PETERSON’S M ASTER AP C ALCULUS AB & BC 2nd Edition W. Michael Kelley Mark Wilding, Contributing Author AnARCOBook 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. 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ISBN 13: 978-0-7689-2470-1 ISBN 10: 0-7689-2470-7 Printed in the United States ofAmerica 10 9 8 7 6 5 4 3 2 1 09 08 07 Second Edition Petersons.com/publishing CheckoutourWebsiteatwww.petersons.com/publishingtoseeifthereisanynewinformationregardingthetestand anyrevisionsorcorrectionstothecontentofthisbook.We’vemadesuretheinformationinthisbookisaccurateand up-to-date;however,thetestformatorcontentmayhavechangedsincethetimeofpublication. 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 1 All About the AP Calculus AB & BC Tests............... 3 Frequently Asked Questions About the AP Calculus Tests .. 3 Summing It Up ......................................... 9 PART II: AP CALCULUS AB & BC REVIEW 2 Calculus Prerequisites ................................ 13 Functions and Relations ................................ 13 Exercise 1 ............................................. 19 Answers and Explanations ............................. 21 Function Properties .................................... 22 Exercise 2 ............................................. 24 Answers and Explanations ............................. 25 Inverse Functions ..................................... 26 Exercise 3 ............................................. 29 Answers and Explanations ............................. 30 Hands-On Activity 2.1: Transforming Functions .......... 31 Selected Solutions to Hands-on Activity 2.1 .............. 34 Exercise 4 ............................................. 35 Answers and Explanations ............................. 37 Trigonometry .......................................... 39 Exercise 5 ............................................. 45 Answers and Explanations ............................. 46 v vi Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parametric Equations (BC Topic Only) .................. 48 . . Exercise 6 ............................................. 51 . . Answers and Explanations ............................. 51 . . Polar Equations (BC Topic Only) ........................ 53 . Exercise 7 ............................................. 56 . . Answers and Explanations ............................. 57 . . Vectors and Vector Equations (BC Topic Only) ........... 59 . . Exercise 8 ............................................. 63 . Answers and Explanations ............................. 63 . . Technology: Solving Equations with a Graphing . . Calculator ............................................. 63 . Exercise 9 ............................................. 66 . . Answers and Explanations ............................. 67 . . Summing It Up ........................................ 70 . . .3 Limits and Continuity ................................. 71 . . Hands-On Activity 3.1: What Is a Limit? ................ 71 . Selected Solutions to Hands-On Activity 3.1 ............. 74 . . Exercise 1 ............................................. 76 . . Answers and Explanations ............................. 77 . . Evaluating Limits Analytically ......................... 78 . Exercise 2 ............................................. 80 . . Answers and Explanations ............................. 80 . . Continuity ............................................ 82 . . Exercise 3 ............................................. 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 4 ............................................. 92 . Answers and Explanations ............................. 92 . . Hands-On Activity 3.3: The Intermediate Value Theorem . 93 . . Solutions to Hands-On Activity 3.3 ...................... 95 . . Exercise 5 ............................................. 97 . Answers and Explanations ............................. 97 . . Limits Involving Infinity ............................... 99 . . Exercise 6 ............................................ 102 . . Answers and Explanations ............................ 102 . Special Limits ........................................ 104 . . Exercise 7 ............................................ 107 . . Answers and Explanations ............................ 107 . . Technology: Evaluating Limits with a Graphing . Calculator ............................................ 108 . . Exercise 8 ............................................ 111 . . Answers and Explanations ............................ 113 . Summing It Up ....................................... 117 . . . . . . . .www.petersons.com . . . . . . . Contents vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Differentiating...................................... 119 . Derivative as a Rate of Change ........................ 119 . . Exercise 1 ............................................ 123 . . Answers and Explanations ............................ 124 . . The Power Rule ...................................... 125 . Exercise 2 ............................................ 127 . . Answers and Explanations ............................ 127 . . Derivatives to Memorize .............................. 128 . . Exercise 3 ............................................ 131 . Answers and Explanations ............................ 132 . . The Chain Rule ...................................... 133 . . Exercise 4 ............................................ 135 . . Answers and Explanations ............................ 136 . The Product Rule ..................................... 137 . . Exercise 5 ............................................ 139 . . Answers and Explanations ............................ 139 . . The Quotient Rule .................................... 140 . Exercise 6 ............................................ 142 . . Answers and Explanations ............................ 142 . . A Word About Respecting Variables .................... 143 . . Exercise 7 ............................................ 146 . Answers and Explanations ............................ 146 . . Implicit Differentiation ................................ 146 . . Exercise 8 ............................................ 149 . . Answers and Explanations ............................ 149 . Hands-On Activity 4.1: Approximating Derivatives ...... 152 . . Selected Solutions to Hands-On Activity 4.1 ............ 153 . . Exercise 9 ............................................ 154 . . Answers and Explanations ............................ 154 . Technology: Finding Numerical Derivatives with the . . Graphing Calculator .................................. 155 . . Exercise 10 ........................................... 157 . Answers and Explanations ............................ 160 . . Summing It Up ....................................... 164 . . . 5 Advanced Topics in Differentiation ................. 165 . . The Derivative of an Inverse Function ................. 165 . . Exercise 1 ............................................ 168 . Answers and Explanations ............................ 168 . . Hands-On Activity 5.1: Linear Approximations .......... 170 . . Solutions to Hands-On Activity 5.1 ..................... 171 . . Exercise 2 ............................................ 172 . Answers and Explanations ............................ 172 . . L’Hôpital’s Rule (BC Topic Only) ....................... 173 . . Exercise 3 ............................................ 175 . . Answers and Explanations ............................ 175 . Parametric Derivatives (BC Topic Only) ................ 177 . . . . www.petersons.com . . . . . . . . viii Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercise 4 ............................................ 180 . . Answers and Explanations ............................ 180 . . Polar Derivatives (BC Topic Only) ..................... 182 . . Exercise 5 ............................................ 184 . Answers and Explanations ............................ 184 . . Technology: Finding Polar and Parametric Derivatives . . with Your Calculator (BC Topic Only) .................. 187 . Exercise 6 ............................................ 189 . . Answers and Explanations ............................ 190 . . Summing It Up ....................................... 194 . . .6 Applications of the Derivative ...................... 195 . . Related Rates ........................................ 195 . Exercise 1 ............................................ 200 . . Answers and Explanations ............................ 200 . . Hands-On Activity 6.1: Rolle’s and Mean Value . Theorems ............................................ 203 . . Selected Solutions to Hands-On Activity 6.1 ............ 205 . . Exercise 2 ............................................ 207 . . Answers and Explanations ............................ 208 . Hands-On Activity 6.2: The First Derivative Test ........ 210 . . Selected Solutions to Hands-On Activity 6.2 ............ 211 . . Exercise 3 ............................................ 214 . . Answers and Explanations ............................ 215 . Concavity ............................................ 217 . . Exercise 4 ............................................ 220 . . Answers and Explanations ............................ 221 . . Motion ............................................... 222 . Exercise 5 ............................................ 225 . . Answers and Explanations ............................ 225 . . Motion in the Plane (BC Topic Only) ................... 226 . . Exercise 6 ............................................ 228 . Answers and Explanations ............................ 228 . . Optimization ......................................... 230 . . Exercise 7 ............................................ 234 . . Answers and Explanations ............................ 234 . Technology: Modeling a Particle’s Movement with a . . Graphing Calculator .................................. 237 . . Exercise 8 ............................................ 240 . Answers and Explanations ............................ 243 . . Summing It Up ....................................... 249 . . . 7 Integration ......................................... 251 . . Basic Antiderivatives ................................. 251 . . Exercise 1 ............................................ 256 . Answers and Explanations ............................ 256 . . . . . . .www.petersons.com . . . . . . . Contents ix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hands-On Activity 7.1: Approximating Area with . Riemann Sums ....................................... 258 . . Selected Solutions to Hands-On Activity 7.1 ............ 260 . . Exercise 2 ............................................ 263 . Answers and Explanations ............................ 264 . . The Trapezoidal Rule ................................. 264 . . Exercise 3 ............................................ 269 . . Answers and Explanations ............................ 269 . The Fundamental Theorem of Calculus ................. 270 . . Exercise 4 ............................................ 276 . . Answers and Explanations ............................ 277 . . Hands-On Activity 7.2: Accumulation Functions ......... 281 . Solutions to Hands-On Activity 7.2 ..................... 282 . . Exercise 5 ............................................ 284 . . Answers and Explanations ............................ 285 . . The Mean Value Theorem for Integration, Average . Value of a Function ................................... 287 . . Exercise 6 ............................................ 290 . . Answers and Explanations ............................ 290 . U-Substitution ....................................... 292 . . Exercise 7 ............................................ 299 . . Answers and Explanations ............................ 299 . . Integrating Inverse Trigonometric Functions ............ 301 . Exercise 8 ............................................ 304 . . Answers and Explanations ............................ 304 . . Technology: Evaluating Definite Integrals with Your . Graphing Calculator .................................. 305 . . Exercise 9 ............................................ 307 . . Answers and Explanations ............................ 308 . . Summing It Up ....................................... 314 . . . 8 Advanced Methods of Integration.................. 315 . Miscellaneous Methods of Integration .................. 315 . . Exercise 1 ............................................ 319 . . Answers and Explanations ............................ 319 . . Parts (BC Topic Only) ................................. 321 . Exercise 2 ............................................ 325 . . Answers and Explanations ............................ 325 . . Powers of Trigonometric Functions (BC Topic Only) ..... 326 . . Exercise 3 ............................................ 330 . Answers and Explanations ............................ 330 . . Partial Fractions (BC Topic Only) ...................... 332 . . Exercise 4 ............................................ 335 . . Answers and Explanations ............................ 335 . Improper Integrals (BC Topic Only) .................... 337 . . Exercise 5 ............................................ 341 . . Answers and Explanations ............................ 341 . . . . www.petersons.com . . . . . . . . x Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Technology: Drawing Derivative and Integral Graphs . . with Your Calculator .................................. 343 . . Exercise 6 ............................................ 346 . Answers and Explanations ............................ 346 . . Summing It Up ....................................... 352 . . . 9 Applications of Integration ......................... 353 . . Hands-On Activity 9.1: Area Between Curves ........... 353 . . Selected Solutions to Hands-On Activity 9.1 ............ 355 . Exercise 1 ............................................ 358 . . Answers and Explanations ............................ 358 . . The Disk and Washer Methods ........................ 362 . . Exercise 2 ............................................ 368 . Answers and Explanations ............................ 369 . . The Shell Method ..................................... 372 . . Exercise 3 ............................................ 375 . . Answers and Explanations ............................ 375 . Finding the Volume of Regions with Known Cross . . Sections .............................................. 378 . . Exercise 4 ............................................ 382 . Answers and Explanations ............................ 382 . . Arc Length (BC Topic Only) ........................... 385 . . Exercise 5 ............................................ 387 . . Answers and Explanations ............................ 387 . Polar Area (BC Topic Only) ............................ 389 . . Exercise 6 ............................................ 392 . . Answers and Explanations ............................ 392 . . Technology: Using Your Calculator Efficiently ........... 396 . Exercise 7 ............................................ 398 . . Answers and Explanations ............................ 398 . . Summing It Up ....................................... 406 . . .10 Differential Equations............................... 407 . . Hands-On Activity 10.1: Separation of Variables ........ 407 . Solutions to Hands-On Activity 10.1 .................... 409 . . Exercise 1 ............................................ 411 . . Answers and Explanations ............................ 411 . . Hands-On Activity 10.2: Slope Fields ................... 413 . Selected Solutions to Hands-On Activity 10.2 ........... 415 . . Exercise 2 ............................................ 417 . . Answers and Explanations ............................ 418 . . Euler’s Method (BC Topic Only) ........................ 420 . Exercise 3 ............................................ 425 . . Answers and Explanations ............................ 425 . . Exponential Growth and Decay ........................ 428 . . Exercise 4 ............................................ 431 . Answers and Explanations ............................ 431 . . . . .www.petersons.com . . . . . . .