the th degree Taylor polynomial.
if and only if
We would like to estimate with . We need
an estimate for .
For example, if , this says that
which should look familiar from a previous class (Mean Value Theorem).
- 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
, we have
For example, if we assume that
, we get
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.