The Implicit Function theorem— technique versus understanding

Abstract

In teaching catastrophe theory to undergraduates it is necessary to sketch families of polynomial graphs that depend on several parameters. The qualitative features of these graphs must be deduced using the Implicit Function theorem, which students find difficult both to comprehend and to apply. This article discusses the specific example of cubic and quartic families from the point of view of developing in students the intuitive feeling that roots of polynomial equations depend (smoothly, in general) on the coefficients. We also analyse the skills that must necessarily be built up in students, so that they can handle the technical problems involved.