20.500.12556/DKUM-22596 LEIBNIZ-NEWTONOVA FORMULA IN NJENE POSPLOŠITVE LEIBNIZ-NEWTON FORMULA AND ITS GENERALIZATIONS V diplomskem delu obravnavamo Leibniz-Newtonovo formulo in njene posplošitve. Pojem Riemannovega oz. določenega integrala je vpeljan s pomočjo Darbouxovih in Riemannovih vsot, pri čemer je poudarjena ekvivalentnost omenjenih pristopov. V tretjem poglavju so predstavljeni potrebni pogoji za integrabilnost funkcij, v četrtem pa ena izmed povezav med določenimi integrali in primitivnimi funkcijami. Sledi pomemben matematični rezultat Leibniza in Newtona, t. i. Leibniz-Newtonova formula, poznana tudi kot osnovni izrek analize. V nadaljevanju obravnavamo posplošitve te formule; pri prvi obliki nadomestimo obojestranski odvod z desnim odvodom, v drugi nastopa Schwarzov odvod, tretja posplošitev Leibniz-Newtonove formule pa se nanaša na funkcije, ki izpolnjujejo Lipschitzev pogoj. V zadnjem poglavju je navedenih nekaj primerov, kjer postane računanje določenega integrala s pomočjo Leibniz-Newtonove formule precej lažje kot računanje po definiciji določenega integrala. Ključna sta primera, ki ponazarjata napake, ki nastanejo pri uporabi rešitev iz tablic nedoločenih integralov na neustreznih intervalih. Pokazali smo, kako se tem napakam uspešno izogniti. The graduation thesis discusses the Leibniz-Newton formula and its generalizations. The notion of Riemann or the definite integral is introduced by presenting Darboux and Riemann sums, whereby the equivalence of these approaches is emphasised. The third chapter presents the necessary conditions for integrability of functions and in the fourth chapter a connection of the definite integral and the primitive function is presented. This is followed by an important mathematical result, the Leibniz-Newton formula, also known as the fundamental theorem of calculus. Several generalizations of this formula are then discussed; in the first form we assumed that f is only the right-hand derivative of F, the second form uses the Schwarz derivative, and the third generalization of the Leibniz-Newton formula refers to the functions that satisfy the Lipschitz condition. In the last chapter some examples for calculating the definite integrals using the Leibniz-Newton formula are given, and it is much easier to do so, than to calculate them by using the definition of the definite integral. The most important among them are two examples that illustrate mistakes that occur when using solutions from the table of indefinite integrals at unsuitable intervals. We have shown how to successfully avoid making such mistakes. Darbouxove vsote Riemannove vsote Riemannov integral določeni integral primitivna funkcija desni odvod Schwarzov odvod Lipschitzev pogoj Leibniz-Newtonova formula napake pri uporabi Leibniz-Newtonove formule. Darboux sums Riemann sums Riemann integral definite integral primitive function right-hand derivative Schwarz derivative Lipschitz condition Leibniz-Newton formula mistakes made by using Leibniz-Newton formula. true true false [S. Topler] Slovenski jezik Angleški jezik Diplomsko delo 2012-04-05 15:09:40 2012-04-23 13:57:53 2022-04-14 11:56:33 2023-12-30 03:28:23 2012 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 51(043.2) 19087368 URN:SI:UM:DK:PPLSQHLX UNI_Topler_Sara_2012.pdf UNI_Topler_Sara_2012.pdf 1 6728882FACA3DB96467DA2ADB5CAAD49 2625188d3daf9ae80737ef938bba78235f9b0d8c106d4b8c4cd2984117b3f652 8687e661-7c06-11eb-bb7a-00155d0001ca 20.500.12556/dkum/a36a460e-378a-48e1-b70a-cda1d1bed9f2 https://dk.um.si/Dokument.php?lang=slv&id=29223 Fakulteta za naravoslovje in matematiko 0 0 0