Predicting Sales Refer to Example(a) Compute S(10) and S_(10).(b) Use the data in part (a)...

Predicting Sales Refer to Example

(a) Compute S(10) and S_(10).

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(b) Use the data in part (a) to estimate the total sales on January 11. Compare your estimate to the actual value given by S(11).

Example

Declining sales At the end of the holiday season in January, the sales at a department store are expected to fall (Fig. 1). It is estimated that for the x day of January the sales will be

thousand dollars.

(a) Compute S(2) and S′(2) and interpret your results.

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(b) Estimate the value of sales on January 3 and compare your result with the exact value S(3).

SOLUTION (a) We have

thousand dollars.

Figure 1 The sales decrease starting the first day of January.

To compute S′(2), write  as 9 • (x + 1)−2; then

The equations S(2) = 4 and S′(2) = −.667 tell us that on January 2 the sales are $4000 and are falling at the rate of .667 thousand dollars per day, or $667 per day.

(b) To estimate S(3), we use (3):

S(3) ≈ S(2) + S′(2).

Thus, our estimate of the sales on January 3 is 4000−667 = $3333. To compare with the exact value of sales on January 3, we compute S(3) from the formula:

 3.5625 thousand dollars or $3562.5.

This is close to our estimate of $3333.

The Marginal Concept in Economics

For the sake of this discussion, suppose that C(x) is a cost function (the cost of producing x units of a commodity), measured in dollars. A problem of interest to economists is to approximate the quantity C(a + 1) − C(a), which is the additional cost that is incurred if the production level is increased by one unit from x = a to x = a + 1. Note that C(a + 1) − C(a) is also the cost of producing the (a + 1) unit. Taking f(x) = C(x) in (2), we find that

additional cost = C(a + 1) − C(a) ≈ C′(a),

where the symbol ≈ is used to indicate that, in general, we have an approximation and not an equality (Fig. 2). Economists refer to the derivative C′(a) as the marginal cost at production level a or the marginal cost of producing a units of a commodity.

DEFINITION Marginal Cost If C(x) is a cost function, then the marginal cost function is C′(x). The marginal cost of producing a units, C′(a), is approximately equal to C(a + 1) − C(a), which is the additional cost that is incurred when the production level is increased by 1 unit from a to a + 1.

Before we give an example, let us note that if C(x) is measured in dollars, where x is the number of items, then C′(x), being a rate of change, is measured in dollars per item.