Calculus: Analytic Geometry and Calculus, with Vectors

194 Functions, limits, derivatives

lies in the fact that small relative errors in the large weights can produce

huge relative errors in the small difference. The obvious way to find

the order of magnitude of the number Ay defined by

(3.91) Ay =f(x+Ax) -f(x)

is to calculate the terms on the right side and subtract. However, when

the numbers Ax and Ay are very small in comparison to x and f(x) and

f(x + Ax), this calculation can be thoroughly tedious and impractical.

The following alternative way of estimating Ay is very often used. If

f and x are such that f'(x) exists, we can define O(x,Ax) by the formula

(3.92)

Ay=f(x + Ax) - f(x)

= f'(x) + -b(x,Ax)

Ax Ax

and conclude that

(3.921) lim A(x,AX) = 0.

Multiplying (3.92) by Ax gives the formula

(3.922) Ay = f'(x) Ax + 0(x,Ax) Ax,

which separates Ay into the sum of two "parts." In case f'(x) s 0 and

Ax is near 0, the "part" 4(x,Ax) Ax is small in comparison to the "part"

f'(x) Ax and the number f'(x) Axis the "principal part" of Ay. Therefore,

when f' (x) 0 0, we can write the formula

(3.93) Ay r.. f'(x) Ax

to mean that the numbers Ay and f'(x) Ax have the same order of mag-

nitude when Ax is near 0, that is,

lim

Ay = 1.

Am-Of (x) Ax

In any case, it is a common practice to use the number f'(x) Ax as an

approximation to Ay when f, x, and Ax are given and JAxJ is judged to

be small enough to make the approximation useful. In some cases it is

equally useful to use the number Ay/f(x) as an approximation to Ax

when f, x, and Ay are given.

We have seen that, in appropriate circumstances, the numbers Ay

(an increment of y) and Ax (an increment of x) are such that Ay and f (x) Ax

are nearly equal in the sense that their ratio is nearly 1. While it may be

difficult to see why we should become excited about the matter, it is

worthwhile to think about and even use pairs of numbers dy and dx for

which dy is exactly (not merely approximately) equal to f' (x) dx so that

(3.94) dy = f'(x) dx.