In the prior section, Exponential Growth and Decay: Introduction,
we learned that most people use the function $\,P(t) = P_{\,0}\,{\text{e}}^{rt}\,$ to model exponential growth or decay, where:

This section gives additional information about the family of functions, $P(t) = P_{\,0}\,{\text{e}}^{rt}\,$,
and also explores the relative growth rate.

Increasing/Decreasing Properties of $\,{\text{e}}^{rt}\,$

As illustrated below:

$r > 0\,$:
$y = {\text{e}}^{rt}\,$ is increasing
(exponential growth)

$r < 0\,$:
$y = {\text{e}}^{rt}\,$ is decreasing
(exponential decay)

pairs like
$\,\color{green}{y = {\text{e}}^{2t}}\,$ and $\,\color{blue}{y = {\text{e}}^{-2t}}\,$
are symmetric about the $\,y\,$-axis

How Fast does an Exponentially Growing Population Increase?

For exponential growth:

The relative growth rate will make this observation mathematically precise.

Relative Growth Rate

First, we borrow some Calculus results:

Putting this all together, for $\,P(t) = P_{\,0}{\text{e}}^{rt} \,$:

The rate at which the population is changing (i.e., the growth rate)
is proportional to the current population size,
and the proportionality constant is $\,r\,$.
$$ \overbrace{P'(t)\strut }^{\text{the rate at which the population is changing}} = \overbrace{r\strut }^{\text{is proportional to}}\cdot \overbrace{P(t)\strut}^{\text{the current population size}} $$


The Rate at which the Population Grows/Shrinks Depends on its Current Size

The rate at which the population $\,P(t) = P_{\,0}{\text{e}}^{rt} \,$ is growing or shrinking depends on its current size!
In other words, the growth rate is relative to the current population.
For this reason, the proportionality constant $\,r\,$ is called the relative growth rate.

Graphical Understanding of the Relative Growth Rate

Let $\,(t,\color{red}{y})\,$ be any point on the graph of $\,P(t) = P_{\,0}{\text{e}}^{\color{red}{r}t}\,$, as shown below.
The slope of the tangent line at this point is $\,\color{red}{ry}\,$.
This slope depends on two things:

Units of the Relative Growth Rate

What are the units of the relative growth rate, $\,r\,$?
Answer: $$ \text{units of $\,r\,$} = \frac{1}{\text{units of time}} $$ Here are some examples:


Suppose that $\,P(t) = 100{\text{e}}^{0.12t}\,$ gives the number of people at time $\,t\,$, where $\,t\,$ is measured in years.

How fast is the population growing at time zero?
$\,rP(0) = (0.12)(100) = 12$
At $\,t = 0\,$, the population is growing at the rate of $\,12\,$ people per year.

Based on the calculation above, about how many people do you expect to have after one year?
$100 + 12 = 112\,$
After one year, we expect to have about $\,112\,$ people.

What is the exact population after one year?
$P(1) = 100{\text{e}}^{0.12(1)} \approx 112.7 \approx 113$

Why is the exact population slightly bigger than the estimate?
As soon as you move away from a point, the growth rate changes!
In other words, the growth rate changes as the population changes.
As soon as the population increases (say, from $\,100\,$ to $\,101\,$ people), then the growth rate increases, too!

Careful with language: ‘Relative Growth Rate’ versus ‘Growth Rate’

Be careful to distinguish between two similar-sounding concepts:

The word ‘relative’ is important!

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Solving Exponential Equations
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
1 2 3 4 5 6 7 8 9 10 11

(MAX is 11; there are 11 different problem types.)