Question

Given that

$x$	$10$	$15$	$20$
$f\left(x\right)$	$14$	$18$	$28$

the estimated value of $f\left(12\right)$ is

• A. $15.520$
• B. $14.88$
• C. $15.37$
• D. $14.78$

Answer 1

The correct option is B $14.88$

The difference table is as follows

$x$	$f\left(x\right)$	$\Delta f\left(x\right)$	${\Delta }^{2}f\left(x\right)$
$10$	$14$
$15$	$18$	$4$
$20$	$28$	$10$	$6$

Here, $a=10,x=12,h=5$
$m=\frac{12-10}{5}=\frac{2}{5}=0.4$
By Newton-Gregory forward interpolation formula
$f\left(x\right)=f\left(a\right)+m\Delta f\left(a\right)+\frac{m(m-1)}{2!}{\Delta }^{2}f\left(a\right)$
$=f\left(10\right)+0.4\Delta f\left(10\right)+\frac{\left(0.4\right)(0.4-1)}{2!}{\Delta }^{2}f\left(10\right)$
$=14+\left(0.4\right)\left(4\right)+\frac{\left(0.4\right)(-0.6)}{2}\times 6=14+1.6-0.72=14.88$