Let the total cost function C(x) be defined as follows.

Question

C(x) = 0.0008x3 - 0.04x2 + 99x + 4400
Find the average cost function C.
C(x) = ??

Find the marginal average cost function C '.
C '(x) = ??

Answer 1

To find the average cost function, we need to divide the total cost function C(x) by the number of units x:

C(x) = (0.0008x^3 - 0.04x^2 + 99x + 4400) / x

Now, we can simplify the function:

C(x) = 0.0008x^2 - 0.04x + 99 + 4400/x

This is the average cost function.

Now, we'll find the marginal average cost function, which is the derivative of the average cost function:

C'(x) = d(C(x))/dx = d(0.0008x^2 - 0.04x + 99 + 4400/x)/dx

To find the derivative, we'll differentiate each term separately:

d(0.0008x^2)/dx = 2 * 0.0008x = 0.0016x
d(-0.04x)/dx = -0.04
d(99)/dx = 0
d(4400/x)/dx = -4400/x^2

Now, we can add these derivatives to find the marginal average cost function:

C'(x) = 0.0016x - 0.04 - 4400/x^2