Models of Growth

Exponential Growth and Decay

Exponential growth and decay happens whenever a rate of change is proportional to how much is already there.

Examples:

Money in the bank (compound interest)
Population growth with unlimited resources
Radioactivity
Heating and cooling (Newton's law)
Number of infected people in the early stages of an epidemic
Toxic waste in Lake Pristine (see previous module)

If the differential equation is $$\frac{dy}{dt} = ky,$$ where $k$ is a constant, then the solution is an exponential $$y(t)=y(0) e^{kt}.$$

If $k>0$, then the doubling time is $\displaystyle\frac{\ln(2)}{k} \approx \frac{.70}{k}$, the "law of 70".

If $k<0$, then we have exponential decay and the half-life is $\displaystyle\frac{\ln(2)}{|k|}$.