Graphics Reference
In-Depth Information
Taylor's Theorem
Taylor's Theorem or nth Mean Value Theorem
35 Thefunction
f
:[
a
,
b
]
1)th derivative on
the closed interval [
a
,
b
] and is differentiable
n
times on the open
interval (
a
,
b
).
Let
R
has a continuous (
n
(
b
a
)
(
n
1)!
f
(
a
)
f
(
b
)
f
(
a
)
(
b
a
)
f
(
a
)
...
K
.
(
b
a
)
Apply Rolle's Theorem to the function
F
:[
a
,
b
]
R given by
(
b
x
)
2!
F
(
x
)
f
(
b
)
f
(
x
)
(
b
x
)
f
(
x
)
f
(
x
)
(
b
x
)
(
n
1)!
f
(
x
)
K
(
b
x
)
,
...
to show that, for some
c
with
a
c
b
,
K
f
(
c
)/
n
!.
So
(
b
a
)
2!
f
(
b
)
f
(
a
)
(
b
a
)
f
(
a
)
f
(
a
)
(
b
a
)
(
n
1)!
f
(
a
)
(
b
a
)
n
!
...
f
(
c
).
36 Rewrite the conclusion of Taylor's Theorem substituting
a
h
for
b
.
Maclaurin's Theorem
37 By putting
a
0 and
h
x
in qn 36 show that, for any function
f
: [0,
x
]
R which is differentiable
n
times,
x
2
f
(0)
...
x
(
n
1)!
f
(0)
x
n
!
f
(
x
),
f
(
x
)
f
(0)
xf
(0)
for some
(0, 1).
38 Use Maclaurin's Theorem to say what you can about
R
(
x
) where
x
2!
...
x
(
n
1)!
R
exp(
x
)
1
x
(
x
),
assuming that exp(0)
1 and exp
(
x
)
exp(
x
).
