The volume and surface area of a sphere are given by v = 4 / 3 πr3 and s = 4πr2. suppose we increase the radius of a
sphere by 0.001 units from r to r + 0.001.
a. use the binomial theorem to write an expression for the increase in volume v(r + 0.001) − v(r) as the sum
of three terms.
b. write an expression for the average rate of change of the volume as the radius increases from r to r + 0.001.
c. simplify the expression in part (b) to compute the average rate of change of the volume of a sphere as the
radius increases from r to r + 0.001.
d. what does the expression from part (c) resemble?
e. why does it make sense that the average rate of change should approximate the surface area? think about
the geometric figure formed by v(r + 0.001) − v(r). what does this represent?
f. how could we approximate the volume of the shell using surface area? and the average rate of change for the
volume?
g. find the difference between the average rate of change of the volume and s(r) when r = 1.