Integraali keskväärtusteoreemid: keskväärtust määravate punktide asümptootiline käitumine

dc.contributor.advisorLeiger, Toivo, juhendaja
dc.contributor.authorMaadik, Inger-Helen
dc.contributor.otherTartu Ülikool. Matemaatika-informaatikateaduskondet
dc.contributor.otherTartu Ülikool. Matemaatika instituutet
dc.date.accessioned2013-06-19T07:58:07Z
dc.date.available2013-06-19T07:58:07Z
dc.date.issued2013
dc.description.abstractThe purpose of this thesis is to study the asymptotic behaviour of intermediate points in mean value theorems for integrals. The most simple mean value theorem states that if f : [a; b] ! R is a continuous function then there exists a number c 2 (a; b) such that Z b a f(t) dt = f(c)(b - a): In the case of the simpler mean value theorems for integrals the intermediate point c(x) asymptotically approaches the midpoint of the interval [a; x] and in addition lim sup x!a c(x) - a x - a 1 e : The simplest weighted mean value theorem for integrals states that if f : [a; b] ! R is a continuous function and g : [a; b] ! [0;1) is an integrable function then there exsists a number c 2 [a; b] such that Z b a f(t)g(t) dt = f(c) Z b a g(t) dt: In the case of the weighted mean value theorems for integrals the intermediate point asymptotically approaches the value a + (x - a) k q 1 k+1, where k 2 N is the number of times the function f di erentiable at the point a. Also when f and g are di erentiable then the intermediate point satis es the equation lim x!a R c(x) a g(t) dt R x a g(t) dt = 1 2 : 38 If the conditions set on the functions in the weighted mean value theorems for integrals are expanded to functions that aren't di erentiable at the point a then the approximate value c(x) a + (x - a) r r s + 1 r + s + 1 ; where r 2 (-1; 0) [ (0;1), s 2 (-1;1) and r +s > -1, can be used to provide close approximiations to certain physics' problems.en
dc.identifier.urihttp://hdl.handle.net/10062/31066
dc.language.isoetet
dc.publisherTartu Ülikoolet
dc.subjectintegraalet
dc.subjectintegraalide keskväärtusteoreemidet
dc.subjectkeskväärtust määrav punktet
dc.subjectkeskväärtust määrava punkti asümptootiline käitumineet
dc.subjectbakalaureusetöödet
dc.titleIntegraali keskväärtusteoreemid: keskväärtust määravate punktide asümptootiline käitumineet
dc.typeThesiset

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