Linear and quadratic approximation. a. Find the linear approximating polynomial for the following functions centered at the given point a.

Khaleesi Herbert 2021-09-19 Answered
Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at a.
c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{\frac{{1}}{{3}}}},{a}={8}\) approximiate \(\displaystyle{7.5}^{{\frac{{1}}{{3}}}}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Margot Mill
Answered 2021-09-20 Author has 10015 answers
Given function:
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{\frac{{1}}{{3}}}},{a}={8}\)
a) The linear approximating polynomial for the functions is given by
\(\displaystyle{p}_{{1}}{\left({x}\right)}={f{{\left({a}\right)}}}+{f}'{\left({a}\right)}{\left({x}-{a}\right)}\)
\(\displaystyle{p}_{{1}}{\left({x}\right)}={8}^{{\frac{{1}}{{3}}}}+{\frac{{{1}}}{{{3}}}}{8}^{{-\frac{{2}}{{3}}}}{\left({x}-{a}\right)}\)
\(\displaystyle{p}_{{1}}{\left({x}\right)}={2}+{\frac{{{1}}}{{{2}}}}{\left({x}-{a}\right)}\)
b) The quadratic approximating polynomial for the function is given by
\(\displaystyle{p}_{{2}}{\left({x}\right)}={f{{\left({a}\right)}}}+{\frac{{{1}}}{{{2}}}}{f}{''}{\left({a}\right)}{\left({x}-{8}\right)}^{{2}}\)
\(\displaystyle{p}_{{2}}{\left({x}\right)}={2}+{\frac{{{1}}}{{{12}}}}{\left({x}-{8}\right)}+{\frac{{{1}}}{{{2}}}}\times{\frac{{-{2}}}{{{9}}}}\times{8}^{{-\frac{{5}}{{3}}}}{\left({x}-{8}\right)}^{{2}}\)
\(\displaystyle{p}_{{2}}{\left({x}\right)}={2}+{\frac{{{1}}}{{{12}}}}{\left({x}-{8}\right)}-{\frac{{{1}}}{{{288}}}}{\left({x}-{8}\right)}^{{2}}\)
c) The polynomials obtained in parts (a) and (b) to approximate the given quantity.
\(\displaystyle{p}_{{1}}{\left({x}\right)}={2}+{\frac{{{1}}}{{{12}}}}{\left({x}-{8}\right)}\)
Substitute the value of \(\displaystyle{x}={7.5}\)
\(\displaystyle{p}_{{1}}{\left({7.5}\right)}={2}+{\frac{{{1}}}{{{12}}}}{\left({7.5}-{8}\right)}\)
\(\displaystyle{p}_{{1}}{\left({7.5}\right)}={1.9583}\)
\(\displaystyle{p}_{{2}}{\left({x}\right)}={2}+{\frac{{{1}}}{{{12}}}}{\left({x}-{8}\right)}-{\frac{{{1}}}{{{288}}}}{\left({x}-{8}\right)}^{{2}}\)
Substitute the value of x=7.5
\(\displaystyle{p}_{{2}}{\left({7.5}\right)}={2}+{\frac{{{1}}}{{{12}}}}{\left({7.5}-{8}\right)}-{\frac{{{1}}}{{{288}}}}{\left({7.5}-{8}\right)}^{{2}}\)
\(\displaystyle{p}_{{2}}{\left({7.5}\right)}={1.95747}\)
Have a similar question?
Ask An Expert
0
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-09-11
Suppose you want to estimate \(\displaystyle\sqrt{{{26}}}\) using a fourth-order Taylor polynomial centered at \(\displaystyle{x}={a}\) for \(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}}}\). Choose an appro-priate value for the center a.
asked 2021-09-16
Consider the function \(\displaystyle{\int_{{0}}^{{1}}}{\left({1}+{x}\right)}{\ln{{\left({1}-{x}\right)}}}{\left.{d}{x}\right.}\)
Find the function’s Taylor polynomials of degrees 1, 2, and 3 centered at c = 0.
asked 2021-09-09
Taylor polynomials for \(\displaystyle{e}^{{x}}\)
a. Find the Taylor polynomials of order \(\displaystyle{n}={0},{1},{2},\) and 3 for \(\displaystyle{f{{\left({x}\right)}}}={e}^{{x}}\) centered at 0.
asked 2021-09-07
What are the taylor polynomials \(\displaystyle{p}_{{4}}\) and \(\displaystyle{p}_{{5}}\) centered at \(\displaystyle{\frac{{\pi}}{{{6}}}}\) for \(\displaystyle{f{{\left({x}\right)}}}={\cos{{\left({x}\right)}}}\)
asked 2021-09-03

Calculate the Taylor polynomials \(T2(x)\) and \(T3(x)\) centered at \(x=9\) for \(f(x)=\ln(x+1)\).

asked 2021-09-05

Calculate the Taylor polynomials \(\displaystyle{T}_{{2}}{\left({x}\right)}\) and \(\displaystyle{T}_{{3}}{\left({x}\right)}\) centered at \(x=9\) for \(\displaystyle{x}={9}\) for \(\displaystyle{f{{\left({x}\right)}}}={\ln{{\left({x}+{1}\right)}}}\)

asked 2021-09-16
Maclaurin and Taylor polynomials: Find third-order Maclaurin or Taylor polynomial for the given function about the indicated point.
\(\displaystyle{{\tan}^{{-{1}}}{x}},{x}_{{0}}={0}\)
...