Chapter 3.6, Problem 255E

For the following exercises, use the information in the following table to find $h\text{'}\left(a\right)$ at the given value for a.

$x$	$f\left(x\right)$	$f\text{'}\left(x\right)$	$g\left(x\right)$	$g\text{'}\left(x\right)$
0	2	5	0	2
1	1	-2	3	0
2	4	4	1	-1
3	3	-3	2	3

255. [T] The total cost to produce x boxes of Thin mint Girl Scout cookies is C dollars, where $C=0.0001x^3-0.02x^2+3x+300$ . In t weeks production is estimated to be $x=1600+100t$ boxes.

a. Find the marginal cost $C\text{'}\left(x\right)$ .

b. Use Leibniz’s notation for the chain rule, $\frac{dC}{dt}=\frac{dC}{dx}\cdot \frac{dx}{dt}$ , to find the rate with respects to time t that the cost is changing.

c. Use b. to determine how fast costs are increasing when t = 2 weeks. Include units with the answer.