Engr/Math/Physics 25
Chp11:
MuPAD Misc
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering/Math/Physics 25: Computational Methods
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Using Greek Letters
 Can only do ONE
letter at time
 Not ALL std Ltrs
convert to Greek
•
Also Use
Ctrl+G
Engineering/Math/Physics 25: Computational Methods
2
 Some
Letters do
NOT have
conversions
 Spaces do
NOT
Convert
• Select
ONLY
letters;
NOT
letters and
a space
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
TypeSetting Symbols
Engineering/Math/Physics 25: Computational Methods
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Greek from Command Bar
 Make Expression
h  A cost   
 Use Assignment
Operator → :=
 Next Pick-off the
Greek from the
COMMAND
BAR
 Click the Down
Arrow
 Now type
A*cos( *t+ )
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Greek from Command Bar
 Then pick off omega
& phi from the pulldown list with cursor
in the right spot in
the “h” expression
 Some Other
Expressions with
Greek Pulled From
the Command Bar
 Then hit Enter to
create symbolic
expression
Engineering/Math/Physics 25: Computational Methods
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
“HashTag” PlaceHolders
 PlaceHolder for
items from the
Command Bar look
Something like: #f,
or #x
• Sort of Like
“HashTag” in Twitter
 Let take an AntiDerviative, and
Calculate some
Integrals
Engineering/Math/Physics 25: Computational Methods
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1
 1  y 
2 2
dy

1
 1  y 
2 2
dy
7
1
 1  y 
3
 Use the
Command
Bar Integral
Pull-Down
 Pick first one to
expose Place
Holders for fcn & var
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
2 2
dy
1
 1  y 
“HashTag” PlaceHolders
 Replace“HashTags”
 For Variable EndPoint Definite
Integral
2 2
dy

1
 1  y 
7
dy
 The symbolic
Definite Integral
 The NUMERIC
Definite Integral(s)
 The HastTags
Engineering/Math/Physics 25: Computational Methods
2 2
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
7
1
 1  y 
3
2 2
dy
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 →
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Quick Plot by Command Bar
 Find
Plot
Icon
 The
Template
 The Result after
filling in HashTag
 Then Fill in the
HashTag the the
desired Function;
say
y  x sin x
Engineering/Math/Physics 25: Computational Methods
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Adjust Plot
 MuPad picks the
InDep Var limits ±5
 Write out Function
to set other limits
Engineering/Math/Physics 25: Computational Methods
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 2X-Clik the Plot to
Fine Tune Plot
formatting Using
the Object Browser
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Object Brower (2X Clik Plot)
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
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
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
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
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
TYU 11.2-1
 For a A very Good Exercise See file
• ENGR25_TYU11_2_1_Expressions_Functi
ons_1204.mn
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
TYU11.3
 Another Good Exercise
• ENGR25_TYU11_3_Expressions_Function
s_1204.mn
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
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
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
TYU11.5 → Derivatives
 Take Some Derivatives
• ENGR25_TYU11_5_Derivatives_1204.mn
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
TYU11.5 → AntiDerivatives
 Do Some Integration
• ENGR25_TYU11_5_Integration_1204.mn
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Power Series
 General Power Series:
n 
a0 x  a1 x  a2 x  a3 x   ak x     an x n
0
1
2
3
k
n 0
• 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
Engineering/Math/Physics 25: Computational Methods
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Functions as Power Series
 Many Functions can be represented as
Infinitely Long PolyNomials
 Consider this Function and Domain
1
f x  
for x  1
1 x
 The Geometric Series form of f(x)
n 
1
 1x 0  1x1  1x 2  1x 3  1x k    1  x n  f  x 
1 x
 Thus
n 0
n 
1
f x  
  xn
1  x n 0
Engineering/Math/Physics 25: Computational Methods
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for
x 1
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Taylor Series
 Consider some general Function, f(x),
that might be Represented by a Power
n 
2
3
Series f x   a0  a1 x  a2 x  a3 x     an x n
n 0
 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:
n 
converges
n
a
x

 n  f x  for
x R
n 0
Engineering/Math/Physics 25: Computational Methods
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Taylor Series
 If x = 0 and if f(0) is KNOWN then
f 0  a0  a1 0  a2 0 2  a3 03    a0  0  a0  f 0
• a0 done, 1→∞ to go….
 Next Differentiate Term-by-Term
n 
df  x 
2
3
dx
 a1  2a2 x  3a3 x  4a4 x    nan x n 1
n 1
 Now if the First Derivative (the Slope) is
KNOWN when x = 0, then
df
dx
df
 a1  2a2 0  3a3 0  4a4 0   a1  0  a1 
dx
2
x 0
Engineering/Math/Physics 25: Computational Methods
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3
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
x 0
Taylor Series
 Again Differentiate Term-by-Term
n 
d2 f
2
n2



2
a

3

2
a
x

4

3
a
x


n
n

1
a
x

2
3
4
n
2
dx
n2
 Now if the 2nd Derivative (the Curvature)
is KNOWN when x = 0, then
d2 f
dx 2
x 0
2
d
f
2
 2a2  3  2a3 0  4  3a4 0   2a2  0  2a2  2
dx
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
x 0
Taylor Series

Another
Differentiation
3
n 
d f
2
n 3




3

2
a

4

3

2
a
x

5

4

3
a
x


n
n

1
n

2
a
x

3
4
4
n
3
dx
n 3
d3 f
dx 3
 Again if the 3rd Derivative is KNOWN at
x=0
x 0
3
d
f
2
 3  2a3  4  3  2a4 0  5  4  3a4 0   6a3  0  6a3  3
dx
 Recognizing
the Pattern:
d n  f
n!an  n 
dx
Engineering/Math/Physics 25: Computational Methods
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 an 
x 0
d n  f
dx n 
n!
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
x 0
x 0
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
n 
• Calculate the Values of the
Taylor Series CoEfficients by an 
• Finally Construct the
Power Series from
the CoEfficients
Engineering/Math/Physics 25: Computational Methods
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d f
dx  n 
n!
n 
f  x    an x n
n 0
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
x 0
Example  Taylor Series for ln(e+x)
 Calculate the Derivatives
df dx
d
1
lne  x  
dx
e x
d 2 f dx 2
d  1 
1



dx  e  x  e  x 2
d 3 f dx 3
d  1 
2


2
dx  e  x   e  x 3
 Find the Values of the Derivatives at 0
df
dxx 0
1
1

e0 e
d
2
Engineering/Math/Physics 25: Computational Methods
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
f dx 2 x 0
1
1
 2
2
e  0 e
d
3

f dx 3 x 0
2
2
 3
3
e  0 e
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Example  Taylor Series for ln(e+x)
 Generally d n  f dx n  x 0   1 nn  1! for n  1
e
 Then the CoEfficients
n 1
an

d

n 
f dx
n!
n 

x 0
 1n1 n  1!

n
e
n!

 1

n 1
ne
n
for n  1
 The 1st four CoEfficients
ln(e + 0)
1 (e + 0) 1
a0 =
=1
a1 =
=
0!
1!
e
-1 (e + 0)2
1
2 (e + 0)3
1
a2 =
= - 2 a3 =
= 3
2!
2e
3!
3e
Engineering/Math/Physics 25: Computational Methods
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Example  Taylor Series for ln(e+x)
 Then the Taylor Series

ln e  x   a0 x 0   an x n
n 1
n 1

 n



1
0
 a0 x   
x
n 
n 1  n  e


(1) n 1 x n
ln(e  x)  1  
n
ne
n 1

Engineering/Math/Physics 25: Computational Methods
28
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Taylor Series at x ≠ 0
 The Taylor Series “Expansion” can
Occur at “Center” Values other than 0
 Consider a function
n 
n




f
x

a
x

b
stated in a series

n
n 0
centered at b, that is:
 Now the Radius of Convergence for the
function is the SAME as the Zero Case:
x  b   R

 R  x b  R

 R b  x bb  R b   R b  x  R b
Engineering/Math/Physics 25: Computational Methods
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Taylor Series at x ≠ 0
 To find the CoEfficients
n 
d
f
need (x−b) = 0 which
dx n  x b f n  b 
requires x = b, Then the an 

n!
n!
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
Engineering/Math/Physics 25: Computational Methods
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Example  Expand x½ about 4
 Expand about b = 4: f x  
x
 The 1st four Taylor CoEfficients
4
a0 =
=2
0!
1 -1/2
(4)
1
2
a1 =
=
1!
4
1 -3/2
- (4)
1
4
a2 =
=2!
64
3 -5/2
(4)
1
8
a3 =
=3!
512
Engineering/Math/Physics 25: Computational Methods
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Example  Expand x½ about 4
 SOLUTION:
 Use the CoEfficients to Construct the
Taylor Series centered at b = 4

x   an ( x  b) n
n 0
1
1
1
5
2
3
= 2 + (x - 4) - (x - 4) +
(x - 4) (x - 4)4 +...
4
64
512
16384
Engineering/Math/Physics 25: Computational Methods
32
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Example  Expand x½ about 4
 Use the Taylor Series centered at b = 4
to Find the Square Root of 3

4
n 0
n 0
3   an (3  b) n   an (3  b) n
1
1
1
5
2
3
 2  (3  4)  (3  4) 
(3  4) 
(3  4) 4
4
64
512
16384
1
1
1
5
2
3
 2  (1)  (1) 
(1) 
(1) 4
4
64
512
16384
 2  0.25  0.0156  0.0020  0.0003
 2  0.2679  1.7321 By MATLAB  1.7320508
Engineering/Math/Physics 25: Computational Methods
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Expand About b=1, ln(x)/1
 Da1 := diff(ln(x)/x, x)

df
dx
1
0
2
1

x 1
 Db2 := diff(Da1, x)
d2 f

dx 2
 0
x 1
3
15
ReCall that
ln(1) = 0
 Dc3 := diff(Db2, x)
d3 f

dx 3

x 1
11
0
4
1
 Dd4 := diff(Dc3, x)
d4 f

dx 4
Engineering/Math/Physics 25: Computational Methods
34
 0
x 1
50
15
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
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)
Engineering/Math/Physics 25: Computational Methods
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
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)
d5 f

dx 5

x 1
274
0
6
1
0 x  1  3 1  x  1 11 1  x  1  50 1  x  1 274 1  x  1
lnx   




0!
1!
2!
3!
4!
5!
1
2
Engineering/Math/Physics 25: Computational Methods
36
3
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
5
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])
Engineering/Math/Physics 25: Computational Methods
37
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Engineering/Math/Physics 25: Computational Methods
38
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Engineering/Math/Physics 25: Computational Methods
39
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
Engineering/Math/Physics 25: Computational Methods
40
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
TYU 11.5 → Sums & Series
 Exercise Taylor’s Series & Sums
• ENGR25_TYU11_5_6789_Taylor_Sums_L
imits_1204.mn
Engineering/Math/Physics 25: Computational Methods
41
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
TYU11.6 → ODEs
 Do an ODE Solution
• file = ENGR25_TYU11_6_ODE_1204.mn
– By: File → Export → PDF
Engineering/Math/Physics 25: Computational Methods
42
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx
All Done for Today
It’s All
GREEK
to me…
Engineering/Math/Physics 25: Computational Methods
43
Bruce Mayer, PE
BMayer@ChabotCollege.edu ENGR-25_Lec-27_MuPAD_Miscellaneous-n-TYUs.pptx