Calculus in economics (2)

Question

If the current production level is x items with cost C(x), then the cost of computing h additionial items is C(x+h). The average cost of those h items is (C(x+h)−C(x))h . As we analyze the cost of just the last item produced, this can be made into a mathematical model by taking the limit as h→0, i.e. the derivative C′(x). Use this function in the model below for the Marginal Cost function MC(x).

 

Problem Set question:

 

The cost, in dollars, of producing x units of a certain item is given by

C(x)=0.02x3−10x+500.

 

(a) Find the marginal cost function.

 MC(x)=

 

 __________

 

(b) Find the marginal cost when 70 units of the item are produced.

 

The marginal cost when 70 units are produced is $____________.

 

(c) Find the actual cost of increasing production from 70 units to 71 units.

 

The actual cost of increasing production from 70 units to 71 units is $____________.

Answer 1

(a) Marginal cost function MC(x) = C'(x) = 0.06x² - 10.

(b) If x = 70 then MC(70) = 0.06·70² - 10 = $284 per item

(c) ΔC = C(71) - C(70) = 6948.22 - 6660 = $288.22, so ΔC ≈ MC(70)

Answer 2

C(x) = 0.02x³ - 10x + 500

Marginal cost function = C'(x) = 0.06x² - 10

Marginal cost when x = 70 is C'(70) = $284

Actual cost of increasing production from 70 to 71 units is C(71) - C(70) = 6948.22 - 6660 = $288.22