Defining average and instantaneous rates of change at a point
Newton, Leibniz, and Usain Bolt
This is a picture of Isaac Newton, super famous British mathematician and physicist. This is a picture of a Gottfried Leibnitz, super famous, or maybe not as famous, but maybe should be, famous German philosopher and mathematician, and he was a contemporary of Isaac Newton. These two gentlemen together were really the founding fathers of calculus. And they did some of their-- most of their major work in the late 1600s. And this right over here is Usain Bolt, Jamaican sprinter, whose continuing to do some of his best work in 2012. And as of early 2012, he's the fastest human alive, and he's probably the fastest human that has ever lived. And you might have not made the association with these three gentleman. You might not think that they have a lot in common. But they were all obsessed with the same fundamental question. And this is the same fundamental question that differential calculus addresses. And the question is, what is the instantaneous rate of change of something? And in the case of Usain Bolt, how fast is he going right now? Not just what his average speed was for the last second, or his average speed over the next 10 seconds. How fast is he going right now? And so this is what differential calculus is all about. Instantaneous rates of change. Differential calculus. Newton's actual original term for differential calculus was the method of fluxions, which actually sounds a little bit fancier. But it's all about what's happening in this instant. And to think about why that is not a super easy problem to address with traditional algebra, let's draw a little graph here. So on this axis I'll have distance. I'll say y is equal to distance. I could have said d is equal to distance, but we'll see, especially later on in calculus, d is reserved for something else. We'll say y is equal to distance. And in this axis, we'll say time. And I could say t is equal to time, but I'll just say x is equal to time. And so if we were to plot Usain Bolt's distance as a function of time, well at time zero he hasn't gone anywhere. He is right over there. And we know that this gentleman is capable of traveling 100 meters in 9.58 seconds. So after 9.58 seconds, we'll assume that this is in seconds right over here, he's capable of going 100 meters. And so using this information, we can actually figure out his average speed. Let me write it this way, his average speed is just going to be his change in distance over his change in time. And using the variables that are over here, we're saying y is distance. So this is the same thing as change in y over change in x from this point to that point. And this might look somewhat familiar to you from basic algebra. This is the slope between these two points. If I have a line that connects these two points, this is the slope of that line. The change in distance is this right over here. Change in y is equal to 100 meters. And our change in time is this right over here. So our change in time is equal to 9.58 seconds. We started at 0, we go to 9.58 seconds. Another way to think about it, the rise over the run you might have heard in your algebra class. It's going to be 100 meters over 9.58 seconds. So this is 100 meters over 9.58 seconds. And the slope is essentially just rate of change, or you could view it as the average rate of change between these two points. And you'll see, if you even just follow the units, it gives you units of speed here. It would be velocity if we also specified the direction. And we can figure out what that is, let me get the calculator out. So let me get the calculator on the screen. So we're going 100 meters in the 9.58 seconds. So it's 10.4, I'll just write 10.4, I'll round to 10.4. So it's approximately 10.4, and then the units are meters per second. And that is his average speed. And what we're going to see in a second is how average speed is different than instantaneous speed. How it's different than what the speed he might be going at any given moment. And just to have a concept of how fast this is, let me get the calculator back. This is in meters per second. If you wanted to know how many meters he's going in an hour, well there's 3,600 seconds in an hour. So he'll be able to go this many meters 3,600 times. So that's how many meters he can, if he were able to somehow keep up that speed in an hour. This is how fast he's going meters per hour. And then, if you were to say how many miles per hour, there's roughly 1600-- and I don't know the exact number, but roughly 1600 meters per mile. So let's divide it by 1600. And so you see that this is roughly a little over 23, about 23 and 1/2 miles per hour. So this is approximately, and I'll write it this way-- this is approximately 23.5 miles per hour. And relative to a car, not so fast. But relative to me, extremely fast. Now to see how this is different than instantaneous velocity, let's think about a potential plot of his distance relative to time. He's not going to just go this speed immediately. He's not just going to go as soon as the gun fires, he's not just going to go 23 and 1/2 miles per hour all the way. He's going to accelerate. So at first he's going to start off going a little bit slower. So the slope is going to be a little bit lot lower than the average slope. He's going to go a little bit slower, then he's going to start accelerating. And so his speed, and you'll see the slope here is getting steeper and steeper and steeper. And then maybe near the end he starts tiring off a little bit. And so his distance plotted against time might be a curve that looks something like this. And what we calculated here is just the average slope across this change in time. What we could see at any given moment the slope is actually different. In the beginning, he has a slower rate of change of distance. Then over here, then he accelerates over here, it seems like his rate of change of distance, which would be roughly-- or you could view it as the slope of the tangent line at that point, it looks higher than his average. And then he starts to slow down again. When you average it out, it gets to 23 and 1/2 miles per hour. And I looked it up, Usain Bolt's instantaneous velocity, his peak instantaneous velocity, is actually closer to 30 miles per hour. So the slope over here might be 23 whatever miles per hour. But the instantaneous, his fastest point in this 9.58 seconds is closer to 30 miles per hour. But you see it's not a trivial thing to do. You could say, OK, let me try to approximate the slope right over here. And you could do that by saying, OK, well, what is the change in y over the change of x right around this? So you could say, well, let me take some change of x, and figure out what the change of y is around it, or as we go past that. So you get that. But that would just be an approximation, because you see that the slope of this curve is constantly changing. So what you want to do is see what happens as your change of x gets smaller and smaller and smaller. As your change of x get smaller and smaller and smaller, you're going to get a better and better approximation. Your change of y is going to get smaller and smaller and smaller. So what you want to do, and we're going to go into depth into all of this, and study it more rigorously, is you want to take the limit as delta x approaches 0 of your change in y over your change in x. And when you do that, you're going to approach that instantaneous rate of change. You could view it as the instantaneous slope at that point in the curve. Or the slope of the tangent line at that point in the curve. Or if we use calculus terminology, we would view that as the derivative. So the instantaneous slope is the derivative. And the notation we use for the derivative is a dy over dx. And that's why I reserved the letter y. And then you say, well, how does this relate to the word differential? Well, the word differential is relating-- this dy is a differential, dx is a differential. And one way to conceptualize it, this is an infinitely small change in y over an infinitely small change in x. And by getting super, super small changes in y over change in x, you're able to get your instantaneous slope. Or in the case of this example, the instantaneous speed of Usain Bolt right at that moment. And notice, you can't just put a 0 here. If you just put change in x is zero, you're going to get something that's undefined. You can't divide by 0. So we take the limit as it approaches 0. And we'll define that more rigorously in the next few videos.
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