write-the-formula-for-newtons-method-and-use-the-given-initial-approximation-to-compute-the-approx-4

Question

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

Nick Johnson · Accepted Answer

all right. So the function were given here is X cube is F of X is equal to X cubed minus two. So in room here is obviously the cube root of two. Right. But Nunes Method says that if we have an basically initial guests at a root which were given were given an initial guests are ex, not is equal to two says that except M plus one is equal to x A been minus f of x A been divided by f prime of X a bit. So in this case, our initial x seven Rx zero is equal to two. So to find, except one well except one x Someone is equal to x urban in this case too minus f of two. So to cube is two times two times two, which is eight minus two sets to minus well, six divided by F prime of two. So what is a crime? We need to find a crime? Yes, So are derivative of prime of acts is going to be, well three X square case. We take f prime of ex urban in his case except Joe. It&#x27;s two f prime of to is three times two square. So three times, four trough. So to minus six twelves or two minus 1/2 So except one, therefore, is one point five. All right, so what about except two or exit tube might become Well, let me more involved. But except to is equal to X sub one. Um, in this case you just found, except one at being 1.5. So except you is 1.5 minus f of X event. So 1.5 into our function and get 1.5 cubed minus two, which should evaluate to 1.375 So 1.5 minus 1.375 divided by well, f prime off X event. So in this case, we find, except tubes for X sub ends except one. When she found 1.5, we take 1.5, we plug it into our derivative function, so f prime off 1.5 is three times 1.5 squared. So that&#x27;s evaluate to 6.75 Yeah, 1.5, minus this fraction here. And you worked this out. That should give you one point to nine six. But this if you Look, it&#x27;s gonna keep on going 1.296 to 9 69 69 6 So we can put, um, a line here and say that BP forever. So they&#x27;re one point 296 Repeating is therefore our X sub chief. So here&#x27;s except one and except to, and we are done with the question.

Nick Johnson · Answer

All right, so here are function. F of X given to us is X squared minus two x minus three, and our initial value X not is equal to So what does Newton&#x27;s method say? Well, Nunes Method says that Xsa M plus one is equal to x of n minus f of x A been divided by f prime of x event. So to use that formula, we&#x27;re gonna have to find while the derivative off affects so f of X is X squared minus two x minus tree, then F prime vax. A crime of X is just equal to well X squared is just two X and the derivative of minus two. X is minus two. So and fruit of a constant zero salt are derivative is two x minus two and then, well, gonna find exit one. And except to so, x of one is just equal to except one X. Someone is equal to well except end in this case, except zero, which is to accept one is equal to two minus well f off to take to plug it in f of X, we get what two squared is four minus four us that zero minus streets that&#x27;s minus three. So we have two minus a minus three. So two minus a minus three divided by F prime of to so are derivative. Function is to X minus two. So we take our to implement plug it in there and we get two times two is four minus two is two. So we&#x27;ve got except one. You go two to minus a negative three halves or two plus three halves. Which would be, um, two plus 1.5 sets. 3.5, right? Yeah. So 3.5 there is our ex sub one And now rest defying exit too, except to So it&#x27;s x two or X. It, too, is just again, except M plus one is good, except and so except two is equal to X. Someone just found was 3.5. So, except to is equal to three point five minus what? We&#x27;re minus f of X event. So now our exit bend is, except one, which is 3.5. So we&#x27;ve got, um, except to is even a 3.5 minus f off 3.5. They 3.5. We plug it back into our again our original function after VAX, which is X squared minus two X minus three. So plug in 3.5. There he is, 3.5 square minus two times 3.5 minus three. And you work that out and you should get, I believe, 2.25 So 3.5 minus 2.25 divided by F prime of X event. So again are derivative. Function is to X minus two and you plug in 3.5. There you get 3.5, um, times two minus two, 3.3 times two is ah, seven writes that stuff minus two, which is five to get 3.5, divided by 2.25 Divided by five in this 2.2 5/5. Very weight. Teoh 0.45 3.5 minus 0.45 is equal to three points. 05 There you go. So are with our initial approximation of to, um we can see that, um, we get to it our better approximation with 3.5, and then we use that and plug that right back into our formula for a murder iterated former here. And we get 3.5 as our ex ah to

Sam Sohn · Answer

we need to find except one. And except to given that EPA backs is equal to X cubed minus two and accept zero is equal to two. We can see that the derivative of Apple Macs is want to be equal to three X squared. Now let&#x27;s write down the news formula, which says X of and plus one is equal to x seven minus f of exit pin, divided by F prime of X event so that someone there&#x27;s gonna be equal to two minus two cubed minus two is six and three times two squared is equal to 12. So that&#x27;s going to be two miners, 1/2 or 1.5. Now, exit too is gonna be 1.5 minus and 1.5 cubed minus two is exactly 11/8 and three times 1.5 squared is going exactly 27/4. So after some calculation, that gives us exactly 35/27 or approximately 1.296 So those are our answers, except one is exactly 1.5, and X itude is approximately 1.296

Sam Sohn · Answer

we want to use Newman&#x27;s method to find except one. And except to when fo Max is he going to X squared minus two eggs minus three and x of zero. There is equal to two. Now F Prime of X is going to be two X minus two. And let&#x27;s remind ourselves of munis formula, which says x of n plus one is equal to X of em minus f uh, except end divided by have prime of X. A band now except one is going to be except zero, which is two minus after two divided by Have primary too effort to is going to be negative three and have private to is going to be, too. So then we get to minus negative, free or two, which is two plus 3/2, which is 3.5 now. We can find exit to by plugging in that 3.5 so active two is equal to 3.5 minus F of 3.5, divided by F crime of 3.5. So then, for F 3.5, we get 3.5 squared, which is 12.25 minutes two times 3.5, which is seven minus 3 12.25 minus seven minus three is equal to 2.25 and then have prime of 3.5 is 3.5 times two minus two. So that&#x27;s seven minutes to or just five. So then we get this, and then the 2.25 divided by five is gonna be just 0.45 So we have 3.5, minus 0.45 or just three 0.5 So those are answers, except one is equal to 3.5 and accept to visit with the 3.5

John Gehad · Answer

put the number 11. We know that the formula for the Newton&#x27;s method is x n plus one equals xn minus f off xn over s dash of xn. In our case, we have f o X equals power Negative, X minus X and f Dash of wax equals negative Eve Our negative X minus one The four x one equals X note minus f of X note over f dash off X note. Have lateral X nor which means that X one equals Len off to minus. Keep our negative. Lim toh minus learned to over negative if our negative land to minus one before X one is approximately equal 0.5 six. Fourth and next to using the same equation is approximately because 0.64 this is not approximation but the actual equation. If our 0.56 four minus 40.564 over negative evil 25 64 minus one the fourth XTO with approximately equal 0.56 That&#x27;s you

Nick Johnson · Answer

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&#x27;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&#x27;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&#x27;s so starting with the initial value of three. Well, we&#x27;re gonna need to evaluate the function at three and then evaluate the derivative at three. So we&#x27;re gonna need to find first, while what&#x27;s the derivative of our function? Our function is X squared. Money, sex, the derivative of Prime of X is just what? Well, we&#x27;re not a twos. They&#x27;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&#x27;re playing three and we&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;re done. But we see that. I mean, every time we&#x27;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.

Problem

Finding roots with Newton's method For the given …

Problem 12 Medium Difficulty

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

Answer

$x_{1}=1.5, x_{2} \approx 1.296$

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$x_{1}=1.5, x_{2} \approx 1.296$

Calculus Early Transcendentals

Grace H.

Anna Marie V.

Kayleah T.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

William Briggs, Lyle Cochran, Bernard GilletCalculus Early Transcendentals

Grace H.

Anna Marie V.

Kayleah T.

Samuel H.

William Briggs, Lyle Cochran, Bernard Gillet

Calculus Early Transcendentals