Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number. f(x)=1x with a=4, approximate 13

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Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.  f(x)=1x with a=4, approximate 13

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Consider the provided function,f(x)=1x with a=4, approximate 13Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.The Taylor series as a form of ∑k=0∞ck(x−a)k, where ck=fk(a)k!,k=0,1,2,…We evaluate the derivative of the function at point a=4f(x)=1x⇒f(4)=12f′(x)=−12x32⇒f′(4)=−116f″(x)=34x52⇒f″(4)=3128f‴(x)=−158x72⇒f‴(4)=−151024Hence, the first four term of the series is shown below.12−116(x−4)+3128(x−4)2−151024(x−4)3So,13=12−116(13−4)+3128(13−4)2−151024(13−4)3=12−1−43163+493−241283−5(145−2043)10243=12+1668−32731024Hence.

Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number. f(x)=frac{1}{sqrt{x}} with a=4, approximate frac{1}{sqrt{3}}

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