Question

The management of ThermoMaster Company, whose Mexican subsidiary manufactures an indoor-outdoor thermometer, has estimated that the total weekly cost (in dollars) for producing x thermometers is represented by the following function.

C(x) = 3100 + 6x

(a) Find the average cost function C.
C(x) =

(b) Find the marginal average cost function C ' .
C ' (x) =

Answer 1

The average cost function is
`\bar{C}(x)=6+(3100)/(x)`.

The marginal average cost function is
`(d)/(dx) \bar{C}(x)=-(3100)/(x^2)`.

Explanation:

If
`C(x)` is the cost function for some item then the average cost function is,

`\bar{C}(x)=(C(x))/(x)`

The marginal average cost function is the derivative of the average cost function.

We have that the total weekly cost function is

`C(x) = 3100 + 6x`

Applying the above definitions, we get that:

The average cost function is

`\bar{C}(x)=(3100 + 6x)/(x) =6+(3100)/(x)`

The marginal average cost function is

`(d)/(dx) \bar{C}(x)=(d)/(dx)(6+(3100)/(x))\\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\(d)/(dx) \bar{C}(x)=(d)/(dx)\left(6\right)+(d)/(dx)\left((3100)/(x)\right)\\\\(d)/(dx) \bar{C}(x)=0-(3100)/(x^2)\\\\(d)/(dx) \bar{C}(x)=-(3100)/(x^2)`