AB Practice Test 2

Section I

Part A

TIME: 60 MINUTES

The use of calculators is not permitted for this part of the examination.
There are 30 questions in Part A, for which 60 minutes are allowed. Because there is no deduction for wrong answers, you should answer every question, even if you need to guess.

DIRECTIONS: Choose the best answer for each question.

 

1. 

(A) –1
(B) 
(C) 0
(D) nonexistent

2. 

(A) 
(B) –1
(C) 0
(D) ∞

3. If , then  equals

4. Using the line tangent to  at x = 0, an estimate of f(0.06) is

(A) 0.02
(B) 2.98
(C) 3.01
(D) 3.02

5. The rate at which schoolchildren are dropped off to school on a certain day is modeled by the function S, where S(t) is the number of students per minute and t is the number of minutes since the school doors were opened that morning. What is the approximate number of students dropped off at the school in the first 30 minutes the doors were open? Use a trapezoidal sum with three subintervals indicated by the data in the table.

(A) 470
(B) 535
(C). 550
(D) 1070

6. If y = sin³(1 – 2x), then  is

(A) 3 sin²(1 – 2x)
(B) –2 cos³(1 – 2x)
(C) –6 sin²(1 – 2x)
(D) –6 sin²(1 – 2x) cos(1 – 2x)

7. If y = x²e^1/x (x ≠ 0), then  is

(A) xe^1/x (x + 2)
(B) e^1/x (2x – 1)
(C) 
(D) e^–x (2x – x²)

8. A point moves along the curve y = x² + 1 so that the x-coordinate is increasing at the constant rate of  units per second. The rate, in units per second, at which the distance from the origin is changing when the point has coordinates (1,2) is equal to

9. 

10. The base of a solid is the first-quadrant region bounded by . Each cross section perpendicular to the x-axis is a square with one edge in the xy-plane. The volume of the solid is

11. 

12. 

13. 

14. Given f′ as graphed, which could be a graph of f?

 

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

15. The first woman officially timed in a marathon was Violet Piercy of Great Britain in 1926. Her record of 3:40:22 stood until 1963, mostly because of a lack of women competitors. Soon after, times began dropping rapidly, but lately they have been declining at a much slower rate. Let M(t) be the curve that best represents winning marathon times in year t. Which of the following is (are) positive for t > 1963?

I. M(t)
II. M′(t)
III. M″(t)

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

16. The graph of f is shown above. Let  and . Which of the following is true?

(A) G(x) = H(x)
(B) G′(x) = H′(x + 2)
(C) G(x) = H(x + 2)
(D) G(x) = H(x) + 3

17. The minimum value of  on the interval  is

18. Which function could be a particular solution of the differential equation whose slope field is shown above?

19. Which of the following functions could have the graph sketched below?

Questions 20–22. Use the graph below, consisting of two line segments and a quarter-circle. The graph shows the velocity of an object during a 6-second interval.

20. For how many values of t in the interval 0 < t < 6 is the acceleration undefined?

(A) none
(B) one
(C) two
(D) three

21. During what time interval (in seconds) is the speed increasing?

(A) 0 < t < 3
(B) 3 < t < 5
(C) 5 < t < 6
(D) never

22. What is the average acceleration (in units/sec²) during the first 5 seconds?

23. The curve of  has

(A) two horizontal asymptotes and one vertical asymptote
(B) two vertical asymptotes but no horizontal asymptote
(C) one horizontal and one vertical asymptote
(D) one horizontal and two vertical asymptotes

24. Suppose

Which statement is true?

(A) f is continuous everywhere.
(B) f is discontinuous only at x = 1.
(C) f is discontinuous at x = –2 and at x = 1.
(D) If f(–2) is defined to be 4, then f will be continuous everywhere.

25. The function f(x) = x⁵ + 3x – 2 passes through the point (1,2). Let f^–1 denote the inverse of f. Then (f^–1)′(2) equals

26. 

27. Which of the following statements is (are) true about the graph of y = ln (4 + x²)?

I. It is symmetric to the y-axis.
II. It has a local minimum at x = 0.
III. It has inflection points at x = ±2.

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

28. Let . Then f(3) =

(A) –3π
(B) –1
(C) –3
(D) –π

29. Choose the integral that is the limit of the Riemann Sum:

30. The region bounded by y = e^x, y = 1, and x = 2 is rotated about the x-axis. The volume of the solid generated is given by the integral:

 

Part B

TIME: 45 MINUTES

Some questions in this part of the examination require the use of a graphing calculator. There are 15 questions in Part B, for which 45 minutes are allowed. Because there is no deduction for wrong answers, you should answer every question, even if you need to guess.

DIRECTIONS: Choose the best answer for each question. If the exact numerical value of the correct answer is not listed as a choice, select the choice that is closest to the exact numerical answer.

 

 

31. A particle moves on a straight line so that its velocity at time t is given by , where s is its distance from the origin. If s = 1 when t = 0, then, when t = 1, s equals

(A) 6
(B) 7
(C) 36
(D) 49

32. The sketch show the graphs of f(x) = x² – 4x – 5 and the line x = k. The regions labeled A and B have equal areas if k =

(A) 7.899
(B) 8
(C) 8.144
(D) 11

33. Bacteria in a culture increase at a rate proportional to the number present. An initial population of 200 triples in 10 hours. If this pattern of increase continues unabated, then the approximate number of bacteria after 1 full day is

(A) 1056
(B) 1440
(C) 2793
(D) 3240

34. When the substitution x = 2t – 1 is used, the definite integral  may be expressed in the form , where {k,a,b} =

35. The curve defined by x³ + xy – y² = 10 has a vertical tangent line when x =

(A) 1.037
(B) 1.087
(C) 2.074
(D) 2.096

Questions 36 and 37. Use the graph of f shown on [0,7]. Let .

36. G′(1) is

(A) 1
(B) 2
(C) 3
(D) 6

37. G has a local maximum at x =

(A) 1
(B) 
(C) 2
(D) 8

38. You are given two thrice-differentiable functions, f(x) and g(x). The table above gives values for f(x) and g(x) and their first and second derivatives at x = 2. Find .

(A) 0
(B) 1
(C) 6
(D) nonexistent

39. Using the left rectangular method and four subintervals of equal width, estimate , where f is the function graphed below.

(A) 4
(B) 8
(C) 12
(D) 16

40. Suppose f(3) = 2, f′(3) = 5, and f″(3) = –2. Then  at x = 3 is equal to

(A) –20
(B) –4
(C) 10
(D) 42

41. The velocity of a particle in motion along a line (for t ≥ 0) is v(t) = ln (2 – t²). Find the acceleration when the object is at rest.

42. Suppose , x > 0 and x is increasing. The value of x for which the rate of increase of f is 10 times the rate of increase of x is

43. The rate of change of the surface area, S, of a balloon is inversely proportional to the square of the surface area. Which equation describes this relationship?

44. Two objects in motion from t = 0 to t = 3 seconds have positions x₁(t) = cos(t² + 1) and , respectively. How many times during the 3 seconds do the objects have the same velocity?

(A) 0
(B) 1
(C) 3
(D) 4

45. After t years, A(t) = 50e^–0.015t pounds of a deposit of a radioactive substance remain. The average amount of the radioactive substance that remains during the time from t = 100 to t = 200 years is

(A) 2.9 pounds
(B) 5.8 pounds
(C) 6.8 pounds
(D) 15.8 pounds

 

Section II

Part A

TIME: 30 MINUTES
2 PROBLEMS

A graphing calculator is required for some of these problems. See instructions on page 8.

 

1. Let function f be continuous and decreasing, with values as shown in the table:

(a) Use a trapezoidal sum to estimate the area between f and the x-axis on the interval 2.5 ≤ x ≤ 5.0.
(b) Find the average rate of change of f on the interval 2.5 ≤ x ≤ 5.0.
(c) Estimate the instantaneous rate of change of f at x = 2.5.
(d) If g(x) = f^–1(x), estimate the slope of g at x = 4.

2. The curve  divides a first-quadrant rectangle into regions A and B, as shown in the figure above.

(a) Region A is the base of a solid. Cross sections of this solid perpendicular to the x-axis are rectangles. The height of each rectangle is 5 times the length of its base in region A . Find the volume of this solid.
(b) The other region, B, is rotated around the y-axis to form a different solid. Set up, but do not evaluate, an integral for the volume of this solid.

 

Part B

TIME: 60 MINUTES
4 PROBLEMS

No calculator is allowed for any of these problems.

If you finish Part B before time has expired, you may return to work on Part A, but you may not use a calculator.

 

3. Consider the first-quadrant region bounded by the curve , the coordinate axes, and the line x = k, as shown in the figure above.

(a) For what value of k will the area of this region equal π?
(b) What is the average value of the function on the interval 0 ≤ x ≤ k?
(c) What happens to the area of the region as the value of k increases?

 

4. Given the following differential equation .

(a) Sketch the slope field for the differential equation at the nine indicated points on the axes provided.

(b) Find the second derivative,, in terms of x and y. The region in the xy-plane where all the solution curves to the differential equation are concave down can be expressed as a linear inequality. Find this region.
(c) The function y = f(x) is the solution to the differential equation with initial condition f(–1) = 0. Determine whether f has a local maximum, local minimum, or neither at x = –1. Justify your answer.
(d) For which values of m and b is the line y = mx + b a solution to the differential equation?

 

5. A bungee jumper has reached a point in her exciting plunge where the taut cord is 100 feet long with a 1/2-inch radius and is stretching. She is still 80 feet above the ground and is now falling at 40 feet per second. You are observing her jump from a spot on the ground 60 feet from the potential point of impact, as shown in the diagram above.

(a) Assuming the cord to be a cylinder with volume remaining constant as the cord stretches, at what rate is its radius changing when the radius is 1/2 inch?
(b) From your observation point, at what rate is the angle of elevation to the jumper changing when the radius is 1/2 inch?

 

6. The figure above shows the graph of f, whose domain is the closed interval [–2,6]. Let .

(a) Find F(–2) and F(6).
(b) For what value(s) of x does F(x) = 0?
(c) On what interval(s) is F increasing?
(d) Find the maximum value and the minimum value of F.
(e) At what value(s) of x does the graph of F have points of inflection? Justify your answer.