Population growth Consider the following population functions....

Question

Population growth Consider the following population functions.

e.Use a graphing utility to graph the population and its growth rate.

p(t) = 600 (t²+3/t²+9)

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says the concentration of a certain chemical in a reaction over time can be described by the function D of T is equal to 500 multiplied by the quantity of 2 T2 + 5 in quantity, divided by the quantity of T2 + 7, where T is the time and hour since the reaction started. We need to draw the graph of the concentration and its growth rate with the help of a graphing calculator. Now, we need to draw both the graph of the concentration and the growth rate. So, the concentration is just going to be graphing our function CFT, but we need to determine its growth rate, and to get the growth rate, we need to take a derivative of our function CFT with respect to time. So that means we're going to be looking for CT. And that's gonna be equal to the derivative with respect to T. Of the quantity of 500. Multiplied by the quantity of 2 T squared plus 5 in quantity, divided by the quantity of T2 plus 7. So, taking a derivative of this function, this is going to be equal to 500 is a constant, so we can move through the derivative, so we'll have 500. That's gonna be multiplied by the quantity, and here we'll be taking a derivative of a fraction, so we will need to use our quotient rule. So we call for the quotient rule that we're going to have our denominator, which is going to be the quantity of T2 + 7. In quantity, and that gets multiplied by the derivative of the numerator, which in this case is going to be 140. Density derivative with respect to T of 2 T 2 + 5 is 4T. Then we will have minus our numerator, which is the quantity of 2 T 2 + 5 in quantity, and that gets multiplied by the derivative of the denominator, which is going to be the quantity of 2 T. Since the derivative of T2 + 7 with respect to T is to T. And this whole quantity is going to be divided by our denominator squared, which is going to be the quantity of T2 + 7 in quantity squared. So now we just need to simplify this expression. So this is going to be equal to 500 multiplied by the quantity, and here we will simplify our numerator by multiplying everything out. And so doing so, we'll get 4 T cubed plus 28 E minus 4 T cubed. Minus 10. That whole quantity is still being divided by the quantity of T2 + 7 in quantity squared. So now we can collect like terms in the numerator, and so we see that we have 40 cubed minus 4 T cubed, so those two will add up to 0. Then we'll have 28T minus 10 T, so that'll give me 18T. So this is going to be equal to 500 multiplied by 18T. That is divided by the quantity of T2 plus 7 in quantity squared. So finally, simplifying to our final answer, this is going to be equal to 9000. Divided by the quantity of T2 + 7 in quantity squared. So this is the function that we need to plot on our graph. In addition to our original function CFT. So doing so, With the help of our graphing calculator, we will get the following graph where we have um in the diagram here we have Y is equal to CFT in the black curve and it's near the top of the graph, and then we have Y is equal to C T as the red line and that's the curve near the bottom of the graph. So this is actually the answer to the problem that we were looking for. So I hope that this explanation has been useful, and I'll see you in the next video.

Population growth Consider the following population functions.
e.Use a graphing utility to graph the population and its growth rate.
p(t) = 600 (t²+3/t²+9)

Key Concepts

Population Functions

Graphing Utilities

Growth Rate

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