How to Use This Book ix
Barron's Essential 5 x
Introduction 1
The Courses 1
Topic Outline for the AB and BC Calculus Exams 1
The Examinations 7
The Graphing Calculator: Using Your Graphing Calculator on the AP Exam 8
Grading the Examinations 12
Diagnostic Tests
Diagnostic Test Calculus AB 17
Diagnostic Test Calculus BC 41
Topical Review and Practice
1. Functions 63
A Definitions 63
B Special Functions 66
C Polynomial and Other Rational Functions 69
D Trigonometric Functions 69
E Exponential and Logarithmic Functions 72
F Parametrically Defined Functions 73
G Polar Functions 76
Practice Exercises 79
2. Limits and Continuity 89
A Definitions and Examples 89
B Asymptotes 94
C Theorems on Limits 96
D Limit of a Quotient of Polynomials 98
E Other Basic Limits 99
F Continuity 99
Practice Exercises 105
3. Differentiation 113
A Definition of Derivative 113
B Formulas 115
C The Chain Rule: The Derivative of a Composite Function 116
D Differentiability and Continuity 121
E Estimating a Derivative 122
E1 Numerically 122
E2 Graphically 125
F Derivatives of Parametrically Defined Functions 126
G Implicit Differentiation 128
H Derivative of the Inverse of a Function 130
I The Mean Value Theorem 131
J Indeterminate Forms and L'Hospital's Rule 133
K Recognizing a Given Limit as a Derivative 136
Practice Exercises 139
4. Applications of Differential Calculus 159
A Slope; Critical Points 159
B Tangents to a Curve 161
C Increasing and Decreasing Functions 163
Case I Functions with Continuous Derivatives 163
Case II Functions Whose Derivatives Have Discontinuities 163
D Maximum, Minimum, Concavity, and Inflection Points: Definitions 164
E Maximum, Minimum, and Inflection Points: Curve Sketching 165
Case I Functions That Are Everywhere Differentiable 165
Case II Functions Whose Derivatives May Not Exist Everywhere 169
F Global Maximum or Minimum 171
Case I Differentiable Functions 171
Case II Functions That Are Not Everywhere Differentiable 171
G Further Aids in Sketching 171
H Optimization: Problems Involving Maxima and Minima 173
I Relating a Function and Its Derivatives Graphically 177
J Motion Along a Line 180
K Motion Along a Curve: Velocity and Acceleration Vectors 182
L Tangent-Line Approximations 186
M Related Rates 188
N Slope of a Polar Curve 190
Practice Exercises 193
5. Antidifferentiation 213
A Antiderivatives 213
B Basic Formulas 213
C Integration by Partial Fractions 220
D Integration by Parts 221
E Applications of Antiderivatives; Differential Equations 224
Practice Exercises 227
6. Definite Integrals 241
A Fundamental Theorem of Calculus (FTC); Evaluation of Definite integrals 241
B Properties of Definite Integrals 241
C Definition of Definite integral as the Limit of a Riemann Sum 246
D The Fundamental Theorem Again 247
E Approximations of the Definite Integral; Riemann Sums 248
E1 Using Rectangles 248
E2 Using Trapezoids 250
E3 Comparing Approximating Sums 252
F Graphing a Function from Its Derivative; Another Look 253
G Interpreting In x as an Area 260
H Average Value 261
Practice Exercises 270
7. Applications of Integration to Geometry 283
A Area 283
A1 Area Between Curves 285
A2 Using Symmetry 286
A3 Region Bounded by Polar Curve 288
B Volume 290
B1 Solids with Known Cross Sections 290
B2 Solids of Revolution 292
C Length of Curve (Arc Length) 297
D Improper Integrals 299
Practice Exercises 309
8. Further Applications of Integration 337
A Motion Along a Straight Line 337
B Motion Along a Plane Curve 339
C Other Applications of Riemann Sums 342
D FTC: Definite Integral of a Rate Is Net Change 344
Practice Exercises 346
9. Differential Equations 357
A Basic Definitions 357
B Slope Fields 358
C Euler's Method 363
D Solving First-Order Differential Equations Analytically 367
E Exponential Growth and Decay 369
Case I Exponential Growth 369
Case II Restricted Growth 373
Case III Logistic Growth 376
Practice Exercises 381
10. Sequences and Series 399
A Sequences of Real Numbers 399
B Infinite Series 400
B1 Definitions 400
B2 Theorems About Convergence or Divergence of Infinite Series 402
B3 Tests for Convergence of Infinite Series 403
B4 Tests for Convergence of Nonnegative Series 404
B5 Alternating Series and Absolute Convergence 408
C Power Series 411
C1 Definitions; Convergence 411
C2 Functions Defined by Power Series 413
C3 Finding a Power Series for a Function: Taylor and Maclaurin Series 415
C4 Approximating Functions with Taylor and Maclaurin Polynomials 418
C5 Taylor's Formula with Remainder; Lagrange Error Bound 423
C6 Computations with Power Series 425
C7 Power Series over Complex Numbers 429
Practice Exercises 431
11. Miscellaneous Multiple-Choice Practice Questions 443
12. Miscellaneous Free-Response Practice Exercises 475
AB Practice Tests
AB Practice Test 1 505
AB Practice Test 2 529
AB Practice Test 3 555
BC Practice Tests
BC Practice Test 1 583
BC Practice Test 2 605
BC Practice Test 3 625
Appendix: Formulas and Theorems for Reference 645
Index 653