Successive approximations

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Find all root of ln(4-x^2)=x correct to 8 decimal places. List all successive approximations.

  • Successive approximations -

    by definition:
    4 - x^2 = e^x
    by making a rough sketch of
    y = 4-x^2 and y = e^x, I can see that there are roots at appr. 1 and -2

    setting up for Newton's Method:
    let y = 4 - x^2 - e^x
    y' = -2x - e^x
    His method says:

    newx = x - y/y' = x - (4-x^2 - e^x)/(-2x - e^x)
    = x + (4-x^2 - e^x)/(2x+e^x)
    so start with x = 1

    newx ---- x
    ???? ----- 1
    1.059707788 ---1.059707788
    1.058007813 --- 1.058007813
    1.058006401 --- 1.058006401
    1.058006401 YEahhhhh! correct to 9 decimal places

    start with x = -2

    ------ -1.9649813...
    -1.964635631 --- -1.964635631
    -1.964635597 --- -1.964635597
    -1.964635597 Yeah again,

    x = 1.058006401 or x = -1.964635597

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