Exponential increase

1 Exponential increase

Task!

Look back at Section 6.2 to review the definitions of $a n$ exponential function and $t h e$ exponential function.
List examples in this Workbook of contexts in which exponential functions are appropriate.

$A n$ exponential function has the form $y = a^{x}$ where $a > 0$ . $T h e$ 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

$P = 12 e^{0.1 t} (0 \leq t \leq 25)$

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 $(0 \leq t \leq 25)$ 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.
$P (10) = 12 e^{1} \approx$ 33 million. This is getting very large for a city but might be attainable in 10 years and just about sustainable.
$P (25) = 12 e^{2.5} \approx$ 146 million. This is unrealistic for a city.

Note that exponential population growth of the form $P = P_{0} e^{k t}$ 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 $e^{k t}$ is positive for all values of $t$ . The parameter $k$ is called the instantaneous fractional growth rate .

Figure 6 :

${ The function $P = 12e^{0.01t}$}$

For the model $P = 12 e^{k t}$ 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 :

${ Comparison of the functions $P = 12 e^{0.01t}$ and $P = 12e^{0.02t}$}$

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

$y = a e^{b x} a > 0, b > 0, c \leq x \leq d$

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.

1 Exponential increase

Task!

Answer

Task!

Answer