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Differentiation and Applications

Differentiation and Applications. Module 5 Lecture 3. Topic: Series and Approximations. Plot of first 2 terms. Plot of first 6 terms. Converges to a triangular wave. Electronic signals can always be represented by infinite series like this.

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Differentiation and Applications

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  1. Differentiation and Applications Module 5 Lecture 3

  2. Topic: Series and Approximations

  3. Plot of first 2 terms Plot of first 6 terms

  4. Converges to a triangular wave Electronic signals can always be represented by infinite series like this Crucial in signal processing and anything in the form of a wave

  5. The exponential series constant linear quadratic cubic

  6. The exponential series constant linear quadratic cubic

  7. The exponential series constant linear quadratic cubic

  8. The exponential series constant linear quadratic cubic

  9. The exponential series constant linear quadratic cubic

  10. A few terms very accurate for small x

  11. Another use of series – A simple 2 or 3 – term series can often be used instead of a more complicated function. This makes the maths a lot simpler

  12. Deriving the exponential series Where does the exponential series come from ? If we didn’t know what the series was, we could first write … The question now is – how do we find the a’s ?

  13. To get the first (a0) term, put in x = 0 To get the second (a1) term, differentiate and then let x = 0

  14. etc. etc. …….. To get the third (a2) term, differentiate again and let x = 0

  15. General Formula This is called a Maclaurin Series

  16. General Method for getting these series Initial Step: Put x = 0 to get a0 Loop: Differentiate Put x = 0 and divide by n! for an Example:

  17. ( We’ll come back to this … ) NO 4-term series is …. ( + Error term ) Nice pattern …. Does the series work for any value of x ?

  18. Initial Step: Put x = 0 to get a0 Loop: Differentiate Put x = 0 and divide by n! for an Example: 3-term series is ….

  19. Initial Step: Put x = 0 to get a0 Loop: Differentiate Put x = 0 and divide by n! for an Example: 3-term series is ….

  20. Initial Step: Put x = 0 to get a0 Loop: Differentiate Put x = 0 and divide by n! for an Example: 3-term series is ….

  21. Initial Step: Put x = 0 to get a0 Loop: Differentiate Put x = 0 and divide by n! for an Example: 3-term series is ….

  22. Initial Step: Put x = 0 to get a0 Loop: Differentiate Put x = 0 and divide by n! for an Example: 4-term series is ….

  23. Class Exercise Get the 3-term series for Initial Step: Put x = 0 to get a0 Loop: Differentiate Put x = 0 and divide by n! for an Challenge Question: Get the 3-term series for

  24. “converges” “diverges”

  25. Important: there are values of x for which series work and there are values for which it does not work There are special mathematical tests which we can do on series to see whether they converge (e.g. the “ratio test”)

  26. Some famous series and valid ranges of x

  27. Interlude … going complex … The exponential series Let us replace x by ix

  28. and so Euler’s Formula When , Euler’s formula turns into

  29. The Binomial Series The following series can be derived using the same methods as described … This expansion is valid for This series can be used to get approximations to, for example,

  30. valid for |x| < 1 valid for |x| < 1 valid for |x| < 1 valid for |-x| < 1 |x| < 1 valid for |3x| < 1 |x| < 1/3 |x| < 1 valid for |-x3| < 1

  31. valid for |x| < 1 Class Exercise Use the Binomial series to get a 3-term series for Challenge Question: Get the 3-term series for What are the valid values of x ?

  32. Series and approximations … what you need to know • Know what a linear, quadratic and cubic approximation of a function is • Derive the first three terms of the exponential series • Use the “Initial Step & Loop” method to derive series • Be aware that series converge only for certain values of x • Use the Binomial Series formula to derive series

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