Name:
Date:
Block:
Chapter 2 Revision Packet
1) Determine whether the slope at the indicated point is positive, negative, or zero.
2) Represent the derivative of f ( x)
x2
x using the definition of the derivative.
3) If f (1) 4 and f ' (1) 2 find an equation of the tangent line when x 1 .
4) Find the derivative of y
2
5) Find the slope of the tangent line to the graph of f ( x) 3x 1 when x
1.
6) At each point on the graph, determine whether the value of the derivative is positive, negative, zero, of
the function has no derivative.
A)
B)
C)
D)
E)
7) Consider f ( x)
x
Use the definition of the derivative to calculate the derivative of f .
Find the slope of the tangent line to the graph at the point (4,2)
Write an equation of the tangent line to the graph at the point (4,2).
8) Find f ' ( x) : f ( x)
4 x 4 5 x3 2 x 3
9) Let g ( x) 9 f ( x) and let f ' ( 6)
6 . Find g ' ( 6) .
10) Find the instantaneous rate of change of w with respect to y for w
11) Find an equation of the tangent line to the graph of f ( x) 3 x 3
12) Find the values of x for all points on the graph of f ( x)
tangent line is 4.
1
y
y
.
2
x when x 1 .
x 3 2 x 2 5 x 16 at which the slope of the
13) Suppose the position equation for a moving object is given by s (t ) 3t 2 2t 5 where s is measured in
meters and t is measured in seconds. Find the velocity of the object when t 2 .
14) The position function for a particular object is s
a)
b)
c)
d)
e)
The velocity is a constant
The velocity at time t 1 is 74.5
The initial position is 33
The initial velocity is –29
None of these
25 2
t 58t 91 . Which statement is true?
2
15) Find the average rate of change of y with respect to x on the interval [1, 4] where y
x2
x 1.
10t 2 4 , where s is measured in feet and t is
16) The velocity function for an object is given by s ' (t )
measured in seconds. What is the instantaneous velocity when t 2 ?
17) Find
dy
: y
dx
4sin x 5cos x x
18) Differentiate: f ( x)
19) Calculate
d2y
if y
dx 2
x2
2 tan x
1 x
2 x
20) Find the derivative of x 2 f ( x)
21) Let f ( 5) 0, f ' ( 5)
10, g ( 5) 1 , and g ' ( 5)
22) If f " ( x) 5 x 2 9 x 8, find f (4) ( x)
23) Find f ' ( ) : f ( )
24) Find
dy
for y
dx
cot
x3 x 1
1
. Find h ' ( 5) if h( x)
5
f ( x)
.
g ( x)
2(1 x) 2 9(1 x) 6
25) Find the derivative of f ( x)
26) Find the derivative: s(t ) csc
t
2
27) The position equation for the movement of a particle is given by s (t 2 1)3 when s is measured in
feet and t is measured in seconds. Find the acceleration at two seconds.
1
28) Find the derivative: f ( x)
3
30) Find
dy
if y 2 3 xy x 2
dx
32) Find
dy
: x3 2 x 2 y 3xy 2
dx
3 x
7
38
3
29) Find f " ( x) : f ( x)
31) Find
dy
: 2 x2
dx
33) Find
dy
: x
dx
2x2 5
xy 3 y 2
tan( x y)
0
34) Find an equation of the tangent line to the graph of x 2 3 y 2
35) Use implicit differentiation to find
dy
for x 2
dx
xy
36) Find the slope of the normal line to the graph of 4 y 2
y2
4 at the point (1,1) .
5
xy
3 at the point
1,
3
4
37) A machine is rolling a metal cylinder under pressure. The radius of the cylinder is decreasing at a
constant rate of 0.05 inches per second and the volumes is 128 cubic inches. At what rate is the length
h changing when the radius r is 1.8 inches? ( V
r 2h )
38) The formula for the volume of a tank is V 2 r 3 where r is the radius of the tank. If the radius is
3
increasing at the rate of feet per minutes, find the rate at which the volume is increasing when the
2
radius is 3 feet.
39) A 5-meter-long ladder is leaning against the side of a house. The foot of the ladder is pulled away
from the house at a rate of 0.4 m/sec. Determine how fast the top of the ladder is descending when the
foot of the ladder is 3 meters from the house.
40) After reviewing all of this material, what was the easiest topic in this unit and what was the most
difficult topic in this unit? Explain your response.