Question

Suppose that the cost (in dollars) for a company to produce x pairs of a new line of jeans is given by
C(x)=2000+3x+0.01x^2 +0.0002x^3
(a) find the marginal cost funtion
(b) findC'(100) and explain its meaning. What does it predict?

Answer 1

Answer:

The marginal cost function is $C'(x)=0.0006x^2+0.02x+3$

The marginal cost at the production level of 100 pairs is

$C'(100)=\ 11/pair$. This gives the rate at which costs are increasing with respect to the production level when x = 100 and predicts the cost of the 101st pair of jeans.

Step-by-step explanation:

(a) The marginal cost function is the derivative of the cost function.

$\:{Marginal \:Cost} = \frac{d}{dx}(C(x))=C'(x)\\\\C'(x)=\frac{d}{dx}(2000+3x+0.01x^2 +0.0002x^3)\\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\C'(x)=\frac{d}{dx}\left(2000\right)+\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(0.01x^2\right)+\frac{d}{dx}\left(0.0002x^3\right)\\\\C'(x)=0.0006x^2+0.02x+3$

(b) The marginal cost at the production level of 100 pairs is

$C'(x)=0.0006x^2+0.02x+3\\C'(100)= 0.0006(100)^2+0.02(100)+3\\C'(100)=\ 11/pair$

This gives the rate at which costs are increasing with respect to the production level when x = 100 and predicts the cost of the 101st pair of jeans.