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Question

Derive a suitable numerical differentiation formula of 0(h² ) to find f'(2,4) with h = 0.1 given the table

x	0.1	1.2	2.4	3.9
f(x)	3.41	2.68	1.37	-1.48

14 Feb 2018

Answer :

Word Count : 325

We are required to find $f'(2.4)$ with $O(h^2)$ accuracy and step size $h=0.1$. From the table, we have values of $f(x)$ at $x=0.1, 1.2, 2.4, 3.9$. Since $f'(x)$ is required at the interior point $x=2.4$, the central difference formula of order $O(h^2)$ is most suitable: $$ f'(x) \approx \frac{f(x+h) - f(x-h)}{2h} $$ But here the spacing between given points is not uniform. The gap from 1.2 to 2.4 is $1.2$, and from 2.4 to 3.9 is $1.5$. Thus, the normal central difference with equal $h$ cannot be applied. Instead, we use the interpolating polynomial approach (Newton’s divided difference

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