The AP Calculus Problem Book - Mater Academy Charter Middle/ High

216 Pages·2012·2.27 MB·English

The AP Calculus
Problem Book
￿
Chuck Garner, Ph.D.
TheAPCalculusProblemBook
Publicationhistory:
Firstedition,2002
Secondedition,2003
Thirdedition,2004
ThirdeditionRevisedandCorrected,2005
Fourthedition,2006,EditedbyAmyLanchester
FourtheditionRevisedandCorrected,2007
Fourthedition,Corrected,2008
Thisbookwasproduceddirectlyfromtheauthor’sLATEXfiles.
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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’Hˆopital’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
CONTENTS 5
7: Indeterminate Limits and l’Hˆopital’s Rule . . . . . . . . . . . . . . . . . . . . . . . 267
8: Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
9: Approximating Functions by Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 272
10: Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
12 TI-CALCULATOR LABS 277
Before You Start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
1: Useful Stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
2: Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
3: Maxima, Minima, Inflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
4: Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
5: Approximating Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
6: Approximating Integrals II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
7: Applications of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
8: Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
9: Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
13 CHALLENGE PROBLEMS 295
Set A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Set B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Set C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Set D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
Set E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Set F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
A FORMULAS 309
Formulas from Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
B SUCCESS IN MATHEMATICS 315
Calculus BC Syllabus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
C ANSWERS 329
Answers to Last Year’s Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
6 The AP CALCULUS PROBLEM BOOK
1
CHAPTER
LIMITS
7
8 The AP CALCULUS PROBLEM BOOK
1.1 Graphs of Functions
Describe the graphs of each of the following functions using only one of the
following terms: line, parabola, cubic, hyperbola, semicircle.
1. y = x3+5x2 x 1 7. y = −3
− −
x 5
1 −
2. y = x 8. y = 9 x2
−
3. y = 3x+2 9. y = 3x3
−
4. y = x3+500x 10. y =34x 52
− −
5. y = √9 x2 11. y =34x2 52
− −
6. y = x2+4 12. y =√1 x2
−
Graph the following functions on your calculator on the window 3 x 3,
− ≤ ≤
2 y 2. Sketch what you see. Choose one of the following to describe what
− ≤ ≤
happens to the graph at the origin: A) goes vertical; B) forms a cusp; C) goes
horizontal; or D) stops at zero.
13. y = x1/3 17. y =x1/4
14. y = x2/3 18. y =x5/4
15. y = x4/3 19. y =x1/5
16. y = x5/3 20. y =x2/5
21. Based on the answers from the problems above, finda pattern for the behavior of functions
with exponents of the following forms: xeven/odd, xodd/odd, xodd/even.
Graph the following functions on your calculator in the standard window and
sketch what you see. At what value(s) of x are the functions equal to zero?
22. y = x 1 25. y = 4+x2
| − | | |
23. y = x2 4 26. y = x3 8
| − | | |−
24. y = x3 8 27. y = x2 4x 5
| − | | − − |
Inthecompanyoffriends,writerscandiscusstheirbooks,economiststhestateoftheeconomy,lawyerstheir
latest cases, and businessmen their latest acquisitions, butmathematicians cannot discuss theirmathematics at
all. Andthemore profound their work, theless understandableit is. —Alfred Adler