Изворни код странице Infinitezimalni račun

'''Infinitezimalni račun''' je grana [[matematika|matematike]] koja se bavi [[funkcija]]ma, [[izvod]]ima, [[integral]]ima, [[limes]]ima i [[niz|beskonačnim nizovima]]. Proučava razumevanje i opisivanje promena merljivih [[varijabla|varijabli]]. Središnji koncept kojim se opisuje promena varijable je funkcija. Dve glavne grane su [[diferencijalni račun]] i [[integralni račun]]. Infinitezimalni račun je osnova [[matematička analiza|matematičke analize]].<ref>{{cite book|title=Calculus Concepts: An Applied Approach to the Mathematics of Change |author1=Donald R. Latorre |author2=John W. Kenelly |author3=Iris B. Reed |last4=Biggers|first4=Sherry|publisher=Cengage Learning |year=2007|isbn=0-618-78981-2 |url=http://books.google.com/books?id=bQhX-3k0LS8C}}</ref><ref>{{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 }}</ref> Koristi se u [[nauka|nauci]], [[ekonomija|ekonomiji]], [[inženjerstvo|inženjerstvu]] itd. Služi za rešavanje mnogih matematičkih problema, koji se ne mogu rešiti [[algebra|algebrom]] ili [[geometrija|geometrijom]]. Infinitezimalni račun se na [[latinski jezik|latinskom jeziku]] kaže -{„calculus infinitesimalis"}- i iz toga je proizašao naziv „kalkulus", koji se koristi u jednom delu [[svet]]a. Reč -{„infinitesimalis"}- znači "beskrajno mala veličina".

Infinitezimalni račun su nezavisno razvili krajem 17. veka [[Isaac Newton|Isak Njutn]] i [[Gottfried Wilhelm Leibniz|Gotfrid Vilhelm Lajbnic]].<ref>{{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 }}</ref><ref>{{cite book |first=Jason Socrates |last=Bardi |title=The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time |url=https://archive.org/details/calculuswarsnewt0000bard |location=New York |publisher=Thunder's Mouth Press |year=2006 |isbn=1-56025-706-7 }}</ref> Kasniji rad, uključujući [[(ε, δ)-definition of limit|kodifikaciju ideje o granicama]], stavio je ovaj razvoj na čvršće konceptualne osnove. Danas, račun ima široku upotrebu u [[science|nauci]], [[engineering|inženjerstvu]] i [[social science|društvenim naukama]].<ref>{{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 }}</ref>

== Istorija ==
[[Датотека:GodfreyKneller-IsaacNewton-1689.jpg|180px|desno|thumb|[[Isak Njutn]]]]
[[Датотека:Gottfried Wilhelm von Leibniz.jpg|180px|desno|thumb|[[Gotfrid Vilhelm Lajbnic]]]]
U [[antika|antičkom]] razdoblju bilo je ideja sličnih infinitezimalnom računu. [[Egipćani]] su računali zapreminu zarubljene [[piramida|piramide]]. [[Grci]] [[Eudoks]] i [[Arhimed]] koristili su metodu iscrpljivanja kojom se [[površina]] nekog oblika izračunava tako što se u njega ubacuje niz [[mnogougao|mnogouglova]] čije površine konvergiraju prema površini celog oblika. Tu metodu koristio je i Kinez [[Liu Hui]] u [[3. vek]]u, da bi izračunao površinu kruga. U [[5. vek]]u [[Ču Čungdži]] koristio je metodu koja će kasnije biti nazvana [[Kavalijerijev princip]] za zapreminu lopte.

Godine [[499]]. indijski matematičar [[Ariabhata I.]] je računao infinitezimalnim računom i zapisao [[astronomija|astronomski]] problem u obliku diferencijalne jednačine. Na osnovu te jednačine je u [[12. vek]]u [[Bhaskara]] razvio neku vrstu izvoda. Oko [[1000]]. godine [[Ibn al-Haitam]] je osmislio formulu za sve vrste četvrtih [[stepenovanje|stepena]] i time pripremio put za integralni račun. U 12. veku [[Iran|persijski]] matematičar Šaraf al-Din al-Tusi otkrio je pravilo za odvajanje kubnog [[polinom]]a. U [[17. vek]]u japanski matematičar Šinsuke Seki Kova došao je do osnovnih spoznaja infinitezimalnog računa.

Infinitezimalni račun otkrili su nezavisno jedan od drugog u otprilike isto vreme [[Isak Njutn]] i [[Gotfrid Vilhelm Lajbnic]]. Oni su otkrili zakone diferencijalnog i integralnog računa, izvoda (derivacije) i aproksimacija polinomnih nizova. Njihov rad nastavili su matematičari [[Ogisten Luj Koši]], [[Bernhard Riman]], [[Karl Vajerštras]], Henri Lion Lebesk i dr.

=== Drevni prethodnici ===

==== Egipat ====

Proračun [[volume|zapremine]] i [[area|površine]], jedan od ciljeva integralnog računa, može se naći u [[Egyptian mathematics|egipatskom]] [[Moscow Mathematical Papyrus|moskovskom papirusu]] (oko 1820 pne), ali formule su jednostavna uputstva, bez naznaka kako su dobijene.<ref>{{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 }}</ref><ref>{{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}}</ref>

== Glavna poglavlja ==

=== Izvod ===
[[Izvod]] (derivacija) funkcije <math> f </math> je granična vrednost koeficijenta porasta funkcije i prirasta argumenta kada prirast argumenta teži nuli.

=== Integral ===
Za datu [[funkcija (matematika)|funkciju]] -{''f''(''x'')}- realne [[promenljiva|promenljive]] ''x'' i [[interval (matematika)|interval]] -{[''a'',''b'']}- na pravcu [[realni broj|realnih brojeva]], [[integral]]

: <math>\int_a^b f(x)dx </math>

predstavlja [[površina|površinu]] područja u ravni ''xy'' ograničenog [[graf]]om od -{''f''}-, ''x''-osom i vertikalnim crtama ''x''=''a'' i ''x''=''b''.

=== Limes ===

Poglavlje [[granična vrednost|limesa funkcije]] razvilo se iz problema kako izračunati vrednost funkcije u slučajevima kada funkcija nije dobro definisana, npr. deljenje nulom. Limes funkcije -{''f''}- u tački -{''a''}- je [[broj]] kome se pridružuje funkcijska vrednost -{''f(x)''}-, kada vrednost -{''x''}- teži -{''a''}-.

<math>\lim_{x\to a} f(x)</math>

npr.

<math>\lim_{x\to 0} \frac {sin(x)}{x}\ = 1 </math>

== Svojstva limesa ==

:<math>\begin{matrix}
\lim\limits_{x \to p} & (f(x) + g(x)) & = & \lim\limits_{x \to p} f(x) + \lim\limits_{x \to p} g(x) \\
\lim\limits_{x \to p} & (f(x) - g(x)) & = & \lim\limits_{x \to p} f(x) - \lim\limits_{x \to p} g(x) \\
\lim\limits_{x \to p} & (f(x)\cdot g(x)) & = & \lim\limits_{x \to p} f(x) \cdot \lim\limits_{x \to p} g(x) \\
\lim\limits_{x \to p} & (f(x)/g(x)) & = & {\lim\limits_{x \to p} f(x) / \lim\limits_{x \to p} g(x)}
\end{matrix}</math>

== Pogledajte još ==
* [[Lajbnic-Njutn računska kontroverza]]

== Reference ==
{{reflist|30em}}

== Literatura ==
{{Литература|30em}}
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{{Литература крај}}

== Spoljašnje veze ==
{{Commonscat|Integral functions}}
{{Литература|30em}}
* {{springer|title=Calculus|id=p/c020050}}
* {{MathWorld | urlname=Calculus | title=Calculus}}
* {{PlanetMath | urlname=TopicsOnCalculus | title=Topics on Calculus | id=7592}}
* [http://djm.cc/library/Calculus_Made_Easy_Thompson.pdf Calculus Made Easy (1914) by Silvanus P. Thompson] Full text in PDF
* {{In Our Time|Calculus|b00mrfwq|Calculus}}
* [http://www.calculus.org Calculus.org: The Calculus page] at University of California, Davis&nbsp;– contains resources and links to other sites
* [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]
* [http://www.ericdigests.org/pre-9217/calculus.htm The Role of Calculus in College Mathematics] {{Wayback|url=http://www.ericdigests.org/pre-9217/calculus.htm |date=20210726234750 }} from ERICDigests.org
* [https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/ OpenCourseWare Calculus] from the [[Massachusetts Institute of Technology]]
* [http://www.encyclopediaofmath.org/index.php?title=Infinitesimal_calculus&oldid=18648 Infinitesimal Calculus]&nbsp;– an article on its historical development, in ''Encyclopedia of Mathematics'', ed. [[Michiel Hazewinkel]].
* {{cite web |url=http://math.mit.edu/~djk/calculus_beginners/ |title=Calculus for Beginners and Artists |author=Daniel Kleitman, MIT}}
* [http://www.imomath.com/index.php?options=277 Calculus training materials at imomath.com] {{Wayback|url=http://www.imomath.com/index.php?options=277 |date=20230709175857 }}
* [http://www.wdl.org/en/item/4327/ The Excursion of Calculus], 1772
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