Taylor polynomials and Taylor series aren't just for
approximating numerical values of known functions. They also
help us represent integrals
(that are otherwise too hard for us) and solutions
to differential equations as power series.

Example: There is no difficulty computing $e^{-2x^2}$ –
any scientific calculator will do it. But how do you compute$\displaystyle \int_{0.1}^{0.2} e^{-2x^2} \,dx$? A good
solution is to find a Taylor polynomial from the Taylor series for
$e^{-2x^2}$ and integrate it term-by-term.

Example: How do you solve differential equations like
$\frac{dy}{dx} = y + x$ that aren't separable? A good
solution is to write $y$ as a power series and compare the two sides
of the differential equation. This allows us to recursively figure
out all of the coefficients.

Both of these examples are discussed in the video.