Diagnostic Test Calculus BC - 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 particle moves along the parametric curve given by and y(t) = (t³ + 2)². Which of the following is the velocity vector at time t = 1?

(A)〈2e³, 18〉
(B)〈2e³, 6〉
(C)〈e³, 6〉
(D)〈e³, 18〉

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 = 1
(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. The region in the first quadrant bounded by the x-axis, the y-axis, and the curve of y = e^–x is rotated about the x-axis. The volume of the solid obtained is equal to

8. Of the following, which is the Maclaurin series for x² sin(x²)?

9. Which series diverges?

10. 

(A) 
(B) 4
(C) 8
(D) divergent

11. The table above gives values of differentiable functions f and g. If H(x) = f^–1(x), then H′(3) equals

12. 

(A) 1
(B) –1
(C) 
(D) e – 1

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 length of the curve y = 2x^3/2 between x = 0 and x = 1 is equal to

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 above. 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 particle moves along the curve given parametrically by x = tan t and y = 2 sin t. At the instant when , the particle’s speed equals

19. Suppose  and y = 2 when x = 0. Use Euler’s method with two steps to estimate y at x = 1.

(A) 1
(B) 2
(C) 3
(D) 5

20. The graph above consists of a quarter-circle and two line segments and represents the velocity of an object during a 6-second interval. The object’s average speed (in units/sec) during the 6-second interval is

21. Which of the following is the interval of convergence for the series 

(A) (− 3,3)
(B) [− 5,−1)
(C) [− 1,5)
(D) (− ∞,∞)

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. For the function f shown in the graph, which has the smallest value on the interval 2 ≤ x ≤ 6?

(A) 
(B) the left Riemann Sum with 8 equal subintervals
(C) the midpoint Riemann Sum with 8 equal subintervals
(D) the trapezoidal approximation with 8 equal subintervals

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. What is the radius of the Maclaurin series for ?

29. 

(A) 3 ln|1 − x| − 100 ln|5x − 7| + C
(B) 3 ln|1 − x| − 4 ln|5x − 7| + C
(C) − 3 ln|1 − x| − 4 ln|5x − 7| + C
(D) − 3 ln|1 − x| − 20 ln|5x − 7| + C

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.

 

31. The series  converges

(A) for all real x
(B) if 0 ≤ x < 2
(C) if 0 < x ≤ 2
(D) only if x = 1

32. If f(x) is continuous at the point where x = a, which of the following statements may be false?

33. A Maclaurin polynomial is to be used to approximate y = sin x on the interval –π ≤ x ≤ π. What is the least number of terms needed to guarantee no error greater than 0.1?

(A) 3
(B) 4
(C) 5
(D) 6

34. The region S in the figure above is bounded by y = sec x and y = 4. What is the volume of the solid formed when S is rotated about the x-axis?

(A) 11.385
(B) 23.781
(C) 53.126
(D) 108.177

35. Values of f′(x) are given in the table above. Using Euler’s method with a step size of 0.5, approximate f(3), if f(2) = 6.

(A) 7.90
(B) 7.45
(C) 4.55
(D) 4.10

36. If x = 2t – 1 and y = 3 – 4t², then  is

(A) 4t
(B) –4t
(C) 
(D) –8t

37. For a function, g, it is known that g(3) = − 2, g′(3) = 5, g″(3) = 4, and g″(3) = 9. The function has derivatives of all orders. Find the third-degree Taylor polynomial for g about x = 3, and use it to approximate g(3.2).

(A) –0.768
(B) –0.896
(C) –0.908
(D) –0.920

38. The coefficient of x³ in the Taylor series of ln (1 – x) about x = 0 (the Maclaurin series) is

39. The rate at which a rumor spreads across a campus of college students is given by , where P(t) represents the number of students who have heard the rumor after t days. If 200 students heard the rumor today (t = 0), how many will have heard it by midnight the day after tomorrow (t = 2)?

(A) 320
(B) 474
(C) 726
(D) 1,015

40. Given function. f, defined by. . Find the average rate of change of. f on the interval [–1,1].

(A) –1.433
(B) 0
(C) 0.264
(D) 0.693

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. How many gallons will leak out during the first day?

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

45. The first-quadrant area inside the rose r = 3 sin 2θ is approximately

(A) 1.5
(B) 1.767
(C) 3
(D) 3.534

 

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. A particle is moving in the plane with position (x(t),y(t)) at time t. It is known that  and . The position at time t = 0 is x(0) = 4 and y(0) = 3.

(a) Find the speed of the particle at time t = 2, and find the acceleration vector at time t = 2.
(b) Find the slope of the tangent line to the path of the particle at t = 2.
(c) Find the position of the particle at t = 2.
(d) Find the total distance traveled by the particle on the interval 0 ≤ t ≤ 2.

 

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. Let f be the function satisfying the differential equation  and passing through (0,–1).

(a) Sketch the slope field for this differential equation at the points shown.

 

(b) Use Euler’s method with a step size of 0.5 to estimate f(1).
(c) Solve the differential equation, expressing f as a function of x.

 

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 solve, an integral for the volume of the solid generated when R is rotated around the x-axis.

 

6. The function p is given by the series

(a) Find the interval of convergence for p. Justify your answer.
(b) The series that defines p is the Taylor series about x = 2. Find the sum of the series for p.
(c) Let . Find , if it exists, or explain why it cannot be determined.
(d) Let r be defined as r(x) = p(x³ + 2). Find the first three terms and the general term for the Taylor series for r centered at x = 0, and find .