Math 108 - Practice Problems for the Final Exam
Instructions: Show all of your work. No notes or books are allowed. Only non-graphing
calculators are allowed. Give both exact and approximate answers wherever possible. Enjoy!
1. Evaluate  10x 6  3x dx
2. Evaluate  5e 6x dx
3. Evaluate  5e 6 dx
4. Evaluate 
3
x
5. Evaluate 
4
dx
x 2/3
1x
dx
x
x
e 1
dx
e 2x
1
2
2
x x  x dx

3x
6. Evaluate 
7. Evaluate 
8. Evaluate 
dx
9. Find fx if f x  e  x and f0  2
10. Find fx if f  x  x and the graph of f passes through 1, 5
11. Determine the cost function, Cx, if the marginal cost is 5  0. 2x and the fixed cost is
$200
12. Determine the revenue function, Rx, if the marginal revenue is 10  e 0.02x
13. Determine the profit function, Px, if the marginal profit is 50  0. 3 x and
P0  $130
4
14. Compute 
n0
n
n2
15. Use summation notation to express the sum
2  3  4  5    100
7
5
11
201
9
16. Use summation notation to express the sum
a 0  a 1 e x 1  a 2 e 2x 2  a 3 e 3x 3    a 100 e 100x 100
4
17. Compute  fx i x if fx  x 3  x, x 1  0, x 2  0. 5, x 3  1, x 4  1. 5, and x  0. 5
i1
18. Use five rectangles (equal width, right hand endpoints) to approximate the area bounded
by the graphs of fx  1  x 2 , x  0, x  2, and the x-axis
6
19. Evaluate  5t  t 3 dt
2
1
20. Evaluate   3 x  e x  dx
0
21. Evaluate 
2 1x
1 x
dx
22. Determine the area under the graph of fx  1x from x  1 to x  6
23. Determine the area of the shaded region under the graph of fx  4x  x 2 shown
24. A stone is dropped on the moon from a high cliff and falls with velocity vt  5. 3t
ft/sec. How far does the stone travel during the first 5 seconds?
25. Find the average value of fx  2x over the interval 1. . e 2 
26. Find the volume of the solid of revolution produced by revolving about the x-axis the
region bounded by fx  x  1 , y  0, x  1, and x  8.
27. If the supply and demand functions are Sx  3 x  1 and Dx  11  x
respectively
a. Find the equilibrium point
b. Find the consumer’s surplus
c. Find the producer’s surplus
28. Determine the area between fx  2x  1 and gx  1x  x 2 from x  1 to x  3
29. Determine the area of the shaded region between the graphs of fx  e x and gx  x 2
shown:
30. Determine the area of the finite region enclosed by the graphs of y  x 3 and y  x
31. Determine the area of the finite region enclosed by the graphs of y  x and
y  13 x  23
32. Determine the area of the finite region enclosed by the graphs of y  1x , x  2, and
y3
33. Determine the area of the region between the graph of y  x 2  4 and the x-axis from
x  4 to x  4
34. Evaluate  2x x 2  3 dx
x 1/3 1
35. Evaluate 
36. Evaluate 
3
10
x 2/3
1
0 5x2 2
dx
dx
37. Evaluate 
38. Evaluate 
39. Evaluate 
1 1/x
e dx
x2
ex
dx
1e x
e ln x 3
dx
x
1
40. Evaluate 
1
x
1
x
10
ln
dx
41. Evaluate  xx  3 dx
lnx
x
42. Evaluate 
dx
e2
43. Evaluate  lnx dx
1
44. Evaluate  x 2 e x dx
45. Evaluate  x x  1 dx
46. Write the following expression in AIM notation:
ln
e 2x1  1  1
4x 2  1
1 1
47. Write the following AIM expression in standard mathematical notation:
exp(x/2)exp(x)/2/x^2/3  4  1/sqrt6  x
1
48. Compute the value of  xe x dx
0
a. exactly by the Fundamental Theorem of Calculus
b. by the trapezoidal rule with n  6 (use four decimal places, approximate value only)
c. by Simpson’s rule with n  6 (use four decimal places, approximate value only)
49. Evaluate lim
3 lnn
n
2
en
2
lim
n n 3
 x
e dx
0
0
1
 x1 2

3
x 2 e x

50. Evaluate n
lim
51. Evaluate
52. Evaluate 
53. Evaluate 
dx
54. Evaluate 
dx
Compute f2, 3 if fx, y  x 2 y  y 2  x  1
Compute ge, 0 if gx, y  lnx 2   e y  1
3x2y
Compute f1, 2, 3 if fx, y, z  z
A producer makes two sizes of flags. Large flags cost $4 each to produce, and Small
flags cost $3 each to produce. The fixed cost is $2000. Determine the cost function Cx, y
for making x small flags and y large flags.
3x
59. Determine and plot the domain of fx, y  y1
55.
56.
57.
58.
60. Determine and plot the domain of fx, y  x  y  1
61. Determine and plot the domain of fx, y  lnxy
62. Draw the level curves of fx, y  25  x 2  y 2 for the values c  0, 9, 16
63.
64.
65.
66.
67.
y
Draw the level curves of fx, y  x for the values c  0, 1, 2, 3
Draw the level curves of fx, y  2  x  y for the values c  1, 2, 4
Use the formal definition of partial derivative to determine g x and g y if gx, y  xy 2  1
Compute g x and g y for gx, y  lnx 2  y 3 
Compute g x , g y , and g z for gx, y, z  xy12 z 3
Compute g x 2, 3 if gx, y  e xy  1
Find all four second partial derivatives of gx, y  xy 3 lnx
Find the critical points for the function hx, y  2x 3  y 2  3x 2  8y  5
Find the critical points for the function hx, y  x 2  y 3  3xy  3
Find the relative maximum and minimum values if any, for the function
fx, y  4x 3  y 3  12x  3y  5
73. The US Postal Service insists that the length plus girth of a package to be mailed cannot
exceed 84 inches. The girth of a package is the total distance around the middle of the
package. What are the dimensions of the box of length y, width x, and height z that has the
largest volume that can be mailed?
74. Use Lagrange multipliers to maximize fx, y  2xy  4y subject to x  y  3
75. A farmer has 600 meters of fence. Use Lagrange multipliers to determine the largest
rectangular area he can enclose as a pig pen.
76. Determine the equation of the least squares regression line through the points
0, 2, 1, 0, 4, 1, 6, 3 and plot both the data points and the line. Hint:
68.
69.
70.
71.
72.
n
m
n
n  xkyk
  xk
n
n
k1
n  x 2k
k1
k1
  xk
n
 yk
k1
2
k1
and
n
n
 yk  m  xk
b
k1
n
k1
77. Determine df if fx, y  x 3 y  lnxy
78. Determine the value of df if fx, y  e x y 3 , x  2, y  3, dx  0. 1, and dy  0. 2
79. Use the total differential to approximate the change in f corresponding to the changes in x
and y if fx, y  3y lnx  1, x changes from 2 to 2. 01, and y changes from 4 to 3. 97.
Given an approximate answer only, accurate to two decimal places.
80. Compute the actual change in f in the previous problem and compare your answer to the
approximation obtained in the previous problem.
81. The Cobb-Douglas production function is fx, y  40x 1/3 y 2/3 where x is the number of
units of labor, y is the number of units of capital, and fx, y is the number of units
produced. Use differentials to estimate the change in production if the number of units of
labor is increased from 27 to 28 and the number of units of capital is increased from 64 to
66.
3
82. Evaluate  2xy  y 2 dy
0
3
83. Evaluate  2xy  y 2 dx
0
5 3
84. Evaluate   e y dx dy
0 1
1 0
85. Evaluate   2xy 2 dy dx
0 1
86. Determine the volume between the graph of fx, y  xy over the rectangular region
R : 0  x  2, 1  y  2
87. Determine the average value of fx, y  10x 2 y over the rectangular region
R : 1  x  1, 0  y  2
88. Set up an integral of fx, y  3xe xy over the region shown. The blue function (on top) is
x while the red function (on the bottom) is lnx. (Do not evaluate the integral.)
1 x 2 x
89. Evaluate  
0 x
3
y
90. Evaluate  
1 y
3x  y dy dx
4xy dx dy