EXAMPLE 35
Suppose the diameter of a cylinder is 8 centimeters. If its circumference is increased by 2
centimeters, how much larger, approximately, are
(a) the diameter, and
(b) the area of a cross section?
SOLUTIONS:
(a) Let D and C be respectively the diameter and circumference of the cylinder. Here, D plays
the role of f, and C that of x, in the linear approximation equation (1) a previous page. The
approximate increase in diameter, when C = 8π, is therefore equal to D ′(C) times (the change
in C). Since C = πD, and (which is constant for all C). The change in C is given
as 2 cm; so the increase in diameter is equal approximately to 2. 0.6366 cm.
(b) The approximate increase in the area of a (circular) cross section is equal to
A ′(C) · (change in C),
where the area Therefore,Since the change in C is 2 cm, the area of a cross section increases by approximately 4 · 2 = 8
cm^2.M. RELATED RATES
If several variables that are functions of time t are related by an equation, we can obtain a relation
involving their (time) rates of change by differentiating with respect to t.
EXAMPLE 36
If one leg AB of a right triangle increases at the rate of 2 inches per second, while the other leg
AC decreases at 3 inches per second, find how fast the hypotenuse is changing when AB = 6 feet
and AC = 8 feet.FIGURE N4–21
SOLUTION: See Figure N4–21. Let u, v, and z denote the lengths respectively of AB, AC, and
BC. We know that Since (at any time) z^2 = u^2 + v^2 , then