Final ExaminationSample
Math 0120 Final Examination
Sample
Name (Print) ID # .
Signature .
Instructor (circle one):
Lecture time (circle one):
Instructions:
1. Show your University of Pittsburgh ID if requested.
2. Clearly print your name and PeopleSoft number and sign your name in the space
above. Circle the name of your lecturer and the time of your lecture.
3. Work each problem in the space provided. Extra space is available on the back of
each exam sheet. Clearly identify the problem for which the space is required when
using the backs of sheets.
4. Show all calculations and display answers clearly. Unjustified answers will receive no
credit.
5. Write neatly and legibly. Cross out any work that you do not wish to be considered
for grading.
6. No tables, books, notes, earphones, calculators, or computers may be used. All
derivatives and integrals are to be found by learned methods of calculus.
DO NOT WRITE BELOW THIS LINE
______
Problem / Points / Score / Problem / Points / Score1 / 6
2 / 7
3 / 8
4 / 9
5 / 10
Total / 200
1. (40 pts.) Find Do not simplify.
(a)
(b) f(x) =
(c)
(d) f(x) =
(e) f(x) = e2x ln(x2 + 7)
2. (a) (12 pts.) Use this definition to find the derivative of f(x) =
b. (5 pts.) Find an equation of the tangent line to f(x) = at x = 4
3. (10 pts.)
4. (40 pts.) Find the following integrals:
(a)
(b) dx
(c) dx
(d)dx
5. (10 pts.) Assume that the demand function for yams is given by D(p) = 5000 – p2 where D(p) is the
quantity in pounds of yams and p is the price in dollars of a pound of yams.
(a) If the current price of yams is $3 per pound, how many yams will be sold?
(b) At $3 per pound, is the demand elastic or inelastic?
(c) Is it more accurate to say “People must have their yams and will buy them no matter what the
price” or “Yams are a luxury item and people will stop buying them if the price gets too high”?
6. (15 pts.) It costs the Pitt Motor Company $10,000 to produce each automobile, and its fixed costs
are $16,000 per week. The company’s price function is p(x) = 25,000 – 25x, where p(x) is the price
at which exactly x automobiles will be sold in a week. How many automobiles should the company
produce each week to maximize its profit? (Show your work)
7. (20 pts.) Given , do the following:
(a) Make a sign diagram for the first derivative of f(x).
(b) Make a sign diagram for the second derivative of f(x).
(c) State the open intervals on which f(x) is increasing, decreasing, concave up and concave
down.
(d ) What are the critical points and inflection points?
(e) Sketch the graph of y =f(x) by hand, labeling all relative extreme points and inflection
points.
8. (15 pts.) Find the area between the curves y = 2x and y = x2 from x = -1 to x = 3. (SHOW YOUR
WORK).
9. (12 pts.) Find all relative extreme values of f(x,y) =. (SHOW YOUR
WORK).
10. (18 pts.) Use the method of Lagrange multipliers to find the maximum and minimum values of
f(x,y) = 4xy subject to the constraint (Both extreme values exist.) (SHOW YOUR
WORK).
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