Infinitezimalni račun — разлика између измена
м Нова страница: '''Infinitezimalni račun''' je grana matematike, koja se bavi funkcijama, derivacijama, integralima, [[granična vrednost funkcije|limesima fun... |
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==Literatura== |
==Literatura== |
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=== Dodatna literatura === |
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{{Refbegin|2}} |
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* Larson, Ron, Bruce H. Edwards (2010). "Calculus", 9th ed., Brooks Cole Cengage Learning. ISBN 9780547167022 |
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* McQuarrie, Donald A. (2003). ''Mathematical Methods for Scientists and Engineers'', University Science Books. ISBN 9781891389245 |
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* Stewart, James (2008). ''Calculus: Early Transcendentals'', 6th ed., Brooks Cole Cengage Learning. ISBN 9780495011668 |
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* Thomas, George B., Maurice D. Weir, Joel Hass, Frank R. Giordano (2008), "Calculus", 11th ed., Addison-Wesley. ISBN 0-321-48987-X |
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* Courant, Richard ISBN 978-3540650584 ''Introduction to calculus and analysis 1.'' |
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* Edmund Landau. ISBN 0-8218-2830-4 ''Differential and Integral Calculus'', American Mathematical Society. |
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* Robert A. Adams. (1999). ISBN 978-0-201-39607-2 ''Calculus: A complete course''. |
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* Albers, Donald J.; Richard D. Anderson and Don O. Loftsgaarden, ed. (1986) ''Undergraduate Programs in the Mathematics and Computer Sciences: The 1985-1986 Survey'', Mathematical Association of America No. 7. |
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* John Lane Bell: ''A Primer of Infinitesimal Analysis'', Cambridge University Press, 1998. ISBN 978-0-521-62401-5. Uses synthetic differential geometry and nilpotent infinitesimals. |
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* Florian Cajori, "The History of Notations of the Calculus." ''Annals of Mathematics'', 2nd Ser., Vol. 25, No. 1 (Sep., 1923), pp. 1–46. |
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* Leonid P. Lebedev and Michael J. Cloud: "Approximating Perfection: a Mathematician's Journey into the World of Mechanics, Ch. 1: The Tools of Calculus", Princeton Univ. Press, 2004. |
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* Cliff Pickover. (2003). ISBN 978-0-471-26987-8 ''Calculus and Pizza: A Math Cookbook for the Hungry Mind''. |
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* Michael Spivak. (September 1994). ISBN 978-0-914098-89-8'' Calculus''. Publish or Perish publishing. |
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* Tom M. Apostol. (1967). ISBN 9780471000051 ''Calculus, Volume 1, One-Variable Calculus with an Introduction to Linear Algebra''. Wiley. |
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* Tom M. Apostol. (1969). ISBN 9780471000075 ''Calculus, Volume 2, Multi-Variable Calculus and Linear Algebra with Applications''. Wiley. |
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* Silvanus P. Thompson and Martin Gardner. (1998). ISBN 978-0-312-18548-0 ''Calculus Made Easy''. |
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* Mathematical Association of America. (1988). ''Calculus for a New Century; A Pump, Not a Filter'', The Association, Stony Brook, NY. ED 300 252. |
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* Thomas/Finney. (1996). ISBN 978-0-201-53174-9 ''Calculus and Analytic geometry 9th'', Addison Wesley. |
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* Weisstein, Eric W. [http://mathworld.wolfram.com/SecondFundamentalTheoremofCalculus.html "Second Fundamental Theorem of Calculus."] From MathWorld—A Wolfram Web Resource. |
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{{Refend}} |
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=== Onlajn knjige === |
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{{Refbegin|2}}-{ |
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* Crowell, B. (2003). "''Calculus''" Light and Matter, Fullerton. Retrieved 6 May 2007 from [http://www.lightandmatter.com/calc/calc.pdf http://www.lightandmatter.com/calc/calc.pdf] |
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* Garrett, P. (2006). "''Notes on first year calculus''" University of Minnesota. Retrieved 6 May 2007 from [http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf <nowiki>http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf</nowiki>] |
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* Faraz, H. (2006). "''Understanding Calculus''" Retrieved 6 May 2007 from Understanding Calculus, URL [http://www.understandingcalculus.com/ http://www.understandingcalculus.com/] (HTML only) |
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* Keisler, H. J. (2000). "''Elementary Calculus: An Approach Using Infinitesimals''" Retrieved 29 August 2010 from [http://www.math.wisc.edu/~keisler/calc.html <nowiki>http://www.math.wisc.edu/~keisler/calc.html</nowiki>] |
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* Mauch, S. (2004). "''Sean's Applied Math Book''" California Institute of Technology. Retrieved 6 May 2007 from [http://www.cacr.caltech.edu/~sean/applied_math.pdf <nowiki>http://www.cacr.caltech.edu/~sean/applied_math.pdf</nowiki>] |
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* Sloughter, Dan (2000). "''Difference Equations to Differential Equations: An introduction to calculus''". Retrieved 17 March 2009 from [http://synechism.org/drupal/de2de/ http://synechism.org/drupal/de2de/] |
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* Stroyan, K.D. (2004). "''A brief introduction to infinitesimal calculus''" University of Iowa. Retrieved 6 May 2007 from [http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm] (HTML only) |
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* Strang, G. (1991). "''Calculus''" Massachusetts Institute of Technology. Retrieved 6 May 2007 from [http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm] |
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* Smith, William V. (2001). "''The Calculus''" Retrieved 4 July 2008 [http://www.math.byu.edu/~smithw/Calculus/] (HTML only).}- |
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{{Refend}} |
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[[Категорија:Математичка анализа]] |
[[Категорија:Математичка анализа]] |
Верзија на датум 4. децембар 2010. у 06:36
Infinitezimalni račun je grana matematike, koja se bavi funkcijama, derivacijama, integralima, limesima funkcije i graničnim vrednostima. Proučava razumevanje i opisivanje promena merljivih varijabli. Središnji koncept kojim se opisuje promena varijable je funkcija. Dve glavne grane su diferencijalni račun i integralni račun. Infinitezimalni račun je osnova matematičke analize.[1]
Koristi se u nauci, ekonomiji, inženjerstvu itd. Služi za rešavanje mnogih matematičkih problema, koji se ne mogu rešiti algebrom ili geometrijom.
Infinitezimalni račun se na latinskom jeziku kaže "calculus infinitesimalis" i iz toga je proizašao naziv "kalkulus", koji se koristi u dijelu sveta. Reč "infinitesimalis" znači "beskrajno mala količina".
Istorija
U antičkom razdoblju bilo je ideja sličnih infinitezimalnom računu. Egipćani su računali volumen piramide bez vrha. Grci Eudoks i Arhimed koristili su metodu ekshaustacije, koja je metoda izračunavanja površine nekog oblika tako što se u njega ubacuje niz poligona, čije površine konvergiraju prema površini celog oblika. Tu metodu koristio je i Kinez Liu Hui u 3. veku, da bi izračunao površinu kruga. U 5. veku Ču Čungdži koristio je metodu, koja će se kasnije nazvati Cavalierov princip za volumen sfere.
Godine 499. indijski je matematičar Aryabhata I. računao infinitezimalanim računom i zapisao astronomski problem u obliku diferencijalne jednačine. Na temelju te jednačine, u 12. veku Bhaskara je razvio neku vrstu derivacije. Oko 1000. godine Ibn al-Haitam osmislio je formulu za sve vrste četvrtih potencija i time pripremio put za integralni račun. U 12. veku perzijski matematičar Šaraf al-Din al-Tusi otkrio je pravilo za odvajanje kubičnog polinoma. U 17. veku japanski matematičar Šinsuke Seki Kova došao je do osnovnih spoznaja infinitezimalnoga računa.
Infinitezimalni račun otkrili su nezavisno jedan o drugog u otprilike isto vreme Isak Njutn i Gottfried Wilhelm Leibniz. Oni su otkrili zakone diferencijalnog i integralnog računa, derivacije i približne polinomske serije. Njihov rad nastavili su matematičari Augustin Louis Cauchy, Bernhard Riemann, Karl Weierstrass, Henri Léon Lebesgue i dr.
Glavna poglavlja
Derivacija
Derivacija funkcije je granična vrednost koeficijenta porasta funkcije i prirasta argumenta kada prirast argumenta teži nuli.
Integral
Za danu funkciju f(x) realne varijable x i interval [a,b] na pravcu realnih brojeva, integral
predstavlja površinu područja u xy-ravnini ograničenu grafom od f, x-osi, i vertikalnim crtama x=a i x=b.
Limes funkcije
Poglavlje limesa funkcije razvilo se iz problema, kako izračunati vrednost funkcije u slučajevima, kada funkcija nije dobro definisana npr.: deljenje s nulom. Limes funkcije f u tački a je broj, kojemu se pridružuje funkcijska vrednost f(x), kada se vrednost x približuje a.
npr.
Svojstva limesa
Literatura
- ^ Donald R. Latorre, John W. Kenelly, Iris B. Reed, Sherry Biggers (2007). Calculus Concepts: An Applied Approach to the Mathematics of Change. Cengage Learning. ISBN 0-618-78981-2.
Dodatna literatura
- Larson, Ron, Bruce H. Edwards (2010). "Calculus", 9th ed., Brooks Cole Cengage Learning. ISBN 9780547167022
- McQuarrie, Donald A. (2003). Mathematical Methods for Scientists and Engineers, University Science Books. ISBN 9781891389245
- Stewart, James (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning. ISBN 9780495011668
- Thomas, George B., Maurice D. Weir, Joel Hass, Frank R. Giordano (2008), "Calculus", 11th ed., Addison-Wesley. ISBN 0-321-48987-X
- Courant, Richard ISBN 978-3540650584 Introduction to calculus and analysis 1.
- Edmund Landau. ISBN 0-8218-2830-4 Differential and Integral Calculus, American Mathematical Society.
- Robert A. Adams. (1999). ISBN 978-0-201-39607-2 Calculus: A complete course.
- Albers, Donald J.; Richard D. Anderson and Don O. Loftsgaarden, ed. (1986) Undergraduate Programs in the Mathematics and Computer Sciences: The 1985-1986 Survey, Mathematical Association of America No. 7.
- John Lane Bell: A Primer of Infinitesimal Analysis, Cambridge University Press, 1998. ISBN 978-0-521-62401-5. Uses synthetic differential geometry and nilpotent infinitesimals.
- Florian Cajori, "The History of Notations of the Calculus." Annals of Mathematics, 2nd Ser., Vol. 25, No. 1 (Sep., 1923), pp. 1–46.
- Leonid P. Lebedev and Michael J. Cloud: "Approximating Perfection: a Mathematician's Journey into the World of Mechanics, Ch. 1: The Tools of Calculus", Princeton Univ. Press, 2004.
- Cliff Pickover. (2003). ISBN 978-0-471-26987-8 Calculus and Pizza: A Math Cookbook for the Hungry Mind.
- Michael Spivak. (September 1994). ISBN 978-0-914098-89-8 Calculus. Publish or Perish publishing.
- Tom M. Apostol. (1967). ISBN 9780471000051 Calculus, Volume 1, One-Variable Calculus with an Introduction to Linear Algebra. Wiley.
- Tom M. Apostol. (1969). ISBN 9780471000075 Calculus, Volume 2, Multi-Variable Calculus and Linear Algebra with Applications. Wiley.
- Silvanus P. Thompson and Martin Gardner. (1998). ISBN 978-0-312-18548-0 Calculus Made Easy.
- Mathematical Association of America. (1988). Calculus for a New Century; A Pump, Not a Filter, The Association, Stony Brook, NY. ED 300 252.
- Thomas/Finney. (1996). ISBN 978-0-201-53174-9 Calculus and Analytic geometry 9th, Addison Wesley.
- Weisstein, Eric W. "Second Fundamental Theorem of Calculus." From MathWorld—A Wolfram Web Resource.
Onlajn knjige
- Crowell, B. (2003). "Calculus" Light and Matter, Fullerton. Retrieved 6 May 2007 from http://www.lightandmatter.com/calc/calc.pdf
- Garrett, P. (2006). "Notes on first year calculus" University of Minnesota. Retrieved 6 May 2007 from http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf
- Faraz, H. (2006). "Understanding Calculus" Retrieved 6 May 2007 from Understanding Calculus, URL http://www.understandingcalculus.com/ (HTML only)
- Keisler, H. J. (2000). "Elementary Calculus: An Approach Using Infinitesimals" Retrieved 29 August 2010 from http://www.math.wisc.edu/~keisler/calc.html
- Mauch, S. (2004). "Sean's Applied Math Book" California Institute of Technology. Retrieved 6 May 2007 from http://www.cacr.caltech.edu/~sean/applied_math.pdf
- Sloughter, Dan (2000). "Difference Equations to Differential Equations: An introduction to calculus". Retrieved 17 March 2009 from http://synechism.org/drupal/de2de/
- Stroyan, K.D. (2004). "A brief introduction to infinitesimal calculus" University of Iowa. Retrieved 6 May 2007 from http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm (HTML only)
- Strang, G. (1991). "Calculus" Massachusetts Institute of Technology. Retrieved 6 May 2007 from http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm
- Smith, William V. (2001). "The Calculus" Retrieved 4 July 2008 [1] (HTML only).