# Read e-book online L'essentiel en théorie des probabilités (Licence Master) PDF

By J.Jacob, Ph.Protter

ISBN-10: 2842250508

ISBN-13: 9782842250508

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Ifaxm/n = y; it shall be _^_x(m+n)/n m +n = area ΑβΓ) NEWTONS CALCULUS (PART 1) A B Fig. 1 A typical curve. As Newton's comments toward the end of the tract indicate, he was fully aware of the relationship between the area under a curve and the antiderivative and of the difference between them. This statement about areas should be interpreted as one about antiderivatives, or indefinite integrals, whose modern equivalent is / xrdx=-^—xr+1+C, r + 1 where r is any real number different from — 1. It is clear from Newton's examples that he knew exactly how to handle situations where the curve dips below the x-axis, or where A is not the origin.

If so, then Fermat is to be counted amongst these giants, and his differentiation and integration methods are precursors of the calculus of Leibniz and Newton. 1 FERMAT'S CALCULUS The lawyer and part time mathematician Pierre de Fermat (16017-1665) is justly famed for his pioneering and influential work in number theory. His contributions to the evolution of calculus are less well known, yet important. His invention of the coordinate system, now known as Cartesian coordinates, predates Rene Descartes' (1596-1650) work on the same topic by eight years.

Newton did offer some arguments to support his new mathematics, but these remain ultimately unsatisfactory. This page intentionally left blank L- Newton's Calculus (Part 2) As Newton recognized and stressed, power series provide a powerful technique for solving a variety of algebraic and differential equations. 1 THE SOLUTION OF DIFFERENTIAL EQUATIONS An ordinary differential equation is an equation that relates x, y, y' and possibly higher derivatives. Such equations are y' = x + y, y' = 1 - 3x 4- y + x2 + xy, (1 + x2)y" + 2xy' + Ax2y = 0.