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UAE_Math_Grade_12_Vol_1_SE_718383_ch4
October 22, 2019
20:1
CHAPTER 4 • •
4-76
286
Applications of Differentiation
In economics, the term marginal is used to indicate a rate. Thus, marginal cost is
the derivative of the cost function, marginal profit is the derivative of the profit function
and so on.
Suppose that you are manufacturing an item, where your start-up costs are AED
4000 and production costs are AED 2 per item. The total cost of producing x items
would then be 4000 + 2x. Of course, the assumption that the cost per item is constant
is unrealistic. Efficient mass-production techniques could reduce the cost per item, but
machine maintenance, labor, plant expansion and other factors could drive costs up as
production (x) increases. In example 9.1, a quadratic cost function is used to take into
account some of these extra factors.
Whenthecostperitemisnotconstant,animportantquestionformanagerstoanswer
is how much it will cost to increase production. This is the idea behind marginal cost.
EXAMPLE 9.1 Analyzing the Marginal Cost of Producing
a Commercial Product
Suppose that
2
C(x) = 0.02x + 2x + 4000
is the total cost (in AED) for a company to produce x units of a certain product.
Compute the marginal cost at x = 100 and compare this to the actual cost of
producing the 100th unit.
Solution The marginal cost function is the derivative of the cost function:
′
C (x) = 0.04x + 2
′
and so, the marginal cost at x = 100 is C (100) = 4 + 2 = 6 AED per unit. On the
other hand, the actual cost of producing item number 100 would be C(100) − C(99).
(Why?) We have
C(100) − C(99) = 200 + 200 + 4000 − (196.02 + 198 + 4000)
= 4400 − 4394.02 = 5.98 AED.
Note that this is very close to the marginal cost of AED 6. Also notice that the
marginal cost is easier to compute.
Another quantity that businesses use to analyze production is average cost. You
can easily remember the formula for average cost by thinking of an example. If it costs a
( )
total of AED 120 to produce 12 items, then the average cost would be AED 10 AED 120
12
per item. In general, the total cost is given by C(x) and the number of items by x, so
average cost is defined by
C(x)
C(x) = .
x
Business managers want to know the level of production that minimizes average cost.

EXAMPLE 9.2 Minimizing the Average Cost of Producing
a Commercial Product

Suppose that
2
C(x) = 0.02x + 2x + 4000
is the total cost (in dollars) for a company to produce x units of a certain product.
Find the production level x that minimizes the average cost.
Solution The average cost function is given by
2
−1
C(x) = 0.02x + 2x + 4000 = 0.02x + 2 + 4000x .
x

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