1. ## Error Bounds

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

$\displaystyle f^{(2)}(x)=-sinx$

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

$\displaystyle \frac{-sin(1)} {(3!)}(1^3)$

Is this right? Thanks for the help.

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

$\displaystyle f^{(2)}(x)=-sinx$

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

$\displaystyle \frac{-sin(1)} {(3!)}(1^3)$

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 $\displaystyle \sin(x)$:

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

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

CB