12/10/2023 0 Comments Calculus 2 differential equations![]() ![]() 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. $$4(gal/min)\cdot0.005 lb/gal\cdot t (min)=0.Look up calculus in Wiktionary, the free dictionary. ![]() Why can't we just do it without differential equations, like this?īecause the solution flowing in is the same no matter what time, we can say that the pounds of salt coming in are However, the solution to that doesn't really fit the logic of the problem that well. Where $V=100-t$ because of the way the tank drains. My calculus teacher says that we should do it using differential equations. We call a variable $Q$ as the pounds of salt in the tank, and we want to find the function that describes $Q$. We have a pipe flowing into the tank at 4 gal/min delivering solution that has 0.005 lb/gal of salt, and we have an output that is flowing out of the tank at 5 gal/min, with salt content same as that of the tank the instant it drained out. We have the following scenario: There is a 100 gallon tank with 20 lb of salt dissolved in it. ![]()
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