Chapter 3.6, Problem 256E

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

256. [T] The formula for the area of a circle is $A=\pi r^2$ , where r is the radius of the circle. Suppose a circle is expanding, meaning that both the area A and the radius r (in inches) are expanding.

a. Suppose $r=2-\frac{100}{{\left(t+7\right)}^2}$ where t is time in seconds. Use the chain rule $\frac{dA}{dt}=\frac{dA}{dr}\cdot \frac{dr}{dt}$ to find the rate at which the area is expanding.

b. Use a. to find the rate at which the area is expanding t = 4 s