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Explainer Video

Tutor: Dylan

Summary

Exponential modelling

In a nutshell

There is a wide range of situations wherein you will want to study quantities for which the rate of growth or decay is proportional to the quantity itself. Functions of the form $f(t) = Ae^{kt}$ , where $A$ and $k$ are constants, can be used to model such quantities, as they have the property that their rate of change is proportional to the original function; in particular, the derivative is $f'(t) = kAe^{kt} = kf(t)$ .

Modelling exponential growth

{P = Ae^{kt}}

$P$	A quantity which changes with respect to $t$ at a rate proportional to the function itself
$A$	A constant; the value of $P$ when $t = 0$
$k$	A constant; the constant of proportionality between $P$ and $\dfrac{dP}{dt}$
$t$	A variable; in situations like population growth this could be time measured in days, but it does not necessarily have to be time.

In many cases, $A$ is positive. Notice that if $A$ is positive and $k$ is positive, $P$ will grow as $t$ increases, and if $A$ is positive but $k$ is negative, $P$ decays as $t$ increases. This is referred to as exponential growth, or exponential decay, respectively.

In situations where $t$ is referring to time (in some specified unit, e.g. days) then $A$ is the initial value of $P$ : when $t = 0$ , you have that $e^{kt} = e^{k \times 0} = 1$ , so $P = Ae^{kt} = A$ .

Example 1

A petri dish contains a tiny colony of bacteria. Two scientists approximate the population of the colony, and then decide to model the population in the dish going forward by the equation $P = 1000000e^{0.02t}$ , where $t$ is the time in minutes since the scientists initially approximated the population of the bacteria. What is the number $1000000$ in this model? Approximately what will be the population in the dish $6$ hours after the scientists make their approximation? What is $\dfrac{dP}{dt}$ ? Sketch the graph of $P$ against $t$ (choosing the scales of your axes appropriately).

The number $1000000$ is the constant $A$ in this model - it refers to the initial value, i.e. the value of $P$ when $t = 0$ , which here is the approximation made by the scientists.

There are $60 \times 6 = 360$ minutes in $6$ hours. Using your calculator, find that the model predicts the population at time $t = 360$ to be $1000000e^{0.02 \times 360}\approx1,339,430,764$ .

$\dfrac{dP}{dt}$ is equal to:

$\begin{aligned}\dfrac{dP}{dt} &= \dfrac{d}{dt}(1000000e^{0.02t})\\&= 1000000\dfrac{d}{dt}(e^{0.02t})\\&= 1000000 \times 0.02e^{0.02t}\\&= 20000e^{0.02t}\end{aligned}$

The graph of this function looks something like this:

Maths; Exponentials and logarithms; KS5 Year 12; Exponential modelling

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y = e^x

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