Answer
(a)
$C(x)$ = $2000+3x+0.01x^{2}+0.0002x^{3}$
$C'(x)$ = $3+0.02x+0.0006x^{2}$
(b)
$C'(100)$ = $3+0.02(100)+0.0006(100)^{2}$ = $11$ $dollars/pair$
$C'(100)$ is the rate at which the cost is increasing as the 100th pair of jeans is produced. It predicts the (approximate) cost of the 101st pair.
(c)
The cost of manufacturing the 101st pair of jeans is
$C(101)-C(100)$ = $2611.07-2600$ $\approx$ $11.07$ $dollars$
This is close to the marginal cost from part (b).
Work Step by Step
(a)
$C(x)$ = $2000+3x+0.01x^{2}+0.0002x^{3}$
$C'(x)$ = $3+0.02x+0.0006x^{2}$
(b)
$C'(100)$ = $3+0.02(100)+0.0006(100)^{2}$ = $11$ $dollars/pair$
$C'(100)$ is the rate at which the cost is increasing as the 100th pair of jeans is produced. It predicts the (approximate) cost of the 101st pair.
(c)
The cost of manufacturing the 101st pair of jeans is
$C(101)-C(100)$ = $2611.07-2600$ $\approx$ $11.07$ $dollars$
This is close to the marginal cost from part (b).