字幕列表 影片播放 列印英文字幕 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses visit MIT OpenCourseWare at ocw.mit.edu. Okay so I'd like to begin the second lecture by reminding you what we did last time. So last time, we defined the derivative as the slope of a tangent line. So that was our geometric point of view and we also did a couple of computations. We worked out that the derivative of 1 / x was -1 / x^2. And we also computed the derivative of x to the nth power for n = 1, 2, etc., and that turned out to be x, I'm sorry, nx^(n-1). So that's what we did last time, and today I want to finish up with other points of view on what a derivative is. So this is extremely important, it's almost the most important thing I'll be saying in the class. But you'll have to think about it again when you start over and start using calculus in the real world. So again we're talking about what is a derivative and this is just a continuation of last time. So, as I said last time, we talked about geometric interpretations, and today what we're gonna talk about is rate of change as an interpretation of the derivative. So remember we drew graphs of functions, y = f(x) and we kept track of the change in x and here the change in y, let's say. And then from this new point of view a rate of change, keeping track of the rate of change of x and the rate of change of y, it's the relative rate of change we're interested in, and that's delta y / delta x and that has another interpretation. This is the average change. Usually we would think of that, if x were measuring time and so the average and that's when this becomes a rate, and the average is over the time interval delta x. And then the limiting value is denoted dy/dx and so this one is the average rate of change and this one is the instantaneous rate. Okay, so that's the point of view that I'd like to discuss now and give you just a couple of examples. So, let's see. Well, first of all, maybe some examples from physics here. So q is usually the name for a charge, and then dq/dt is what's known as current. So that's one physical example. A second example, which is probably the most tangible one, is we could denote the letter s by distance and then the rate of change is what we call speed. So those are the two typical examples and I just want to illustrate the second example in a little bit more detail because I think it's important to have some visceral sense of this notion of instantaneous speed. And I get to use the example of this very building to do that. Probably you know, or maybe you don't, that on Halloween there's an event that takes place in this building or really from the top of this building which is called the pumpkin drop. So let's illustrates this idea of rate of change with the pumpkin drop. So what happens is, this building-- well let's see here's the building, and here's the dot, that's the beautiful grass out on this side of the building, and then there's some people up here and very small objects, well they're not that small when you're close to them, that get dumped over the side there. And they fall down. You know everything at MIT or a lot of things at MIT are physics experiments. That's the pumpkin drop. So roughly speaking, the building is about 300 feet high, we're down here on the first usable floor. And so we're going to use instead of 300 feet, just for convenience purposes we'll use 80 meters because that makes the numbers come out simply. So we have the height which starts out at 80 meters at time 0 and then the acceleration due to gravity gives you this formula for h, this is the height. So at time t = 0, we're up at the top, h is 80 meters, the units here are meters. And at time t = 4 you notice, 5 * 4^2 is 80. I picked these numbers conveniently so that we're down at the bottom. Okay, so this notion of average change here, so the average change, or the average speed here, maybe we'll call it the average speed, since that's-- over this time that it takes for the pumpkin to drop is going to be the change in h divided by the change in t. Which starts out at, what does it start out as? It starts out as 80, right? And it ends at 0. So actually we have to do it backwards. We have to take 0 - 80 because the first value is the final position and the second value is the initial position. And that's divided by 4 - 0; times 4 seconds minus times 0 seconds. And so that of course is -20 meters per second. So the average speed of this guy is 20 meters a second. Now, so why did I pick this example? Because, of course, the average, although interesting, is not really what anybody cares about who actually goes to the event. All we really care about is the instantaneous speed when it hits the pavement and so that's can be calculated at the bottom. So what's the instantaneous speed? That's the derivative, or maybe to be consistent with the notation I've been using so far, that's d/dt of h. All right? So that's d/dt of h. Now remember we have formulas for these things. We can differentiate this function now. We did that yesterday. So we're gonna take the rate of change and if you take a look at it, it's just the rate of change of 80 is 0, minus the rate change for this -5t^2, that's minus 10t. So that's using the fact that d/dt of 80 is equal to 0 and d/dt of t^2 is equal to 2t. The special case... Well I'm cheating here, but there's a special case that's obvious. I didn't throw it in over here. The case n = 2 is that second case there. But the case n = 0 also works. Because that's constants. The derivative of a constant is 0. And then the factor n there's 0 and that's consistent. And actually if you look at the formula above it you'll see that it's the case of n = -1. So we'll get a larger pattern soon enough with the powers. Okay anyway. Back over here we have our rate of change and this is what it is. And at the bottom, at that point of impact, we have t = 4 and so h', which is the derivative, is equal to -40 meters per second. So twice as fast as the average speed here, and if you need to convert that, that's about 90 miles an hour. Which is why the police are there at midnight on Halloween to make sure you're all safe and also why when you come you have to be prepared to clean up afterwards. So anyway that's what happens, it's 90 miles an hour. It's actually the buildings a little taller, there's air resistance and I'm sure you can do a much more thorough study of this example. All right so now I want to give you a couple of more examples because time and these kinds of parameters and variables are not the only ones that are important for calculus. If it were only this kind of physics that was involved, then this would be a much more specialized subject than it is. And so I want to give you a couple of examples that don't involve time as a variable. So the third example I'll give here is-- The letter T often denotes temperature, <