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.

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

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

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