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Finite Difference Approximations of the First Derivative of a Function

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

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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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