# 5 Steps to a 5: AP Calculus BC 2023 Elite Student Edition

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## Book Preface

You are an AP Calculus student. Not too shabby! As you know, AP Calculus is one of the most challenging subjects in high school. You are studying mathematical ideas that helped change the world. Not that long ago, calculus was taught at the graduate level. Today, smart young people like yourself study calculus in high school. Most colleges will give you credit if you score a 3 or more on the AP Calculus BC Exam.

So how do you do well on the AP Calculus BC Exam? How do you get a 5? Well, you’ve already taken the first step. You’re reading this book. The next thing you need to do is to make sure that you understand the materials and do the practice problems. In recent years, the AP Calculus exams have gone through many changes. For example, today the questions no longer stress long and tedious algebraic manipulations. Instead, you are expected to be able to solve a broad range of problems including problems presented to you in the form of a graph, a chart, or a word problem. For many of the questions, you are also expected to use your calculator to find the solutions.

After having taught AP Calculus for many years and having spoken to students and other calculus teachers, we understand some of the difficulties that students might encounter with the AP Calculus exams. For example, some students have complained about not being able to visualize what the question was asking and other students said that even when the solution was given, they could not follow the steps. Under these circumstances, who wouldn’t be frustrated? In this book, we have addressed these issues. Whenever possible, problems are accompanied by diagrams, and solutions are presented in a step-by-step manner. The graphing calculator is used extensively whenever it is permitted. The book also begins with a chapter on limits and continuity. These topics are normally taught in a pre-calculus course. If you’re familiar with these concepts, you might skip this chapter and begin with Chapter 6.

So how do you get a 5 on the AP Calculus BC Exam?

Step 1: Set up your study program by selecting one of the three study plans in Chapter 2 of this book.

Step 2: Determine your test readiness by taking the Diagnostic Exam in Chapter 3.

Step 3: Develop strategies for success by learning the test-taking techniques offered in Chapter 4.

Step 4: Review the knowledge you need to score high by studying the subject materials in Chapter 5 through Chapter 14.

Step 5: Build your test-taking confidence by taking the Practice Exams provided in this book.

As an old martial artist once said, “First you must understand. Then you must practice.” Have fun and good luck!

### How Is This Book Organized?

This book begins with an introduction to the Five-Step Program followed by 14 chapters reflecting the 5 steps.

•   Step 1 provides an overview of the AP Calculus BC Exam, and offers three study plans for preparing for the Exam.

•   Step 2 contains a diagnostic test with answers and explanations.

•   Step 3 offers test-taking strategies for answering both multiple-choice and free-response questions, and for using a graphing calculator.

•   Step 4 consists of 10 chapters providing a comprehensive review of all topics covered on the AP Calculus BC Exam. At the end of each chapter (beginning with Chapter 5), you will find a set of practice problems with solutions, a set of cumulative review problems with solutions, and a Rapid Review section giving you the highlights of the chapter.

•   Step 5 provides three full practice AP Calculus BC Exams with answers, explanations, and worksheets to compute your score.

The book concludes with a summary of math formulas and theorems needed for the AP Calculus BC Exam. (Please note that the exercises in this book are done with the TI-89 Graphing Calculator.)

### Introducing the Five-Step Preparation Program

This book is organized as a five-step program to prepare you to succeed in the AP Calculus BC Exam. These steps are designed to provide you with vital skills, strategies, and the practice that can lead you to that perfect 5. Here are the 5 steps.

#### Step 1: Set Up Your Study Plan

In this step you will read an overview of the AP Calculus BC Exam, including a summary of topics covered in the exam and a description of the format of the exam. You will also follow a process to help determine which of the following preparation programs is right for you:

•   Full school year: September through May.

•   One semester: January through May.

•   Six weeks: Basic training for the exam.

In this step you will take a diagnostic multiple-choice exam in calculus. This pre-test should give you an idea of how prepared you are to take the real exam before beginning to study for the actual AP Calculus BC Exam.

#### Step 3: Develop Strategies for Success

In this step you will learn strategies that will help you do your best on the exam. These strategies cover both the multiple-choice and free-response sections of the exam.

•   Learn to read multiple-choice questions.

•   Lean how to answer multiple-choice questions.

•   Learn how to plan and write answers to the free-response questions.

#### Step 4: Review the Knowledge You Need to Score High

In this step you will learn or review the material you need to know for the test. This review section takes up the bulk of this book. It contains:

•   A comprehensive review of AP Calculus BC.

•   A set of practice problems.

•   A set of cumulative review problems beginning with Chapter 5.

•   A rapid review summarizing the highlights of the chapter.

#### Step 5: Build Your Test-Taking Confidence

In this step you will complete your preparation by testing yourself on practice exams. We have provided you with three complete practice exams in AP Calculus BC with solutions and scoring guides. Although these practice exams are not reproduced questions from the actual AP calculus exam, they mirror both the material tested by AP and the way in which it is tested.

Finally, at the back of this book you will find additional resources to aid your preparation. These include:

•   A brief bibliography.

•   A list of websites related to the AP Calculus BC exam.

•   A summary of formulas and theorems related to the AP Calculus BC exam.

## CONTENTS

Dedication and Acknowledgments

Preface

Introduction: The Five-Step Program

STEP 1   Set Up Your Study Plan

1   What You Need to Know About the AP Calculus BC Exam

1.1   What Is Covered on the AP Calculus BC Exam?

1.2   What Is the Format of the AP Calculus BC Exam?

How Is the AP Calculus BC Exam Grade Calculated?

1.4   Which Graphing Calculators Are Allowed for the Exam?

Calculators and Other Devices Not Allowed for the AP Calculus BC Exam

Other Restrictions on Calculators

2   How to Plan Your Time

2.1   Three Approaches to Preparing for the AP Calculus BC Exam

Overview of the Three Plans

2.2   Calendar for Each Plan

Summary of the Three Study Plans

3   Take a Diagnostic Exam

3.1   Getting Started!

3.2   Diagnostic Test

3.4   Solutions to Diagnostic Test

AP Calculus BC Diagnostic Exam

STEP 3   Develop Strategies for Success

4   How to Approach Each Question Type

4.1   The Multiple-Choice Questions

4.2   The Free-Response Questions

4.3   Using a Graphing Calculator

4.4   Taking the Exam

What Do I Need to Bring to the Exam?

Tips for Taking the Exam

STEP 4   Review the Knowledge You Need to Score High

Big Idea 1: Limits

5   Limits and Continuity

5.1   The Limit of a Function

Definition and Properties of Limits

Evaluating Limits

One-Sided Limits

Squeeze Theorem

5.2   Limits Involving Infinities

Infinite Limits (as xa)

Limits at Infinity (as x → ±∞)

Horizontal and Vertical Asymptotes

5.3   Continuity of a Function

Continuity of a Function at a Number

Continuity of a Function over an Interval

Theorems on Continuity

5.4   Rapid Review

5.5   Practice Problems

5.6   Cumulative Review Problems

5.7   Solutions to Practice Problems

5.8   Solutions to Cumulative Review Problems

Big Idea 2: Derivatives

6   Differentiation

6.1   Derivatives of Algebraic Functions

Definition of the Derivative of a Function

Power Rule

The Sum, Difference, Product, and Quotient Rules

The Chain Rule

6.2   Derivatives of Trigonometric, Inverse Trigonometric, Exponential, and Logarithmic Functions

Derivatives of Trigonometric Functions

Derivatives of Inverse Trigonometric Functions

Derivatives of Exponential and Logarithmic Functions

6.3   Implicit Differentiation

Procedure for Implicit Differentiation

6.4   Approximating a Derivative

6.5   Derivatives of Inverse Functions

6.6   Higher Order Derivatives

L’Hôpital’s Rule for Indeterminate Forms

6.7   Rapid Review

6.8   Practice Problems

6.9   Cumulative Review Problems

6.10 Solutions to Practice Problems

6.11 Solutions to Cumulative Review Problems

7   Graphs of Functions and Derivatives

7.1   Rolle’s Theorem, Mean Value Theorem, and Extreme Value Theorem

Rolle’s Theorem

Mean Value Theorem

Extreme Value Theorem

7.2   Determining the Behavior of Functions

Test for Increasing and Decreasing Functions

First Derivative Test and Second Derivative Test for Relative Extrema

Test for Concavity and Points of Inflection

7.3   Sketching the Graphs of Functions

Graphing without Calculators

Graphing with Calculators

7.4   Graphs of Derivatives

7.5   Parametric, Polar, and Vector Representations

Parametric Curves

Polar Equations

Types of Polar Graphs

Symmetry of Polar Graphs

Vectors

Vector Arithmetic

7.6   Rapid Review

7.7   Practice Problems

7.8   Cumulative Review Problems

7.9   Solutions to Practice Problems

7.10 Solutions to Cumulative Review Problems

8   Applications of Derivatives

8.1   Related Rate

General Procedure for Solving Related Rate Problems

Common Related Rate Problems

Inverted Cone (Water Tank) Problem

Angle of Elevation Problem

8.2   Applied Maximum and Minimum Problems

General Procedure for Solving Applied Maximum and Minimum Problems

Distance Problem

Area and Volume Problem

8.3   Rapid Review

8.4   Practice Problems

8.5   Cumulative Review Problems

8.6   Solutions to Practice Problems

8.7   Solutions to Cumulative Review Problems

9   More Applications of Derivatives

9.1   Tangent and Normal Lines

Tangent Lines

Normal Lines

9.2   Linear Approximations

Tangent Line Approximation (or Linear Approximation)

Estimating the nth Root of a Number

Estimating the Value of a Trigonometric Function of an Angle

9.3   Motion Along a Line

Instantaneous Velocity and Acceleration

Vertical Motion

Horizontal Motion

9.4   Parametric, Polar, and Vector Derivatives

Derivatives of Parametric Equations

Position, Speed, and Acceleration

Derivatives of Polar Equations

Velocity and Acceleration of Vector Functions

9.5   Rapid Review

9.6   Practice Problems

9.7   Cumulative Review Problems

9.8   Solutions to Practice Problems

9.9   Solutions to Cumulative Review Problems

Big Idea 3: Integrals and the Fundamental Theorems of Calculus

10   Integration

10.1   Evaluating Basic Integrals

Antiderivatives and Integration Formulas

Evaluating Integrals

10.2   Integration by U-Substitution

The U-Substitution Method

U-Substitution and Algebraic Functions

U-Substitution and Trigonometric Functions

U-Substitution and Inverse Trigonometric Functions

U-Substitution and Logarithmic and Exponential Functions

10.3   Techniques of Integration

Integration by Parts

Integration by Partial Fractions

10.4   Rapid Review

10.5   Practice Problems

10.6   Cumulative Review Problems

10.7   Solutions to Practice Problems

10.8   Solutions to Cumulative Review Problems

11   Definite Integrals

11.1   Riemann Sums and Definite Integrals

Sigma Notation or Summation Notation

Definition of a Riemann Sum

Definition of a Definite Integral

Properties of Definite Integrals

11.2   Fundamental Theorems of Calculus

First Fundamental Theorem of Calculus

Second Fundamental Theorem of Calculus

11.3   Evaluating Definite Integrals

Definite Integrals Involving Algebraic Functions

Definite Integrals Involving Absolute Value

Definite Integrals Involving Trigonometric, Logarithmic, and Exponential Functions

Definite Integrals Involving Odd and Even Functions

11.4   Improper Integrals

Infinite Intervals of Integration

Infinite Discontinuities

11.5   Rapid Review

11.6   Practice Problems

11.7   Cumulative Review Problems

11.8   Solutions to Practice Problems

11.9   Solutions to Cumulative Review Problems

12   Areas, Volumes, and Arc Lengths

12.1   The Function F(x) = ∫xaf(t)dt

12.2   Approximating the Area Under a Curve

Rectangular Approximations

Trapezoidal Approximations

12.3   Area and Definite Integrals

Area Under a Curve

Area Between Two Curves

12.4   Volumes and Definite Integrals

Solids with Known Cross Sections

The Disc Method

The Washer Method

12.5   Integration of Parametric, Polar, and Vector Curves

Area, Arc Length, and Surface Area for Parametric Curves

Area and Arc Length for Polar Curves

Integration of a Vector-Valued Function

12.6   Rapid Review

12.7   Practice Problems

12.8   Cumulative Review Problems

12.9   Solutions to Practice Problems

12.10   Solutions to Cumulative Review Problems

13   More Applications of Definite Integrals

13.1   Average Value of a Function

Mean Value Theorem for Integrals

Average Value of a Function on [a, b]

13.2   Distance Traveled Problems

13.3   Definite Integral as Accumulated Change

Temperature Problem

Leakage Problem

Growth Problem

13.4   Differential Equations

Exponential Growth/Decay Problems

Separable Differential Equations

13.5   Slope Fields

13.6   Logistic Differential Equations

13.7   Euler’s Method

Approximating Solutions of Differential Equations by Euler’s Method

13.8   Rapid Review

13.9   Practice Problems

13.10 Cumulative Review Problems

13.11 Solutions to Practice Problems

13.12 Solutions to Cumulative Review Problems

Big Idea 4: Series

14   Series

14.1   Sequences and Series

Convergence

14.2   Types of Series

p-Series

Harmonic Series

Geometric Series

Decimal Expansion

14.3   Convergence Tests

Divergence Test

Integral Test

Ratio Test

Comparison Test

Limit Comparison Test

Informal Principle

14.4   Alternating Series

Error Bound

Absolute and Conditional Convergence

14.5   Power Series

14.6   Taylor Series

Taylor Series and MacLaurin Series

Common MacLaurin Series

14.7   Operations on Series

Substitution

Differentiation and Integration

Error Bounds

14.8   Rapid Review

14.9   Practice Problems

14.10 Cumulative Review Problems

14.11 Solutions to Practice Problems

14.12 Solutions to Cumulative Review Problems

STEP 5   Build Your Test-Taking Confidence

AP Calculus BC Practice Exam 1

AP Calculus BC Practice Exam 2

ELITE STUDENT EDITION 5 Minutes to a 5

180 Activities and Questions in 5 Minutes a Day

Formulas and Theorems

Bibliography