Finite Difference Approximations of the First Derivative of a Function

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Derivatives of functions can be approximated by finite difference formulas. In this Demonstration, we compare the various difference approximations with the exact value.

Contributed by: Vincent Shatlock and Autar Kaw (April 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The first derivative of a function is defined as

.

The simplest finite difference formulas for the first derivative of a function are:

(forward difference) (central difference) (backward difference)

Both forward and backward difference formulas have error , while the central difference formula has error .

In this Demonstration, we show the difference in values calculated from the three difference formulas and the exact value.

Questions: 1. Does the true error increase proportionally with the step size, , for the forward and backward difference formulas? 2. Does the true error increase proportionally with the square of the step size, , for the central difference formula?

Reference

[1] A. A. Kaw, D. Nguyen, and E. E. Kalu, Numerical Methods with Applications. http://numericalmethods.eng.usf.edu/publications_book.html.



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