOMMENDED TITLES Master 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 xv How This Book Is Organized xv Special Study Features xv Appendix xvi You’re Well on Your Way to Success xvi Give Us Your Feedback xvi Mike Kelley’s How to Use This Book xvii Quick Reference Guide xx Top 10 Strategies to Raise Your Score xxiv PART I: AP CALCULUS AB & BC BASICS All About the AP Calculus AB & BC Tests Frequently Asked Questions About the AP Calculus Tests Summing It Up PART II: AP CALCULUS AB & BC REVIEW Calculus Prerequisites Functions and Relations Exercise Answers and Explanations Function Properties Exercise Answers and Explanations Inverse Functions Exercise Answers and Explanations Hands-On Activity 2.1: Transforming Functions Selected Solutions to Hands-on Activity 2.1 Exercise Answers and Explanations Trigonometry Exercise Answers and Explanations v 13 13 19 21 22 24 25 26 29 30 31 34 35 37 39 45 46 vi Contents Parametric Equations (BC Topic Only) Exercise Answers and Explanations Polar Equations (BC Topic Only) Exercise Answers and Explanations Vectors and Vector Equations (BC Topic Only) Exercise Answers and Explanations Technology: Solving Equations with a Graphing Calculator Exercise Answers and Explanations Summing It Up 48 51 51 53 56 57 59 63 63 63 66 67 70 Limits and Continuity 71 Hands-On Activity 3.1: What Is a Limit? 71 Selected Solutions to Hands-On Activity 3.1 74 Exercise 76 Answers and Explanations 77 Evaluating Limits Analytically 78 Exercise 80 Answers and Explanations 80 Continuity 82 Exercise 86 Answers and Explanations 86 Hands-On Activity 3.2: The Extreme Value Theorem 88 Selected Solutions to Hands-On Activity 3.2 90 Exercise 92 Answers and Explanations 92 Hands-On Activity 3.3: The Intermediate Value Theorem 93 Solutions to Hands-On Activity 3.3 95 Exercise 97 Answers and Explanations 97 Limits Involving Infinity 99 Exercise 102 Answers and Explanations 102 Special Limits 104 Exercise 107 Answers and Explanations 107 Technology: Evaluating Limits with a Graphing Calculator 108 Exercise 111 Answers and Explanations 113 Summing It Up 117 www.petersons.com Contents vii Differentiating Derivative as a Rate of Change Exercise Answers and Explanations The Power Rule Exercise Answers and Explanations Derivatives to Memorize Exercise Answers and Explanations The Chain Rule Exercise Answers and Explanations The Product Rule Exercise Answers and Explanations The Quotient Rule Exercise Answers and Explanations A Word About Respecting Variables Exercise Answers and Explanations Implicit Differentiation Exercise Answers and Explanations Hands-On Activity 4.1: Approximating Derivatives Selected Solutions to Hands-On Activity 4.1 Exercise Answers and Explanations Technology: Finding Numerical Derivatives with the Graphing Calculator Exercise 10 Answers and Explanations Summing It Up 119 119 123 124 125 127 127 128 131 132 133 135 136 137 139 139 140 142 142 143 146 146 146 149 149 152 153 154 154 Advanced Topics in Differentiation The Derivative of an Inverse Function Exercise Answers and Explanations Hands-On Activity 5.1: Linear Approximations Solutions to Hands-On Activity 5.1 Exercise Answers and Explanations L’Hôpital’s Rule (BC Topic Only) Exercise Answers and Explanations Parametric Derivatives (BC Topic Only) 165 165 168 168 170 171 172 172 173 175 175 177 155 157 160 164 www.petersons.com viii Contents Exercise Answers and Explanations Polar Derivatives (BC Topic Only) Exercise Answers and Explanations Technology: Finding Polar and Parametric Derivatives with Your Calculator (BC Topic Only) Exercise Answers and Explanations Summing It Up 180 180 182 184 184 187 189 190 194 Applications of the Derivative Related Rates Exercise Answers and Explanations Hands-On Activity 6.1: Rolle’s and Mean Value Theorems Selected Solutions to Hands-On Activity 6.1 Exercise Answers and Explanations Hands-On Activity 6.2: The First Derivative Test Selected Solutions to Hands-On Activity 6.2 Exercise Answers and Explanations Concavity Exercise Answers and Explanations Motion Exercise Answers and Explanations Motion in the Plane (BC Topic Only) Exercise Answers and Explanations Optimization Exercise Answers and Explanations Technology: Modeling a Particle’s Movement with a Graphing Calculator Exercise Answers and Explanations Summing It Up 195 195 200 200 Integration Basic Antiderivatives Exercise Answers and Explanations 251 251 256 256 www.petersons.com 203 205 207 208 210 211 214 215 217 220 221 222 225 225 226 228 228 230 234 234 237 240 243 249 Contents ix Advanced Methods of Integration Miscellaneous Methods of Integration Exercise Answers and Explanations Parts (BC Topic Only) Exercise Answers and Explanations Powers of Trigonometric Functions (BC Topic Only) Exercise Answers and Explanations Partial Fractions (BC Topic Only) Exercise Answers and Explanations Improper Integrals (BC Topic Only) Exercise Answers and Explanations 258 260 263 264 264 269 269 270 276 277 281 282 284 285 287 290 290 292 299 299 301 304 304 305 307 308 314 315 315 319 319 321 325 325 326 330 330 332 335 335 337 341 341 www.petersons.com Hands-On Activity 7.1: Approximating Area with Riemann Sums Selected Solutions to Hands-On Activity 7.1 Exercise Answers and Explanations The Trapezoidal Rule Exercise Answers and Explanations The Fundamental Theorem of Calculus Exercise Answers and Explanations Hands-On Activity 7.2: Accumulation Functions Solutions to Hands-On Activity 7.2 Exercise Answers and Explanations The Mean Value Theorem for Integration, Average Value of a Function Exercise Answers and Explanations U-Substitution Exercise Answers and Explanations Integrating Inverse Trigonometric Functions Exercise Answers and Explanations Technology: Evaluating Definite Integrals with Your Graphing Calculator Exercise Answers and Explanations Summing It Up x Contents 10 Technology: Drawing Derivative and Integral Graphs with Your Calculator Exercise Answers and Explanations Summing It Up 343 346 346 352 Applications of Integration Hands-On Activity 9.1: Area Between Curves Selected Solutions to Hands-On Activity 9.1 Exercise Answers and Explanations The Disk and Washer Methods Exercise Answers and Explanations The Shell Method Exercise Answers and Explanations Finding the Volume of Regions with Known Cross Sections Exercise Answers and Explanations Arc Length (BC Topic Only) Exercise Answers and Explanations Polar Area (BC Topic Only) Exercise Answers and Explanations Technology: Using Your Calculator Efficiently Exercise Answers and Explanations Summing It Up 353 353 355 358 358 362 368 369 372 375 375 Differential Equations Hands-On Activity 10.1: Separation of Variables Solutions to Hands-On Activity 10.1 Exercise Answers and Explanations Hands-On Activity 10.2: Slope Fields Selected Solutions to Hands-On Activity 10.2 Exercise Answers and Explanations Euler’s Method (BC Topic Only) Exercise Answers and Explanations Exponential Growth and Decay Exercise Answers and Explanations 407 407 409 411 411 413 415 417 418 420 425 425 428 431 431 www.petersons.com 378 382 382 385 387 387 389 392 392 396 398 398 406 658 APPENDIX Credits Granted Test Required Score BC 4–5 University of San Diego (CA) AB/BC 3–5 The University of Scranton (PA) AB/BC 4–5 AB 3–5 MATH 141 BC 3–5 MATH 141 & MATH 142 AB/BC 3–5 AB 3–5 MATH 151 & MATH 152 BC 3–5 MATH 151, 152, 161, & 162 AB M 408C, B AB 4–5 M 408C, A BC M 408C, B or M 408K, 408L, B BC M 408C, A or M 408K, A; M 408L, B BC M 408C, A or M 408K, 408L, A AB MATH 2312 AB 4–5 MATH 2312 & MATH 1325 BC MATH 2312 & MATH 1325 BC MATH 2312 2417 or MATH 2312 1325 School Name University of St Thomas— continued University of South Carolina (SC) University of Southern California (CA) The University of Tennessee at Chattanooga (TN) The University of Texas at Austin (TX) The University of Texas at Dallas (TX) www.petersons.com Course Waived Stipulations MATH 1431 The Mathematics Department offers a & MATH collection of placement exams for 1432 entry-level students An applicant with a score of or on an AB portion of the Advanced Placement exam in Mathematics will earn four hours of credit for MATH 1431 3–6 Students who score a or but receive a or above on the Calculus AB subscore will earn credit for MATH 141 College-by-College Guide to AP Credit and Placement 659 Credits Granted Course Waived Test Required Score The University of Texas at Dallas—continued BC University of the Pacific (CA) AB 4–5 MATH 51 BC 4–5 MATH 53 University of the Sciences in Philadelphia (PA) AB/BC 3–5 School Name University of Tulsa (OK) University of Utah (UT) University of Virginia (VA) University of Washington (WA) Stipulations MATH 2312 2417 or MATH 1325 1326[] AB 3–5 MATH 2014 BC 3–5 MATH 2014 & MATH 2024 AB 3–5 QA/QB Placement by Department A score of Requirement in the Calculus AB or Calculus BC will waive the QA Requirement but no credit hours will be awarded BC 3–5 QA/QB A score of in the Calculus AB or Requirement Calculus BC will waive the QA Requirement but no credit hours will be awarded Placement by Department AB 4–5 MATH 131 BC 4–5 MATH 131 & MATH 132 AB AB 3–4 MATH 124 AB 10 MATH 124 & MATH 125 BC w/AB 3–5 MATH 124 First quarter of calculus Counts toward Natural World general education requirement for graduation Also satisfies Quantitative and Symbolic Reasoning graduation requirement BC MATH 124 First quarter of calculus Counts toward Natural World general education requirement for graduation Also satisfies Quantitative and Symbolic Reasoning graduation requirement BC 4–5 10 MATH 124 & MATH 125 First two quarters of calculus Counts toward Natural World general education requirement for graduation Also satisfies Quantitative and Symbolic Reasoning graduation requirement Placement only into MATH 124 Placement into first quarter of calculus First quarter of calculus www.petersons.com 660 APPENDIX School Name University of Wisconsin–La Crosse (WI) University of Wisconsin– Madison (WI) University of Wisconsin–River Falls (WI) Ursinus College (PA) Valparaiso University (IN) Test Required Score Credits Granted Course Waived AB/BC 3–5 MATH 207 AB 3 AB 4–5 MATH 221 BC MATH 221 BC 3–5 10 MATH 221 & MATH 222 AB 3–5 MATH 166 BC MATH 166 BC 4–5 MATH 166 & MATH 167 AB/BC 4–5 Mathematics Exempt from Quantitative Reasoning Part A Electives AB 4 MATH 131 AB MATH 131 & MATH 132 BC MATH 131 BC 4–5 MATH 131 & MATH 132 AB 4–5 MATH 150a & MATH 150b BC w/AB 4–5 MATH 150a & MATH 150b BC 4–5 MATH 155a & MATH 155b Vassar College (NY) AB/BC 4–5 Villanova University (PA) AB/BC 4–5 AB 3–5 MA 115 BC 3–5 MA 115 & MA 116 AB 3–5 MATH 1205 BC 3 MATH 1205 BC 4–5 MATH 1205 & MATH 1206 Vanderbilt University (TN) Virginia Military Institute (VA) Virginia Polytechnic Institute and State University (VA) www.petersons.com Stipulations College-by-College Guide to AP Credit and Placement 661 School Name Wabash College (IN) Wagner College (NY) Wake Forest University (NC) Wartburg College (IA) Washington & Jefferson College (PA) Washington and Lee University (VA) Washington College (MD) Credits Granted Course Waived Test Required Score AB 4–5 BC 4–5 AB/BC 4–5 AB 4–5 MTH 555 BC MTH 555 MTH 111 The exception being for English Composition, Mathematics, and Modern and Ancient languages, where the department may grant or deny Advanced Placement credit based on proficiency exam performance and may waive the additional course requirement BC 4–5 AB/BC MA 201 AB/BC 4–5 MA 201 & MA 202 AB/BC 4–5 AB MATH 101 BC MATH 101 & MATH 102 AB 3–5 MAT 201 BC 3–5 MAT 201 & MAT 202 AB AB BC BC AB 4–5 MATH 115 BC MATH 115 BC 4–5 MATH 115 & 116 or MATH 115 & 120 Wells College (NY) AB/BC 4–5 Wesleyan College (GA) AB/BC 4–5 AB 4–5 Washington University in St Louis (MO) Wellesley College (MA) Wesleyan University (CT) Stipulations Take Math Placement test You will probably be placed into Math 132 MATH 131 You will be placed in MATH 132 Take Math Placement test You will probably be placed into Math 233 MATH 131 & MATH 132 You will be placed into Math 233 An AB Subscore of will earn units of credit for Math 131 Wesleyan courses required for credit: Student must complete any 200-level mathematics course with a grade of C- or better www.petersons.com 662 APPENDIX Credits Granted Test Required Score BC Wesleyan courses required for credit: The student passes Math 122 or a 200-level Mathematics course with a grade of C- or better BC 4–5 Wesleyan courses required for credit: Student must complete any 200-level mathematics course with a grade of C- or better AB/BC 3–5 MATH 240 & Quantitative & Symbolic Reasoning AB MATH 141 & MATH 100T AB 4–5 MATH 141 & MATH 201 BC MATH 141 & MATH 201 BC 4–5 MATH 201 & MATH 202 Westmont College (CA) AB/BC 4–5 Wheaton College (MA) AB/BC 4–5 Whitman College (WA) AB 4–5 MATH 125 BC 4–5 MATH 125 & MATH 126 AB 3–4 MA 110 AB MA 110 & MA 111 BC 3–5 MA 110 & MA 111 AB 4–5 MATH 141 Credit for these courses may be applied toward the Quantitative and Analytical Reasoning requirement BC MATH 141 Credit for these courses may be applied toward the Quantitative and Analytical Reasoning requirement BC MATH 141 & MATH 142 Credit for these courses may be applied toward the Quantitative and Analytical Reasoning requirement AB 4–5 GEN 104 BC 4–5 GEN 104 & MAT 200 School Name Wesleyan University—continued Western Washington University (WA) Westminster College (UT) Whitworth College (WA) Willamette University (OR) William Jewell College (MO) www.petersons.com Course Waived Stipulations Student may receive credit for either AB or BC, but not both unless recommended by Mathematics Department after placement College-by-College Guide to AP Credit and Placement 663 Credits Granted Course Waived Test Required Score AB 2–3 MATH 104 If a score of or is earned on the & STAT 101 Calculus–AB test, the student is eligible to reduce the requirements in the Math major by one course If course credit is blank, there is no reduction in the number of courses required for that major AB 4–5 MATH 104, 105, 251, & STAT 201 BC 1–2 MATH 104 If a score of or is earned on the & STAT 101 Calculus–AB test, the student is eligible to reduce the requirements in the Math major by one course If course credit is blank, there is no reduction in the number of courses required for that major BC 3–5 MATH 104, 106, 251 & STAT 201 Winona State University (MN) AB/BC 3–5 Wittenberg University (OH) AB/BC 4–5 School Name Williams College (MA) Wofford College (SC) Worcester Polytechnic Institute (MA) Xavier University (OH) AB 4–5 MATH 181 BC 3 MATH 181 BC 4–5 MATH 181 & MATH 182 AB 4–5 MA 1021 & MA 1022 BC 4–5 MA 1021, 1022, & 1023 AB 3–5 MATH 170 BC MATH 170 BC 4–5 MATH 170 & MATH 171 Stipulations www.petersons.com NOTES NOTES NOTES NOTES NOTES NOTES Peterson’s Book Satisfaction Survey Give Us Your Feedback Thank you for choosing Peterson’s as your source for personalized solutions for your education and career achievement Please take a few minutes to answer the following questions Your answers will go a long way in helping us to produce the most user-friendly and comprehensive resources to meet your individual needs When completed, please tear out this page and mail it to us at: Publishing Department Peterson’s, a Nelnet company 2000 Lenox Drive Lawrenceville, NJ 08648 You can also complete this survey online at www.petersons.com/booksurvey What is the ISBN of the book you have purchased? (The ISBN can be found on the book’s back cover in the lower right-hand corner ) Where did you purchase this book? ❑ Retailer, such as Barnes & Noble ❑ Online reseller, such as Amazon.com ❑ Petersons.com ❑ Other (please specify) _ If you purchased this book on Petersons.com, please rate the following aspects of your online purchasing experience on a scale of to (4 = Excellent and = Poor) Comprehensiveness of Peterson’s Online Bookstore page ❑ ❑ ❑ ❑ Overall online customer experience ❑ ❑ ❑ ❑ Which category best describes you? ❑ High school student ❑ Parent of high school student ❑ College student ❑ Graduate/professional student ❑ Returning adult student ❑ ❑ ❑ ❑ Teacher Counselor Working professional/military Other (please specify) _ Rate your overall satisfaction with this book Extremely Satisfied Satisfied Not Satisfied ❑ ❑ ❑ Rate each of the following aspects of this book on a scale of to (4 = Excellent and = Poor) Comprehensiveness of the information ❑ ❑ ❑ ❑ Accuracy of the information ❑ ❑ ❑ ❑ Usability ❑ ❑ ❑ ❑ Cover design ❑ ❑ ❑ ❑ Book layout ❑ ❑ ❑ ❑ Special features (e.g., CD, flashcards, charts, etc.) ❑ ❑ ❑ ❑ Value for the money ❑ ❑ ❑ ❑ This book was recommended by: ❑ Guidance counselor ❑ Parent/guardian ❑ Family member/relative ❑ Friend ❑ Teacher ❑ Not recommended by anyone—I found the book on my own ❑ Other (please specify) _ Would you recommend this book to others? Yes Not Sure No ❑ ❑ ❑ Please provide any additional comments Remember, you can tear out this page and mail it to us at: Publishing Department Peterson’s, a Nelnet company 2000 Lenox Drive Lawrenceville, NJ 08648 or you can complete the survey online at www.petersons.com/booksurvey Your feedback is important to us at Peterson’s, and we thank you for your time! If you would like us to keep in touch with you about new products and services, please include your e-mail here: _ ... OTHER RECOMMENDED TITLES Master AP English Language & Composition Master AP English Literature & Composition Master AP Chemistry Master AP U.S Government & Politics Master AP U.S History Contents... best to be neat and organized www .petersons. com PART I AP CALCULUS AB & BC BASICS CHAPTER All About the AP Calculus AB & BC Tests All About the AP Calculus AB & BC Tests ... site www .petersons. com SUMMING IT UP PART II AP CALCULUS AB & BC REVIEW CHAPTER Calculus Prerequisites CHAPTER Limits and Continuity CHAPTER Differentiating CHAPTER