Math 117
Exam 2
_______________________________________________
Name
Instructor: Nancy Goodisman
1. Let f ( x)  2 x  e x .
a. Find f (x) .
b. Find the critical value where f (x) = 0.
c. What is the interval on which f is increasing.
d. What is the interval on which f is decreasing.
e. Find a relative extreme point and identify it as a relative maximum or a relative
minimum.
f. Is the relative extreme point an absolute maximum or minimum? How can you be
sure?
2. Use implicit differentiation to find
the tangent line at the point (5, 0).
dy
if xy  2 y  x 2  25 . Then find the slope of
dx
3. The demand function, p  1000  4q , gives the demand q at price p. The cost of
producing q units is C  q 2  40q  1000 .
a. What is the revenue function? ( R  p  q )
b. What is the profit function? ( P  R  C )
c. Find the marginal profit.
d. Find the maximum profit.
e. How do you know it is and absolute maximum?
f. What are the quantity and price for a maximum profit?
dR dC
dR
dC
when the profit is maximized. Hint. Find
and
and

dq dq
dq
dq
evaluate them at the value for q results in a maximum profit.
g. Show that
h. Find the maximum revenue and corresponding quantity and price.
Maximum revenue = __________
Quantity = _________
Price = ________
4. If the demand function, p  1000  4q , gives the demand q at price p.
p
q
a. Find the point elasticity of demand,  
, when q = 50.
dp
dq
b. When the demand is 50 units, a 2% increase in price would result in what
decrease in demand?
5. Let W be the weekly sales of a product and t be weeks from the start of an advertizing
campaign.
a. What conclusion can you draw if W (8)  0 and W (8)  5 ?
b. What does this tell you about the effectiveness of the ad campaign at week 8?