THE DEVELOPMENT OF DIFFERENTIAL CALCULUS

After the advent of algebra in the sixteenth century, a flood of mathematical discoveries swept through Europe. The most important were differential calculus and integral calculus, bold new methods for attacking a host of problems that had challenged the world's best minds for more than 2000 years. Differential calculus deals with ideas such as speed, rate of growth, tangent lines, and curvature, whereas integral calculus treats topics such as area, volume, arc length, and centroids.

Work begun by Archimedes in the third century b.c. led ultimately to the birth of integral calculus in the seventeenth century a.d. This development has a long and fascinating history which we will explore in more detail later.

Differential calculus has a relatively short history. The concept of derivative was first formulated early in the seventeenth century when the French mathematician Pierre de Fermat tried to devise a way of finding the smallest and largest values of a given function. He imagined the graph of a function having, at each of its points, a direction given by a tangent line, as suggested by the points labeled in Fig. 3.1.