Math 142-102
Exam #3
Name (printed):
UIN:
“An Aggie does not lie, cheat, or steal or tolerate those who do.”
Signature:
Instructions:
• You must clear your calculator:
TI-83 Plus & TI-84 Plus: MEM (2nd +), Reset (7), cursor right to ALL, All Memory (1),
Reset (2).
TI-83: MEM (2nd +), Reset (5), All Memory (1), Reset (2).
• There are 18 questions and 8 pages to this exam including the cover sheet. The true-false questions are worth
3 points each, the multiple choice questions are worth 5 points each. The point value of work out problems
is indicated in the question. Partial credit will only be given, if deserved, in the work out section.
• In order to receive full credit on the work out problems, you must show appropriate, legible work.
• Round your answer to 2 decimal places unless stated otherwise in the problem.
• You must box or circle your final answer in the work out section.
• Your grade on the exam will be written on the back of the last page.
GOOD LUCK!
For questions 1–10, circle the correct answer. No partial credit will be given in this section.
Use the graph of the function f (x) shown to the right
to answer questions 1–3. A1 and A2 denote the areas
of the respective shaded regions.
Zc
1. (3 pts)
2f (x) dx = 2A2
b
TRUE
FALSE
Zb
2. (3 pts)
f (x) dx = A1
a
TRUE
FALSE
Zc
3. (5 pts) If A1 = A2 , then
f (x) dx =
a
A. A1 + A2
B. A1
C. 0
D. 2A1
E. None of the above
4. (3 pts) If F (x) is an antiderivative of f (x), then F (x) + 1 is an antiderivative of f (x) + 1.
TRUE
FALSE
Z
4
5. (5 pts) Write
Z
−2
Z
−2
f (x) dx +
Z
f (x) dx +
−4
8
f (x) dx as a single integral.
4
8
A.
f (x) dx
−2
Z −2
B.
f (x) dx
−4
Z 8
C.
f (x) dx
−4
Z
D. −
Z
8
f (x) dx
2
4
E.
f (x) dx
−4
2
6. (5 pts) What is the AVERAGE VALUE of the function f (x) = x3 − 1 over the interval [0, 4]?
A. 15
B. 255
C. 63
D. 60
E. None of the above
7. (5 pts) Given the marginal cost function C 0 (x) = 2x + 5, find the cost function.
A. x2 /2 + 5x + c
B. x2 + 5x
C. x2 + 5 + c
D. x2 + 5x + c
E. None of the above
8. (5 pts) Let f be a function on the interval [−4, 2]. For the partition of this interval into 3 subintervals of equal
length, the left Riemann sum R3 of f is given by
A. 2f (−2) + 2f (0) + 2f (2)
B. 3f (−4) + 3f (−1) + 3f (2)
C. 2f (−4) + 2f (−2) + 2f (0)
D. 2f (−3) + 2f (−1) + 2f (1)
E. None of the above
9. (5 pts) To the right, the graphs of a price-demand equation p = D(x) and a price-supply equation p = S(x) are
shown for x items at price level p. The consumer’s surplus at equilibrium price level p̄ is given by which one of
the shaded areas?
A. A1
B. A2
C. A3
D. A4
E. None of the above
10. (5 pts) Given f (x, y) = x2 y 4 , find fy (2, 1)
A. 8
B. 32
C. 4
D. 16
E. None of the above
3
Work Out part: For the following questions, you must show appropriate, legible work to receive full
credit. Remember to box your final answers.
11. Use Calculus to find the following indefinite integrals. DO NOT SIMPLIFY YOUR RESULT!
Z
(ln(x − 1))3
dx
x−1
Z
2x3
dx
(x4 − 3)2
(a) (5 pts)
(b) (5 pts)
Z
12. (6 pts) Use a right Riemann sum to approximate the definite integral
1
Round your answer to two decimal places.
4
3
(2x2 − 3x − 1) dx with n = 4.
13. Use Calculus to find the following definite integral. Round your answer to two decimal places.
Z
2
(6x2 − 2)ex
(a) (4 pts)
3
−x
dx
1
Z
(b) (4 pts)
1
3
2x4 + x − 2
dx
x2
5
14. (5 pts) A marginal cost function is C 0 (x) = 0.02x + 50 and the total cost for the first 100 units is $6000. Find
the cost function.
15. (8 pts) Find the area between the graphs of f (x) = 2x and g(x) = 3 − x2 . Include a sketch of the graphs
with the relevant area(s) shaded! If you use the calculator, outline your steps.
6
16. (5 pts) Find fxy if f (x, y) =
x
. SIMPLIFY YOUR RESULT!
y2
2x + 5y
2x − 3y
(a) (3 pts) Find the domain of the function.
17. Given the function f (x, y) =
(b) (3 pts) Find fy . SIMPLIFY YOUR RESULT!
7
18. The demand and supply functions are given as the following:
D(x)
=
60 − x2
S(x)
=
4x
(a) (3 pts) Find the equilibrium point.
(b) (5 pts) Determine the consumers’ surplus when the market place is acting at equilibrium price.
Question:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Total
Points:
3
3
5
3
5
5
5
5
5
5
10
6
8
5
8
5
6
8
100
Score:
8