fundamental theorem of calculus

fundamental theorem of calculus

Basic principle of calculus. It relates the derivative to the integral and provides the principal method for evaluating definite integrals (see differential calculus; integral calculus). In brief, it states that any function that is continuous (see continuity) over an interval has an antiderivative (a function whose rate of change, or derivative, equals the function) on that interval. Further, the definite integral of such a function over an interval a < x < b is the difference F(b) − F(a), where F is an antiderivative of the function. This particularly elegant theorem shows the inverse function relationship of the derivative and the integral and serves as the backbone of the physical sciences. It was articulated independently by Isaac Newton and Gottfried Wilhelm Leibniz.

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fundamental theorem of calculus [¦fən·də¦ment·əl ¦thir·əm əv ′kal·kyə·ləs]

(mathematics)

Given a continuous function ƒ(x) on the closed interval [a,b] the functionalis differentiable on [a,b] andF′(x)= ƒ(x) for everyxin [a,b], and ifGis any function on [a,b] such thatG′(x) = ƒ(x) for allxin [a,b], then