ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 1 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engr/Math/Physics 25 Chp11: MuPAD Misc

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 2 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Using Greek Letters  Can only do ONE letter at time  Not ALL std Ltrs convert to Greek Also Use Ctrl+G  Some Letters do NOT have conversions  Spaces do NOT Convert Select ONLY letters; NOT letters and a space

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 3 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods TypeSetting Symbols

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 4 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 5 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 6 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods “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

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 7 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods “HashTag” PlaceHolders  Replace“HashTags”  For Variable End- Point Definite Integral  The HastTags  The symbolic Definite Integral  The NUMERIC Definite Integral(s)

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 8 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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 →

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 9 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 10 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 11 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Object Brower (2X Clik Plot)

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 12 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 13 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 14 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods TYU  For a A very Good Exercise See file ENGR25_TYU11_2_1_Expressions_Functi ons_1204.mn

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 15 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods TYU11.3  Another Good Exercise ENGR25_TYU11_3_Expressions_Function s_1204.mn

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 16 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Inserting Images into MuPAD  Unlike the MATLAB Command Window, IMAGES can be imported into Text Regions of a MuPAD WorkBook  Copy the Image then  See File Insert-Graphic_1204.mn –Contains some other “tips” on MuPAD as well

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 17 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods TYU11.5 → Derivatives  Take Some Derivatives ENGR25_TYU11_5_Derivatives_1204.mn

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 18 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods TYU11.5 → AntiDerivatives  Do Some Integration ENGR25_TYU11_5_Integration_1204.mn

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 19 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Power Series  General Power Series: A form of a GENERALIZED POLYNOMIAL  Power Series Convergence Behavior Exclusively ONE of the following holds True a)Converges ONLY for x = 0 (Trivial Case) b)Converges for ALL x c)Has a Finite “Radius of Convergence”, R

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 20 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 21 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Taylor Series  Consider some general Function, f ( x ), that might be Represented by a Power Series  Thus need to find all CoEfficients, a n, such that the Power Series Converges to f ( x ) over some interval. Stated Mathematically Need a n so that:

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 22 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Taylor Series  If x = 0 and if f (0) is KNOWN then a 0 done, 1→∞ to go….  Next Differentiate Term-by-Term  Now if the First Derivative (the Slope) is KNOWN when x = 0, then

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 23 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Taylor Series  Again Differentiate Term-by-Term  Now if the 2 nd Derivative (the Curvature) is KNOWN when x = 0, then

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 24 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Taylor Series  Another Differentiation  Again if the 3 rd Derivative is KNOWN at x = 0  Recognizing the Pattern:

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 25 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 26 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Taylor Series for ln(e+x)  Calculate the Derivatives  Find the Values of the Derivatives at 0

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 27 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Taylor Series for ln(e+x)  Generally  Then the CoEfficients  The 1 st four CoEfficients

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 28 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Taylor Series for ln(e+x)  Then the Taylor Series

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 29 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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:

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 30 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 31 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Expand x½ about 4  Expand about b = 4:  The 1 st four Taylor CoEfficients

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 32 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Expand x ½ about 4  SOLUTION:  Use the CoEfficients to Construct the Taylor Series centered at b = 4

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 33 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Example  Expand x ½ about 4  Use the Taylor Series centered at b = 4 to Find the Square Root of 3

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 34 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 35 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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)

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 36 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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)

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 37 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods 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])

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 38 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 39 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 40 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 41 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods TYU 11.5 → Sums & Series  Exercise Taylor’s Series & Sums ENGR25_TYU11_5_6789_Taylor_Sums_L imits_1204.mn

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 42 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods TYU11.6 → ODEs  Do an ODE Solution file = ENGR25_TYU11_6_ODE_1204.mn –By: File → Export → PDF

ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 43 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods All Done for Today It’s All GREEK to me…