The exponential e^x

Special properties

Its slope is equal to the function. (reminder - slope : rate of change over a very small distance)

Climbs much faster than y=x^100 for instance. The highest y gets, the higher the slope, so it keeps increasing.

Where does this happen? Interest. The more money you have the more interest, so more money, more interest, etc.

Let's construct it:

y(x) = dy/dx by definition of e^x. Also what we're adding becomes smaller and smaller because n! grows faster than x^n. So it doesn't keep growing, it stops at some point.

Now let's check why e^xe^X = e^(x+X)

It works.

So with this we can find out what e (euler's number) is:for x=1 we have

It's not really 2.7, 2,71828......

If we graph it:

Let's have an example: computing compound interest.

Say we have 1 dollar, and the bank gives you 100% interest after 1 year. we you get, 2, 4, 8 ... $.

If you ask the bank to give you interest after each month, you'd get 1+(1/12)(1/12)^2..., every day = 1/365. But we want continuous interest because we're doing calculus, so N.

When N tends towards infinity, (1+1/N)^N is e.

If we add c constant -

(property: The slope of the tangent line to any point of the graph below equals the height of that point above the horizontal axes. This means that y=e^x = y'=e^x too.

These are equivalent:

2^t = e^ln(2)*t. That is why we almost never see anywhere a^t, we can always write it e^ct.