audio read-through Exponential Growth and Decay: Relative Growth Rate

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:

exponential growth

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

exponential decay

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

pairs of exponential functions

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 $\,r\,.$

$$ \begin{gather} \cssId{s34}{\overbrace{P'(t)\strut }^{\text{the rate at which the population is changing}}}\cr \cssId{s35}{\ \ =\ \ \strut}\cr \cssId{s36}{\overbrace{r\strut }^{\text{is proportional to}\ \ \ \ }\cdot } \cssId{s37}{\overbrace{P(t)\strut}^{\text{the current population size}}} \end{gather} $$


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:

understanding the relative growth rate

Units of the Relative Growth Rate

What are the units of the relative growth rate, $\,r\,$?


$$ \text{units of $\,r\,$} = \frac{1}{\text{units of time}} $$

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. Then:

How fast is the population growing at time zero?

$$\cssId{s70}{rP(0) = (0.12)(100) = 12}$$

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?

$$\cssId{s73}{100 + 12 = 112}$$

After one year, we expect to have about $\,112\,$ people.

What is the exact population after one year?

$$\cssId{s76}{P(1) = 100{\text{e}}^{0.12(1)} \approx 112.7 \approx 113}$$

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!

Concept Practice