EXAMPLE 8 Suppose C(x) is the total cost that a company incurs in producing x units of a certain commodity. The function C is called a cost function. If the number of items produced is increased from x₁ to x₂, then the additional cost is AC = C(x₂) - C(x₁), and the average rate of change of the cost is AC C(x₂) C(x₁) C(x₁ + Ax) - C(x₁). X2-X1 Ax Ax The limit of this quantity as Ax→ 0, that is, the instantaneous rate of change of cost with respect to the number of items produced, is called the marginal cost b economists: = marginal cost = AC lim Ax-0 Ax = [Since x often takes on as in this example.] Taking Ax = 1 and n large (so that Ax is small compared to n), we have C'(n) C(n + 1) - C(n). Thus the marginal cost of producing n units is approximately equal to the cost of producing one more unit [the (n+1)st unit]. It is often appropriate to represent a total cost function by a polynomial C(x) = a + bx + cx² + dx³ only integer values, it may not make literal sense to let Ax approach 0, but we can always replace C(x) by a smooth approximating functio dc dx Then the marginal cost function is C'(x) = where a represents the overhead cost (rent, heat, maintenance) and other terms represent the cost of raw materials, labor, and so on. (The cost of raw materials may be proportional to x, but labor costs might depend partly on higher powers of x because of overtime costs and inefficiencies involved in large-scale operations.) For instance, suppose a company has estimated that the cost (in dollars) of producing x items is C(x) = 7000 + 10x + 0.05x². The marginal cost at the production level of 700 items is C'(700) = 10 + (700) = dollars/item. This gives the rate at which costs are increasing with respect to the production level when x = 700 and predicts the cost of the 701st item. The actual cost of producing the 701st item is +100 C(701) - C(700) = [7000 + -0.05( - [7000 + 10(700) + 0.05(700)²] = $ Notice that C'(700) C(701) - C(700).