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In-Depth Information
Z
b
j
g.x/
j
dx
.b
a/ max
axb
j
g.x/
j
:
(1.35)
a
-
Yet Another Integral Inequality.
A more general result can also be useful; Let
f
D
f.x/ and g
D
g.x/ be bounded functions defined on the interval Œa; b.
Then,
Z
b
j
g.x/
j
Z
b
a
j
f .x/g.x/
j
dx
max
axb
j
f.x/
j
dx:
(1.36)
a
-
The Triangle Inequality.
For any real numbers x and y,wehave
j
x
C
y
j j
x
j C j
y
j
:
(1.37)
Generally, for real numbers x
1
;x
2
;:::;x
n
,wehave
ˇ
ˇ
n
X
n
X
x
i
j
x
i
j
:
(1.38)
i D1
i D1
-
The Taylor Series.
It is hard to find a more important tool for the development
and analysis of numerical methods than the Taylor series. For future reference,
we shall state it here in several versions. Let g
D
g.x/ be a function with n
C
1
continuous derivatives. Then we have the following expansion
10
:
1
2
˛
2
g
00
.x/
C
1
g.x
C
˛/
D
g.x/
C
˛g
0
.x/
C
nŠ
˛
n
g
.n/
.x/
C
R
nC1
;
(1.39)
where
R
nC1
D
1
.n
C
1/Š
˛
nC1
g
.nC1/
.x
C
/;
for some in the interval bounded by x and x
C
˛. This can be written more
compactly as (recall that 0Š
D
1)
n
X
˛
m
mŠ
g
.m/
.x/
C
R
nC1
:
g.x
C
˛/
D
(1.40)
mD0
The Taylor series expansion can also be written as
10
The notation g
.n/
may seem a bit unfamiliar, but it simply means
d
n
g
dx
n
i.e., g
.n/
is the nth order derivative of the function g with respect to x.
