Find formulas for the rates of change of area and volume of a circle and sphere

by RoRi

September 11, 2015

Show that the rate at which the area of a circle changes is equal to the circumference of the circle.
Show that the rate at which the volume of a sphere changes is equal to the surface area of the sphere.

The area of a circle of radius $r$ is $\pi r^2$ . Therefore, the rate of change of the area with respect to $r$ is the derivative,
$a'(r) = 2\pi r = \text{circumference}. \qquad \blacksquare$
The volume of a sphere of radius $r$ is $\frac{4}{3} \pi r^3$ . Therefore, the rate of change of the volume with respect to 4r$ is the derivative,
$v'(r) = 4 \pi r^2 = \text{ surface area}. \qquad \blacksquare$