Calculus Word Problem

Question

 The monthly cost function, in dollars, for a coffee maker factory is C(x) where x is the number of coffee makers produced. The financial model used by management predicts that C(1000)=4000 and C′(1000)=20.

How much would you expect monthly cost to increase if production were increased from 1000 to 1010 coffeemakers?

ΔC≈ 200 dollars

Find the average cost per coffee maker at a production level of 1000 coffeemakers per month.

AC(1000)=  ? dollars per coffee maker

Find the marginal average cost at a production level of 1000 coffee makers.

MAC(1000)= ?

Answer 1

How much would you expect monthly cost to increase if production were increased from 1000 to 1010 coffeemakers?

The derivative at 1000 represents the local "rate of change" or "slope" so you could use linear interpolation: C′(1000) * (1010-1000) = 20*10 = 200 is the change

Find the average cost per coffee maker at a production level of 1000 coffeemakers per month.

AC(1000)= C(1000) / 1000 = 4. The average cost per coffeemaker is the total cost divided by the number of coffeemakers.

Find the marginal average cost at a production level of 1000 coffee makers.

MAC(1000)= C'(1000) = 20. The marginal cost is the change when the next coffeemaker is added, which is 20.