| How to Use This Book | ix |
| Barron's Essential 5 | x |
| Introduction | 1 |
| Content Areas | 1 |
| Exam Format | 1 |
| Scoring of the Exams | 2 |
| Using Your Graphing Calculator on the AP Exam | 2 |
| Diagnostic Tests | |
| Diagnostic Test Calculus AB | 11 |
| Answer Explanations | 21 |
| Diagnostic Test Calculus BC | 35 |
| Answer Explanations | 46 |
| Topical Review and Practice | |
| Functions | 53 |
| Definitions | 53 |
| Special Functions | 56 |
| Polynomial and Other Rational Functions | 59 |
| Trigonometric Functions | 59 |
| Exponential and Logarithmic Functions | 61 |
| Parametrically Defined Functions | 62 |
| Polar Functions | 66 |
| Practice Exercises | 68 |
| Answer Explanations | 72 |
| Limits and Continuity | 75 |
| Definitions and Examples | 75 |
| Asymptotes | 80 |
| Theorems on Limits | 81 |
| Limit of a Quotient of Polynomials | 83 |
| Other Basic Limits | 84 |
| Continuity | 85 |
| Practice Exercises | 90 |
| Answer Explanations | 95 |
| Differentiation | 97 |
| Definition of Derivative | 97 |
| Formulas | 99 |
| The Chain Rule: The Derivative of a Composite Function | 100 |
| Differentiability and Continuity | 105 |
| Estimating a Derivative | 106 |
| Numerically | 106 |
| Graphically | 108 |
| Derivatives of Parametrically Defined Functions | 109 |
| Implicit Differentiation | 111 |
| Derivative of the Inverse of a Function | 113 |
| The Mean Value Theorem | 114 |
| Indeterminate Forms and L'Hospital's Rule | 116 |
| Recognizing a Given Limit as a Derivative | 119 |
| Practice Exercises | 121 |
| Answer Explanations | 132 |
| Applications of Differential Calculus | 139 |
| Slope; Critical Points | 139 |
| Tangents to a Curve | 141 |
| Increasing and Decreasing Functions | 143 |
| Functions with Continuous Derivatives | 143 |
| Functions Whose Derivatives Have Discontinuities | 143 |
| Maximum, Minimum, Concavity, and Inflection Points: Definitions | 144 |
| Maximum, Minimum, and Inflection Points: Curve Sketching | 145 |
| Functions That Are Everywhere Differentiable | 145 |
| Functions Whose Derivatives May Not Exist Everywhere | 149 |
| Global Maximum or Minimum | 151 |
| Differentiable Functions | 151 |
| Functions That Are Not Everywhere Differentiable | 151 |
| Further Aids in Sketching | 151 |
| Optimization: Problems Involving Maxima and Minima | 153 |
| Relating a Function and Its Derivatives Graphically | 157 |
| Motion Along a Line | 160 |
| Motion Along a Curve: Velocity and Acceleration Vectors | 162 |
| Tangent-Line Approximations | 166 |
| Related Rates | 168 |
| Slope of a Polar Curve | 170 |
| Practice Exercises | 173 |
| Answer Explanations | 184 |
| Antidifferentiation | 191 |
| Antiderivatives | 191 |
| Basic Formulas | 191 |
| Integration by Partial Fractions | 198 |
| Integration by Parts | 199 |
| Applications of Antiderivatives; Differential Equations | 202 |
| Practice Exercises | 204 |
| Answer Explanations | 212 |
| Definite Integrals | 217 |
| Fundamental Theorem of Calculus (FTC); Evaluation of Definite integrals | 217 |
| Properties of Definite Integrals | 217 |
| Definition of Definite Integral as the Limit of a Riemann Sum | 222 |
| The Fundamental Theorem Again | 222 |
| Approximations of the Definite Integral; Riemann Sums | 224 |
| Using Rectangles | 224 |
| Using Trapezoids | 225 |
| Comparing Approximating Sums | 227 |
| Graphing a Function from Its Derivative; Another Look | 229 |
| Interpreting In x as an Area | 235 |
| Average Value | 237 |
| Practice Exercises | 244 |
| Answer Explanations | 251 |
| Applications of Integration to Geometry | 255 |
| Area | 255 |
| Area Between Curves | 257 |
| Using Symmetry | 258 |
| Region Bounded by Polar Curve | 260 |
| Volume | 262 |
| Solids with Known Cross Sections | 262 |
| Solids of Revolution | 264 |
| Length of Curve (Arc Length) | 269 |
| Improper integrals | 271 |
| Practice Exercises | 281 |
| Answer Explanations | 289 |
| Further Applications of Integration | 307 |
| Motion Along a Straight Line | 307 |
| Motion Along a Plane Curve | 309 |
| Other Applications of Riemann Sums | 312 |
| FTC: Definite Integral of a Rate Is Net Change | 313 |
| Practice Exercises | 316 |
| Answer Explanations | 321 |
| Differential Equations | 325 |
| Basic Definitions | 325 |
| Slope Fields | 326 |
| Euler's Method | 331 |
| Solving First-Order Differential Equations Analytically | 335 |
| Exponential Growth and Decay | 337 |
| Exponential Growth | 337 |
| Restricted Growth | 341 |
| Logistic Growth | 343 |
| Practice Exercises | 348 |
| Answer Explanations | 356 |
| Sequences and Series | 361 |
| Sequences of Real Numbers | 361 |
| Infinite Series | 362 |
| Definitions | 362 |
| Theorems About Convergence or Divergence of Infinite Series | 364 |
| Tests for Convergence of Infinite Series | 365 |
| Tests for Convergence of Nonnegative Series | 366 |
| Alternating Series and Absolute Convergence | 370 |
| Power Series | 373 |
| Definitions; Convergence | 373 |
| Functions Defined by Power Series | 375 |
| Finding a Power Series for a Function: Taylor and Maclaurin Series | 377 |
| Approximating Functions with Taylor and Maclaurin Polynomials | 380 |
| Taylor's Formula with Remainder; Lagrange Error Bound | 384 |
| Computations with Power Series | 386 |
| Power Series over Complex Numbers | 390 |
| Practice Exercises | 391 |
| Answer Explanations | 397 |
| Miscellaneous Multiple-Choice Practice Questions | 401 |
| Answer Explanations | 416 |
| Miscellaneous Free-Response Practice Exercises | 429 |
| Answer Explanations | 436 |
| AB Practice Tests | |
| AB Practice Test 1 | 453 |
| Answer Explanations | 464 |
| AB Practice Test 2 | 477 |
| Answer Explanations | 488 |
| BC Practice Tests | |
| BC Practice Test 1 | 503 |
| Answer Explanations | 514 |
| BC Practice Test 2 | 525 |
| Answer Explanations | 536 |
| Appendix: Formulas and Theorems for Reference | 543 |
| Index | 551 |
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