Fundamental Theorem of Calculus Fundamental Theorem of Calculus The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the integral. The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions. The second part, sometimes called the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals. The first published statement and proof of a restricted version of the fundamental theorem was by James Gregory (1638–1675). Isaac Barrow (1630–1677) proved a more generalized version of the theorem, while Barrow's student Isaac Newton (1643–1727) completed the development of the surrounding mathematical theory. Know More About Prime Factorization Calculator
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Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities, and introduced the notation used today. Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time (or over some other quantity) adds up to the net change in the quantity. In the case of a particle traveling in a straight line, its position, x, is given by x(t) where t is time and x(t) means that x is a function of t. The derivative of this function is equal to the infinitesimal change in quantity, dx, per infinitesimal change in time, dt (of course, the derivative itself is dependent on time). This change in displacement per change in time is the velocity v of the particle. In Leibniz's notation: By the logic above, a change in x (or Δx) is the sum of the infinitesimal changes dx. It is also equal to the sum of the infinitesimal products of the derivative and time. This infinite summation is integration; hence, the integration operation allows the recovery of the original function from its derivative. It can be concluded that this operation works in reverse; the result of the integral can be differentiated to recover the original function. For a continuous function y = f(x) whose graph is plotted as a curve, each value of x has a corresponding area function A(x), representing the area beneath the curve between 0 and x. The function A(x) may not be known, but it is given that it represents the area under the curve. The area under the curve between x and x + h could be computed by finding the area between 0 and x + h, then subtracting the area between 0 and x. In other words, the area of this “sliver” would be A(x + h) − A(x). Learn More :- How To Find The Area Of A Triangle
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There is another way to estimate the area of this same sliver. h is multiplied by f(x) to find the area of a rectangle that is approximately the same size as this sliver. It is intuitive that the approximation improves as h becomes smaller. It can thus be shown, in an informal way, that f(x) = A′(x). That is, the derivative of the area function A(x) is the original function f(x); or, the area function is simply the antiderivative of the original function. Computing the derivative of a function and “finding the area” under its curve are "opposite" operations. This is the crux of the Fundamental Theorem of Calculus. Most of the theorem's proof is devoted to showing that the area function A(x) exists in the first place, under the right conditions
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