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

Below there today we want to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Consider the graph below where F represents the fish population in a lake and F is its growth rate. Sketch the graph of the population function F. Versus T with an initial population of F subscript 0 is greater than 0. Awesome. So it appears for this particular problem we're ultimately asked to create a graph of the population function F versus time T with an initial population F subscript 0 or F0 is greater than 0. Awesome. So looking at our graph that we are given by the problem itself, we have our curve, which is represented by this green line, which appears to be horizontal with the F-axis, which would be where the X-axis would be. So the X axis is represented by F and the Y axis, the vertical axis, is denoted as F. And we have this horizontal line. Awesome. So, now that we know what we're ultimately trying to solve for and we have a visualization of the graph for F. Represents the fish population in F is its growth rate. We can now use this information to help us create our sketch for the population function versus time with an initial population where F 0 is greater than 0. So with that in mind, our first step that we need to take is we need to recognize that F is a positive constant, meaning we could write that F is equal to K where some constant is represented by K. Where, once again, we need to note that where K is greater than 0, because it is a positive constant. Awesome. So now, with that in mind, we now need to integrate F with respect to time T in order to obtain the population function. So you can go ahead and write that F of T is equal to the integral. Of F DT, which is equal to the integral of KDT which is equal to K multiplied by T plus C. Fantastic. So now we need to recall and use the initial condition. F of 0 is equal to F subscript zero, where once again we're told that F subscript 0 is greater than 0. Fantastic. So with that in mind, we could write that F subscript 0 is equal to K multiplied by 0 plus C. Fantastic. So that will mean when we rearrange this expression to isolate and solve for C, you will find that C is simply equal to F0 or F subscript 0. So with that in mind, we can officially sketch our graph, so we can sketch the graph as a as a straight line, similar to how we have a straight line in the graph that is given to us, but we need to draw our straight line with a positive slope starting at F subscript 0 is greater than 0. So when we do that, we will be able to achieve the following. The graph which will look as follows, I've gone ahead and put together a digitized version of my sketch. So our new curve, or curve for F subscript zero will be represented in purple, and it will be a straight line with a positive increasing slope, so it will resemble a diagonal line, and we can't forget that our X axis, our horizontal axis is denoted as time T. And our Y axis or vertical axis is denoted as F, our population function F. And that's it. That's how you solve for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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.

Key Concepts

Population Growth Rate Function

Relationship Between Derivative and Function Shape

Initial Conditions in Differential Equations

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