Shadowing Practice: Big Picture: Derivatives - Learn English Speaking with Video
Ders oluşturuluyor...
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OK, hi.
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This is the second in my videos about the main ideas, the big picture of calculus.
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And this is an important one because I want to introduce and compute some derivatives.
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And you remember the overall situation is we have pairs of functions,
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distance and speed, Function one and function two, height of a graph, slope of the graph,
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height of a mountain, slope of a mountain.
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And it's the connection between those two that calculus is about.
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And so our problem today is you could imagine we have an airplane climbing.
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Its height is y as it covers a distance x.
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And its flight recorder will, well, probably it has two flight recorders.
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Let's suppose it has.
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Or your car has two recorders.
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One records the distance, the height, the total amount achieved up to that moment,
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up to that time t or that point x.
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The second recorder would tell you at every instant what the speed is.
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So it would tell you the speed at all times.
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Do you see the difference?
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The speed is like what's happening at an instant.
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The distance or the height y is the total accumulation of how far you've gone, how high you've gone.
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And now I'm going to suppose that this speed, this second function, the recorder is lost.
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But the information is there and how to recover it.
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So that's the question.
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How if I have a total record, say, of height?
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Let me, I'll stay mostly with y of x.
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I write these two so that you realize that letters are not what calculus is about.
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It's ideas.
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And here is the central idea.
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If I know the height as I go along, I know the height.
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It could go down.
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How can I recover from that height what the slope is at each point?
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So here's something rather important.
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That's the notation that Leibniz created, and it was a good idea for the derivative.
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And you'll see where it comes from.
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But somehow I'm dividing height distance up by distance across.
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And that ratio of up to across is a slope.
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So let me develop what we're doing.
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So one thing we can do and now will do is,
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for the great functions of calculus, a few very special, very important functions,
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we will actually figure out what the slope is.
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These are given by formulas, and I'll find a formula for the slope, dy dx equals.
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And I won't write it in yet, let me keep a little suspense,
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But this short list of the great functions is tremendously valuable.
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The process that we go through takes a little time, but once we do it, it's done.
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Once we write in the answer here, we know it.
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And the point is that other functions of science,
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of engineering, of economics, of life,
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come from these functions by multiplying.
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I could multiply that times that.
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And then I need a product rule, a rule for the derivative, the slope of a product.
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I could divide that by that.
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And so I need a quotient rule.
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I could do a chain, and you'll see that's maybe the best and most valuable, e to the sine x.
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So I'm putting e to the x together with sine x in a chain of functions, e to the sine of x.
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Then we need a chain rule.
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That's all coming.
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Let me start, well, let me even start by giving away the main facts for these three examples.
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Because there are three you want to remember and might as well start now.
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The x to the nth, so that's something, if n is positive, x to the nth climbs up.
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Let me draw a graph here of y equal x squared.
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Because that's one we'll work out in detail, y equal x squared.
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So this direction is x, this direction is y, and I want to know the slope,
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and the point is that that slope is changing as I go up.
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So the slope depends on x.
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The slope will be different here, so it's getting steeper and steeper.
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I'll figure out that slope.
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For this example, x squared when n is 2 and it's climbing.
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If n was minus 2, just because that's also on our list,
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n could be negative, the function would be dropping.
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You remember x to the minus 2, that negative exponent means divide by x squared.
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x to the minus 2 is 1 divided by x squared, and it'll be dropping.
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Here, so n could be positive or negative here.
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So I tell you the derivative.
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The derivative is easy to remember.
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It's that number n.
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It's another power of x, and that power is one less.
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One down.
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You lose one power.
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So I'm going to go through the steps here for n equal to 2.
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So I hope to get the answer 2 times x to the 2 minus 1 will just be 1, 2x.
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But what does the slope mean?
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That's what this lecture is really telling you.
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I'll tell you the answer for if it's sine x going in beautifully,
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the derivative of sine x is cos x, the cosine.
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The derivative of the sine curve is the cosine curve.
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You couldn't hope for more than that.
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And then we'll also at the same time find the derivative of the cosine curve, which is minus the sine curve.
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It turns out a minus sine comes in because the cosine curve drops at the start.
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OK.
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And would you like to know this one?
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e to the x, which I will introduce in a lecture coming very soon, because it's the function of calculus.
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And the reason it's so terrific is the connection between the function,
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whatever it is, whatever this number e is to whatever the x power means,
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the slope is the same as the function, e to the x.
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That's amazing.
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The slope, as the function changes, the slope changes, and they stay equal.
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OK, that's just, really my help is just to say, if you know those three, you're really off to a good start,
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plus the rules that you need.
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All right, now I'll tackle this particular one and say, what does slope mean?
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OK.
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So I'm given the recorder that I have.
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This is function one here.
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This is function one, the one I know.
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And I know it at every point.
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If I only had, like, the trip meter after an hour or two hours or three hours,
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well, calculus can't do the impossible.
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It can't know if I only knew the distance reached after each hour,
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I couldn't tell what it did over that hour, how often you had to brake, how often you accelerated.
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I could only find like an average speed over that hour.
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That would be easy.
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So averages don't need calculus.
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It's instant stuff.
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What happens at a moment?
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What is that speedometer reading at the moment, say, x equal 1?
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What is the slope?
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Yeah.
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Let me put in x equal 1 on this graph and x equal 2, say.
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And now x squared is going to be at height 1.
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If x is 1, then x squared is also 1.
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If x is 2, x squared would be 4.
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What's the average?
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Let me just come back one second to that average business.
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The average slope there would be in a distance across of 1, I went up by how much?
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3. I went up from 1 to 4.
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I have to do a subtraction.
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Differences, derivatives, that's the connection here.
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So it's 4 minus 1, that is 3.
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So I would say the average is 3 over 1. But that's not calculus.
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Calculus is looking at the instant thing.
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And let me begin at this instant, at that point.
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What does the slope look like to you at that point?
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At x equals 0, here's x equals 0, and here's y equals 0.
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We're very much 0.
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You see it's climbing, but at that moment,
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it's like it just started from a red light, whatever.
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The speed is 0 at that point.
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And I want to say the slope is 0.
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That's flat right there.
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That's flat.
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If I continued the curve, if I continued the x squared curve for x negative, it would be the same as for x positive.
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Whoa, it doesn't look very the same.
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Let me improve that.
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It would start up the same way.
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It would be completely symmetric.
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Everybody sees that at that zero position, the curve has hit bottom.
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Actually, this will be a major, major application of calculus.
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You identify the bottom of a curve by the fact that the slope is zero.
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It's not going up, it's not going down.
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It's zero at that point.
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But now what do I mean by slope at a point?
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OK.
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Here comes the new idea.
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If I go way over to one, that's too much.
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I just want to stay near this point.
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I'll go over a little bit.
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And I call that a little bit delta x.
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So that letter delta signals to our minds small.
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Some small, actually probably smaller than I drew out there.
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And then, so what's the average?
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I'd like to just find the average speed or average slope.
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If I go over by delta x, and then how high do I go up?
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Well, the curve is y equal x squared.
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So how high is this?
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So the average is up first divided by across.
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Across is our usual delta x.
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How far did it go up?
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Well, if our curve is x squared and I'm at the point delta x, then it's delta x squared.
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That's the average over the first piece,
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over the first piece of the curve.
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Out from here, out to delta x.
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OK.
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Now, again, that's still only an average because delta x might have been short.
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I want to push it to 0.
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That's where calculus comes in.
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Taking the limit of shorter and shorter and shorter pieces in order to zoom in on that instant,
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that moment, that spot where we're looking at the slope and where we're expecting the answer 0 in this case.
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And you see that the average in this happens to be especially simple.
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Delta x squared over delta x is just delta x.
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So the average slope is extremely small.
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And I'll just complete that thought as so the instant slope at 0,
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at x equals 0.
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I let this delta x get smaller and smaller.
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I get the answer 0, which is just what I expected.
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And you could say, well, not too exciting.
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But it was an easy one to do.
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It was the first time that we actually went through the steps of computing a, this is like a delta y.
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This is the delta x.
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Instead of 3 over 1, starting here I had delta x squared over delta x.
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That was easy to see, was delta x.
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And if I move in closer, that average slope is smaller and smaller, and the slope at that instant is zero.
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No problem.
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The travel, the climbing, began from rest, but it picked up speed.
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The slope here is certainly not zero.
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We'll find that slope.
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We need now to find the slope at every point.
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OK.
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So we've got, that's a good start.
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Now I'm ready to find the slope at any point.
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Instead of just x equals 0,
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which we've now done, I better draw a new graph of the same picture climbing up.
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Now I'm putting in a little climb.
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At some point x here, I'm up at a height x squared, and I'd like to know at that,
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I'm at that point on the current climb, I'd like to know the slope there at that point.
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How am I going to do it?
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I will follow this as the central dogma of calculus,
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of differential calculus, function one to function two.
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Take a little delta x.
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Go as far as x plus delta x.
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That will take you to a higher point on the curve.
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That's now the point x plus delta x squared, because our curve is still y equal x squared.
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We're not this nice, simple parabola.
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OK.
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So now I look at distance across and distance up.
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So delta y is the change up.
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Delta x is the across.
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And I have to put, what is delta y?
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I have to write in what is delta y.
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It's this distance up.
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It's x plus delta x squared.
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That's this height minus this height.
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So it's only, I'm not counting this bit, of course.
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It's that that I want.
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That's the delta y, is this piece.
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So it's up to this, subtract x squared.
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That's delta y.
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Dun-dun.
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That's important.
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Now I divide by delta x.
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This is all algebra now.
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Calculus is going to come in a moment, but not yet.
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For algebra, what do I do?
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I multiply this out.
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I see that thing squared.
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I remember that x is squared.
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And then I have this times this twice, 2x delta x's.
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And then I have delta x squared.
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And then I'm subtracting x squared.
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So that's delta y written out in full glory.
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I wrote it out because now I can simplify by canceling the x squared.
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I'm not surprised.
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Now, in this case, I can actually do the division.
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Delta x is just there, so it leaves me with a 2x.
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Delta x over delta x is 1.
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And then here's a delta x squared over a delta x, so that leaves me with 1 delta x.
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People who get, as you get the hang of calculus, you see the important things,
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the important things is like this first order, delta x to the first power.
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Delta x squared, that, when divided by delta x, gives us this, which is going to disappear.
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That's the point.
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This was the average over a short but still not instant range distance.
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Now what happens?
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Now dy dx.
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So if this is short,
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short over short, this is darn short over darn short.
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That's, that d is the, is, well, it's too short to see.
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So I don't actually now try to separate a distance dy.
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This isn't a true division.
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Because dy, because it's effectively 0 over 0.
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And you might say, well, 0 over 0, what's the meaning?
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Well, the meaning of 0 over 0 in this situation is I take the limit of this one,
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which does have a meaning.
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Because those are true numbers.
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They're little numbers, but they're numbers.
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And this was this.
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So now here's the big step, leaving algebra behind, going to calculus.
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In order to get what's happening at a point, I let delta x go to zero, and what is that?
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So delta y over delta x is this.
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What is dy dx?
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So in the limit, ah, it's not hard.
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Here's the 2x.
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It's there.
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Here's the delta x in the limit.
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It disappears.
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So the conclusion is that the derivative is 2x.
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OK.
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I should really, so that's function 2.
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That's function 2 here.
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That's the slope function.
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That's the speed function.
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Maybe I should draw it.
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Can I draw it above, and then I'll put the board back up?
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So here's a picture of function 2, the derivative, or the slope, which I was calling s.
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So that's the s function against x x is still the thing that's varying, or it could be t, or it could be whatever letter we've got.
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And we want to, and the answer was 2x for this function.
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So if I graph it, it starts at 0, and it climbs steadily with slope 2.
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So that's a graph of s of x.
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And for example, yeah, so take a couple of points on that graph.
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At x equals 0, the slope is 0, right?
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And we did that first.
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And we actually got it right.
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The slope is 0 at the start, at the bottom of the curve.
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At some other point on the curve, what's the slope here?
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Huh, yeah, tell me the slope there.
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At that point on the curve, an average slope was 3 over 1, but that was the slope of this,
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like, you know, sometimes called a chord.
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That's over a big jump of 1.
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Then I did it over a small jump of delta x, and then I let delta x go to 0, so it was an instant, infinitesimal jump.
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So the actual slope, the way to visualize it is that it's more like that.
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That's the line that's really giving the slope at that point.
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Do you see my best picture?
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It's not Rembrandt, but it's got it.
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And what is the slope at that point?
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Well, that's what our calculation was.
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It found the slope at that point and at the particular point x equal 1.
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The height was 2.
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The slope is 2.
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The actual tangent line is only there.
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You see it?
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It's up, up.
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Oh, wait a minute.
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Yeah, well, the slope is 2.
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I don't know.
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Yeah, maybe, yeah, it goes up to 3.
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It's not Rembrandt.
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But the math is OK.
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So what have we done?
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We've taken the first small step, and literally I could say small step almost a play on words because that's the point,
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this step is so small, to getting these great functions.
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Before I close this lecture, can I draw this pair, function one and function two.
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And just see that the movement of the curves is what we would expect.
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So let me just for one more good example, great example actually, is let me draw.
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Here goes x.
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I'm going to draw.
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In fact, maybe I already drew in the first lecture that bit out to 90 degrees.
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Only if we want a nice formula, we better call that pi over 2 radians.
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And here is a graph of sine x.
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This is y.
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This is function 1 sine x.
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And what's function 2?
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What can we see about function 2?
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Again, x.
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We see a slope.
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This is not the same as x squared.
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This starts with a definite slope.
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And it turns out this will be one of the most important limits we'll find.
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We'll discover that the first little delta x, which goes up by sine of delta x,
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has a slope that gets closer and closer to 1.
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Good.
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Luckily, cosine does start at 1.
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So we're OK so far.
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Now the slope is dropping.
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And what's the slope at the top of the sine curve?
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It's a maximum, but we identify that by the fact that the slope is 0,
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because we know the thing is going to go down here and go somewhere else.
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The slope there is zero.
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The tangent line is horizontal.
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And that is that point.
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It passes through zero.
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The slope is dropping.
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So this is the slope curve.
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And the great thing is that it's the cosine of x.
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And what I'm doing now is not proving this fact.
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I'm not doing my delta x's.
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That's the job I have to do once, and it won't be today, but I only have to do it once.
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And it will tell me, but today I'm just saying it makes sense.
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The slope is dropping.
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The slope, in that first part I'm going up, so the slope is positive, but the slope is dropping.
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OK, and then at this point it hits zero, and that's this point, and then the slope turns negative.
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I'm falling.
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So the slope goes negative, and actually it follows the cosine.
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So I go along here to that point,
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and then I can continue on to this point where it bottoms out again and then starts up.
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So where is that on this curve?
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Well, I better draw a little further out.
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This bottom here would be the, this is our pi over 2.
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This is our pi, 180 degrees, everybody would say.
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So what's happening on that curve?
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The function is dropping, and actually it's dropping its fastest.
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It's dropping its fastest at that point, which is the slope is minus 1.
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And then the slope is still negative, but it's not so negative, and it comes back up to 0 at 3 pi over 2.
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So this is the point 3 pi over 2.
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And this has come back to 0 at that point.
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And then it finishes the whole thing at 2 pi.
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This finishes up here back at 1 again.
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It's climbing.
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All right, climbing, dropping, faster,
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slower, maximum, minimum, those are the words that make derivatives important and useful to learn.
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And we've done in detail the first of our great list of functions.
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Okay, thanks.
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This has been a production of MIT OpenCourseWare and Gilbert Strang.
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Funding for this video was provided by the Lord Foundation.
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To help OCW continue to provide free and open access to MIT courses,
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please make a donation at ocw.mit.edu slash donate.
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