11DIFFERENTIATION - Department of Mathematics

734 11 DIFFERENTIATION
11.4 Marginal Functions in Economics
EXAMPLE 1
2. a. The life expectancy at birth of a female born at the beginning of 1980 is given
by
g(80) 50.02[1 1.09(80)] 0.1 78.29
or approximately 78 yr. Similarly, the life expectancy at birth of a female born at
the beginning of the year 2000 is given by
g(100) 50.02[1 1.09(100)] 0.1 80.04
or approximately 80 yr.
b. The rate of change of the life expectancy at birth of a female born at any time
t is given by g(t). Using the general power rule, we have
g(t) 50.02 d
(1 1.09t)0.1
dt
0.9 d
(50.02)(0.1)(1 1.09t) (1 1.09t)
dt
(50.02)(0.1)(1.09)(1 1.09t) 0.9
5.45218(1 1.09t) 0.9
5.45218
(1 1.09t) 0.9
Marginal analysis is the study of the rate of change of economic quantities.
For example, an economist is not merely concerned with the value of an
economy’s gross domestic product (GDP) at a given time but is equally concerned
with the rate at which it is growing or declining. In the same vein, a
manufacturer is not only interested in the total cost corresponding to a certain
level of production of a commodity but is also interested in the rate of change
of the total cost with respect to the level of production, and so on. Let’s begin
with an example to explain the meaning of the adjective marginal, as used
by economists.
C OST F UNCTIONS
Suppose the total cost in dollars incurred per week by the Polaraire Company
for manufacturing x refrigerators is given by the total cost function
C(x) 8000 200x 0.2x 2 (0 x 400)
a. What is the actual cost incurred for manufacturing the 251st refrigerator?
b. Find the rate of change of the total cost function with respect to x when
x 250.
c. Compare the results obtained in parts (a) and (b).