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

Marks : 3

Year : MAY 2014

By definition

$$\triangledown f(x)=f(x)-f(x-h)$$ $ f(x-h)=f(x)-\triangledown f(x)---- (1)\\ similarly\\ \triangle f(x)=f(x+h)-f(x)\\ i.e \\ \triangle f(x-h)=f(x-h+h)-f(x-h)\\ =f(x)- f(x-h)\\ \triangle f(x-h)+ f(x+h)=f(x) ---- (2)\\ consider \\ (1+\triangle)(1-\triangledown) f(x)= (1+\triangle)[f(x)-\triangledown f(x)] \\ = (1+\triangle)f(x+h)\hspace {1cm} from 1 \\ =f(x+h)+\triangle f(x-h)\\ =f(x) \\ \therefore (1+\triangle)(1-\triangledown) f(x)=f(x)\\ \therefore (1+\triangle)(1-\triangledown)=1$