The AP Calculus Problem Book ∫, Slides of Calculus

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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 Vicky Johnson Kayla Koch 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

AB Class of 2005 Lacey Avery Alicia Bellis Mollie Bogle Carin Godemann Shawn Kumar Joe Madsen Julie Matthews Jazmine Reaves Sarah Singh Andrew Vanstone Jawaan Washington Michael Westbury Jeremy Wilkerson

BC Class of 2006 John Barnett Chelsea Britt Sam Brotherton Justin Carlin Ryan Ceciliani Nicole Fraute Carin Godemann Megan Harris Zack Higbie Nayoon Kim Amy Kovac Julie Leber Salman Makhani Julie Matthews Chris Meador Sophia Newton Chris Randall Nicole Richardson Britt Schneider Michaela Simoes Jesse Smith

AB Class of 2006 Shail Amin Lucky Baker Savannah Barrus Taylor Boggus Justin Clemons Kim Dang Will Gibson Latiria Hill John-Lee Hughey Michael Hyman Cassandra Lohmeyer Chris Long Jason Long Joseph Long Cason Lowe Cassie Lowury Victoria Nesmith Dion Roseberry Cassie Smith Aniya Watson

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

BC Class of 2008 Layla Bouzoubaa Kevin Brawley Ashley Chackalayil Nate Coursey Michelle Dang Kathryn Daniel Justin Easley Samantha Girardot Jessie Holmes Johnathan Johnson Kathryn Johnson 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

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 www.miktex.org, which also includes TEXnicCenter, a free and easy-to-use user-interface.

Contents

6 The AP CALCULUS PROBLEM BOOK

CHAPTER 1

LIMITS

CHAPTER 1. LIMITS 9

1.2 The Slippery Slope of Lines

The point-slope form of a line is

m(x − x 1 ) = y − y 1.

In the first six problems, find the equation of the line with the given properties.

28. slope: 23 ; passes through (2, 1) 29. slope: − 14 ; passes through (0, 6) 30. passes through (3, 6) and (2, 7) 31. passes through (− 6 , 1) and (1, 1) 32. passes through (5, −4) and (5, 9) 33. passes through (10, 3) and (− 10 , 3) 34. A line passes through (1, 2) and (2, 5). Another line passes through (0, 0) and (− 4 , 3). Find the point where the two lines intersect. 35. A line with slope − 25 and passing through (2, 4) is parallel to another line passing through (− 3 , 6). Find the equations of both lines. 36. A line with slope −3 and passing through (1, 5) is perpendicular to another line passing through (1, 1). Find the equations of both lines. 37. A line passes through (8, 8) and (− 2 , 3). Another line passes through (3, −1) and (− 3 , 0). Find the point where the two lines intersect. 38. The function f (x) is a line. If f (3) = 5 and f (4) = 9, then find the equation of the line f (x). 39. The function f (x) is a line. If f (0) = 4 and f (12) = 5, then find the equation of the line f (x). 40. The function f (x) is a line. If the slope of f (x) is 3 and f (2) = 5, then find f (7). 41. The function f (x) is a line. If the slope of f (x) is 23 and f (1) = 1, then find f ( 32 ). 42. If f (2) = 1 and f (b) = 4, then find the value of b so that the line f (x) has slope 2. 43. Find the equation of the line that has x-intercept at 4 and y-intercept at 1. 44. Find the equation of the line with slope 3 which intersects the semicircle y =

25 − x^2 at x = 4.

I hope getting the nobel will improve my credit rating, because I really want a credit card. —John Nash

10 The AP CALCULUS PROBLEM BOOK

1.3 The Power of Algebra

Factor each of the following completely.

45. y^2 − 18 y + 56 46. 33 u^2 − 37 u + 10 47. c^2 + 9c − 8 48. (x − 6)^2 − 9 49. 3(x + 9)^2 − 36(x + 9) + 81 50. 63 q^3 − 28 q 51. 2 πr^2 + 2πr + hr + h 52. x^3 + 8 53. 8 x^2 + 27 54. 64 x^6 − 1 55. (x + 2)^3 + 125 56. x^3 − 2 x^2 + 9x − 18 57. p^5 − 5 p^3 + 8p^2 − 40

Simplify each of the following expressions.

58. 3(x − 4) + 2(x + 5) 6(x − 4)

59.

x − y

y − x

60. 3 x −

5 x − 7 4

61.

9 x^2 5 x^3 3 x

62.

y 1 −

y

63.

x 1 −

y

y 1 −

x

Rationalize each of the following expressions.

64.

65.

66. 2 x + 8 √ x + 4

67.

68. x − 6 √ x − 3 +

69.

2 x + 3 −

2 x

70.

5 x √ x + 5 −

71.

72. x √ x + 3 −

Incubation is the work of the subconscious during the waiting time, which may be several years. Illumination, which can happen in a fraction of a second, is the emergence of the creative idea into the conscious. This almost always occurs when the mind is in a state of relaxation, and engaged lightly with ordinary matters. Illumination implies some mysterious rapport between the subconscious and the conscious, otherwise emergence would not happen. What rings the bell at the right moment? —John E. Littlewood

12 The AP CALCULUS PROBLEM BOOK

1.5 Take It to the Limit

Evaluate each limit.

93. (^) xlim→− 2 (3x^2 − 2 x + 1) 94. lim x→ 5

95. lim x→− 3 (x^3 − 2) 96. (^) zlim→ 8

z^2 − 64 z − 8

97. lim t→ 1 / 4

4 t − 1 1 − 16 t^2

98. lim x→− 2

x^2 + 5x + 6 x^2 − 4

99. lim x→ 1 / 3

3 x^2 − 7 x + 2 − 6 x^2 + 5x − 1

100. lim p→ 4

p^3 − 64 4 − p

101. lim k→− 1

3

3 k − 5 25 k − 2

102. lim x→ 2

x^2 − 4 2 x^2 + x − 6

103. lim x→ 0

x √ x + 3 −

104. lim y→ 0

3 y + 2 −

y

105. Let F (x) =

3 x − 1 9 x^2 − 1

. Find lim x→ 1 / 3 F (x). Is this the same as the value of F

3

106. Let G(x) = 4 x^2 − 3 x 4 x − 3 . Find lim x→ 3 / 4

G(x). Is this the same as the value of G

4

107. Let P (x) =

3 x − 2 x 6 = (^13) 4 x = 13.

Find lim x→ 1 / 3

P (x). Is this the same as the value of P

3

108. Let Q(x) =

x^2 − 16 x − 4 x 6 = 4 3 x = 4.

Find lim x→ 4 Q(x). Is this the same as the value of Q(4)?

Solve each system of equations.

109.

2 x − 3 y = − 4 5 x + y = 7

110.

6 x + 15y = 8 3 x − 20 y = − 7

111. If F (x) =

2 x − 5 x > (^12) 3 kx − 1 x < (^12)

then find the value of k such that lim x→ 1 / 2

F (x) exists.

CHAPTER 1. LIMITS 13

1.6 One-Sided Limits

Find the limits, if they exist, and find the indicated value. If a limit does not exist, explain why.

112. Let f (x) =

4 x − 2 x > 1 2 − 4 x x ≤ 1.

a) lim x→ 1 +^

f (x) b) lim x→ 1 −^

f (x) c) lim x→ 1 f (x) d) f (1)

113. Let a(x) =

3 − 6 x x > 1 − 1 x = 1 x^2 x < 1.

a) lim x→ 1 +^

a(x) b) lim x→ 1 −^

a(x) c) lim x→ 1 a(x) d) a(1)

114. Let h(t) =

3 t − 1 t > 2 − 5 t = 2 1 + 2t t < 2.

a) lim t→ 2 +^

h(t) b) lim t→ 2 −^

h(t) c) lim t→ 2 h(t) d) h(2)

115. Let c(x) =

x^2 − 9 x < 3 5 x = 3 9 − x^2 x > 3.

a) lim x→ 3 +^

c(x) b) lim x→ 3 −^

c(x) c) lim x→ 3 c(x) d) c(3)

116. Let v(t) = | 3 t − 6 |.

a) lim t→ 2 +^

v(t) b) lim t→ 2 −^

v(t) c) lim t→ 2 v(t) d) v(2)

117. Let y(x) = | 3 x| x

a) lim x→ 0 +^

y(x) b) lim x→ 0 −^

y(x) c) lim x→ 0 y(x) d) y(0)

118. Let k(z) = | − 2 z + 4| − 3.

a) lim z→ 2 +^

k(z) b) lim z→ 2 −^

k(z) c) lim z→ 2 k(z) d) k(2)

Explain why the following limits do not exist.

119. (^) xlim→ 0

x |x|

120. lim x→ 1

x − 1

CHAPTER 1. LIMITS 15

1.8 Limits Determined by Graphs

Refer to the graph of h(x) to evaluate the following limits.

142. lim x→− 4 +^ h(x) 143. lim x→− 4 −^

h(x)

144. (^) xlim→∞ h(x) 145. lim x→−∞ h(x)

h(x)

Refer to the graph of g(x) to evaluate the following limits.

146. lim x→a+^

g(x)

147. lim x→a−^

g(x)

148. lim x→ 0 g(x) 149. lim x→∞ g(x) 150. lim x→b+^ g(x) 151. lim x→b−^

g(x)

a

c

d b

g(x)

Refer to the graph of f (x) to determine which statements are true and which are false. If a statement is false, explain why.

152. lim x→− 1 +^

f (x) = 1

153. lim x→ 0 −^

f (x) = 0

154. lim x→ 0 −^ f (x) = 1 155. lim x→ 0 −^

f (x) = lim x→ 0 +^

f (x)

156. lim x→ 0 f (x) exists 157. (^) xlim→ 0 f (x) = 0 158. lim x→ 0 f (x) = 1 159. lim x→ 1 f (x) = 1 160. lim x→ 1 f (x) = 0 161. lim x→ 2 −^ f (x) = 2 162. lim x→− 1 −^

f (x) does not exist

163. lim x→ 2 +^

f (x) = 0

f (x)

If your experiment needs statistics, you ought to have done a better experiment. —Ernest Rutherford

16 The AP CALCULUS PROBLEM BOOK

1.9 Limits Determined by Tables

Using your calculator, fill in each of the following tables to five decimal places. Using the information from the table, determine each limit. (For the trigonometric functions, your calculator must be in radian mode.)

164. lim x→ 0

x + 3 −

x

x − 0. 1 − 0. 01 − 0. 001 0. 001 0. 01 0. 1 √x+3−√ 3 x

165. lim x→− 3

1 − x − 2 x + 3

x − 3. 1 − 3. 01 − 3. 001 − 2. 999 − 2. 99 − 2. 9 √ 1 −x− 2 x+

166. lim x→ 0

sin x x

x − 0. 1 − 0. 01 − 0. 001 0. 001 0. 01 0. 1

sin x

167. lim x→ 0

1 − cos x x

x − 0. 1 − 0. 01 − 0. 001 0. 001 0. 01 0. 1

1 −cos x x

168. lim x→ 0 (1 + x)^1 /x

x − 0. 1 − 0. 01 − 0. 001 0. 001 0. 01 0. 1

(1 + x)^1 /x

169. lim x→ 1 x^1 /(1−x)

x 0. 9 0. 99 0. 999 1. 001 1. 01 1. 1

x^1 /(1−x)

Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house. —Henri Poincar´e