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Learn how to derive the exponential function for population growth and explore a modified model with a carrying capacity. Solve differential equations using integration and discover the logistic function.
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Math 180 Packet #29 Exponential Growth and Decay (3.8 in Stewart book)
Recall: In Math 160, we modeled population growth with the exponential function: How can we derive this from basic principles?
Recall: In Math 160, we modeled population growth with the exponential function: How can we derive this from basic principles?
Suppose the birth rate () and death rate () are both constant (ex: and ).
Then the rate of change of the population depends on the population size as follows:
Then the rate of change of the population depends on the population size as follows:
Then the rate of change of the population depends on the population size as follows:
is an example of a differential equation (an equation that involves a function and its derivative(s)). We can solve differential equations using integration.
If we want to modify our model to include a carrying capacity (), then approach zero as approaches . Here’s a modified model:
Here’s a modified model: If you solve it via integration, then you get the logistic function:
