Suppose C(x) is the total cost that a company incurs in producing x units of a c

Question

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 x_1 to x_2, then the additional cost is Delta C = C(x_2) - C(x_1), and the average rate of change of the cost is Delta C/Delta x = C(x_2) - C(x_1/x_2 - x_1) = C(x_1 + Delta x) - C(x_1)Delta x The limit of this quantity as Delta x rightarrow 0, that is, the instantaneous rate of change of cost with respect to the number of items produced, is called the marginal cost by economists: marginal cost = lim_Delta x rightarrow 0 Delta C/Delta x = dC/dx. [Since x often takes on only integer values, it may not make literal sense to let Delta x approach 0, but we can always replace C(x) by a smooth approximating function as in this example.] Taking Delta x = 1 and n large (so that Delta x is small compared to n), we have C'(n) almostequalto 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 It is often appropriate to represent a total cost function by a polynomial C(x) = a + bx + cx^2 + dx^3 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) = 6000 + 15x + 0.05x^2. Then the marginal cost function is C'(x) = The marginal cost at the production level of 400 items is C'(400) = 15 + (400) = dollars/item. This gives the rate at which costs are increasing with respect to the production level when x = 400 and predicts the cost of the 401st item. The actual cost of producing the 401st item is C(401) - C(400) = [6000 + 15() + 0.05 ()^2] - [6000 + 15(400) + 0.05(400)^2] = exist. Notice that C'(400) almostequalto C(401) - C(400).

Explanation / Answer

C(x) = 6000 + 15x + 0.05*x^2

Marginal cost function will be:

C'(x) = 0 + 15*1 + 0.05*2*x

C'(x) = 15 + 0.1x

C'(400) = 15 + 0.1*400 = 55 dollars/item

the actual cost of producing the 401st item is

C(401) - C(400) = [6000 + 15*401 + 0.05*401^2] - [6000 + 15*400 + 0.05*400^2]

C(401) - C(400) = 20055.05 - 20000

C(401) - C(400) = 55.05

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