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.
 A: A negative example: show that an approximation derived from  a Taylor series is often not what you want. It will be very good at one point and will get progressivly worse as you move away from that point. In applications, you very often want approximations that match derivatives, too (Hermite methods), are good at a specific set of points (Lagrange methods), or are uniformly good throughout an entire interval (Chebyshev methods). If you can stop your students from jumping immediately to Taylor series whenever they have to approximate anything, you will have done them a great service (in my opinion).
A: I think that deriving an explicit formula for the $n^{th}$ fibonnaci number by using generating functions is a pretty cool "mathy" application.  Check out the book by Wilf ``generatingfunctionology''
A: Detecting maxima/minima/influx of a function at a point. 
Actually computing values of functions like $e^x$ etc. 
Proving that $e$ is irrational. 
A: Calculation of some series, say the Leibniz series for $\pi$.
