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

$$x_{1}=3.5, x_{2}=3.05$$

Calculus 1 / AB

Chapter 4

Applications of the Derivative

Section 8

Newton's Method

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we want to use Newman'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'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'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

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