Question

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

Answer 1

A logistic function is a mathematical function that models population growth. The parameters A, M, and k in the function have different effects on the graph. A determines the carrying capacity, M determines the growth rate, and k determines the steepness of the curve.

Step-by-step explanation:

A logistic function is defined as \(P(t)=\frac{A}{1+Me^{-kt}}\), where \(A\), \(M\), and \(k\) are parameters with slider ranges: \(0.01\leq A\leq 10\), \(0.01\leq M\leq 10\), and \(0.01\leq k\leq 5\). The graph of a logistic function is an S-shaped curve. The parameter \(A\) determines the carrying capacity, which is the maximum population size the environment can sustain. The parameter \(M\) determines the growth rate, with larger values of \(M\) leading to faster growth. The parameter \(k\) determines the steepness of the curve, with larger values of \(k\) resulting in a steeper curve.

Answer 2

a) Using Desmos, the logistic function
`\( P(t) = (A)/(1+M e^(-kt)) \)` with sliders for
`\( A, M, \) and \( k \) in the ranges \( 0.01 \leq A \leq 10, \ 0.01 \leq M \leq 10, \) and \( 0.01 \leq k \leq 5 \)` can be graphed. The parameters A, M and k affect the graph as follows: A represents the carrying capacity or maximum population, M influences the initial growth rate, and k controls the rate at which the population approaches the carrying capacity. b) On a typical logistic graph, the population appears to grow most rapidly at the initial stages of growth, where the curve is steepest. This value is connected to the carrying capacity A, as it represents the maximum population the environment can support. c) As t increases without bound, the function
`\( 1+M e^(-kt) \)`approaches 1. This behavior occurs because as t goes to infinity, the exponential term
`\( e^(-kt) \)` approaches zero, making the whole expression tend towards 1.

Step-by-step explanation:

a) A,M and k. Adjusting these sliders illustrates the impact of each parameter on the shape of the logistic curve. The carrying capacity A determines the maximum population the environment can sustain, M influences the initial growth rate, and k controls how quickly the population approaches the carrying capacity.

b) On a logistic graph, the population grows most rapidly initially when the curve is steepest. This initial growth rate is connected to the carrying capacity A. The steepness of the curve depends on the values of A,M and k, and it decreases as the population approaches A.

c) As t increases without bound, the function
`\( 1+M e^(-kt) \)` approaches 1. This behavior occurs because the exponential term
`\( e^(-kt) \)` tends to zero as t goes to infinity, making the entire expression tend towards 1. The logistic function reaches a stable equilibrium, and the population stops growing. d) To find a logistic function that fits the given criteria
`(\( P(0) = 2, P(2) = 4, \) and \( P(t) \) approaches 9 as \( t \)` increases without bound), one can use Desmos sliders to adjust
`\( A, M, \) and \( k \)`. Fine-tune the sliders until the desired properties are met, obtaining approximate values for
`\( A, M, \) and \( k \)` that satisfy the given conditions.