M 112 Short Course in Calculus

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M 112 Short Course in Calculus

• Chapter 2 Rate of Change The Derivative
• Sections 2.5 Marginal Cost and Revenue
• V. J. Motto

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Cost and Revenue

• Management decisions within a particular firm or
  industry usually depend on the costs and revenues
  involved. In this section we look at the cost
  and revenue functions.

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Example 1 If cost, C, and revenue, R, are given
by the graph in Figure 2.47, for what production
quantities does the firm make a profit?
Figure 2.47 Costs and revenues
Solution
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Marginal Analysis
Let C(q) be the function giving the cost of
running q flights. If the airline had originally
planned to run 100 flights, its costs would be
C(100). With the additional flight, its costs
would be C(101). Therefore, Marginal cost
C(101) C(100). Now and this quantity is the
average rate of change of cost between 100 and
101 flights.
Figure 2.48 Marginal cost Slope of one of
these lines
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In Figure 2.48 the average rate of change is the
slope of the secant line. If the graph of the
cost function is not curving too fast near the
point, the slope of the secant line is close to
the slope of the tangent line there. Therefore,
the average rate of change is close to the
instantaneous rate of change. Since these rates
of change are not very different, many economists
choose to define marginal cost, MC, as the
instantaneous rate of change of cost with respect
to quantity
Figure 2.48 Marginal cost Slope of one of
these lines
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Sample Problem (page 120 12) Cost and revenue
functions for a charter bus company are shown in
Figure 2.57. Should the company add a 50th bus?
How about a 90th ? Explain your answers using
marginal revenue and marginal cost.
Solution At q 50, the slope of the revenue is
larger than the slope of the cost. Thus, at q
50, marginal revenue is greater than marginal
cost and the 50th bus should be added. At q 90
the slope of revenue is less than the slope of
cost. Thus, at q 90 the marginal revenue is
less than marginal cost and the 90th bus should
not be added.
Figure 2.57