If is differentiable at
, then the function
where
and
is the ``best'' linear approximation
to
near
.
For we have
.
Note that
and
.
If is a function which is
times differentiable at
, then the
Taylor polynomial of
at
is the polynomial
of degree (at most
) for which
for all
.
Compute the third Taylor polynomial of at
.
Write
. We need to find
,
,
, and
so that
for
,
,
, and
.
In our case
for all
and
. So,
for all
.
We compute
,
,
and
. Thus,
.
.
so that
.
Finally,
so that
.
Thus, the third Taylor polynomial of at
is
.
Find the third Taylor polynomial of at
.
As before, we write
and we find
,
,
and
.
Differentiating,
,
,
and
. Thus,
Solving these equations, we find
,
,
, and
.
That is, the third Taylor polynomial of at
is
.
We may write any polynomial of degree three as
.
Differentiating, we have
,
, and
.
So, ,
,
, and
.
Hence, if is the third Taylor polynomial of
at
,
we have
,
,
, and
.
That is, the third Taylor polynomial of at
is
.
If we write
,
then
where
. (We define
and
.)
In particular,
. So, if
is the
Taylor polynomial of
at
, we have
.
Thus,
or to put it another way, the
Taylor polynomial of
at
is
.
We compute
,
,
,
, and
. Thus,
,
,
,
,
, and
.
We compute the first few factorials: ,
,
,
,
, and
.
Therefore, the fifth
Taylor polynomial of
at
is
.
If is
times differentiable between on the interval
(or
if
) and
is the
Taylor polynomial
of
at
, then there is a number
so that
.
So, if we can find so that
whenever
, we would know that
.
Find a decimal approximation to valid to the hundredths place.
We will find so that if
is the
Taylor polynomial for
at
, then
.
We know that
and that on the interval
from zero to one this function is bounded by
.
Thus,
.
So, we want so that
or
what is the same thing
. If
, then
.
Now,
. So,
.
In point of fact,
.