Exponential growth and decay
Some examples
The key property of exponential functions is that the rate of
growth (or decay) is proportional to how much is already there. As
a result, the following realworld situations (and others!) are
modeled by exponential functions:
 The population of a colony of bacteria can double every 20
minutes, as long as there is enough space and food. The more
bacteria you already have, the more new bacteria you get. This
is modeled by the function $P(t) = P_0 2^{t/20}$, where $P_0$ is
the number of bacteria you start with and $t$ is the time,
measured in minutes.
 The amount of money in a bank account grows exponentially,
since the amount of interest you earn is proportional to the
amount of money you have. If your annual interest rate is $r$,
then the law of compound interest says that $A(t) = A_0
(1+r)^t$, where $t$ is the elapsed time in years and $A_0$ is
the amount of money you start with.
 In radioactive decay, a certain fraction of the atoms decay
every second, so the rate of shrinkage is proportional to how
much is there. The law is $Y(t) = Y_0 2^{t/\tau}$, where the
constant $\tau$ is called the halflife of the material.
The number $e$
The numerical value of $e$ is approximately 2.718281828;
since $e$ is irrational, the decimal part neither terminates or
repeats.
$e$ shows up in many places in mathematics. Keep an eye out
throughout calculus for this fascinating number!
For example (for those of you who know the concept of the
proportionality constant): The exact proportionality
constant for the function $a^x$ depends on the number $a$. When
$a=2$, the constant is around 0.69. When $a=3$, the constant is
around 1.1. $e$ is the special number for which the
proportionality constant is exactly 1. That is, the rate at which $e^x$ grows is exactly $e^x$.
We will see this rate when we differentiate $e^x$ this semester.
