## Sümmeetriline tuletis ja sellega seotud keskväärtusteoreemid

##### Abstract

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].