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Problem 25 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.

$ \dfrac{x}{x^2 + 1} = \sqrt{1 - x} $

Answer

$x \approx 0.76682579$

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Video Transcript

Okay, here is the graph of the function, and I subtracted. And this we're route of one minus X right away. And then this is the graph. So we can pretty much see that what we're looking for answers right here. And it's approximately 0.8. So, using Newton's method, we're gonna hone in on that answer now to use the Newton's method. First we need are derivative of our function. So the derivative of our function of a prime of X is going to be, um So you have ex quick, let me write my function. And, um, thanks by my ex were last one, my news one minus x 1/2 hour. So for the first part, I have to use the questionable, which is f prime of G minus g prime of f divided by G square. So crime is one times g is ex cleared Last one minus g prime you two x times f, which is times excellent. Just make that's where All over G square. It's one we're and then I have to take the derivative of one minus extra 1/2 power. So I'm gonna end up with 1/2 and then when I subtract one, I'm gonna get the one minus X with negative 1/2 who is gonna go in my denominator. And then I have two times it. By the derivative of the inside, I have negative X, which would be negative one. So that would make this a positive. All right. So now of Newton's says that I need to do, um, um X to the end plus one. Yeah, equal next to the end minus and, uh, and over crime of n. And then we need to do our generations. So I'm gonna put my crying over my function. So my f crime when I combined X squared minus two X square, I get negative x squared. So I have That's my my function is X uh, thanks. Weird plus one minus. That's where route of one minus X over F crying, which is one minus X squared over. Thanks. Where? Plus one square. Yes, over too. Won't my ex So put all of that into your graphing calculator. And from the graph, we saw that our answer was approximately 0.8. So I'm gonna start at putting in. Except one equals 0.8 in my graphing calculator. for ask, and it returns to me. 0.7675 75 81 I put that in for X of three, and it returns to me. 0.766 82 610 I put that in for X and poor. These are all of approximates, of course. Um, 0.7668 to 5, 79 And when I put that in for exit five, it's the same thing. So that would be my answer. It's converging on that answer.