 Welcome to the GVSU Calculus Screencasts. In this edition, we're going to talk about a differential equation to model exponential growth. Populations of things like people or viruses typically spread exponentially at first if there's no quarantine to contain them or no immunization for them. In this screencast, we'll uncover a differential equation that can be used to model the growth of a virus. We need to start with some initial assumptions. We'll assume that a virus grows at a rate of 11% per day. And we'll assume that we're going to start with a virus population of 100. So pause the video for a moment and consider the following question. We have a population of 100 and the virus grows by 11% per day. What would the population be after day one? Resume the video when you're ready. Well, if we have 100 virus and the growth rate is 11%, there will be an additional 11% of 100 or 11 virus the next day. Add that to the original 100 virus and we'd end up with 111 virus after day one. Now we have 111 virus. So pause the video and answer the same kind of question. With 111 virus and an 11% growth rate, what will the virus population be after two days? Resume the video when you're ready. With an assumption of a virus population of 111 and a growth rate of 11%, then there will be an additional 11% of 111 virus or about 12.21 virus that we add the next day for a total population of 111 after one day plus 12 virus the next day or 123 virus. And we're not going to count fractions of virus. Now this example illustrates an important idea that the rate of growth of the virus population depends on how many virus we have. The larger the population of virus, assuming an 11% growth rate, the faster the population is going to grow. In other words, as the population increases, so does the rate of growth of the population. And this should be expected. When you have a large population, there are more individuals to reproduce. And in general, when we're modeling population growth, we have to make the assumption that the rate of growth of the population is proportional to the population. To make this more clear from a mathematical perspective, we'll want some notation. So let's let p equal p of t be the population of the virus at time t. So p is a function of time. The differential equation that we're going to build to model this population growth is going to tell us how the population changes over time. So we're going to need a function that tells us how the population changes. Pause the video for a moment and write down such a function. And then resume the video when you're ready. Now we know a function that tells us the rate of change of a function. So the function that's going to tell us how the population changes is the derivative dp dt. So the function p of t equals some constant times e to the point 11 t has the property that the derivative of p of t is 0.11 times p. This means that our virus population can be modeled by an exponential function because p of t is an exponential function. If we start with an initial population of 100 virus, then notice that means that p of 0 is 100 where p of t is the population at time t. If we use our function p of t equals c times e to the point 11 t, then p of 0 is c times e to the 0, and if we evaluate the 0 as 1, so p of 0 is c, that makes c equal to 100. And so our model for our population growth of our virus is 100 e to the point 11 t at time t. Now if you evaluate this function p at 1, you get about 111. And if you evaluated it at 2, you get about 124. So this solution p actually does a pretty good job of approximating our earlier calculations. To summarize, as our example demonstrates, an exponential function y of t equals c e to the kt is a solution to a differential equation of the form dy dt equals k times y. And this means that any function that satisfies a differential equation of the form dy dt equals k times y is said to grow exponentially and grow if k is positive and decay if k is negative. That concludes our screencast on differential equations modeling exponential growth. We hope to see you back again soon.