Question

Find the marginal average cost function Cave′(x):

Given the cost function:
C(x)=3000+10x−0.05x2
The average cost function is:
Cave(x)=xC(x)=x3000+10x−0.05x2

Answer 1

To obtain the marginal average cost function Cave'(x) from the given average cost function, one must differentiate the average cost function with respect to x. The result is Cave'(x) = (10 - 0.10x) - (3000/x^2), which indicates how the average cost changes with each additional unit of output produced.

Step-by-step explanation:

The student has asked to find the marginal average cost function Cave'(x), given the total cost function C(x) = 3000 + 10x - 0.05x2 and the average cost function Cave(x) = (3000 + 10x - 0.05x2)/x. The marginal average cost function represents the change in average cost associated with producing one additional unit of output.

To find Cave'(x), we need to differentiate the average cost function Cave(x) with respect to x. The differentiation yields:

Cave'(x) = d/dx [(3000 + 10x - 0.05x2)/x]

Using the quotient rule, we get:

Cave'(x) = (10 - 0.10x) - (3000/x2)

Therefore, the marginal average cost function is Cave'(x) = (10 - 0.10x) - (3000/x2), which shows how the average cost changes as the quantity of output x changes.