Error Bounds

**Spudwad** · March 16th 2010, 08:05 AM

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.

**CaptainBlack** · March 16th 2010, 02:36 PM

Originally Posted by Spudwad

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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