1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Engr/Math/Physics 25
Chp11: MuPAD Misc
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. Using Greek Letters Can only do ONE letter at time
Some Letters do NOT have conversions
Spaces do NOT Convert
Select ONLY letters; NOT letters and a space
Can only do ONE letter at time
Not ALL std Ltrs convert to Greek
Also Use Ctrl+G

3. TypeSetting Symbols

4. Greek from Command Bar Make Expression Use Assignment Operator → :=
Now type A*cos( *t+ )
Next Pick-off the Greek from the COMMAND BAR
Click the Down Arrow

5. Greek from Command Bar
Then pick off omega & phi from the pull-down list with cursor in the right spot in the “h” expression
Then hit Enter to create symbolic expression
Some Other Expressions with Greek Pulled From the Command Bar

6. “HashTag” PlaceHolders
PlaceHolder for items from the Command Bar look Something like: #f, or #x
Sort of Like “HashTag” in Twitter
Let take an Anti-Derviative, and Calculate some Integrals
Use the Command Bar Integral Pull-Down
Pick first one to expose Place Holders for fcn & var

7. “HashTag” PlaceHolders
Replace“HashTags”
For Variable End-Point Definite Integral
The HastTags
The symbolic Definite Integral
The NUMERIC Definite Integral(s)

8. Assignment vs. Procedure
:= does NOT Create a function
It assigns a complex expression to an Abbreviation
To Create A Function (MuPad “Procedure”) include characters ->
Comparing →

9. Quick Plot by Command Bar
Find Plot Icon
Then Fill in the HashTag the the desired Function; say
The Template
The Result after filling in HashTag

10. Adjust Plot MuPad picks the InDep Var limits ±5
Write out Function to set other limits
2X-Clik the Plot to Fine Tune Plot formatting Using the Object Browser

11. Object Brower (2X Clik Plot)

12. delete → early & often
In MuPAD there is NO WorkSpace Browser to see if a variable has been evaluated and currently contains a value
Use “delete(p)”, where “p” is the variable to be cleared in a manner similar to using “clear” in MATLAB
When in Doubt, DELETE if ReUsing a variable symbol

13. delete → early & often
BOOBY PRIZE → A Variable defined in one WorkBook will CARRY OVER into OTHER WorkBooks
The Deleted Assignment in the original WorkBook can be Recovered by using Evaluate
When in doubt → DELETE
See File: Multiple_Assigns_Deletions_1204

14. TYU 11.2-1 For a A very Good Exercise See file
ENGR25_TYU11_2_1_Expressions_Functions_1204.mn

15. TYU11.3 Another Good Exercise
ENGR25_TYU11_3_Expressions_Functions_1204.mn

16. Inserting Images into MuPAD
Unlike the MATLAB Command Window, IMAGES can be imported into Text Regions of a MuPAD WorkBook
Use Menu Path: Insert→Image
See File
Insert-Graphic_into_MuPAD_1608.mn
Contains some other “tips” on MuPAD as well

17. TYU11.5 → Derivatives Take Some Derivatives
ENGR25_TYU11_5_Derivatives_1204.mn

18. TYU11.5 → AntiDerivatives Do Some Integration
ENGR25_TYU11_5_Integration_1612.mn

19. Power Series General Power Series: Power Series Convergence Behavior
A form of a GENERALIZED POLYNOMIAL
Power Series Convergence Behavior
Exclusively ONE of the following holds True
Converges ONLY for x = 0 (Trivial Case)
Converges for ALL x
Has a Finite “Radius of Convergence”, R

20. Functions as Power Series
Many Functions can be represented as Infinitely Long PolyNomials
Consider this Function and Domain
The Geometric Series form of f(x)
Thus

21. Taylor Series
Consider some general Function, f(x), that might be Represented by a Power Series
Thus need to find all CoEfficients, an, such that the Power Series Converges to f(x) over some interval. Stated Mathematically Need an so that:

22. Taylor Series If x = 0 and if f(0) is KNOWN then
a0 done, 1→∞ to go….
Next Differentiate Term-by-Term
Now if the First Derivative (the Slope) is KNOWN when x = 0, then

23. Taylor Series Again Differentiate Term-by-Term
Now if the 2nd Derivative (the Curvature) is KNOWN when x = 0, then

24. Taylor Series Another Differentiation
Again if the 3rd Derivative is KNOWN at x = 0
Recognizing the Pattern:

25. Taylor Series
Thus to Construct a Taylor (Power) Series about an interval “Centered” at x = 0 for the Function f(x)
Find the Values of ALL the Derivatives of f(x) when x = 0
Calculate the Values of the Taylor Series CoEfficients by
Finally Construct the Power Series from the CoEfficients

26. Example  Taylor Series for ln(e+x)
Calculate the Derivatives
Find the Values of the Derivatives at 0

27. Example  Taylor Series for ln(e+x)
Generally
Then the CoEfficients
The 1st four CoEfficients

28. Example  Taylor Series for ln(e+x)
Then the Taylor Series

29. Taylor Series at x ≠ 0
The Taylor Series “Expansion” can Occur at “Center” Values other than 0
Consider a function stated in a series centered at b, that is:
Now the Radius of Convergence for the function is the SAME as the Zero Case:

30. Taylor Series at x ≠ 0
To find the CoEfficients need (x−b) = 0 which requires x = b, Then the CoEfficient Expression
The expansion about non-zero centers is useful for functions (or the derivatives) that are NOT DEFINED when x=0
For Example ln(x) can NOT be expanded about zero, but it can be about, say, 2

31. Example  Expand x½ about 4
Expand about b = 4:
The 1st four Taylor CoEfficients

32. Example  Expand x½ about 4
SOLUTION:
Use the CoEfficients to Construct the Taylor Series centered at b = 4

33. Example  Expand x½ about 4
Use the Taylor Series centered at b = 4 to Find the Square Root of 3

34. Expand About b=1, ln(x)/1 Da1 := diff(ln(x)/x, x) Db2 := diff(Da1, x)
Dc3 := diff(Db2, x)
Dd4 := diff(Dc3, x)
ReCall that ln(1) = 0

35. Expand About b=1, ln(x)/1 ln(x)/x, x f0 := taylor(ln(x)/x, x = 1, 0)

36. Expand About b=1, ln(x)/1 f3 := taylor(ln(x)/x, x = 1, 3)
d6 := diff(ln(x)/x, x $ 5)

37. Expand About b=1, ln(x)/1
plot(f0, f1, f2, f3, f4, f5, x =0.5..3, GridVisible = TRUE, LineWidth = 0.04*unit::inch, Width = 320*unit::mm, Height = 180*unit::mm, AxesTitleFont = ["sans-serif", 24], TicksLabelFont=["sans-serif", 16])

41. TYU 11.5 → Sums & Series Exercise Taylor’s Series & Sums
ENGR25_TYU11_5_6789_Taylor_Sums_Limits_1204.mn

42. TYU11.6 → ODEs Do an ODE Solution file = ENGR25_TYU11_6_ODE_1204.mn
By: File → Export → PDF

43. All Done for Today
It’s All GREEK to me…
8.[35,116,107]