Problem

Write the formula for Newton's method and use the…

05:01

Problem 9 Medium Difficulty

Write the formula for Newton's method and use the given initial approximation to compute the approximations $x_{1}$ and $x_{2}$.
$$f(x)=x^{2}-6 ; x_{0}=3$$

Answer

$x_{1}=2.5, x_{2}=2.45$

Related Courses

Calculus 1 / AB

Calculus Early Transcendentals

Chapter 4

Applications of the Derivative

Section 8

Newton's Method

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Video Transcript

all right. So new inspected tells us that given some initial value called X urban, we confined except M plus one that equals x m n minus the function evaluated at that point. So f of x a been divided by the derivative evaluated at that point. So accept M plus one equals Xa Been minus f of X urban divided by F prime of x a bit It's our function. Here is, um is F backs equal to X squared minus six. Okay, so we got if X is equal to X squared minus six. Okay, an initial point, our X not accept zero is going to be three. So basically, we could look at this and say, Well, I mean, if we set f of X equal to zero and Software X, whether X is going to be the square root of sex, right? Okay, So basically, what Newton's method can do is kind of how a calculator tells what exporters sixes is. We can basically get an approximation for the squared of sex. It's so starting with the initial value of three. Well, we're gonna need to evaluate the function at three and then evaluate the derivative at three. So we're gonna need to find first, while what's the derivative of our function? Our function is X squared. Money, sex, the derivative of Prime of X is just what? Well, we're not a twos. They've got to acts. And then the derivative of a constant of 60 So the derivative of F of X is just two X. Okay? And then you have to evaluate well, our function at at three. First, find what is exa been evaluated at our initial point of three. So you're playing three and we've got while three squared is 99 96 is three. So f of three is equal to three, right? Okay. And then what is what is f prime off? Um, ex urban or ex urban again is three. And F crime is two X With this plug, our initial value of three into our derivative and then two times three is six. So f prime of except not in this case, three f prime of three. It's just equal to two times three or six cam. So then how we find X of M plus one? This case are X of one, right? If we If X sub zero is three than X of one except one, it just equal to well except zero, which is three minus f of three. And he said, well, F of three was equal to three. So three minus three, divided by F prime of three. And of Primary three a six. So that's just three minus 3/6 or three minus 1/2 which is 2.5 so x of one this equal to two point five. OK, and there is 2.5 is except one and then were asked to find, except to So we just do an innovative process for taking except to is just equal to now, except one right, because in one breath except M plus one is equal to x ub n So except two is equal except one, which is 2.5. So except you getting used to this new, um, tablet and are hopefully his videos to get better soon gave it, except two is equal to accept one, which is 2.5 and then minus minus were f of x event. So we now take X event now is 2.5. So we go back to original function right f of X urban. So are functions of X. Is X squared minus six. So you plug in 2.5 back into our original function. So 2.5 squared is What is that? That 6.25 and then minus six. So that's, um, what 25 Divided by f prime of ex urban so that we take again. Our ex urban in this case is 2.5 to plug that into our derivative. And then, um, you evaluate this, we get 2.5 minus, um, 0.5 So we get we take f off 2.5 divided by F prime of 2.5. So 2.5 minus, um, point, your five is gonna give us once and give us two point 45 So our X sub to is going to be two point 45 So we see that do you go to point 45? So there is our except to and we're going way could We wanted to find exit three. Just take our except to then, and plug that into our door for my hair and take 2.45 minus F off 2.45 divided by a crime of 2.445 But where will we ask to find except one and accept too? So we're done. But we see that. I mean, every time we're gonna basically tracked off, um, less and less and getting closer and closer to the actual value. And you plug into a calculator with the square root of sex is equal. Teoh square root of sex is equal to 2.449489 something something. So 2.45 at the two iterations we see that were already, I mean fairly, fairly close.

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