Integraali keskväärtusteoreemid: keskväärtust määravate punktide asümptootiline käitumine
Zusammenfassung
The 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
:
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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.