Question

The value of a function f(x) are tabulated below

x	0	1	2	3
f(x)	1	2	1	10

Using Newton's forward difference formula, the cubic polynomial that can be fitted to the above data, is

• A. $2x^3+7x^2-6x+2$
• B. $2x^3-7x^2+6x-2$
• C. $x^3-7x^2-6x+1$
• D. $2x^3-7x^2+6x+1$

Answer 1

The correct option is D $2x^3-7x^2+6x+1$

The difference table is shown as:

By Newton's forward difference formula,

$f(a+nh)=f\left(a\right)+n\Delta f\left(a\right)+\frac{n(n-1)}{2!}{\Delta }^{2}f\left(a\right)+\frac{n(n-1)(n-2)}{3!}{\Delta }^{3}f\left(a\right)+...$

$\Rightarrow f\left(x\right)=f\left(0\right)+x\Delta f\left(0\right)+\frac{x(x-1)}{2!}{\Delta }^{2}f\left(0\right)+\frac{x(x-1)(x-2)}{3!}{\Delta }^{3}f\left(0\right)+........\left[\begin{array}{c}a+nh=x\\ 0+n\times 1=x\\ n=x\end{array}\right]$

$=1+x\times 1+\frac{x(x-1)}{2!}\times (-2)+\frac{x(x-1)(x-2)}{3!}\times 12$

$=1+x-(x^2-x)+2\left[x\right(x^2-3x+2x\left)\right]$

$=1+x-x^2+x+2x^3-6x^2+4x$

$=2x^3-7x^2+6x+1$