- Mean value theorem
- Mean value theorem (divided differences)
- Central limit theorem
- Divided differences
- Logarithmic mean
- Fundamental theorem of calculus
- Expected value
- Errors and residuals
- Finite difference
- Difference quotient
mean value theorem divided differences
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In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.
Statement of the theorem
For any n + 1 pairwise distinct points x0, ..., xn in the domain of an n-times differentiable function f there exists an interior point
ξ
∈
(
min
{
x
0
,
…
,
x
n
}
,
max
{
x
0
,
…
,
x
n
}
)
{\displaystyle \xi \in (\min\{x_{0},\dots ,x_{n}\},\max\{x_{0},\dots ,x_{n}\})\,}
where the nth derivative of f equals n ! times the nth divided difference at these points:
f
[
x
0
,
…
,
x
n
]
=
f
(
n
)
(
ξ
)
n
!
.
{\displaystyle f[x_{0},\dots ,x_{n}]={\frac {f^{(n)}(\xi )}{n!}}.}
For n = 1, that is two function points, one obtains the simple mean value theorem.
Proof
Let
P
{\displaystyle P}
be the Lagrange interpolation polynomial for f at x0, ..., xn.
Then it follows from the Newton form of
P
{\displaystyle P}
that the highest order term of
P
{\displaystyle P}
is
f
[
x
0
,
…
,
x
n
]
x
n
{\displaystyle f[x_{0},\dots ,x_{n}]x^{n}}
.
Let
g
{\displaystyle g}
be the remainder of the interpolation, defined by
g
=
f
−
P
{\displaystyle g=f-P}
. Then
g
{\displaystyle g}
has
n
+
1
{\displaystyle n+1}
zeros: x0, ..., xn.
By applying Rolle's theorem first to
g
{\displaystyle g}
, then to
g
′
{\displaystyle g'}
, and so on until
g
(
n
−
1
)
{\displaystyle g^{(n-1)}}
, we find that
g
(
n
)
{\displaystyle g^{(n)}}
has a zero
ξ
{\displaystyle \xi }
. This means that
0
=
g
(
n
)
(
ξ
)
=
f
(
n
)
(
ξ
)
−
f
[
x
0
,
…
,
x
n
]
n
!
{\displaystyle 0=g^{(n)}(\xi )=f^{(n)}(\xi )-f[x_{0},\dots ,x_{n}]n!}
,
f
[
x
0
,
…
,
x
n
]
=
f
(
n
)
(
ξ
)
n
!
.
{\displaystyle f[x_{0},\dots ,x_{n}]={\frac {f^{(n)}(\xi )}{n!}}.}
Applications
The theorem can be used to generalise the Stolarsky mean to more than two variables.