Obtain a four-term Taylor polynomial approximation valid near x=,0 for each (1+x)^-1/3. Estimate the range of x over which three term polynomials will give two decimal accuracy
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Expert's answer
2021-04-26T17:47:11-0400
set f(x)=(1+x)−1/3
for a four term Taylor polynomial centered at x=0
f(x)=(1+x)−1/3⟹f(0)=1
f′(x)=−31(1+x)−4/3⟹f′(0)=−21
f′′(x)=−94(1+x)−7/3⟹f′′(0)=94
f′′′(x)=−2728(1+x)−10/3⟹f′′′(0)=−2728
now the four term Taylor polynomial approximation to f(x) valid near x=0 is;
f(x)=(1+x)−1/3=1−31x+2!94x2−3!2728x3
=1−31x+92x2−8114x3
for the range of values of x over which these term polynomials will give a two decimal accuracy we may use the Lagrange error bound;
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