Find forward difference numeric function b and the backward difference numeric function c of the function a, where

Question

a^r = 0 at 0<r<2 and 2^r+5 at r>3

Answer 1

The difference quotients for A^r=0 will be zero apparently...

For the latter function, the forward difference is:

2^(r+h) +5 - 2^(r) - 5 = 2^(r+h) - 2^r = 2^r * 2^h - 2^r = 2^r( 2^h - 1)

The backwards difference shall be:

2^r - 2^(r-h) <---- since the 5 cancels

= 2^r - 2^r*2^h = 2^r ( 1 - 2^h)

Therefore, the forward difference is merely the opposite of the backwards difference as the

quantity inside the parenthesis gets flipped