Calculus AB2008 ExamSection I

Part A, Questions 1-28, 55 minutes, calculators are not allowed.Part B, Questions 29-45, 50 minutes, calculators allowed.

1. is

(A) -3 (B) -2 (C) 2 (D) 3 (E) nonexistent

2.

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

3. If , then

(A) (B) (C)

(D) (E)

4.

(A) (B)

(C) (D)

(E)

5. is

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

6. Let f be the function defined below. Which of the following statements below about f are true?

I. f has a limit at x = 2.II. f is continuous at x = 2.III. f is differentiable at x = 2.

(A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III

7. A particle moves along the x-axis with velocity given by for time . If the

particle is at position x = 2 at time t = 0, what is the position of the particle at time t = 1?

(A) 4 (B) 6 (C) 9 (D) 11 (E) 12

8. If

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

9. The graph of the piecewise linear function f is shown in the figure below. If ,

which of the following values is greatest?

Graph of f

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

10. If , then

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

11. The graph of the function f is shown below for . Of the following, which has the least value?

Graph of f

(A)

(B) Left Riemann sum approximation of with 4 subintervals of equal length.

(C) Right Riemann sum approximation of with 4 subintervals of equal length.

(D) Midpoint Riemann sum approximation of with 4 subintervals of equal length.

(E) Trapezoidal sum approximation of with 4 subintervals of equal length.

12. The graph of a function f is shown below. Which of the following could be the graph of , the derivative of f ?

Graph of f

(A) (B) (C)

(D) (E)

13. If , then

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

14. The polynomial function f has selected values of its second derivative given in the table below. Which of the following statements must be true?

x 0 1 2 3

5 0 -7 4

(A) f is increasing on the interval .

(B) f is decreasing on the interval .

(C) f has a local maximum at x = 1.(D) The graph of f has a point of inflection at x = 1.

(E) The graph of f changes concavity in the interval .

15.

(A) (B) (C)

(D) (E)

16. If , then

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

17. In the xy-plane, the line , where k is a constant, is tangent to the graph of . What is the value of k?

(A) -3 (B) -2 (C) -1 (D) 0 (E) 1

18. The graph of the function f shown below has horizontal tangents at x = 2 and x = 5. Let g be the

function defined by . For what values of x does the graph of g have a point of

inflection?

Graph of f

(A) 2 only (B) 4 only (C) 2 and 5 only (D) 2, 4, and 5 (E) 0, 4, and 6

19. What are all horizontal asymptotes of the graph of in the xy-plane?

(A) y = -1 only (B) y = 0 only (C) y = 5 only(D) y = -1 and y = 0 (E) y = -1 and y = 5

20. Let f be a function with a second derivative given by . What are the x-

coordinates of the points of inflection of the graph of f ?

(A) 0 only (B) 3 only (C) 0 and 6 only (D) 3 and 6 only (E) 0, 3, and 6

21. A particle moves along a straight line. The graph of the particle’s position at time t is shown

below for . The graph has horizontal tangents at t = 1 and t = 5 and a point of inflection at t = 2. For what values of t is the velocity of the particle increasing?

(A) (B) (C) (D) only(E) and

22. The function f is twice differentiable with , , and . What is the value

of the approximation of using the line tangent to the graph of f at x = 2?

(A) 0.4 (B) 0.6 (C) 0.7 (D) 1.3 (E) 1.4

23. A rumor spreads among a population of N people at a rate proportional to the product of the number of people who have heard the rumor and the number of people who have not heard the rumor. If p denotes the number of people who have hear the rumor, which of the following differential equations could be used to model this situation with respect to time t, where k is a positive constant?

(A) (B) (C)

(D) (E)

24. Which of the following is the solution to the differential equation with the initial

condition ?

(A) (B) (C)

(D) (E)

25. Let f be the function defined below, where c and d are constants. If f is differentiable at x = 2, what is the value of c + d?

(A) -4 (B) -2 (C) 0 (D) 2 (E) 4

26. What is the slope of the line tangent to the curve at the point at which ?

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

27. Let f be a differentiable function such that , , , and .

The function g is differentiable and for all x. What is the value of ?

(A) (B) (C) (D) (E) Cannot be determined

28. Shown below is a slope field for which of the following differential equations?

(A)

(B)

(C)

(D)

(E)

THIS IS THE END OF PART A

PART B (Calculators allowed)

29. The graph of , the derivative of f , is shown below for . On what intervals is f increasing?

(A) only

(B)

(C) only

(D) and

(E) , and

30. The figure below shows the graph of a function f with domain . Which of the following statements are true?

I. exists.

II. exists.

III. exists.

(A) I only (B) II only(C) I and II only (D) I and III only(E) I, II, and III

31. The first derivative of the function f is defined by for . On what

intervals is f increasing?

(A) only(B) (C) (D) and (E) and

32. If and , what is the value of ?

(A) -21 (B) -13 (C) 0 (D) 13 (E) 21

33. The derivative of the function f is given by . How many points of inflection

does the graph of f have on the open interval ?

(A) One (B) Two (C) Three (D) Four (E) Five

34. If is an antiderivative for and , then

(A) (B) (C)

(D) (E)

35. A particle moves along a straight line with velocity given by at time .

What is the acceleration of the particle at time t = 3?

(A) -0.914 (B) 0.055 (C) 5.486 (D) 6.086 (E) 18.087

36. What is the area enclosed by the curves and ?

(A) 10.667 (B) 11.833 (C) 14.583 (D) 21.333 (E) 32

37. An object traveling in a straight line has position at time t. If the initial position is

and the velocity of the object is , what is the position of the object at time t = 3?

(A) 0.431 (B) 2.154 (C) 4.512 (D) 6.512 (E) 17.408

38. The graph of the derivative of a function f is shown in the figure below. The graph has horizontal tangent lines at x = -1, x = 1, and x = 3. At which of the following values of x does f have a relative maximum?

(A) -2 only(B) 1 only(C) 4 only(D) -1 and 3 only(E) -2, 1, and 4

39. The table below give values of a function f and its derivative at selected values of x. If is

continuous on the interval , what is the value of ?

x -4 -3 -2 -1

0.75 -1.5 -2.25 -1.5

-3 -1.5 0 1.5

(A) -4.5 (B) -2.25 (C) 0 (D) 2.25 (E) 4.5

40. The radius of a sphere is decreasing at a rate of 2 centimeters per second. At the instant when the radius of the sphere is 3 centimeters, what is the rate of change, in square centimeters per second, of the surface area of the sphere? (The surface are S of a sphere with radius r is .)

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

41. The function f is continuous for and . If there is no c, where

, for which , which of the following statements must be true?

(A) For , .

(B) For , .

(C) For , exists.

(D) For , exists, but is not continuous.

(E) For some k, where , does not exist.

42. What is the average value of on the closed interval ?

(A) -0.085 (B) 0.090 (C) 0.183 (D) 0.244 (E) 0.732

43. The table gives selected values of the velocity, , of a particle moving along the x-axis. At time

t = 0, the particle is at the origin. Which of the following could be the graph of the position, ,

of the particle for ?

t 0 1 2 3 4

-1 2 3 0 -4

(A) (B) (C)

(D) (E)

44. The function f is continuous on the closed interval and twice differentiable on the open

interval . If and on the open interval , which of the following

could be a table of values for f ?

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

45. A city located beside a river has a rectangular boundary as shown below. The population density

of the city at any point along a strip x miles from a river’s edge is persons per square mile.

Which of the following expressions gives the population of the city?

(A) (B) (C)

(D) (E)

2008 MC KEY

1 B 1 B 1 B2 D 2 D 2 D3 D 3 D 3 D4 B 4 B 4 B5 A 5 A 5 A6 A 6 A 6 A7 B 7 B 7 B8 E 8 E 8 E9 D 9 D 9 D10 D 10 D 10 D11 C 11 C 11 C12 B 12 B 12 B13 A 13 A 13 A14 E 14 E 14 E15 C 15 C 15 C16 D 16 D 16 D17 A 17 A 17 A18 C 18 C 18 C19 E 19 E 19 E20 D 20 D 20 D21 A 21 A 21 A22 B 22 B 22 B23 B 23 B 23 B24 E 24 E 24 E25 B 25 B 25 B26 A 26 A 26 A27 A 27 A 27 A28 C 28 C 28 C29 B 29 B 29 B30 C 30 C 30 C31 B 31 B 31 B32 B 32 B 32 B33 E 33 E 33 E34 E 34 E 34 E35 B 35 B 35 B36 B 36 B 36 B37 D 37 D 37 D38 C 38 C 38 C39 B 39 B 39 B40 C 40 C 40 C41 E 41 E 41 E42 C 42 C 42 C43 C 43 C 43 C44 A 44 A 44 A45 B 45 B 45 B