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Slopes, Rates of Change, and Derivatives

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Slopes, Rates of Change, and Derivatives 2.2 Slopes, Rates of Change, and Derivatives APPLICATION PREVIEW The Confused Creation of Calculus* Calculus evolved from consideration of four major problems that were current in 17th century Europe: how to define instantaneous velocity, how to define tangent lines to curves, how to find maxi- mum and minimum values of functions, and how to calculate areas and volumes, all of which will be discussed in this and the follow- ing chapters. While calculus is now presented as a logical mathe- matical system, it began instead as a collection of self-contradictory ideas and poorly understood techniques, accompanied by much controversy and criticism. It was developed by two people of dia- metrically opposed temperaments, Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716), each while in his twenties (Newton at age 22 and Leibniz at about 29). Newton attended mediocre schools in England, entered Cam- bridge University (with a deficiency in Euclidean geometry), im- mersed himself in solitary studies, and graduated with no particular distinction. He then returned to his family’s small farm where, during a 2-year period, he singlehandedly developed calcu- lus, in addition to the science of optics and the theory of universal gravitation, all of which he kept to himself. He then returned to Cambridge for a master’s degree and secured an appointment as a professor, after which he became even more solitary and intro- verted. A contemporary described him as never taking “any recre- ation or pastime either in riding out to take the air, walking, bowling, or any other exercise whatever, thinking all hours lost that was not spent in his studies.” He was often seen “with shoes down at heels, stockings untied,..., and his head scarcely combed.” He was not a popular teacher, sometimes lecturing to empty class- rooms, and his mathematical writings were very difficult to under- stand (he told one friend that he wrote this way intentionally “to avoid being bated by little smatterers in mathematics”). Leibniz, on the other hand, was a diplomat, a lawyer, and a world traveler, corresponding with people as far away as China and Ceylon. He scorned universities as “monkish,” believing that real knowledge came from more practical activities. While in Paris and London on political missions for his native Germany, he met scientists who stimulated his interest in mathematics, and he began to develop the ideas of calculus. Leibniz’s writings, however, were as obscure as Newton’s. His paper explaining the rules of calculus was deemed by another mathematician to be “an enigma rather than an explication.” To define quantities approaching zero, Leibniz said they were to ordinary variables as the radius of the earth is to that of the heavens, clearly a statement with no mathematical meaning. Criticism of calculus was not limited to mathematicians. One of its severest critics was Bishop George Berkeley, who, in The Analyst, Or ADiscourse Addressed to an Infidel Mathematician, wrote that vari- ables approaching zero were “neither finite quantities, nor quanti- ties infinitely small, nor yet nothing,” ridiculing them as “the ghosts of departed quantities.” Toward the end of the 18th century, Voltaire summed up the study of calculus as “the art of numbering and measuring exactly a Thing whose Existence cannot be con- ceived.” In spite of such inauspicious beginnings, calculus was eventu- ally placed on sound foundations, through the works of mathemati- cians such as the Bernoullis, Cauchy, Lagrange, and Weierstrass, and finally achieved logical status in the 19th century. The improve- ment in the explanation and understanding of calculus can be ap- preciated by its progress from a subject usable by only a few brilliant experts to one that is now routinely taught in colleges and understood by undergraduates. * For more information, see Mathematical Thought from Ancient to Modern Times (Oxford University Press, 1972), by Morris Kline..
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