Question

First, construct the sixth degree Taylor polynomial P_6(x) for function f(x) = sin(x^2) about x_0 = 0 The use int_{0}^{1} P_6 (x) dx to approximate the integral int_{0}^{1} sin(x^2)dx. Use 4-digit rounding arithmetic in all calculations. What is the approximate value?

Polynomial arithmetic
ANSWERED
asked 2021-02-06
First, construct the sixth degree Taylor polynomial \(\displaystyle{P}_{{6}}{\left({x}\right)}\) for function \(\displaystyle{f{{\left({x}\right)}}}={\sin{{\left({x}^{{2}}\right)}}}\) about \(\displaystyle{x}_{{0}}={0}\) The use \(\displaystyle{\int_{{{0}}}^{{{1}}}}{P}_{{6}}{\left({x}\right)}{\left.{d}{x}\right.}\) to approximate the integral \(\displaystyle\ {\int_{{{0}}}^{{{1}}}}{\sin{{\left({x}^{{2}}\right)}}}{\left.{d}{x}\right.}.\) Use 4-digit rounding arithmetic in all calculations. What is the approximate value?

Answers (1)

2021-02-07

Taylor series \(\displaystyle{f{{\left({x}\right)}}}={\sin{{\left({x}^{{2}}\right)}}}\) is \(\displaystyle={f{{\left({x}_{{0}}\right)}}}+{\frac{{{\left({x}-{x}_{{0}}\right)}}}{{{1}!}}}{f}'{\left({x}_{{0}}\right)}+{\frac{{{\left({x}-{x}_{{0}}\right)}^{{2}}}}{{{2}!}}}{f}\text{}{\left({x}_{{0}}\right)}\pm---\) Note that \(\displaystyle{x}_{{0}}={0}\) So, the polynomial is
\(f(0) + x f'(0) + \frac{x^2}{2} f"(0)\)
\(+\frac{x^{3}}{6}f'(0)+\frac{x^{4}}{24}f'(0)\)
\(\displaystyle+{\frac{{{x}^{{5}}}}{{{120}}}}{f}\text{}'{\left({0}\right)}\)
\(\displaystyle+{\frac{{{x}^{{6}}}}{{{720}}}}\cdot{{f}^{{6}}{\left({0}\right)}}+--\) (up to sixth degree) Now \(\displaystyle{f{{\left({x}\right)}}}={\sin{{\left({x}^{{2}}\right)}}}\)
\(\displaystyle{f}'{\left({x}\right)}={\cos{{\left({x}^{{2}}\right)}}}\cdot{2}{x}\)
\(\displaystyle{f}\text{}{x}{)}={2}{\left[{{\cos{{x}}}^{{2}}+}{x}\cdot{\sin{{\left({x}^{{2}}\right)}}}{\left(-{2}{x}\right)}\right]}\)
\(\displaystyle={2}{\left[{{\cos{{x}}}^{{2}}-}{2}{x}^{{2}}{\sin{{x}}}^{{2}}\right]}\)
\(\displaystyle{f}\text{}{\left({x}\right)}={2}{\left(-{\sin{{\left({x}^{{2}}\right)}}}\cdot{2}{x}\right)}\)
\(\displaystyle-{4}{\left[{x}^{{2}}{{\cos{{x}}}^{{2}}\cdot}{2}{x}+{{\sin{{x}}}^{{2}}\cdot}{2}{x}\right]}\)
\(\displaystyle=-{4}{x}{\sin{{\left({x}^{{2}}\right)}}}-{8}{x}^{{3}}{{\cos{{x}}}^{{2}}-}{8}{x}{{\sin{{x}}}^{{2}}}\)
\(\displaystyle=-{12}{x}{{\sin{{x}}}^{{2}}-}{8}{x}^{{3}}{{\cos{{x}}}^{{2}}}\)
\(\displaystyle{{f}^{{4}}{\left({x}\right)}}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({f}\text{}{\left({x}\right)}\right)}\)
\(\displaystyle=-{12}{\left({{\sin{{x}}}^{{2}}+}{x}\cdot{{\cos{{x}}}^{{2}}\ }\cdot{2}{x}\right)}-{8}{\left({{\cos{{x}}}^{{2}}\cdot}{3}{x}^{{2}}-{x}^{{3}}\cdot{{\sin{{x}}}^{{2}}\cdot}{2}{x}\right)}\)
\(\displaystyle=-{12}{{\sin{{x}}}^{{2}}-}{24}{x}^{{2}}{{\cos{{x}}}^{{2}}-}{24}{x}^{{2}}{{\cos{{x}}}^{{2}}+}{16}{x}^{{4}}{{\sin{{x}}}^{{2}}}\)
\(\displaystyle-{12}{{\sin{{x}}}^{{2}}-}{48}{x}^{{2}}{{\cos{{x}}}^{{2}}+}{16}{x}^{{4}}{{\sin{{x}}}^{{2}}}\)
\(\displaystyle{{f}^{{5}}{\left({x}\right)}}=-{12}{{\cos{{x}}}^{{2}}\cdot}{2}{x}-{48}{\left({{\cos{{x}}}^{{2}}\cdot}{2}{x}-{x}^{{2}}\cdot{{\sin{{x}}}^{{2}}\cdot}{2}{x}\right)}+{16}{\left({4}{x}^{{3}}{{\sin{{x}}}^{{2}}+}{x}^{{4}}\cdot{{\cos{{x}}}^{{2}}\cdot}{2}{x}\right)}\)
\(\displaystyle=-{24}{x}{{\cos{{x}}}^{{2}}-}{96}{x}{{\cos{{x}}}^{{2}}+}{96}{x}^{{3}}{{\sin{{x}}}^{{2}}+}{64}{x}^{{3}}{{\sin{{x}}}^{{2}}+}{32}{x}^{{5}}{{\cos{{x}}}^{{2}}}\)
\(\displaystyle=-{120}{x}{{\cos{{x}}}^{{2}}+}{160}{x}^{{3}}{{\sin{{x}}}^{{2}}+}{32}{x}^{{5}}{{\cos{{x}}}^{{2}}}\)
\(\displaystyle{{f}^{{6}}{\left({x}\right)}}=-{120}{\left({x}-{{\sin{{x}}}^{{2}}\cdot}{2}{x}+{\cos{{x}}}^{{2}}\right)}+{160}{\left({3}{x}^{{2}}{{\sin{{x}}}^{{2}}+}{x}^{{3}}\cdot{{\cos{{x}}}^{{2}}\cdot}{2}{x}\right)}+{32}{\left({5}{x}^{{4}}{{\cos{{x}}}^{{2}}-}{x}^{{5}}{{\sin{{x}}}^{{2}}\cdot}{2}{x}\right)}\)
\(f(0)=0\)

\(f'(0)=0f(0)=2\cdot1=2f'(0)=0\)

\(f(0)=0\)

\(f^{5}(0)=0\)

\(f^{6}(0)=-120\)
\(\displaystyle\therefore{{\sin{{x}}}^{{2}}\sim}{e}{q}{0}+{\frac{{{x}^{{2}}}}{{{2}!}}}\cdot{2}+{0}+{\frac{{{\left(-{120}\right)}{x}^{{6}}}}{{{720}}}}={x}^{{2}}-{\frac{{{x}^{{6}}}}{{{6}}}}\)
\(\displaystyle\therefore{\int_{{{0}}}^{{{1}}}}{{\sin{{x}}}^{{2}}{\left.{d}{x}\right.}}\approx{\int_{{{0}}}^{{{1}}}}{x}^{{2}}{\left.{d}{x}\right.}-{\frac{{{1}}}{{{6}}}}{\int_{{{0}}}^{{{1}}}}{x}^{{6}}{\left.{d}{x}\right.}={{\left[{\frac{{{x}^{{3}}}}{{{3}}}}\right]}_{{{0}}}^{{{1}}}}-{\frac{{{1}}}{{{6}}}}\cdot{{\left[{\frac{{{x}^{{7}}}}{{{7}}}}\right]}_{{{0}}}^{{{1}}}}\)
\(=\frac{1}{3}=\frac{1}{42}\)

\(\frac{14-1}{42}\)

\(\frac{13}{42}\)

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