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Math Help - Error Bounds

  1. #1
    Junior Member
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    Error Bounds

    I am supposed to find the error bound for sin(x) which we are to approximate with x on the interval of [-1,1]. Because x is the first degree Taylor Polynomial, would the upper bound be the second derivative of sin(x):

    f^{(2)}(x)=-sinx

    and then the maximum value on this interval for sin(x) is at 1, so the error bound would be:

    <br />
\frac{-sin(1)} {(3!)}(1^3)

    Is this right? Thanks for the help.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Spudwad View Post
    I am supposed to find the error bound for sin(x) which we are to approximate with x on the interval of [-1,1]. Because x is the first degree Taylor Polynomial, would the upper bound be the second derivative of sin(x):

    f^{(2)}(x)=-sinx

    and then the maximum value on this interval for sin(x) is at 1, so the error bound would be:

    <br />
\frac{-sin(1)} {(3!)}(1^3)

    Is this right? Thanks for the help.
    If you are not sure of what you are doing don't miss out steps, first write out the Taylor series for \sin(x):

    \sin(x) = \sin(0) + x \cos(0) - \frac{1}{2!} x^2 \sin(0) - \frac{1}{3!} x^3 \cos(0)+...=x-\frac{1}{3!}x^3+...

    and as the series is alternating the error is bound by the absolute value of the first neglected term.

    CB
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