The AP Calculus Problem Book

The AP Calculus

Problem Book

?

Chuck Garner, Ph.D.

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

1.1 Graphs of Functions . . . . . . . . . . .

1.2 The Slippery Slope of Lines . . . . . . .

1.3 The Power of Algebra . . . . . . . . . .

1.4 Functions Behaving Badly . . . . . . . .

1.5 Take It to the Limit . . . . . . . . . . .

1.6 One-Sided Limits . . . . . . . . . . . . .

1.7 One-Sided Limits (Again) . . . . . . . .

1.8 Limits Determined by Graphs . . . . . .

1.9 Limits Determined by Tables . . . . . .

1.10 The Possibilities Are Limitless... . . . .

1.11 Average Rates of Change: Episode I . .

1.12 Exponential and Logarithmic Functions

1.13 Average Rates of Change: Episode II . .

1.14 Take It To the Limit¡ªOne More Time .

1.15 Solving Equations . . . . . . . . . . . .

1.16 Continuously Considering Continuity . .

1.17 Have You Reached the Limit? . . . . . .

1.18 Multiple Choice Questions on Limits . .

1.19 Sample A.P. Problems on Limits . . . .

Last Year¡¯s Limits Test . . . . . . . . . . . .

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7

8

9

10

11

12

13

14

15

16

17

18

18

19

20

21

22

23

24

26

27

2 DERIVATIVES

2.1 Negative and Fractional Exponents

2.2 Logically Thinking About Logic . .

2.3 The Derivative By Definition . . .

2.4 Going Off on a Tangent . . . . . .

2.5 Six Derivative Problems . . . . . .

2.6 Trigonometry: a Refresher . . . . .

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35

36

37

38

39

40

41

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2

The AP CALCULUS PROBLEM BOOK

2.7

2.8

2.9

2.10

2.11

2.12

2.13

2.14

2.15

2.16

2.17

2.18

2.19

Last

Continuity and Differentiability . . . . . . . . .

The RULES: Power Product Quotient Chain .

Trigonometric Derivatives . . . . . . . . . . . .

Tangents, Normals, and Continuity (Revisited)

Implicit Differentiation . . . . . . . . . . . . . .

The Return of Geometry . . . . . . . . . . . . .

Meet the Rates (They¡¯re Related) . . . . . . .

Rates Related to the Previous Page . . . . . . .

Excitement with Derivatives! . . . . . . . . . .

Derivatives of Inverses . . . . . . . . . . . . . .

De?rive?, Derivado, Ableitung, Derivative . . . .

Sample A.P. Problems on Derivatives . . . . . .

Multiple-Choice Problems on Derivatives . . . .

Year¡¯s Derivatives Test . . . . . . . . . . . . . .

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42

43

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46

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49

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51

52

54

56

58

3 APPLICATIONS of DERIVATIVES

3.1 The Extreme Value Theorem . . . . . . . . . . . . . . .

3.2 Rolle to the Extreme with the Mean Value Theorem . .

3.3 The First and Second Derivative Tests . . . . . . . . . .

3.4 Derivatives and Their Graphs . . . . . . . . . . . . . . .

3.5 Two Derivative Problems . . . . . . . . . . . . . . . . .

3.6 Sketching Functions . . . . . . . . . . . . . . . . . . . .

3.7 Problems of Motion . . . . . . . . . . . . . . . . . . . .

3.8 Maximize or Minimize? . . . . . . . . . . . . . . . . . .

3.9 More Tangents and Derivatives . . . . . . . . . . . . . .

3.10 More Excitement with Derivatives! . . . . . . . . . . . .

3.11 Bodies, Particles, Rockets, Trucks, and Canals . . . . .

3.12 Even More Excitement with Derivatives! . . . . . . . . .

3.13 Sample A.P. Problems on Applications of Derivatives . .

3.14 Multiple-Choice Problems on Applications of Derivatives

Last Year¡¯s Applications of Derivatives Test . . . . . . . . . .

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67

68

69

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84

86

89

92

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101

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. 118

4 INTEGRALS

4.1 The ANTIderivative! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Derivative Rules Backwards . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Method of Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 Using Geometry for Definite Integrals . . . . . . . . . . . . . . . . . . . . .

4.5 Some Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6 The MVT and the FTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7 The FTC, Graphically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8 Definite and Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . .

4.9 Integrals Involving Logarithms and Exponentials . . . . . . . . . . . . . . .

4.10 It Wouldn¡¯t Be Called the Fundamental Theorem If It Wasn¡¯t Fundamental

4.11 Definite and Indefinite Integrals Part 2 . . . . . . . . . . . . . . . . . . . . .

4.12 Regarding Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.13 Definitely Exciting Definite Integrals! . . . . . . . . . . . . . . . . . . . . . .

4.14 How Do I Find the Area Under Thy Curve? Let Me Count the Ways... . . .

4.15 Three Integral Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3

CONTENTS

4.16

4.17

4.18

4.19

Last

Trapezoid and Simpson . . . . . . . .

Properties of Integrals . . . . . . . . .

Sample A.P. Problems on Integrals . .

Multiple Choice Problems on Integrals

Year¡¯s Integrals Test . . . . . . . . . .

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5 APPLICATIONS of INTEGRALS

5.1 Volumes of Solids with Defined Cross-Sections . . . . .

5.2 Turn Up the Volume! . . . . . . . . . . . . . . . . . . .

5.3 Volume and Arc Length . . . . . . . . . . . . . . . . .

5.4 Differential Equations, Part One . . . . . . . . . . . .

5.5 The Logistic Curve . . . . . . . . . . . . . . . . . . . .

5.6 Differential Equations, Part Two . . . . . . . . . . . .

5.7 Slope Fields and Euler¡¯s Method . . . . . . . . . . . .

5.8 Differential Equations, Part Three . . . . . . . . . . .

5.9 Sample A.P. Problems on Applications of Integrals . .

5.10 Multiple Choice Problems on Application of Integrals

Last Year¡¯s Applications of Integrals Test . . . . . . . . . .

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119

120

121

124

127

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135

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. 137

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. 139

. 140

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. 147

. 150

6 TECHNIQUES of INTEGRATION

6.1 A Part, And Yet, Apart... . . . . . . . . . . . . . . . . . . . . .

6.2 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3 Trigonometric Substitution . . . . . . . . . . . . . . . . . . . .

6.4 Four Integral Problems . . . . . . . . . . . . . . . . . . . . . . .

6.5 L¡¯Ho?pital¡¯s Rule . . . . . . . . . . . . . . . . . . . . . . . . . .

6.6 Improper Integrals! . . . . . . . . . . . . . . . . . . . . . . . . .

6.7 The Art of Integration . . . . . . . . . . . . . . . . . . . . . . .

6.8 Functions Defined By Integrals . . . . . . . . . . . . . . . . . .

6.9 Sample A.P. Problems on Techniques of Integration . . . . . .

6.10 Sample Multiple-Choice Problems on Techniques of Integration

Last Year¡¯s Techniques of Integration Test . . . . . . . . . . . . . . .

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159

160

161

162

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168

170

173

175

7 SERIES, VECTORS, PARAMETRICS and POLAR

7.1 Sequences: Bounded and Unbounded . . . . . . . . . . . . . . . .

7.2 It is a Question of Convergence... . . . . . . . . . . . . . . . . . .

7.3 Infinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.4 Tests for Convergence and Divergence . . . . . . . . . . . . . . .

7.5 More Questions of Convergence... . . . . . . . . . . . . . . . . . .

7.6 Power Series! . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.7 Maclaurin Series . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.8 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.9 Vector Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.10 Calculus with Vectors and Parametrics . . . . . . . . . . . . . . .

7.11 Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . .

7.12 Motion Problems with Vectors . . . . . . . . . . . . . . . . . . .

7.13 Polar Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.14 Differentiation (Slope) and Integration (Area) in Polar . . . . . .

7.15 Sample A.P. Problems on Series, Vectors, Parametrics, and Polar

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