Suppose you want to estimate \sqrt{26} using a fourth-order Taylor polynomial centered at x=a for f(x)=\sqrt{x}

Reggie

Reggie

Answered question

2021-09-11

Suppose you want to estimate 26 using a fourth-order Taylor polynomial centered at x=a for f(x)=x. Choose an appro-priate value for the center a.

Answer & Explanation

smallq9

smallq9

Skilled2021-09-12Added 106 answers

Given:
The Taylor polynomial f(x)=x centered at x=a
Estimate 26
The n-th order Taylor polynomials centered at a is:
pn=f(a)+f(a)(xa)+f(a)2!(xa)2+...+fn(x)n!(xa)n
Let f(x)=x
Now, we can choose a near to 26 which is a=25
Substitute a=25 in the n-th order Taylor polynomials for fourth-order,
p4=f(25)+f(25)(x25)+f(25)2!(x25)2+f(25)3!(x25)3+f4(25)4!(x25)4
Now, we have to find derivatives of f(x)=x
f(x)=x,f(25)=25=5
f(x)=12x,f(25)=1225=110
f(x)=14x3,f(25)=14253=1500
f(x)=38x5,f(25)=38255=325000
f4(x)=1516x7,f4(25)=1516257=3250000
Therefore, substitute this derivative in the Taylor polynomials for fourth-order,
p4=5+110(x25)12×500(x25)2+36×25000

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