Derivative
Practice
Definition
Family of f:
Of the
f & f’ & f’’ Derivative
Applied
Derivatives
NO Calc
Derivatives
WITH Calc
Applied
100
100
100
100
100
200
200
200
200
200
300
300
300
300
300
400
400
400
400
400
500
500
500
500
500
1
d
3 2
cos x 
dx
(A)
(B)
(C)
(D)
(E)
2
6 x cos x
3
sin x
2
2
2
2
6 x sin x cos x
2
3sin x cos x
2
2 2
6 x sin x cos x
2
2
2
2
Solution (100 pts)
E.
6 x sin x cos x
2
2
2
3
If
f ( x)  x 4x 1
then
f '( x)
is
(A)
6x 1
4x 1
(B)
2x
4x 1
(C)
1
4x 1
(D)
6 x  2
4x 1
(E)
9x  2
2 4x 1
4
Solution (200 pts)
A.
6x 1
4x 1
5
If
then
1
f ( x)   x  x  2
x
5
f '(1) 
(A) 8
(B) 2
(C) -2
(D) -3
(E) -8
6
Solution (300 pts)
C.
-2
7
If
x  25  y
2
2
what is the value of
2
d y at (3, 4)
dx
2
(A) 25
(B) 7
64
64
7
64
(D) 25
(C)
(E)
4
3
64
8
Solution (400 pts)
A.
25
64
9
Let f and g be differentiable functions with
the following properties:
I. f ( x)  0 for all x
II. g (5)  2
f '( x)
f ( x)
If h( x)  g ( x) and h '( x)  g ( x) then g ( x) 
(A)
1
f '( x)
(B)
f ( x)
(C)
 f ( x)
(D)
0
(E) 2
10
Solution (500 pts)
E.
2
11
(A)
1
(B)
3x
3x
e
If f ( x) 
3x
then f ' ( x) 
(C)
e
e 3 x (1  3 x)
3x 2
e 3 x (3 x  1)
(D)
2
3x
(E)
e 3 x (3 x  1)
3x 2
12
Solution (100 pts)
E
e (3 x  1)
2
3x
3x
13
The graph of the function
1 3
2
y  x  x  5 x  3 sin x
3
changes concavity at x =
(A) 3.29
(B) 2.21
(C) 1.34
(D) 0.41
(E) -0.39
14
Solution (200 pts)
B
2.21
15
Let f be a function such that f ''( x)  0
For all x in the closed interval [3,4] with
selected values shown in the table. Which
of the following must be true about f '(3.3)?
X
3.2
3.3
3.4
3.5
f(x)
2.48
2.68
2.86
3.03
(A) f '(3.3)  0
(B) 0  f '(3.3)  1.6
(C) 1.6  f '(3.3)  1.8
(D) 1.8  f '(3.3)  2.0
(E) f '(3.3)  2.0
16
Solution (300 pts)
D
1.8  f '(3.3)  2.0
17
y  5x  24x  24x  17
4
3
2
is concave down for
(A) x  0
(B) x  0
(C)
2
x  2 or x  
5
(D)
2
x  2 or x 
5
(E)
2
x2
5
18
Solution (400 pts)
E
2
x2
5
19
What are all the values of x for
which the function f defined by
f ( x)  ( x 15)e
2
x
is increasing?
(A) There are no such values of x
(B) x  3 or x  5
5  x  3
(C)
3  x  5
(D)
(E) All values of x
20
Solution (500 pts)
D.
3  x  5
21
Let
( x  h)  x
f ( x)  lim
h 0
h
2
For what value of x does
(A)
(B)
(C)
(D)
(E)
2
f ( x)  4
-2
-1
1
2
4
22
Solution (100 pts)
D
2
23
Let f be a function such that
f (7  h)  f (7)
lim
4
h 0
h
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)
(B)
(C)
(D)
(E)
I only
II only
I and II only
I and III only
II and III only
24
Solution (200 pts)
C
I and II only
25
(10  h)  1000
lim
h 0
h
3
(A)
(B)
(C)
(D)
(E)
0
1
30
300
3000
26
Solution (300 pts)
D. 300
27
x 2
lim
x 8 x  8
3
(A)
(B)
(C)
(D)
(E)
0
1/12
1/3
4/3
nonexistent
28
Solution (400 pts)
B. 1/12
29

 1
cos   h  
3
2


lim
h 0
h
(A)
-1
(B)
3

2
(C)
1

2
(D)
1
2
(E)
3
2
30
Solution (500 pts)
B.
3

2
31
An equation of the line tangent to the
graph of
y  cos3x
at
x

is
6
(B) y  ( x   )
(A) y  3( x   )
6
6


(D) y  1  ( x  )
(C) y  3( x  )
6
6

(E) y  1  2( x  6 )
32
Solution (100 pts)
C.
y  3( x 

6
)
33
Let
f ( x)  x  3
then
f '(1)
(A)
(B)
(C)
(D)
(E)
-1
0
1
2
nonexistent
34
Solution (200 pts)
A. -1
35
The minimum acceleration attained on
the interval 0 ≤ t ≤ 4
by the particle whose velocity is given
by
is
3
2
v(t )  t  4t  3t  2
(A)
(B)
(C)
(D)
(E)
-16
-10
-8
-25/3
-3
36
Solution (300 pts)
D. -25/3
37
The maximum value of
f ( x)  x  3x  9x  2
3
2
on the interval [0, 2] is
(A)
(B)
(C)
(D)
(E)
25
-7
-2
0
2
38
Solution (400 pts)
D. 0
39
A particle is moving on the x-axis with
position given by
x(t )  t  sin(2t ) for 0  t  2
Then the particle is at rest only when t =

2
(A)
(C)
2
4
and
3
3
(E)  , 2 , 4
3
3
3
(B)

2
(D)

5
and
3
and
3
2
2
and
3
3
40
Solution (500 pts)
E.
 2 4
3
,
3
,
3
5
and
3
41
Which of the following is an equation
of the line tangent to the graph of
f ( x)  x  x
6
at the point where
4
f '( x)  1
(A) y   x  1.031 (B) y   x  0.836
(C) y   x  0.836 (D) y   x  0.934
(E) y   x  1.031
42
Solution (100 pts)
A.
y   x  1.031
43
The side of a cube is increasing at a
constant rate of 0.2 centimeter per
second. In terms of the surface area
S, what is the rate of change of the
volume of the cube, in cubic
centimeters per second?
(A)
(B)
(C)
(D)
(E)
0.1S
0.2S
0.6S
0.04S
0.008S
44
Solution (200 pts)
A. 0.1S
45
Let f be a function that is differentiable on
the open interval (-3, 7).
If f(-1) = 4, f(2) = -5 and f(6) = 8,
which of the following must be true?
I. f has at least 2 zeroes
II. f has a relative minimum at x = 2
III. For some c, 2 < c < 6, f(c) = 4
(A) I only
(B)
II only
(C) I and II only
(D) I and III only
(E) I, II, and III
46
Solution (300 pts)
D.
I and III only
47
If f is continuous on [2, 5] and
differentiable on (2, 5) with f(2) = -4
and f(5) = 14, which of the following
statements must be true?
I. f(x) = 6 has a solution in (2, 5)
II. f’(x) = 6 has a solution in (2, 5)
III. f’’(x) = 6 has a solution in (2, 5)
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III
48
Solution (400 pts)
C. I and II only
49
At the moment that a rectangle is 8 feet
long and 3 feet wide, its length is
increasing at 0.5 feet/minute and its
width is decreasing at 1.5 feet/minute.
The area is
(A)
(B)
(C)
(D)
(E)
decreasing at 10.5 square feet/minute
increasing at 13.5 square feet/minute
increasing at 8.5 square feet/minute
decreasing at 0.5 square feet/minute
decreasing at 0.75 square feet/minute
50
Solution (500 pts)
(A) decreasing at
10.5 square
feet/minute
51