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.

Question
Polynomial arithmetic
asked 2020-12-03
Solve the given information Let \(\displaystyle{f{{\left({x}\right)}}}={\sin{{x}}}.\) Determine an error bound for the approxiamtion \(\displaystyle{f{{\left({0.34}\right)}}}={\sin{{0.34}}}.\) and compare it to the actual error.

Answers (1)

2020-12-04
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 \(\displaystyle{\left({0.33348709214081}-{0.33349}\right)}={2.91}\times{10}^{{-{6}}}\) The bound for the error on [0.30, 0.35] is given by, \(\displaystyle{\left|{f{{\left({x}\right)}}}-{H}_{{5}}{\left({x}\right)}\right|}={\left|{\frac{{{{f}^{{6}}{\left(\xi\right)}}}}{{{6}!}}}{\left({x}={0.30}\right)}^{{2}}{\left({x}-{0.32}\right)}^{{2}}{\left({x}-{0.35}\right)}^{{2}}\right|}{N}{S}{K}={\left|{\frac{{-{\sin{{\left(\xi\right)}}}}}{{{720}}}}{\left({x}-{0.30}\right)}^{{2}}{\left({x}-{0.32}\right)}^{{2}}{\left({x}-{0.35}\right)}^{{2}}\right|}\) When \(\displaystyle\xi\in{\left[{0.30},{0.35}\right]}.\) 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|
\leq |\frac{-sin(0.35)}{720}(0.04)^2 (0.02)^2 (-0.01)^2|
\leq 3.05 \times 10^{-14}ZSK This bound is not inconsistent with the actual error, because the approximation was computed using five-digit rounding arithmetic.
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