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.

We to find an error bound for the approximation and compare it to the actual error. The exact value of ${\displaystyle \sin \left(0.34\right)=0.33348709214081.}$ So, the actual error is ${\displaystyle (0.33348709214081-0.33349)=2.91\times {10}^{-6}}$ The bound for the error on [0.30, 0.35] is given by, ${\displaystyle |f\left(x\right)-H_5\left(x\right)|=\left|\frac{f^6\left(\xi \right)}{6!}{(x=0.30)}^2{(x-0.32)}^2{(x-0.35)}^2\right|=\left|\frac{-\sin \left(\xi \right)}{720}{(x-0.30)}^2{(x-0.32)}^2{(x-0.35)}^2\right|}$ When ${\displaystyle \xi \in [0.30,0.35].}$ Evaluating this error term at x = 0.34 yields,

$|\sin (0.34)-H_5(0.34)|=|\frac{-sin(\xi )}{720}(0.04{)}^2(0.02{)}^2(-0.01{)}^2|$
$\le |\frac{-\sin (0.35)}{720}(0.04{)}^2(0.02{)}^2(-0.01{)}^2|$
 $\le 3.05\times {10}^{-14}$

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