use-newtons-method-with-the-specified-initial-approximation-x_1-to-find-x_3-the-third-approximatio-4

Question

Use Newton's method with the specified initial approximation $x_{1}$ to find $x_{3},$ the third approximation to the root of the given equation. (Give your answer to four decimal places.)
$x^{3}+2 x-4=0, \quad x_{1}=1$

Amrita Bhasin · Accepted Answer

this question asked us to use Newton&#x27;s method to solve. So first things first. Let&#x27;s take the derivative if f of X is x cubed plus two x minus for the derivative of X Cubed is three are exponent becomes on new coefficient. We subtract, exploited by one and then two acts. The derivative is simply too, Which means now, except to is gonna be one minus one cubed plus two times one minus four. Divide by three times one squared plus two, which is 1.2 and then exit three is gonna be 1.2 minus 1.2 cubed plus two times 1.2 minus four over three times 1.2 squared plus two, which is 1.1797

Doruk Isik · Answer

in this function. Where in this pub away Rusty&#x27;s Newton&#x27;s method to find the third approximation of the boot today function that where our issue dez is negative free. So let&#x27;s remember the intense method. It&#x27;s a practice is dark. The next up information could&#x27;ve rude ISS plus one, the previous of pollution minus the function relates that the previous point divided by their toe, functional. It&#x27;s at the service or waiting to find the derrick of functional, which is, um, express Plus. So the apartment, your guess is negative. Three. Then our next test of the extra is swollen. Miners have fallen over X prime one and this is negative. Three minus nine negative time bus then plus three divided by very ministry. And this gives us negative 2.75 Then the burger Pricks mission to read off dysfunction this text to minus that true over a crime of two. And this is the Linus native to form 75 minus. No one heard O negative 2.7 Fire Jew boasts one or 2.75 Spring plus three What were, um, negative 2.7 Fire squared minus 247 and he did this cognition. This gives us the third approximation to Group B negative. 2.7186

Amrita Bhasin · Answer

given off of exes. Jack&#x27;s Cubans tricks cripples to weaken. Differentiate of prime attacks is six X squared minus six bucks. So now we know that if axe is negative one then we know using the formula X The N plus one is excellent. My eyes off of xn over a promise x n We end up with negative 0.75 and then when x is negative 0.75 we have negative 0.68 25

Amrita Bhasin · Answer

the first thing you can do is differentiate the equation we&#x27;ve been given, which gives us negative to overact squared minus two acts. Okay, now that we&#x27;ve got that, we know that when acts is two, we have 1.55 because we&#x27;re using the formula Axe Marchioness to over acts minus X squared plus one divided by negative to over X squared minus two acts. Therefore, when X is 1.55 we have acts over here would be 1.5215

Nick Johnson · Answer

Okay, so we would like to find the root of X to the fifth minus X minus one is equal to zero. Okay. Find the value of X. That makes this true. Um, well, Nunes method right. Tells us that Well, X sub and plus one is equal to well, exit been minus f of exit. Ben, um, divided by the derivative F prime off. Excellent. Okay, so let&#x27;s start out by finding the derivative. So, um, what is f prime of X? So, given that f of X right is equal, chew Excellent. Fifth, minus X minus one. Therefore, F prime of X is just, um, down the 55 x to the fourth, um, minus one. Okay, so here&#x27;s how derivative Now we have that x of one is equal to one. And, um, I want to find, um well, f of x of one rights were given, except what we need f of except one and f prime of X of one. Okay, so, um, while giving the X of one is equal to one we have f of one is just equal to one, um minus one minus one is equal to minus one and asked prime off one. Well, you could 25 minus one, which is equal to four. Okay, so we plug in, um, Frexit, one fx of want and F prime Brexit born into our equation for Nunes Method That tells us that ex up to then is equal to just one minus. Um well, minus 1/4, which is equal to one plus 1/4 which is equal to five course. Okay, so, um, now that we know except to we must find f of except you and f prime of X obtuse. Except to is find force so f of X up to is f of five force, which is equal to, um, five fours to, um, perhaps minus five fours. Um, minus one, which is approximately equal to, um, 0.8 01 75 78 Okay. And then we&#x27;ll f prime a prime of except two, which is a crime of five. Force is equal. Chew Well, five times five. Force shoot 1/4 minus one. And that is approximately equal. Jew, um, 11.2 0703 13 Okay, so now, um, right to find, except to what we just plug in except to f of X of Jew and F prime of except you to find except three. So we&#x27;re trying to find Exit three. So except three in Newman&#x27;s method says that except three is equal to well, except two, which is five force, um, minus right, um, zero point eight And some stuff, right? Divided by 11.2. Right or seven. And some stuff. And, um, that is going to be approximately equal. Chu to four decimal places. 1.7, um, 1.171 point one seven 85 All right, There you go. Take care.

Amrita Bhasin · Answer

The first thing we&#x27;re gonna want to do is find the derivative of what we&#x27;ve been given. We&#x27;re just gonna be seven x the sex. Which means that when x one is negative one, the X two is negative. One minus negative. One to the seventh plus four over seven times negative 16 which gives us access to is negative 1.4 two, which gives us Exit three is negative. 1.42 minus negative, one point for two to the seventh post. Four over 70 native, one point for two to the sex and continuing onwards. We got negative 1.29 17

Problem

Use Newton's method with the specified initial ap…

Problem 21 Medium Difficulty

Use Newton's method with the specified initial approximation $x_{1}$ to find $x_{3},$ the third approximation to the root of the given equation. (Give your answer to four decimal places.)
$x^{3}+2 x-4=0, \quad x_{1}=1$

Answer

1.1797

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