AP CALCULUS AB SECTION I, Part A Time – 55 Minutes

Number of questions – 28 Time Began: _______________ Time Ended: ___________

A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM

Directions: Solve each of the following problems, using the available space for scratchwork. Do not spend too much time on any one problem. In this assignment:

1. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which is a real number.

2. The inverse of a trigonometric function f may be indicated using the inverse function notation or with the

prefix “arc” (e.g. ).

1.

(A) 10 (B) (C) 40 (D) 80 (E)

2. limx→∞

5x2 − 3x + 14x2 + 2x + 5

is

(A) 0 (B) 45

(C) 3

11 (D)

54

(E)

3.

(A) 1 (B) (C) (D) (E)

4. If the function f is continuous for all real number and if f x( ) = x2 − 7x + 12

x − 4when x ≠ 4, then f 4( ) =

(A) 1 (B) (C) (D) 0 (E) undefined

5.

(A)

(B)

(C)

(D)

(E)

6. Which of the following integrals correctly corresponds to the area of the shaded region in the figure above?

(A) (B) (C) (D) (E)

7.

(A) 0

(B)

(C)

(D)

(E)

8. An equation of the line normal to the graph of y = 3x2 + 2x at is

(A) (B) (C) (D) (E)

9.

41+ x2

dx =−1

1

∫

(A) 0 (B) (C) 1 (D) (E) 2

10. If f x( ) = cos2 x , then f " π( ) =

(A) -2 (B) 0 (C) 1 (D) 2 (E)

11.

(A) 1

105x2 − 4( )

3

2 + C (B) 1

155x2 − 4( )

3

2 + C (C) −

15

5x2 − 4( )3

2 + C

(D) 203

5x2 − 4( )3

2 + C (E)

320

5x2 − 4( )3

2 + C

12. The slope of the line tangent to the graph of at is

(A) (B) (C) (D) 12 (E)

13. If f x( ) = x2 + 5 if x < 2

7x − 5 if x ≥ 2

⎧⎨⎩

, for all real number x, which of the following must be true

I. f x( ) is continuous everywhere.

II. f x( ) is differentiable everywhere.

III. f x( ) has a local minimum at x = 2. (A) I only (B) I and II only (C) II and III only (D) I and III only (E) I, II, and III 14.

The graph of a piecewise linear function f, for , is shown above. What is the value of ?

(A) 1 (B) 4 (C) 8 (D) 10 (E) 13

15. If , then

(A)

(B)

(C)

(D)

(E)

16. What is the instantaneous rate of change at of the function f, if f t( ) = t3 + t

4t + 1?

(A) 129

(B) 49

(C) −

209

(D) −

49

(E) −

129

17.

(A) 4 (B) 4e (C) 0 (D) (E)

18. limh→0

tanπ6+ h

⎛⎝⎜

⎞⎠⎟− tan

π6

⎛⎝⎜

⎞⎠⎟

h=

(A) (B) (C) (D) 0 (D)

19. The average value of the function f x( ) = x − 1( )2 on the interval from to is

(A) −

163

(B) 163

(C) 643

` (D) 663

(E) 256

3

20. A solid is generated when the region in the first quadrant enclosed by the graph of , the line , the x-

axis, and the y-axis is revolved about the x-axis. Its volume is found by evaluating which of the following integrals?

(A)

(B)

(C)

(D)

(E)

21. If and when , then when ,

(A) 18 (B) (C) 58 (D) (E)

22. If then

(A)

(B)

(C)

(D)

(E) 0

23. A particle moves along the x-axis so that its position at time t, in seconds, is given by . For what

value(s) of t is the velocity of the particle zero?

(A) 1 (B) 6 (C) 1 or 6 (D) 3.5 (E) 1 or 3.5 or 6

24. If , then

(A) (B) (C) (D) (E) 0

25. Find the area of the region bounded by the parabolas and .

(A) 9 (B) 27 (C) 6 (D) (E)

26. The function f is given by On which intervals is f decreasing? (A) −3,0( ) (B) (C) (D) −∞,−3( ) (E)

27. Find the value of c that satisfies the Mean Value Theorem on the interval for the function

(A) (B) 0 (C) 1 (D) (E)

28. The graph of is shown in the figure above. Which of the following could be the graph of .

STOP

END OF PART A SECTION 1

AP CALCULUS AB SECTION I, Part B Time – 50 Minutes

Number of Questions – 17 Time Began: _______________ Time Ended:_____________

A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM

Directions: Solve each of the following problems, using the available space for your work. Place your answer on the line in the bottom right hand corner of the space. Do not spend too much time on any one problem In this assignment:

1. The exact numerical value of a correct answer may not always be found. Your answer should then be given as the best approximation, correct to three decimal places.

2. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which

is a real number.

3. The inverse of a trigonometric function f may be indicated using the inverse function notation or with the prefix “arc” (e.g. ).

76.

sin x dx0

π

4∫ + cos x dx−π

4

0

∫ =

(A) − 2 (B) (C) 0 (D) 1 (E) 2

77. ddx

tan2 4x( ) =

(A) (B) (C) (D) (E)

78.

(A) (B) (C) 0 (D) (E) 20 79. If , and , then could be

(A) 59 (B) (C) (D) 11 (E)

80. The rate at which water is sprayed on a field of vegetables is given by R t( ) = 2 1+ 5t3 , where t is in minutes at

is in gallons per minute. During the time interval , what is the average rate of water flow, in gallons per minute?

(A) 8.458 (B) 13.395 (C) 14.691 (D) 18.916 (E) 35.833

81. If the function f x( ) = ax3 − 6x; if x ≤ 1

bx2 + 4; if x > 1

⎧⎨⎩⎪

is continuous and differentiable at all real numbers, then a =

(A) 0 (B) 1 (C) −14 (D) −24 (E) 26

82.

(A) tan7 x

7+ C

(B) tan7 x

7+

sec3 x3

+ C

(C) tan7 x sec3 x

21+ C

(D) 7 tan7 x + C

(E) 27

tan7 x sec x + C

83. A 20 foot ladder slides down a wall at 5 ft/sec. At what rate is the bottom of the ladder sliding out when the top is 10

feet from the floor?

(A) 0.346 (B) 2.887 (C) 0.224 (D) 5.774 (E) 4.472

84. Find the distance traveled (to three decimals ) from to seconds, for a particle whose velocity is given by

. (A) 6.000 (B) 1.609 (C) 16.047 (D) 0.800 (E) 148.413

85. : If , find .

(A) (B) (C) (D) 80 (E) 86. The second derivative of a function f is given by . How many points of inflection does f have on

the interval ?

(A) Zero (B) Two (C) Four (D) Six (E) Eight

87. Find the total area of the region between the curve and the x-axis from to radians.

(A) 0 (B) 0.068 (C) 0.249 (D) 1.751 (E) 2.592 88. The tangent line to the curve at the point has an x-intercept at what point?

(A) (B) (C) (D) (E)

x f g f’ g’

a -4 c 8 b

c 15 10 6 5

89. The graph of , the derivative of f, is shown above. On which interval(s) is the graph of f decreasing?

(A)

(B)

(C)

(D)

(E) 90. A particle moves along the x-axis so that at any time its velocity is given by . What is the

acceleration of the particle at time ?

(A) 1.500 (B) 20.453 (C) 29.453 (D) 74.860 (E) 133.417

91. A left Riemann sum, a right Riemann sum, and a trapezoidal sum are used to approximate the value of ,

each using the same number of subintervals. The graph of the function f is shown in the figure above. Which of the

sums give an underestimate of the value ?

I. Left sum II. Right sum III. Trapezoidal sum

(A) I only (B) II only (C) III only (D) I and III only (E) II and III only 92. Let f be the function with the first derivative defined by for . At what value of x does f

attain its absolute maximum value on the closed interval ?

(A) 0 (B) 1.162 (C) 1.468 (D) 1.845 (E) 2

STOP

END OF SECTION I

IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART B ONLY.

DO NOT GO TO SECTION II UNTIL YOU ARE TOLD TO DO SO.