November 13th, 2008, 10:47 AM  #1 
Newbie Joined: Nov 2008 Posts: 8 Thanks: 0  Newton's Method
I need help on a Newton's Method problem. I have to "Use Newton's method to approximate the given number correct to eight decimal places". The number is the hundredth root of 100 (100^(1/100)). When done on calculator, the result is 1.04712855. I've tried the approximation but am not able to get close. The method is Xn+1=Xn(f(n)/f'(n)) Thanks 
November 13th, 2008, 12:15 PM  #2  
Senior Member Joined: Jul 2008 Posts: 895 Thanks: 0  Re: Newton's Method Quote:
[You might want to multiply your calculator answer by 100?] First, what equation did you set up to use for the calculations? Do note also, from the Wicki: "Newton's method makes no guarantee on convergence, and depending on the shape of the function and the starting point it may or may not converge."  
November 13th, 2008, 12:26 PM  #3 
Newbie Joined: Nov 2008 Posts: 8 Thanks: 0  Re: Newton's Method
From the examples I've seen on the book, f(x) = x^(100)100 and f'(x) = 100x^99 I've tried using n=1, n=2, n=3, also tried big numbers and cannot seem to come up with the true approximation. 
November 13th, 2008, 12:50 PM  #4  
Global Moderator Joined: May 2007 Posts: 6,835 Thanks: 733  Re: Newton's Method Quote:
 
November 13th, 2008, 01:08 PM  #5 
Newbie Joined: Nov 2008 Posts: 8 Thanks: 0  Re: Newton's Method
For the initial guess I use the Intermediate Value Theorem. But in this case it isn't helping much. Someone else told me to to try 1.1 and I think it is the best choice in this case, even though I have to repeat more than 9 times to get the 8 decimal places.

November 13th, 2008, 01:24 PM  #6 
Senior Member Joined: Jul 2008 Posts: 895 Thanks: 0  Re: Newton's Method
Anyone know how to delete a message already sent? I am logged in and have cookies on.

November 15th, 2008, 12:32 PM  #7 
Newbie Joined: Oct 2008 Posts: 16 Thanks: 0  Re: Newton's Method
To find the first approximation, you can use the value from the calculator. If you do not want to "cheat", you can repeatedly take the square root of 100 9 times until you get 100^(1/512)=100^.001953125=1.009035, raising it to the fifth power gives you 100^(.009765625)= 1.045999, which is close enough for an initial "guess". 

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