Graphics Reference
In-Depth Information
R
Apply Rolle's Theorem to the function
F
:[
a
,
b
]
given by
(
b
x
)
2!
F
(
x
)
f
(
b
)
f
(
x
)
(
b
x
)
f
(
x
)
f
(
x
)
(
b
x
)
...
1)!
f
(
x
)
K
(
b
x
)
(
n
to show that, for some
c
with
a
c
b
,
(
b
c
)
(
n
1)!
f
(
c
).
K
(
b
a
)
2!
So
f
(
b
)
f
(
a
)
(
b
a
)
f
(
a
)
f
(
a
)
...
(
b
a
)
(
n
1)!
f
(
a
)
(
b
c
)
(
b
a
)
(
n
1)!
f
(
c
).
This last term is called
Cauchy
'
s form of the remainder
.
(46) Use Cauchy's form of the remainder to prove that the Taylor series
for log(1
x
) converges to the function when
1
x
1.
Summary
-
Taylor's Theorem
Taylor's Theorem with Lagrange's form of the remainder, or nth
Mean Value Theorem
qn 35
If
f
:[
a
,
b
]
R
(i) has a continuous (
n
1)th derivative on
[
a
,
b
],
(ii) is differentiable
n
times on (
a
,
b
), then
(
b
a
)
2!
f
(
b
)
f
(
a
)
(
b
a
)
f
(
a
)
f
(
a
)
(
b
a
)
(
n
1)!
f
(
a
)
(
b
a
)
n
!
...
f
(
c
)
for some
c
, with
a
c
b
.
Taylor's Theorem with Cauchy's form of the remainder
qn 45
If
f
:[
a
,
b
]
R
(i) has a continuous (
n
1)th derivative on
[
a
,
b
],
(ii) is differentiable
n
times on (
a
,
b
), then
