Introduction

This is the mathematical tale of a cusp in the shape of a bird's beak. Although precalculus and calculus courses must stress the idea of function over that of equation, they nevertheless include a number of important topics concerning polynomial equations in two variables, including implicit differentiation and the study of conic sections. Whereas polynomial functions of one variable have very simple graphs, the graphs of polynomial equations in x and y — even those of relatively low degree — can exhibit wonderfully exotic features.

The story of the bird's beak can be used to enrich a course in analytic geometry, precalculus or calculus. For students who know some calculus, it also provides insight into continuous nondifferentiable functions. There is also a connection to power series representations, although this will not be discussed in this chapter (Euler treats them in §5–9 of [1, 2]).

For further reading on these topics, see [3, 4].

Historical Background

In the 18th century, calculus and the related branches of mathematics gradually changed their perspective from the geometric to the algebraic. When Renée Descartes (1596–1650) and Pierre de Fermat (1601–1665) invented analytic geometry, for example, mathematicians were already familiar with a large assortment of curves, given by a variety of geometric constructions. Analytic geometry gave them a means of associating equations with these curves. With passing time, the study of equations took primacy, so that the graph came to be seen as an attribute of the equation.