|
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral. The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function〔More exactly, the theorem deals with definite integration with variable upper limit and arbitrarily selected lower limit. This particular kind of definite integration allows us to compute one of the infinitely many antiderivatives of a function (except for those that do not have a zero). Hence, it is almost equivalent to indefinite integration, defined by most authors as an operation that yields any one of the possible antiderivatives of a function, including those without a zero.〕 is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions. The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals. ==History== The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that these two operations were related. Ancient Greek mathematicians knew how to compute area via infinitesimals, an operation that we would now call integration. The origins of differentiation likewise predate the Fundamental Theorem of Calculus by hundreds of years; for example, in the fourteenth century the notions of ''continuity'' of functions and ''motion'' were studied by the Oxford Calculators and other scholars. The historical relevance of the Fundamental Theorem of Calculus is not the ability to calculate these operations, but the realization that the two seemingly distinct operations (calculation of geometric areas, and calculation of velocities) are actually closely related. The first published statement and proof of a restricted version of the fundamental theorem was by James Gregory (1638–1675).〔 See, e.g., Marlow Anderson, Victor J. Katz, Robin J. Wilson, ''Sherlock Holmes in Babylon and Other Tales of Mathematical History'', Mathematical Association of America, 2004, (p. 114 ). 〕 Isaac Barrow (1630–1677) proved a more generalized version of the theorem〔https://archive.org/details/geometricallectu00barruoft〕 while Barrow's student Isaac Newton (1643–1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「fundamental theorem of calculus」の詳細全文を読む スポンサード リンク
|