Interesting applications of Taylor's theorem I am assistant for a real analysis course (kind of a TA, holding a couple of hours complementary to the lecture). I have to treat Taylor's theorem this Monday, and I'd like to give a few examples of where it is useful. The only thing I can think of right now is approximations in the context of physics, specifically the example I would give them is: take the pendulum equation:
$$\ddot{\varphi}=-\sin\varphi\approx-\varphi$$
by Taylor's theorem, thus obtaining the harmonic motion equation for small oscillations.
Does anyone have other interesting examples of where this theorem can be useful?
Additional informations: The class is for $1$st year students in mathematics and physics. it is proof based, so they shouldn't be afraid by a bit of formality or "hard" arguments.

Detecting maxima/minima/influx of a function at a point. 
Actually computing values of functions like $e^x$ etc. 
Proving that $e$ is irrational. 
