Chapter 7
1.
Suppose g is the inverse function of a differentiable function f and G( x)  1 / g ( x). If f (4)  3 and
1
f (4)  , find G (3).
16
2.
If h( x)  x  x , find h 1 (30).
Select the correct answer.
a.
23
b.
22
3.
Find the inverse function of f ( x) 
4.
Find f
c.
26
d.
25
e.
28
x 1
.
2x 1
  a .
1
f ( x)  x 3  x 2  x  6 , a  3
5.
Evaluate the integral.
6
 x ln x
dx
e
Select the correct answer.
a. 0.194
0.094
6.
0.533
c.
0.183
d.
0.583
e.
Evaluate the integral.

7.
b.
2
xe 2 x dx
2
0
Use transformations to sketch the graph of the function.
y  3ln( x  2)
8.
Find the volume of the solid obtained by rotating about the y-axis the region bounded by the curves.
y  e  x , y  0, x  0, x  7
2
9.
Find the solution of the equation correct to four decimal places.
e1 2 x  190
10. Find the limit.
lim ln 5  x  ln 4  x
x
11. Suppose that the graph of y =log 2 x is drawn on a coordinate grid where the unit of measurement is an
inch. How many miles to the right of the origin do we have to move before the height of the curve
reaches 2 ft.
Select the correct answer.
a. 66.99 mi
e. 609.99 mi
b.
409.37 mi
12. Find the inverse function.
y
2  ex
5  ex
13. Solve the inequality.


ln x 2  4 x  4  0
Select the correct answer.
a.
b.
c.
d.
e.
x   1, 2  2 2

x   1, 2  2 2

x  2  2 2, 5

x [1, 5]


x  2  2 2, 5

14. Evaluate the integral.

 /2
0
 2  2

2 cos x
1  sin 2 x
dx
2, 5

c. 202.56 mi
d.
264.79 mi
15. Differentiate the function.
y
ln x
5 x
Select the correct answer.
a.
y 
b.
y 
c.
y 
d.
e.
16.
5  x  x ln x
x5  x 2
5  x  x ln x
5  x 2
5  x  x ln x
x5  x 2
1
y 
x5  x 
1
y 
5  x 
If f ( x) 
x
, find f e.
ln x
17. If a bacteria population starts with 150 bacteria and doubles every 7 hours, then the number of bacteria
after t hours is f (t )  150 2t / 7 .
How many hours will it take for the population to reach 10,000?
18. Use logarithmic differentiation to find the derivative of the function.
y  x 4x
Select the correct answer.
a.
b.
c.
d.
e.
19.
y   4x 4 x 4 ln x  1
y   4x 4 x ln x  1
y   4ln x  1
y   ln x  4
y   4 xln x  1
Find the derivative of f (x).
f ( x)  x sinh x
20.
Find the values of  for which y  e x satisfies the equation 3 y  3 y   y .
1.
-1
2.
d
3.
f 1 ( x)  
4.
1
5.
d
6.
1
1
 e8 
4
4
x 1
2x 1
7.

1 

e 49 
8.
 1 
9.
x  2.1235

10. 0
11. d
 5x  2 

12. y  ln 
 x 1 
13. d
14.

2
15. c
16. 0
17. 42.41
18. b
19. sinh x  x cosh x
20.
3  21
2
1.
Solve each equation for x.
(a)
2.
x
(b) ee  3
ln x  5
Differentiate the function.
f ( )  ln cos 4 
3.
Differentiate the function.

y  ln x 4 sin 2 x
4.

Find the volume of the solid obtained by rotating about the y-axis the region bounded by the curves.
y  e  x , y  0, x  0, and x  9
2
Select the correct answer.
a.
 1

V    81  1
e

e.
 1 
V    81 
e 
b.
1 

V   1  81 
e 

c.


V   e 81  1
d.
V  e 81
  (6) .
1
5.
If f ( x)  5  3x  e x , find f
6.
Use logarithmic differentiation to find the derivative of the function.
y  x 2x
Select the correct answer.
a.
y   2x 2 x ln x  4
b.
c.
y   2x 2 x ln x  1
y   2ln x  1
d.
y   x x ln x  4
e.
7.
y   4x 2 x ln x  4
Find the limit.
lim e 5 x
5
x 
Select the correct answer.
a.
1
b.
0
c.

d.

e.
e5
8.
Find an equation of the tangent line to the curve at the given point.
y  7e 2 x cosx,
9.
0,7
Find the absolute minimum value of the function.
g ( x) 
ex
, x0
x
10. Evaluate the integral.

ln x 2 dx
x
Select the correct answer.
ln x 3  C
a.
b.

1
ln x 3  C
3
c.
 ln x 3  C
0
d.
e.
1
ln x 3  C
3
11. Find the limit.
lim x  9 tan
x9
x
18
Select the correct answer.
a.
0
b.
1
c.

e. 
d. 
18

12. Find g ( x).
g ( x) 
x

1
es
ds
s
Select the correct answer.
g ( x) 
a.
g ( x)  2 xe
e x
2x
b.
g ( x ) 
e2 x
x
c.
g ( x ) 
e
x
d.
g ( x) 
e
2 x
e.
x
13. Find the solution of the equation correct to four decimal places.


ln e x  2  7
14. If a bacteria population starts with 150 bacteria and doubles every three hours, then the number of
bacteria after t hours is n  f (t )  150 2t / 3 . When will the population reach 45,000?
 
15. Suppose that the graph of y  log 2 x is drawn on a coordinate grid where the unit of measurement is
an inch. How many miles to the right of the origin do we have to move before the height of the curve
reaches 3 ft?
16. If f ( x) 
 
x
, find f  e 4 .
ln x
17. Evaluate the integral.
6
sin
cos d
Select the correct answer.
6 cos
ln sin  
none of these
a.
b.
6 cos 
C
ln 6 
c.
6 sin
C
ln 6
18. Use logarithmic differentiation to find the derivative of the function.

y  4 x  14 x 4  6

2
19. Find the derivative of the function.
 
y  5 sin 1 x 2
20. Evaluate the integral.

2
 1 y 

 dy
 y 
Select the correct answer.
a.
b.
c.
d.
e.
2 y ln | y | C
y  2ln | y | 1/ y  C
2 y ln | y | 1/ y  C
y  2ln | y | 1/ y  C
2 y  ln | y | 1/ y  C
d.
6 sin
C
ln sin  
e.
1.
(a) x  e5 , (b) x  ln  ln 3
2.
 4 tan4 
3.
y ( x) 
4.
b
5.
1/4
6.
b
7.
b
8.
y  14x  7
9.
g (1)  e
4 sin x  2 x cos x
x sin x
10. e
11. e
12. a
13. x  7.0018
14. 24.69
15. 1084588
16. 0.1875
17. c
18.
4 x  14 x 4  6
19.
20. b
2
10 x
1 x4
3
 16  8 x
 4 x  1 x 4  6






