AP Calculus Midterm

AP Calculus Midterm

December 17, 2009

Section 1 Part A

Number of Questions: 10

25 Minutes

A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given to anything written in the test book. Do not spend too much time on any one problem

In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers.

1. If , find

(A) -72
(B) -32
(C) -24
(D) 24
(E) 32

2. is

(A) 0
(B) 10
(C) –10
(D) 5
(E) the limit does not exist

3. If , find

(A)
(B)
(C)
(D)
(E)

4. Iff is continuous on [− 2,4] and f (−2) = 5, f (0) = − 3, and f (4) = 711, then according to the Intermediate Value Theorem, how many zeroes are guaranteed on the closed interval [−2, 4]?

(A) none

(B) one

(C) two

(D) three

(E) four

5.

(A)
(B)
(C)
(D)
(E)

1. Find k so that is continuous for all x.

(A) All real values of k make f(x) continuous for all x
(B) 0
(C) 16
(D) 8
(E) There is no real value of k that makes f(x) continuous

1. An equation of the line tangent to is

(A)
(B)
(C)
(D)
(E)

1. Find a positive value c, for x, that satisfies the conclusion of the Mean Value Theorem for Derivatives for on the interval [2,5].

(A) 1
(B)
(C)
(D)
(E)

1. The graph of has a horizontal asymptote at y = 2 and a vertical asymptote at x=-3.

What is a + c = ?

(A)-5

(B)–1

(C)0

(D)1

(E)5

10.

(A)-1

(B)0

(C)1

(D)

(E) undefined

STOP

END OF PART A SECTION I

IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO.

AP Calculus Midterm

December 17, 2009

Section 1 Part B

Number of Questions: 6

20 Minutes

A CALCULATOR MAY BE USED ON THIS PART OF THE EXAMINATION

Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given to anything written in the test book. Do not spend too much time on any one problem

In this test:

1. The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select among the choices the best number that approximates the exact numerical value.
2. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers.

11. What are all values of k for which

(A)–3
(B)0
(C)3
(D)–3 and 3
(E)–3, 0, and 3

1. If f is the antiderivative of such that f(1) = 0 then the f(4) =?

(A)127.5

(B)0

(C)-0.5

(D).5

(E)128.5

13. Find the value(s) of of

(A) only
(B) only
(C) only
(D)
(E)

14. The graph of has an inflection point(s) at

(A) x=0 only
(B) x=3 only
(C) x=0,3
(D) x=-3 only
(E) x=0,-3
15. Find the average rate of change of the function f(x) on [0, 2] if
(A)–1
(B)–4
(C)0
(D)1
(E)4

16. The data for acceleration a(t) of a car from 0 to 6 seconds are given in the table above. If the velocity at t=0
is 11 feet per second, the approximate value of the velocity at t = 6, computed using a left-hand Riemann sum
with three subintervals of equal length, is:
(A)26 ft / sec
(B)30 ft/sec
(C)37 ft/sec
(D)39 ft/sec
(E)41 ft/sec

STOP

END OF SECTION I

IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART B ONLY. DO NOT GO ON TO SECTION II UNTIL YOU ARE TOLD TO DO SO.

AP Calculus Midterm

December 17, 2009

Section II Part A

Number of Questions: 2

20 Minutes

A CALCULATOR MAY BE USED ON THIS PART OF THE EXAMINATION

Directions: On Part A, you are permitted to use your calculator to solve an equation, find the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the set up of your problem, namely the equation, function, or integral you are using. If you use other built in features or programs, you must show the mathematical steps necessary to produce your results.

17. Consider the curve defined by

1. Find the equation of the tangent line to the curve at x=-1.
2. Find the coordinates of the absolute minimum.
3. Find the coordinates of the point(s) of inflection.

18. Given the following table:

x / / / /
1 / 1 / 3 / -2 / -1
-2 / -2 / -5 / 1 / 7

Find the following derivatives.

a. at x = - 2 / d. at x = 1
b. at x = - 2 / e. at x = -2
c. at x = - 2 / f. at x = -2

AP Calculus Midterm

December 17, 2009

Section II Part B

Number of Questions: 1

15 Minutes

A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

During the timed portion for Part B, you may continue to work on problems in Part A without the use of a calculator.

A table of values for the velocity v(t) in ft/sec of a car traveling on a straight road for 0 t 50 seconds is shown below.

t (secs) / v(t) ft/sec
0 / 0
5 / 10
10 / 20
15 / 32
20 / 60
25 / 82
30 / 90
35 / 103
40 / 91
45 / 62
50 / 70

1. During what intervals of time between 0 t 50 seconds is the acceleration of the car negative?

Give a reason for your answer.

Interval(s):

Justification:

b. Find the average acceleration of the car, in ft/sec2, over the interval 5 t 25 seconds.

c. Find one approximation for the acceleration of the car, in ft/sec2, at t = 20 seconds. Show the computation you used to arrive at your answer.

d. Approximate with a left hand Reimann sum, using three subintervals of equal length. Use correct units.

e. Explain the meaning of the integral in part d using correct units.

END OF EXAMINATION