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 real-world 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 half-life 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.