# Solve the given information Let f(x) = sin x. Determine an error bound for the approxiamtion f(0.34) = sin 0.34. and compare it to the actual error.

Solve the given information Let $f\left(x\right)=\mathrm{sin}x.$ Determine an error bound for the approxiamtion $f\left(0.34\right)=\mathrm{sin}0.34.$ and compare it to the actual error.
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Jozlyn

We to find an error bound for the approximation and compare it to the actual error. The exact value of $\mathrm{sin}\left(0.34\right)=0.33348709214081.$ So, the actual error is $\left(0.33348709214081-0.33349\right)=2.91×{10}^{-6}$ The bound for the error on [0.30, 0.35] is given by, $|f\left(x\right)-{H}_{5}\left(x\right)|=|\frac{{f}^{6}\left(\xi \right)}{6!}{\left(x=0.30\right)}^{2}{\left(x-0.32\right)}^{2}{\left(x-0.35\right)}^{2}|=|\frac{-\mathrm{sin}\left(\xi \right)}{720}{\left(x-0.30\right)}^{2}{\left(x-0.32\right)}^{2}{\left(x-0.35\right)}^{2}|$ When $\xi \in \left[0.30,0.35\right].$ Evaluating this error term at x = 0.34 yields,

$|\mathrm{sin}\left(0.34\right)-{H}_{5}\left(0.34\right)|=|\frac{-sin\left(\xi \right)}{720}\left(0.04{\right)}^{2}\left(0.02{\right)}^{2}\left(-0.01{\right)}^{2}|$
$\le |\frac{-\mathrm{sin}\left(0.35\right)}{720}\left(0.04{\right)}^{2}\left(0.02{\right)}^{2}\left(-0.01{\right)}^{2}|$
$\le 3.05×{10}^{-14}$

This bound is not inconsistent with the actual error, because the approximation was computed using five-digit rounding arithmetic.

Jeffrey Jordon