The AP Calculus Problem Book

[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

Rochelle Dunlap Daniel Eisenman Kylene Farmer

Nathan Garcia Brandon Jackson

Drew Keenan Amin Makhani Patrick McGahee Rachel Meador Richard Moss Trent Phillips

John Powell Blake Serra Jon Skypek David Thompson

AB Class of 2003 Ryan Boyd R. T. Collins

Holly Ellington Stephen Gibbons

Aly Mawji Chris McKnight Franklin Middlebrooks Kelly Morrison

Julien Norton Jurod Russell

BC Class of 2004 Anita Amin

Anushka Amin Rachel Atkinson Max Bernardy Lindsey Broadnax Andy Brunton Mitch Costley Caitlin Dingle

Krista Firkus Justin Gilstrap Casey Haney Kendra Heisner Daniel Hendrix Candace Hogan Luke Hotchkiss Shawn Hyde Whitney Irwin Garett McLaurin C. K. Newman Matt Robuck Melissa Sanders Drew Sheffield Carsten Singh Andrea Smith Frankie Snavely Elizabeth Thai

Ray Turner Timothy Van Heest

Josh Williams Ben Wu Jodie Wu

Michael Wysolovski Drew Yaun

AB Class of 2004 Brooke Atkinson Kevin Dirth Jaimy Lee Robert Parr Garion Reddick Andre Russell

Megan Villanueva

BC Class of 2005 Jonathan Andersen

James Bascle Eryn Bernardy

Joesph Bost William Brawley Alex Hamilton Sue Ann Hollowell

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Amy Lanchester Dana McKnight Kathryn Moore

Ryan Moore Candace Murphy Hannah Newman

Bre'Ana Paige Lacy Reynolds Jacob Schiefer

Kevin Todd Amanda Wallace Jonathan Wysolovski

Keeli Zanders

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Carin Godemann Shawn Kumar Joe Madsen Julie Matthews Jazmine Reaves Sarah Singh

Andrew Vanstone Jawaan Washington Michael Westbury Jeremy Wilkerson

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Sam Brotherton Justin Carlin Ryan Ceciliani Nicole Fraute

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BC Class of 2007 Betsey Avery Aaron Bullock Cecily Bullock Daniel Chen

Raymond Clunie Jim Creager Katie Dugan

Mitchell Granade Allyse Keel Jacob Kovac Jan Lauritsen Ally Long Chris Long Nick Macie Will Martin

Justin McKithen Ruhy Momin Steven Rouk Tyler Sigwald Lauren Troxler Ryan Young

AB Class of 2007 Melanie Allen

Briana Brimidge Michelle Dang

Kyle Davis Holly Dean Shaunna Duggan Chris Elder Kevin Gorman Jessie Holmes Gary McCrear Faith Middlebrooks D'Andra Myers Brandi Paige Miriam Perfecto Thomas Polstra Torri Preston Heather Quinn Andrew Smith Nicole Thomas

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Tyler Kelly Rochelle Lobo Monica Longoria April Lovering Kevin Masters David McCalley Gary McCrear Patti Murphy

Sarah Pace Bijal Patel Kunal Patel Miriam Perfecto Thomas Polstra James Rives Khaliliah Smith Andrew Stover Aswad Walker Ashley Williams

AB Class of 2008 Omair Akhtar John Collins Rachel Delevett Kelsey Hinely Sarah Kustick Bianca Manahu

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

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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

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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

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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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