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Problem 14 Medium Difficulty

For a fixed value of $ M $ (say $ M = 10), $ the family of logistic functions given by Equation & depends on the initial value $ P_o $ and the proportionality constant $ k. $ Graph several members of this family. How does the graph change when $ P_o $ varies? How does it changes when $ k $ varies?


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Video Transcript

Okay. This question asked about graphing logistic functions. So I'll write down what that is. The logistic function. It looks like this P of T equals M divided by one plus A. E. To the negative K. T. Where A is equal to m minus peanut divided by peanut. Okay, so you can think of em as the capacity. Peanut is the initial population. K. Is this scaling factor. And so when we are looking at these graphs, we're gonna want to compare what happens when we change some of these values. So I have grabbed a couple of examples below here and I'll say that read is when we have P. Not Equals two. I have a blue here. Blue is for one. p. not equals 0.5. And the last one is an orange. That's when He not equals .01. So for the red graphs, I'm going to look first at this first graph here, the red graph starts off above the cure carrying capacity. So you would expect it to go down to M which is one in this case. Uh The other two graphs are below the carrying capacity. So you would expect them to go up to em which is one. All right, M. equals one for all these. Okay, so if you start off with a smaller population, it's going to take more time for you to get to the caring capacity. If you change K. That's what the next two graphs show That determines how quickly you get to that caring capacity. So when K equals five we get there pretty quickly. It takes less than equals two to get to that care and capacity. But if you have K equals 20.5, it's going to take much longer to get to that point. So K. Represents the scale and that peanut is the initial, and that is how the logistic function changes as you change K, and peanut.