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, .