Math 151 – Review Exam #2
Spring 2008
1.
Find the slope of the tangent line to the graph of
f ( x)  5 x 3  x 2
f
at the given point.
at (1,4)
What is the equation of the tangent line?
2.
Find the Instantaneous Rate of Change for the function at the given number.
Use the formula
f ( x )  f (c )
lim
x c
xc
2
f ( x)  3x  5 x at 4
3.
Find the derivative using the difference quotient.
f ( x)  4 x 2  5 x  6
4.
Give the derivative of each of the following:
3
4
a.
f ( x)  
b.
f ( x)  x 3
c.
f ( x)  5 x 6
d.
f ( x)  3x 2  5 x 4  7
e.
f ( x)  2 x 3
f.
f ( x) 
3
x4
5.
Use the Product Rule to find the derivative of
6.
Use the Quotient Rule to find the derivative of
7.
If
f ( x) 
1
( x  4) 5
3
then
f ( x)  (4 x 2  5)(3x 3  7)
f ( x) 
f ' ( x)  ______________ .
3x 2  5
4x  5
If
f ( x)  [ x( x  5)] 4
9.
If
 3x 2  4 

f ( x)  
 2x 
10.
If
f ( x) 
11.
Give the derivative of each:
8.
12.
then
2
then
(3x 2  4) 2
2x
a.
f ( x)  3 e 2 x
b.
5 ex
f ( x) 
x
c.
f ( x)  ln x 5
d.
f ( x)  5 x
e.
f ( x)  log 3 x
f ' ( x)  ______________ .
then
f ' ( x)  ______________ .
f ' ( x)  ______________ .
2
Give the following for
f ( x)  x 6  2 x 5  3x 2  2 x  1
f ' ( x)  ______________
f ' ' ( x)  ______________
f ' ' ' ( x)  ______________
f ( 4) ( x)  ______________
f (5) ( x)  ______________
f (6) ( x)  ______________
f (7 ) ( x)  ______________
13.
Use the Chain Rule to find the derivative:
if
y
5
u3
and
u  2x  3
14.
15.
Find the derivative by implicit differentiation:
a.
5x 2  6 y 3  2 x
b.
x  ln ( x 2  y 2 )
Give the derivative of each:
3
a.
16.
f ( x)  x 4
2
3
b.
f ( x)  3 x
c.
f ( x)  3 2 x  1
Given the cost demand functions for a certain product:
C ( x)  60 x  7200
and
p  2 x  300
Find:
17.
a.
The revenue function
b.
The marginal revenue
c.
The marginal cost
d.
The break-even point
e.
The x where marginal cost = marginal revenue
A stone is dropped from a bridge that is 100 feet above the water. Its height h after t seconds is
h(t )  16t 2  100
a.
How much time elapses before the stone hits the water?
b.
What is the instantaneous velocity as the stone hits the water?
c.
What is the average rate of change from 1 second to 2 seconds?
ANSWERS:
1.
13
y = 13x + 9
2.
19
3.
8x – 5
4.
a.
b.
c.
d.
0
3x2
– 30x5
6x – 20x3
e.
 6 x 4 
f.
 12 x 5
6
x4
 12
 5
x
5.
60x4 – 45x2 + 56x
6.
12 x 2  30 x  20
(4 x  5) 2
7.
 15 x 2 ( x 3  4) 4
 15 x 2

( x 3  4)10
( x 3  4) 6
8.
4[x(x - 5)]3[(x - 5)+ x] = 4x3(2x – 5)(x – 5)3
9.
48 x 3 (3x 2  4)  8 x(3x 2  4) 2 (3x 2  4)(3x 2  4)

16 x 4
2x 3
10.
24 x 2 (3x 2  4)  2(3x 2  4) 2 (3x 2  4)(9 x 2  4)

4x 4
2x 2
11.
a.
6e 2 x
2
b.
c.
10 x 2 e x  5e x
x2
5
x
d.
5 x ln 5
e.
1
x ln 3
2
12.
f ' ( x)  6 x 5  10 x 4  6 x  2
f ' ' ( x)  30 x 4  40 x 3  6
f ' ' ' ( x)  120 x 3  120 x 2
f
( 4)
( x )  360 x 2  240 x
f
(5)
( x)  720 x  240
f
(6)
( x )  720
f
(7)
( x)  0
13.
 30
(2 x  3) 4
14.
a.
 10 x  2  5 x  1

18 y 2
9y2
b.
x 2  y 2  2x
2y
a.
3 41
3
x  1
4
4x 4
b.
-2x-5/3 =  2
15.
x
c.
17.
2(2 x  1)
3
2
3
5
3
 2
3(2 x  1)
2
3
R( x)  2 x 2  300 x
R' ( x)  4 x  300
C ' ( x)  60
60
60
16.
a.
b.
c.
d.
e.
a.
b.
c.
2.5 seconds
-80 ft./sec.
-48 ft./sec.