7FExpGrowth


 * Exponential Growth and Decay**

This topic is related to populations (of people, animals, microbes, etc). Also temperatures and radioactive isotopes.

__**Growth**__ means population is __**increasing**__:

math . \qquad \dfrac{dP}{dt}>0 math

__**Decay**__ means population is __**decreasing**__:

math . \qquad \dfrac{dP}{dt}<0 math

Proportion

If y is __**proportional**__ to x (or y **__varies directly__** with x)

math \text{Then } \, y \propto x \quad \text{so } \, y = kx math

where **k** is a constant {called the __**constant of proportionality**__}

If y is **__inversely proportional__** to x (or y **__varies inversely__** with x)

math \text{Then } \, y \propto \dfrac{1}{x} \quad \text{so } \, y = \dfrac{k}{x} math

where k is constant {the constant of proportionality}

Example

... ... Speed, **v**, is inversely proportional to the square of time, **t**.

math . \qquad \text{Would be written as } \; v \propto \dfrac{1}{t^2} \quad \text{or } \; v = \dfrac{k}{t^2} math

Exponential Growth

Many populations grow proportional to the size of the population.

math \dfrac{dP}{dt} \propto P \quad \text{or} \; \dfrac{dP}{dt} = kP math

where k is a constant (different for each population)

Note that if k > 0, the the population is increasing (ie growth).

If we solve the differential equation, we get:

math . \qquad \dfrac{dP}{dt} = kP, \quad k>0 math

Take the reciprocal of both sides math . \qquad \dfrac{dt}{dP} = \dfrac{1}{kP} math

Integrate both sides math \\ . \qquad t = \displaystyle{\int} \dfrac{1}{kP} dP \\. \\ . \qquad t = \dfrac{1}{k} \log_e P + c \\. \\ . \qquad kt = \log_e P + kc \\. \\ . \qquad \log_e P = kt - kc math

Using log laws math \\ . \qquad P = e^{kt-kc} \\. \\ . \qquad P = e^{kt} \times e^{-kc} math

Notice that k, c and e are constants, so e -kc is a constant:

math . \qquad \text{Let } P_0 = e^{-kc} math

Thus **The equation for exponential population growth**

math . \qquad P = P_0 e^{kt}, \quad k>0 math

Notice math . \qquad \text{When } t = 0, \, e^{kt} = 1, \, \text{ so } P = P_0 math

This means that P 0 is the __**Initial Population**__ (or population at time = 0)

Exponential Decay

Solving the similar equation: math . \qquad \dfrac{dP}{dt}=-kP math

gives the equation for exponential decay.

math . \qquad P=P_0e^{-kt}, \quad k>0,\, P_0=P(0) math

Example

Half Life A common type of exponential decay is that of radioactive substances. in this case the half life is often used. A substances half life is the time taken for half of the substance to decay. So in 1 half life you have half the original, in 2 half lives you have a quarter, in three you have an eighth etc. In other words at math t=t_{\frac{1}{2}} \\ P=\frac{1}{2} P_0 \\ math Subbing this into the equation above and simplifying gives: math \frac{1}{2} =e^{-kt_{\frac{1}{2}}} \\ \text{ or rearranged for t} \\ t_{\frac{1}{2}} = \frac{1}{k} log_e(2) \\ math

**See also:** www.mathsonline.com.au Y12Advanced --> Calculus --> Calculus & the Physical World --> Lesson 4 .