AB Calc Midterm Review 08

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Midterm Exam Review Questions Free Response – Non Calculator Directions: Solve each of the following problems. Choose the BEST answer choice from those given. A calculator may not be used. Do not spend too much time on any one problem. In this test: Unless otherwise specified, the domain of the function f is assumed to be the set of all real numbers x for which f(x) is a real number. 1. The graph of y = 3x2 – x3 has a relative maximum at (A) (0,0) only (B) (1,2) only (C) (2,4) only (D) (4, −16) only (E) (0,0) and (2,4)

2.

lim x

10 8 x 5 10 6 x 4 10 4 x 2 10 9 x 6 10 7 x 5 10 5 x 3

(A) 0 (B) 1 (C) −1 (D) 1/10 (E) −1/10

y

B B

A x

C

E D

3. The figure above shows the graph of the velocity of a moving object as a function of time. At which of the marked points is the speed the greatest? (A) (B) (C) (D) (E)

A B C D E

AP Calculus AB Midterm Exam Revision

2012-2013

4. What are all values of x for which the graph of y = (A) (B) (C) (D) (E)

4

x

is concave downward?

No values of x x–4 x4

5. The equation of the tangent line to the curve (A) (B) (C) (D) (E)

2

x2 + y2 = 169 at the point (5, −12) is

5y – 12x = −120 5x – 12y = 119 5x – 12y = 169 12x + 5y = 0 12x + 5y = 169

6. If the graph of f ( x)

2 x2

(A) −2 (B) −1 (C) 0 (D) 1 (E) 2

k x

has a point of inflection at x

1 , then the value of k is

7. A particle moves along the x-axis in such a way that its position at time t is given by x(t )

1 t . What is 1 t

the acceleration of the particle at time t = 0? (A) −4 (B) −2 (C) −3/5 (D) 2 (E) 4

8. If

dy dx

x 2 y 2 , then

d2y = dx 2

(A) 2xy 2 (B) 4x 3 y 3 (C) 2 x 2 x 2 y 3 (D) 2 x 2 y 2 xy 2 (E) 2 x 4 y 3 2 xy 2

AP Calculus AB Midterm Exam Revision

2012-2013

y

9.

g(x)

g(x) f (x)

x

f (x) Piecewise functions f and g are shown above. If h(x) = f ( x) ● g ( x) , then h (3) = (A) −8/3 (B) −1/3 (C) 0 (D) 2/3 (E) 8/3 10. The average rate of change of the function

f ( x) cos

1 x on the closed interval [−4, 0] is 2

1 sin(2) 2 1 − sin(2) 4 1 cos(2) 4 1 cos(2) 4 1 sin(2) 4

(A) − (B) (C) (D) (E)

11. If

6 0

( x2

2 x 2)dx is approximated by three inscribed rectangles of equal width on the x-axis, then the

approximation is (A) (B) (C) (D) (E)

24 26 28 48 76

AP Calculus AB Midterm Exam Revision

2012-2013

Name:

Date:

Block:

Midterm Exam Review Questions Free Response – Calculator Active Directions: Solve each of the following problems. Choose the BEST answer choice from those given. A calculator may be used. Do not spend too much time on any one problem. In this test: (1) The exact numerical value of the correct answer may not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. (2)

Unless otherwise specified, the domain of the function f is assumed to be the set of all real numbers x for which f(x) is a real number.

12. Let f be the function given by f(x) = tan x and let g be the function given by g(x) = x2. At what value of x in the interval 0 ≤ x ≤ do the graphs of f and g have parallel tangent lines? (A) 0 (B) 0.660 (C) 2.083 (D) 2.194 (E) 2.207

1 for t > 0. For what value of t is f  (t) equal to the average rate of change t of f on the closed interval [a, b]?

13. Let

f (t )

(A)

ab

(B)

ab

(C)

1 ab

(D)

1 ab

(E)

1 1 2 b

1 a

AP Calculus AB Midterm Exam Revision

2012-2013

R(x)

C B x

A

500

1000

1500

2000

14. The figure above shows a road running in the shape of a parabola from the bottom of a hill at A to point B. At B, it changes to a line and continues to on to C. The equation of the road is

R( x)

ax 2 , From A to B bx c, From B to C

B is 1,000 feet from A and 100 feet higher. Since the road is smooth, R (x) is continuous. What is the value of b? (A) 0.2 (B) 0.02 (C) 0.002 (D) 0.0002 (E) 0.00002

y

f ( x)

15. The figure above shows the graph of the derivative of a function f. How many points of inflection does f have in the interval shown? (A) (B) (C) (D) (E)

None One Two Three Four

AP Calculus AB Midterm Exam Revision

2012-2013

16. The amount A(t) of a certain item produced in a factory is given by A(t) = 4000 + 48(t – 3) – 4(t – 3)3 Where t is the number of hours of production since the beginning of the workday at 8:00 a.m. At what time is the rate of the production increasing most rapidly? (A) (B) (C) (D) (E)

8:00 a.m. 10:00 a.m. 11:00 a.m. 12:00 noon 1:00 p.m.

17. At how many points on the curve y the origin? (A) (B) (C) (D) (E)

4 x 5 3x 4 15 x 2

6 will the line tangent to the curve pass through

One Two Three Four Five

18. Suppose that f ( x) is an even function and let

1 0

f ( x)dx

5 and

7 0

f ( x)dx 1 . What is

1 7

f ( x )dx ?

(A) −5 (B) −4 (C) 0 (D) 4 (E) 5

19. The graph of the derivative of a twice differentiable function is shown below. y

y

f ( x) x

1

2

3

If f (1) = −2, which of the following must be true? (A) (B) (C) (D) (E)

f (2) < f ′(2) < f ′′(2) f ′′(2) < f ′(2) < f (2) f ′(2) < f (2) < f ′′(2) f (2) < f ′′(2) < f ′(2) f ′ (2) < f ′′(2) < f (2)

AP Calculus AB Midterm Exam Revision

2012-2013

20. Let f be a function that is everywhere differentiable. The value of f ′(x) is given for several values of x in the table below. x

−10

−5

0

5

10

f ′(x)

−2

−1

0

1

2

If f ′(x) is always increasing, which statement about f (x) must be true? (A) (B) (C) (D) (E)

f (x) f (x) f (x) f (x) f (x)

has a relative min at x = 0. is concave down for all x. has a point of inflection at (0, f (0)) passes through the origin is an odd function

21. The table below gives the values of a differentiable function f. what is the approximate value of f ′ (4)? x 3.99800 3.99900 4.00000 4.00100 4.00200 (A) (B) (C) (D) (E)

f (x) 1.15315 1.15548 1.15782 1.16016 1.16250

0.00234 0.289 0.427 2.340 f ′ (4) can not be approximated from the information given.

22. The function f (x) = tan(3x) has a zero in [0, 1.4]. The derivative at this point is (A) (B) (C) (D) (E)

0.411 1.042 3.451 3.763 undefined

23. The edge of a cube is increasing at the rate of 0.05 centimeters per second. In terms of the edge of the cube, s, what is the rate of change of the volume of the cube, in cubic centimeters per second? (A) (B) (C) (D) (E)

0.053 0.05s2 0.05s3 0.15s2 3s2

AP Calculus AB Midterm Exam Revision

2012-2013

24. Which graph best represents the position of a particle, s(t), as a function of time, if the particle’s velocity and acceleration are both positive? (A)

s(t)

s(t)

(B)

(C)

t

(D)

s(t)

t

t

s(t)

(E)

t

t

25. Let f be a function such that

s(t)

lim h

0

f (7 h) h

f (7)

12 . Which of the following must be true?

I. f is continuous at x = 7 II. f is differentiable at x = 7 III. The derivative of f is continuous at x = 7 (A) I only (B) II only (C) I and II only (D) I and III only (E) II and III only

MC Answer Key

1. C 2. A 3. D 4. E 5. C

Non-Calculator 6. E 11. B 7. E 8. E 9. C 10. D

12. C 13. B 14. A 15. E 16.C

AP Calculus AB Midterm Exam Revision

Calculator 17. A 18. B 19. A 20. A 21. D

22. C 23. D 24. C 25. C

2012-2013