Introduction to Calculus in Economics

Question

Problem Set question:

 

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

C(x)=0.03x^3−5x+300.

 

 Find the marginal cost function.

 

MC(x)=_____

 

  

 

(b) Find the marginal cost when 30 units of the item are produced.

 

The marginal cost when 30 units are produced is $___ .

 

(c) Find the actual cost of increasing production from 30 units to 31 units.

 

The actual cost of increasing production from 30 units to 31 units is $____ .

Answer 1

C(x) = .03x^3 - 5x + 300

Marginal Cost = C'(x)= .09x^2 -5

C'(30) = .09(30^2) -5 = 81-5 = 76

C(30) = .03(30^3)-5(30) +300 = 810 -150+300 = 960

C(31) = .03(31^3) -5(31) +300 = 893.73 -155+300= 1038.73

C(31) - C(30) = 1038.73 - 960 = 78.73

which appears to be different from Marginal Cost = C'(30) =76

but take the average of C'(30) and C'(31) = (76+81.49)/2 = 78.745 which is fairly close to 78.73 within 1.5 hundredth

C'(30.5) = .09(30.5)^2 -5 = 78.7225 is even closer, within .0075 or within 0.75 hundredth

C(31)-C(30) is the marginal cost over an interval

C'(30.5) is the marginal cost at a point. The smaller the interval the closer C'(x) is to the difference in total cost.

Answer 2

(a)

MC(x)= dC/dx = 0.09x^2 - 5

(b)

MC(30) = 0.09*900-5 = 81-5=76

(c)

C(31) = 0.03*31*31*31 - 5*31+300

= 1,038.73

C(30) = 0.03*30*30*30 - 5*30 + 300

= 960

c(31)-c(30) = 78.73