While we can certainly find the exact change in the average cost by taking the difference of the average cost function at the two production values specified, we can also approximate the change using the marginal average cost function. Like other functions in economics, the marginal average cost function can be found by differentiating the average cost function.

The average cost function given is a rational function, so we need to apply the quotient rule if we wish to differentiate it. Let's first define the numerator, denominator, and their derivatives clearly.

f=18x+1250f′=18g=xg′=1f=18x+1250f′=18g=xg′=1

Let's now apply the quotient rule to this average cost function.

¯C′(x)=f′g−fg′g2=82x−(18x+1250)x2=−1250x2=−1250x2C¯′(x)=f′g−fg′g2=82x−(18x+1250)x2=−1250x2=−1250x2

Evaluating this marginal average cost function at 250 will give an estimate for how the average cost will change when the company increases production to 251 units.

¯C′(250)=−1250(250)2=−0.02C¯′(250)=−1250(250)2=−0.02

Thus, the average cost decreases by 2 cents per item when production is increased from 250 to 251 items.