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I seem to be having a hard time understanding the growth function. If I begin with

$x(t)=x(0)e^{-\lambda t}$

I can work backwards until I reach

$x'(t) = -\lambda x(t)$

Which is telling me that the rate of growth is the current amount times some constant. For some reason, my brain has a hard time making sense of that. Can someone explain that idea to me?

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All this means is that the rate of growth is proportional to the value right now. As an example, say that you had an initial large amount of "stuff". At every hour, half of your stuff disappears, so if you had $1$ ton of stuff, after one hour you $1/2$ ton, after two $1/4$ ton and so on. If you look at the rate: $x'(t) \simeq \frac{x(t+\Delta t)-x(t)}{\Delta t}=-1/2, -1/4, -1/8...$ That is, the rate is constantly decreasing, but relative to the value you have: $x'(t) = -1/2 x(t)$ This is called exponential decay - you lose a lot very quickly, and then the rate slows down, as you no longer have very much left.

N.B. George Carlin had A classic example of exponential decay of stuff..

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    Exponential growth and decay is common in nature. Consider the concept of [half-life](http://en.wikipedia.org/wiki/Half-life) in radioactive decay.2012-09-02