Sümmeetriline tuletis ja sellega seotud keskväärtusteoreemid

Ordinary differentiable functions are known from base course of calculus. In this bachelor thesis they are presented in a generalized meaning. The function f : D !R, where D " R, is said to be symmetrically differentiable at a point x #D if the limit lim h!0 f (x +h)− f (x −h) 2h := f s (x) exists. f s (x) is called the symmetric derivative of the function f . It is assumed that with some ! > 0 (x −!,x)%D &=! and (x,x +!)%D &=!. Denote that symmetrical differention is weaker condition than ordinary differention so, if f (a) exists then f s (a) = f (a). It is also seen that symmetrically differentiable function may not be continuous. In this paper mean value theorems of symmetrically differentiable functions are examined. They are derived from Lagrange’s, Rolle’s and Cauchy’s mean value theorems. This work consists of four chapters. In the first chapter there are necessary notions and results. Symmetrical continuity and differentiability are defined, connections with ordinary continuous and differentiation are found. It is proven that equations of symmetrically differentiable functions are similar to the equations of orinary differentiable functions. In the second chapter the main result — the mean value theorem for symmetrically differentiable functions — is presented. In addition the necessary conditions for this quasi–mean value theorem to manifest as classical one are found. In the second part of the chapter some applications of the quasi–mean value theorem are shown, the conditionswhen symmetrically differentiable functions are differentiable in ordinary sense, investigated. The third chapter introduces different versions of the quasi–mean value theorem. They are analogous to theorems for symmetrically differentiable functions of Flett, Trahan and others . In the last, fourth, chapter there is an overview of Cauchy like mean value theorems for symmetrically differentiable functions. They are based onWachnicks result, which is one of the versions of mean value theorem of Cauchy. This bachelor thesis is referable, it is drawn on the next articles: C. E. Aull [1], R. M.Davitt, R. C. Powers, T. Riedel, P. K. Sahoo [2], T. M. Flett [3], S. Reich [4], P. K. Sahoo [5], P. K. Sahoo [6], D. H. Trahan [8], E.Wachnick [9].