Question

The total cost of producing x units of a certain commodity is C(x) thousand dollars, where C(x)=x3+158x+293

(a) Find A′(x), where A(x)=C(x)/x is the average cost function. A′(x)=

Answer 1

The question involves calculating the derivative of the average cost function A(x), given the total cost function C(x) for producing a commodity. By differentiating A(x), we find the rate of change of the average cost with respect to the number of units produced.

Step-by-step explanation:

The student's question involves finding the derivative of the average cost function, A'(x), where A(x) is defined as the average cost of producing x units of a commodity. Given the total cost function C(x) = x^3 + 158x + 293, we can express the average cost function A(x) as C(x)/x = x^2 + 158 + 293/x. To find A'(x), we take the derivative of A(x) with respect to x.

Here are the steps:

1. Apply the quotient rule, if necessary, to differentiate terms involving division by x.
2. Simplify the resulting expression to find A'(x).

The derivative of A(x) represents the rate of change of the average cost with respect to the quantity of units produced and is an essential concept in understanding the economics of production. This process involves calculus, specifically differentiation.