 The revenue realized by a small town from the collection of fines associated to like parking tickets and other traffic violations is given by the following function you can see on the screen right here. R of N equals 8,000 N over N plus two where in this situation, N is the number of work hours each day that can be devoted to the parking patrol. So if there's no actual like a meter made or traffic cop or what have you going around checking the parking meters and other things, you're not gonna get any tickets if you don't have someone looking around. So we have to dedicate a certain number of work hours to calculate that. So if N is the number of work hours, then R of N will be the revenue the city collects by those work, by those work hours there. Well, suppose an outbreak of flu epidemic is spreading throughout this town. And so while usually there are 30 work hours dedicated to parking duty, parking patrol for this small town, because of the epidemic of the flu that's spreading around this town, it turns out that the officers who normally would be able to be looking for these tickets are getting sick. They're not able to come into duty. And so we see that there is a, while we started off with 30 work hours a day, they're decreasing at a rate of six work hours per day. And so then it's begs the question, how fast is revenue from the parking fees decreasing at the outbreak of the epidemic? All right, so this question is something we're gonna talk about a lot more in the future, but when you see things like how fast, when you hear a word fast, that cues in our mind, something like speed or better yet velocity. We've talked about velocity before in this lecture series. What is velocity? Velocity is the change of position over time or better yet, it's the derivative of position. So when you ask questions about how fast or how quickly something's happening, that is a question about a rate of change, AKA it's a question about a derivative. How fast, when you ask someone how fast, you're asking about a derivative. Also words like decreasing very much suggests to us that we're talking about a rate of change. Rate of change are derivatives, derivatives are instantaneous rate of change. How fast is the revenue decreasing when the outbreak starts out? Which I can say something about that. That's when we've had T equals zero. We're here, T equals the number of days you know, of this epidemic going on in our town. So if we're asking for what's the, how fast is revenue decreasing at the start, then we're asking about this T here. But wait a second, what's T? There's no T on our side of our function right here. We have R of N, N is the number of work hours. So can we somehow derive information on revenue with respect to time? So this is the important thing to focus on this problem right here. What they're asking for, what they're asking for is the following. We wanna find the change of revenue with respect to time at the moment that time equals zero. That is what's being asked for this problem right here. But how are we gonna figure that out when revenue is a function of N? This is where the chain rule comes into play here because the change of revenue with respect to time is equal to the change of revenue with respect to N, the work hours, times DN over DT. The chain rule tells you that basically you can cancel these things out or in this case it allows us to factor, I could take the derivative of R of N with respect to N and I could take the derivative of N with respect to T which I already know what that is. The derivative of N with respect to T right here is this six work hours per day. So we know that DN, DT, it's gonna be decreasing six work hours per day. So the change of work hours with respect to time is gonna be negative six. So we can plug that in right here, okay? Well then we still have to do this DRDT business, DRDT. This is gonna equal the derivative of revenue with respect to N, which the revenue function you can still see on the top of the screen, just barely 8,000 N over N plus two, squint to see it, we get that. And then the change of the number of work hours with respect to time, that was the negative six that we have right there. So we have to take the derivative of R of N with respect to N, which by the usual derivative rules we can use the quotient rule in this example right here. So we're gonna get low, D high, which the derivative with respect to N of 8,000 N would just be 8,000, minus high, D low, which take the derivative of the bottom, it's gonna go to one, square the bottom, here we go. And then also make sure you don't forget this factor of negative six that's floating around right here. So this is gonna give us our DRDT. This is the general function. Now we don't necessarily want the general function, we want it when it is, remember, we need to know what the change is at zero. So when we're at zero, so we can just plug in T equals zero, but wait a second, where there's no T's to plug in T equals zero. Oh, okay, well that's because, like we observed earlier, at the outbreak of this epidemic, there were 30 work hours. So if T equals zero, that means N equals 30. We could plug 30 into this formula, that's great there. And so let's try that. We're going to get, why don't we just get the negative six in front so I don't lose track of it anymore. So we're gonna get 30 plus two times that by 8,000. And then we're gonna subtract from that 8,000 times 30. And this all sits above the 30 plus two quantity squared. Also to make it like a little bit easier, I noticed that the numerator, there's a common factor of 8,000, so let's factor it out. We're gonna get negative six times 8,000, like so. And then what's left behind, you have a 32 minus a 30, this sits above 32 squared, which of course 30 take away from 32 gives you a two, that's easy enough, 8,000 times two, and then 32 squared is gonna be 1024, 1024 right there. So which of course, two goes into, we can simplify this a little bit. So two goes into 1024, 512 times, and we can keep on going with this. I'm just gonna put this in a calculator. Don't worry about the arithmetic too much, but we're gonna get a negative six. If we take 8,000 by 512, we end up with 15.625, just pause there for a moment because I want us to understand what is this measuring. So the way that our function is factored, remember this was our DRDN when N was equal to 30. So notice this right here, so this number right here is the change of revenue with respect to work hours. So as the work hours are dropping, this tells you how the revenue is dropping, but we didn't ask how does revenue change as work hours drop. We wanna see how does revenue decrease with respect to time. So as this epidemic goes longer and longer and longer, how's that affecting things? So that's what this number over here was, right? This was our DNDT. So as the work hours change over time and the revenue changes with respect to work hours, their product will give you how the revenue changes with respect to time. 15.625 times that by negative six, that'll give us negative 93.75, which we then can round that to be the city loses about $94 per day at the start of the epidemic. Of course that changes at different times of the epidemic, but at the outbreak, the revenue is decreasing at a rate of $94 per day. And so this example here, using the chain rule in this story problem setting, teaches us a very important lesson. And that lesson I hope that you learn here is that the derivative is always with respect to some variable, right? And you need to pay close attention to which variable is changing here, is R changing with the number of work hours, is R changing with respect to time. Those are related, but not the same thing. Is the work hours changing with respect to time? The variables matter. And so when we talk about the rates of change, you need to make sure you're paying attention to which variables are changing with respect to each other. If you can take care of that, you're gonna be just fine.