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