Activity 3.6.3.
In Desmos, define \(P(t) = \frac{A}{1 + Me^{-kt}}\) and accept sliders for \(A\text{,}\) \(M\text{,}\) and \(k\text{.}\) Set the slider ranges for these parameters as follows: \(0.01 \le A \le 10\text{;}\) \(0.01 \le M \le 10\text{;}\) \(0.01 \le k \le 5\text{.}\)
(a)
Sketch a typical graph of \(P(t)\) on the axes provided and write several sentences to explain the effects of \(A\text{,}\) \(M\text{,}\) and \(k\) on the graph of \(P\text{.}\)
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(b)
On a typical logistic graph, where does it appear that the population is growing most rapidly? How is this value connected to the carrying capacity, \(A\text{?}\)
(c)
How does the function \(1 + Me^{-kt}\) behave as \(t\) decreases without bound? What is the algebraic reason that this occurs?
(d)
Use your Desmos worksheet to find a logistic function \(P\) that has the following properties: \(P(0) = 2\text{,}\) \(P(2) = 4\text{,}\) and \(P(t)\) approaches \(9\) as \(t\) increases without bound. What are the approximate values of \(A\text{,}\) \(M\text{,}\) and \(k\) that make the function \(P\) fit these criteria?
