Calculus: Analytic Geometry and Calculus, with Vectors

196 Functions, limits, derivatives

and then multiply by dx. A little experience with these things makes us
realize that if y = sin x, we can write the formula dy = cos x dx without
bothering to write the intermediate step dy/dx = cos x.
In most situations where increments Ay, Ax and differentials dy, dx
simultaneously appear, it is convenient to suppose that dx = Ax. In
such cases, glances at figures more or less like Figure 3.951 can fortify
the idea that Ay can easily be twice dy when Ax and dx are equal but
not small, but that Ay and dy must have the same orderof magnitude
when f'(x) v 0 and the equal numbers Ax and dx are near 0. Thus a
useful cookbook modus operandi runs as follows.
When f, x, and Ax are given such that f' (x) s' 0 and we want an approxi-
mation to the number Ay defined by

(3.96) Ay = AX + Ax) - AX),

we put Ax = dx, calculate the number (ordifferential) dy defined by

(3.961) dy = f'(x) dx,

and use dy as an approximation to Ay.

It is instructive to consider a thoroughly simple example in which all

of the details are easily understood. Letting x and Ax be numbers which

could be 38.27 and 0.05, we can determine the increment Ay in the area

of a square when the lengths of its sides are increased from x to x + Ax.

Letting y = x2 and y + Ay = (x + Ax)2, we find that

(3.97) Ay = (x + Ax)2 - x2 = 2x Ax + Axe.

When Ax > 0, the number 2x Ax is the sum of the areas of the two

rectangles of Figure 3.971 which have dimensions x and Ax. The number

AX

Figure 3.971

0x2 is the area of the smaller square in the upper right-hand corner of

the figure. The differential dy is, when dx = Ax,

(3.972) dy = 2x dx = 2x Ax,

and it is easily seen that this is a good approximation to Ay when Ax is

small in comparison to x.

It is sometimes convenient to solve problems more or less like the fol-

lowing one in order to determine the accuracy of measurements required

to produce required accuracy of results computed from the measurements.