sodni3

## Answered question

2020-12-03

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.

### Answer & Explanation

Jozlyn

Skilled2020-12-04Added 85 answers

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

Expert2021-12-16Added 2605 answers

Answer is given below (on video)

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?