AP® Calculus AB & BC All Access Book + Online

Research & Education Association

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AP® CALCULUS AB & BC ALL ACCESS®, 2nd EDITION

Copyright © 2017 by Research & Education Association, Inc. Prior edition copyright © 2014 by Research & Education Association, Inc. All rights reserved. No part of this book may be reproduced in any form without permission of the publisher.

Printed in the United States of America

Library of Congress Control Number 2016948519

ISBN-13: 978–0–7386–1220–1

ISBN-10: 0–7386–1220–0

LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: Publication of this work is for the purpose of test preparation and related use and subjects as set forth herein. While every effort has been made to achieve a work of high quality, neither Research & Education Association, Inc., nor the authors and other contributors of this work guarantee the accuracy or completeness of or assume any liability in connection with the information and opinions contained herein and in REA’s software and/or online materials. REA and the authors and other contributors shall in no event be liable for any personal injury, property or other damages of any nature whatsoever, whether special, indirect, consequential or compensatory, directly or indirectly resulting from the publication, use or reliance upon this work.

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AP® CALCULUS AB & BC

ALL ACCESS®

Access your Online Tools

by following the instructions

found ot the back of this book.

Contents

About Our Author

About REA

Acknowledgments

Chapter 1: Welcome to REA’s All Access for AP Calculus AB & BC

Chapter 2: Strategies for the Exams

Content of the AP Calculus AB & BC Exams

Format of the Exams

What’s the Score?

The Focus of Our Review Chapters

General Strategies for the AP Calculus Exams

Section I: Strategies for the Multiple-Choice Section

Section II: Strategies for the Free-Response Section

Calculators

Using the Calculator

Chapter 3: A Precalculus Review

1. Functions

2. Domain and Range

3. The Function Toolbox

4. Even and Odd Functions

5. Transformation of Graphs

6. Special Factorization

7. Linear Equations

8. Solving Quadratic Equations

9. Asymptotes

10. Inverses

11. Negative and Fractional Exponents

12. Eliminating Complex Fractions

13. Solving Fractional Equations

14. Solving Absolute Value Equations

15. Solving Inequalities

16. Exponential Functions and Logs

17. Right Angle Trigonometry

18. Special Angles

19. Trigonometric Identities

20. Solving Trigonometric Equations or Inequalities

Quiz 1 available online at www.rea.com/studycenter

AP Calculus AB

Chapter 4: Basic Calculus Concepts and Limits

Introduction

Limits

Graphical Approach to Limits

Algebraic Approach to Limits

Continuity

Quiz 2 available online at www.rea.com/studycenter

Chapter 5: Derivatives

Average and Instantaneous Rates of Change

The Derivative

Derivatives of Algebraic Functions

The Chain Rule

Tangent Line Approximations

Derivatives of Trigonometric Functions

Derivatives of Exponential, Logarithmic, and Inverse Trig Functions

Implicit Differentiation

Derivatives of Inverse Functions

Horizontal and Vertical Tangent Lines

Differentiability

Quiz 3 available online at www.rea.com/studycenter

Chapter 6: Applications of Differentiation

Related Rates

Procedure for Solving Related-Rates Problems

Straight-Line Motion

3 Important Theorems

The Intermediate Value Theorem

The Mean-Value Theorem and Rolle’s Theorem

Function Analysis

Increasing/Decreasing

Concavity

Relative (or Local) Extrema

Absolute Extrema

Optimization

Procedure for Solving an Optimization Problem

L’Hospital’s Rule

Quiz 4 available online at www.rea.com/studycenter

Mini-Test 1 available online at www.rea.com/studycenter

Chapter 7: Integration

Antidifferentiation and Indefinite Integrals

Area and the Definite Integral

The Accumulation Function

Riemann Sums and the Trapezoidal Rule

The Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus

Quiz 5 available online at www.rea.com/studycenter

Chapter 8: Applications of Integration

Area Between Curves

Average Value of a Function

Accumulated Change Problems

Straight-Line Motion Revisited

Volume

Solving Differential Equations

Growth and Decay

Quiz 6 available online at www.rea.com/studycenter

Mini-Test 2 available online at www.rea.com/studycenter

AP Calculus BC

Chapter 9: Additional Limit and Integration Problems

L’Hospital’s Rule Revisited

Integration by Parts

Integration Using Partial Fractions

Improper Integrals

Quiz 7 available online at www.rea.com/studycenter

Chapter 10: Additional BC Calculus Applications

Euler’s Method

Logistic Growth

Arc Length

Parametric Functions

Vector-Valued Functions

Polar Equations

Quiz 8 available online at www.rea.com/studycenter

Chapter 11: Sequences and Series

Basic Series and Convergence Tests

Power Series and Interval of Convergence

Taylor Polynomial Approximations

Error Bounds

Taylor and Maclaurin Series

Functions Defined as Power Series

Quiz 9 available online at www.rea.com/studycenter

Mini-Test 3available online at www.rea.com/studycenter

AP Calculus AB Practice Exam

(also available online at www.rea.com/studycenter)

Answer Key

Calculus AB Practice Exam Topic-Level Grid

Detailed Explanations of Answers

AP Calculus BC Practice Exam

(also available online at www.rea.com/studycenter)

Answer Key

Calculus BC Practice Exam Topic-Level Grid

Detailed Explanations of Answers

Answer Sheets

Index

About Our Author

Stu Schwartz has been teaching mathematics since 1973. For 35 years he taught in the Wissahickon School District, in Ambler, Pennsylvania, specializing in AP Calculus AB and BC. He has also taught at Chestnut Hill College in Philadelphia and Broward College in Fort Lauderdale. Mr. Schwartz received his B.S. degree in Mathematics from Temple University, Philadelphia.

Mr. Schwartz was a 2002 recipient of the Presidential Award for Excellence in Mathematics Teaching and also won the 2007 Outstanding Educator of the Year Award for the Wissahickon School District.

Mr. Schwartz is well-known in the AP community, having developed the website www.mastermathmentor.com, which is geared toward helping educators teach AP Calculus, AP Statistics, and other math courses. Mr. Schwartz is always looking for ways to provide teachers with new and innovative teaching materials, believing that it should be the goal of every math teacher not only to teach students mathematics, but also to find joy and beauty in math as well.

Mr. Schwartz lives with his cat, Newton, in Pompano Beach, Florida. He loves to bike and swim.

Thanks to my brother, Jerry, who originally kindled my interest in math and always manages to spot small errors in my work. Thanks also to my special friend and business partner, Ted Tyree, for always inspiring me and helping me feel that retiring provided me with an opportunity to share my enthusiasm for learning mathematics with the world.

About REA

Founded in 1959, Research & Education Association (REA) is dedicated to publishing the finest and most effective educational materials—including study guides and test preps—for students of all ages.

Today, REA’s wide-ranging catalog is a leading resource for students, teachers, and other professionals. Visit www.rea.com to see a complete listing of all our titles.

Acknowledgments

We would like to thank Pam Weston, Publisher, for setting the quality standards for production integrity and managing the publication to completion; John Cording, Vice President, Technology, for coordinating the design and development of the REA Study Center; Larry B. Kling, Vice President, Editorial, for overall direction; Diane Goldschmidt, Managing Editor, for coordinating development of this edition; Kathy Caratozzolo of Caragraphics for typesetting this edition.

In addition, we thank Jim Sears of Cape Cod Academy, and Mel Friedman, M.S., for technically reviewing the manuscript.

Chapter 1

Welcome to REA’s All Access for AP Calculus AB & BC

A new, more effective way to prepare for your AP exam

There are many different ways to prepare for an AP exam. What’s best for you depends on how much time you have to study and how comfortable you are with the subject matter. To score your highest, you need a system that can be customized to fit you: your schedule, your learning style, and your current level of knowledge.

This book, and the free online tools that come with it, will help you personalize your AP prep by testing your understanding, pinpointing your weaknesses, and delivering flashcard study materials unique to you.

Let’s get started and see how this system works.

How to Use REA’s AP All Access

The REA AP All Access system allows you to create a personalized study plan through three simple steps: targeted review of exam content, assessment of your knowledge, and focused study in the topics where you need the most help.

Here’s how it works:

Finding Your Strengths and Weaknesses: The REA Study Center

The best way to personalize your study plan and truly focus on the topics where you need the most help is to get frequent feedback on what you know and what you don’t. At the online REA Study Center, you can access three types of assessment: end-of-chapter quizzes, mini-tests, and full-length practice tests. Each of these tools delivers a detailed score report that follows the topics set by the College Board.

✓ 9 End-of-Chapter Quizzes

Short online quizzes are available throughout the review and are designed to test your immediate grasp of the topics just covered.

✓ 3 Mini-Tests

Two online mini-tests for Calculus AB students and one online mini-test for Calculus BC students cover what you’ve studied in each major section of the book. These tests are like the actual AP exam, only shorter, and will help you evaluate your overall understanding of the subject material.

✓ 2 Full-Length Practice Tests

After you’ve finished reviewing the book, take our full-length exam for Calculus AB or BC (or both) to practice under test-day conditions. Available both in this book and online at the REA Study Center (www.rea.com/studycenter), these practice tests give you the most complete picture of your strengths and weaknesses. We strongly recommend that you take the online version of the exam for the added benefits of timed testing, automatic scoring, and a detailed score report.

Improving Your Score: e-Flashcards

With your score reports from our online quizzes and practice tests in hand, you’ll be able to see exactly which topics you need to review. Use this information to create your own flashcards for the areas where you are weak. And, because you will create these flashcards through the online REA Study Center, you’ll be able to access them from any computer or smartphone.

Not quite sure what to put on your flashcards? Start with the 100 free cards (60 for Calculus AB; 40 for Calculus BC) included when you buy this book.

Need More Review?

Pick up a copy of REA’s Crash Course® for AP Calculus AB & BC, the fastest way to raise your score during the last few weeks before the exam. Use your All Access® score reports to focus on the topics where you still need extra review.

REA’s Suggested 8-Week Study Plan for AP Calculus AB

Depending on how much time you have until test day, you can expand or condense our eight-week study plan as you see fit.

To score your highest, use our suggested study plan and customize it to fit your schedule, targeting the areas where you need the most review.

REA’s Suggested 8-Week Study Plan for AP Calculus BC

Depending on how much time you have until test day, you can expand or condense our eight-week study plan as you see fit. You may be learning some of these topics at the same time you are doing your review. That’s okay. The more exposure to these topics that you get, the easier it will be for you to do the problems.

To score your highest, use our suggested study plan and customize it to fit your schedule, targeting the areas where you need the most review.

Most schools offer Calculus AB and Calculus BC as separate courses to be taken in consecutive years. If you are taking Calculus AB and BC in one course, you will need more study time. It is suggested that you consolidate the time spent on the AB topics in order to allow yourself additional time to spend on the BC topics.

Test-Day Checklist

Chapter 2

Strategies for the Exams

Content of the AP Calculus AB & BC Exams

Although the content of the Calculus AB and BC exams changes slightly from year to year, the following outline shows the topics found on the exams.

I. Functions and Limits

Limits (graphical and algebraic)

Continuity

Parametric functions*

Polar functions*

Vector-valued functions*

L’Hospital’s Rule (added to the AB curriculum in 2017 with more complicated

problems remaining in the BC curriculum)

II. Derivatives

Rates of change

Derivative definition

Differentiation techniques for algebraic functions

Chain rule

Implicit differentiation

Derivatives of inverse functions

Differentiability

III. Applications of Differentiation

Related rates

Straight-line motion

Function analysis

The Intermediate and Mean Value Theorems

Absolute extrema

Optimization

IV. Integrals

Anti-differentiation rules

u-substitution

The definite integral as area

The accumulation function

Riemann sums and the trapezoidal rule

Fundamental theorem of calculus

Integration by parts*

Integration using partial fractions*

Improper integrals*

V. Application of Integration

Area

Average value

Accumulated change

Straight line motion

Volume

Arc length*

Solving differential equations by separation of variables

Growth and decay

Slope fields

Euler’s method*

Logistic growth*

VI. Infinite Series

Convergence tests*

Radius/interval of convergence*

Taylor polynomial approximations*

Error bounds*

Taylor and Maclaurin series*

Functions defined as power series*

Format of the Exams

The Multiple-Choice Section

Each exam begins with multiple-choice questions. Section I, Part A contains 30 multiple-choice questions to be completed in 60 minutes. No calculator is allowed on this part. Section I, Part B contains 15 multiple-choice questions to be completed in 45 minutes. A graphing calculator is required for this part. The problems in Part A are numbered 1–30. However, for unknown reasons, the problems in Part B are numbered 76–90. Each multiple-choice question is followed by four answer choices, (A) through (D).

The Free-Response Section

After time is called on the multiple-choice section, you’ll get a short break before diving into the free-response section. The section requires you to produce a total of six written responses, usually with multiple parts, in 90 minutes. Part A contains 2 problems and these require a graphing calculator; this part must be completed in 30 minutes. Part B contains 4 problems and no calculator is allowed. The time allotted for Part B is 1 hour.

The entire exam is summarized in the table below:

* Just because a calculator is required doesn’t mean it will need to be used on all problems.

** If you finish Section II, Part B early, you may return to work on the questions in Section II, Part A. However, you will not be able to use your calculator.

What’s the Score?

Although the scoring process for the AP exam may seem quite complex, it boils down to two simple components: your multiple-choice score plus your free-response score. The multiple-choice section accounts for 50% of your overall score and is generated by awarding one point toward your raw score for each question you’ve answered correctly.

The free-response section accounts for the remaining 50% of your total score. Trained graders read students’ written responses and assign points according to grading rubrics. Each free-response problem is worth 9 points and the number of points you accrue out of the total points possible will make up your score on the free-response section.

The test maker reports AP scores on a scale of 1 to 5. Although individual colleges and universities determine what credit or advanced placement, if any, is awarded to students at each score level, these are the assessments typically associated with each numeric score:

5 Extremely well qualified

4 Well qualified

3 Qualified

2 Possibly qualified

1 No recommendation

Calculus AB Subscore for the Calculus BC Exam

The College Board provides AP Calculus BC test-takers with two scores: the score you receive for the Calculus BC exam and the grade you would have received if you had taken the Calculus AB exam. This is called the AB subscore and is based on your performance on the portion of the exam devoted to Calculus AB topics. If you are familiar with the more complex BC topics, you have nothing to lose by taking the BC exam because of the AB subscore.

The Focus of Our Review Chapters

This book contains an AP Calculus AB and BC course review, which is meant to complement your AP Calculus textbook, and while it covers the full spectrum of problems you will see in the AP Calculus exams, it is by no means exhaustive. You will, however, be well prepared for the AP exam by using our review along with your textbook.

Your calculus textbook most likely gives a lot of practice problems on individual calculus skills. Many problems give specific instructions: e. g., take the derivative, find the integral, find relative extrema, determine the limit, etc. The review problems in this book concentrate more on problems where calculus skills are combined and where you are not told specifically what skills to use. For example, a word problem that asks about the time when the value of a house is increasing the fastest, uses calculus skills taught by a textbook, but requires students to understand the underlying concepts behind them. In this case, you need to know that the underlying idea is the derivative of the function describing the cost of the house and maximizing it requires you to set it equal to zero. This makes the overall theme of the problem function analysis.

So, while other books may focus on computational skills in calculus, this book was specifically written with an eye towards what typical AP exam questions ask.

Our AB & BC Practice Exams

This book provides two full-length practice exams, one for Calculus AB and one for Calculus BC.

Our AB practice exam is modeled after a typical AB exam. Our BC practice exam specifically emphasizes the BC curriculum. We do this because the BC exam is quite different from any other AP exam. After all, Calc BC is not so much the next course after AB as it is an extension, or continuation, of AB. Because BC embeds all AB topics in addition to its own unique content, you’ll actually get an AB subscore when you sit for the BC exam. What’s taught in the BC course, and thus what you need to know for the BC exam, equates with what’s typically covered in a college Calc II course.

Here’s a boiled-down snapshot of what AB versus BC looks like:

* indicates topic includes both basic and advanced material

What the topic list in itself can’t effectively show, however, is the special challenge BC students face in having to tackle more advanced topics in integration, sequences, and function approximation. This, then, sets the stage for them to make the leap to