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

[email protected] ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx 1

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Chp11: MuPAD

Misc



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



TypeSetting Symbols



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

tAh cos



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



“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

3 222222 11

11

11 dy

ydy

ydy

y



“HashTag” PlaceHolders Replace“HashTags”

For Variable End-Point Definite Integral

The HastTags

The symbolic Definite Integral

The NUMERIC Definite Integral(s)

7

3 222222 11

11

11 dy

ydy

ydy

y



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 →



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

xxy sin



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



Object Brower (2X Clik Plot)



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



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



TYU 11.2-1 For a A very Good Exercise See file

• ENGR25_TYU11_2_1_Expressions_Functions_1204.mn



TYU11.3 Another Good Exercise

• ENGR25_TYU11_3_Expressions_Functions_1204.mn



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



TYU11.5 → Derivatives Take Some Derivatives

• ENGR25_TYU11_5_Derivatives_1204.mn



TYU11.5 → AntiDerivatives Do Some Integration

• ENGR25_TYU11_5_Integration_1204.mn



Power Series General Power Series:

• A form of a GENERALIZED POLYNOMIAL Power Series Convergence Behavior

• Exclusively ONE of the following holds Truea) Converges ONLY for x = 0 (Trivial Case)b) Converges for ALL x c) Has a Finite “Radius of Convergence”, R

n

n

nn

kk xaxaxaxaxaxa

0

33

22

11

00



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

1for1

1

x

xxf

xfxxxxxxx

n

n

nk

0

3210 1111111

1

1for1

10

xxx

xfn

n

n



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:

Rxxfxan

n

nn

forconverges

0

n

n

nn xaxaxaxaaxf

0

33

2210



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

000000 003

32

210 faaaaaaf

n

n

nn xnaxaxaxaa

dxxdf

1

134

2321 432

011

34

2321

0

0040302

xx dx

dfaaaaaadxdf



Taylor Series Again Differentiate Term-by-Term

Now if the 2nd Derivative (the Curvature) is KNOWN when x = 0, then

n

n

nn xannxaxaa

dxfd

2

224322

2

134232

02

2

222

4320

2

2

2020340232

xx

dxfdaaaaa

dxfd



Taylor Series Another Differentiation

Again if the 3rd Derivative is KNOWN at x = 0

Recognizing the Pattern:

n

n

nn xannnxaxaa

dxfd

3

324433

3

2134523423

03

3

332

4430

3

3

6060345023423

xx dx

fdaaaaadx

fd

!! 0

0n

dxfd

adx

fdan xn

n

nx

n

n

n



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

!0

ndx

fd

a xn

n

n

n

n

nn xaxf

0



Example Taylor Series for ln(e+x) Calculate the Derivatives

Find the Values of the Derivatives at 0

322

3322

21111lnxexedx

dxexedx

dxe

xedxd

dxfddxfddxdf

3322

033

022

02

021

011

01

eeeeee

dxfddxfddxdf xxx



Example Taylor Series for ln(e+x) Generally Then the CoEfficients

The 1st four CoEfficients

1for!11 1

0

ne

ndxfd n

n

xnn

1for1!

!11

!

1

1

0

nenn

en

ndxfd

a n

nn

n

xnn

n



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

1

1)1(1)ln(n

n

nn

enxxe

n

nn

nn

nn x

enxaxaxaxe

1

10

01

00

1ln



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:

n

n

nn bxaxf

0

bRxbRbRbbxbR

RbxRRbx



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

!! nbf

ndx

fd

an

bxn

n

n



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

xxf



Example Expand x½ about 4 SOLUTION: Use the CoEfficients to Construct the

Taylor Series centered at b = 4

0

)(n

nn bxax



Example Expand x½ about 4 Use the Taylor Series centered at b = 4

to Find the Square Root of 3

4

00

)3()3(3n

nn

n

nn baba

432 )43(16384

5)43(5121)43(

641)43(

412

432 )1(16384

5)1(5121)1(

641)1(

412

0003.00020.00156.025.02

1.7320508 MATLABBy 7321.12679.02



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

0112

1

xdx

df

51

4

4

1500

xdx

fd

0

111

41

3

3

xdx

fd

51

2

2

130

xdx

fd



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)



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)

0

1274

61

5

5

xdx

fd

!5

11274!4

1150!3

1111!2

113!11

!00ln

54321

xxxxxx



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])



TYU 11.5 → Sums & Series Exercise Taylor’s Series & Sums

• ENGR25_TYU11_5_6789_Taylor_Sums_Limits_1204.mn



TYU11.6 → ODEs Do an ODE Solution

• file = ENGR25_TYU11_6_ODE_1204.mn– By: File → Export → PDF



All Done for Today

It’s AllGREEKto me…

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

Documents

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