Download Practice Exam 2

Math 117
Exam 2 Practice
_______________________________________________
Name
Instructor: Nancy Goodisman
1. Let f ( x)  ln( 8 x  1) . (18 points)
a. Find f (x) .
b. Find f (5) .
c. Find f (x) .
d. Find f (5) .
e. What do parts b. and d. tell you about the function?
2. The demand function, p  0.5q  450 , gives the demand q at price p. (20 points)
dR
a. Find the marginal revenue,
when q = 500.
dq
b. What does it mean if the marginal revenue is negative?
c. Find the maximum revenue and corresponding quantity and price.
Maximum revenue = __________
Quantity = _________
Price = ________
3. If the demand function, p  0.5q  450 , gives the demand q at price p. (12 points)
p
q
a. Find the point elasticity of demand,  
, as a function of q.
dp
dq
b. What quantity and price correspond to unit elasticity,   1 ?
4. If R (q ) gives the revenue from the sale of a product as a function of the demand, q,
what conclusion can you draw if R (200)  0 and R (200)  5 ? (6 points)
5. Let S be the weekly sales of a product t weeks after the start of an advertizing
campaign. (20 points)
S  20t  e
a.
(
t
)
10
Find the derivative, S (t ) .
b. Evaluate the derivative at t = 3.
c. What does this tell you about the effectiveness of the add campaign at week 3?
d. Evaluate the derivative at t = 15.
e. What does this tell you about the effectiveness of the add campaign at week 15?
f. At what point would you want either a new product or new ad campaign?
6.
In researching the best way to grow apples, agricultural scientists found that the
density of trees per acre affects the yield per tree. At 20 trees per acre, the yield was
30 boxes of apples per tree, and if the density of trees is increased by 2 trees, the yield
decreases by 1 box. This gives a formula for the total yield per acre, since
yield yield trees


. So the yield per acre, Y = (30 – x)(20 + 2x) where x is the
acre
tree acre
increase in trees per acre above 20 trees. (24 points)
a.
Find Y  .
b. Find the extreme point.
c. Find Y 
d. How does Y  insure that your extreme point is a maximum?
e. What is the maximum yield per acre and what is the corresponding number of
trees per acre?