IGNOU MTE 10 2018 - A table of values is to be constructed for the function f (x) given by

Question

A table of values is to be constructed for the function f (x) given by
$f(x)=\frac{1}{1+x}$ in the interval [1, 4] with equal step length. Determine the spacing h such that quadratic interpolation gives result with accuracy 1x10^-6.

14 Feb 2018

Answer :

Word Count : 255

We are asked to determine the spacing $h$ for quadratic interpolation of $f(x) = \frac{1}{1+x}$ in $[1,4]$ such that the interpolation error is at most $10^{-6}$. Let's solve this step by step. The error formula for quadratic interpolation (Lagrange polynomial of degree 2) is: $$ E(x) = \frac{f^{(3)}(\xi)}{3!}(x - x_0)(x - x_1)(x - x_2), \quad \xi \in [x_0,x_2] $$ 1. Compute the third derivative of $f(x)$: $$ f(x) = (1+x)^{-1} \quad \Rightarrow \quad f'(x) = - (1+x)^{-2}, \quad f''(x) = 2 (1+x)^{-3}, \quad f'''(x) = -6 (1+x)^{-4} $$ So $|f^{(3)}(x)| = 6 (1+x)^{-4}$. The

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x	− 3	− 2	− 1	0	1	2	3
f ( x)	13	7	3	1	1	3	7

x	1	2	3	4	5
$\sqrt\frac{{1+x^4}}{x^4}$	1.414	1.031	1.007	1.002	1.001