Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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

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

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 InDepVar 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 MuPADWorkBook • Copy the Image then • See File • Insert-Graphic_1204.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_1204.mn

19. Power Series • General Power Series: • 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 CoEfficientsneed (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 thatln(1) = 0

35. Expand About b=1, ln(x)/1 • ln(x)/x, x • f0 := taylor(ln(x)/x, x = 1, 0) • f1 := taylor(ln(x)/x, x = 1, 1) • f2 := taylor(ln(x)/x, x = 1, 2)

36. Expand About b=1, ln(x)/1 • f3 := taylor(ln(x)/x, x = 1, 3) • f4 := taylor(ln(x)/x, x = 1, 4) • 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 AllGREEKto me…