M ATH 1325 – B USINESS C ALCULUS
S ECTION 9.8 M ARGINAL F UNCTIONS IN E CONOMICS
Marginal analysis is the study of the rate of change of economic quantities.
Examples:
Economist observes the value of an economy’s gross domestic product (GDP) at a given time and is
equally concerned with the rate at which it is growing or declining.
A manufacturer watches total cost at a certain level of production but is equally concerned with how fast
the total cost is changing with respect to the level of production
Let C(x) denote the total cost to produce x items. C0 (x) is the marginal cost function.
When you see marginal, think derivative.
Units of marginal cost are the same as the units of cost per item.
The marginal cost function, C0 (x), is interpreted to be the approximate cost of one more item
st item
Marginal cost ≈ Exact cost of the (x + 1)
C0 (x) ≈ C(x + 1) −C(x)
Likewise,
R(x) = total revenue
R0 (x) = marginal revenue
P(x) = R(x) −C(x) = total profit
P0 (x) = R0 (x) −C0 (x) = marginal profit
Ex: Suppose that the weekly cost in dollars to manufacture x industrial barrels is given by
C(x) = 10, 000 + 90x − 0.05x2
(a) Find the exact cost of manufacturing the 501st barrel.
(b) Find the marginal cost function.
(c) Use your result from (b) to estimate the cost of manufacturing the 501st barrel.
(d) How do the results from (a) and (c) compare?
Math 1325
Section 9.8 Continued
Looking at the graph of C(x), we can see marginal
cost very easily. Looking at the tangent lines, we
can see that the slope of the tangent lines decrease
as production increases. As we produce more barrels, the cost of the subsequent barrel decreases.
Question: Would a good businessperson prefer for
the cost of subsequent barrels to increase or decrease?
The average cost function is C̄(x) =
Marginal average cost:
NOTE:
C(x)
. We read this as “C bar of x”
x
Likewise,
R(x)
R̄(x) =
average revenue
x
0
R̄ (x) = marginal average revenue
P(x)
= average profit
x
0
P̄ (x) = marginal average profit
P̄(x) =
Ex: Revisiting our first example:
Suppose that the weekly cost in dollars to manufacture x industrial barrels is given by
C(x) = 10, 000 + 90x − 0.05x2
(a) Find the average cost function.
(b) Find the marginal average cost function.
(c) If 501 barrels are manufactured, find the average cost per barrel.
(d) If 501 barrels are manufactured, find the marginal average cost per barrel.
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Math 1325
Section 9.8 Continued
x
, where x is the number of computers that can be sold
30
at $p per computer. The cost function is C(x) = 72, 000 + 60x, where C(x) is the cost in dollars of producing
x computers.
Ex: The demand function for a computer is p = 200 −
(a) Find the revenue function.
(b) Find the profit function.
(c) Find the marginal profit function.
(d) Find and interpret P0 (1500) and P0 (3000).
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