a) \(\displaystyle{f{{\left({x}\right)}}}={\arcsin{{x}}}\)

\(\displaystyle{P}_{{{3}}}{\left({x}\right)}={\sum_{{{k}={0}}}^{{{3}}}}{\frac{{{{f}^{{{\left({k}\right)}}}{\left({0}\right)}}}}{{{k}!}}}{x}^{{{k}}}\)

\(\displaystyle{P}_{{{3}}}{\left({x}\right)}={x}+{\frac{{{x}^{{{3}}}}}{{{6}}}}\) \(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{x}&-{0.75}&-{0.50}&-{0.25}&-{0.0}&{0.25}&{0.50}&{0.75}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&-{0.848}&-{0.524}&-{0.253}&{0}&{0.253}&{0.524}&{0.848}\backslash{h}{l}\in{e}{P}_{{{3}}}{\left({x}\right)}&-{0.820}&-{0.521}&-{0.253}&{0}&{0.253}&{0.521}&{0.820}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\) On the graph below: Purple curve = \(\displaystyle{f{{\left({x}\right)}}}\) Orange curve = \(\displaystyle{P}_{{{3}}}{\left({x}\right)}\) Step 3

\(\displaystyle{P}_{{{3}}}{\left({x}\right)}={\sum_{{{k}={0}}}^{{{3}}}}{\frac{{{{f}^{{{\left({k}\right)}}}{\left({0}\right)}}}}{{{k}!}}}{x}^{{{k}}}\)

\(\displaystyle{P}_{{{3}}}{\left({x}\right)}={x}+{\frac{{{x}^{{{3}}}}}{{{6}}}}\) \(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{x}&-{0.75}&-{0.50}&-{0.25}&-{0.0}&{0.25}&{0.50}&{0.75}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&-{0.848}&-{0.524}&-{0.253}&{0}&{0.253}&{0.524}&{0.848}\backslash{h}{l}\in{e}{P}_{{{3}}}{\left({x}\right)}&-{0.820}&-{0.521}&-{0.253}&{0}&{0.253}&{0.521}&{0.820}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\) On the graph below: Purple curve = \(\displaystyle{f{{\left({x}\right)}}}\) Orange curve = \(\displaystyle{P}_{{{3}}}{\left({x}\right)}\) Step 3