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Math Help - [SOLVED] Finding zeropoint with Newton's method

  1. #1
    mr_pj
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    Exclamation [SOLVED] Finding zeropoint with Newton's method

    Explain why
    f(x) = ln x - 1/x - 1 , x>0
    have one and only one zero point
    Use Newton's method to find approximately value for the zero-point. The answer must state with 2 correct decimals.

    How do I solve this?
    Why?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by mr_pj
    Explain why
    f(x) = ln x - 1/x - 1 , x>0
    have one and only one zero point
    Use Newton's method to find approximately value for the zero-point. The answer must state with 2 correct decimals.

    How do I solve this?
    Why?
    It would help pitch answers at the correct level if
    you tell us what course/level you are studying.

    Look at the definition of f(x), it is the
    difference of two positive functions of x:

    ln(x), \mbox{ and }\ 1/x+1
    ,

    f(x) is zero when these two functions are
    equal. Now ln(x) is an increasing function and
     1/x+1 is a decreasing function of x.
    So it they are equal at some point, this is the only point at
    which they are ever equal. This is clear if you sketch the
    two functions (see attachment).

    Look at your notes to find what the Newton's Method
    is. You will find it gives you a new estimate of the
    root of a function in terms of the old estimate.

    You start with a guess, get a new estimate and feed
    that back, repeating the procedure until it has converged
    sufficiently for your purposes.

    In this case starting with a guess of 2 I get:

    >itert(10,2)


    n\ \ \ \ \ \ \ x_n
    0\ \ \ \ \ \ 2.000000
          1\ \ \ \ \ \      3.075804
          2\ \ \ \ \ \     3.543637
          3\ \ \ \ \ \     3.590737
         4\ \ \ \ \ \    3.591121
          5\ \ \ \ \ \      3.591121
          6\ \ \ \ \ \     3.591121
          7\ \ \ \ \ \     3.591121
          8\ \ \ \ \ \     3.591121
          9\ \ \ \ \ \     3.591121
         10\ \ \ \    3.591121
    >
    Attached Thumbnails Attached Thumbnails [SOLVED] Finding zeropoint with Newton's method-newton1.png  
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