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Problem 2 Easy Difficulty

Follow the instructions for Exercise 1(a) but use $ x_1 = 1 $ as the starting approximation for finding the root $ r $.

Answer

$$
x_{2} \approx 3.4 \quad x_{3} \approx 2.6
$$

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Video Transcript

All right, let's go ahead and solve this problem. So Newton's method to approximate a route. So basically what the Newton methods do is this you're randomly first pick a point. For example, if I pick a point right here, the Y value is going to be located right there. Now, what I do is I draw a tangent line at that point and that is going to be my guest off the roof. Let's call it our one. Okay. Now, if that doesn't seem like it actually is the route I can actually continue now, what do I do? I actually plug in that point again. And now I get this height. Let's draw the tangent line going to look something like this. So here I can call it, are to. So as you can see, the second guess seems like it's going to be a better approximation to the actual word. Right now, this method seems like it worked really well for my choice of Let's call it are not, for example, I could have also done something like this. Now, this is a little bit artificial, but if I chose a point to be a root right here. Can you see that? This is the horizontal line right there. If I happen to choose a point where the tangent at the slope is going to be completely horizon tal, I will actually never reach a route anymore. So I won't be able to approximate anything. So that's one of the things about the Newton's method. If you don't get a good guess in the very first try, um, you might not actually get a really good estimation. However, I'm just going to still use this process in order to get to the solution or trying to answer the question that the book gives us. Okay, Now they are telling you what if my initial guess is one I'm going to call that X Not if X not is equal to one. I will start at this height Tangent line here. Looks like it's going to be something like this. So my guess is going to be when you actually draw that a little bit straight. It like this. Uh, There you go. That's a little better. Ah, still not the best drawing. But I can guess that the next point is going to be some around here that's avoiding the horizontal tangent, so it'll be pretty good. As you can see, it is going to be 123 maybe 3.2 or so. So that could be my first guess. And of course, my drawing is not really the most perfect. And there's no precise answer to this particular question because it's not like we're doing an analytical, um, calculation. So it's just all an estimate. Okay, so X one could be 3.2 Now. If I draw a tangent line from here, I can see that this is going to be my next guests x two. It's going to look like it's going to be about 1.8 or so if I want X three. Now I use this point. This one looks like it's gonna be a very good guess. So somewhere around there, maybe 2.4 or so let's call that X three. And when you plug in that value, you will see that the why is going to be very close to zero so you can decide to stop right there, or you can keep on moving and trying to find the next tangent line. So long story short. What you do is plug in the initial guests into the Y value, find the tangent line, and this point is going to be your guests for the for the route. And if you're not satisfied with that, you just continue on with the process. Okay, and that's the Newton's method.