The AP Calculus Problem Book - crunchy math
[Pages:354]The AP Calculus Problem Book
Chuck Garner, Ph.D.
Dedicated to the students who used previous editions of this book!
BC Class of 2003 Will Andersen Kenny Baskett Amanda Dugan
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BC Class of 2004 Anita Amin
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Megan Villanueva
BC Class of 2005 Jonathan Andersen
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Shonette McCalmon Ansley Mitcham Tesia Olgetree Lauren Powell Xan Reynolds Lauren Stewart Matt Wann Ryan Young
The AP Calculus Problem Book
Chuck Garner, Ph.D. Rockdale Magnet School for Science and Technology
Fourth Edition, Revised and Corrected, 2008
The AP Calculus Problem Book
Publication history: First edition, 2002 Second edition, 2003 Third edition, 2004 Third edition Revised and Corrected, 2005 Fourth edition, 2006, Edited by Amy Lanchester Fourth edition Revised and Corrected, 2007 Fourth edition, Corrected, 2008
This book was produced directly from the author's LATEX files. Figures were drawn by the author using the TEXdraw package. TI-Calculator screen-shots produced by a TI-83Plus calculator using a TI-Graph Link.
LATEX (pronounced "Lay-Tek") is a document typesetting program (not a word processor) that is available free from , which also includes TEXnicCenter, a free and easy-to-use user-interface.
Contents
1 LIMITS
7
1.1 Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 The Slippery Slope of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 The Power of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Functions Behaving Badly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Take It to the Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 One-Sided Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 One-Sided Limits (Again) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8 Limits Determined by Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.9 Limits Determined by Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.10 The Possibilities Are Limitless... . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.11 Average Rates of Change: Episode I . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.12 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . 18
1.13 Average Rates of Change: Episode II . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.14 Take It To the Limit--One More Time . . . . . . . . . . . . . . . . . . . . . . . . 20
1.15 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.16 Continuously Considering Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.17 Have You Reached the Limit? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.18 Multiple Choice Questions on Limits . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.19 Sample A.P. Problems on Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Last Year's Limits Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 DERIVATIVES
35
2.1 Negative and Fractional Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Logically Thinking About Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 The Derivative By Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Going Off on a Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5 Six Derivative Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.6 Trigonometry: a Refresher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1
2
The AP CALCULUS PROBLEM BOOK
2.7 Continuity and Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.8 The RULES: Power Product Quotient Chain . . . . . . . . . . . . . . . . . . . . 43 2.9 Trigonometric Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.10 Tangents, Normals, and Continuity (Revisited) . . . . . . . . . . . . . . . . . . . 45 2.11 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.12 The Return of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.13 Meet the Rates (They're Related) . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.14 Rates Related to the Previous Page . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.15 Excitement with Derivatives! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.16 Derivatives of Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.17 D?eriv?e, Derivado, Ableitung, Derivative . . . . . . . . . . . . . . . . . . . . . . . 52 2.18 Sample A.P. Problems on Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.19 Multiple-Choice Problems on Derivatives . . . . . . . . . . . . . . . . . . . . . . . 56 Last Year's Derivatives Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3 APPLICATIONS of DERIVATIVES
67
3.1 The Extreme Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2 Rolle to the Extreme with the Mean Value Theorem . . . . . . . . . . . . . . . . 69
3.3 The First and Second Derivative Tests . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Derivatives and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Two Derivative Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6 Sketching Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.7 Problems of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.8 Maximize or Minimize? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.9 More Tangents and Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.10 More Excitement with Derivatives! . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.11 Bodies, Particles, Rockets, Trucks, and Canals . . . . . . . . . . . . . . . . . . . 82
3.12 Even More Excitement with Derivatives! . . . . . . . . . . . . . . . . . . . . . . . 84
3.13 Sample A.P. Problems on Applications of Derivatives . . . . . . . . . . . . . . . . 86
3.14 Multiple-Choice Problems on Applications of Derivatives . . . . . . . . . . . . . . 89
Last Year's Applications of Derivatives Test . . . . . . . . . . . . . . . . . . . . . . . . 92
4 INTEGRALS
101
4.1 The ANTIderivative! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2 Derivative Rules Backwards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 The Method of Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4 Using Geometry for Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . 105
4.5 Some Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.6 The MVT and the FTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.7 The FTC, Graphically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.8 Definite and Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.9 Integrals Involving Logarithms and Exponentials . . . . . . . . . . . . . . . . . . 110
4.10 It Wouldn't Be Called the Fundamental Theorem If It Wasn't Fundamental . . . 111
4.11 Definite and Indefinite Integrals Part 2 . . . . . . . . . . . . . . . . . . . . . . . . 113
4.12 Regarding Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.13 Definitely Exciting Definite Integrals! . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.14 How Do I Find the Area Under Thy Curve? Let Me Count the Ways... . . . . . . 117
4.15 Three Integral Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
CONTENTS
3
4.16 Trapezoid and Simpson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.17 Properties of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.18 Sample A.P. Problems on Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.19 Multiple Choice Problems on Integrals . . . . . . . . . . . . . . . . . . . . . . . . 124 Last Year's Integrals Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 APPLICATIONS of INTEGRALS
135
5.1 Volumes of Solids with Defined Cross-Sections . . . . . . . . . . . . . . . . . . . . 136
5.2 Turn Up the Volume! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.3 Volume and Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4 Differential Equations, Part One . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.5 The Logistic Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.6 Differential Equations, Part Two . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.7 Slope Fields and Euler's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.8 Differential Equations, Part Three . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.9 Sample A.P. Problems on Applications of Integrals . . . . . . . . . . . . . . . . . 144
5.10 Multiple Choice Problems on Application of Integrals . . . . . . . . . . . . . . . 147
Last Year's Applications of Integrals Test . . . . . . . . . . . . . . . . . . . . . . . . . 150
6 TECHNIQUES of INTEGRATION
159
6.1 A Part, And Yet, Apart... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.2 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.3 Trigonometric Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.4 Four Integral Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.5 L'H^opital's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.6 Improper Integrals! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.7 The Art of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.8 Functions Defined By Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.9 Sample A.P. Problems on Techniques of Integration . . . . . . . . . . . . . . . . 170
6.10 Sample Multiple-Choice Problems on Techniques of Integration . . . . . . . . . . 173
Last Year's Techniques of Integration Test . . . . . . . . . . . . . . . . . . . . . . . . . 175
7 SERIES, VECTORS, PARAMETRICS and POLAR
183
7.1 Sequences: Bounded and Unbounded . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.2 It is a Question of Convergence... . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.3 Infinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.4 Tests for Convergence and Divergence . . . . . . . . . . . . . . . . . . . . . . . . 187
7.5 More Questions of Convergence... . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.6 Power Series! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.7 Maclaurin Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.8 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.9 Vector Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.10 Calculus with Vectors and Parametrics . . . . . . . . . . . . . . . . . . . . . . . . 193
7.11 Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.12 Motion Problems with Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.13 Polar Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.14 Differentiation (Slope) and Integration (Area) in Polar . . . . . . . . . . . . . . . 197
7.15 Sample A.P. Problems on Series, Vectors, Parametrics, and Polar . . . . . . . . . 198
4
The AP CALCULUS PROBLEM BOOK
7.16 Sample Multiple-Choice Problems on Series, Vectors, Parametrics, and Polar . . 201 Last Year's Series, Vectors, Parametrics, and Polar Test . . . . . . . . . . . . . . . . . 203
8 AFTER THE A.P. EXAM
211
8.1 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.2 Surface Area of a Solid of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.3 Linear First Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . 214
8.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.5 Newton's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
9 PRACTICE and REVIEW
217
9.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
9.2 Derivative Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
9.3 Can You Stand All These Exciting Derivatives? . . . . . . . . . . . . . . . . . . . 220
9.4 Different Differentiation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9.5 Integrals... Again! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
9.6 Int?egrale, Integrale, Integraal, Integral . . . . . . . . . . . . . . . . . . . . . . . . 225
9.7 Calculus Is an Integral Part of Your Life . . . . . . . . . . . . . . . . . . . . . . . 226
9.8 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
9.9 Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
9.10 The Deadly Dozen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
9.11 Two Volumes and Two Differential Equations . . . . . . . . . . . . . . . . . . . . 230
9.12 Differential Equations, Part Four . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
9.13 More Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.14 Definite Integrals Requiring Definite Thought . . . . . . . . . . . . . . . . . . . . 233
9.15 Just When You Thought Your Problems Were Over... . . . . . . . . . . . . . . . 234
9.16 Interesting Integral Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
9.17 Infinitely Interesting Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . 238
9.18 Getting Serious About Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
9.19 A Series of Series Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
10 GROUP INVESTIGATIONS
241
About the Group Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
10.1 Finding the Most Economical Speed for Trucks . . . . . . . . . . . . . . . . . . . 243
10.2 Minimizing the Area Between a Graph and Its Tangent . . . . . . . . . . . . . . 243
10.3 The Ice Cream Cone Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.4 Designer Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
10.5 Inventory Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
10.6 Optimal Design of a Steel Drum . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
11 CALCULUS LABS
247
About the Labs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
1: The Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
2: Local Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
3: Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
4: A Function and Its Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
5: Riemann Sums and Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
6: Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
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