Introduction to Calculus in Economics

Question

Calculus is a powerful tool used in economics. One of the initial applications areas is the study of a firm, a topic in microeconomics. An important function is the cost function function C(x), the cost of producing x items (of whatever they are selling). This question deals with just the cost function C(x).

 

Problem Set question:

 

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

C(x)=√10x−4x−8.

 

Find the production level that minimizes the average cost per unit.

 

 

The number of units that minimizes the average cost is ?

Answer 1

C(x)= sqr10x -4x -8

take the derivative and set = 0

C'(x) = -1/2sqr10x - 4 = 0

-1 -4(2sqr10x) = 0

-2sqr10x = 4

sq10x = 4/-2=-2

10x = 4

x = 4/10 = 0.4 minimizes Total Cost = C(.4)

Average Cost =AC= Total Cost divided by x= per unit cost

AC= C(x)/x = (sqr10x)/x- 4 - 8/x= sqr10/sqrx -4 -8/x

AC'(x) = -sqr10/2sqrx +8/x^2 = 0

-(sqr10)x^2 +16sqrx = 0

10x^4 = 256x

5x^4 - 128x = 0

x(5x^3 -128) = 0

5x^3 = 128

x^3 = 128/5 = 25.6

x = cube root of 25.6 = a little less than 3 as 3 cubed =27, about 2.9 maybe

minimum per unit cost = a little less than 3

x= 25.6^(1/3) = about 2.947

unless you really meant

Cost = sqr(10x-4x-8) then

C(x)/x = AC = sqr(6x-8)/x

AC' = -sqr(6x-8)/x^2 =0

6x-8=0

x = 8/6= 4/3