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master's thesis
Eksponencijalna funkcija i derivabilnost
Zagreb: University of Zagreb, Faculty of Science, 2020. urn:nbn:hr:217:942023
Trajbar, Željka
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Trajbar, Ž. (2020). Eksponencijalna funkcija i derivabilnost (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:942023

Trajbar, Željka. "Eksponencijalna funkcija i derivabilnost." Master's thesis, University of Zagreb, Faculty of Science, 2020. https://urn.nsk.hr/urn:nbn:hr:217:942023

Trajbar, Željka. "Eksponencijalna funkcija i derivabilnost." Master's thesis, University of Zagreb, Faculty of Science, 2020. https://urn.nsk.hr/urn:nbn:hr:217:942023

Trajbar, Ž. (2020). 'Eksponencijalna funkcija i derivabilnost', Master's thesis, University of Zagreb, Faculty of Science, accessed 27 March 2025, https://urn.nsk.hr/urn:nbn:hr:217:942023

Trajbar Ž. Eksponencijalna funkcija i derivabilnost [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2020 [cited 2025 March 27] Available at: https://urn.nsk.hr/urn:nbn:hr:217:942023

Ž. Trajbar, "Eksponencijalna funkcija i derivabilnost", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2020. Available at: https://urn.nsk.hr/urn:nbn:hr:217:942023

Metadata
TitleEksponencijalna funkcija i derivabilnost
Title (english)Exponential function and derivability
AuthorŽeljka Trajbar
MentorZvonko Iljazović (mentor)
Committee memberZvonko Iljazović (predsjednik povjerenstva)
Committee memberVedran Čačić (član povjerenstva)
Committee memberMarko Horvat (član povjerenstva)
Committee memberMladen Vuković (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2020-09-29, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Ukratko, u ovom diplomskom radu proučavali smo neka svojstva eksponencijalne funkcije s naglaskom na njenu derivabilnost. Najprije smo definirali eksponencijalnu funkciju s bazom većom od 1, prvo smo za a > 0 definirali ax za sve x∈N, potom za sve x∈Z, onda i za x∈Q, pri čemu smo dokazali da za svaki a > 0 i n∈N, postoji jedinstveni n√a. Nadalje, definirali smo za a > 1 ax za sve... More x∈R, proučavali smo konvergenciju nizova realnih brojeva te omeđene nizove u R. Potom smo proučavali neprekidnost eksponencijalne funkcije te posljedice koje ona ima na svojstva eksponencijalne funkcije. Također, dokazali smo da je eksponencijalna funkcija neprekidna, zatim smo došli do međuvrijednosti neprekidnih funkcija te koristeći dobivene rezultate dokazali smo da je eksponencijalna funkcija bijekcija, proučavali smo i pojmove limesa i derivacije funkcija te smo dokazali da vrijedi tzv. eksponencijalni zakon i definirali broj e. Definirali smo logaritamsku funkciju s bazom većom od 1, proučavali smo derivabilnost eksponencijalne funkcije pa smo potom definirali eksponencijalnu funkciju s bazom manjom od 1, zatim logaritamsku funkciju s bazom manjom od 1 te smo dokazali neka svojstva tih funkcija. Na kraju smo dokazali da je funkcija ln derivabilna u točki 1 te, pomoću toga, došli smo do glavnog rezultata, a to je da je eksponencijalna funkcija derivabilna. Less
Abstract (english)
In short, in this thesis we have studied some properties of the exponential function, with an emphasis on its derivability. Foremost, we defined an exponential function with a base greater than 1, we first defined ax fora > 0 for all x∈N, then for all x∈Z, and then for x∈Q, by which we have proved that for every a > 0 and n∈N, there exists a unique n√a. Furthermore, we defined for a > 1... More ax for all x∈R, we studied the convergence of sequences of real numbers, as well as bounded sequences in R. Then we studied the continuity of the exponential function and consequences which it has on the properties of the exponential functions. Also, we proved that the exponential function is continuous, then we came to the intermediate value of continuous functions and, using the obtained results, proved that the exponential function is a bijection, we studied the concepts of limes and derivatives, proved that the so-called exponential law is valid and defined the number e. We defined the logarithmic function with a base greater than 1, studied the derivability of the exponential function and then defined an exponential function with a base less than 1, then a logarithmic function with a base less than 1 and proved some properties of those functions. In the end, we proved that the function ln is derivable in the point 1 and, using that, came to the main result, which is that the exponential function is derivable. Less
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:942023
Study programmeTitle: Mathematics Education; specializations in: Mathematics Education
Course: Mathematics Education
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2021-03-11 13:42:25
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