The exponential function $e^{x}$

The exponential function $exp(x)$ , also written as $e^{x}$ , occurs very commonly in many applications. It is used to model population growth, radioactive decay, heating and cooling, compound interest, chemical reactions, metabolic processes, and numerous other situations.

In what follows, we'll consider the function as modeling a system which changes with time, and write it as a function of the independent variable $t$ .

We'll see later that the exponential function has many interesting and remarkable properties, but for now, we'll make use of just a few.

Algebraic properties

The notation $e^{x}$ suggests that the exponential function should behave as follows:

\displaystyle e^{(s+t)}=e^{s}e^{t}

We'll see this is in fact valid for the exponential function.

For the notation to be consistent, it had better be true that

\displaystyle e^{t}=e^{(0+t)}=e^{0}e^{t}

Hence we need

\displaystyle e^{0}=1

\displaystyle 1=e^{0}=e^{(t-t)}=e^{(t+{-t})}=e^{t}e^{-t}

So

{\frac {1}{e^{t}}}=e^{-t}

$exp(t)$ is an increasing function of $t$

We'll see later (from looking at models for continuously compounded interest) that

e=\lim _{n\to \infty }{(1+{\frac {1}{n}})}^{n}

From this, we can show that the number $e$ is a positive constant, about equal to 2.71828...,

This is consistent with viewing the exponential function $e^{t}$ as a raising the number $e$ to the power $t$ (we have to make sense of doing this for arbitrary real numbers $t$ ).

From this we see that the exponential function is an increasing function of $t$ , and hence $e^{-t}$ is a decreasing function of $t$ .

Rate of growth

Consider

\displaystyle e^{2t}=e^{(t+t)}=e^{t}e^{t}=(e^{t})^{2}

So multiplying $t$ by the constant 2 causes the function to increase as the square of $e^{t}$ . Generalizing this observation, we'll think of the parameter $k$ as determining the growth rate of $e^{kt}$ .

Summary: modeling with the exponential function

If we have a model which starts at time $t=0$ with an initial size of $c$ , and which grows exponentially at a rate of $k$ , it will be described by

\displaystyle y(t)=ce^{kt}

If $k$ is positive, the function will increase with $t$ , else it will decrease.

Note that $e^{t}$ is positive for all values of $t$

Notes

This is still a fairly rough draft. The organization of this will likely change, and there's various things that are stated without proof; we'll address this later in readings, lectures, and homework.