Infinitezimalni račun — разлика између измена
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=== Dodatna literatura === |
=== Dodatna literatura === |
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* Larson, Ron, Bruce H. Edwards (2010). "Calculus", 9th ed., Brooks Cole Cengage Learning. ISBN 978-0-547-16702-2 |
* Larson, Ron, Bruce H. Edwards (2010). "Calculus", 9th ed., Brooks Cole Cengage Learning. ISBN 978-0-547-16702-2. |
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* McQuarrie, Donald A. (2003). ''Mathematical Methods for Scientists and Engineers'', University Science Books. ISBN 978-1-891389-24-5 |
* McQuarrie, Donald A. (2003). ''Mathematical Methods for Scientists and Engineers'', University Science Books. ISBN 978-1-891389-24-5. |
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* {{Cite book |ref= harv|last= Stewart|first= James|title= Calculus: Early Transcendentals|year= 2008|url= |publisher= 6th ed., Brooks Cole Cengage Learning|location= | |
* {{Cite book |ref= harv|last= Stewart|first= James|title= Calculus: Early Transcendentals|year= 2008|url= |publisher= 6th ed., Brooks Cole Cengage Learning|location= |isbn=978-0-495-01166-8}} |
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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 |
* 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 978-3-540-65058-4 ''Introduction to calculus and analysis 1.'' |
* Courant, Richard. ISBN 978-3-540-65058-4. ''Introduction to calculus and analysis 1.'' |
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* Edmund Landau. ISBN 0-8218-2830-4 ''Differential and Integral Calculus'', American Mathematical Society. |
* 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''. |
* 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. |
* 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. {{page|1998|978-0-521-62401-5|pages=}} Uses synthetic differential geometry and nilpotent infinitesimals. |
* John Lane Bell: ''A Primer of Infinitesimal Analysis'', Cambridge University Press. {{page|1998|978-0-521-62401-5|pages=}} 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. |
* 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. |
* 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''. |
* 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. |
* Michael Spivak. (September 1994). ISBN 978-0-914098-89-8.'' Calculus''. Publish or Perish publishing. |
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* Tom M. Apostol. (1967). ISBN 978-0-471-00005-1 ''Calculus, Volume 1, One-Variable Calculus with an Introduction to Linear Algebra''. Wiley. |
* Tom M. Apostol. (1967). ISBN 978-0-471-00005-1. ''Calculus, Volume 1, One-Variable Calculus with an Introduction to Linear Algebra''. Wiley. |
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* Tom M. Apostol. (1969). ISBN 978-0-471-00007-5 ''Calculus, Volume 2, Multi-Variable Calculus and Linear Algebra with Applications''. Wiley. |
* Tom M. Apostol. (1969). ISBN 978-0-471-00007-5. ''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''. |
* 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. |
* 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. |
* 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. |
* Weisstein, Eric W. [http://mathworld.wolfram.com/SecondFundamentalTheoremofCalculus.html "Second Fundamental Theorem of Calculus."] From MathWorld—A Wolfram Web Resource. |
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Верзија на датум 26. септембар 2014. у 02:31
Infinitezimalni račun je grana matematike koja se bavi funkcijama, izvodima, integralima, limesima i beskonačnim nizovima. 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 jednom delu sveta. Reč „infinitesimalis" znači "beskrajno mala veličina".
Istorija
U antičkom razdoblju bilo je ideja sličnih infinitezimalnom računu. Egipćani su računali zapreminu zarubljene piramide. Grci Eudoks i Arhimed koristili su metodu iscrpljivanja kojom se površina nekog oblika izračunava tako što se u njega ubacuje niz mnogouglova 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 kasnije biti nazvana Kavalijerijev princip za zapreminu lopte.
Godine 499. indijski matematičar Ariabhata I. je računao infinitezimalanim računom i zapisao astronomski problem u obliku diferencijalne jednačine. Na osnovu te jednačine je u 12. veku Bhaskara razvio neku vrstu izvoda. Oko 1000. godine Ibn al-Haitam je osmislio formulu za sve vrste četvrtih stepena i time pripremio put za integralni račun. U 12. veku persijski matematičar Šaraf al-Din al-Tusi otkrio je pravilo za odvajanje kubnog polinoma. U 17. veku 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.
Glavna poglavlja
Izvod
Izvod (derivacija) funkcije je granična vrednost koeficijenta porasta funkcije i prirasta argumenta kada prirast argumenta teži nuli.
Integral
Za datu funkciju f(x) realne promenljive x i interval [a,b] na pravcu realnih brojeva, integral
predstavlja površinu područja u ravni xy ograničenog grafom od f, x-osom i vertikalnim crtama x=a i x=b.
Limes
Poglavlje 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.
npr.
Svojstva limesa
Reference
- ^ 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 978-0-547-16702-2.
- McQuarrie, Donald A. (2003). Mathematical Methods for Scientists and Engineers, University Science Books. ISBN 978-1-891389-24-5.
- Stewart, James (2008). Calculus: Early Transcendentals. 6th ed., Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8.
- 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 978-3-540-65058-4. 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. 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 978-0-471-00005-1. Calculus, Volume 1, One-Variable Calculus with an Introduction to Linear Algebra. Wiley.
- Tom M. Apostol. (1969). ISBN 978-0-471-00007-5. 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., Приступљено 6. 5. 2007. from http://www.lightandmatter.com/calc/calc.pdf
- Garrett, P. (2006). "Notes on first year calculus" University of Minnesota., Приступљено 6. 5. 2007. from http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf
- Faraz, H. (2006). "Understanding Calculus", Приступљено 6. 5. 2007. from Understanding Calculus, URL http://www.understandingcalculus.com/ (HTML only)
- Keisler, H. J. (2000). "Elementary Calculus: An Approach Using Infinitesimals", Приступљено 29. 8. 2010. from http://www.math.wisc.edu/~keisler/calc.html
- Mauch, S. (2004). "Sean's Applied Math Book" California Institute of Technology., Приступљено 6. 5. 2007. from http://www.cacr.caltech.edu/~sean/applied_math.pdf
- Sloughter, Dan (2000). "Difference Equations to Differential Equations: An introduction to calculus"., Приступљено 17. 3. 2009. from http://synechism.org/drupal/de2de/
- Stroyan, K.D. (2004). "A brief introduction to infinitesimal calculus" University of Iowa., Приступљено 6. 5. 2007. from http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm (HTML only)
- Strang, G. (1991). "Calculus" Massachusetts Institute of Technology., Приступљено 6. 5. 2007. from http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm
- Smith, William V. (2001). "The Calculus", Приступљено 4. 7. 2008. [1] (HTML only).