AP Calculus BC: Extra Credit Pd 2&3

The History of Calculus

Calculus is the study of how things change. This branch of mathematics mainly focuses on limits, functions, derivatives, integrals and infinite series. When discussing the history of calculus, we often give Newton all the credit for the discovery, however it was actually developed independently by two different men in the seventeenth century, Gottfried Wilhelm Leibniz, a self-taught German mathematician, and Isaac Newton, an English scientist. They established the basic principles of calculus, and, with the help of other mathematicians, it was refined using the concept of the limit. Although both men were fundamental in the creation of calculus, they had different views. Newton considered variables changing with time while Leibniz thought of the variables x and y as ranging over sequences of infinitely close values. Leibniz developed dx and dy as the difference between values of these sequence to find the tangent. Newtown used x’ and y’ as finite velocities to compute the tangent. Neither Leibniz nor Newtown thought in terms of functions but instead in terms of graphs. Newton had a geometrical viewpoint of calculus, relating it to the physical world while Leibniz had an analytical viewpoint. In their development of calculus, both used “infinitesimals”, which are quantities that are infinitely small but nonzero. Although it was helpful with their computations and derivations, mathematicians disapprove of this concept of “infinitesimals” because they do not really exist. Lord Bishop Berkeley referred to them as “the ghost of departed quantities”. Berkeley’s criticisms played an important role in the development of calculus because later on, Cauchy, Weierstrass, and Riemann replaced infinitesimals with limits, the notion of quantities being “close” to others. The development of calculus goes through a timeline of three periods: Anticipation, Development, and Rigorization. During the Anticipation stage mathematicians used techniques that involved infinite processes to find areas under curves. In the Development stage Newton and Leibniz created the foundations of calculus and tied all the techniques together under derivative and integral. The ideas were sloppy and not always logically correct. It took many years during the Rigorization stage to refine and clean up those ideas and finally finalize the mathematical foundation of calculus. Newton built his theory of calculus on earlier works by René Descartes, Pierre de Fermat, and other Continental mathematicians. In 1664 he laid the foundations of the differential calculus, which he described as the "method of fluxions. ”The initial problem Newton dealt with was calculation the exact slope…