Why are Taylor series useful? I think I have a good grasp on the fundamentals of Taylor series (what they do and how they approximate the functions), but I just don't understand how these can be useful.
For example, lets look at the following Taylor series:
$$e^x\approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +\frac{x^4}{4!}+ \frac{x^5}{5!}\dotsb.$$
Why would you want to use the approximation when you have the actual equation $e^x$. It not only look simpler, but also gives you the true value of this function for any value of $x$; so why use Taylor series for which gives you just an approximation. 
Maybe the point I am missing is that Taylor Series can give you an approximation on unknown functions, i.e. $f(x) = ???$.
But then you wouldn't be able to find derivatives of this function? 
Could someone please help me see Taylor Series as an actual tool that can be used to solve real life problems (with an example ideally)? 

Without using a calculator, a set of tables, etc, how would find the value of $e^x$?  For some people, that series actually is the definition of $e^x$.  
A more general use is expanding the domain of a function e,g. from $\mathbb{R}$ to $\mathbb{C}$.
Another is integration of a function for which there is no anti-derivative.
