Page 42 - How To Solve Word Problems In Calculus

A(x + h) − A(x) A(7) − A(4) 49 − 16 33
= = = = 11
h 3 3 3
(b) We determine the instantaneous rates of change by com-
puting the derivative of the area function and evaluating
it at x = 4, 5, 6, and 7.

A(x) = x 2

A (x) = 2x

A (4) = 8

A (5) = 10

A (6) = 12
A (7) = 14

Observe that the area grows at a faster rate as x increases.

EXAMPLE 2
Find the rate of change of the volume of a sphere with respect
to its radius when its radius is 5.

Solution
4
3
The volume of a sphere of radius r is V(r) = πr .
3
V (r) = 4πr 2

V (5) = 100π

EXAMPLE 3
Let f (x) = 1/x. Determine (a) the rate of change of f at x = 1
and (b) the average rate of change of f from x = 1to x = 2.
Interpret the results graphically.

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