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STEP 1 Set Up Your Study Plan 1 What You Need to Know About the AP Calculus AB Exam 3 1.1 What Is Covered on the AP Calculus Exam? 4 1.2 What Is the Format of the AP Calculus AB Exam? 4 1.3 What Are the Advanced Placement Exam Grades? 5 How Is the AP Calculus AB 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 AB Exam 7 Other Restrictions on Calculators 7 2 How to Plan Your Time 8 2.1 Three Approaches to Preparing for the AP Calculus AB 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! 20 3.2 Diagnostic Test 20 3.3 Answers to Diagnostic Test 26 3.4 Solutions to Diagnostic Test 27 3.5 Calculate Your Score 35 Short-Answer Questions 35 AP Calculus AB Diagnostic Exam 35 STEP 3 Develop Strategies for Success 4 How to Approach Each Question Type 39 4.1 The Multiple-Choice Questions 40 4.2 The Free-Response Questions 40 4.3 Using a Graphing Calculator 41 4.4 Taking the Exam 42 What Do I Need to Bring to the Exam? 42 Tips for Taking the Exam 43 STEP 4 Review the Knowledge You Need to Score High 5 Review of Precalculus 47 5.1 Lines 48 Slope of a Line 48 Equations of a Line 48 Parallel and Perpendicular Lines 49 5.2 Absolute Values and Inequalities 52 Absolute Values 52 Inequalities and the Real Number Line 53 Solving Absolute Value Inequalities 54 Solving Polynomial Inequalities 55 Solving Rational Inequalities 57 5.3 Functions 59 Definition of a Function 59 Operations on Functions 60 Inverse Functions 62 Trigonometric and Inverse Trigonometric Functions 65 Exponential and Logarithmic Functions 68 5.4 Graphs of Functions 72 Increasing and Decreasing Functions 72 Intercepts and Zeros 74 Odd and Even Functions 75 Shifting, Reflecting, and Stretching Graphs 77 5.5 Rapid Review 80 5.6 Practice Problems 81 5.7 Cumulative Review Problems 82 5.8 Solutions to Practice Problems 82 5.9 Solutions to Cumulative Review Problems 85 6 Limits and Continuity 86 6.1 The Limit of a Function 87 Definition and Properties of Limits 87 Evaluating Limits 87 One-Sided Limits 89 Squeeze Theorem 92 6.2 Limits Involving Infinities 94 Infinite Limits (as x → a) 94 Limits at Infinity (as x → ±∞) 96 Horizontal and Vertical Asymptotes 98 6.3 Continuity of a Function 101 Continuity of a Function at a Number 101 Continuity of a Function over an Interval 101 Theorems on Continuity 101 6.4 Rapid Review 104 6.5 Practice Problems 105 6.6 Cumulative Review Problems 106 6.7 Solutions to Practice Problems 107 6.8 Solutions to Cumulative Review Problems 109 7 Differentiation 111 7.1 Derivatives of Algebraic Functions 112 Definition of the Derivative of a Function 112 Power Rule 115 The Sum, Difference, Product, and Quotient Rules 116 The Chain Rule 117 7.2 Derivatives of Trigonometric, Inverse Trigonometric, Exponential, and Logarithmic Functions 118 Derivatives of Trigonometric Functions 118 Derivatives of Inverse Trigonometric Functions 120 Derivatives of Exponential and Logarithmic Functions 121 7.3 Implicit Differentiation 123 Procedure for Implicit Differentiation 123 7.4 Approximating a Derivative 126 7.5 Derivatives of Inverse Functions 128 7.6 Higher Order Derivatives 130 7.7 Rapid Review 131 7.8 Practice Problems 132 7.9 Cumulative Review Problems 132 7.10 Solutions to Practice Problems 133 7.11 Solutions to Cumulative Review Problems 136 8 Graphs of Functions and Derivatives 138 8.1 Rolle’s Theorem, Mean Value Theorem, and Extreme Value Theorem 138 Rolle’s Theorem 139 Mean Value Theorem 139 Extreme Value Theorem 142 8.2 Determining the Behavior of Functions 143 Test for Increasing and Decreasing Functions 143 First Derivative Test and Second Derivative Test for Relative Extrema 146 Test for Concavity and Points of Inflection 149 8.3 Sketching the Graphs of Functions 155 Graphing without Calculators 155 Graphing with Calculators 156 8.4 Graphs of Derivatives 158 8.5 Rapid Review 163 8.6 Practice Problems 165 8.7 Cumulative Review Problems 168 8.8 Solutions to Practice Problems 168 8.9 Solutions to Cumulative Review Problems 175 9 Applications of Derivatives 177 9.1 Related Rate 177 General Procedure for Solving Related Rate Problems 177 Common Related Rate Problems 178 Inverted Cone (Water Tank) Problem 179 Shadow Problem 180 Angle of Elevation Problem 181 9.2 Applied Maximum and Minimum Problems 183 General Procedure for Solving Applied Maximum and Minimum Problems 183 Distance Problem 183 Area and Volume Problems 184 Business Problems 187 9.3 Rapid Review 188 9.4 Practice Problems 189 9.5 Cumulative Review Problems 191 9.6 Solutions to Practice Problems 192 9.7 Solutions to Cumulative Review Problems 199 10 More Applications of Derivatives 202 10.1 Tangent and Normal Lines 202 Tangent Lines 202 Normal Lines 208 10.2 Linear Approximations 211 Tangent Line Approximation (or Linear Approximation) 211 Estimating the nth Root of a Number 213 Estimating the Value of a Trigonometric Function of an Angle 213 10.3 Motion Along a Line 214 Instantaneous Velocity and Acceleration 214 Vertical Motion 216 Horizontal Motion 216 10.4 Rapid Review 218 10.5 Practice Problems 219 10.6 Cumulative Review Problems 220 10.7 Solutions to Practice Problems 221 10.8 Solutions to Cumulative Review Problems 225 11 Integration 227 11.1 Evaluating Basic Integrals 228 Antiderivatives and Integration Formulas 228 Evaluating Integrals 230 11.2 Integration by U-Substitution 233 The U-Substitution Method 233 U-Substitution and Algebraic Functions 233 U-Substitution and Trigonometric Functions 235 U-Substitution and Inverse Trigonometric Functions 236 U-Substitution and Logarithmic and Exponential Functions 238 11.3 Rapid Review 241 11.4 Practice Problems 242 11.5 Cumulative Review Problems 243 11.6 Solutions to Practice Problems 244 11.7 Solutions to Cumulative Review Problems 246 12 Definite Integrals 247 12.1 Riemann Sums and Definite Integrals 248 Sigma Notation or Summation Notation 248 Definition of a Riemann Sum 249 Definition of a Definite Integral 250 Properties of Definite Integrals 251 12.2 Fundamental Theorems of Calculus 253 First Fundamental Theorem of Calculus 253 Second Fundamental Theorem of Calculus 254 12.3 Evaluating Definite Integrals 257 Definite Integrals Involving Algebraic Functions 257 Definite Integrals Involving Absolute Value 258 Definite Integrals Involving Trigonometric, Logarithmic, and Exponential Functions 259 Definite Integrals Involving Odd and Even Functions 261 12.4 Rapid Review 262 12.5 Practice Problems 263 12.6 Cumulative Review Problems 264 12.7 Solutions to Practice Problems 265 12.8 Solutions to Cumulative Review Problems 268 13 Areas and Volumes 270 13.1 The Function F(x) =fxaf (t)dt 271 13.2 Approximating the Area Under a Curve 275 Rectangular Approximations 275 Trapezoidal Approximations 279 13.3 Area and Definite Integrals 280 Area Under a Curve 280 Area Between Two Curves 285 13.4 Volumes and Definite Integrals 289 Solids with Known Cross Sections 289 The Disc Method 293 The Washer Method 298 13.5 Rapid Review 301 13.6 Practice Problems 303 13.7 Cumulative Review Problems 305 13.8 Solutions to Practice Problems 305 13.9 Solutions to Cumulative Review Problems 312 14 More Applications of Definite Integrals 315 14.1 Average Value of a Function 316 Mean Value Theorem for Integrals 316 Average Value of a Function on [a, b] 317 14.2 Distance Traveled Problems 319 14.3 Definite Integral as Accumulated Change 322 Business Problems 322 Temperature Problem 323 Leakage Problems 324 Growth Problem 324 14.4 Differential Equations 325 Exponential Growth/Decay Problems 325 Separable Differential Equations 327 14.5 Slope Fields 330 14.6 Rapid Review 334 14.7 Practice Problems 335 14.8 Cumulative Review Problems 337 14.9 Solutions to Practice Problems 338 14.10 Solutions to Cumulative Review Problems 342 STEP 5 Build Your Test-Taking Confidence AP Calculus AB Practice Exam 1 347 AP Calculus AB Practice Exam 2 373 AP Calculus AB Practice Exam 3 401 Appendix 427 Bibliography and Websites 431