Find the linearization L(x) of the function at a. f(x)=x^4-+3x^2, a=-1Z

Brennan Flores 2021-09-16 Answered
Find the linearization L(x) of the function at a. \(\displaystyle{f{{\left({x}\right)}}}={x}^{{4}}\mp{3}{x}^{{2}}\), \(\displaystyle{a}=-{1}\)

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Expert Answer

wheezym
Answered 2021-09-17 Author has 13491 answers

Linearization through tangent line approximation is achieved with equation:
\(\displaystyle{L}{\left({x}\right)}\approx{f{{\left({a}\right)}}}+{f}`{\left({a}\right)}\cdot{\left({x}-{a}\right)}\)
The derivative of f(x) is:
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{4}}}+{3}{x}^{{{2}}}\)
\(\displaystyle{f}`{\left({x}\right)}={\left({x}^{{{4}}}+{3}{x}^{{{2}}}\right)}`\)
\(\displaystyle={\left({x}^{{{4}}}\right)}`+{\left({3}{x}^{{{2}}}\right)}`\)
\(\displaystyle={4}{x}^{{{3}}}+{2}\cdot{3}{x}^{{{1}}}\)
\(\displaystyle={4}{x}^{{{3}}}+{6}{x}\)
And with \(\displaystyle{f{{\left({a}=-{1}\right)}}}={4}\) and \(f`(a=-1)=-10\) our linearization model is:
\(\displaystyle{L}{\left({x}\right)}={4}-{10}\cdot{\left({x}+{1}\right)}\)
\(\displaystyle={4}-{10}\cdot{x}-{10}\)
\(\displaystyle{L}{\left({x}\right)}=-{6}-{10}\cdot{x}\)

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