11DIFFERENTIATION - Department of Mathematics

746 11 DIFFERENTIATION
11.4 Exercises
1. P RODUCTION C OSTS The graph of a typical total cost function
C(x) associated with the manufacture of x units of
a certain commodity is shown in the following figure.
a. Explain why the function C is always increasing.
b. As the level of production x increases, the cost per
unit drops so that C(x) increases but at a slower pace.
However, a level of production is soon reached at which
the cost per unit begins to increase dramatically (due
to a shortage of raw material, overtime, breakdown of
machinery due to excessive stress and strain) so that
C(x) continues to increase at a faster pace. Use the graph
of C to find the approximate level of production x 0 where
this occurs.
y ($)
1 2 3 4 5 6 7 8
y = C(x)
x (thousands)
A calculator is recommended for Exercises
2–33.
2. M ARGINAL C OST The total weekly cost (in dollars) incurred
by the Lincoln Record Company in pressing x
long-playing records is
C(x) 2000 2x 0.0001x 2 (0 x 6000)
a. What is the actual cost incurred in producing the
1001st and the 2001st record?
b. What is the marginal cost when x 1000 and 2000?
3. M ARGINAL C OST Adivision of Ditton Industries manufactures
the Futura model microwave oven. The daily cost
(in dollars) of producing these microwave ovens is
C(x) 0.0002x 3 0.06x 2 120x 5000
where x stands for the number of units produced.
a. What is the actual cost incurred in manufacturing the
101st oven? The 201st oven? The 301st oven?
b. What is the marginal cost when x 100, 200, and 300?
4. M ARGINAL A VERAGE C OST The Custom Office Company
makes a line of executive desks. It is estimated that the
total cost for making x units of their Senior Executive
Model is
C(x) 100x 200,000
dollars per year.
a. Find the average cost function C.
b. Find the marginal average cost function C.
c. What happens to C(x) when x is very large? Interpret
your results.
5. M ARGINAL A VERAGE C OST The management of the
ThermoMaster Company, whose Mexican subsidiary
manufactures an indoor–outdoor thermometer, has estimated
that the total weekly cost (in dollars) for producing
x thermometers is
C(x) 5000 2x
a. Find the average cost function C.
b. Find the marginal average cost function C.
c. Interpret your results.
6. Find the average cost function C and the marginal average
cost function C associated with the total cost function
C of Exercise 2.
7. Find the average cost function C and the marginal average
cost function C associated with the total cost function
C of Exercise 3.
8. M ARGINAL R EVENUE The Williams Commuter Air Service
realizes a monthly revenue of
R(x) 8000x 100x 2
dollars when the price charged per passenger is x dollars.
a. Find the marginal revenue R.
b. Compute R(39), R(40), and R(41). What do your
results imply?
9. M ARGINAL R EVENUE The management of the Acrosonic
Company plans to market the Electro-Stat, an electrostatic
speaker system. The marketing department has
determined that the demand for these speakers is
p 0.04x 800 (0 x 20,000)
where p denotes the speaker’s unit price (in dollars) and
x denotes the quantity demanded.
a. Find the revenue function R.
b. Find the marginal revenue function R.
c. Compute R(5000) and interpret your result.