Array ( [0] => {{Short description|Branch of mathematics}} [1] => {{About|the branch of mathematics}} [2] => {{pp-semi-indef}} [3] => {{pp-move-indef}} [4] => {{Use dmy dates|date=May 2016}} [5] => {{Calculus}} [6] => {{Math topics TOC}} [7] => '''Calculus''' is the [[mathematics|mathematical]] study of continuous change, in the same way that [[geometry]] is the study of shape, and [[algebra]] is the study of generalizations of [[arithmetic operations]]. [8] => [9] => Originally called '''infinitesimal calculus''' or "the calculus of [[infinitesimal]]s", it has two major branches, [[differential calculus]] and [[integral calculus]]. The former concerns instantaneous [[Rate of change (mathematics)|rates of change]], and the [[slope]]s of [[curve]]s, while the latter concerns accumulation of quantities, and [[area]]s under or between curves. These two branches are related to each other by the [[fundamental theorem of calculus]]. They make use of the fundamental notions of [[convergence (mathematics)|convergence]] of [[infinite sequence]]s and [[Series (mathematics)|infinite series]] to a well-defined [[limit (mathematics)|limit]].{{cite book |first1=Henry F. |last1=DeBaggis |first2=Kenneth S. |last2=Miller |title=Foundations of the Calculus |location=Philadelphia |publisher=Saunders |year=1966 |oclc=527896 }} [10] => [11] => Infinitesimal calculus was developed independently in the late 17th century by [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]].{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=The History of the Calculus and its Conceptual Development |url=https://archive.org/details/historyofcalculu0000boye |url-access=registration |location=New York |publisher=Dover |year=1959 |oclc=643872 }}{{cite book |first=Jason Socrates |last=Bardi |title=The Calculus Wars : Newton, Leibniz, and the Greatest Mathematical Clash of All Time |location=New York |publisher=Thunder's Mouth Press |year=2006 |isbn=1-56025-706-7 }} Later work, including [[(ε, δ)-definition of limit|codifying the idea of limits]], put these developments on a more solid conceptual footing. Today, calculus has widespread uses in [[science]], [[engineering]], and [[social science]].{{cite book |last1=Hoffmann |first1=Laurence D. |last2=Bradley |first2=Gerald L. |title=Calculus for Business, Economics, and the Social and Life Sciences |location=Boston |publisher=McGraw Hill |year=2004 |edition=8th |isbn=0-07-242432-X }} [12] => [13] => ==Etymology== [14] => {{Wiktionary}} [15] => In [[mathematics education]], ''calculus'' denotes courses of elementary [[mathematical analysis]], which are mainly devoted to the study of [[Function (mathematics)|functions]] and limits. The word ''calculus'' is [[Latin]] for "small pebble" (the [[diminutive]] of ''[[wikt:calx|calx]],'' meaning "stone"), a meaning which still [[Calculus (medicine)|persists in medicine]]. Because such pebbles were used for counting out distances,See, for example: [16] => * {{Cite web|title=History – Were metered taxis busy roaming Imperial Rome?|url=https://skeptics.stackexchange.com/questions/8841/were-metered-taxis-busy-roaming-imperial-rome|access-date=2022-02-13|date=2020-06-17|website=Skeptics Stack Exchange|archive-date=25 May 2012|archive-url=https://web.archive.org/web/20120525035132/https://skeptics.stackexchange.com/questions/8841/were-metered-taxis-busy-roaming-imperial-rome|url-status=live}} [17] => * {{Cite book|last=Cousineau|first=Phil|url=https://books.google.com/books?id=m8lJVgizhbQC&q=Ancient+Roman+taximeter+calculus&pg=PT80|title=Wordcatcher: An Odyssey into the World of Weird and Wonderful Words|year=2010|publisher=Simon and Schuster|isbn=978-1-57344-550-4|oclc=811492876|pages=58|language=en|access-date=15 February 2022|archive-date=1 March 2023|archive-url=https://web.archive.org/web/20230301150357/https://books.google.com/books?id=m8lJVgizhbQC&q=Ancient+Roman+taximeter+calculus&pg=PT80|url-status=live}} tallying votes, and doing [[abacus]] arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years before the publications of Leibniz and Newton.{{cite OED|calculus}} [18] => [19] => In addition to differential calculus and integral calculus, the term is also used for naming specific methods of calculation and related theories that seek to model a particular concept in terms of mathematics. Examples of this convention include [[propositional calculus]], [[Ricci calculus]], [[calculus of variations]], [[lambda calculus]], [[sequent calculus]], and [[process calculus]]. Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems as [[Jeremy Bentham|Bentham's]] [[felicific calculus]], and the [[ethical calculus]]. [20] => [21] => == History == [22] => [25] => {{Main|History of calculus}} [26] => [27] => Modern calculus was developed in 17th-century Europe by [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]] (independently of each other, first publishing around the same time) but elements of it first appeared in ancient Egypt and later Greece, then in China and the Middle East, and still later again in medieval Europe and India. [28] => [29] => === Ancient precursors === [30] => [31] => ==== Egypt ==== [32] => Calculations of [[volume]] and [[area]], one goal of integral calculus, can be found in the [[Egyptian mathematics|Egyptian]] [[Moscow Mathematical Papyrus|Moscow papyrus]] ({{circa|1820}} BC), but the formulae are simple instructions, with no indication as to how they were obtained.{{Cite book |last=Kline |first=Morris |url=https://books.google.com/books?id=wKsYrT691yIC |title=Mathematical Thought from Ancient to Modern Times: Volume 1 |year=1990 |publisher=Oxford University Press |isbn=978-0-19-506135-2 |pages=15–21 |language=en |author-link=Morris Kline |access-date=20 February 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301150420/https://books.google.com/books?id=wKsYrT691yIC |url-status=live }}{{Cite book |last=Imhausen |first=Annette |title=Mathematics in Ancient Egypt: A Contextual History |title-link=Mathematics in Ancient Egypt: A Contextual History |date=2016 |publisher=Princeton University Press |isbn=978-1-4008-7430-9 |page=112 |oclc=934433864 |author-link=Annette Imhausen}} [33] => [34] => ==== Greece ==== [35] => {{See also|Greek mathematics}} [36] => [[File:Parabolic segment and inscribed triangle.svg|thumb|upright|right|Archimedes used the [[method of exhaustion]] to calculate the area under a parabola in his work ''[[Quadrature of the Parabola]]''.]] [37] => Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician [[Eudoxus of Cnidus]] ({{circa|390}} – 337 BC) developed the [[method of exhaustion]] to prove the formulas for cone and pyramid volumes. [38] => [39] => During the [[Hellenistic period]], this method was further developed by [[Archimedes]] ({{circa|287}} – {{circa|212 BC}}), who combined it with a concept of the [[Cavalieri's principle|indivisibles]]—a precursor to [[Archimedes use of infinitesimals|infinitesimals]]—allowing him to solve several problems now treated by integral calculus. In ''[[The Method of Mechanical Theorems]]'' he describes, for example, calculating the [[center of gravity]] of a solid [[Sphere|hemisphere]], the center of gravity of a [[frustum]] of a circular [[paraboloid]], and the area of a region bounded by a [[parabola]] and one of its [[secant line]]s.See, for example: [40] => * {{Cite web |last=Powers |first=J. |date=2020 |title="Did Archimedes do calculus?" |url=https://www.maa.org/sites/default/files/images/upload_library/46/HOMSIGMAA/2020-Jeffery%20Powers.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.maa.org/sites/default/files/images/upload_library/46/HOMSIGMAA/2020-Jeffery%20Powers.pdf |archive-date=2022-10-09 |url-status=live |website=[[Mathematical Association of America]] }} [41] => * {{cite book |last=Jullien |first=Vincent |chapter=Archimedes and Indivisibles |date=2015 |doi=10.1007/978-3-319-00131-9_18 |title=Seventeenth-Century Indivisibles Revisited |pages=451–457 |place=Cham |publisher=Springer International Publishing |series=Science Networks. Historical Studies |volume=49 |isbn=978-3-319-00130-2 }} [42] => * {{Cite web |last=Plummer |first=Brad |date=2006-08-09 |title=Modern X-ray technology reveals Archimedes' math theory under forged painting |url=http://news.stanford.edu/news/2006/august9/arch-080906.html |access-date=2022-02-28 |website=Stanford University |language=en |archive-date=20 January 2022 |archive-url=https://web.archive.org/web/20220120065134/https://news.stanford.edu/news/2006/august9/arch-080906.html |url-status=live }} [43] => * {{cite book|author=Archimedes |title=The Works of Archimedes, Volume 1: The Two Books On the Sphere and the Cylinder |isbn=978-0-521-66160-7 |translator-first=Reviel |translator-last=Netz |publisher=Cambridge University Press |year=2004}} [44] => * {{Cite journal |last1=Gray |first1=Shirley |last2=Waldman |first2=Cye H. |date=2018-10-20 |title=Archimedes Redux: Center of Mass Applications from The Method |journal=The College Mathematics Journal |volume=49 |issue=5 |pages=346–352 |doi=10.1080/07468342.2018.1524647 |issn=0746-8342 |s2cid=125411353}} [45] => [46] => ==== China ==== [47] => The method of exhaustion was later discovered independently in [[Chinese mathematics|China]] by [[Liu Hui]] in the 3rd century AD to find the area of a circle.{{cite book|series=Chinese studies in the history and philosophy of science and technology|volume=130|title=A comparison of Archimdes' and Liu Hui's studies of circles |first1=Liu|last1=Dun|first2=Dainian |last2=Fan |first3=Robert Sonné|last3=Cohen|year=1966|isbn=978-0-7923-3463-7|page=279|publisher=Springer |url=https://books.google.com/books?id=jaQH6_8Ju-MC|access-date=15 November 2015|archive-date=1 March 2023|archive-url=https://web.archive.org/web/20230301150353/https://books.google.com/books?id=jaQH6_8Ju-MC|url-status=live}},[https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 pp. 279ff] {{Webarchive |url=https://web.archive.org/web/20230301150353/https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 |date=1 March 2023 }} In the 5th century AD, [[Zu Gengzhi]], son of [[Zu Chongzhi]], established a method{{cite book|last1=Katz |first1=Victor J.|title=A history of mathematics|date=2008|location=Boston, MA|publisher=Addison-Wesley|isbn=978-0-321-38700-4 |edition=3rd|pages=203|author-link=Victor J. Katz}}{{cite book|title=Calculus: Early Transcendentals|first1=Dennis G. |last1=Zill |first2=Scott|last2=Wright|first3=Warren S.|last3=Wright |publisher=Jones & Bartlett Learning|year=2009 |edition=3rd |isbn=978-0-7637-5995-7|page=xxvii |url=https://books.google.com/books?id=R3Hk4Uhb1Z0C|access-date=15 November 2015|archive-date=1 March 2023|archive-url=https://web.archive.org/web/20230301150357/https://books.google.com/books?id=R3Hk4Uhb1Z0C|url-status=live}} [https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27 Extract of page 27] {{Webarchive |url=https://web.archive.org/web/20230301150353/https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27 |date=1 March 2023 }} that would later be called [[Cavalieri's principle]] to find the volume of a [[sphere]]. [48] => [49] => === Medieval === [50] => ====Middle East==== [51] => [[File:Hazan (cropped).png|thumb|upright|Ibn al-Haytham, 11th-century Arab mathematician and physicist]] [52] => In the Middle East, [[Ibn al-Haytham|Hasan Ibn al-Haytham]], Latinized as Alhazen ({{c.|lk=no|965|1040}}  AD) derived a formula for the sum of [[fourth power]]s. He used the results to carry out what would now be called an [[Integral|integration]] of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a [[paraboloid]].{{Cite journal |last=Katz |first=Victor J. |author-link=Victor J. Katz |date=June 1995 |title=Ideas of Calculus in Islam and India |journal=[[Mathematics Magazine]] |volume=68 |issue=3 |pages=163–174 |doi=10.1080/0025570X.1995.11996307 |issn=0025-570X |jstor=2691411}} [53] => [54] => ====India==== [55] => [[Bhāskara II]] was acquainted with some ideas of differential calculus and suggested that the "differential coefficient" vanishes at an extremum value of the function.{{cite journal |last=Shukla |first=Kripa Shankar |year=1984 |title=Use of Calculus in Hindu Mathematics |journal=Indian Journal of History of Science |volume=19 |pages=95–104}} In his astronomical work, he gave a procedure that looked like a precursor to infinitesimal methods. Namely, if x \approx y then \sin(y) - \sin(x) \approx (y - x)\cos(y). This can be interpreted as the discovery that [[cosine]] is the derivative of [[sine]].{{cite book |first=Roger |last=Cooke |title=The History of Mathematics: A Brief Course |publisher=Wiley-Interscience |year=1997 |chapter=The Mathematics of the Hindus |pages=[https://archive.org/details/historyofmathema0000cook/page/213 213–215] |isbn=0-471-18082-3 |chapter-url=https://archive.org/details/historyofmathema0000cook/page/213}} [56] => 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]] stated components of calculus, but 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". [57] => [58] => === Modern === [59] => [[Johannes Kepler]]'s work ''Stereometrica Doliorum'' formed the basis of integral calculus.{{cite web |title=Johannes Kepler: His Life, His Laws and Times |date=24 September 2016 |publisher=NASA |url=https://www.nasa.gov/kepler/education/johannes |accessdate=2021-06-10 |archive-url=https://web.archive.org/web/20210624003856/https://www.nasa.gov/kepler/education/johannes/ |archive-date=24 June 2021 |url-status=live}} 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.{{cite EB1911 |wstitle=Infinitesimal Calculus/History |display=Infinitesimal Calculus § History |volume=14 |page=537}} [60] => [61] => Significant work was a treatise, the origin being Kepler's methods, written by [[Bonaventura Cavalieri]], who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in ''[[The Method of Mechanical Theorems|The Method]]'', but this treatise is believed to have been lost in the 13th century and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. [62] => [63] => 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.{{cite book|author-link=André Weil |last=Weil |first=André |title=Number theory: An approach through History from Hammurapi to Legendre |location=Boston |publisher=Birkhauser Boston |year=1984 |isbn=0-8176-4565-9 |page=28}} The combination was achieved by [[John Wallis]], [[Isaac Barrow]], and [[James Gregory (astronomer and mathematician)|James Gregory]], the latter two proving predecessors to the [[Fundamental theorem of calculus|second fundamental theorem of calculus]] around 1670.{{Cite journal|last=Hollingdale|first=Stuart |date=1991 |title=Review of Before Newton: The Life and Times of Isaac Barrow|journal=[[Notes and Records of the Royal Society of London]] |volume=45|issue=2|pages=277–279|doi=10.1098/rsnr.1991.0027|issn=0035-9149|jstor=531707 |s2cid=165043307|quote=The most interesting to us are Lectures X–XII, in which Barrow comes close to providing a geometrical demonstration of the fundamental theorem of the calculus... He did not realize, however, the full significance of his results, and his rejection of algebra means that his work must remain a piece of mid-17th century geometrical analysis of mainly historic interest.}}{{Cite journal|last=Bressoud |first=David M.|author-link=David Bressoud|date=2011|title=Historical Reflections on Teaching the Fundamental Theorem of Integral Calculus |journal=[[The American Mathematical Monthly]]|volume=118|issue=2|pages=99 |doi=10.4169/amer.math.monthly.118.02.099|s2cid=21473035}} [64] => [65] => The [[product rule]] and [[chain rule]],{{cite book |title=Calculus: Single Variable, Volume 1 |edition=Illustrated |first1=Brian E. |last1=Blank |first2=Steven George |last2=Krantz |publisher=Springer Science & Business Media |year=2006 |isbn=978-1-931914-59-8 |page=248 |url=https://books.google.com/books?id=hMY8lbX87Y8C&pg=PA248 |access-date=31 August 2017 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301150354/https://books.google.com/books?id=hMY8lbX87Y8C&pg=PA248 |url-status=live }} the notions of [[higher derivative]]s and [[Taylor series]],{{cite book |title=The Rise and Development of the Theory of Series up to the Early 1820s |edition=Illustrated |first1=Giovanni |last1=Ferraro |publisher=Springer Science & Business Media |year=2007 |isbn=978-0-387-73468-2 |page=87 |url=https://books.google.com/books?id=vLBJSmA9zgAC&pg=PA87 |access-date=31 August 2017 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301150355/https://books.google.com/books?id=vLBJSmA9zgAC&pg=PA87 |url-status=live }} and of [[analytic function]]s{{cite book|last=Guicciardini|first=Niccolò|chapter=Isaac Newton, Philosophiae naturalis principia mathematica, first edition (1687)|date=2005|title=Landmark Writings in Western Mathematics 1640–1940|pages=59–87|publisher=Elsevier |doi=10.1016/b978-044450871-3/50086-3|isbn=978-0-444-50871-3|quote=[Newton] immediately realised that quadrature problems (the inverse problems) could be tackled via infinite series: as we would say nowadays, by expanding the integrand in power series and integrating term-wise.}} were used by [[Isaac Newton]] in an idiosyncratic notation which he applied to solve problems of [[mathematical physics]]. In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a [[cycloid]], and many other problems discussed in his ''[[Philosophiæ Naturalis Principia Mathematica|Principia Mathematica]]'' (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the [[Taylor series]]. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable. [66] => [67] => {{multiple image [68] => |direction = horizontal [69] => |total_width = 330 [70] => |image1=Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg [71] => |caption1=[[Gottfried Wilhelm Leibniz]] was the first to state clearly the rules of calculus. [72] => |image2=GodfreyKneller-IsaacNewton-1689.jpg [73] => |caption2=[[Isaac Newton]] developed the use of calculus in his [[Newton's laws of motion|laws of motion]] and [[Newton's law of universal gravitation|universal gravitation]]. [74] => }} [75] => [76] => These ideas were arranged into a true calculus of infinitesimals by [[Gottfried Wilhelm Leibniz]], who was originally accused of [[plagiarism]] by Newton.{{cite book |last=Leibniz |first=Gottfried Wilhelm |title=The Early Mathematical Manuscripts of Leibniz |publisher=Cosimo, Inc. |year=2008 |page=228 |url=https://books.google.com/books?id=7d8_4WPc9SMC&pg=PA3 |isbn=978-1-605-20533-5 |access-date=5 June 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301150355/https://books.google.com/books?id=7d8_4WPc9SMC&pg=PA3 |url-status=live }} He is now regarded as an [[Multiple discovery|independent inventor]] of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the [[product rule]] and [[chain rule]], in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation.{{cite book|first=Joseph |last=Mazur |author-link=Joseph Mazur |title=Enlightening Symbols / A Short History of Mathematical Notation and Its Hidden Powers|year=2014|publisher=Princeton University Press |isbn=978-0-691-17337-5 |page=166 |quote=Leibniz understood symbols, their conceptual powers as well as their limitations. He would spend years experimenting with some—adjusting, rejecting, and corresponding with everyone he knew, consulting with as many of the leading mathematicians of the time who were sympathetic to his fastidiousness.}} [77] => [78] => Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general [[physics]]. Leibniz developed much of the notation used in calculus today.{{Rp|pages=51–52}} The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series. [79] => [80] => When Newton and Leibniz first published their results, there was [[Newton v. Leibniz calculus controversy|great controversy]] over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be published in his ''[[Method of Fluxions]]''), but Leibniz published his "[[Nova Methodus pro Maximis et Minimis]]" first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the [[Royal Society]]. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics.{{Cite journal|last=Schrader|first=Dorothy V.|date=1962|title=The Newton-Leibniz controversy concerning the discovery of the calculus|journal=The Mathematics Teacher|volume=55|issue=5|pages=385–396 |doi=10.5951/MT.55.5.0385|jstor=27956626 |issn=0025-5769}} A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "[[Method of fluxions|the science of fluxions]]", a term that endured in English schools into the 19th century.{{cite book|first=Jacqueline |last=Stedall |author-link=Jackie Stedall |title=The History of Mathematics: A Very Short Introduction |title-link=The History of Mathematics: A Very Short Introduction |year=2012 |isbn=978-0-191-63396-6 |publisher=Oxford University Press}}{{Rp|100}} The first complete treatise on calculus to be written in English and use the Leibniz notation was not published until 1815.{{Cite journal |last=Stenhouse |first=Brigitte |date=May 2020 |title=Mary Somerville's early contributions to the circulation of differential calculus |journal=[[Historia Mathematica]] |volume=51 |pages=1–25 |doi=10.1016/j.hm.2019.12.001 |s2cid=214472568|url=http://oro.open.ac.uk/68466/1/accepted_manuscript.pdf }} [81] => [82] => [[File:Maria Gaetana Agnesi.jpg|thumb|upright|right|[[Maria Gaetana Agnesi]]]] [83] => Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and [[integral calculus]] was written in 1748 by [[Maria Gaetana Agnesi]].{{cite book |title=A Biography of Maria Gaetana Agnesi, an Eighteenth-century Woman Mathematician |first1=Antonella |last1=Cupillari |author-link=Antonella Cupillari |location=[[Lewiston, New York]] |publisher=[[Edwin Mellen Press]] |year=2007 |isbn=978-0-7734-5226-8 |page=iii |title-link=A Biography of Maria Gaetana Agnesi |contributor-last=Allaire |contributor-first=Patricia R.|contribution=Foreword}}{{cite web| url=http://www.agnesscott.edu/lriddle/women/agnesi.htm| title=Maria Gaetana Agnesi| first=Elif| last=Unlu| date=April 1995| publisher=[[Agnes Scott College]]| access-date=7 December 2010| archive-date=3 December 1998| archive-url=https://web.archive.org/web/19981203075738/http://www.agnesscott.edu/lriddle/women/agnesi.htm| url-status=live}} [84] => [85] => === Foundations === [86] => In calculus, ''foundations'' refers to the [[Rigorous#Mathematical rigor |rigorous]] development of the subject from [[axiom]]s and definitions. In early calculus, the use of [[infinitesimal]] quantities was thought unrigorous and was fiercely criticized by several authors, most notably [[Michel Rolle]] and [[George Berkeley|Bishop Berkeley]]. Berkeley famously described infinitesimals as the [[ghosts of departed quantities]] in his book ''[[The Analyst]]'' in 1734. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today.{{cite web |url=https://plato.stanford.edu/entries/continuity/ |title=Continuity and Infinitesimals |date=2013-09-06 |website=[[Stanford Encyclopedia of Philosophy]] |first=John L. |last=Bell |access-date=2022-02-20 |author-link=John Lane Bell |archive-date=16 March 2022 |archive-url=https://web.archive.org/web/20220316170134/https://plato.stanford.edu/entries/continuity/ |url-status=live }} [87] => [88] => Several mathematicians, including [[Colin Maclaurin|Maclaurin]], tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of [[Augustin-Louis Cauchy|Cauchy]] and [[Karl Weierstrass|Weierstrass]], a way was finally found to avoid mere "notions" of infinitely small quantities.{{Cite book |last=Russell |first=Bertrand |author-link=Bertrand Russell |year=1946 |title=History of Western Philosophy |location=London |publisher=[[George Allen & Unwin Ltd]] |page=[https://archive.org/stream/westernphilosoph035502mbp#page/n857/mode/2up 857] |quote=The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibniz believed in actual infinitesimals, but although this belief suited his metaphysics it had no sound basis in mathematics. Weierstrass, soon after the middle of the nineteenth century, showed how to establish calculus without infinitesimals, and thus, at last, made it logically secure. Next came Georg Cantor, who developed the theory of continuity and infinite number. "Continuity" had been, until he defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word and showed that continuity, as he defined it, was the concept needed by mathematicians and physicists. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated. |title-link= A History of Western Philosophy }} 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 [[continuous function|continuity]] in terms of infinitesimals, and a (somewhat imprecise) prototype of an [[(ε, δ)-definition of limit]] in the definition of differentiation.{{cite book |first=Judith V. |last=Grabiner |author-link=Judith Grabiner |title=The Origins of Cauchy's Rigorous Calculus |url=https://archive.org/details/originsofcauchys00judi |url-access=registration |location=Cambridge |publisher=MIT Press |year=1981 |isbn=978-0-387-90527-3 }} In his work Weierstrass formalized the concept of [[Limit of a function|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.{{cite book|first=Tom |last=Archibald |chapter=The Development of Rigor in Mathematical Analysis |pages=117–129 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}} It was also during this period that the ideas of calculus were generalized to the [[complex plane]] with the development of [[complex analysis]].{{cite book|first=Adrian |last=Rice |chapter=A Chronology of Mathematical Events |pages=1010–1014 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}} [89] => [90] => In modern mathematics, the foundations of calculus are included in the field of [[real analysis]], which contains full definitions and [[mathematical proof|proofs]] of the theorems of calculus. The reach of calculus has also been greatly extended. [[Henri Lebesgue]] invented [[measure theory]], based on earlier developments by [[Émile Borel]], and used it to define integrals of all but the most [[Pathological (mathematics)|pathological]] functions.{{cite book|first=Reinhard |last=Siegmund-Schultze |chapter=Henri Lebesgue |pages=796–797 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}} [[Laurent Schwartz]] introduced [[Distribution (mathematics)|distributions]], which can be used to take the derivative of any function whatsoever.{{Cite journal |last1=Barany |first1=Michael J. |last2=Paumier |first2=Anne-Sandrine |last3=Lützen |first3=Jesper |date=November 2017 |title=From Nancy to Copenhagen to the World: The internationalization of Laurent Schwartz and his theory of distributions |journal=[[Historia Mathematica]] |volume=44 |issue=4 |pages=367–394 |doi=10.1016/j.hm.2017.04.002|doi-access=free }} [91] => [92] => Limits are not the only rigorous approach to the foundation of calculus. Another way is to use [[Abraham Robinson]]'s [[non-standard analysis]]. Robinson's approach, developed in the 1960s, uses technical machinery from [[mathematical logic]] to augment the real number system with [[infinitesimal]] and [[Infinity|infinite]] numbers, as in the original Newton-Leibniz conception. The resulting numbers are called [[hyperreal number]]s, and they can be used to give a Leibniz-like development of the usual rules of calculus.{{cite book|first=Joseph W. |last=Daubin |chapter=Abraham Robinson |pages=822–823 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}} There is also [[smooth infinitesimal analysis]], which differs from non-standard analysis in that it mandates neglecting higher-power infinitesimals during derivations. Based on the ideas of [[F. W. Lawvere]] and employing the methods of [[category theory]], smooth infinitesimal analysis views all functions as being [[continuous function|continuous]] and incapable of being expressed in terms of [[Discrete mathematics|discrete]] entities. One aspect of this formulation is that the [[law of excluded middle]] does not hold. The law of excluded middle is also rejected in [[constructive mathematics]], a branch of mathematics that insists that proofs of the existence of a number, function, or other mathematical object should give a construction of the object. Reformulations of calculus in a constructive framework are generally part of the subject of [[constructive analysis]]. [93] => [94] => === Significance === [95] => While many of the ideas of calculus had been developed earlier in [[Greek mathematics|Greece]], [[Chinese mathematics|China]], [[Indian mathematics|India]], [[Islamic mathematics|Iraq, Persia]], and [[Japanese mathematics|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.{{Cite book|title=Chinese studies in the history and philosophy of science and technology|date=1996 |publisher=Kluwer Academic Publishers|author1=Dainian Fan|author2=R. S. Cohen|isbn=0-7923-3463-9|location=Dordrecht|oclc=32272485}}{{Cite book|title=Landmark writings in Western mathematics 1640–1940|date=2005 |publisher=Elsevier|editor-first1=I.|editor-last1=Grattan-Guinness|editor-link1=Ivor Grattan-Guinness |isbn=0-444-50871-6 |location=Amsterdam |oclc=60416766}}{{Cite book|last=Kline |first=Morris|author-link=Morris Kline|title=Mathematical thought from ancient to modern times |volume=3|date=1990 |publisher=Oxford University Press|isbn=978-0-19-977048-9 |location=New York|oclc=726764443}} The Hungarian polymath [[John von Neumann]] wrote of this work, [96] => {{blockquote|The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.{{cite book|last=von Neumann |first=J. |author-link=John von Neumann |chapter=The Mathematician |editor-last=Heywood |editor-first=R. B. |title=The Works of the Mind |publisher=University of Chicago Press |year=1947 |pages=180–196}} Reprinted in {{cite book|editor-last1=Bródy |editor-first1=F. |editor-last2=Vámos |editor-first2=T. |title=The Neumann Compendium |publisher=World Scientific Publishing Co. Pte. Ltd. |year=1995 |isbn=981-02-2201-7 |pages=618–626}}}} [97] => [98] => Applications of differential calculus include computations involving [[velocity]] and [[acceleration]], the [[slope]] of a curve, and [[Mathematical optimization|optimization]].{{Rp|pages=341–453}} Applications of integral calculus include computations involving area, [[volume]], [[arc length]], [[center of mass]], [[work (physics)|work]], and [[pressure]].{{Rp|pages=685–700}} More advanced applications include [[power series]] and [[Fourier series]]. [99] => [100] => Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving [[division by zero]] or sums of infinitely many numbers. These questions arise in the study of [[Motion (physics)|motion]] and area. The [[ancient Greek]] philosopher [[Zeno of Elea]] gave several famous examples of such [[Zeno's paradoxes|paradoxes]]. Calculus provides tools, especially the [[Limit (mathematics)|limit]] and the [[infinite series]], that resolve the paradoxes.{{cite book|first=Eugenia |last=Cheng |author-link=Eugenia Cheng |title=Beyond Infinity: An Expedition to the Outer Limits of Mathematics |title-link=Beyond Infinity (mathematics book) |pages=206–210 |publisher=Basic Books |year=2017 |isbn=978-1-541-64413-7 |oclc=1003309980}} [101] => [102] => == Principles == [103] => [104] => === Limits and infinitesimals === [105] => {{Main|Limit of a function|Infinitesimal}} [106] => Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by [[infinitesimal]]s. 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 dx and dy were taken to be infinitesimal, and the derivative dy/dx was their ratio. [107] => [108] => The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. In the late 19th century, infinitesimals were replaced within academia by the [[epsilon, delta]] approach to [[Limit of a function|limits]]. Limits describe the behavior of a [[function (mathematics)|function]] at a certain input in terms of its values at nearby inputs. They capture small-scale behavior using the intrinsic structure of the [[real number|real number system]] (as a [[metric space]] with the [[least-upper-bound property]]). In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by sequences of smaller and smaller numbers, and the infinitely small behavior of a function is found by taking the limiting behavior for these sequences. Limits were thought to provide a more rigorous foundation for calculus, and for this reason, they became the standard approach during the 20th century. However, the infinitesimal concept was revived in the 20th century with the introduction of [[non-standard analysis]] and [[smooth infinitesimal analysis]], which provided solid foundations for the manipulation of infinitesimals. [109] => [110] => === Differential calculus === [111] => {{Main|Differential calculus}} [112] => [[File:Tangent line to a curve.svg|thumb|upright=1.35 |Tangent line at {{math|(''x''0, ''f''(''x''0))}}. The derivative {{math|''f′''(''x'')}} of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point.]] [113] => [114] => Differential calculus is the study of the definition, properties, and applications of the [[derivative]] of a function. The process of finding the derivative is called ''differentiation''. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the ''derivative function'' or just the ''derivative'' of the original function. In formal terms, the derivative is a [[linear operator]] which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. The function produced by differentiating the squaring function turns out to be the doubling function.{{Cite book |last1=Frautschi |first1=Steven C. |title=The Mechanical Universe: Mechanics and Heat |title-link=The Mechanical Universe |last2=Olenick |first2=Richard P. |last3=Apostol |first3=Tom M. |last4=Goodstein |first4=David L. |date=2007 |publisher=Cambridge University Press |isbn=978-0-521-71590-4 |edition=Advanced |location=Cambridge [Cambridgeshire] |oclc=227002144 |author-link=Steven Frautschi |author-link3=Tom M. Apostol |author-link4=David L. Goodstein}}{{Rp|32}} [115] => [116] => In more explicit terms the "doubling function" may be denoted by {{math|''g''(''x'') {{=}} 2''x''}} and the "squaring function" by {{math|''f''(''x'') {{=}} ''x''2}}. The "derivative" now takes the function {{math|''f''(''x'')}}, defined by the expression "{{math|''x''2}}", as an input, that is all the information—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to output another function, the function {{math|''g''(''x'') {{=}} 2''x''}}, as will turn out. [117] => [118] => In [[Lagrange's notation]], the symbol for a derivative is an [[apostrophe]]-like mark called a [[prime (symbol)|prime]]. Thus, the derivative of a function called {{math|''f''}} is denoted by {{math|''f′''}}, pronounced "f prime" or "f dash". For instance, if {{math|''f''(''x'') {{=}} ''x''2}} is the squaring function, then {{math|''f′''(''x'') {{=}} 2''x''}} is its derivative (the doubling function {{math|''g''}} from above). [119] => [120] => If the input of the function represents time, then the derivative represents change concerning time. For example, if {{math|''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 {{math|''f''}} is how the position is changing in time, that is, it is the [[velocity]] of the ball.{{Rp|18–20}} [121] => [122] => If a function is [[linear function|linear]] (that is if the [[Graph of a function|graph]] of the function is a straight line), then the function can be written as {{math|''y'' {{=}} ''mx'' + ''b''}}, where {{math|''x''}} is the independent variable, {{math|''y''}} is the dependent variable, {{math|''b''}} is the ''y''-intercept, and: [123] => [124] => :m= \frac{\text{rise}}{\text{run}}= \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}. [125] => [126] => This gives an exact value for the slope of a straight line.{{Cite book |last1=Salas |first1=Saturnino L. |title=Calculus; one and several variables |last2=Hille |first2=Einar |date=1971 |publisher=Xerox College Pub. |location=Waltham, MA |oclc=135567}}{{Rp|page=6}} If the graph of the function is not a straight line, however, then the change in {{math|''y''}} divided by the change in {{math|''x''}} varies. Derivatives give an exact meaning to the notion of change in output concerning change in input. To be concrete, let {{math|''f''}} be a function, and fix a point {{math|''a''}} in the domain of {{math|''f''}}. {{math|(''a'', ''f''(''a''))}} is a point on the graph of the function. If {{math|''h''}} is a number close to zero, then {{math|''a'' + ''h''}} is a number close to {{math|''a''}}. Therefore, {{math|(''a'' + ''h'', ''f''(''a'' + ''h''))}} is close to {{math|(''a'', ''f''(''a''))}}. The slope between these two points is [127] => [128] => :m = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h}. [129] => [130] => This expression is called a ''[[difference quotient]]''. A line through two points on a curve is called a ''secant line'', so {{math|''m''}} is the slope of the secant line between {{math|(''a'', ''f''(''a''))}} and {{math|(''a'' + ''h'', ''f''(''a'' + ''h''))}}. The second line is only an approximation to the behavior of the function at the point {{math|'' a''}} because it does not account for what happens between {{math|'' a''}} and {{math|'' a'' + ''h''}}. It is not possible to discover the behavior at {{math|'' a''}} by setting {{math|''h''}} to zero because this would require [[dividing by zero]], which is undefined. The derivative is defined by taking the [[limit (mathematics)|limit]] as {{math|''h''}} tends to zero, meaning that it considers the behavior of {{math|''f''}} for all small values of {{math|''h''}} and extracts a consistent value for the case when {{math|''h''}} equals zero: [131] => [132] => :\lim_{h \to 0}{f(a+h) - f(a)\over{h}}. [133] => [134] => Geometrically, the derivative is the slope of the [[tangent line]] to the graph of {{math|''f''}} at {{math|'' 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 {{math|''f''}}.{{Rp|pages=61–63}} [135] => [136] => Here is a particular example, the derivative of the squaring function at the input 3. Let {{math|''f''(''x'') {{=}} ''x''2}} be the squaring function. [137] => [138] => [[File: Sec2tan.gif|thumb|upright=1.35|The derivative {{math|''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 {{math|''f''(''x'') {{=}} ''x''3 − ''x''}}. The tangent line (in green) which passes through the point {{nowrap|(−3/2, −15/8)}} has a slope of 23/4. The vertical and horizontal scales in this image are different.]] [139] => [140] => :\begin{align}f'(3) &=\lim_{h \to 0}{(3+h)^2 - 3^2\over{h}} \\ [141] => &=\lim_{h \to 0}{9 + 6h + h^2 - 9\over{h}} \\ [142] => &=\lim_{h \to 0}{6h + h^2\over{h}} \\ [143] => &=\lim_{h \to 0} (6 + h) \\ [144] => &= 6 [145] => \end{align} [146] => [147] => [148] => The slope of the tangent line to the squaring function at the point (3, 9) is 6, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the ''derivative function'' of the squaring function or just the ''derivative'' of the squaring function for short. A computation similar to the one above shows that the derivative of the squaring function is the doubling function.{{Rp|page=63}} [149] => [150] => === Leibniz notation === [151] => {{Main|Leibniz's notation}} [152] => [153] => A common notation, introduced by Leibniz, for the derivative in the example above is [154] => : [155] => \begin{align} [156] => y&=x^2 \\ [157] => \frac{dy}{dx}&=2x. [158] => \end{align} [159] => [160] => In an approach based on limits, the symbol {{math|{{sfrac|''dy''|'' dx''}}}} is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above.{{Rp|page=74}} Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, {{math|''dy''}} being the infinitesimally small change in {{math|''y''}} caused by an infinitesimally small change {{math|'' dx''}} applied to {{math|''x''}}. We can also think of {{math|{{sfrac|''d''|'' dx''}}}} as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example: [161] => : [162] => \frac{d}{dx}(x^2)=2x. [163] => [164] => [165] => In this usage, the {{math|''dx''}} in the denominator is read as "with respect to {{math|''x''}}".{{Rp|page=79}} Another example of correct notation could be: [166] => :\begin{align} [167] => g(t) &= t^2 + 2t + 4 \\ [168] => {d \over dt}g(t) &= 2t + 2 [169] => \end{align} [170] => [171] => [172] => Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like {{math|'' dx''}} and {{math|''dy''}} as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the [[total derivative]]. [173] => [174] => === Integral calculus === [175] => {{Main|Integral}} [176] => {{multiple image| total_width = 300px | direction = vertical [177] => | image1 = Integral as region under curve.svg [178] => | caption1 = Integration can be thought of as measuring the area under a curve, defined by {{math|''f''(''x'')}}, between two points (here {{math|'' a''}} and {{math|''b''}}). [179] => | image2 = Riemann integral regular.gif [180] => | caption2 = A sequence of midpoint Riemann sums over a regular partition of an interval: the total area of the rectangles converges to the integral of the function. [181] => }} [182] => ''Integral calculus'' is the study of the definitions, properties, and applications of two related concepts, the ''indefinite integral'' and the ''definite integral''. The process of finding the value of an integral is called ''integration''.{{Cite book |last1=Herman |first1=Edwin |url=https://openstax.org/details/books/calculus-volume-1 |title=Calculus |volume=1 |last2=Strang |first2=Gilbert |date=2017 |publisher=OpenStax |isbn=978-1-938168-02-4 |location=Houston, Texas |oclc=1022848630 |display-authors=etal |author-link2=Gilbert Strang |access-date=26 July 2022 |archive-date=23 September 2022 |archive-url=https://web.archive.org/web/20220923230919/https://openstax.org/details/books/calculus-volume-1 |url-status=live }}{{Rp|page=508}} The indefinite integral, also known as the ''[[antiderivative]]'', is the inverse operation to the derivative.{{Rp|pages=163–165}} {{math|''F''}} is an indefinite integral of {{math|''f''}} when {{math|''f''}} is a derivative of {{math|''F''}}. (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.) The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the [[x-axis]]. The technical definition of the definite integral involves the [[limit (mathematics)|limit]] of a sum of areas of rectangles, called a [[Riemann sum]].{{Cite book |last1=Hughes-Hallett |first1=Deborah |title=Calculus: Single and Multivariable |last2=McCallum |first2=William G. |last3=Gleason |first3=Andrew M. |last4=Connally |first4=Eric |date=2013 |publisher=Wiley |isbn=978-0-470-88861-2 |edition=6th |location=Hoboken, NJ |oclc=794034942 |display-authors=3 |author-link=Deborah Hughes Hallett |author-link2=William G. McCallum|author-link3=Andrew M. Gleason}}{{Rp|page=282}} [183] => [184] => A motivating example is the distance traveled in a given time.{{Rp|pages=153}} If the speed is constant, only multiplication is needed: [185] => :\mathrm{Distance} = \mathrm{Speed} \cdot \mathrm{Time} [186] => But if the speed changes, a more powerful method of finding the distance is necessary. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a [[Riemann sum]]) of the approximate distance traveled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled. [187] => [188] => When velocity is constant, the total distance traveled over the given time interval can be computed by multiplying velocity and time. For example, traveling a steady 50 mph for 3 hours results in a total distance of 150 miles. Plotting the velocity as a function of time yields a rectangle with a height equal to the velocity and a width equal to the time elapsed. Therefore, the product of velocity and time also calculates the rectangular area under the (constant) velocity curve.{{rp|535}} This connection between the area under a curve and the distance traveled can be extended to ''any'' irregularly shaped region exhibiting a fluctuating velocity over a given period. If {{math|''f''(''x'')}} represents speed as it varies over time, the distance traveled between the times represented by {{math|'' a''}} and {{math|''b''}} is the area of the region between {{math|''f''(''x'')}} and the {{math|''x''}}-axis, between {{math|''x'' {{=}} ''a''}} and {{math|''x'' {{=}} ''b''}}. [189] => [190] => To approximate that area, an intuitive method would be to divide up the distance between {{math|'' a''}} and {{math|''b''}} into several equal segments, the length of each segment represented by the symbol {{math|Δ''x''}}. For each small segment, we can choose one value of the function {{math|''f''(''x'')}}. Call that value {{math|''h''}}. Then the area of the rectangle with base {{math|Δ''x''}} and height {{math|''h''}} gives the distance (time {{math|Δ''x''}} multiplied by speed {{math|''h''}}) traveled in that segment. Associated with each segment is the average value of the function above it, {{math|''f''(''x'') {{=}} ''h''}}. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. A smaller value for {{math|Δ''x''}} will give more rectangles and in most cases a better approximation, but for an exact answer, we need to take a limit as {{math|Δ''x''}} approaches zero.{{rp|512–522}} [191] => [192] => The symbol of integration is \int , an [[long s|elongated ''S'']] chosen to suggest summation.{{Rp|pages=529}} The definite integral is written as: [193] => [194] => :\int_a^b f(x)\, dx. [195] => [196] => and is read "the integral from ''a'' to ''b'' of ''f''-of-''x'' with respect to ''x''." The Leibniz notation {{math|'' dx''}} is intended to suggest dividing the area under the curve into an infinite number of rectangles so that their width {{math|Δ''x''}} becomes the infinitesimally small {{math|'' dx''}}.{{Rp|44}} [197] => [198] => The indefinite integral, or antiderivative, is written: [199] => [200] => :\int f(x)\, dx. [201] => [202] => Functions differing by only a constant have the same derivative, and it can be shown that the antiderivative of a given function is a family of functions differing only by a constant.{{Rp|page=326}} Since the derivative of the function {{math|''y'' {{=}} ''x''2 + ''C''}}, where {{math|''C''}} is any constant, is {{math|''y′'' {{=}} 2''x''}}, the antiderivative of the latter is given by: [203] => :\int 2x\, dx = x^2 + C. [204] => The unspecified constant {{math|'' C''}} present in the indefinite integral or antiderivative is known as the [[constant of integration]].{{cite book|first1=William |last1=Moebs |first2=Samuel J. |last2=Ling |first3=Jeff |last3=Sanny |display-authors=etal |title=University Physics, Volume 1 |publisher=OpenStax |year=2022 |isbn=978-1-947172-20-3 |oclc=961352944}}{{rp|135}} [205] => [206] => === Fundamental theorem === [207] => {{Main|Fundamental theorem of calculus}} [208] => The [[fundamental theorem of calculus]] states that differentiation and integration are inverse operations.{{Rp|page=290}} More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the fundamental theorem of calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration. [209] => [210] => The fundamental theorem of calculus states: If a function {{math|''f''}} is [[continuous function|continuous]] on the interval {{math|[''a'', ''b'']}} and if {{math|''F''}} is a function whose derivative is {{math|''f''}} on the interval {{math|(''a'', ''b'')}}, then [211] => [212] => :\int_{a}^{b} f(x)\,dx = F(b) - F(a). [213] => [214] => Furthermore, for every {{math|''x''}} in the interval {{math|(''a'', ''b'')}}, [215] => [216] => :\frac{d}{dx}\int_a^x f(t)\, dt = f(x). [217] => [218] => This realization, made by both [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]], was key to the proliferation of analytic results after their work became known. (The extent to which Newton and Leibniz were influenced by immediate predecessors, and particularly what Leibniz may have learned from the work of [[Isaac Barrow]], is difficult to determine because of the priority dispute between them.See, for example: [219] => * {{cite book|last=Mahoney |first=Michael S. |year=1990 |chapter=Barrow's mathematics: Between ancients and moderns |title=Before Newton |editor-first=M. |editor-last=Feingold |pages=179–249 |publisher=Cambridge University Press |isbn=978-0-521-06385-2}} [220] => * {{Cite journal |first=M. |last=Feingold |date=June 1993 |title=Newton, Leibniz, and Barrow Too: An Attempt at a Reinterpretation |journal=[[Isis (journal)|Isis]] |language=en |volume=84 |issue=2 |pages=310–338 |doi=10.1086/356464 |bibcode=1993Isis...84..310F |s2cid=144019197 |issn=0021-1753}} [221] => * {{cite book|first=Siegmund |last=Probst |chapter=Leibniz as Reader and Second Inventor: The Cases of Barrow and Mengoli |title=G.W. Leibniz, Interrelations Between Mathematics and Philosophy|editor-first1=Norma B. |editor-last1=Goethe |editor-first2=Philip |editor-last2=Beeley |editor-first3=David |editor-last3=Rabouin |publisher=Springer |isbn=978-9-401-79663-7 |pages=111–134 |year=2015 |series=Archimedes: New Studies in the History and Philosophy of Science and Technology |volume=41}}) The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulae for [[antiderivative]]s. It is also a prototype solution of a [[differential equation]]. Differential equations relate an unknown function to its derivatives and are ubiquitous in the sciences.{{Cite book |last1=Herman |first1=Edwin |url=https://openstax.org/details/books/calculus-volume-2 |title=Calculus. Volume 2 |last2=Strang |first2=Gilbert |date=2017 |publisher=OpenStax |isbn=978-1-5066-9807-6 |location=Houston |oclc=1127050110 |display-authors=etal |access-date=26 July 2022 |archive-date=26 July 2022 |archive-url=https://web.archive.org/web/20220726140351/https://openstax.org/details/books/calculus-volume-2 |url-status=live }}{{Rp|pages=351–352}} [222] => [223] => == Applications == [224] => [[File: NautilusCutawayLogarithmicSpiral.jpg|thumb|right|The [[logarithmic spiral]] of the [[Nautilus|Nautilus shell]] is a classical image used to depict the growth and change related to calculus.]] [225] => Calculus is used in every branch of the physical sciences,{{Cite book |last=Baron |first=Margaret E. |title=The origins of the infinitesimal calculus |date=1969 |isbn=978-1-483-28092-9 |location=Oxford |publisher=Pergamon Press |oclc=892067655 |author-link=Margaret Baron}}{{Rp|page=1}} [[actuarial science]], [[computer science]], [[statistics]], [[engineering]], [[economics]], [[business]], [[medicine]], [[demography]], and in other fields wherever a problem can be [[mathematical model|mathematically modeled]] and an [[optimization (mathematics)|optimal]] solution is desired.{{cite news |last1=Kayaspor |first1=Ali |date=28 August 2022 |title=The Beautiful Applications of Calculus in Real Life |url=https://ali.medium.com/the-beautiful-applications-of-calculus-in-real-life-81331dc1bc5a |access-date=26 September 2022 |work=Medium |archive-date=26 September 2022 |archive-url=https://web.archive.org/web/20220926011601/https://ali.medium.com/the-beautiful-applications-of-calculus-in-real-life-81331dc1bc5a |url-status=live }} It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other.{{Cite book |last=Hu |first=Zhiying |title=2021 2nd Asia-Pacific Conference on Image Processing, Electronics, and Computers |chapter=The Application and Value of Calculus in Daily Life |date=2021-04-14 |series=Ipec2021 |location=Dalian China |publisher=ACM |pages=562–564 |isbn=978-1-4503-8981-5 |s2cid=233384462 |doi=10.1145/3452446.3452583}} Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with [[linear algebra]] to find the "best fit" linear approximation for a set of points in a domain. Or, it can be used in [[probability theory]] to determine the [[expectation value]] of a continuous random variable given a [[probability density function]].{{cite book|first=Mehran |last=Kardar |author-link=Mehran Kardar |title=Statistical Physics of Particles |title-link=Statistical Physics of Particles |year=2007 |publisher=[[Cambridge University Press]] |isbn=978-0-521-87342-0 |oclc=860391091}}{{Rp|37}} In [[analytic geometry]], the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, [[Concave function|concavity]] and [[inflection points]]. Calculus is also used to find approximate solutions to equations; in practice, it is the standard way to solve differential equations and do root finding in most applications. Examples are methods such as [[Newton's method]], [[fixed point iteration]], and [[linear approximation]]. For instance, spacecraft use a variation of the [[Euler method]] to approximate curved courses within zero gravity environments. [226] => [227] => [[Physics]] makes particular use of calculus; all concepts in [[classical mechanics]] and [[electromagnetism]] are related through calculus. The [[mass]] of an object of known [[density]], the [[moment of inertia]] of objects, and the [[potential energy|potential energies]] due to gravitational and electromagnetic forces can all be found by the use of calculus. An example of the use of calculus in mechanics is [[Newton's laws of motion|Newton's second law of motion]], which states that the derivative of an object's [[momentum]] concerning time equals the net [[force]] upon it. Alternatively, Newton's second law can be expressed by saying that the net force equals the object's mass times it's [[acceleration]], which is the time derivative of velocity and thus the second time derivative of spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path.{{Cite book|first=Elizabeth|last=Garber|title=The language of physics: the calculus and the development of theoretical physics in Europe, 1750–1914|date=2001|publisher=Springer Science+Business Media|isbn=978-1-4612-7272-4 |oclc=921230825}} [228] => [229] => Maxwell's theory of [[electromagnetism]] and [[Albert Einstein|Einstein]]'s theory of [[general relativity]] are also expressed in the language of differential calculus.{{Cite journal|last=Hall|first=Graham|date=2008|title=Maxwell's Electromagnetic Theory and Special Relativity|journal=Philosophical Transactions: Mathematical, Physical and Engineering Sciences|volume=366|issue=1871 |pages=1849–1860|doi=10.1098/rsta.2007.2192|jstor=25190792|pmid=18218598 |bibcode=2008RSPTA.366.1849H|s2cid=502776|issn=1364-503X}}{{Cite book |last=Gbur|first=Greg|title=Mathematical Methods for Optical Physics and Engineering|date=2011 |publisher=Cambridge University Press |isbn=978-0-511-91510-9|location=Cambridge|oclc=704518582|author-link=Greg Gbur}}{{Rp|pages=52–55}} Chemistry also uses calculus in determining reaction rates{{Cite book|last1=Atkins|first1=Peter W. |title=Chemical principles: the quest for insight|last2=Jones|first2=Loretta|date=2010|publisher=W.H. Freeman|isbn=978-1-4292-1955-6|edition=5th|location=New York |oclc=501943698}}{{Rp|page=599}} and in studying radioactive decay.{{Rp|page=814}} In biology, population dynamics starts with reproduction and death rates to model population changes.{{Cite book|last=Murray|first=J. D. |title=Mathematical biology. I, Introduction|date=2002 |publisher=Springer|isbn=0-387-22437-8 |edition=3rd|location=New York |oclc=53165394}}{{Cite book|last=Neuhauser|first=Claudia|title=Calculus for biology and medicine|date=2011 |publisher=Prentice Hall|isbn=978-0-321-64468-8|edition=3rd|location=Boston|oclc=426065941|author-link=Claudia Neuhauser}}{{Rp|page=631}} [230] => [231] => [[Green's theorem]], which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a [[planimeter]], which is used to calculate the area of a flat surface on a drawing.{{Cite journal |first=R. W. |last=Gatterdam |title=The planimeter as an example of Green's theorem |journal=[[The American Mathematical Monthly]] |volume=88 |year=1981 |issue=9 |pages=701–704 |doi= 10.2307/2320679|jstor=2320679 }} For example, it can be used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool when designing the layout of a piece of property. [232] => [233] => In the realm of medicine, calculus can be used to find the optimal branching angle of a [[blood vessel]] to maximize flow.{{Cite journal|last=Adam|first=John A.|date=June 2011|title=Blood Vessel Branching: Beyond the Standard Calculus Problem |journal=[[Mathematics Magazine]] |volume=84|issue=3|pages=196–207 |doi=10.4169/math.mag.84.3.196|s2cid=8259705|issn=0025-570X}} Calculus can be applied to understand how quickly a drug is eliminated from a body or how quickly a [[cancer]]ous tumor grows.{{cite journal |url=https://archive.siam.org/pdf/news/203.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://archive.siam.org/pdf/news/203.pdf |archive-date=2022-10-09 |url-status=live |title=Mathematical Modeling and Cancer |journal=[[SIAM News]] |date=2004 |volume=37 |number=1 |first=Dana |last=Mackenzie}} [234] => [235] => In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both [[marginal cost]] and [[marginal revenue]].{{Cite book|last=Perloff|first=Jeffrey M.|title=Microeconomics: Theory and Applications with Calculus |date=2018|isbn=978-1-292-15446-6|edition=4th global|location=Harlow |publisher=Pearson|oclc=1064041906}}{{Rp|page=387}} [236] => [237] => == See also == [238] => {{Main|Outline of calculus}} [239] => * [[Glossary of calculus]] [240] => * [[List of calculus topics]] [241] => * [[List of derivatives and integrals in alternative calculi]] [242] => * [[List of differentiation identities]] [243] => * [[List of publications in mathematics#Calculus|Publications in calculus]] [244] => * [[Table of integrals]] [245] => [246] => == References == [247] => {{Reflist}} [248] => [249] => ==Further reading== [250] => {{refbegin|30em}} [251] => * {{cite book|first=Robert A. |last=Adams |year=1999 |isbn=978-0-201-39607-2 |title=Calculus: A complete course|publisher=Addison-Wesley }} [252] => * {{cite book|editor-last1=Albers |editor-first1=Donald J. |editor-first2=Richard D. |editor-last2=Anderson |editor-first3=Don O. |editor-last3=Loftsgaarden |year=1986 |title=Undergraduate Programs in the Mathematics and Computer Sciences: The 1985–1986 Survey |publisher=Mathematical Association of America}} [253] => * {{cite book|first1=Howard |last1=Anton |first2=Irl |last2=Bivens |first3=Stephen |last3=Davis |year=2002 |isbn=978-81-265-1259-1 |title=Calculus |publisher=John Wiley and Sons Pte. Ltd.}} [254] => * {{cite book|first=Tom M. |last=Apostol |author-link=Tom M. Apostol |year=1967 |isbn=978-0-471-00005-1 |title=Calculus, Volume 1, One-Variable Calculus with an Introduction to Linear Algebra |publisher=Wiley}} [255] => * {{cite book|first=Tom M. |last=Apostol |author-link=Tom M. Apostol |year=1969 |isbn=978-0-471-00007-5 |title=Calculus, Volume 2, Multi-Variable Calculus and Linear Algebra with Applications |publisher=Wiley}} [256] => * {{cite book|last=Bell |first=John Lane |author-link=John Lane Bell |title=A Primer of Infinitesimal Analysis |publisher=Cambridge University Press |year=1998 |isbn=978-0-521-62401-5}} Uses [[synthetic differential geometry]] and nilpotent infinitesimals. [257] => * {{cite book |author=Boelkins, M. |year=2012 |title=Active Calculus: a free, open text |access-date=1 February 2013 |url=http://gvsu.edu/s/km |archive-url=https://web.archive.org/web/20130530024317/http://faculty.gvsu.edu/boelkinm/Home/Download_files/Active%20Calculus%20ch1-8%20%28v.1.1%20W13%29.pdf |archive-date=30 May 2013 |url-status=dead |df=dmy-all }} [258] => * {{cite book |author-link=Carl Benjamin Boyer |last=Boyer |first=Carl Benjamin |orig-year=1949 |url=https://books.google.com/books?id=KLQSHUW8FnUC |title=The History of the Calculus and its Conceptual Development |publisher=Hafner |edition=Dover |year=1959 |isbn=0-486-60509-4 }} [259] => * {{cite journal|first=Florian |last=Cajori |author-link=Florian Cajori |title=The History of Notations of the Calculus |journal=Annals of Mathematics |series=2nd Series |volume=25 |number=1 |date=September 1923 |pages=1–46 |doi=10.2307/1967725 |jstor=1967725|hdl=2027/mdp.39015017345896 |hdl-access=free }} [260] => * {{cite book|author-link=Richard Courant|last=Courant |first=Richard |isbn=978-3-540-65058-4 |title=Introduction to calculus and analysis 1.|date=3 December 1998 }} [261] => * {{cite book|first=Larry |last=Gonick |author-link=Larry Gonick |title=The Cartoon Guide to Calculus |isbn=978-0-061-68909-3 |oclc=932781617 |publisher=William Morrow |year=2012}} [262] => * Keisler, H.J. (2000). ''Elementary Calculus: An Approach Using Infinitesimals''. Retrieved 29 August 2010 from [http://www.math.wisc.edu/~keisler/calc.html http://www.math.wisc.edu/~keisler/calc.html] {{Webarchive|url=https://web.archive.org/web/20110501113944/http://www.math.wisc.edu/~keisler/calc.html |date=1 May 2011 }} [263] => * {{cite book|author-link=Edmund Landau |first=Edmund |last=Landau |isbn=0-8218-2830-4 |title=Differential and Integral Calculus |year=2001 |publisher=[[American Mathematical Society]]}} [264] => * {{cite book|first1=Leonid P. |last1=Lebedev |first2=Michael J. |last2=Cloud |title=Approximating Perfection: a Mathematician's Journey into the World of Mechanics |chapter=The Tools of Calculus |publisher=Princeton University Press |year=2004|bibcode=2004apmj.book.....L }} [265] => * {{cite book|author-link1=Ron Larson (mathematician) |last1=Larson |first1=Ron |first2=Bruce H. |last2=Edwards |year=2010 |title=Calculus |edition=9th |publisher=Brooks Cole Cengage Learning |isbn=978-0-547-16702-2}} [266] => * {{cite book|last=McQuarrie |first=Donald A. |year=2003 |title=Mathematical Methods for Scientists and Engineers |publisher=University Science Books |isbn=978-1-891389-24-5}} [267] => * {{cite book|first=Cliff |last=Pickover |author-link=Cliff Pickover |year=2003 |isbn=978-0-471-26987-8 |title=Calculus and Pizza: A Math Cookbook for the Hungry Mind|publisher=John Wiley & Sons }} [268] => * {{Cite book |last1=Salas |first1=Saturnino L. |last2=Hille |first2=Einar |author2-link=Einar Hille |last3=Etgen |first3= Garret J. |year=2007 |title=Calculus: One and Several Variables |edition=10th |publisher=[[John Wiley & Sons|Wiley]] |isbn=978-0-471-69804-3 }} [269] => * {{cite book|author-link=Michael Spivak |first=Michael |last=Spivak |date=September 1994 |isbn=978-0-914098-89-8 |title=Calculus |publisher=Publish or Perish publishing}} [270] => * {{cite book|editor-last=Steen |editor-first=Lynn Arthur |editor-link=Lynn Steen |year=1988 |isbn=0-88385-058-3 |title=Calculus for a New Century; A Pump, Not a Filter |publisher=[[Mathematical Association of America]]}} [271] => * {{cite book|author-link=James Stewart (mathematician)|last=Stewart |first=James |year=2012 |title=Calculus: Early Transcendentals |edition=7th |publisher=Brooks Cole Cengage Learning |isbn=978-0-538-49790-9}} [272] => * {{cite book|first1=George Brinton |last1=Thomas |author-link1=George B. Thomas |first2=Ross L. |last2=Finney |first3=Maurice D. |last3=Weir |year=1996 |isbn=978-0-201-53174-9 |title=Calculus and Analytic Geometry, Part 1 |publisher=Addison Wesley}} [273] => * {{cite book|author-link1=George B. Thomas |last1=Thomas |first1=George B. |first2=Maurice D. |last2=Weir |author-link3=Joel Hass |first3=Joel |last3=Hass |first4=Frank R. |last4=Giordano |year=2008 |title=Calculus |edition=11th |publisher=Addison-Wesley |isbn=978-0-321-48987-6}} [274] => * {{cite book|author-link1=Silvanus P. Thompson |author-link2=Martin Gardner |first1=Silvanus P. |last1=Thompson |first2=Martin |last2=Gardner |year=1998 |isbn=978-0-312-18548-0 |title=Calculus Made Easy |title-link=Calculus Made Easy}} [275] => {{Refend}} [276] => [277] => == External links == [278] => {{Sister project links|Calculus}} [279] => * {{springer|title=Calculus|id=p/c020050}} [280] => * {{MathWorld | urlname=Calculus | title=Calculus}} [281] => * {{PlanetMath | urlname=TopicsOnCalculus | title=Topics on Calculus | id=7592}} [282] => * [http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf Calculus Made Easy (1914) by Silvanus P. Thompson] Full text in PDF [283] => * {{In Our Time|Calculus|b00mrfwq|Calculus}} [284] => * [http://www.calculus.org Calculus.org: The Calculus page] at University of California, Davis – contains resources and links to other sites [285] => * [http://www.economics.soton.ac.uk/staff/aldrich/Calculus%20and%20Analysis%20Earliest%20Uses.htm Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis] [286] => * [http://www.ericdigests.org/pre-9217/calculus.htm The Role of Calculus in College Mathematics] {{Webarchive|url=https://web.archive.org/web/20210726234750/http://www.ericdigests.org/pre-9217/calculus.htm |date=26 July 2021 }} from ERICDigests.org [287] => * [https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/ OpenCourseWare Calculus] from the [[Massachusetts Institute of Technology]] [288] => * [http://www.encyclopediaofmath.org/index.php?title=Infinitesimal_calculus&oldid=18648 Infinitesimal Calculus] – an article on its historical development, in ''Encyclopedia of Mathematics'', ed. [[Michiel Hazewinkel]]. [289] => * {{cite web |url=http://math.mit.edu/~djk/calculus_beginners/ |title=Calculus for Beginners and Artists |author=Daniel Kleitman, MIT}} [290] => * [http://www.imomath.com/index.php?options=277 Calculus training materials at imomath.com] [291] => * {{in lang|en|ar}} [http://www.wdl.org/en/item/4327/ The Excursion of Calculus], 1772 [292] => [293] => {{Integral}} [294] => {{Infinitesimals}} [295] => {{Calculus topics}} [296] => {{Analysis-footer}} [297] => {{Areas of mathematics}} [298] => {{Isaac Newton}} [299] => {{Glossaries of science and engineering}} [300] => [301] => {{Authority control}} [302] => [303] => [[Category:Calculus| ]] [] => )
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Calculus

Calculus is a branch of mathematics that encompasses the study of limits, derivatives, integrals, and infinite series. It is a fundamental tool in various fields, including physics, engineering, economics, and computer science.

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It is a fundamental tool in various fields, including physics, engineering, economics, and computer science. Developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz, calculus provides methods for analyzing and quantifying change, such as calculating velocities and accelerations. The two main branches of calculus are differential calculus, which focuses on rates of change and slopes, and integral calculus, which focuses on accumulation and area under curves. The concepts and techniques of calculus have wide-ranging applications, from optimizing functions to modeling complex systems.

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