1. Differentiation-Discrete Functions
Electric Engineering Majors
Authors: Autar Kaw, Sri Harsha Garapati
Transforming Numerical Methods Education for STEM Undergraduates
7/4/2018

2. Differentiation –Discrete Functions http://numericalmethods. eng. usf

3. Forward Difference Approximation
For a finite

4. Graphical Representation Of Forward Difference Approximation
Figure 1 Graphical Representation of forward difference approximation of first derivative.

5. Example 1
To increase the reliability and life of a switch, one needs to turn the switch off as close to zero crossing as possible. To find this time of zero crossing, the value of is to be found at all times given in Table 1, where
is the voltage and is the time. To keep the problem simple, you are asked to find the approximate value of at
See Table 1 for voltage as a function of time data.
Use Forward Divided Difference approximation of the first derivative to calculate
at Use a step size of

6. Table 1 Voltage as a function of time.
Example 1 Cont.
Table 1 Voltage as a function of time.
Time,
Voltage, 1 13 − 2 14 3 15 4 − 16 5 − 17 6 − 18 7 − 19 8 − 20 9 − 21 10 − 22 11 − 23 12 − 24

7. Example 1 Cont.
Solution

8. Example 1 Cont.

9. Direct Fit Polynomials
In this method, given
data points
one can fit a
order polynomial given by
To find the first derivative,
Similarly other derivatives can be found.

10. Example 2-Direct Fit Polynomials
To increase the reliability and life of a switch, one needs to turn the switch off as close to zero crossing as possible. To find this time of zero crossing, the value of is to be found at all times given in Table 2, where
is the voltage and is the time. To keep the problem simple, you are asked to find the approximate value of at
See Table 2 for voltage as a function of time data.
Using the third order polynomial interpolant for Voltage, find the value of at

11. Example 2-Direct Fit Polynomials cont.
Table 2 Voltage as a function of time.
Time,
Voltage, 1 13 − 2 14 3 15 4 − 16 5 − 17 6 − 18 7 − 19 8 − 20 9 − 21 10 − 22 11 − 23 12 − 24

12. Example 2-Direct Fit Polynomials cont.
Solution
For the third order polynomial (also called cubic interpolation), we choose the velocity given by
Since we want to find the voltage at , and we are using a third order polynomial, we need
to choose the four points closest to
and that also bracket
to evaluate it.
The four points are , , and

13. Example 2-Direct Fit Polynomials cont.
such that
Writing the four equations in matrix form, we have

14. Example 2-Direct Fit Polynomials cont.
Solving the above four equations gives
Hence

15. Example 2-Direct Fit Polynomials cont.
Figure 2 Graph of voltage of the switch vs. time.

16. Example 2-Direct Fit Polynomials cont.
The derivative of voltage at t=10 is given by
Given that ,

17. Example 2-Direct Fit Polynomials cont.

18. Lagrange Polynomial In this method, given , one can fit a given by
order Lagrangian polynomial
given by
where ‘
’ in
stands for the
order polynomial that approximates the function
given at
data points as
, and
a weighting function that includes a product of
terms with terms of
omitted.

19. Lagrange Polynomial Cont.
Then to find the first derivative, one can differentiate
once, and so on
for other derivatives.
For example, the second order Lagrange polynomial passing through is
Differentiating equation (2) gives

20. Lagrange Polynomial Cont.
Differentiating again would give the second derivative as

21. Example 3
To increase the reliability and life of a switch, one needs to turn the switch off as close to zero crossing as possible. To find this time of zero crossing, the value of is to be found at all times given in Table 3, where
is the voltage and is the time. To keep the problem simple, you are asked to find the approximate value of at
See Table 3 for voltage as a function of time data.
Use the second order Lagrangian polynomial interpolation to calculate the value of at

22. Table 2 Voltage as a function of time.
Example 3 Cont.
Table 2 Voltage as a function of time.
Time,
Voltage, 1 13 − 2 14 3 15 4 − 16 5 − 17 6 − 18 7 − 19 8 − 20 9 − 21 10 − 22 11 − 23 12 − 24

23. Example 3 Cont. Solution:
For second order Lagrangian polynomial interpolation, we choose the voltage given by
Since we want to find the voltage at , and we are using a second order Lagrangian polynomial, we need to choose the three points closest to
that also bracket to evaluate it.
The three points are , , and
Differentiating the above equation gives

24. Example 3 Cont.
Hence

25. Additional Resources
For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

26. THE END