Calculus in Economics

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).

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

C(x)=0.02x3−20x+350.

(a) Find the marginal cost function.

 

MC(x)=

(b) Find the marginal cost when 60 units of the item are produced.

 

The marginal cost when 60 units are produced is $____

(c) Find the actual cost of increasing production from 60 units to 61 units.

 

The actual cost of increasing production from 60 units to 61 units is $___

Answer 1

C(x) = 0.02x^3 -20x +350

a) MC(x) = C'(x) = 0.06x^2 -20

b) MC(60) = C'(60) = 0.06(60)^2 -20 = 216-20 = 196

c) C(60) = 0.02(60)^3 -20(60) +350 = 4320-1200+350= 3470

C(61) = 0.02(61)^3 -20(61) +350 = 4539.62 - 1220 +350 = 3669.62

C(61)-C(60) =199.62 = cost of the 61st

MC(61) = 0.06(61)^2 -20 = 203.26

average of MC(61) and MC(60) = 199.63, virtually the same as C(61)-C(60)=199.62