AP Calculus Midterm
December 17, 2009
Name:__________________________
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 g ( x ) 
(A)
(B)
(C)
(D)
(E)
2.
1 4
x  5x 2 , find g' (4)
32
-72
-32
-24
24
32
x 2  25
is
lim
x 5
x 5
(A) 0
(B) 10
(C) –10
(D) 5
(E) the limit does not exist
x5  x  2
, find f ' ( x )
x3  7
5x 4  1
(A)
3x 2
5x 4  1  3x 2
(B)
x3  7
3. If f ( x) 
c hc h
c h
 x  x  2h
cx  7hc5x  1hc
c3x h
cx  7h
 x  7h
cx  x  2hc3x hc
c5x  1h
cx  7h
 x  x  2h
cx  7hc5x  1hc
c3x h
cx  7h
3
(C)
4
2
3
3
(E)
2
3
5
(D)
5
4
3
3
5
3
4
2
2
4. If f 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
z
5.
x 3xdx
(A)
(B)
(C)
(D)
(E)
2 3 52
x C
5
5 3 25
x C
2
3 21
x C
2
2 3x  C
5 3 23
x C
2
R
x  16
;x  4
|
Find k so that f ( x )  Sx  4
|Tk ; x  4
2
6.
(A)
(B)
(C)
(D)
(E)
is continuous for all x.
All real values of k make f(x) continuous for all x
0
16
8
There is no real value of k that makes f(x) continuous
7. An equation of the line tangent to y  4 x 3  7 x 2 at x = 3 is
y  45  66( x  3)
y  45  66( x  3)
y  66 x
y  66( x  3)
1
(E) y  45  ( x  3)
66
(A)
(B)
(C)
(D)
8.
Find a positive value c, for x, that satisfies the conclusion of the Mean Value Theorem for Derivatives for
f ( x)  3x 2  5x  1 on the interval [2,5].
(A) 1
13
(B)
6
11
(C)
6
23
(D)
6
7
(E)
2
ax  b
has a horizontal asymptote at y = 2 and a vertical asymptote at x=-3.
xc
What is a + c = ?
9. The graph of y 
(A)
(B)
(C)
(D)
(E)
-5
–1
0
1
5




 
cos   h  cos 
2
2 ?
10. lim
h 0
h
(A) -1
(B) 0
(C) 1
(D) cos   h
2
(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.
k
11. What are all values of k for which
x
2
dx  0
3
(A) –3
(B) 0
(C) 3
(D) –3 and 3
(E) –3, 0, and 3
12.
If f is the antiderivative of 2x 3 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
3
only
2
2
 only
3
2
only
3
2

3
3

2
(A) 
(B)
(C)
(D)
(E)
dy
of x 2 y  y 2  5 at y  1
dx
14. The graph of y  5x 4  x 5 has an inflection point(s) at
(A)
(B)
(C)
(D)
(E)
x=0 only
x=3 only
x=0,3
x=-3 only
x=0,-3
15. Find the average rate of change of the function f(x) on [0, 2] if f (x)  2x 2  2
(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)
(B)
(C)
(D)
(E)
26 ft / sec
30 ft/sec
37 ft/sec
39 ft/sec
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 y  x 4  4 x 3
a. Find the equation of the tangent line to the curve at x=-1.
b. Find the coordinates of the absolute minimum.
c. Find the coordinates of the point(s) of inflection.
18. Given the following table:
x
1
-2
f ( x)
1
-2
f '( x)
3
-5
g ( x)
-2
1
g '( x )
-1
7
Find the following derivatives.
a.
d
 f ( x)  g ( x) at x = - 2
dx
d. ( f ( x)  g ( x))' at x = 1
b.
d
 f ( g ( x)) at x = - 2
dx
 f ( x)  '
e. 
 at x = -2
 g ( x) 
c.
d
 g ( f ( x)) at x = - 2
dx
 1 '
f. 
 at x = -2
 g ( x) 
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
v(t)
(secs) ft/sec
0
0
5
10
10
20
15
32
20
60
25
82
30
90
35
103
40
91
45
62
50
70
a. 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.
30
d. Approximate
 v(t )dt with a left hand Reimann sum, using three subintervals of equal length.
0
e. Explain the meaning of the integral in part d using correct units.
END OF EXAMINATION
Use correct units.