(a) find the Maclaurin polynomial P_{3}(x) for f(x), (b) complete the following x: -0.75, -0.50, -0.25, 0, 0.25, 0.50, 0.75 for f(x) and P_{3}(x), and (c) sketch the graphs of f(x) and P_{3}(x) on the same set of coordinate axes. f(x) = arcsin x

Question
Polynomial graphs
asked 2020-12-15
(a) find the Maclaurin polynomial \(\displaystyle{P}_{{{3}}}{\left({x}\right)}\) for f(x), (b) complete the following \(\displaystyle{x}:-{0.75},-{0.50},-{0.25},{0},{0.25},{0.50},{0.75}{f}{\quad\text{or}\quad}{f{{\left({x}\right)}}}\) and \(\displaystyle{P}_{{{3}}}{\left({x}\right)}\), and (c) sketch the graphs of f(x) and \(\displaystyle{P}_{{{3}}}{\left({x}\right)}\) on the same set of coordinate axes. \(\displaystyle{f{{\left({x}\right)}}}={\arcsin{{x}}}\)

Answers (1)

2020-12-16
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 image
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