Question

In the second derivative using newton's forward difference formula, what is the coefficient of Δ^3 f(a)​

Answer 1

Answer:

The coefficient of Δ³f(a) is -1/h².

Step-by-step explanation:

To find,

The coefficient of Δ³f(a).

Calculation,

Newton's forward difference formula is a finite difference identity giving an interpolated value between tabulated points $f_p$ in terms of the first value $f_o$ and the powers of the forward difference Δ. For p in [0,1], the formula states:
$f(a) = f_o + p\Delta f_o +\frac{p(p-1)}{2!}\Delta^2 f_o + \frac{p(p-1)(p-2)}{3!} \Delta^3f_o+....$

Now the 2nd derivate of Newton's forward difference formula is:

$f''(a)_{x = x_o} = \frac{1}{h^2} (\Delta^2f_o- \Delta^3f_o + \frac{11}{12} \Delta^4f_o+ ..... )$

Where h is called the interval of difference.

Therefore, the coefficient of Δ³f(a) is -1/h².

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