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Contents
Beginnings of Calculus
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Beginnings of Calculus
Mathematicians, over the centuries, had a few problems they had difficulty solving: finding maximum and minimum, area of regions bounded by curves, finding tangents as well as volumes. Mathematicians have tackled them, however, each solution was unique to the problem; a general solution for the infinitely possible problems could not be found. Although the various problems were tackled by different mathematicians, Issac Newton and Gottfried Wilhelm Leibniz were credited as the inventors of calculus. They developed general concepts which relate to the main problems of calculus, as well as introduced notations.
Differentiation Concepts In the late 1620s, Pierre de Fermat found a general procedure for finding maximum and minimum values of a function. His idea came from Johann Kepler, who proved that the largest parallelpiped that can be inscribed in a sphere is a cube. He worked with various radii of spheres and altitudes, eventually discovering that near the maximum volume, the decrements in altitude were so small that they were virtually zero. However, Fermat's method was geometric, and thus included many assumptions which will not stand when expressed in algebraic terms. He also failed to consider if there were two or more soltuions. Later, he modified his methods, as well as extended it to finding tangents. This was before the advent of analytic geometry,this was mainly geometric, and was criticised by Rene Descartes. Descartes had found a method of finding normals, which in turn could be used to find tangents. This was one reason Descartes was critical of Fermat's work. His method was developed from the idea that the radii of a circle are always normal to the circumference. Gilles Persone de Roberval later discovered a kinematic method of determining tangents, by considering the curve to be generated by a moving point. However, the method was complicated and tedious. Johann Hudde simplified these methods and eventually found what we now call the derivative, that the tangent of y = xn is nxn-1. Rene Francois de Sluse also developed his own methods. With these, a general procedure for finding tangents have been found.
Integration Concepts The works during the Greek and Islamic times provided little information about finding the areas and volumes of regions bounded by curves. However, the idea that sustained was that the region needed to be broken up into smaller regions of known areas and volumes. Kepler developed the method of infinitesimals, where he divided circles into very small triangles and volumes into very thin disks. Galilee Galileo used the method of indivisibles instead, where a geometric object is said to e made up of objects of one dimension lower. However, it was Bonaventura Cavalieri was developed a complete theory of infinitesimals. Using his methodology, he eventually derived what is now known as the definite integral, that the area under the curve y = xk is given by:
Evangelista Torricelli also worked on the abovementioned problems, but his most prominent was regarding volumes of revolution. He proved that when the hyperbola xy = k2 is rotated around the y-axis from y = a to infinity, the volume generated was finite. Cavalieri's discovery was actually also discovered by Fermat, Pascal, Roberval and Torricelli around the same time. Fermat used a method involving summation and inequalities. He also developed methods to find the areas under hyperbolas. John Wallis further explored this ideas, introducing the use of fractional exponents, finding the area under the curve y = xp/q. What stumped him, however, was the curve y = x-k, as the area under the hyperbola y = x-1 could not be defined using his method. The problem of the area under the rectangular hyperbola was further investigated by Gregory of St. Vincent, and eventually solved by Alfonso Antonio de Sarasa, who related it to logarithms.
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