Problem

Use Newton's method to find all the roots of the …

Problem 18 Easy Difficulty

Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.
$x^{2}\left(4-x^{2}\right)=\frac{4}{x^{2}+1}$

Answer

$x=-1.9438254, \quad-0.84310821,1.9438254,0.84310821$

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Problem 9

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Problem 18

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Video Transcript

Yeah. In this problem we are asked to use Newton's wanted to find the roots of dysfunction. Um So let's remember what matters. Have you have just said that approximation? Uh to the roots of a function is echo. Two previous approximation minus the function evaluated that. The previous approximation divided by the derivative of this function relates that the previous approximation. So initially this is what we have. So solving for this is equal to solving forward, defining the function F. Where F X X square multiplied by four minus X squared minus for our expert plus one. And this should be equal to zero since we're sewing since we're searching for it roots. And from here, you see that at each step you need a previous step, right? So initially to start this procedure, you need initial guesses and to find those initial guess is uh we're gonna applaud this function F. So this is what function F uh looks like. And as you can see this curve intersects X. Accessing for points where those points are plus -1.9 or posting negative 0.8. And one thing you realize from this um Sure is that it is symmetric with respect to the y axis or whatever you have on the positive side, the same values will be here, but only with minus in phone. So instead of using for initial guesses will just use to So we're gonna start with .8 And we're gonna say that X two will be X one minus F at X one divided by F prime of F. Prime at X one. Now, as you can see we also need the there were 12 dysfunction. So let's find the derivative of f. prime of X. is HX -4. execute Plus eight eggs forward. Um Express plus one square. No 42nd guessed if you plug 0.8 F. N. F. In F. And F. Prime. You find the second guessed to be point 84287645 cents. The question since the problem wants us to get the result in a small place. No for the target. If you plug this number in F the next time you find Turkey has to be 0.8 31 0.84 3108- zero. And or four guys. Now you need to plug the third. Cast this number. This number indefinite prime. And you find the forecast to be .843108-0. Which is the same as the third guest. So it means that this is one of our roots now uh We also have 1.9 is our initial guess. Um So be Use this, apply the Newton's method. Same procedure. But this time plug 1.9 in FNF prime. You find the second guessed to be 1.94 689103. And now if you apply your second guest to find the third guess, you find the third guest to be 1.94 3 8. Um 3891. Now use the star guests to find the forecast and that will be 1.94-8. 2538. Yeah. Let's check. 45 50 guys. Now plug the forecast in F&F prime. You find fifth cast to be 1.94382538. So which is same as the forecast. It means that this is also one of our roots. So since this function is symmetric with respect to Y axis than all, truth will be plus minus 0.84 31 08 to zero and plus minus one point 943, 8- 5- eight. It wasn't possible.

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