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پاورپوینت کامل History of calculus 33 اسلاید در PowerPoint

اسلاید ۴: Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See Archimedes Quadrature of the Parabola, The Method, Archimedes on Spheres & Cylinders.) It was not until the time of Newton that these methods were made obsolete. It should not be thought that infinitesimals were put on rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true.In the third century Liu Hui wrote his Nine Chapters and also Haidao suanjing (Sea Island Mathematical Manual), which dealt with using the Pythagorean theorem (already stated in the Nine Chapters), known in China as the Gougu theorem, to measure the size of things. He discovered the usage of Cavalieris principle to find an accurate formula for the volume of a cylinder, showing a grasp of elementary concepts associated with the differential and integral calculus. In the 11th century, the Chinese polymath, Shen Kuo, developed packing equations that dealt with integration.

اسلاید ۵: Indian mathematicians produced a number of works with some ideas of calculus. The formula for the sum of the cubes was first written by Aryabhata circa 500 AD, in order to find the volume of a cube, which was an important step in the development of integral calculus.The next major step in integral calculus came in the 11th century, when Ibn al-Haytham (known as Alhacen in Europe), an Iraqi mathematician working in Egypt, devised what is now known as Alhazens problem, which leads to an equation of the fourth degree, in his Book of Optics. While solving this problem, he was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a paraboloid, and was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.

اسلاید ۶: Parallel developmentIn the 17th century, Pierre de Fermat, among other things, is credited with an ingenious trick for evaluating the integral of any power function directly, thus providing a valuable clue to Newton and Leibniz in their development of the fundamental theorem of calculus. Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature.At around the same time, there was also a great deal of work being done by Japanese mathematicians, particularly Kowa Seki. He made a number of contributions, namely in methods of determining areas of figures using integrals, extending the method of exhaustion. While these methods of finding areas were made largely obsolete by the development of the fundamental theorems by Newton and Leibniz, they still show that a sophisticated knowledge of mathematics existed in 17th century Japan.

اسلاید ۷: Development of differential calculusThe Greek mathematician Archimedes was the first to find the tangent to a curve, other than a circle, in a method akin to differential calculus. While studying the spiral, he separated a points motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together thereby finding the tangent to the curve.The Indian mathematician-astronomer Aryabhata in 499 used a notion of infinitesimals and expressed an astronomical problem in the form of a basic differential equation. Manjula, in the 10th century, elaborated on this differential equation in a commentary. This equation eventually led Bhskara II in the 12th century to develop the concept of a derivative representing infinitesimal change, and he described an early form of Rolles theorem.

اسلاید ۸: Concept of functionIn the late 12th century, the Persian mathematician, Sharaf al-Dn al-Ts, introduced the idea of a function. In his analysis of the equation x3 + d = bx2 for example, he begins by changing the equations form to x2(b x) = d. He then states that the question of whether the equation has a solution depends on whether or not the “function” on the left side reaches the value d. To determine this, he finds a maximum value for the function. Sharaf al-Din then states that if this value is less than d, there are no positive solutions; if it is equal to d, then there is one solution; and if it is greater than d, then there are two solutions. However, his work was never followed up on in either Europe or the Islamic world.

اسلاید ۹: Development of derivativeSharaf al-Dn was also the first to discover the derivative of cubic polynomials. His Treatise on Equations developed concepts related to differential calculus, such as the derivative function and the maxima and minima of curves, in order to solve cubic equations which may not have positive solutions. For example, in order to solve the equation x3 + a = bx, al-Tusi finds the maximum point of the curve . He uses the derivative of the function to find that the maximum point occurs at , and then finds the maximum value for y at by substituting back into . He finds that the equation has a solution if , and al-Tusi thus deduces that the equation has a positive root if , where D is the discriminant of the equation.

اسلاید ۱۰: Pre-Newton-LeibnizIn the 15th century, an early version of the mean value theorem was first described by Parameshvara (1370–۱۴۶۰) from the Kerala school of astronomy and mathematics in his commentaries on Govindasvmi and Bhaskara II.In 17th century Europe, Isaac Barrow, Pierre de Fermat, Blaise Pascal, John Wallis and others discussed the idea of a derivative. In particular, in Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed a method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation. Isaac Newton would later write that his own early ideas about calculus came directly from Fermats way of drawing tangents.The first proof of Rolles theorem was given by Michel Rolle in 1691 after the founding of modern calculus. The mean value theorem in its modern form was stated by Augustin Louis Cauchy (1789-1857) also after the founding of modern calculus.

اسلاید ۱۱: Mathematical analysisGreek mathematicians such as Eudoxus and Archimedes made informal use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. In India, the 12th century mathematician Bhaskara II gave examples of the derivative and differential coefficient, along with a statement of what is now known as Rolles theorem.Mathematical analysis has its roots in work done by Madhava of Sangamagrama in the 14th century, along with later mathematician-astronomers of the Kerala school of astronomy and mathematics, who described special cases of Taylor series, including the Madhava-Gregory series of the arctangent, the Madhava-Newton power series of sine and cosine, and the infinite series of . Yuktibhasa, which some consider to be the first text on calculus, summarizes these results.[19][20][21]In the 15th century, a German cardinal named Nicholas of Cusa argued that rules made for finite quantities lose their validity when applied to infinite ones, thus putting to rest Zenos paradoxes.

اسلاید ۱۲: Modern calculusJames Gregory was able to prove a restricted version of the second fundamental theorem of calculus in the mid-17th century.Newton and Leibniz are usually credited with the invention of modern calculus in the late 17th century. Their most important contributions were the development of the fundamental theorem of calculus. Also, Leibniz did a great deal of work with developing consistent and useful notation and concepts. Newton was the first to organize the field into one consistent subject, and also provided some of the first and most important applications, especially of integral calculus.Important contributions were also made by Barrow, Descartes, de Fermat, Huygens, Wallis and many others.

اسلاید ۱۳: Newton and LeibnizBefore Newton and Leibniz, the word “calculus” was a general term used to refer to any body of mathematics, but in the following years, calculus became a popular term for a field of mathematics based upon their insights. The purpose of this section is to examine Newton and Leibniz’s investigations into the developing field of calculus. Specific importance will be put on the justification and descriptive terms which