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STEP 1 Set Up Your Study Plan 3

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 17

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 41

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 49

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

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

6.7 Indeterminate Forms 95

L'H？pital's Rule for Indeterminate Forms 95

6.8 Rapid Review 95

6.9 Practice Problems 97

6.10 Cumulative Review Problems 98

6.11 Solutions to Practice Problems 98

6.12 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

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 and Volumes 257

12.1 The Function F（x）=？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 Problems 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

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

Integral Test 350

Ratio Test 351

Comparison Test 351

Limit Comparison Test 352

14.4 Alternating Series 353

Error Bound 354

Absolute Convergence 354

14.5 Power Series 354

Radius and Interval of Convergence 355

14.6 Taylor Series 355

Taylor Series and MacLaurin Series 355

Common MacLaurin Series 357

14.7 Operations on Series 357

Substitution 357

Differentiation and Integration 358

Error Bounds 359

14.8 Rapid Review 360

14.9 Practice Problems 362

14.10 Cumulative Review Problems 363

14.11 Solutions to Practice Problems 363

14.12 Solutions to Cumulative Review Problems 366

STEP 5 Build Your Test-Taking Confidence 371

AP Calculus BC Practice Exam 1 371

AP Calculus BC Practice Exam 2 401

AP Calculus BC Practice Exam 3 433

Formulas and Theorems 463

Bibliography and Websites 471