4. e

This is one of those magical numbers like $\pi$. We see it in continuous compounding - exponential growth - entities that grow like $e^{r\,t}$. Sometimes called the ``base of natural logarithms'', $e \sim 2.718282$.

Look at a graph. of e to the power t, e^t = exp(t).

Notes

Defined for all values of t, even negative ones.

Crosses that y-axis at y = 1, why?

This relationship really grows fast, hence the term exponential growth.

As t heads to negative infinity, e^t goes to 0.

As t heads to positive infinity, e^t goes to infinity.

Why don't we call this a power function?

Because the object we are changing, t, is in the exponent, not the base.

The more general form of the function:

$$\fbox{$C \, e^{r \, t}.$ }$$

Here

C is a multiplicative constant (how much money at the start).

e is the magical number.

r is a fixed constant (the interest rate).

t is what we are changing (how long the investment is held for).

Example.

C = 100.

r = 0.05 is a fixed constant (the interest rate).

t is time in years.

Rules for exponential functions (same as power functions):

$e^x \, e^y = e ^{x + y}$.

$e^{-x} = \frac{1}{e^x}$ Exponential decay.

A useful modeling function: the logistic:

$$\fbox{$y = \frac{e^x}{1 + e^x}.$ }$$

This is in fact the formula for the graph.