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 39
How to Score Practice Test 1 65
Part III About the AP Calculus BC Exam 67
AB Calculus vs. BC Calculus 68
Structure of the AP Calculus BC Exam 68
How the AP Calculus BC Exam is Scored 69
Past AP Calculus BC Score Distributions 69
Overview of Content Topics 70
General Overview of This Book 73
How AP Exams Are Used 74
Other Resources 75
Designing Your Study Plan 75
Part IV Test-Taking Strategies for the AP Calculus BC Exam 77
1 How to Approach Multiple-Choice Questions 79
2 How to Approach Free-Response Questions 83
Part V Content Review for the AP Calculus BC Exam 87
3 Limits and Continuity 89
Introducing Calculus: Can Change Occur at an Instant? 90
Defining Limits and Using Limit Notation 90
Estimating Limit Values from Graphs 92
Estimating Limit Values from Tables 93
Determining Limits Using Algebraic Properties of Limits 94
Determining Limits Using Algebraic Manipulation 95
Selecting Procedures for Determining Limits 97
Determining Limits Using the Squeeze Theorem 101
Connecting Multiple Representations of Limits 103
Exploring Types of Discontinuities 105
Defining Continuity at a Point 109
Confirming Continuity over an Interval 111
Removing Discontinuities 112
Connecting Infinite Limits and Vertical Asymptotes 113
Connecting Limits at infinity and Horizontal Asymptotes 115
Working with the Intermediate Value Theorem (IVT) 116
End of Chapter 3 Drill 119
4 Differentiation: Definition and Basic Derivative Rules 121
Defining Average and Instantaneous Rates of Change at a Point 122
Defining the Derivative of a Function and Using Derivative Notation 123
Estimating Derivatives of a Function at a Point 129
Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist 132
Applying the Power Rule 133
Derivative Rules: Constant, Sum, Difference, and Constant Multiple 134
Derivatives of cos x, sin x, ex, and In x 136
The Product Rule 143
The Quotient Rule 144
Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions 145
End of Chapter 4 Drill 148
5 Differentiation: Composite, Implicit, and Inverse Functions 149
The Chain Rule 150
Implicit Differentiation 154
Differentiating Inverse Functions 160
Differentiating Inverse Trigonometric Functions 164
Selecting Procedures for Calculating Derivatives 166
Calculating Higher-Order Derivatives 168
End of Chapter 5 Drill 171
6 Contextual Applications of Differentiation 173
Interpreting the Meaning of the Derivative in Context 174
Straight-Line Motion: Connecting Position, Velocity, and Acceleration 174
Rates of Change in Applied Contexts Other Than Motion 180
Introduction to Related Rates 180
Solving Related Rates Problems 181
Approximating Values of a Function Using Local Linearity and Linearization 187
Using L'Hospital's Rule for Determining Limits of Indeterminate Forms 197
End of Chapter 6 Drill 202
7 Analytical Applications of Differentiation 203
Using the Mean Value Theorem 204
Extreme Value Theorem, Global Versus Local Extrema, and Critical Points 209
Determining Intervals on Which a Function Is Increasing or Decreasing 210
Using the First Derivative Test to Determine Relative (Local) Extrema 212
Using the Candidates Test to Determine Absolute (Global) Extrema 214
Determining Concavity of Functions over Their Domains 216
Using the Second Derivative Test to Determine Extrema 219
Sketching Graphs of Function and Their Derivatives 222
Connecting a Function, Its First Derivative, and Its Second Derivative 237
Introduction to Optimization Problems 243
Solving Optimization Problems 244
Exploring Behaviors of Implicit Relations 254
End of Chapter 7 Drill 255
8 Integration and Accumulation of Change 257
Exploring Accumulations of Change 258
Approximating Areas with Riemann Sums 259
Riemann Sums, Summation Notation, and Definite Integral Notation 272
The Fundamental Theorem of Calculus and Accumulation Functions 273
Interpreting the Behavior of Accumulation Functions Involving Area 275
Applying Properties of Definite Integrals 279
The Fundamental Theorem of Calculus and Definite Integrals 280
Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation 281
Integrating Using Substitution 290
Integrating Functions Using Long Division and Completing the Square 313
Integration Using Integration by Parts 317
Using Linear Partial Fractions 323
Evaluating Improper Integrals 326
Selecting Techniques for Antidifferentiation 330
End of Chapter 8 Drill 332
9 Differential Equations 335
Modeling Situations with Differential Equations 336
Verifying Solutions for Differential Equations 336
Sketching Slope Fields 337
Reasoning Using Slope Fields 341
Approximating Solutions Using Euler's Method 343
Finding General Solutions Using Separation of Variables 349
Finding Particular Solutions Using Initial Conditions and Separation of Variables 350
Exponential Models with Differential Equations 354
Logistic Models with Differential Equations 356
End of Chapter 9 Drill 360
10 Applications of Integration 361
Finding the Average Value of a Function on an Interval 362
Connecting Position, Velocity, and Acceleration Functions Using Integrals 364
Using Accumulation Functions and Definite Integrals in Applied Contexts 365
Finding the Area Between Curves Expressed as Functions of x 366
Finding the Area Between Curves Expressed as Functions of y 368
Finding the Area Between Curves That Intersect at More Than Two Points 371
Volumes with Cross-Sections: Squares and Rectangles 372
Volumes with Cross-Sections: Triangles and Semicircles 374
Volume with Disc Method: Revolving Around the x- or y-Axis 375
Volume with Disc Method: Revolving Around Other Axes 378
Volume with Washer Method: Revolving Around the x- or y-Axis 379
Volume with Washer Method: Revolving Around Other Axes 382
The Arc Length of a Smooth, Planar Curve and Distance Traveled 385
End of Chapter 10 Drill 388
11 Parametric Equations, Polar Coordinates, and Vector-Valued Functions 391
Defining and Differentiating Parametric Equations 392
Second Derivatives of Parametric Equations 394
Finding Arc Lengths of Curves Given by Parametric Equations 395
Defining and Differentiating Vector-Valued Functions 397
Integrating Vector-Valued Functions 398
Solving Motion Problems Using Parametric and Vector-Valued Functions 400
Defining Polar Coordinates and Differentiating in Polar Form 402
Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve 403
Finding the Area of the Region Bounded by Two Polar Curves 405
End of Chapter 11 Drill 407
12 Infinite Sequences and Series 409
Defining Convergent and Divergent Infinite Series 410
Working with Geometric Series 413
The nth Term Test for Divergence 415
Integral Test for Convergence 416
Harmonic Series and p-Series 417
Comparison Tests for Convergence 419
Alternating Series Test for Convergence 422
Ratio Test for Convergence 423
Determining Absolute or Conditional Convergence 424
Alternating Series Error Bound 425
Finding Taylor Polynomial Approximations of Functions 426
Lagrange Error Bound 428
Radius and Interval of Convergence of Power Series 429
Finding Taylor or Maclaurin Series for a Function 430
Representing Functions as Power Series 432
End of Chapter 12 Drill 437
13 Answers to Practice Problem Sets 439
14 Answers to End of Chapter Drills 601
Part VI Practice Tests 2 and 3 635
15 Practice Test 2 637
16 Practice Test 2: Answers and Explanations 667
How to Score Practice Test 2 691
17 Practice Test 3 693
18 Practice Test 3: Answers and Explanations 725
How to Score Practice Test 3 743
About the Author 745
Practice Test 4 online
Practice Test 4: Answers and Explanations online
Practice Test 5 online
Practice Test 5: Answers and Explanations online
