Question

The values of a function f(x) are tabulaed 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 for the given data

x 0 1 2 3

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

${\Delta }^2$f(x) 1=$\Delta$f(0) -1 9

${\Delta }^2$f(x) -2=${\Delta }^2$f(0) 10

${\Delta }^3$f(x) 12=${\Delta }^3$f(0)

$x_0=0$
Newton's forward difference formula

Diagram

$f\left(x\right)=1+x\left(\right)+\left(\frac{x^2-x}{2}\right)(-2)+\left(\frac{(x^3-3x^2+2x)}{6}\right)\times 12$
$f\left(x\right)=1+x-x^2+x+2x^3-6x^2+4x$
$f\left(x\right)=2x^3-7x^2+6x+1$
Note: In examination, the students are advised to check options by putting values of x.