5 Steps to a 5: AP Calculus BC 2020
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!
CONTENTS
Dedication and Acknowledgments
Preface
About the Authors
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?
1.3 What Are the Advanced Placement Exam Grades?
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
STEP 2 Determine Your Test Readiness
3 Take a Diagnostic Exam
3.1 Getting Started!
3.2 Diagnostic Test
3.3 Answers to Diagnostic Test
3.4 Solutions to Diagnostic Test
3.5 Calculate Your Score
Short-Answer Questions
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 x → a)
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
Shadow 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
Business Problems
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) = f (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
Business Problems
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
Radius and Interval of Convergence
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
Formulas and Theorems
Bibliography
Websites
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