### 1 Exponential increase

##### Task!

- Look back at Section 6.2 to review the definitions of $an$ exponential function and $the$ exponential function.
- List examples in this Workbook of contexts in which exponential functions are appropriate.

- $An$ exponential function has the form $y={a}^{x}$ where $a>0$ . $The$ exponential function has the form $y={e}^{x}$ where $e=2.718282......$
- It is stated that exponential functions are useful when modelling the shape of a hanging chain or rope under the effect of gravity or for modelling exponential growth or decay.

We will look at a specific example of the exponential function used to model a population increase.

##### Task!

Given that

$\phantom{\rule{2em}{0ex}}P=12{e}^{0.1t}\phantom{\rule{2em}{0ex}}\left(0\le t\le 25\right)$

where
$P$
is the number in the population of a city
**
in millions
**
at time
$t$
**
in years
**
answer these questions.

- What does this model imply about $P$ when $t$ = 0?
- What is the stated upper limit of validity of the model?
- What does the model imply about values of $P$ over time?
- What does the model predict for $P$ when $t=10$ ? Comment on this.
- What does the model predict for $P$ when $t=25$ ? Comment on this.

- At $t$ = 0, $P$ = 12 which represents the initial population of 12 million. (Recall that ${e}^{0}=1$ .)
- The time interval during which the model is valid is stated as $\left(0\le t\le 25\right)$ so the model is intended to apply for 25 years.
- This is exponential growth so $P$ will increase from 12 million at an accelerating rate.
- $P\left(10\right)=12{\text{e}}^{1}\approx $ 33 million. This is getting very large for a city but might be attainable in 10 years and just about sustainable.
- $P\left(25\right)=12{\text{e}}^{2.5}\approx $ 146 million. This is unrealistic for a city.

Note that exponential population growth of the form
$P={P}_{0}{e}^{kt}$
means that as
$t$
becomes large and positive,
$P$
becomes very large. Normally such a population model would be used to predict values of
$P$
for
$t>0$
, where
$t=0$
represents the present or some fixed time when the population is known. In Figure 6, values of
$P$
are shown for
$t<0$
. These correspond to extrapolation of the model into the past. Note that as
$t$
becomes increasingly negative,
$P$
becomes very small but is never zero or negative because
${e}^{kt}$
is positive for all values of
$t$
. The parameter
$k$
is called the
**
instantaneous fractional growth rate
**
.

**
Figure 6
**
:

For the model $P=12{\text{e}}^{kt}$ we see that $k=0.1$ is unrealistic, and more realistic values would be $k=0.01$ or $k=0.02$ . These would be similar but $k$ =0.02 implies a faster growth for $t>0$ than $k=0.01$ . This is clear in the graphs for $k=0.01$ and $k=0.02$ in Figure 7. The functions are plotted up to 200 years to emphasize the increasing difference as $t$ increases.

**
Figure 7
**
:

The exponential function may be used in models for other types of growth as well as population growth. A general form may be written

$\phantom{\rule{2em}{0ex}}y=a{e}^{bx}\phantom{\rule{2em}{0ex}}a>0,\phantom{\rule{1em}{0ex}}b>0,\phantom{\rule{1em}{0ex}}c\le x\le d$

where $a$ represents the value of $y$ at $x=0$ . The value $a$ is the intercept on the $y$ -axis of a graphical representation of the function. The value $b$ controls the rate of growth and $c$ and $d$ represent limits on $x$ .

In the general form,
$a$
and
$b$
represent the
**
parameters
**
of the exponential function which can be selected to fit any given modelling situation where an exponential function is appropriate.