Find the third-order Taylor polynomial for f centered at 2 and use this polynomial to estimate f(1.9).

tearstreakdl
2021-12-18
Answered

Suppose f(2) = 1, f'(2) = 1, f"(2) = 0, and f(3)(2) = 12.

Find the third-order Taylor polynomial for f centered at 2 and use this polynomial to estimate f(1.9).

Find the third-order Taylor polynomial for f centered at 2 and use this polynomial to estimate f(1.9).

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Debbie Moore

Answered 2021-12-19
Author has **43** answers

Step 1:Concept

Taylor polynomial is

$f\left(x\right)=f\left(a\right)+{f}^{\prime}\left(a\right)(x-a)+\frac{f{}^{\u2033}\left(a\right){(x-a)}^{2}}{2!}+\frac{f{}^{\u2034}\left(a\right){(x-a)}^{3}}{3!}+\dots$

Step 2:Solution

Plugging the values at a=2

$f\left(x\right)=f\left(2\right)+{f}^{\prime}\left(2\right)(x-2)+\frac{f{}^{\u2033}\left(2\right){(x-2)}^{2}}{2!}+\frac{f{}^{\u2034}\left(2\right){(x-2)}^{3}}{3!}$

$=1+1(x-2)+\frac{0{(x-2)}^{2}}{2}+\frac{12{(x-2)}^{3}}{3!}$

$=1+x-2+2{(x-2)}^{3}$ At x=1.9

$=1+1.9-2+2{(1.9-2)}^{3}f\left(1.9\right)=0.898$

Taylor polynomial is

Step 2:Solution

Plugging the values at a=2

eskalopit

Answered 2021-12-20
Author has **31** answers

Given,

$f\left(2\right)=1,{f}^{\prime}\left(2\right)=1,f{}^{\u2033}\left(2\right)=0,{f}^{3}\left(2\right)=12$

$f\left(n\right)=f\left(a\right)+{f}^{\prime}\left(a\right)(n-a)+f{}^{\u2033}\left(a\right)\frac{{(n-a)}^{2}}{2!}+f{}^{\u2034}\left(a\right)\frac{{(n-a)}^{3}}{3!}$

$f\left(n\right)=f\left(2\right)+{f}^{\prime}\left(2\right)(n-2)+f{}^{\u2033}\left(2\right)\frac{{(n-2)}^{2}}{2!}+f{}^{\u2034}\left(2\right)\frac{{(n-a)}^{3}}{3!}$

$=1+1(n-2)+\left(0\right)\frac{{(n-2)}^{2}}{2}+12\frac{{(n-2)}^{3}}{6}$

$=1+n-2+0+2{(n-2)}^{3}$

$f\left(n\right)=2{(n-2)}^{3}+n-1$

$f\left(1.9\right)=2{(1.9-2)}^{3}+1.9-1$

$=2{(-0.1)}^{3}+0\cdot 9=-2\left(0.001\right)+0.9$

=0.9-0.002

=0.878

$\therefore f\left(1.9\right)=0.878$

=0.9-0.002

=0.878

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