 Welcome back to our lecture series Math 1220, Calculus II for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Miseldine. In this lecture, we're going to be in section 9.4 of James Stewart's Calculus textbook. And I want to talk about models of growth. How we can use differential equations to make predictions about how a population grows over time. And here, the idea of a population can actually be sort of a loose definition. In the previous lecture, for example, we saw or we did an example of which we were looking at the amount of salt in a tank of water over time. And one could think of that function as a population, how many salt molecules are living inside the tank, so to speak. And as such, we can use differential equations to come up with functions to try to model how something grows or decays over time. And this is a very relevant type of question to ask ourselves. Like we saw in that salt example, like we saw in examples here. We can use differential equations to study how populations of people grow. Populations of bacteria or plants or animals grow over time. How populations of money, a.k.a. our bank account, how does that grow over time? We can use differential equations to try to model the growth or spread of rumors, trends, fads, infectious diseases amongst so many other things. Now, the way this is going to work is that a differential equation is going to be created by making assumptions about the growth rate. If we know how the quantity grows or spreads, which is a statement about its derivative, then we can then solve those differential equations to help us make some functions to model these things. Like, for example, in the year 2020 when the coronavirus was very prevalent spreading amongst everyone, right? There was a lot of mathematical models, statistical models released by doctors, statisticians, you name it, to talk about how the disease would spread in a certain state or in a certain country or around the world. Lots of them had some problems, some of them did pretty good. And what was the difference between them? Why are there so many different models to model the spread of the coronavirus? That is the effects of COVID-19. It really comes down to these models are based upon assumptions. No one has a crystal ball that they can rub to see the future. Instead, we have to make assumptions on how the virus will spread, how this population will grow, how this money will be invested. And based upon those assumptions, we can make predictions on how it's going to grow. So, for example, if you're a medical professional and you make the assumption that people will be hygienic, they'll wear a mask, they'll social distance, that will affect how the coronavirus spreads. And when they find out that no one wants to wash their hands, no one wants to wear a mask, no one wants to show some distance, people are going around coughing on each other at the store as a game, then you can't really blame the statisticians for being wrong with their models because they made the assumption that people would be wise and people didn't exactly use that wisdom. And so, these models of growth are going to be based upon assumptions. So, let me give you three examples in this lecture here. So, the first model for growth of a population, and when I say growth, of course, these can also be decay models. We could lose quantity over time. For example, if we were trying to model how much radioactive material we have for a specific isotope, perhaps it loses mass over time and thus decays. That's a possibility. But it could be growing like it's a bank account that's incurring interest over time anyways. The simplest of all growth models is based upon a very simple assumption. Assume that the rate at which a population P grows is proportional to the population's current size. Basically, what that means is the more of it you have, the faster it will grow. This kind of makes sense from an interest point of view, a finance point of view. The more money you have in your savings account, the bigger percentage of interest you get each month and so the more you have, the faster it grows. That's also true for most biological organisms. The more there is in the population, generally speaking, means that reproduction is more available for the culture and therefore it's easier for them to grow, the bigger they are. And so when we use the word rate here, remember, rate is describing derivatives. So when we talk about the rate of a population P, we're talking about its derivative P prime or DP over DT to be more specific, the change of population with respect to time. And so if we're saying that the derivative is proportional to the population, that becomes a differential equation of the following form. P prime over P equals K. The ratio between the derivative and the population is a constant, this constant growth factor K right here. And this would depend on the population here. Now, of course, if we times both sides of the equation by P, we end up with the equation here on the left, DP over DT is equal to KP. That is the derivative P prime is equal to KP. So in order to model this, and this is going to be a first order differential equation right here, right? In order to solve this equation here, we're looking for a function whose derivative is a multiple of the original function. Now, some things I should mention, of course, is that when K is positive, this would represent some type of growth model. When K is negative, this would represent some type of decay model. Things are getting smaller over time. Now, if we take the very special case that K equals 1, it turns out we've already solved this differential equation. We'd be looking for a function whose derivative is equal to the function itself. And Bob Jeronco, we know such a function, P would equal E to the T. That is, the natural exponential is a function which is equal to its own derivative. And how do you compensate for this factor of K? It turns out it's not too hard to do that. If we compensate correctly, we just have to take P to equal E to the KT. Because by the chain rule, the derivative of this will be K E to the KT. So you're going to get K times the original function. But by derivative rules as well, go back here a little bit, we could take any constant multiple of E to the KT. And its derivative would be K C E to the KT, which is just going to be K times P again. And so we see that the general solution to this very simple differential equation is going to be P equals C times E to the KT. That you see right here. And so as a model of population growth, well this right here is often referred to as the law of natural growth. The law of natural growth. Because it's quite natural that when things get bigger, they grow faster. Or sometimes this is referred to in uninhibited growth for reasons that we will see in the next example, uninhibited growth here. Now some other things I should mention of course is that, oh it says right down there at the bottom, the law of natural growth. Some other things to mention is the C value right here that we see in the formula, where does it come from? Well if we were to take this and plug in T equals zero, so the initial value P equals zero, this equals C times E times K times zero. K times zero of course will be zero, E to the zero is equal to one. And so we're going to see that C is equal to this initial value P naught. And so whatever our initial population is, we're going to get the population after T units of time will be the initial population times E to the KT. Where K is our growth factor and P naught is our initial population. We've seen this formula before with just biological growth that one studies in like a college algebra class, or with radioactive decay in a chemistry class, or continuously compounded interest in a finance class. This model we have, this is a very natural population growth model and it's pretty good. As long as we only focus on a short amount of time. And we'll talk about this more in the next example, but this model right here, P equals P naught E to the KT is a classic growth model. And I want to emphasize that this equation that we use is followed from this very simple assumption that if things grow proportional to their size, then this is how we model the growth.