What is Newton's Method?

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  • 0:06 Predicting the Path of…
  • 2:13 Newton's Method
  • 5:21 Linearization
  • 8:42 Lesson Summary
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Lesson Transcript
Instructor: Erin Lennon

Erin has taught math and science from grade school up to the post-graduate level. She holds a Ph.D. in Chemical Engineering.

Hang gliding can be perilous, especially if you think you might land in a stinky pigpen. In this lesson, find out how Newton's Method might help you determine whether you'll be covered in mud when you land.

Predicting the Path of a Hang Glider

Root finding: also known as 'what my foot does when I'm walking through a park.'

Graphing the hang glider height as a function of distance along the ground
Hang Glider Path Graph

I've always wanted to learn how to hang glide. It seems a little bit safer than parachuting, but I'm not exactly sure. Here's my big fear, other than heights: I'm hang gliding and oh, say there's some place I don't want to land but it looks like I'm going to land, like a pigpen. Now as I'm hang gliding, I'm trying to land, and I'm trying to determine whether or not I'm going to land smack-dab in the pigpen. So let's plot my height as a function of distance along the Earth.

I'm traveling along this line here. When I get, oh let's say 100 yards up, I look down, and I say, 'Oh no! On my current trajectory, I'm going to land in the pigpen!' How did I determine that? Well, I looked at how fast my height was changing as I was moving forward. Really, I looked at the slope of my height as a function of distance line. Unfortunately, right now it looks like I'm going to land in the pigpen, so I close my eyes for a few minutes. When I open them, I'm a lot farther that I thought I would be, and I haven't hit the ground yet. I thought I would hit right here in the middle of the pigpen, but now I'm that far over. I'm right above the spot that I thought I'd land in and still a good hundred feet up. So now I want to estimate where I'm going to land. I'm beyond the pigpen, but maybe I can figure out exactly how far over in this field I'm going to land.

Again, I take my current trajectory - that straight line that's a tangent of my height as a function of distance - and I extrapolate out. This says that I'm going to land just beyond the pigpen. That's much better. So I'm going to mark that right here. When I get there, I'm still not quite there, but I'm really close to the ground. Actually, I hit just a few feet beyond that. I've landed, and I'm not in the pigpen.

Newton's Method

So what did we do? Well, we actually used what, in math, is known as Newton's Method. We're going to use information about the derivatives - that is, my current trajectory - to find roots, where things go to zero.

In the case of my hang gliding, that's where my altitude gets to zero. So let's look at a mathematical example where we've got y as some function f(x). Now let's graph it. My goal here is to find where this function hits the x-axis. So I want to know what value of x will give me y=0. That is, I'm trying to solve 0=f(x).

Trying to solve 0=f(x)
Newtons Method Example Graph

Now, if I were a magician, I might be able to just come up with off the top of my head whatever value of x gives me that. But, in the real world, we don't know. So we're going to make a guess.

My initial guess is this x sub 0 spot. If I plug in x sub 0, I find out that y is way up here, nowhere close to zero. But that's okay; it's just an initial guess. I'm going to then calculate the derivative - that is my trajectory - at that point. So y sub 0 might be when I was behind the pigpen. I was calculating out my trajectory, and I was going to see how far over I was when I hit the ground. In this case, I'm going to use the derivative - that's my trajectory - and I'm going to extrapolate out where I'll be when y gets to zero. What value of x will give me y=0? Here I've got the derivative at x sub 0 is f(x sub 0). So that gives me another estimate - that gives me x sub 1, let's call it. At x sub 1, though, if I plug in x sub 1 to y=f(x), I still find that I'm not at y=0; I'm at some number greater than zero. In this case, I'm all the way up here. So I'm not quite there.

I'm going to repeat the process. I'm again going to look down and say, 'Am I going to land in the pigpen?' So I'm going to look at my trajectory. Where am I going to be when y gets to zero, when my altitude gets to zero? Okay, well, that's going to give me another guess. I'm going to call that x sub 2. Now, in this case, x sub 2 might actually be close enough to the actual value of x that will give me y=0. Let's say that if I plug in x sub 2 to f(x), I get 1 inch. Now if I'm hang gliding, 1 inch from the ground is close enough. I can put my foot down a little further to make that up. On the other hand, let's say that we have an ant hang gliding. That ant needs to get a lot closer than 1 inch. I mean, 1 inch is still many stories above the ground for an ant. So that might not be close enough for an ant, but it's close enough for me. I'm going to say that my root, where y=0, is at f(x sub 2) and call it done.

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