CALC BC SEQUENCES AND SERIES SERIES
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Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The ideas were similar to Archimedes' in The Method, but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Ī significant work was a treatise, the origin being Kepler's methods, written by Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections.

Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse. Johannes Kepler's work Stereometrica Doliorum formed the basis of integral calculus. However, they were not able to 'combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today'.

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A complete theory encompassing these components is now well known in the Western world as the Taylor series or infinite series approximations. Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics thereby stated components of calculus. In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. 1040 AD) derived a formula for the sum of fourth powers.

In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen ( c. Medieval Middle East Ibn al-Haytham, 11th-century Arab mathematician and physicist In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method that would later be called Cavalieri's principle to find the volume of a sphere. The method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In The Method of Mechanical Theorems he describes, for example, calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. 212 BC), who combined it with a concept of the indivisibles-a precursor to infinitesimals-allowing him to solve several problems now treated by integral calculus. 390 – 337 BC) developed the method of exhaustion to prove the formulas for cone and pyramid volumes.ĭuring the Hellenistic period, this method was further developed by Archimedes ( c. Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus ( c. See also: Greek mathematics Archimedes used the method of exhaustion to calculate the area under a parabola in his work Quadrature of the Parabola.

Furthermore, the term 'calculus' has variously been applied in ethics and philosophy, for such systems as Bentham's felicific calculus, and the ethical calculus. Examples of this convention include propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus. In addition to the differential calculus and integral calculus, the term is also used for naming specific methods of calculation and related theories which seek to model a particular concept in terms of mathematics. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton. Because such pebbles were used for counting out distances, tallying votes, and doing abacus arithmetic, the word came to mean a method of computation. The word calculus is Latin for 'small pebble' (the diminutive of calx, meaning 'stone'), a meaning which still persists in medicine. In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits.

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Calc bc sequences and series