I still need help to splve the following problems:
1. Let f(x)=tan^-1(x) a) Obtain the 3rd degree Taylor polynomial P3(x) and the Taylor remainder R3 (x) about c=1 for the given function.
this is what I have tried:
The first derivative is
f'(x) = 1/(x^2 + 1)
factoring (1 + x^2) within the complex numbers:
(x^2 + 1) = (x + i)(x - i)
So f'(x) = 1/(x+i)(x-1) f'(x) = (i/2)/(x+i) - (i/2)/(x-i)
f'(0) = 1
f"(x) = -(i/2)/(x+i)^2 + (i/2)/(x-i)^2
f"(0) = 0
f'"(x) = i/(x+i)^3 - i/(x-i)^3
f"'(0) = -1/2
That's all I could do. Am I on the right track? Could you please help to find: 1. the remainder R3(X) R3 (x) about c=1 for the given function? 2. Use the Use P3(x) to estimate tan^-1(0.7) and use R3(x) to estimate the error in the approximation.
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Posted - 09/13/2007 : 08:05:34
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