Mumbai University > First Year Engineering > sem 2 > Applied Maths 2

Marks : 3

Year : DEC 2013

By definition

$E f(x) = f(x+h) \to (1)\\ \triangle f(x)=f(x+h)-f(x)-----\gt (1)\\ \triangledown f(x)=f(x)-f(x-h) -----\gt (2)\\ Consider, E\triangledown f(x)=E[f(x)-f(x-h)] \hspace {1cm} from (2)\\ E\triangledown f(x)=Ef(x)-Ef(x-h) \\ =f(x+h)-f[(x-h)+h] \hspace {1cm} from (1)\\ =f(x+h)-f(x)\\ =\triangle f(x)----\gt from (2)\\ Similarly \triangledown f(x)=\triangledown f(x+h)-----\gt from(1)\\ =f(x+h)-f[(x+h)-h]\hspace {1cm}from 2\\ =f(x+h)-f(x)\\ =\triangle f(x)\hspace {1cm} from 2\\ from \space (4) \space and \space (5)\\ E\triangledown f(x)=\triangle f(x)-\triangledown Ef(x)\\ E\triangledown =\triangle=\triangledown E$