Find the linearization L(x) of the function at a. f(x)=\sqrt{x}, a=4

Dottie Parra 2021-09-09 Answered
Find the linearization L(x) of the function at a.
\(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}}},{a}={4}\)

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Expert Answer

brawnyN
Answered 2021-09-10 Author has 10121 answers
\(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}}}\)
\(\displaystyle{f{{\left({4}\right)}}}=\sqrt{{{4}}}={2}\)
\(\displaystyle{f}'{\left({x}\right)}={\frac{{{1}}}{{{2}\sqrt{{{x}}}}}}\)
\(\displaystyle{f}'{\left({4}\right)}={\frac{{{1}}}{{{2}\sqrt{{{4}}}}}}={\frac{{{1}}}{{{4}}}}\)
The linearization L(x) of f at a=4:
L(x)=f(a)+f'(a)(x-a)
\(\displaystyle{L}{\left({x}\right)}={2}+{\frac{{{1}}}{{{4}}}}{\left({x}-{4}\right)}\)
\(\displaystyle{L}{\left({x}\right)}={\frac{{{1}}}{{{4}}}}{x}+{1}\)
Results:
\(\displaystyle{L}{\left({x}\right)}={\frac{{{1}}}{{{4}}}}{x}+{1}\)
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