18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 14
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PROFESSOR: What we're going to talk about today is a continuation of last time. I
want to review Newton's method because I want to talk to you about its accuracy.
So if you remember, the way Newton's method works is this. If you have a curve and
you want to know whether it crosses the axis. And you don't know where this point
is, this point which I'll call x here, what you do is you take a guess. Maybe you take
a point x0 here. And then you go down to this point on the graph. and you draw the
tangent line. I'll draw these in a couple of different colors so that you can see the
difference between them.
So here's a tangent line. It's coming out like that. And that one is going to get a little
closer to our target point. But now the trick is, and this is rather hard to see because
the scale gets small incredibly fast, is that if you go right up from that, and you do
this same trick over again. That is, this is your second guess, x1, and now you draw
the second tangent line. Which is going to come down this way. That's really close.
You can see here on the chalkboard, it's practically the same as the dot of x. So
that's the next guess. Which is x2. And I want to analyze, now, how close it gets.
And just describe to you how it works. So let me just remind you of the formulas,
too. It's worth having them in your head.
So the formula for the next one is this. And then the idea is just to repeat this
process. Which has a fancy name, which is in algorithms, which is to iterate, if you
like. So we repeat the process. And that means, for example, we generate x2 from
x1 by the same formula. And we did this last time. And, more generally, the n + 1st
is generated from the nth guess, by this formula here. So what I'd like to do is just
draw the picture of one step a little bit more closely. So I want to blow up the
picture, which is above me there. That's a little too high. Where are my erasers? Got
to get it a little lower than that, since I'm going to depict everything above the line
here. So here's my curve coming down. And suppose that x1 is here, so this is