Increments, Differentials, and Linear Approximations in Edwards & Penney

presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

We might want an easy estimate of the change in f (x) resulting after a given change in x. We will write y for f (x) and presume first that the change in the independent variable is the increment Δx, so that x changes from its original value to the new value x + Δx. This change of

y 's value is the increment Δy, which we compute by subtracting the old value of y from its new value:

Δy = f (x + Δx) - f (x).

The increments Δx and Δy are represented geometrically in the diagram below:

When we compare the actual increment Δy with the change that would occur in the value of y if it continued to change at the fixed rate f ' (x) while the value of the independent variable changes from x to x + Δx. This hypothetical change in y is the differential

dy = f ' (x) Δx.

The figure below shows how dy is the change in height of a point that moves along the tangent line at the point (x, f (x)) rather than along the curve y = f (x).

So we must think of x as fixed, and thus the equation

dy = f ' (x) Δx

shows us that the differential dy is a linear function of the increment Δx. Thus, dy is called the linear approximation to the increment Δy. We can approximate f (x + Δx) by substituting dy for Δy:

Because y = f (x) and dy = f ' (x)Δx, this gives the linear approximation formula

This approximation is good if when Δx is relatively small. If we combine the above formulas we get

Thus the differential dy = f ' (x) Δx is a good approximation for the increment Δy = f (x + Δx) - f (x).

If we replace x with a in the equation

we obtain

And if we then write Δx = x - a, so that x = a + Δx, then the result would be

And because the right hand side of the above equation is a linear function of x,

we then call it the linear approximation L (x) to the function f (x) near the point x = a. As illustrated below, the graph y = L (x) is the straight line tangent to the graph y = f (x) at the point (a, f (a)):

For example, we might want to find the linear approximation for the function

near the point a = 0.

We take note firstly that f (0) = 1 and that

[because the square root of 1 + x is rendered as 1 + x to the 1/2 power. We subtract one from the exponent, leaving -1/2, and make the coefficient one value of the exponent, 1/2.]

and thus f ' (0) = 1/2, because if we substitute 0 for the x, then we get 1/2. Hence if we take the equation

and make a = 0, we obtain

when we substitute the proper values for the two derivatives. Thus our desired linear approximation is

The graph below illustrates the close approximation near x = 0 of the nonlinear function

by its linear approximation

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.204a-205b.