let q(x) =bo+b1(x-a)+b2(x-a)square be a quadratic approximation to f(x)at x=a wi

Question

let q(x) =bo+b1(x-a)+b2(x-a)square be a quadratic approximation to f(x)at x=a with the the properties; i q(a)=f(a) ii q'(a)=f'(a) iii ''(a)=f''(a) find the quadratic approximation to f(x)=1divided by 3+x at x=0

Explanation / Answer

f(x)=1/(x+3) a=0 f(a)=1/3==>q(a)=1/3 now q(x)=bo+b1(x-a)+b2(x-a)^2 q(a)=bo so bo=1/3...(1) now f'(x)=-1/(x+3)^2 f'(0)=-1/9 so q'(0)=-1/9 now q(x)=bo+b1(x-a)+b2(x-a)^2 q'(x)=b1+2*b2*(x-a) q'(a)=b1 so b1=-1/9 now f''(x)=2/(x+3)^3 f''(0)=2/27 now q(x)=bo+b1(x-a)+b2(x-a)^2 q''(x)=2*b2 so 2*b2=2/27 so b2=1/27 hence , q(x)=(1/3)-(1/9)*x+(1/27)*x^2

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