9–14. Growth rate functions Make a sketch of the population fu...

Question

9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.

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Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says consider the graph below where R represents the rabbit population in a field, and our prime is its growth rate. And below this, we're given a graph of our prime versus R. And on this graph, we have a problem drawn on it that is concave down, and this problem begins at the origin, rises to its maximum, and decreases until it crosses the x-axis again at R equal to littler. Now we're asked to sketch the graph of the population of R versus time with an initial population of R not that is greater than 0. And below this, we're given a graph of R versus T on which to plot our function. Now, if we look at the graph that we were given, the R versus R. We noticed that we have a maximum value of our prime. At the apex of our parala, which sits at a value of R equal to littler divided by 2. So this means that we're going to have an inflection point when R is equal to littler divided by 2. So if we come down to our graph, we're going to label a couple of points on the Y axis, which is labeled as R. So first, we'll have our initial R value, which is R0, and so we'll set this very close to 0, but a positive value. So we'll set R0. At that point, we're also going to denote. The R value when R is equal to littler divided by 2, and we're also going to denote the Y value when R is equal to littler. On the y axis. So now we want to do is to draw our graph of R versus T. So, since our knot is a very small value, and based upon our graph of our prime versus R, we know that our prime is also going to be a small value. So it's going to be close to a horizontal slope initially, but that slope is going to increase. And it's going to increase until it reaches a maximum slope at R equal to little R divided by 2. And at this point, it's going to change from being a concave up function to a concave down function. And so the slope is going to begin to decrease. And it's going to asymptotically approach. The value of R equal to little R. And so this here is the graph that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.

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