Newton's Method to approximate the real root of f(x)=2x^7-1 to 3 decimal places.

LANavjeet

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Hi so I have been having some issues with Newton's Method. I understand what you have to do, however, I was solving this one equation, yet no matter what I did I could not arrive at the correct answer.

The question was to use Newton's method to approximate the real root of f(x)=2x^7-1 to 3 decimal places. I see x1=1 and when I do it I get 0.928571429. What I am supposed to get is 0.907343 as my x2. What do I do?

Also this one problem I cannot figure out at all. Use Newton's Method to find the value of squareroot7 until two consecutive terms of the sequence are within 0.0001. (Hint squareroot7 is a root of what function?) I'm very confused on this problem, any help is appreciated.

Edit I have fixed my error in the first problem, I was too stupid to not keep going. However, the second confuses me still, I've tried multiple methods to solve it but cannot come up with anything.
 
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Hi so I have been having some issues with Newton's Method. I understand what you have to do, however, I was solving this one equation, yet no matter what I did I could not arrive at the correct answer.

The question was to use Newton's method to approximate the real root of f(x)=2x^7-1 to 3 decimal places. I see x1=1 and when I do it I get 0.928571429. What I am supposed to get is 0.907343 as my x2. What do I do?

Also this one problem I cannot figure out at all. Use Newton's Method to find the value of squareroot7 until two consecutive terms of the sequence are within 0.0001. (Hint squareroot7 is a root of what function?) I'm very confused on this problem, any help is appreciated.
How do you know that? What is x2?

If x2 is 2nd approximation - then your calculation is correct.

After five iteration the answer will converge.

Hint for the second problem: What are the roots of y = x^2 - b ('b' is a real constant)?
 
Let us not forget that this method relies on a steep slope. If you are too near horizontal, you are out of luck.

Did you use your hint? \(\displaystyle \sqrt{7}\) is a solution to what equation? Use your best algebra.
 
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