Essential Concepts
- Exponential growth and exponential decay are two of the most common applications of exponential functions.
- Systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}.[/latex]
- In exponential growth, the rate of growth is proportional to the quantity present. In other words, [latex]{y}^{\prime }=ky.[/latex]
- Systems that exhibit exponential growth have a constant doubling time, which is given by [latex](\text{ln}2)\text{/}k.[/latex]
- Systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\text{−}kt}.[/latex]
- Systems that exhibit exponential decay have a constant half-life, which is given by [latex](\text{ln}2)\text{/}k.[/latex]
Glossary
- doubling time
- if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by [latex]\frac{(\text{ln}2)}{k}[/latex]
- exponential decay
- systems that exhibit exponential decay follow a model of the form [latex]y={y}_{0}{e}^{\text{−}kt}[/latex]
- exponential growth
- systems that exhibit exponential growth follow a model of the form [latex]y={y}_{0}{e}^{kt}[/latex]
- half-life
- if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by [latex]\frac{(\text{ln}2)}{k}[/latex]
