From relative growth / decay to an exponential function
We now know that any time an initial value increases or decreases by the same factor or percentage over equal increments of time, we can use an exponential model.
The factor or percentage is also known as relative growth: we mostly see it expressed as a percentage in real life models.
For example, if we read that a population of crabs grows by 1.2% every year, this means that for every 1000 crabs in the population, other crabs will be added to the population every year.
Find the equation, given the relative growth r
We can further refine the general model for an exponential growth to an equivalent form that shows visually the main characteristics of the model, and precisely:
where:- is the initial value
- is the relative growth rate (percentage of growth rate), expressed as a positive decimal number
- is the initial population
- is the relative growth rate (percentage of growth rate), expressed as a positive decimal number
- is the time unit
Old Mac Donald HAS a farm...
Use the applet below to get familiar with exponential growth.
Enter your answers (press the Enter key after each entry), then select Check answers to get a visual grading of your answers.
If they are both correct, a button will allow you to create and explore a new model, otherwise you can deselect Check answers and try to enter the correct ones.
Select the Show solution checkbox to view the solution of the problem.
Find the equation, given the relative decay r
If we want to model an exponential decay, the equation will be instead:
Therefore, a population that is exponentially decreasing over time will be modeled by a function of the form . All the variables are defined as above for the exponential growth function.Old MacDonald HAD a farm...
Use the applet below to get familiar with exponential decay.
Enter your answers (press the Enter key after each entry), then select Check answers to get a visual grading of your answers.
If they are both correct, a button will allow you to create and explore a new model, otherwise you can deselect Check answers and try to enter the correct ones.
Select the Show solution checkbox to view the solution of the problem.