How would you explain why "e" is important? (And when it applies?) Possible Duplicate:
  Intuitive Understanding of the constant “e”  




Let's say you want to explain this to your teenage son.  I understand the technical definition of $e$  

$$ e=\lim_{n\to\infty}\left( 1 + \frac1n \right)^n $$

But, I don't want to get lost in technical babble.  While specific examples are welcome, I also want to understand the big picture.   I want to first know the general significance of why and when e is used.  Anyone got a link?    Is there a typical "pattern" that it models?  In what general sorts of situations does "e" arise?

I only know e from the classic continuous compounding example.  But, why does it appear in other applications of growth, science, etc?  Are all these examples just variations on this same limit that defines e?  This increasingly smaller interval when you apply an infinitely smaller percentage growth (1 + 1/n) but infinitely many times (to the power of n)
There is also a difference between an interest in understanding 


the mathematical constant "$e$", which is an irrational, transcendental number, but real number, just as is $\pi$, but a number, nonetheless, which happens to be the value of your limit, and happens to be the value of the infinite sum $$e = \displaystyle \sum_{n=0}^\infty \frac{1}{n!}$$And more:
$$e^x = \sum_{n=0}^\infty{x^n \over n!} = 1 + x + {x^2\over2!}+{x^3\over3!}+{x^4\over4!}+\cdots$$
vs.
exponential functions (base $e$: (e.g., $f(x) = e^x$, or $e^{i\theta} = \cos \theta + i\sin \theta$), functions to which you seem to refer when you speak of its use in representing exponential growth, and other applications which crop up everywhere, so it seems. Certainly, functions involving $e$ tell us something about $e$, but they tell us so much more than that.


That is, the importance of $e$ itself isn't so much for its significance as a particular real number, but its significance as a base for exponential functions, and in terms of the ways that functions involving $e$ as a base appear in surprising ways, are powerful, and have applications in many domains. 

So it's hard to know exactly what you are interested in: the number $e$, or the many functions involving $e$.
$e$ appears in a lot of places in calculus, but before calculus, it has little significance.