11DIFFERENTIATION - Department of Mathematics

Average Cost
Function
EXAMPLE 3
SOLUTION ✔
A VERAGE C OST F UNCTIONS
11.4 MARGINAL FUNCTIONS IN ECONOMICS 737
Let’s now introduce another marginal concept closely related to the marginal
cost. Let C(x) denote the total cost incurred in producing x units of a certain
commodity. Then the average cost of producing x units of the commodity is
obtained by dividing the total production cost by the number of units produced.
This leads to the following definition.
Suppose C(x) is a total cost function. Then the average cost function, denoted
by C(x) (read ‘‘C bar of x’’), is
C(x)
x
The derivative C(x) of the average cost function, called the marginal average
cost function, measures the rate of change of the average cost function with
respect to the number of units produced.
The total cost of producing x units of a certain commodity is given by
dollars.
C(x) 400 20x
a. Find the average cost function C.
b. Find the marginal average cost function C.
c. Interpret the results obtained in parts (a) and (b).
a. The average cost function is given by
C(x) C(x) 400 20x
x x
20 400
x
b. The marginal average cost function is
C(x) 400
x 2
c. Since the marginal average cost function is negative for all admissible values
of x, the rate of change of the average cost function is negative for all x 0;
that is, C(x) decreases as x increases. However, the graph of C always lies
(4)