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There are three ways to numerically approximate the first order derivative of a function:

1-
1^{st}
order forward finite difference:

Consider the 1^{st} order Taylor
series:

_{}

which
yields: _{}

which
can be written as

_{} (1)

where h is called the interval or step
size.

2-
1^{st}
order backward finite difference:

If we write the 1^{st} order Taylor
series backward, we get:

_{}_{}

or:

_{} (2)

3-
1^{st}
order centered finite difference:

The formula can be obtained by subtracting
(2) from (1) to get:

_{} (3)

Approximation for
the second order derivatives can be derived using the following procedure:

Recall the 2^{nd}
order Taylor series:

_{}

which can be written as

_{}

or:

_{} (4)

using (3) in (4),
we can express *f ^{’’}_{i}* in terms of

* *

_{}

which is the second order
centered finite difference. Similar procedure can be used to obtain the second
order backward and forward finite differences.

__Example:__

Given the
following data for the deflection y(x) of a beam of length L = 1m and EI = 1
N.m^{2}, estimate the numerical value for the bending moment at the
middle of the beam.

x |
0 |
0.1 |
0.2 |
0.3 |
0.4 |
0.5 |
0.6 |
0.7 |
0.8 |
0.9 |
1.0 |

Y(x) |
0. |
-2.6 |
-4.9 |
-5.7 |
-7.9 |
-8.3 |
-7.7 |
-5.5 |
-4.8 |
-2.5 |
0. |

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* Important: This handout is only a summery of the lecture. The student take detailed notes during the class and refer to the textbook for more examples and discussion.