As the title says, how come the value e is special? it has functions and rules just dedicated to it, but what sets it apart from another random number?
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Imagine you have a snowball rolling down a hill. As it collects snow, it grows.
Let’s say that the amount of snow it collects depends on how much snow it already has – so it’s getting bigger faster and faster! What would a plot of its size look like?
It turns out that you’ll find an ‘e’ in the equation you’ll need to describe this slope – e is 2.7 and change. The equation y=e^(x) creates a curve whose slope (rate of change) is equal to its value at every point.
So lots of things in nature involve this kind of growth – like a colony of bacteria where the rate of growth depends on the number of bacteria in it. It’s very special in calculus, as y=e^(x) is unaffected by derivatives. The questions ‘what’s the value of this function’ and ‘how fast is this function changing (i.e. what is its derivative?)’ have the same answer!
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