Using the Sign Function to Analyze Graphs

Abstract

The easy availability of graphing calculators brings into question the traditional usage of the first and second derivatives to do curve analysis. It is difficult to convince students that such applications as the first and second derivative tests are useful or even necessary. We illustrate how curve analysis can be implemented on a calculator by using the sign function, thus turning the graphical power of the calculator to powerful advantage. This permits a check of the traditional table used for the first derivative test. A graph that shows the sign of both the first and second derivatives presents a nearly complete analysis of a function. To analyze a curve graphically requires finding a graph that displays all essential features clearly, which is hard to do on a device with limited resolution. Even if the domain is selected so that it contains all critical numbers, determining a usable range is often a chore and beyond the ability of some students. Consider, for example, the function f(x) =x3 2x2 +x 30. The function is flat on (j,l) and its graph appears to be strictly increasing, as shown in Figure 1. Both the realization that something of interest occurs on interval (j, 1) and a bit of experimentation are needed to produce a graph that shows that / is actually decreasing on this interval and that it has two relative extrema.