# 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?

First, construct the sixth degree Taylor polynomial ${P}_{6}\left(x\right)$ for function $f\left(x\right)=\mathrm{sin}\left({x}^{2}\right)$ about ${x}_{0}=0$ The use ${\int }_{0}^{1}{P}_{6}\left(x\right)dx$ to approximate the integral Use 4-digit rounding arithmetic in all calculations. What is the approximate value?
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Clara Reese

Taylor series $f\left(x\right)=\mathrm{sin}\left({x}^{2}\right)$ is $=f\left({x}_{0}\right)+\frac{\left(x-{x}_{0}\right)}{1!}{f}^{\prime }\left({x}_{0}\right)+\frac{{\left(x-{x}_{0}\right)}^{2}}{2!}f\left({x}_{0}\right)±---$ Note that ${x}_{0}=0$ So, the polynomial is
$f\left(0\right)+x{f}^{\prime }\left(0\right)+\frac{{x}^{2}}{2}f"\left(0\right)$
$+\frac{{x}^{3}}{6}{f}^{\prime }\left(0\right)+\frac{{x}^{4}}{24}{f}^{\prime }\left(0\right)$
$+\frac{{x}^{5}}{120}{f}^{\prime }\left(0\right)$
$+\frac{{x}^{6}}{720}\cdot {f}^{6}\left(0\right)+--$ (up to sixth degree) Now $f\left(x\right)=\mathrm{sin}\left({x}^{2}\right)$
${f}^{\prime }\left(x\right)=\mathrm{cos}\left({x}^{2}\right)\cdot 2x$
$fx\right)=2\left[{\mathrm{cos}x}^{2}+x\cdot \mathrm{sin}\left({x}^{2}\right)\left(-2x\right)\right]$
$=2\left[{\mathrm{cos}x}^{2}-2{x}^{2}{\mathrm{sin}x}^{2}\right]$
$f\left(x\right)=2\left(-\mathrm{sin}\left({x}^{2}\right)\cdot 2x\right)$
$-4\left[{x}^{2}{\mathrm{cos}x}^{2}\cdot 2x+{\mathrm{sin}x}^{2}\cdot 2x\right]$
$=-4x\mathrm{sin}\left({x}^{2}\right)-8{x}^{3}{\mathrm{cos}x}^{2}-8x{\mathrm{sin}x}^{2}$
$=-12x{\mathrm{sin}x}^{2}-8{x}^{3}{\mathrm{cos}x}^{2}$
${f}^{4}\left(x\right)=\frac{d}{dx}\left(f\left(x\right)\right)$

Jeffrey Jordon