The function gives the cost to manufacture x items.

C(x) = 20,000 + 5x −
x2
20,000
; x = 20,000
Find the average cost per unit of manufacturing h more items (i.e., the average rate of change of the total cost) at a production level of x, where x is as indicated and h = 10 and 1. (Use smaller values of h to check your estimates.) HINT [See Example 1.] (Round your answers to five decimal places.)
h 10 1
Cave 1 2

Estimate the instantaneous rate of change of the total cost at the given production level x, specifying the units of measurement.
C'(20,000) = 3

To find the average cost per unit of manufacturing h more items, we need to calculate the change in total cost when h items are produced.

The formula to calculate the change in total cost is given by:
ΔC = C(x + h) - C(x)

We are given the function C(x) = 20,000 + 5x - (x^2/20,000) and we need to find the average rate of change for h = 10 and h = 1.

For h = 10:
C(20,000 + 10) = C(20,010) = 20,000 + 5(20,010) - (20,010^2/20,000) = 20,000 + 100,050 - 20,010 = 100,040
ΔC = C(20,010) - C(20,000) = 100,040 - 20,000 = 80,040

For h = 1:
C(20,000 + 1) = C(20,001) = 20,000 + 5(20,001) - (20,001^2/20,000) = 20,000 + 100,005 - 20,000.05 = 100,005.05
ΔC = C(20,001) - C(20,000) = 100,005.05 - 20,000 = 80,005.05

Therefore, for h = 10, the change in total cost is 80,040 and for h = 1, the change in total cost is 80,005.05.

To calculate the average cost per unit, we divide the change in total cost by h:
Average cost for h = 10: 80,040 / 10 = 8,004
Average cost for h = 1: 80,005.05 / 1 = 80,005.05

To estimate the instantaneous rate of change (or the derivative) of the total cost at the given production level x, we can find the derivative of the cost function C(x).

C'(x) = 5 - (x/10,000)

To find the instantaneous rate of change at x = 20,000, we substitute x = 20,000 into the derivative function:

C'(20,000) = 5 - (20,000/10,000) = 5 - 2 = 3

Therefore, the instantaneous rate of change of the total cost at a production level of x = 20,000 is 3. The units of measurement depend on the units used in the function C(x).