4.8 Applications to Economics

 

 Economics:

C (x) = Cost Function

C’ (x)= Marginal Cost

c(x)=  = Average Cost (cost/item)

c’ (x)=  You want to minimize c’(x) so you set it equal to zero; Minimum occurs when c (x) = C’ (x).

Example:

     C (x) = 25,000 + 120x +0.1

  1. c(1000) = 25,000 + 120 (1000) + 0.1()
    1. 245000
  2. C’ (1000)= 120 + .2x = 120+.1(1000)
    1. 320
  3. c(1000)=
    1. 245

 

 

  1. Find Minimum: c(x)=C’(x)      

 = 120+.2x  2500+120x+.1= 120x+.225000=.125000=x=500  

  1. Fine Minimum average cost

c(500) =

c(500)= 220

 

 

p (x) = Price Function (demand)

R (x) = Revenue function = x  p(x)

P (x) = R (x) – C (x) = Profit Function

P’ (x)= R’ (x) – C’ (x)

R’ (x)= Marginal Revenue

Maximum:   R’ (x) –C’ (x) = 0

              R’ (x) = C’ (x)  Profit = Maximum

 

 

 

 

 

Example:

     C (x) = 680 + 4x + .01

     P (x) = 12 -                   C’ (x) = 4 + .02 x

     R (x) = x (12 - ) = 12x -

R’ (x)= 12-(2x) = 12x -

4 + .02x = 12-.004 x

     .004x + .02x-8=0

          .024x = 8

          x= 333