Relations Between Differences And Derivatives

We have,

Neglecting the higher order terms, we get the approximation,

..............................1.7

The error term is given by,

Hence, we call the approximation given in (1.7) as a first order approximation or of order O(h).

Similarly, we have the following results for backward differences.

.........................1.8

..............................1.9

For the central differences, we obtain

Neglecting the higher order terms, we get the approximation

..................................2.0

The error term is given by,

Hence, we call the approximation given in (2.0) as a second order approximation or of order O(h²).

We have,