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calc

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how big must n be so that the absolute value of the error in using the trapezoid rule is less than .00001?

for the integral (square root of (1+x^4))
where a=0 and b=1

  • calc -

    Oh dear me, I would have to get my numerical analysis book out of the attic for this error analysis.
    I guess I would suggest trying it with a few values of n and seeing how quickly it converges, unless you have a text with the error bound rules handy.

  • calc -

    OK, I found it online
    |E| /= (1/[12 n^2])(b-a)^3 df/dx^3
    use the maximum value of the second derivative of the function on the interval a to b, here 0 to 1

  • calc -

    typo repair
    |E| /= (1/[12 n^2])(b-a)^3 d^2f/dx^2

    f = (1+x^4)^.5
    df/dx = .5 (1+x^4)^-.5 (4 x^3)
    = 2 x^3(1+x^4)^-.5
    so
    d^2/dx^2 = 2 x^3 [ -.5)(4x^3)(1+x^4)^-1.5] + (1+x^4)^-.5 [ 6 x^2]

    find the maximum of that between x = 0 and x = 1
    and then your |
    n^2 = (1/|12 E|)1^3 times that maximum

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