Table of Contents
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Part I Using This Book to Improve Your AP Score 1
Preview: Your Knowledge, Your Expectations 2
Your Guide to Using This Book 2
How to Begin 3
Part II Practice Test 1 7
Practice Test 1 9
Practice Test 1: Diagnostic Answer Key and Explanations 37
How to Score Practice Test 1 58
Part III About the AP Calculus AB Exam 59
AB Calculus vs. BC Calculus 60
The Structure of the AP Calculus AB Exam 60
How the AP Calculus AB Exam is Scored 61
Past AP Calculus AB Score Distributions 61
Overview of Content Topics 62
General Overview of This Book 64
How AP Exams Are Used 65
Other Resources 66
Designing Your Study Plan 66
Part IV Test-Taking Strategies for the AP Calculus AB Exam 67
1 How to Approach Multiple-Choice Questions 69
2 How to Approach Free-Response Questions 73
Part V Content Review for the AP Calculus AB Exam 77
3 Limits and Continuity 79
Introducing Calculus: Can Change Occur at an Instant? 80
Defining Limits and Using Limit Notation 80
Estimating Limit Values from Graphs 82
Estimating Limit Values from Tables 83
Determining Limits Using Algebraic Properties of Limits 84
Determining Limits Using Algebraic Manipulation 85
Selecting Procedures for Determining Limits 87
Determining Limits Using the Squeeze Theorem 91
Connecting Multiple Representations of Limits 93
Exploring Types of Discontinuities 95
Defining Continuity at a Point 99
Confirming Continuity Over an Interval 101
Removing Discontinuities 102
Connecting Infinite Limits and Vertical Asymptotes 103
Connecting Limits at Infinity and Horizontal Asymptotes 105
Working with the Intermediate Value Theorem (IVT) 106
End of Chapter 3 Drill 109
4 Differentiation: Definition and Basic Derivative Rules 111
Defining Average and Instantaneous Rates of Change at a Point 112
Defining the Derivative of a Function and Using Derivative Notation 113
Estimating Derivatives of a Function at a Point 119
Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist 122
Applying the Power Rule 123
Derivative Rules: Constant, Sum, Difference, and Constant Multiple 124
Derivatives of cos x, sin x, ex and In x 126
The Product Rule 133
The Quotient Rule 134
Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions 135
End of Chapter 4 Drill 138
5 Differentiation: Composite, Implicit, and Inverse Functions 139
The Chain Rule 140
Implicit Differentiation 144
Differentiating Inverse Functions 150
Differentiating Inverse Trigonometric Functions 154
Selecting Procedures for Calculating Derivatives 156
Calculating Higher-Order Derivatives 158
End of Chapter 5 Drill 161
6 Contextual Applications of Differentiation 163
Interpreting the Meaning of the Derivative in Context 164
Straight-Line Motion: Connecting Position, Velocity, and Acceleration 164
Rates of Change in Applied Contexts Other Than Motion 170
Introduction to Related Rates 170
Solving Related Rates Problems 171
Approximating Values of a Function Using Local Linearity and Linearization 177
Using L'Hospital's Rule for Determining Limits of Indeterminate Forms 187
End of Chapter 6 Drill 192
7 Analytical Applications of Differentiation 193
Using the Mean Value Theorem 194
Extreme Value Theorem, Global Versus Local Extrema, and Critical Points 199
Determining Intervals on Which a Function Is Increasing or Decreasing 200
Using the First Derivative Test to Determine Relative (Local) Extrema 202
Using the Candidates Test to Determine Absolute (Global) Extrema 204
Determining Concavity of Functions over Their Domains 206
Using the Second Derivative Test to Determine Extrema 209
Sketching Graphs of Functions and Their Derivatives 212
Connecting a Function, Its First Derivative, and Its Second Derivative 227
Introduction to Optimization Problems 232
Solving Optimization Problems 233
Exploring Behaviors and Implicit Relations 243
End of Chapter 7 Drill 244
8 Integration and Accumulation of Change 245
Exploring Accumulations of Change 246
Approximating Areas with Riemann Sums 247
Riemann Sums, Summation Notation, and Definite Integral Notation 260
The Fundamental Theorem of Calculus and Accumulation Functions 261
Interpreting the Behavior of Accumulation Functions Involving Area 263
Applying Properties of Definite Integrals 267
The Fundamental Theorem of Calculus and Definite Integrals 268
Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation 269
Integrating Using Substitution 278
Integrating Functions Using Long Division and Completing the Square 301
Selecting Techniques for Antidifferentiation 305
End of Chapter 8 Drill 307
9 Differential Equations 309
Modeling Situations with Differential Equations 310
Verifying Solutions for Differential Equations 310
Sketching Slope Fields 311
Reasoning Using Slope Fields 315
Finding General Solutions Using Separation of Variables 317
Finding Particular Solutions Using Initial Conditions and Separation of Variables 318
Exponential Models with Differential Equations 322
End of Chapter 9 Drill 326
10 Applications of Integration 327
Finding the Average Value of a Function on an Interval 328
Connecting Position, Velocity, and Acceleration of Functions Using Integrals 330
Using Accumulation Functions and Definite Integrals in Applied Contexts 331
Finding the Area Between Curves Expressed as Functions of x 332
Finding the Area Between Curves Expressed as Functions of y 334
Finding the Area Between Curves That Intersect at More than Two Points 337
Volumes with Cross Sections: Squares and Rectangles 338
Volumes with Cross Sections: Triangles and Semicircles 340
Volume with Disc Method: Revolving Around x- or y-Axis 341
Volume with Disc Method: Revolving Around Other Axes 344
Volume with Washer Method: Revolving Around x- or y-Axis 345
Volume with Washer Method: Revolving Around Other Axes 348
End of Chapter 10 Drill 351
11 Answers to Practice Problems Sets 353
12 Answers to End of Chapter Drills 483
Part VI Practice Tests 2 and 3 509
Practice Test 2 511
Practice Test 2: Answers and Explanations 539
How to Score Practice Test 2 563
Practice Test 3 565
Practice Test 3: Answers and Explanations 593
How to Score Practice Test 3 615
Part VII Additional Practice Tests 617
Practice Test 4 619
Practice Test 4: Answers and Explanations 645
How to Score Practice Test 4 663
Practice Test 5 665
Practice Test 5: Answers and Explanations 691
How to Score Practice Test 5 710
About the Author 711
Practice Test A online
Practice Test A: Answers and Explanations online
Practice Test B online
Practice Test B: Answers and Explanations online
Practice Test C online
Practice Test C: Answers and Explanations online
