Table of Contents
Dedication and Acknowledgments xii
Preface xiii
About the Authors xiv
Introduction: The Five-Step Program xv
Step 1 Set Up Your Study Plan
1 What You Need to Know About the AP Calculus BC Exam 3
1.1 What Is Covered on the AP Calculus BC Exam? 4
1.2 What Is the Format of the AP Calculus BC Exam? 4
1.3 What Are the Advanced Placement Exam Grades? 5
How Is the AP Calculus BC Exam Grade Calculated? 5
1.4 Which Graphing Calculators Are Allowed for the Exam? 6
Calculators and Other Devices Not Allowed for the AP Calculus BC Exam 7
Other Restrictions on Calculators 7
2 How to Plan Your Time 8
2.1 Three Approaches to Preparing for the AP Calculus BC Exam 8
Overview of the Three Plans 8
2.2 Calendar for Each Plan 10
Summary of the Three Study Plans 13
Step 2 Determine Your Test Readiness
3 Take a Diagnostic Exam 17
3.1 Getting Started! 21
3.2 Diagnostic Test 21
3.3 Answers to Diagnostic Test 27
3.4 Solutions to Diagnostic Test 28
3.5 Calculate Your Score 38
Short-Answer Questions 38
AP Calculus BC Diagnostic Exam 38
Step 3 Develop Strategies for Success
4 How to Approach Each Question Type 41
4.1 The Multiple-Choice Questions 42
4.2 The Free-Response Questions 42
4.3 Using a Graphing Calculator 43
4.4 Taking the Exam 44
What Do I Need to Bring to the Exam? 44
Tips for Taking the Exam 45
Step 4 Review the Knowledge You Need to Score High
Big Idea 1 Limits
5 Limits and Continuity 49
5.1 The Limit of a Function 50
Definition and Properties of Limits 50
Evaluating Limits 50
One-Sided Limits 52
Squeeze Theorem 55
5.2 Limits Involving Infinities 57
Infinite Limits (as x → a) 57
Limits at Infinity (as x → ±∞) 59
Horizontal and Vertical Asymptotes 61
5.3 Continuity of a Function 64
Continuity of a Function at a Number 64
Continuity of a Function over an Interval 64
Theorems on Continuity 64
5.4 Rapid Review 67
5.5 Practice Problems 69
5.6 Cumulative Review Problems 70
5.7 Solutions to Practice Problems 70
5.8 Solutions to Cumulative Review Problems 73
Big Idea 2 Derivatives
6 Differentiation 75
6.1 Derivatives of Algebraic Functions 76
Definition of the Derivative of a Function 76
Power Rule 79
The Sum, Difference, Product, and Quotient Rules 80
The Chain Rule 81
6.2 Derivatives of Trigonometric, Inverse Trigonometric, Exponential, and Logarithmic Functions 82
Derivatives of Trigonometric Functions 82
Derivatives of Inverse Trigonometric Functions 84
Derivatives of Exponential and Logarithmic Functions 85
6.3 Implicit Differentiation 87
Procedure for Implicit Differentiation 87
6.4 Approximating a Derivative 90
6.5 Derivatives of Inverse Functions 92
6.6 Higher Order Derivatives 94
L'Hôpital's Rule for Indeterminate Forms 95
6.7 Rapid Review 95
6.8 Practice Problems 97
6.9 Cumulative Review Problems 98
6.10 Solutions to Practice Problems 98
6.11 Solutions to Cumulative Review Problems 101
7 Graphs of Functions and Derivatives 103
7.1 Rolle's Theorem, Mean Value Theorem, and Extreme Value Theorem 103
Rolle's Theorem 104
Mean Value Theorem 104
Extreme Value Theorem 107
7.2 Determining the Behavior of Functions 108
Test for Increasing and Decreasing Functions 108
First Derivative Test and Second Derivative Test for Relative Extrema 111
Test for Concavity and Points of Inflection 114
7.3 Sketching the Graphs of Functions 120
Graphing without Calculators 120
Graphing with Calculators 121
7.4 Graphs of Derivatives 123
7.5 Parametric, Polar, and Vector Representations 128
Parametric Curves 128
Polar Equations 129
Types of Polar Graphs 129
Symmetry of Polar Graphs 130
Vectors 131
Vector Arithmetic 132
7.6 Rapid Review 133
7.7 Practice Problems 137
7.8 Cumulative Review Problems 139
7.9 Solutions to Practice Problems 140
7.10 Solutions to Cumulative Review Problems 147
8 Applications of Derivatives 149
8.1 Related Rate 149
General Procedure for Solving Related Rate Problems 149
Common Related Rate Problems 150
Inverted Cone (Water Tank) Problem 151
Shadow Problem 152
Angle of Elevation Problem 153
8.2 Applied Maximum and Minimum Problems 155
General Procedure for Solving Applied Maximum and Minimum Problems 155
Distance Problem 155
Area and Volume Problem 156
Business Problems 159
8.3 Rapid Review 160
8.4 Practice Problems 161
8.5 Cumulative Review Problems 163
8.6 Solutions to Practice Problems 164
8.7 Solutions to Cumulative Review Problems 171
9 More Applications of Derivatives 174
9.1 Tangent and Normal Lines 174
Tangent Lines 174
Normal Lines 180
9.2 Linear Approximations 183
Tangent Line Approximation (or Linear Approximation) 183
Estimating the nth Root of a Number 185
Estimating the Value of a Trigonometric Function of an Angle 185
9.3 Motion Along a Line 186
Instantaneous Velocity and Acceleration 186
Vertical Motion 188
Horizontal Motion 188
9.4 Parametric, Polar, and Vector Derivatives 190
Derivatives of Parametric Equations 190
Position, Speed, and Acceleration 191
Derivatives of Polar Equations 191
Velocity and Acceleration of Vector Functions 192
9.5 Rapid Review 195
9.6 Practice Problems 196
9.7 Cumulative Review Problems 198
9.8 Solutions to Practice Problems 199
9.9 Solutions to Cumulative Review Problems 204
Big Idea 3 Integrals and the Fundamental Theorems of Calculus
10 Integration 207
10.1 Evaluating Basic Integrals 208
Antiderivatives and Integration Formulas 208
Evaluating Integrals 210
10.2 Integration by U-Substitution 213
The U-Substitution Method 213
U-Substitution and Algebraic Functions 213
U-Substitution and Trigonometric Functions 215
U-Substitution and Inverse Trigonometric Functions 216
U-Substitution and Logarithmic and Exponential Functions 218
10.3 Techniques of Integration 221
Integration by Parts 221
Integration by Partial Fractions 222
10.4 Rapid Review 223
10.5 Practice Problems 224
10.6 Cumulative Review Problems 225
10.7 Solutions to Practice Problems 226
10.8 Solutions to Cumulative Review Problems 229
11 Definite Integrals 231
11.1 Riemann Sums and Definite Integrals 232
Sigma Notation or Summation Notation 232
Definition of a Riemann Sum 233
Definition of a Definite Integral 234
Properties of Definite Integrals 235
11.2 Fundamental Theorems of Calculus 237
First Fundamental Theorem of Calculus 237
Second Fundamental Theorem of Calculus 238
11.3 Evaluating Definite Integrals 241
Definite Integrals Involving Algebraic Functions 241
Definite Integrals Involving Absolute Value 242
Definite Integrals Involving Trigonometric, Logarithmic, and Exponential Functions 243
Definite Integrals Involving Odd and Even Functions 245
11.4 Improper Integrals 246
Infinite Intervals of Integration 246
Infinite Discontinuities 247
11.5 Rapid Review 248
11.6 Practice Problems 249
11.7 Cumulative Review Problems 250
11.8 Solutions to Practice Problems 251
11.9 Solutions to Cumulative Review Problems 254
12 Areas, Volumes, and Arc Lengths 257
12.1 The Function F(x) = ∫xa f(t)dt 258
12.2 Approximating the Area Under a Curve 262
Rectangular Approximations 262
Trapezoidal Approximations 266
12.3 Area and Definite Integrals 267
Area Under a Curve 267
Area Between Two Curves 272
12.4 Volumes and Definite Integrals 276
Solids with Known Cross Sections 276
The Disc Method 280
The Washer Method 285
12.5 Integration of Parametric, Polar, and Vector Curves 289
Area, Arc Length, and Surface Area for Parametric Curves 289
Area and Arc Length for Polar Curves 290
Integration of a Vector-Valued Function 291
12.6 Rapid Review 292
12.7 Practice Problems 295
12.8 Cumulative Review Problems 296
12.9 Solutions to Practice Problems 297
12.10 Solutions to Cumulative Review Problems 305
13 More Applications of Definite Integrals 309
13.1 Average Value of a Function 310
Mean Value Theorem for Integrals 310
Average Value of a Function on [a, b] 311
13.2 Distance Traveled Problems 313
13.3 Definite Integral as Accumulated Change 316
Business Problems 316
Temperature Problem 317
Leakage Problem 318
Growth Problem 318
13.4 Differential Equations 319
Exponential Growth/Decay Problems 319
Separable Differential Equations 321
13.5 Slope Fields 324
13.6 Logistic Differential Equations 328
13.7 Euler's Method 330
Approximating Solutions of Differential Equations by Euler's Method 330
13.8 Rapid Review 332
13.9 Practice Problems 334
13.10 Cumulative Review Problems 336
13.11 Solutions to Practice Problems 337
13.12 Solutions to Cumulative Review Problems 343
Big Idea 4 Series
14 Series 346
14.1 Sequences and Series 347
Convergence 347
14.2 Types of Series 348
p-Series 348
Harmonic Series 348
Geometric Series 348
Decimal Expansion 349
14.3 Convergence Tests 350
Divergence Test 350
Integral Test 350
Ratio Test 351
Comparison Test 352
Limit Comparison Test 352
Informal Principle 353
14.4 Alternating Series 354
Error Bound 354
Absolute and Conditional Convergence 355
14.5 Power Series 357
Radius and Interval of Convergence 357
14.6 Taylor Series 358
Taylor Series and MacLaurin Series 358
Common MacLaurin Series 359
14.7 Operations on Series 359
Substitution 359
Differentiation and Integration 360
Error Bounds 361
14.8 Rapid Review 362
14.9 Practice Problems 364
14.10 Cumulative Review Problems 365
14.11 Solutions to Practice Problems 365
14.12 Solutions to Cumulative Review Problems 369
Step 5 Build Your Test-Taking Confidence
AP Calculus BC Practice Exam 1 373
AP Calculus BC Practice Exam 2 403
Elite Student Edition 5 Minutes to a 5
180 Activities and Questions in 5 Minutes a Day 433
Formulas and Theorems 687
Bibliography 695
