 Welcome back to our lecture series Math 1210, Calculus I for students at Southern Intel University. As usual, I'll be a professor today, Dr. Andrew Misaline. What I want to do in the subsequent videos for Chapter 3 here, so 27 and the next couple, is focus really on why we're so interested in computing derivatives in the first place. It's great that we can compute them effectively, but what's the whole point of derivative calculations in the first place? We saw a little bit of this back in Chapter 2, when we first were exposed to the derivative. We saw that the derivative can be interpreted as the slope of a tangent line, and how this has to do with physical problems about motion, velocity, and acceleration. Let me remind you of a little bit of notation, just to make sure we're all on the same page right now. Imagine we have some type of function, y equals f of x. At the moment, I don't want to attach any interpretation to the function. At the moment, it's just an algebraic function, y equals f of x. When we start attaching interpretations to the function, that'll then also attach interpretations to the derivative. We'll do that in just a second. If our function y equals f of x is given, then remember the derivative d y over d x, which we sometimes abbreviate that as f prime of x. The derivative d y over d x, that should be interpreted as the rate of change of the quantity y with respect to the quantity x. As x is changing, how does that relate to the change of y? The derivative measures this rate of change. We're going to see in this video and other videos for lecture 27, is how does this interpretation of the derivative relate to things like, say, in physics, or chemistry, or economics, or some of the other sciences that one would see. I often like to think of calculus very much in the same way as the princess Rapunzel, her experience in the movie Tangled, because for her, she was trapped in a tower her whole life, and when she finally escaped, went to the kingdom, she found this insignia of the royal family, it was this son, and it looks, I could try to draw it, but it looks something like, this is some little star insignia, right? It had some great significance to her, but she didn't know why it was. She didn't yet know that she was the lost princess or anything like that. And so she sees this star, this son, excuse me, and she realizes it's important, but she doesn't know why it's important. Then when she returns to her tower, she looks around her tower and, my goodness, that son is everywhere. She sort of subconsciously had been painting this son over and over and over again. The thing that the, the signia was there the whole time, she just didn't realize the significance of it until she returned to her home. Calculus, in particular in this video, the derivative has a very similar take, right? The derivative is everywhere in the sciences. We just might not realize it until we start looking for it. And so lecture 27 really wants to give us that opportunity to look for the derivative in the sciences. So continuing with our review of notation here, if we talk about a change of X, that means we have two different measurements of X. There's some X1 over here, some X2 over there. The change of X would then be their difference. X2 minus X1 would be the change. How are they different? And that could be a positive change or a negative change. It would be a positive change if you increase from X1 to X2. It would be a negative change if you decrease from X1 to X2. We can do the same thing for Y, where the change of Y, delta Y here, remember that triangle represents the Greek letter delta, d-d-d-d-d-d-difference is what it's measuring there. Delta Y would be the change of the Y coordinates. And since Y is given by this function relationship Y equals F of X, then the Y coordinate attached to X1 would be F of X1. And the Y coordinate attached to X2 would be F of X2. So delta X represents a change of the horizontal, what we often refer to as the run. And delta Y represents a change of the Y coordinate we often refer to as the rise. So when you look at the quotient delta Y over delta X, this has the formula F of X2 minus F of X1, that's the delta Y, over X2 minus X1, that's the delta X. And so you put that together, this looks like your rise over run, which is the typical mnemonic device for remembering the slope. And that's because the average rate of change, that's what this formula is, the delta Y over delta X, the average rate of change, how is Y on average changing as X changes on the interval X1 to X2? This measures the slope of a secant line. Now, we're very much interested, not, I mean, average rate of change is important, but more important is the idea of instantaneous rate of change. How is this quantity changing at this moment of time? Why is it changing, right? How quickly is it changing? How rapidly is it changing? Well, so to compute the instantaneous rate of change, we need to consider what happens, take the limit as the change of X goes to zero. So as the X numbers are closer and closer to closer, almost touching, how does that affect the change? So if you take the limit as delta X approaches zero of delta Y over delta X, then this forms what we call the derivative, the DY over DX. So whenever the function Y equals F of X has a specific interpretation in one of the sciences, and this could be physics, chemistry, biology, whatever, it's derivative that is DY over DX, will have a specific interpretation as the rate of change of that function value. For example, we have talked a lot in this lecture series about a very specific example from physics. In physics, we have already studied the so-called velocity problem. So if S equals F of T, this is the motion function, the motion function of some particle or object, so it's moving, so F of T keeps track of where it's located at any given moment of time. We've seen already that the derivative of position, the derivative of motion with respect to time, this delta S, excuse me, S prime of T, this is gonna equal the so-called velocity function. Velocity being the rate in which position is changing over time. Velocity speed is the first derivative of this function. It's also valuable to consider the second derivative, why is there two S's there, the second derivative, S double prime of T, the so-called acceleration function. These derivatives with acceleration being the rate in which velocity is changing with time, these derivatives have important interpretations with respect to the original question about motion. And in fact, Sir Isaac Newton, one of the founders of our modern notion of calculus, essentially discovered the notion, well, I should say he discovered the fundamental theory of calculus, which is something we'll talk about later on in this, studying these problems with motion. So he's concerned with things like motion, velocity, acceleration, force, a lot of these physical functions that are related to each other by calculus. And so we wanna explore these in this section. And so we'll see some of these in subsequent videos, but just some other topics I wanna throw out there that we're not gonna do concrete examples on. But I just want to be aware that rates of change occur all over in the sciences, physical sciences, life sciences, financial sciences, computer sciences, they're all there, right? For example, a geologist is interested in knowing the rate in which an intruded body of molten rock cools by conduction of heat into surrounding rock. Sounds kinda complicated, right? But it involves calculus. An engineer wants to know the rate in which water flows into or out of a reservoir, the rate in which the water is rising or falling, that's a derivative. Maybe an urban geographer is interested in the rate of change of the population density in a city as the distance from the city center increases, right? So you have some city, right? The far that you get away from downtown area, how does the population density change, right? So think of the variables in play here. You have density, which is a variable, but also distance from the center, which is also a variable. How are those things related to each other? Well, rates of change measure exactly that, it's derivatives. A meteorologist might be concerned with the rate of change of astronomical pressure with respect to height. If you're on planet Earth, I know this horrible drawing now turns into a plant. It's no longer a picture of a city, it's a picture of the Earth, right? As you go higher into the altitude, the pressure changes, right? Is it a linear change? Is it a quadratic change? Maybe it's a logistic, you know, who knows? There's lots of different ways of describing that potentially. A meteorologist would be very interested in as you climb up the atmosphere, how does pressure change with respect to altitude? But these are all examples coming from physical science. I guess the urban geographer, I guess that's more of like a civil science. They're social science. So yeah, let's talk about social science. In psychology, psychologists might be interested in learning theory. That is the so-called learning curve, which if you were to graph it, like how does learning increase with training time? So if you have like your graph here, maybe some initial training has like a big explosion in learning, but eventually there might be a point where, you know, there might be this point where increased time dedicated to learning might not have really great benefits whatsoever. So yeah, that might be encouraged by this. You know, that point would be discovered using the derivative. In sociology, differential calculus is used in analyzing the spread of rumors or fads or fashions. The same thing can also be used for in biology, like the spread of infectious diseases. The rate at which a disease, you know, like the coronavirus is spreading is a derivative of that function with respect to time. How quickly is it spreading? And so this is just an illustration of a few examples that show that as a fact, calculus is more than just some abstract thing. Calculus has real ramifications to everyday life, to everyday science. And so a single abstract mathematical concept such as the derivative can have different interpretations in each of these different scientific settings. And so when we develop the properties of the mathematical concept once and for all, like with the derivative, like the power rule, the product rule, these can be used in every interpretation of the derivative. We can then turn around and apply these results to all of those sciences. Everything we learn about, about derivatives affects what we know about psychology, affects what we know about physics and chemistry and geology and medicine. Just again, to name a few examples, economics included. And for this reason why it's worth diverting away from some of the more mathematical abstract notions of the derivative and focusing in the next couple of lecture videos on specific concrete examples of derivatives in the sciences.