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 Answers and Explanations 37
Part III About the AP Calculus AB Exam 59
AB Calculus versus BC Calculus 60
The Structure of the Calculus Exam 60
How the AP Calculus AB Exam is Scored 61
Overview of Content Topics 62
General Overview of This Book 64
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 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 90
Connecting Multiple Representations of Limits 92
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 102
Connecting Limits at Infinity and Horizontal Asymptotes 105
Working with the Intermediate Value Theorem (IVT) 106
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 136
5 Differentiation: Composite, Implicit, and Inverse Functions 141
The Chain Rule 142
Implicit Differentiation 148
Differentiating Inverse Functions 154
Differentiating Inverse Trigonometric Functions 158
Selecting Procedures for Calculating Derivatives 160
Calculating Higher-Order Derivatives 162
6 Contextual Applications of Differentiation 169
Interpreting the Meaning of the Derivative in Context 170
Straight-Line Motion: Connecting Position, Velocity, and Acceleration 170
Rates of Change in Applied Contexts Other Than Motion 176
Introduction to Related Rates 176
Solving Related Rates Problems 177
Approximating Values of a Function Using Local Linearity and Linearization 183
Using L'Hospital's Rule for Determining Limits of Indeterminate Forms 193
7 Analytical Applications of Differentiation 201
Using the Mean Value Theorem 202
Extreme Value Theorem, Global Versus Local Extrema, and Critical Points 207
Determining Intervals on Which a Function Is Increasing or Decreasing 208
Using the First Derivative Test to Determine Relative (Local) Extrema 210
Using the Candidate Test to Determine Absolute (Global) Extrema 212
Determining Concavity of Functions over Their Domains 214
Using the Second Derivative Test to Determine Extrema 217
Sketching Graphs of Functions and Their Derivatives 220
Connecting a Function, Its First Derivative, and Its Second Derivative 235
Introduction to Optimization Problems 241
Solving Optimization Problems 242
Exploring Behaviors and Implicit Relations 252
8 Integration and Accumulation of Change 255
Exploring Accumulations of Change 256
Approximating Areas with Riemann Sums 257
Riemann Sums, Summation Notation, and Definite Integral Notation 271
The Fundamental Theorem of Calculus and Accumulation Functions 272
Interpreting the Behavior of Accumulation Functions Involving Area 274
Applying Properties of Definite Integrals 278
The Fundamental Theorem of Calculus and Definite Integrals 279
Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation 281
Integrating Using Substitution 289
Integrating Functions Using Long Division and Completing the Square 305
Selecting Techniques for Antidifferentiation 308
9 Differential Equations 313
Modeling Situations with Differential Equations 314
Verifying Solutions for Differential Equations 314
Sketching Slope Fields 315
Reasoning Using Slope Fields 319
Finding General Solutions Using Separation of Variables 321
Finding Particular Solutions Using Initial Conditions and Separation of Variables 322
Exponential Models with Differential Equations 326
10 Applications of Integration 331
Finding the Average Value of a Functions on an Interval 332
Connecting Position, Velocity, and Acceleration of Functions Using Integrals 334
Using Accumulation Functions and Definite Integrals in Applied Contexts 335
Finding the Area Between Curves Expressed as Functions of x 336
Finding the Area Between Curves Expressed as Functions of y 338
Finding the Area Between Curves That intersect at More than Two Points 342
Volumes with Cross Sections: Squares and Rectangles 343
Volumes with Cross Sections: Triangles and Semicircles 345
Volume with Disc Method: Revolving Around x- or y-Axis 346
Volume with Disc Method: Revolving Around Other Axes 349
Volume with Washer Method: Revolving Around x- or y-Axis 351
Volume with Washer Method: Revolving Around Other Axes 354
11 Answers to Practice Problems Sets 359
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
Practice Test 3 563
Practice Test 3: Answers and Explanations 595
Part VII Additional Practice Tests 623
Practice Test 4 625
Practice Test 4: Answers and Explanations 651
Practice Test 5 669
Practice Test 5: Answers and Explanations 695
About the Author 714
Practice Test A online
Practice Test A: Answers and Explanations online
Practice Test B online
Practice Test B: Answers and Explanations online
