Background of the discovery

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The fundamental theorem of calculus relates to differentiation and integration, showing that these 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 gradients) are closely related.

From the conjecture and the proof of the fundamental theorem of calculus, calculus as a unified theory of integration and differentiation is started. The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character,[1] was by James Gregory (1638–1675).[2][3] Isaac Barrow (1630–1677) proved a more generalized version of the theorem,[4] while his student Isaac Newton (1642–1727) 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.

First theorem

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Second theorem

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References

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  1. ^ Malet, Antoni (1993). "James Gregorie on tangents and the "Taylor" rule for series expansions". Archive for History of Exact Sciences. 46 (2). Springer-Verlag: 97–137. doi:10.1007/BF00375656. S2CID 120101519. Gregorie's thought, on the other hand, belongs to a conceptual framework strongly geometrical in character. (page 137)
  2. ^ 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.
  3. ^ Gregory, James (1668). Geometriae Pars Universalis. Museo Galileo: Patavii: typis heredum Pauli Frambotti.
  4. ^ Child, James Mark; Barrow, Isaac (1916). The Geometrical Lectures of Isaac Barrow. Chicago: Open Court Publishing Company.