Derivatives
Name
Supplementary Problems
AP Calculus AB
Date
Calculators allowed for problem 3 only.
1. Use the chart at the right to find:
a. The derivative of f + g at x = 2 .
b. The derivative of f ⋅ g at x = 2 .
c. The derivative of
f
at x = 2 .
g
d. h′(1) where h( x) = f ( g ( x))
1. What is the 50th derivative of cos(x) ?
a.
b.
c.
d.
e.
− cos(x)
cos(x)
sin(x)
− sin( x)
0
x
f (x)
g (x)
f ′(x )
g ′(x)
1
3
2
5
4
2
2
π
6
7
2. If y =
a.
b.
c.
d.
e.
1
− x3
d2y
when x = is
− x 2 then the value of
2
2
3
dx
−1
− 5/ 4
49 / 576
−5
−3
3. If f(x) = sin(2x) then the average rate of change of f on the interval [1, 3] is
a. -0.683
b. -0.594
c. -0.297
d. 0.035
e. 1.376
4. If f ( x ) = 2 x then f ′(2) =
a. 1 / 4
b. 1 / 2
c.
2/2
d. 1
e.
2
5. If f (x) and g (x) are continuous functions such that f ′( x) = g ( x) and g ′( x) = x
then which of the following conclusions are necessarily true?
a. f (x) is a quadratic function
b. f ′( x) = f ′′( x)
c. f ′( x) = g ′( x)
d. f ′′( x) = x
e. g (x) is a linear function
6. If f ( x) = 2 − 4 +
a.
b.
c.
d.
e.
−2
2
0
−8
8
x
then f (x) is not differentiable at x =
2
7. If y = cos 2 x − sin 2 x , then y ′ =
a.
b.
c.
d.
e.
−1
0
− 2 sin( 2 x)
− 2(cos x + sin x)
2(cos x − sin x)
8. Find an equation of the tangent line to the curve y = x 4 − 6 x that is perpendicular
to the line x − 2 y + 6 = 0 .
a. x + 2 y + 6 = 0
b. 2 x + y − 1 = 0
c. x − 2 y + 6 = 0
d. 2 x − y − 1 = 0
e. 2 x + y + 3 = 0
9. If a stone is thrown vertically upward from the ground with an initial velocity of
32 ft/sec, then s = −16t 2 + 32t , where s ft is the distance of the stone from the
starting point at t seconds, and the positive direction is upward. Find the average
velocity of the stone during the time interval 3 / 4, 5 / 4 .
[
a.
b.
c.
d.
e.
]
0
1
8
−8
−1
10. Which of the following statements about even and odd functions is false?
a.
b.
c.
d.
e.
The derivative of every even function is an odd function.
The derivative of every odd function is an even function.
The product of an even function by an even function is an even function.
The product of an odd function by an odd function is an even function.
The product of an odd function by an even function is an even function.
sec( x + h) − sec( x)
=
h →0
h
a. sec x
b. sec 2 x
c. sec x tan x
d. tan 2 x
11. lim
e. does not exist
12. A particle moves on the x -axis in such a way that its position at time t is given by
x(t ) = 3t 5 − 25t 3 + 60t . During which intervals is the particle moving to the left?
a.
b.
c.
d.
e.
− 2 < t < −1 only
− 2 < t < −1 and 1 < t < 2
− 1 < t < 1 and t > 2
1 < t < 2 only
t < −2 and − 1 < t < 1 and t > 2
13. If f ′( x) = x 2 + x − 12 , then f is increasing on
a.
b.
c.
d.
e.
[− 4, 3]
[− 3, 4)
(− ∞, − 0.5]
(− ∞, − 3) ∪ (4, ∞)
(− ∞, − 4) ∪ (3, ∞)
Solutions
1. A. 13
B. 14 + 6π
C.
6π − 14
π2
D. 24
Multiple Choice
1. A 2. E 3.B 4.B 5.D 6.D 7.C 8.E 9.A 10.E 11.C 12.B 13.E