Diagnostic Test Calculus AB - AP Calculus Premium 2024

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.

DIRECTIONS: Choose the best answer for each question.

 

1.  

(A) –3
(B) 0
(C) 3
(D) ∞

2. 

(A) 1
(B) nonexistent
(C) 0
(D) –1

3. If, for all x, f′(x) = (x – 2)⁴(x – 1)³, it follows that the function f has

(A) a relative minimum at x = 1
(B) a relative maximum at x = 1
(C) both a relative minimum at x = 1 and a relative maximum at x = 2
(D) relative minima at x = 1 and at x = 2

4. Let  . Which of the following statements is (are) true?

I. F′(0) = 5
II. F(2) < F(6)
III. F is concave upward

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

5. If f(x) = 10^x and 10^1.04 ≈ 10.96, which is closest to f′(1)?

(A) 0.92
(B) 0.96
(C) 10.5
(D) 24

6. If f is differentiable, we can use the line tangent to f at x = a to approximate values of f near x = a. Suppose that for a certain function f this method always underestimates the correct values. If so, then in an interval surrounding x = a, the graph of f must be

(A) increasing
(B) decreasing
(C) concave upward
(D) concave downward

7. If f(x) = cos x sin 3x, then  is equal to

8. 

9. The graph of f″ is shown below. If f′(1) = 0, then f′(x) = 0 at what other value of x on the interval [0,8]?

(A) 2
(B) 3
(C) 4
(D) 7

Questions 10 and 11. Use the following table, which shows the values of differentiable functions f and g.

10. If P(x) = (g(x))², then P′(3) equals

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

11. If H(x) = f^–1(x), then H′(3) equals

12. The total area of the region bounded by the graph of  and the x-axis is

13. The graph of  is concave upward when

(A) x > 3
(B) 1 < x < 3
(C) x < 1
(D) x < 3

14. As an ice block melts, the rate at which its mass, M, decreases is directly proportional to the square root of the mass. Which equation describes this relationship?

15. The average (mean) value of tan x on the interval from x = 0 to  is

16. If y = x² ln x for x > 0, then y″ is equal to

(A) 3 + ln x
(B) 3 + 2 ln x
(C) 3 + 3 ln x
(D) 2 + x + ln x

17. Water is poured at a constant rate into the conical reservoir shown in the figure. If the depth of the water, h, is graphed as a function of time, the graph is

(A) constant
(B) linear
(C) concave upward
(D) concave downward

18. 

(A) f(x) is not continuous at x = 1
(B) f(x) is continuous at x = 1 but f′(1) does not exist
(C) f′(1) = 2
(D)  does not exist

19. 

(A) –∞
(B) –1
(C) ∞
(D) nonexistent

Questions 20 and 21. The graph below consists of a quarter-circle and two line segments and represents the velocity of an object during a 6-second interval.

20. The object’s average speed (in units/sec) during the 6-second interval is

21. The object’s acceleration (in units/sec²) at t = 4.5 is

22. The slope field shown above is for which of the following differential equations?

23. If y is a differentiable function of x, then the slope of the curve of xy² – 2y + 4y³ = 6 at the point where y = 1 is

24. In the following, L(n), R(n), M(n), and T(n) denote, respectively, left, right, midpoint, and trapezoidal sums with n equal subdivisions. Which of the following is not equal exactly to ?

(A) L(2)
(B) T(3)
(C) M(4)
(D) R(6)

25. The table shows some values of a differentiable function f and its derivative f′:

Find  

(A) 5
(B) 6
(C) 11.5
(D) 14

26. The solution of the differential equation  for which y = –1 when x = 1 is

27. The base of a solid is the region bounded by the parabola y² = 4x and the line x = 2. Each plane section perpendicular to the x-axis is a square. The volume of the solid is

(A) 8
(B) 16
(C) 32
(D) 64

28. Which of the following could be the graph of ?

(A) 

(B)

(C)

(D)

29. If F(3) = 8 and F′(3) = –4, then F(3.02) is approximately

(A) 7.92
(B) 7.98
(C) 8.02
(D) 8.08

30. If , then F′(x) =

 

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.

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.

 

Questions 31 and 32. Refer to the graph of f′ below.

 

31. f has a local maximum at x =

(A) 3 only
(B) 4 only
(C) 2 and 4
(D) 3 and 4

32. The graph of f has a point of inflection at x =

(A) 2 only
(B) 3 only
(C) 2 and 3 only
(D) 2 and 4 only

33. For what value of c on 0 < x < 1 is the tangent to the graph of f(x) = e^x – x² parallel to the secant line on the interval (0,1)?

(A) 0.351
(B) 0.500
(C) 0.693
(D) 0.718

34. Find the volume of the solid generated when the region bounded by the y-axis, y = e^x, and y = 2 is rotated around the y-axis.

(A) 0.386
(B) 0.592
(C) 1.216
(D) 3.998

35. The table below shows the “hit rate” for an Internet site, measured at various intervals during a day. Use a trapezoid approximation with 6 subintervals to estimate the total number of people who visited that site.

 

(A) 5,280
(B) 10,080
(C) 10,440
(D) 10,560

36. The acceleration of a particle moving along a straight line is given by a = 6t. If, when t = 0, its velocity, v, is 1 and its position, s, is 3, then at any time t

37. If y = f(x²) and  then  is equal to

38. Find the area of the first quadrant region bounded by y = x², y = cos (x), and the y-axis.

(A) 0.292
(B) 0.508
(C) 0.547
(D) 0.921

39. If the substitution x = 2t + 1 is used, which of the following is equivalent to ?

40. An object moving along a line has velocity v(t) = t cos t – ln (t + 2), where 0 ≤ t ≤ 10. How many times does the object reverse direction?

(A) one
(B) two
(C) three
(D) four

41. A 26-foot ladder leans against a building so that its foot moves away from the building at the rate of 3 feet per second. When the foot of the ladder is 10 feet from the building, the top is moving down at the rate of r feet per second, where r is

(A) 0.80
(B) 1.25
(C) 7.20
(D) 12.50

42. The functions f(x), g(x), and h(x) have derivatives of all orders. Listed above are values for the functions and their first and second derivatives at x = 3. Find .

43. The graph above shows an object’s acceleration (in ft/sec²). It consists of a quarter-circle and two line segments. If the object was at rest at t = 5 seconds, what was its initial velocity?

(A) –2 ft/sec
(B) 3 – π ft/sec
(C) π – 3 ft/sec
(D) π + 3 ft/sec

44. Water is leaking from a tank at the rate of  gallons per hour, where t is the number of hours since the leak began. To the nearest gallon, how much water will leak out during the first day?

(A) 7
(B) 12
(C) 24
(D) 124

45. Find the y-intercept of the line tangent to  at x = 2.

(A) 0
(B) 2.081
(C) 4.161
(D) 21.746

 

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.When a faulty seam opened at the bottom of an elevated hopper, grain began leaking out onto the ground. After a while, a worker spotted the growing pile below and began making repairs. The following table shows how fast the grain was leaking (in cubic feet per minute) at various times during the 20 minutes it took to repair the hopper.

(a) Estimate L′(15) using the data in the table. Show the computations that lead to your answer. Using correct units, explain the meaning of L′(15) in the context of the problem.
(b) The falling grain forms a conical pile that the worker estimates to be 5 times as far across as it is deep. The pile was 3 feet deep when the repairs had been half-completed. How fast was the depth increasing then?
NOTE: The volume of a cone with height h and radius r is given by: .
(c) Use a trapezoidal sum with seven subintervals as indicated in the table to approximate . Using correct units, explain the meaning of  in the context of the problem.

 

2. An object in motion along the x-axis has velocity v(t) = (t + e^t)sin t² for 1 ≤ t ≤ 3.

(a) At what time, t, is the object moving to the left?
(b) Is the speed of the object increasing or decreasing when t = 2? Justify your answer.
(c) At t = 1 this object’s position was x = 10. What is the position of the object at t = 3?

 

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. The graph of function f consists of the semicircle and line segment shown in the figure below. Define the area function  for 0 ≤ x ≤ 18.

(a) Find A(6) and A(18).
(b) What is the average value of f on the interval 0 ≤ x ≤ 18?
(c) Write an equation of the line tangent to the graph of A at x = 6. Use the tangent line to estimate A(7).
(d) Give the coordinates of any points of inflection on the graph of A. Justify your answer.

 

4. Consider the curve: 2x² – 4xy + 3y² = 16.

(a) Show .
(b) Verify that there exists a point Q where the curve has both an x-coordinate of 4 and a slope of zero. Find the y-coordinate of point Q.
(c) Find  at point Q. Classify point Q as a local maximum, local minimum, or neither. Justify your answer.

 

5. The graph above represents the curve C, given by  for –2 ≤ x ≤ 11.

(a) Let R represent the region between C and the x-axis. Find the area of R.
(b) Set up, but do not solve, an equation to find the value of k such that the line x = k divides R into two regions of equal area.
(c) Set up but do not evaluate an integral for the volume of the solid generated when R is rotated around the x-axis.

 

6. Let y = f(x) be the function that has an x-intercept at (2,0) and satisfies the differential equation .

(a) Write an equation for the line tangent to the graph of f at the point (2,0).
(b) Solve the differential equation, expressing y as a function of x and specifying the domain of the function.
(c) Find an equation of each horizontal asymptote to the graph of y = f(x).