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

$x_{1}=1.5, x_{2} \approx 1.296$

Calculus 1 / AB

Chapter 4

Applications of the Derivative

Section 8

Newton's Method

Derivatives

Differentiation

Campbell University

Harvey Mudd College

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all right. So the function were given here is X cube is F of X is equal to X cubed minus two. So in room here is obviously the cube root of two. Right. But Nunes Method says that if we have an basically initial guests at a root which were given were given an initial guests are ex, not is equal to two says that except M plus one is equal to x A been minus f of x A been divided by f prime of X a bit. So in this case, our initial x seven Rx zero is equal to two. So to find, except one well except one x Someone is equal to x urban in this case too minus f of two. So to cube is two times two times two, which is eight minus two sets to minus well, six divided by F prime of two. So what is a crime? We need to find a crime? Yes, So are derivative of prime of acts is going to be, well three X square case. We take f prime of ex urban in his case except Joe. It's two f prime of to is three times two square. So three times, four trough. So to minus six twelves or two minus 1/2 So except one, therefore, is one point five. All right, so what about except two or exit tube might become Well, let me more involved. But except to is equal to X sub one. Um, in this case you just found, except one at being 1.5. So except you is 1.5 minus f of X event. So 1.5 into our function and get 1.5 cubed minus two, which should evaluate to 1.375 So 1.5 minus 1.375 divided by well, f prime off X event. So in this case, we find, except tubes for X sub ends except one. When she found 1.5, we take 1.5, we plug it into our derivative function, so f prime off 1.5 is three times 1.5 squared. So that's evaluate to 6.75 Yeah, 1.5, minus this fraction here. And you worked this out. That should give you one point to nine six. But this if you Look, it's gonna keep on going 1.296 to 9 69 69 6 So we can put, um, a line here and say that BP forever. So they're one point 296 Repeating is therefore our X sub chief. So here's except one and except to, and we are done with the question.

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