550 AP Calculus AB & BC Practice Questions ( PDFDrive.com ) (2).pdf - Editorial Rob Franek Senior VP Publisher Mary Beth Garrick Director of

550 AP Calculus AB & BC Practice Questions ( PDFDrive.com ) (2).pdf

This preview shows page 1 out of 545 pages.

You've reached the end of your free preview.

Want to read all 545 pages?

Unformatted text preview: Editorial Rob Franek, Senior VP, Publisher Mary Beth Garrick, Director of Production Selena Coppock, Senior Editor Calvin Cato, Editor Kristen O’Toole, Editor Meave Shelton, Editor Alyssa Wolff, Editorial Assistant Random House Publishing Team Tom Russell, Publisher Alison Stoltzfus, Publishing Director Ellen L. Reed, Production Manager Dawn Ryan, Managing Editor Erika Pepe, Associate Production Manager Kristin Lindner, Production Supervisor Andrea Lau, Designer The Princeton Review 111 Speen Street, Suite 550 Framingham, MA 01701 E-mail: [email protected] Copyright © 2013 by TPR Education IP Holdings, LLC Cover art © Jonathan Pozniak All rights reserved. Published in the United States by Random House LLC, New York, a Penguin Random House Company, and in Canada by Random House of Canada Limited, Toronto, Penguin Random House Companies. The Princeton Review is not affiliated with Princeton University. AP and Advanced Placement Program are registered trademarks of the College Board, which does not sponsor or endorse this product. eBook ISBN: 978-0-80412446-1 Trade Paperback ISBN: 978-0-8041-2445-4 The Princeton Review is not affiliated with Princeton University. Editor: Calvin S. Cato Production Editor: Jesse Newkirk Production Coordinator: Deborah A. Silvestrini v3.1 Acknowledgments The Princeton Review would like to give a very special thanks to Bikem Polat and Chris Knuth for their hard work on the creation of this title. Their intimate subject knowledge and thorough fact-checking has made this book possible. In addition, The Princeton Review thanks Jesse Newkirk for his hard work in copy editing the content of this title. About the Authors Bikem Ayse Polat has been teaching and tutoring through The Princeton Review since 2010. She came to TPR as an undergraduate at the University of California, San Diego from which she graduated in 2011 with two Bachelors of Science degrees in Bioengineering: Pre-Med and Psychology. After graduation, she moved to Cincinnati, OH for a year where she certified to tutor online and was promoted to Master Tutor. She is currently residing in Philadelphia, PA where she is a graduate student at Temple University, working on her Ph.D. in Urban Education. She is certified to teach and tutor numerous Advanced Placement courses as well as the PSAT, SAT, ACT, LSAT, GRE, and MCAT. In her spare time, Bikem enjoys spending time with her dog, Gary, and her nieces, Laila and Serra. Chris Knuth have been with the Princeton Review for 9 years and has been teaching Math since the 5th grade through various tutoring organizations and classes. He is certified to teach various Advanced Placement courses as well as SAT, ACT, GMAT, GRE, LSAT, DAT, OAT, PSAT, SSAT, and ISEE. He considers Calculus to be his favorite level of Math. Chris would like to thank his family for their unwavering support and always being the “wind in my sail” and give a very special thank you to the teachers who made math fun, Gai Williams and Anita Genduso. Contents Cover Title Page Copyright Acknowledgments About the Authors Part I: Using This Book to Improve Your AP Score Preview: Your Knowledge, Your Expectations Your Guide to Using This Book How to Begin AB Calculus Diagnostic Test AB Calculus Diagnostic Test Answers and Explanations Answer Key Explanations BC Calculus Diagnostic Test BC Calculus Diagnostic Test Answers and Explanations Answer Key Explanations Part II: About the AP Calculus Exams AB Calculus vs BC Calculus The Structure of the AP Calculus Exams Overview of Content Topics How AP Exams Are Used Other Resources Designing Your Study Plan Part III: Test-Taking Strategies for the AP Calculus Exams How to Approach Multiple Choice Questions How to Approach Free Response Questions Derivatives and Integrals That You Should Know Prerequisite Mathematics Part IV: Drills 1 Limits Drill 1 2 Limits Drill 1 Answers and Explanantions 3 Limits Drill 2 4 Limits Drill 2 Answers and Explanantions 5 Functions and Domains Drill 6 Functions and Domains Drill Answers and Explanantions 7 Derivatives Drill 1 8 Derivatives Drill 1 Answers and Explanations 9 Derivatives Drill 2 10 Derivatives Drill 2 Answers and Explanations 11 Derivatives Drill 3 12 Derivatives Drill 3 Answers and Explanations 13 Applications of Derivatives Drill 1 14 Applications of Derivatives Drill 1 Answers and Explanations 15 Applications of Derivatives Drill 2 16 Applications of Derivatives Drill 2 Answers and Explanations 17 General and Partial Fraction Integration Drill 18 General and Partial Fraction Integration Drill Answers and Explanantions 19 Trigonometric Integration Drill 20 Trigonometric Integration Drill Answers and Explanations 21 Exponential and Logarithmic Integration Drill 22 Exponential and Logarithmic Integration Drill Answers and Explanations 23 Areas, Volumes, and Average Values Drill 24 Areas, Volumes, and Average Values Drill Answers and Explanations 25 AB & BC Calculus Free Response Drill 26 AB & BC Calculus Free Response Drill Answers and Explanantions Part V: Practice Exams 27 AB Calculus Practice Test 28 AB Calculus Practice Test Answers and Explanations 29 BC Calculus Practice Test 30 BC Calculus Practice Test Answers and Explanations Part I Using This Book to Improve Your AP Score • Preview: Your Knowledge, Your Expectations • Your Guide to Using This Book • How to Begin • AB Calculus Diagnostic Test • AB Calculus Diagnostic Test Answers and Explanations • BC Calculus Diagnostic Test • BC Calculus Diagnostic Test Answers and Explanations PREVIEW: YOUR KNOWLEDGE, YOUR EXPECTATIONS Your route to a high score on the AP Calculus Exam depends a lot on how you plan to use this book. Start thinking about your plan by responding to the following questions. 1. Rate your level of confidence about your knowledge of the content tested by the AP Calculus Exam: A. Very confident—I know it all B. I’m pretty confident, but there are topics for which I could use help C. Not confident—I need quite a bit of support D. I’m not sure 2. If you have a goal score in mind, circle your goal score for the AP Calculus Exam: 3. What do you expect to learn from this book? Circle all that apply to you. A. A general overview of the test and what to expect B. Strategies for how to approach the test C. The content tested by this exam D. I’m not sure yet YOUR GUIDE TO USING THIS BOOK This book is organized to provide as much—or as little— support as you need, so you can use this book in whatever way will be most helpful for improving your score on the AP Calculus Exam. The remainder of Part One will provide guidance on how to use this book and help you determine your strengths and weaknesses. Part Two of this book will provide information about the struxcture, scoring, and content of the AP Calculus Exam. help you to make a study plan. point you towards additional resources. Part Three of this book will explore various strategies: how to attack multiple choice questions how to write a high-scoring free response answer how to manage your time to maximize the number of points available to you Part Four of this book contains practice drills covering all of the AB Calculus and BC Calculus concepts you will find on the exams. Part Five of this book contains practice tests. You may choose to use some parts of this book over others, or you may work through the entire book. This will depend on your needs and how much time you have. Let’s now look how to make this determination. HOW TO BEGIN 1. Take a Test Before you can decide how to use this book, you need to take a practice test. Doing so will give you insight into your strengths and weaknesses, and the test will also help you make an effective study plan. If you’re feeling test-phobic, remind yourself that a practice test is a tool for diagnosing yourself—it’s not how well you do that matters but how you use information gleaned from your performance to guide your preparation. So, before you read further, take the AP Calculus AB Diagnostic Test starting at this page of this book or take the AP Calculus BC Diagnostic Test starting on this page. Be sure to do so in one sitting, following the instructions that appear before the test. 2. Check Your Answers Using the answer key on this page (for Calculus AB) or this page (for Calculus BC), count how many multiple choice questions you got right and how many you missed. Don’t worry about the explanations for now, and don’t worry about why you missed questions. We’ll get to that soon. 3. Reflect on the Test After you take your first test, respond to the following questions: How much time did you spend on the multiple choice questions? How much time did you spend on each free response question? How many multiple choice questions did you miss? Do you feel you had the knowledge to address the subject matter of the essays? Do you feel you wrote well organized, thoughtful essays? Circle the content areas that were most challenging for you and draw a line through the ones in which you felt confident/did well. Functions, Graphs, and Limits Differential Calculus Integral Calculus Polynomial Approximations and Series (for BC Calculus Students) Applications of Derivatives Applications of Integrals 4. Read Part Two and Complete the Self-Evaluation As discussed in the Goals section above, Part Two will provide information on how the test is structured and scored. It will also set out areas of content that are tested. As you read Part Two, re-evaluate your answers to the questions above. At the end of Part Two, you will revisit and refine the questions you answered above. You will then be able to make a study plan, based on your needs and time available, that will allow you to use this book most effectively. 5. Engage with the Drills as Needed Notice the word engage. You’ll get more out of this book if you use it intentionally than if you read it passively, hoping for an improved score through osmosis. The drills are designed to give you the opportunity to assess your mastery of calculus concepts through testappropriate questions. 6. Take Test 2 and Assess Your Performance Once you feel you have developed the strategies you need and gained the knowledge you lacked, you should take one of the practice exams at the end of this book. You should do so in one sitting, following the instructions at the beginning of the test. When you are done, check your answers to the multiple choice sections. Once you have taken the test, reflect on what areas you still need to work on, and revisit the drills in this book that address those topics. Through this type of reflection and engagement, you will continue to improve. 7. Keep Working As you work through the drills, consider what additional work you need to do and how you will change your strategic approach to different parts of the test. If you do need more guidance, there are plenty of resources available to you. Our Cracking the AP Calculus AB & BC Exams guide gives you a comprehensive review of all the calculus topics you need to know for the exam and offers 5 practice tests (3 for AB and 2 for BC). In addition, you can go to the AP Central website for more information about exam schedules and calculus concepts. AB Calculus Diagnostic Test (Click here to download a PDF of Diagnostic Test) AP® Calculus AB Exam DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. Instructions Section I of this examination contains 45 multiple-choice questions. Fill in only the ovals for numbers 1 through 45 on your answer sheet. CALCULATORS MAY NOT BE USED IN THIS PART OF THE EXAMINATION. Indicate all of your answers to the multiple-choice questions on the answer sheet. No credit will be given for anything written in this exam booklet, but you may use the booklet for notes or scratch work. After you have decided which of the suggested answers is best, completely fill in the corresponding oval on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely. Here is a sample question and answer. Sample Question Chicago is a (A) state (B) city (C) country (D) continent (E) village Sample Answer Use your time effectively, working as quickly as you can without losing accuracy. Do not spend too much time on any one question. Go on to other questions and come back to the ones you have not answered if you have time. It is not expected that everyone will know the answers to all the multiple-choice questions. About Guessing Many candidates wonder whether or not to guess the answers to questions about which they are not certain. Multiple choice scores are based on the number of questions answered correctly. Points are not deducted for incorrect answers, and no points are awarded for unanswered questions. Because points are not deducted for incorrect answers, you are encouraged to answer all multiple-choice questions. On any questions you do not know the answer to, you should eliminate as many choices as you can, and then select the best answer among the remaining choices. CALCULUS AB SECTION I, Part A Time—55 Minutes Number of questions—28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem. In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. (A) 1 (B) 2 (C) 0 (D) nonexistent (E) 2π (B) 0 (C) ∞ (D) 3 (E) The limit does not exist. At what point does the following function have a removable discontinuity? f(x) = (A) (–5,–1) (B) (–2,–1) (C) (–2,1) (D) (1,1) (E) (1,–1) Which of the following functions is NOT continuous at x = –3? f(x) = g(x) = h(x) = (x) = k(x) = (x + 3)2 Which of the following functions is continuous at x = –3? f(x) = g(x) = h(x) = (x) = (x − 2) k(x) = ( )2 What is ? (B) 0 (D) 1 (E) The limit does not exist. If f(x) = (2x3 + 33)( − 2x), then f′(x) = (A) (2x3 + 33) + 6x2( − 2x) (B) (2x3 + 33) + 6x3( − 2x) (C) (2x3 + 33) + 6x2( − 2x) (D) (2x3 + 33) + 6x2( − 2x) (E) (2x3 + 33) If y = + 6x3( , then − 2x) = Find the second derivative of x2y2 = 2 at (2,1). (A) 1 (B) –2 (C) (D) 2 (E) − If the line y = ax2 + bx + c goes through the point (2,1) and is normal to y = x + 2 at the point (0,2), then a = ? (A) − (B) (C) (D) − (E) 2 If f(x) = 2g(x) and if h(x) = x3, then f(h(x)) = (A) 6x2g(x3) (B) 2g(x3) (C) 2x2g(x3) (D) 6g(x3) (E) 2x3g(x3) Which of the following statements about the function given by f(x) = x5 − 2x3 is true? (A) The function has no relative extrema. (B) The graph has one point of inflection and two relative extrema. (C) The graph has three points of inflection and one relative extremum. (D) The graph has three points of inflection and two relative extrema. (E) The graph has two points of inflection and two relative extrema. ∫(2x − 5)3 dx = + C + C + C + C + C (A) 4x20 + C + C + C + C + C Find the average value of f(x) = 3x2 sin x3 on the interval [0, (B) 2 (E) 1 Find t2 + 4t dt. (A) 9x4 + 12x2 (B) 6x(9x3 + 12x2) . (C) 6x2(9x4 + 12x2) (D) 90x3 (E) 54x5 + 72x3 (A) −ln|cos x| + C (B) ln|sin x| + C (C) ln|sec x + tan x| + C (D) −ln|csc x + cot x| + C (E) −ln|sin x| + C What is the area between y = x3 and y = x? (A) 0 (B) 1 Find the volume of the region bounded by y = (x – 5)3, the x-axis, and the line x = 10 as it is revolved around the line x = 2. Set up, but do not evaluate the integral. (A) 2π x(x − 5)3 dx (B) 2π (x − 2)(x − 5)3 dx (C) 2π x(x − 5)3 dx (D) 2π (x − 2)(x − 5)3 dx (E) 2π (x − 2)(x − 5)3 dx ∫3x2(x3 − 3)7 dx (A) 8(x3 – 3)8 + C + C (C) (x3 – 3)8 + C x3 + C x3(x3 – 3)8 + C Find the equation for the normal line to y = 3x2 – 6x at (2,0). y = −6x − y = − x + y = x = y = 6x + 3 y = − x − (A) 0 (B) –1 (C) −∞ (D) ∞ (E) 1 (2 − t2)dt = (A) 6x2 + 3x8 (B) 6x2 – 9x4 (C) 2 – x2 (D) 6x2 – 3x8 (E) 2 – x6 + C + C + C + C + C Find for f′ (x) = x3 + 2x when x = 1. (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 (x3 + 2x2 − 14) = (A) 6x (B) 6x + 4 (C) 3x2 + 4x (D) 3x2 (E) 4x Find the absolute maximum on the interval [–2, 2] for the curve y = 4x5 – 10x2 – 8. (A) –2 (B) –1 (C) 0 (D) 1 (E) 2 Find at (–1,–2) if 3x3 – 2x2 + x = y3 + 2y2 + 3y. (B) 2 (C) − (D) –2 (E) 0 END OF PART A, SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO. CALCULUS AB SECTION I, Part B Time—50 Minutes Number of questions—17 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem. In this test: 1. The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. 2. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number. If f(x) = x−3 + 3 − 3x4 − 3x−4 (C) 3x−4 + + 5π − e2, then f′(x) = (E) 3x2 + Find the value of c that satisfies Rolle’s theorem for f(x) = 2x4 − 16x on the interval [0,2]. (A) 2 (B) (−2)− (C) 2 (D) (−2) (E) 2 Find the absolute maximum of y = x3 − x2 − 7x on the interval [– 2,2]. (A) (B) (C) 0 (D) (E) − Approximate the area under the curve y = x2 + 2 from x = 1 to x = 2 using four right-endpoint rectangles. (A) 4.333 (B) 3.969 (C) 4.719 (D) 4.344 (E) 4.328 Approximate the area under the curve y = x2 + 2 from x = 1 to x = 2 using four inscribed trapezoids. (A) 4.333 (B) 3.969 (C) 4.719 (D) 4.344 (E) 4.328 Evaluate . (A) 1 (B) 2 (C) –1 (E) 0 Suppose F(x) = 1 to 4. (A) 72 (B) 71.25 (C) 24.75 (D) 6 . What is the change in F(x) as t increases from (E) 0.75 In the xy-plane, 2x + y = k is tangent to the graph of y = 2x2 – 8x + 14. What is the value of k? (C) 5 The function f is continuous on the closed interval [0,4] and twice differentiable over the open interval, (0, 4). If f′(2) = –5 and f″(x) > 0 over the interval (0, 4), which of the following could be a table of values for f? When is the particle whose path is described by x(t) = 2t3 − − 16, from t > 0, slowing down? (A) 0 < t < 3 < t < 3 < t < t > 3 t2 + 9t < t < 3 What is the are enclosed by the curve f(x) = 4x2 − x4 and the x-axis. (E) 0 Which of the following is an asymptote for the curve y = ? x = 7 y = 3 y = –7 x = 3 x = –3 + C (B) 2 ln|x2 − 7| + C (C) ln|x2 − 7| + C ln|x2 − 7| + C (E) ln|x| + C What is the area between the curves y = x3 – 2x2 – 5x + 6 and y = x2 – x – 6 from x = –2 to x = 3? (A) 30 (C) 32 (E) 34 Find the average value of f(x) = (3x − 1)3 on the interval from x = –1 to x = 3. (A) 10.667 (B) 12 (C) 9.333 (D) 15 (E) 18 The curve y = ax2 + bx + c passes through the point (1,5) and is normal to the line –x + 5y = 15 at (0,3). What is the equation of the curve? y = 7x2 – 0.2x + 3 y = 2.2x2 – 0.2x + 3 y = 7x2 + 5x + 3 y = 7x2 – 5x + 3 y = 5x2 – 7x + 3 (A) –1 (B) 0 (C) 1 (D) 8 (E) ∞ STOP END OF PART B, SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART B ONLY. DO NOT GO ON TO SECTION II UNTIL YOU ARE TOLD TO DO SO. SECTION II GENERAL INSTRUCTIONS You may wish to look over the problems before starting to work on them, since it is not expected that everyone will be able to complete all parts of all problems. All problems are given equal weight, but the parts of a particular problem are not necessarily given equal weight. A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THIS SECTION OF THE EXAMINATION. You sh...
View Full Document

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture