Calculus: Analytic Geometry and Calculus, with Vectors

3.9 Increments and differentials 197

If we measure the edges of a cube and decide that, subject to errors in
measurement, each side has length x, we conclude that, subject to con-

sequences of errors in measurement, the volume V of the cube is x3.
If the edges have exact lengths x + Ax, then the exact volume is (x + Ax) 3,
or V + AV, and the number AV is the error in Y produced by the error
Ax in x. In quantitative treatments of this matter, we let Y = x3 and
use the differential dY defined by

(3.98) dV = 3x2 dx

as an approximation to AV. In some practical situations, it is reasonable

to assume that (for some positive constant p that might be j- or 2 or 10),
the error dx in x has a magnitude not exceeding p per cent of x. This
means that

(3.981) Jdxi <

100

jxI.

With this assumption, we find from (3.981) that

(3.982) I dVI 5 3x2

100 x = 00 x3 = 100

This leads us to the idea that if we measure the length of the edge of a

cube with an error not exceeding p per cent, the resulting error in the

computed volume will not exceed 3p per cent.

Problems 3.99

1 Find the increment AA4 and the differential dl of area produced when a

circular disk of radius r is expanded or contracted to a circular disk of radius

r + A. .4ns.: A14 = 7r(2hr + h2) and dll = 2rrrh. Remark: We have another

opportunity to try to understand a formula. Sketch a figure in which Jhi is

small in comparison to r and observe that the difference of the two disks is a

circular ring of thickness JhJ. Since the inner (or outer) boundary of this ring

has length 27rr, it is not surprising that the area of the ring is approximately

21rrihi.

2 The area .4 of a sphere of radius r in Es is 4rrr2; this should seem to be

reasonable because the area of a hemisphere should be about twice the area of an

equatorial disk. The volume Y of the spherical ball bounded by this sphere is

4grrr8. Find the increment AY and the differential dY of volume produced when

the radius changes from r to r + h. Show that the formula for dY can be put

in the form dY = .4h and try to see a geometrical reason why 4h should be a

good approximation to AY when Jhl is small.

3 Use differentials to obtain an approximation to the number of cubic

centimeters of chromium plate that must be applied to the lateral surface of a

circular cylindrical rod 30 centimeters long to increase its radius from 2.34

centimeters to 2.35 centimeters. Ins.: About 4.4 cubic centimeters.