Suppose
Write
We call
the th degree *Taylor polynomial*.
Notice that
if and only if
We would like to estimate with . We need
an estimate for .

**Theorem 6.7.1** (Taylor's theorem)

*If
for
, then
*
for $|x-a|&le#leq;d$.

For example, if , this says that
i.e.,
which should look familiar from a previous class (Mean Value Theorem).

**Applications:**

- One can use Theorem to
prove that functions converge to their Taylor series.

- Returning to the relativity example above, we apply Taylor's theorem
with and . With
and any number such
that
, we have
For example, if we assume that
we use
Using
, we get

Thus for
mph, then the error in throwing away
relativistic factors is
. This is like 200 feet out of
the distance to the sun (93 million miles).
So relativistic and Newtonian kinetic energies are almost the same
for reasonable speeds.

William Stein
2006-03-15