Neke generalizacije Lagrangeovog teorema srednje vrijednosti

master's thesis

• Author: Monika Matika
• Mentor: Ljiljana Arambašić (mentor)
• Committee member: Ljiljana Arambašić (predsjednik povjerenstva)
• Committee member: Damir Bakić (član povjerenstva)
• Committee member: Boris Guljaš (član povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-02-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu smo proučavali teoreme srednje vrijednosti te njihove generalizacije. Prvo smo proučili Lagrangeov, Rolleov, Cauchyjev te Flettov teorem srednje vrijednosti za realne funkcije realne varijable te prikazali njihove geometrijske interpretacije. Osim toga, na primjerima smo ilustrirali kako navedene teoreme koristiti prilikom rješavanja zadataka. Nakon toga, promatrali smo teoreme srednje vrijednosti i njihove generalizacije za funkcije više varijabli i to prvo za realne funkcije više varijabli, zatim za vektorske funkcije jedne realne varijable te na kraju za vektorske funkcije više realnih varijabli. Na primjeru smo pokazali kako se teorem srednje vrijednosti ne može na direktan način proširiti na vektorske funkcije. Za kraj smo diskutirali teoreme srednje vrijednosti za kompleksne funkcije kompleksne varijable.

Abstract (english)

In this paper we studied the mean value theorems and theirs generalizations. First we considered Lagrange’s, Rolle’s, Cauchy’s and Flett’s mean value theorem for real-valued functions of a real variable and showed their geometric interpretations. Furthermore, we gave examples to illustrate how we can use these theorems in solving problems. After that, we considered the mean value theorems and theirs generalizations for the functions of several variables, first for the real-valued functions of several real variables, then for the vector valued functions of one real variable and, finally, for the vector-valued functions of several real variables. In addition, we gave an example to illustrate that the mean value theorem cannot be directly extended in this more general situation. Finally, we discussed the mean value theorem for the complex-valued functions of a complex variable.

• Keywords: teorem srednje vrijednosti; generalizacije teorema srednje vrijednosti
• Keywords (english): mean value theorems; generalizations of mean value theorems
• Language: croatian
• URN:NBN: urn:nbn:hr:217:643387
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2019-01-23 11:37:31