The purpose of this applet is to investigate the nature of the error function error(x) = f(x) - L(x) as x -> a for the specific function drawn. For a fixed value of x from the pull-down menu, the error function is normalized with respect to the error at x, as described below, and is plotted in the inset as the slope of the line changes.

Compared with the drawing in the text, we have written error(x) in the applet as f(x) - L(x) instead of L(x) - f(x) as it is in the book so that the error is positive as the line approaches the tangent line. This is not very important because, normally, we would be interested in the absolute value of the error, which is the same regardless the order in which we write f(x) and L(x).

The inset graph shows the normalized error function error(t). It is the error on the interval a-(x-a) to a+(x-a) normalized so that the error at t = x equals 1. (Remember that x is a fixed value from the pull-down menu.)

Look at the inset. At t = a, the error error(a) is, of course, zero. What is interesting however is the behavior (shape) of the error when t is near a. When the slope m is not equal to f'(a) the error function is linear in a neighborhood of a. When m = f'(a) it is quadratic. This is why the tangent line, of all the lines through the point (a, f(a)), is the "best" linear approximation. It illustrates graphically the fact given in the proof at the end of section 2.14, namely that error(x) not only tends to zero as x -> a, but it does so faster than any line for which m is not equal to f'(a).