Calculs 7+

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The method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD to find the area of a circle. [10] [11] In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method [12] [13] that would later be called Cavalieri's principle to find the volume of a sphere. The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. [22] The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving predecessors to the second fundamental theorem of calculus around 1670. [23] [24]

Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. [47] :341–453 Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. [47] :685–700 More advanced applications include power series and Fourier series. Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like real numbers but which are, in some sense, "infinitely small". For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and thus less than any positive real number. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols d x {\displaystyle dx} and d y {\displaystyle dy} were taken to be infinitesimal, and the derivative d y / d x {\displaystyle dy/dx} was their ratio. [37] There was a high probability of intraoperative and postoperative surgical complication like infection or bleeding In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics thereby stated components of calculus. A complete theory encompassing these components is now well known in the Western world as the Taylor series or infinite series approximations. [18] [19] According to Victor J. Katz 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". [14] Modern A calculus ( pl.: calculi), often called a stone, is a concretion of material, usually mineral salts, that forms in an organ or duct of the body. Formation of calculi is known as lithiasis ( / ˌ l ɪ ˈ θ aɪ ə s ɪ s/). Stones can cause a number of medical conditions.Calculi of the gallbladder and bile ducts are called gallstones and are primarily developed from bile salts and cholesterol derivatives. Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. [15] Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of ' infinitesimals'. [16] There is evidence of an early form of Rolle's theorem in his work. The modern formulation of Rolle's theorem states that if f ( a ) = f ( b ) = 0 {\displaystyle f\left(a\right)=f\left(b\right)=0} , then f ′ ( x ) = 0 {\displaystyle f'\left(x\right)=0} for some x {\displaystyle x} with a < x < b {\displaystyle \ a

The derivative f′( x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of the second lines. Here the function involved (drawn in red) is f( x) = x 3 − x. The tangent line (in green) which passes through the point (−3/2, −15/8) has a slope of 23/4. The vertical and horizontal scales in this image are different. f ′ ( 3 ) = lim h → 0 ( 3 + h ) 2 − 3 2 h = lim h → 0 9 + 6 h + h 2 − 9 h = lim h → 0 6 h + h 2 h = lim h → 0 ( 6 + h ) = 6 {\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{(3+h) Calculi are usually asymptomatic, and large calculi may have required many years to grow to their large size. Calculi in the gastrointestinal tract ( enteroliths) can be enormous. Individual enteroliths weighing many pounds have been reported in horses. Calculi in the stomach are called gastric calculi (Not to be confused with gastroliths which are exogenous in nature). Several mathematicians, including Maclaurin, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of Cauchy and Weierstrass, a way was finally found to avoid mere "notions" of infinitely small quantities. [38] The foundations of differential and integral calculus had been laid. In Cauchy's Cours d'Analyse, we find a broad range of foundational approaches, including a definition of continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an (ε, δ)-definition of limit in the definition of differentiation. [39] In his work Weierstrass formalized the concept of limit and eliminated infinitesimals (although his definition can validate nilsquare infinitesimals). Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus". Bernhard Riemann used these ideas to give a precise definition of the integral. [40] It was also during this period that the ideas of calculus were generalized to the complex plane with the development of complex analysis. [41]

Conclusion

Some common principles (below) apply to stones at any location, but for specifics see the particular stone type in question.

While many of the ideas of calculus had been developed earlier in Greece, China, India, Iraq, Persia, and Japan, the use of calculus began in Europe, during the 17th century, when Newton and Leibniz built on the work of earlier mathematicians to introduce its basic principles. [11] [28] [45] The Hungarian polymath John von Neumann wrote of this work, Johannes Kepler's work Stereometrica Doliorum formed the basis of integral calculus. [20] 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. [21] Geometrically, the derivative is the slope of the tangent line to the graph of f at a. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f. [49] :61–63 From an underlying abnormal excess of the mineral, e.g., with elevated levels of calcium ( hypercalcaemia) that may cause kidney stones, dietary factors for gallstones.

Some stone types (mainly those with substantial calcium content) can be detected on X-ray and CT scan Modification of predisposing factors can sometimes slow or reverse stone formation. Treatment varies by stone type, but, in general: [ citation needed] If the input of the function represents time, then the derivative represents change concerning time. For example, if f is a function that takes time as input and gives the position of a ball at that time as output, then the derivative of f is how the position is changing in time, that is, it is the velocity of the ball. [31] :18–20