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Numerade Educator

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Problem 28 Medium Difficulty

Use Newton's method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.

$ \ln (x^2 + 2) = \dfrac{3x}{\sqrt{x^2 + 1}} $

Answer

$x=0.24852414,4.0501098$

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Video Transcript

28. Well, given the function natural, Juan X squared plus two and I'm gonna see track three X Over this We're of X squared plus one equals zero. So I subtracted it from both sides. There you are My If prime is going to equal two x over X squared plus two because the derivative of natural obvious one of the acts So we have one over X square plus two. And I have the chain of the potion rule would give me three a year x weird less one. My, uh, graph gives me an approximation of thanks equals approximately 0.2 and X equals approximately or so I run those federation X one equals 10.2 or exit too equals approximately points. Two were seven, 33 161 except three is approximately 0.248 52 333 Except for who? For eight I two for one, which is the same as X five. So we stopped there. Let me write my, uh, thanks and plus one exit N minus. Yeah. L and ex were plus minus three x with where we have X. Where one that's my frying to x X where us? Too minus three over expired. That's one. The hardest part is getting all of that in your graphing calculator with all the parentheses. So here we have Green X. That one is approximately four when I plug for in as my next generation except two is approximately 4.0 or 993 for 12 When I plug that in as my ex value you, I get your point zero by he wrote one. He wrote nine e three and except for is approximately is approximately homer 0.5 0109 Aye, for your museum closed. That is Dex. And free. And that's the same except five. So we stopped there.