AP Calculus BC Unit 2 Progress Check MCQ Part A

1.
X | 0 | 1
F(x) | 1 | 2

Let f be the function given by f(x) = 2^(x^2). Selected values of f are given in the table above. If the values in the table are used to approximate f'(0.5), what is the difference between the approximation and the actual value of f'(0.5)?

A. 0
B. 0.176
C. 0.824
D. 1

2. Let f be the function given by f(x) = (1/7)x^7 - (7/6)x^6 + 3x^5 - (5/4)x^4 - (16/3)x^3 + 6x^2. Which of the following statements is true?

A. f'(-1.1) < f'(0.5) < f'(1.4)
B. f'(-1.1) < f'(1.4) < f'(0.5)
C. f'(0.5) < f'(1.4) < f'(-1.1)
D. f'(1.4) < f'(0.5) < f'(-1.1)

3.
X | 1 | 2 | 3 | 4 | 5 | 6
f(x) | -3 | -1 | 1 | 2 | 5 | 10

Selected values of a function f are shown in the table above. What is the average rate of change of f over the interval [1, 6]?

A. (6-1)/[10-(-3)]
B. [10 + (-3)]/(6+1)
C. [10-(-3)]/(6-1)
D. [(-3) + (-1) + 1 + 2 + 5 + 10]/6

4. The graph of the function f, shown above, consists of three line segments. What is the average rate of change of f over the interval 1 < x < 7? (Should be less than or equal to)

A. -2
B. 0
C. 1/3
D. 3

5. The function f is given by f(x) = 1 + 2sin(x). What is the average rate of change of f over the interval [0, pi/2]?

A. 4/pi
B. 6/pi
C. 8/pi
D. 2

6. The derivative of a function f is given by f'(x) = 0.2x + e^(0.15x). Which of the following procedures can be used to determine the value of x at which the line tangent to the graph of f has slope 2?

A. Evaluate 0.2x + e^(0.15x) at x=2
B. Evaluate d/dx (0.2x + e^(0.15x)) at x=2
C. Solve 0.2x + e^(0.15x)= 2 for x
D. Solve d/dx (0.2x + e^(0.15x) = 2) for x

7. The derivative of the function f is given by f'(x) = -2x + 4 for all x, and f(-1) = 5. Which of the following is an equation for the line tangent to the graph of f at x = -1?

A. y = -2x + 3
B. y = -2x + 4
C. y = 6x + 5
D. y = 6x + 11

8. The graph of f', the derivative of a function f, is shown above. The points (2,7) and (4,18.8) are on the graph of f. Which of the following is an equation for the line tangent to the graph of f at x = 2?

A. y = 2x - 1
B. y = 4x - 1
C. y = 4x - 8
D. y = 5.9x - 4.8

9. The graph of the trigonometric function f is shown above for a < x < b. At which of the following points on the graph of f could the instantaneous rate of change of f equal the average rate of change of f on the interval [a,b]? (Should be less than or equal to)

A. A
B. B
C. C
D. D

10. Which of the following statements, if true, can be used to conclude that f(3) exists?

I. lim (x->3) x exists
II. f is continuous at x=3
III. f is differentiable at x=3

A. I only
B. II only
C. II and III only
D. I, II, and III

11. f(x) { 4x + 2 for x < 2 , 3x + 4 for x > 2
Let f be the function defined above. Which of the following statements is true?

A. f is neither continuous nor differentiable at x=2
B. f is continuous but not differentiable at x=2
C. f is differentiable but not continuous at x=2
D. f is both continuous and differentiable at x=2

12. The graph of the function f, shown above, has a vertical tangent at x=3 and horizontal tangents at x=2 and x=4. Which of the following statements is false?

A. f is not differentiable at x = 3 because the graph of f has a vertical tangent at x = 3
B. f is not differentiable at x = -2 and x = 0 because f is not continuous at x = -2 and x = 0
C. f is not differentiable at x=-1 and x=1 because the graph of f has sharp corners at x = -1 and x = 1
D. f is not differentiable at x = 2 and x = 4 because the graph of f has horizontal tangents at x = 2 and x = 4

13. If f(x) = x^6, then f'(x) =

A. x^5
B. 5x^5
C. 6x^5
D. 6x^6

14. If f(x) = 1/(x^6), then f'(x) =

A. 1/(6x^5)
B. -6/(x^5)
C. -1/(6x^7)
D. -6/(x^7)

15. If f is the function defined by f(x) = x^(1/3), what is f'(x)?

A. (1/3)x^(1/3)
B. x^(-2/3)
C. (1/3)x^(-2/3)
D. 3x^(1/2)