Consider the cost function of Example.(a) Graph C(x) in the window [0, 60] by [−300, 1260]...

Consider the cost function of Example.

(a) Graph C(x) in the window [0, 60] by [−300, 1260].

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(b) For what level of production will the cost be $535?

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(c) For what level of production will the marginal cost be $14?

Example

Marginal Cost Suppose that the cost of producing x items is

C(x) = .005x3 − .5x2 + 28x + 300 dollars, and daily production is 50 items.

(a) What is the extra cost of increasing daily production from 50 to 51 items?

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(b) What is the marginal cost when x = 50?

SOLUTION(a) The change in cost when daily production is raised from 50 to 51 items is C(51) − C(50), which equals

(b) The marginal cost at production level 50 is C′(50).

C′(x) = .015x2 − x + 28

C′(50) = 15.5

Notice that 15.5 is close to the actual cost in part (a) of increasing production by one item.Our discussion of cost and marginal cost applies as well to other economic quantities such as profit and revenue. In fact, in economics, the derivatives are often described by the adjective marginal. Here are two more definitions of marginal functions and their interpretations.

DEFINITION Marginal Revenue and Marginal Profit If R(x) is the revenue generated from the production of x units of a certain commodity and P(x) is the corresponding profit, the marginal revenue function is R′(x) and the marginal profit function is P′(x).

The marginal revenue of producing a units, R′(a), is an approximation of the additional revenue that results from increasing the production level by 1 unit, from a to a + 1:

R(a + 1) − R(a) ≈ R′(a).

Similarly, for the marginal profit, we have

P(a + 1) − P(a) ≈ P′(a).

The following example illustrates how marginal functions help in the decision-making process in economics.