105 ppt taylor - walser-h-m.ch€¦ · 14 sichere extremalstelle: stationäre stelle mit...

Modul 105 Taylor

2

Sind wir auf dem Maximum?

x

y

ab

globalesMaximum

lokalesMinimum

globaleMaximalstelle

lokaleMinimalstelle

y = f x( )

3

x

y

a

b

x1 x2 x3 x4 x5 x6 x7

Monotonie-Intervalle zwischen den Extremalstellen

y = f x( )

4

Stationäre Stellen:

! f x0( ) = 0

Extrema

x

y

5


! f x0( ) = 0

Konstante Funktion

x

y

6


! f x0( ) = 0

Terrassenpunkt

x

y

7

Auf der Suche nach Extremstellen:

8


Randextrema

9


(Relative) Randextrema

10


Spitze, nicht differenzierbar

11


Sprungstelle, nicht differenzierbar

12


Klassischer Fall: Stationäre Stelle

13


Vorsicht: Stationäre Stelle als Terrassenpunkt

14

Sichere Extremalstelle:Stationäre Stelle mit Vorzeichenwechsel der Ableitung

x

y

a b!f < 0f fallend

!f > 0f wachsend

!f x0( ) = 0f stationär

y = f x( )

15

Zweite Ableitung

x

y

a bx

y

a b

positiverDrehsinn

negativerDrehsinn

!f nimmt ab,daher ist !!f < 0

!f wächst,daher ist !!f > 0

16

Zweite Ableitung

x

y

a bx

y

a b

positiverDrehsinn

negativerDrehsinn



17

x

y

a b

positiverDrehsinn

Zweite Ableitung

!f x0( ) = 0

UND

!!f x0( ) > 0

"

#$$

%$$

& Minimum


18

Zweite Ableitung

Umkehrung gilt nicht!

x

y

a b

positiverDrehsinn

!f x0( ) = 0

UND

!!f x0( ) > 0

"

#$$

%$$

& Minimum


19

!f x0( ) = 0

UND

!!f x0( ) > 0

"

#$$

%$$

& Minimum Umkehrung gilt nicht!

20


f x( ) = x4

!f x( ) = 4x3 " !f 0( ) = 0

!!f x( ) = 12x2 " !!f 0( ) = 0

Minimum bei x = 0, aber !!f 0( ) = 0

!f x0( ) = 0

UND

!!f x0( ) > 0

"

#$$

%$$

& Minimum

21

f x( ) = x4

!f x( ) = 4x3 " !f 0( ) = 0

!!f x( ) = 12x2 " !!f 0( ) = 0



!f x0( ) = 0

UND

!!f x0( ) > 0

"

#$$

%$$

& Minimum

22

f x( ) = x4

!f x( ) = 4x3 " !f 0( ) = 0

!!f x( ) = 12x2 " !!f 0( ) = 0



!f x0( ) = 0

UND

!!f x0( ) > 0

"

#$$

%$$

& Minimum

23


!f x0( ) = 0

UND

!!f x0( ) > 0

"

#$$

%$$

& Minimum

f x( ) = x4

!f x( ) = 4x3 " !f 0( ) = 0

!!f x( ) = 12x2 " !!f 0( ) = 0


24

f x( ) = x4

!f x( ) = 4x3 " !f 0( ) = 0

!!f x( ) = 12x2 " !!f 0( ) = 0


Vergleich mitStandardparabel

!f x0( ) = 0

UND

!!f x0( ) > 0

"

#$$

%$$

& Minimum

y = x2

25

Zweite Ableitung

x

y

a bx

y

a b

positiverDrehsinn

negativerDrehsinn



!f x0( ) = 0

UND

!!f x0( ) > 0

"

#$$

%$$

& Minimum

26

x

y

a bx

y

a b

positiverDrehsinn

negativerDrehsinn



!f x0( ) = 0

UND

!!f x0( ) > 0

"

#$$

%$$

& Minimum

!f x0( ) = 0

UND

!!f x0( ) < 0

"

#$$

%$$

& Maximum

Zweite Ableitung

27

Taylorpolynome und Taylorreihen

Brook Taylor1685 - 1731

Idee:Lineare Approximationverallgemeinern

28

Beispiel : Exponenzialfunktion y = f x( ) = ex

Approximation durch Konstante

One size fits all.

Entwicklung an der Stelle x0 = ! 12

29


Lineare Approximation (Tangente)


30


Quadratische Approximation (tangentiale Parabel)


31


Approximation dritten Grades


32


Approximation vierten Grades


33


Approximation fünften Grades


34


Approximation sechsten Grades


35


Approximation siebten Grades


36


Approximation achten Grades


37


Approximation neunten Grades


38


Approximation zehnten Grades


39

f x( ) ! f x0( )

f x( ) ! f x0( ) + "f x0( ) x # x0( )

f x( ) ! f x0( ) + "f x0( ) x # x0( ) + ?( ) x # x0( )2

Formales

(Grobe) Approximation durch Konstante

40

f x( ) ! f x0( )

f x( ) ! f x0( ) + "f x0( ) x # x0( )

f x( ) ! f x0( ) + "f x0( ) x # x0( ) + ?( ) x # x0( )2

Formales


Lineare Approximation

41

Formales


Lineare Approximation

Was passt hier?

f x( ) ! f x0( )

f x( ) ! f x0( ) + "f x0( ) x # x0( )

f x( ) ! f x0( ) + "f x0( ) x # x0( ) + ?( ) x # x0( )2

42

p x( ) = a0!

f x0( )+ a1

!

"f x0( )x # x0( ) + a2

!?

x # x0( )2 + a3!?

x # x0( )3 + $ $ $ + an!?

x # x0( )n

Was passt hier?

43

p x( ) = a0 + a1 x ! x0( ) + a2 x ! x0( )2 + a3 x ! x0( )3 + " " " + an x ! x0( )np x0( ) = a0 = f x0( ) (hatten wir schon)

#p x( ) = a1 + 2a2 x ! x0( ) + 3a3 x ! x0( )2 + " " " + nan x ! x0( )n!1

#p x0( ) = a1 = #f x0( ) (hatten wir schon)

##p x( ) = 2a2 + 2 " 3a3 x ! x0( ) + " " " + n !1( )nan x ! x0( )n!2

##p x0( ) = 2a2 =!

##f x0( ) (aus Systemgründen) $ a2 = 12 ##f x0( )

###p x( ) = 2 " 3a3 + " " " + n ! 2( ) n !1( )nan x ! x0( )n!3

###p x0( ) = 2 " 3a3=!

###f x0( ) (aus Systemgründen) $ a3 = 12"3 ###f x0( )

44




##p x( ) = 2a2 + 2 " 3a3 x ! x0( ) + " " " + n !1( )nan x ! x0( )n!2

##p x0( ) = 2a2 =!


###p x( ) = 2 " 3a3 + " " " + n ! 2( ) n !1( )nan x ! x0( )n!3

###p x0( ) = 2 " 3a3=!


45




##p x( ) = 2a2 + 2 " 3a3 x ! x0( ) + " " " + n !1( )nan x ! x0( )n!2

##p x0( ) = 2a2 =!


###p x( ) = 2 " 3a3 + " " " + n ! 2( ) n !1( )nan x ! x0( )n!3

###p x0( ) = 2 " 3a3=!


46




##p x( ) = 2a2 + 2 " 3a3 x ! x0( ) + " " " + n !1( )nan x ! x0( )n!2

##p x0( ) = 2a2 =!


###p x( ) = 2 " 3a3 + " " " + n ! 2( ) n !1( )nan x ! x0( )n!3

###p x0( ) = 2 " 3a3=!


47




##p x( ) = 2a2 + 2 " 3a3 x ! x0( ) + " " " + n !1( )nan x ! x0( )n!2

##p x0( ) = 2a2 =!


###p x( ) = 2 " 3a3 + " " " + n ! 2( ) n !1( )nan x ! x0( )n!3

###p x0( ) = 2 " 3a3=!


48




##p x( ) = 2a2 + 2 " 3a3 x ! x0( ) + " " " + n !1( )nan x ! x0( )n!2

##p x0( ) = 2a2 =!


###p x( ) = 2 " 3a3 + " " " + n ! 2( ) n !1( )nan x ! x0( )n!3

###p x0( ) = 2 " 3a3=!


49




##p x( ) = 2a2 + 2 " 3a3 x ! x0( ) + " " " + n !1( )nan x ! x0( )n!2

##p x0( ) = 2a2 =!


###p x( ) = 2 " 3a3 + " " " + n ! 2( ) n !1( )nan x ! x0( )n!3

###p x0( ) = 2 " 3a3=!


VorsichtFaktor

50




##p x( ) = 2a2 + 2 " 3a3 x ! x0( ) + " " " + n !1( )nan x ! x0( )n!2

##p x0( ) = 2a2 =!


###p x( ) = 2 " 3a3 + " " " + n ! 2( ) n !1( )nan x ! x0( )n!3

###p x0( ) = 2 " 3a3=!


51




##p x( ) = 2a2 + 2 " 3a3 x ! x0( ) + " " " + n !1( )nan x ! x0( )n!2

##p x0( ) = 2a2 =!


###p x( ) = 2 " 3a3 + " " " + n ! 2( ) n !1( )nan x ! x0( )n!3

###p x0( ) = 2 " 3a3=!


52




##p x( ) = 2a2 + 2 " 3a3 x ! x0( ) + " " " + n !1( )nan x ! x0( )n!2

##p x0( ) = 2a2 =!


###p x( ) = 2 " 3a3 + " " " + n ! 2( ) n !1( )nan x ! x0( )n!3

###p x0( ) = 2 " 3a3=!


53ak = 1

2!3!4! !!! !k fk( ) x0( )Allgemein:




##p x( ) = 2a2 + 2 " 3a3 x ! x0( ) + " " " + n !1( )nan x ! x0( )n!2

##p x0( ) = 2a2 =!


###p x( ) = 2 " 3a3 + " " " + n ! 2( ) n !1( )nan x ! x0( )n!3

###p x0( ) = 2 " 3a3=!


54

ak = 12!3!4! !!! !k f

k( ) x0( )

ak = 11!2!3!4! !!! !k f

k( ) x0( )

ak = 1k! f

k( ) x0( )

k-te Ableitung

55

ak = 12!3!4! !!! !k f

k( ) x0( )

ak = 11!2!3!4! !!! !k f

k( ) x0( )

ak = 1k! f

k( ) x0( )

k-te Ableitung

56

k-te Ableitung

1! 2 ! 3 ! 4 ! ! ! ! ! k = k! "k - Fakultät"( )

ak = 12!3!4! !!! !k f

k( ) x0( )

ak = 11!2!3!4! !!! !k f

k( ) x0( )

ak = 1k! f

k( ) x0( )

57

1! 2 ! 3 ! 4 ! ! ! ! ! k = k! "k - Fakultät"( )0! = 1

1! = 1

2! = 1•2 = 2

3! = 1•2•3 = 6

4! = 1•2•3•4 = 24

5! = 120

6! = 720

7! = 5040

•4

•5

•6

•7

•3

•2

•1

58

1! 2 ! 3 ! 4 ! ! ! ! ! k = k! "k - Fakultät"( )0! = 1

1! = 1

2! = 1•2 = 2

3! = 1•2•3 = 6

4! = 1•2•3•4 = 24

5! = 120

6! = 720

7! = 5040

•4

•5

•6

•7

•3

•2

•1

59

1! 2 ! 3 ! 4 ! ! ! ! ! k = k! "k - Fakultät"( )0! = 1

1! = 1

2! = 1•2 = 2

3! = 1•2•3 = 6

4! = 1•2•3•4 = 24

5! = 120

6! = 720

7! = 5040

•4

•5

•6

•7

•3

•2

•1

60

1! 2 ! 3 ! 4 ! ! ! ! ! k = k! "k - Fakultät"( )0! = 1

1! = 1

2! = 1•2 = 2

3! = 1•2•3 = 6

4! = 1•2•3•4 = 24

5! = 120

6! = 720

7! = 5040

•4

•5

•6

•7

•3

•2

•1

61

1! 2 ! 3 ! 4 ! ! ! ! ! k = k! "k - Fakultät"( )0! = 1

1! = 1

2! = 1•2 = 2

3! = 1•2•3 = 6

4! = 1•2•3•4 = 24

5! = 120

6! = 720

7! = 5040

•4

•5

•6

•7

•3

•2

•1

62

1! 2 ! 3 ! 4 ! ! ! ! ! k = k! "k - Fakultät"( )0! = 1

1! = 1

2! = 1•2 = 2

3! = 1•2•3 = 6

4! = 1•2•3•4 = 24

5! = 120

6! = 720

7! = 5040

•4

•5

•6

•7

•3

•2

•1

63

ak = 1k! f

k( ) x0( )

p x( ) = f x0( ) + !f x0( ) x " x0( ) + 12 !!f x0( ) x " x0( )2 ++ 13! !!!f x0( ) x " x0( )3 + # # #+ 1

n! fn( ) x0( ) x " x0( )n

p x( ) = 1k! f

k( ) x0( )k=0

n$ x " x0( )k

f x( ) % 1k! f

k( ) x0( )k=0

n$ x " x0( )k

64

ak = 1k! f

k( ) x0( )


n! fn( ) x0( ) x " x0( )n

p x( ) = 1k! f

k( ) x0( )k=0

n$ x " x0( )k

f x( ) % 1k! f

k( ) x0( )k=0

n$ x " x0( )k

Taylorpolynom n-ten Grades an der Stelle x0

65

ak = 1k! f

k( ) x0( )


n! fn( ) x0( ) x " x0( )n

p x( ) = 1k! f

k( ) x0( )k=0

n$ x " x0( )k

f x( ) % 1k! f

k( ) x0( )k=0

n$ x " x0( )k

66

Taylorreihe

1k! f

k( ) x0( )k=0

!" x # x0( )k

Der kleineUnterschied

67

Examples, examples, examples

68


f x( ) = 11!x

Taylorpolynom an der Stelle x0 = 0

69


f x( ) = 11!x


70

Entwicklung an der Stelle x0 = 0

f x( ) = 11!x = 1! x( )!1 f 0( ) = 1

"f x( ) = !1 1! x( )!2 !1( ) = 1! x( )!2 "f 0( ) = 1

""f x( ) = !2 1! x( )!3 !1( ) = 2 1! x( )!3 ""f 0( ) = 2

"""f x( ) = !3 #2 1! x( )!4 !1( ) = 2 # 3 1! x( )!4 """f 0( ) = 2 # 3

allgemein

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

71


f x( ) = 11!x = 1! x( )!1 f 0( ) = 1

"f x( ) = !1 1! x( )!2 !1( ) = 1! x( )!2 "f 0( ) = 1

""f x( ) = !2 1! x( )!3 !1( ) = 2 1! x( )!3 ""f 0( ) = 2

"""f x( ) = !3 #2 1! x( )!4 !1( ) = 2 # 3 1! x( )!4 """f 0( ) = 2 # 3

allgemein

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

72


f x( ) = 11!x = 1! x( )!1 f 0( ) = 1

"f x( ) = !1 1! x( )!2 !1( ) = 1! x( )!2 "f 0( ) = 1

""f x( ) = !2 1! x( )!3 !1( ) = 2 1! x( )!3 ""f 0( ) = 2

"""f x( ) = !3 #2 1! x( )!4 !1( ) = 2 # 3 1! x( )!4 """f 0( ) = 2 # 3

allgemein

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

73


f x( ) = 11!x = 1! x( )!1 f 0( ) = 1

"f x( ) = !1 1! x( )!2 !1( ) = 1! x( )!2 "f 0( ) = 1

""f x( ) = !2 1! x( )!3 !1( ) = 2 1! x( )!3 ""f 0( ) = 2

"""f x( ) = !3 #2 1! x( )!4 !1( ) = 2 # 3 1! x( )!4 """f 0( ) = 2 # 3

allgemein

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

InnereAbleitung

74


f x( ) = 11!x = 1! x( )!1 f 0( ) = 1

"f x( ) = !1 1! x( )!2 !1( ) = 1! x( )!2 "f 0( ) = 1

""f x( ) = !2 1! x( )!3 !1( ) = 2 1! x( )!3 ""f 0( ) = 2

"""f x( ) = !3 #2 1! x( )!4 !1( ) = 2 # 3 1! x( )!4 """f 0( ) = 2 # 3

allgemein

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

InnereAbleitung

75


f x( ) = 11!x = 1! x( )!1 f 0( ) = 1

"f x( ) = !1 1! x( )!2 !1( ) = 1! x( )!2 "f 0( ) = 1

""f x( ) = !2 1! x( )!3 !1( ) = 2 1! x( )!3 ""f 0( ) = 2

"""f x( ) = !3 #2 1! x( )!4 !1( ) = 2 # 3 1! x( )!4 """f 0( ) = 2 # 3

allgemein

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

InnereAbleitung

76


f x( ) = 11!x = 1! x( )!1 f 0( ) = 1

"f x( ) = !1 1! x( )!2 !1( ) = 1! x( )!2 "f 0( ) = 1

""f x( ) = !2 1! x( )!3 !1( ) = 2 1! x( )!3 ""f 0( ) = 2

"""f x( ) = !3 #2 1! x( )!4 !1( ) = 2 # 3 1! x( )!4 """f 0( ) = 2 # 3

allgemein

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

InnereAbleitung

77


f x( ) = 11!x = 1! x( )!1 f 0( ) = 1

"f x( ) = !1 1! x( )!2 !1( ) = 1! x( )!2 "f 0( ) = 1

""f x( ) = !2 1! x( )!3 !1( ) = 2 1! x( )!3 ""f 0( ) = 2

"""f x( ) = !3 #2 1! x( )!4 !1( ) = 2 # 3 1! x( )!4 """f 0( ) = 2 # 3

allgemein

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

InnereAbleitung

78


f x( ) = 11!x = 1! x( )!1 f 0( ) = 1

"f x( ) = !1 1! x( )!2 !1( ) = 1! x( )!2 "f 0( ) = 1

""f x( ) = !2 1! x( )!3 !1( ) = 2 1! x( )!3 ""f 0( ) = 2

"""f x( ) = !3 #2 1! x( )!4 !1( ) = 2 # 3 1! x( )!4 """f 0( ) = 2 # 3

allgemein

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

InnereAbleitung

79


f x( ) = 11!x = 1! x( )!1 f 0( ) = 1

"f x( ) = !1 1! x( )!2 !1( ) = 1! x( )!2 "f 0( ) = 1

""f x( ) = !2 1! x( )!3 !1( ) = 2 1! x( )!3 ""f 0( ) = 2

"""f x( ) = !3 #2 1! x( )!4 !1( ) = 2 # 3 1! x( )!4 """f 0( ) = 2 # 3

allgemein

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

80


f x( ) = 11!x = 1! x( )!1 f 0( ) = 1

"f x( ) = !1 1! x( )!2 !1( ) = 1! x( )!2 "f 0( ) = 1

""f x( ) = !2 1! x( )!3 !1( ) = 2 1! x( )!3 ""f 0( ) = 2

"""f x( ) = !3 #2 1! x( )!4 !1( ) = 2 # 3 1! x( )!4 """f 0( ) = 2 # 3

allgemein

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

81

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

p x( ) = 1k! f

k( ) x0( )k=0

n" x ! x0( )k = 1

k! k!k=0

n" x ! 0( )k = xk

k=0

n"

p x( ) = xk

k=0

n" = 1+ x + x2 + x3 +!+ xn

82

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

p x( ) = 1k! f

k( ) x0( )k=0

n" x ! x0( )k = 1

k! k!k=0

n" x ! 0( )k = xk

k=0

n"

p x( ) = xk

k=0

n" = 1+ x + x2 + x3 +!+ xn

83

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

p x( ) = 1k! f

k( ) x0( )k=0

n" x ! x0( )k = 1

k! k!k=0

n" x ! 0( )k = xk

k=0

n"

p x( ) = xk

k=0

n" = 1+ x + x2 + x3 +!+ xn

84

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

p x( ) = 1k! f

k( ) x0( )k=0

n" x ! x0( )k = 1

k! k!k=0

n" x ! 0( )k = xk

k=0

n"

p x( ) = xk

k=0

n" = 1+ x + x2 + x3 +!+ xn

85

f k( ) x( ) = k! 1! x( )!k!1 f k( ) 0( ) = k!

p x( ) = 1k! f

k( ) x0( )k=0

n" x ! x0( )k = 1

k! k!k=0

n" x ! 0( )k = xk

k=0

n"

p x( ) = xk

k=0

n" = 1+ x + x2 + x3 +!+ xn

86

f x( ) = 11!x

87

f x( ) = 11!x

p0 x( ) = 1

88

f x( ) = 11!x p1 x( ) = 1+ x

89

f x( ) = 11!x

p2 x( ) = 1+ x + x2

90

f x( ) = 11!x

p3 x( ) = 1+ x + x2 + x3

91

f x( ) = 11!x

p4 x( ) = 1+ x + x2 + x3 + x4

92

f x( ) = 11!x

p x( ) = xk

k=0

n" = 1+ x + x2 + x3 +!+ xn

f x( ) = 11!x # 1+ x + x2 + x3 +!+ xn

Erinnerung: 1+ q + q2 + $ $ $ = 11!q

93


f x( ) = ln x( )Taylorreihe an der Stelle x0 = 1

Warum nicht 0?

94


f x( ) = ln x( )Taylorreihe an der Stelle x0 = 1

Warum nicht 0?

95


f x( ) = ln x( )!f x( ) = 1

x = x"1

!!f x( ) = "1( )x"2

!!!f x( ) = "1( ) "2( )x"3

f 4( ) x( ) = "1( ) "2( ) "3( )x"4

#

#

#

#

#

f 1( ) = 0

!f 1( ) = 1

!!f 1( ) = "1

!!!f 1( ) = "1( ) "2( ) = 2!

f 4( ) 1( ) = "1( ) "2( ) "3( ) = "3!

allgemein: f k( ) 1( ) = "1( )k"1

alternierendesVorzeichen

!"# $# k "1( )!

96


f x( ) = ln x( )!f x( ) = 1

x = x"1

!!f x( ) = "1( )x"2

!!!f x( ) = "1( ) "2( )x"3

f 4( ) x( ) = "1( ) "2( ) "3( )x"4

#

#

#

#

#

f 1( ) = 0

!f 1( ) = 1

!!f 1( ) = "1

!!!f 1( ) = "1( ) "2( ) = 2!

f 4( ) 1( ) = "1( ) "2( ) "3( ) = "3!

allgemein: f k( ) 1( ) = "1( )k"1


!"# $# k "1( )!

97


f x( ) = ln x( )!f x( ) = 1

x = x"1

!!f x( ) = "1( )x"2

!!!f x( ) = "1( ) "2( )x"3

f 4( ) x( ) = "1( ) "2( ) "3( )x"4

#

#

#

#

#

f 1( ) = 0

!f 1( ) = 1

!!f 1( ) = "1

!!!f 1( ) = "1( ) "2( ) = 2!

f 4( ) 1( ) = "1( ) "2( ) "3( ) = "3!

allgemein: f k( ) 1( ) = "1( )k"1


!"# $# k "1( )!

98


f x( ) = ln x( )!f x( ) = 1

x = x"1

!!f x( ) = "1( )x"2

!!!f x( ) = "1( ) "2( )x"3

f 4( ) x( ) = "1( ) "2( ) "3( )x"4

#

#

#

#

#

f 1( ) = 0

!f 1( ) = 1

!!f 1( ) = "1

!!!f 1( ) = "1( ) "2( ) = 2!

f 4( ) 1( ) = "1( ) "2( ) "3( ) = "3!

allgemein: f k( ) 1( ) = "1( )k"1


!"# $# k "1( )!

99


f x( ) = ln x( )!f x( ) = 1

x = x"1

!!f x( ) = "1( )x"2

!!!f x( ) = "1( ) "2( )x"3

f 4( ) x( ) = "1( ) "2( ) "3( )x"4

#

#

#

#

#

f 1( ) = 0

!f 1( ) = 1

!!f 1( ) = "1

!!!f 1( ) = "1( ) "2( ) = 2!

f 4( ) 1( ) = "1( ) "2( ) "3( ) = "3!

allgemein: f k( ) 1( ) = "1( )k"1


!"# $# k "1( )!

100


f x( ) = ln x( )!f x( ) = 1

x = x"1

!!f x( ) = "1( )x"2

!!!f x( ) = "1( ) "2( )x"3

f 4( ) x( ) = "1( ) "2( ) "3( )x"4

#

#

#

#

#

f 1( ) = 0

!f 1( ) = 1

!!f 1( ) = "1

!!!f 1( ) = "1( ) "2( ) = 2!

f 4( ) 1( ) = "1( ) "2( ) "3( ) = "3!

allgemein: f k( ) 1( ) = "1( )k"1


!"# $# k "1( )!

101


f x( ) = ln x( )!f x( ) = 1

x = x"1

!!f x( ) = "1( )x"2

!!!f x( ) = "1( ) "2( )x"3

f 4( ) x( ) = "1( ) "2( ) "3( )x"4

#

#

#

#

#

f 1( ) = 0

!f 1( ) = 1

!!f 1( ) = "1

!!!f 1( ) = "1( ) "2( ) = 2!

f 4( ) 1( ) = "1( ) "2( ) "3( ) = "3!

allgemein: f k( ) 1( ) = "1( )k"1


!"# $# k "1( )!

102


f x( ) = ln x( )!f x( ) = 1

x = x"1

!!f x( ) = "1( )x"2

!!!f x( ) = "1( ) "2( )x"3

f 4( ) x( ) = "1( ) "2( ) "3( )x"4

#

#

#

#

#

f 1( ) = 0

!f 1( ) = 1

!!f 1( ) = "1

!!!f 1( ) = "1( ) "2( ) = 2!

f 4( ) 1( ) = "1( ) "2( ) "3( ) = "3!

allgemein: f k( ) 1( ) = "1( )k"1


!"# $# k "1( )!

103


f x( ) = ln x( )!f x( ) = 1

x = x"1

!!f x( ) = "1( )x"2

!!!f x( ) = "1( ) "2( )x"3

f 4( ) x( ) = "1( ) "2( ) "3( )x"4

#

#

#

#

#

f 1( ) = 0

!f 1( ) = 1

!!f 1( ) = "1

!!!f 1( ) = "1( ) "2( ) = 2!

f 4( ) 1( ) = "1( ) "2( ) "3( ) = "3!

allgemein: f k( ) 1( ) = "1( )k"1


!"# $# k "1( )!

104


f x( ) = ln x( )!f x( ) = 1

x = x"1

!!f x( ) = "1( )x"2

!!!f x( ) = "1( ) "2( )x"3

f 4( ) x( ) = "1( ) "2( ) "3( )x"4

#

#

#

#

#

f 1( ) = 0

!f 1( ) = 1

!!f 1( ) = "1

!!!f 1( ) = "1( ) "2( ) = 2!

f 4( ) 1( ) = "1( ) "2( ) "3( ) = "3!

allgemein: f k( ) 1( ) = "1( )k"1


!"# $# k "1( )!

105


f x( ) = ln x( )!f x( ) = 1

x = x"1

!!f x( ) = "1( )x"2

!!!f x( ) = "1( ) "2( )x"3

f 4( ) x( ) = "1( ) "2( ) "3( )x"4

#

#

#

#

#

f 1( ) = 0

!f 1( ) = 1

!!f 1( ) = "1

!!!f 1( ) = "1( ) "2( ) = 2!

f 4( ) 1( ) = "1( ) "2( ) "3( ) = "3!

allgemein: f k( ) 1( ) = "1( )k"1


!"# $# k "1( )!

106


f k( ) 1( ) = !1( )k!1 k !1( )!ak = 1

k! fk( ) x0( )

"#$

%$&

ak = 1k !1( )k!1 für k > 0

a0 = 0 weil f 1( ) = ln 1( ) = 0

'($

)$

ln x( ) = x !1( ) ! 12 x !1( )2 + 1

3 x !1( )3 ! 14 x !1( )4 ± * * * = !1( )k!1 1

k x !1( )kk=1

+,

107


f k( ) 1( ) = !1( )k!1 k !1( )!ak = 1

k! fk( ) x0( )

"#$

%$&

ak = 1k !1( )k!1 für k > 0

a0 = 0 weil f 1( ) = ln 1( ) = 0

'($

)$

ln x( ) = x !1( ) ! 12 x !1( )2 + 1

3 x !1( )3 ! 14 x !1( )4 ± * * * = !1( )k!1 1

k x !1( )kk=1

+,

108


f k( ) 1( ) = !1( )k!1 k !1( )!ak = 1

k! fk( ) x0( )

"#$

%$&

ak = 1k !1( )k!1 für k > 0

a0 = 0 weil f 1( ) = ln 1( ) = 0

'($

)$

ln x( ) = x !1( ) ! 12 x !1( )2 + 1

3 x !1( )3 ! 14 x !1( )4 ± * * * = !1( )k!1 1

k x !1( )kk=1

+,

109


f k( ) 1( ) = !1( )k!1 k !1( )!ak = 1

k! fk( ) x0( )

"#$

%$&

ak = 1k !1( )k!1 für k > 0

a0 = 0 weil f 1( ) = ln 1( ) = 0

'($

)$

ln x( ) = x !1( ) ! 12 x !1( )2 + 1

3 x !1( )3 ! 14 x !1( )4 ± * * * = !1( )k!1 1

k x !1( )kk=1

+,

110


f k( ) 1( ) = !1( )k!1 k !1( )!ak = 1

k! fk( ) x0( )

"#$

%$&

ak = 1k !1( )k!1 für k > 0

a0 = 0 weil f 1( ) = ln 1( ) = 0

'($

)$

ln x( ) = x !1( ) ! 12 x !1( )2 + 1

3 x !1( )3 ! 14 x !1( )4 ± * * * = !1( )k!1 1

k x !1( )kk=1

+,

111


f k( ) 1( ) = !1( )k!1 k !1( )!ak = 1

k! fk( ) x0( )

"#$

%$&

ak = 1k !1( )k!1 für k > 0

a0 = 0 weil f 1( ) = ln 1( ) = 0

'($

)$

ln x( ) = x !1( ) ! 12 x !1( )2 + 1

3 x !1( )3 ! 14 x !1( )4 ± * * * = !1( )k!1 1

k x !1( )kk=1

+,


112

ln x( ) ! x "1( ) " 12 x "1( )2 + 13 x "1( )3 " 14 x "1( )4 + 15 x "1( )5 " 16 x "1( )6 ± # # #

113

ln x( ) ! x "1( ) " 12 x "1( )2 + 13 x "1( )3 " 14 x "1( )4 + 15 x "1( )5 " 16 x "1( )6 ± # # #

114

ln x( ) ! x "1( ) " 12 x "1( )2 + 13 x "1( )3 " 14 x "1( )4 + 15 x "1( )5 " 16 x "1( )6 ± # # #

115

ln x( ) ! x "1( ) " 12 x "1( )2 + 13 x "1( )3 " 14 x "1( )4 + 15 x "1( )5 " 16 x "1( )6 ± # # #

116

ln x( ) ! x "1( ) " 12 x "1( )2 + 13 x "1( )3 " 14 x "1( )4 + 15 x "1( )5 " 16 x "1( )6 ± # # #

117

ln x( ) ! x "1( ) " 12 x "1( )2 + 13 x "1( )3 " 14 x "1( )4 + 15 x "1( )5 " 16 x "1( )6 ± # # #

118

ln x( ) ! x "1( ) " 12 x "1( )2 + 13 x "1( )3 " 14 x "1( )4 + 15 x "1( )5 " 16 x "1( )6 ± # # #

119

ln x( ) ! x "1( ) " 12 x "1( )2 + 13 x "1( )3 " 14 x "1( )4 + 15 x "1( )5 " 16 x "1( )6 ± # # #

120

CAS

ln x( ) ! x "1( ) " 12 x "1( )2 + 13 x "1( )3 " 14 x "1( )4 + 15 x "1( )5 " 16 x "1( )6 ± # # #

121

CAS


ln x( ) ! x "1( ) " 12 x "1( )2 + 13 x "1( )3 " 14 x "1( )4 + 15 x "1( )5 " 16 x "1( )6 ± # # #

122


f x( ) = ex


123


f x( ) = ex f 0( ) = 1

!f x( ) = ex !f 0( ) = 1

!!f x( ) = ex !!f 0( ) = 1

!!!f x( ) = ex !!!f 0( ) = 1

allgemein

f k( ) x( ) = ex f k( ) 0( ) = 1

124


f x( ) = ex f 0( ) = 1

!f x( ) = ex !f 0( ) = 1

!!f x( ) = ex !!f 0( ) = 1

!!!f x( ) = ex !!!f 0( ) = 1

allgemein

f k( ) x( ) = ex f k( ) 0( ) = 1

125


f x( ) = ex f 0( ) = 1

!f x( ) = ex !f 0( ) = 1

!!f x( ) = ex !!f 0( ) = 1

!!!f x( ) = ex !!!f 0( ) = 1

allgemein

f k( ) x( ) = ex f k( ) 0( ) = 1

126

f k( ) x( ) = ex f k( ) 0( ) = 1

p x( ) = 1k! f

k( ) x0( )k=0

n! x " x0( )k = 1

k!k=0

n! x " 0( )k = 1

k! xk

k=0

n!

p x( ) = 1k! x

k

k=0

n! = 1+ x + 12 x

2 + 16 x3 +!+ 1

n! xn

ex # 1+ x + 12 x2 + 16 x

3 +!+ 1n! x

n

127


Approximation durch Konstante

ex ! 1+ x + 12 x

2 + 16 x3 +!+ 1

n! xn

128


Lineare Approximation (Tangente)

ex ! 1+ x + 12 x

2 + 16 x3 +!+ 1

n! xn

129


Quadratische Approximation (tangentiale Parabel)

ex ! 1+ x + 12 x

2 + 16 x3 +!+ 1

n! xn

130


Approximation dritten Grades

ex ! 1+ x + 12 x

2 + 16 x3 +!+ 1

n! xn

131


Approximation vierten Grades

ex ! 1+ x + 12 x

2 + 16 x3 + 1

24 x4 "!+ 1

n! xn

132

ex = 1+ x + 12 x

2 + 13! x

3 + 14! x

4 + ! !! = 1k! x

k

k=0

"#

Für x = 1 folgt:

e = 1+1+ 12 + 1

3! +14! + ! ! !

e = 1k!

k=0

"#

Taylor-Reihe

133

ex = 1+ x + 12 x

2 + 13! x

3 + 14! x

4 + ! !! = 1k! x

k

k=0

"#

Für x = 1 folgt:

e = 1+1+ 12 + 1

3! +14! + ! ! !

e = 1k!

k=0

"#

Taylor-Reihe

134

Formel von Euler

e = 1+1+ 12 +13! +

14! + ! ! !

e = 1k!

k=0

"#

Leonhard Euler1707 - 1783

135

Formel von Euler e = 1+1+ 12 +13! +

14! + ! !! = 1

k!k=0

"#

k k ! 1/k ! Summe0 1 1 1.00000000001 1 1 2.00000000002 2 0.5 2.50000000003 6 0.166666667 2.66666666674 24 0.041666667 2.70833333335 120 0.008333333 2.71666666676 720 0.001388889 2.71805555567 5040 0.000198413 2.71825396838 40320 2.48016E-05 2.71827876989 362880 2.75573E-06 2.718281525610 3628800 2.75573E-07 2.718281801111 39916800 2.50521E-08 2.718281826212 479001600 2.08768E-09 2.718281828313 6227020800 1.6059E-10 2.718281828414 87178291200 1.14707E-11 2.7182818285

136


f x( ) = sin x( )Taylorreihe an der Stelle x0 = 0

137


f x( ) = sin x( )!f x( ) = cos x( )!!f x( ) = " sin x( )!!!f x( ) = " cos x( )

f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0

allgemein: f k( ) 0( ) =0 falls k gerade

1 falls k : 4 Rest 1 ergibt

"1 falls k : 4 Rest 3 ergibt

$

%&&

'&&

138



f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0




$

%&&

'&&

139



f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0




$

%&&

'&&

140



f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0




$

%&&

'&&

141



f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0




$

%&&

'&&

142



f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0




$

%&&

'&&

Jetzt fängt eswieder von vorne an.

143



f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0




$

%&&

'&&

144



f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0




$

%&&

'&&

145



f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0




$

%&&

'&&

146



f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0




$

%&&

'&&

147



f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0




$

%&&

'&&

148



f 4( ) x( ) = sin x( )

#

#

#

#

#

f 0( ) = 0

!f 0( ) = 1

!!f 0( ) = 0

!!!f 0( ) = "1

f 4( ) 0( ) = 0




$

%&&

'&&

sin x( ) = x ! 13! x

3 + 15! x

5 ! 17! x

7 ± " " "

149

sin x( ) = x ! 13! x

3 + 15! x

5 ! 17! x

7 ± " "" = !1( )k 12k+1( )! x

2k+1

k=0

#$

150

sin x( ) = x ! 13! x

3 + 15! x

5 ! 17! x

7 ± " "" = !1( )k 12k+1( )! x

2k+1

k=0

#$ ?

151

sin x( ) = x ! 13! x

3 + 15! x

5 ! 17! x

7 ± " "" = !1( )k 12k+1( )! x

2k+1

k=0

#$

Hinkendes VorzeichenAlternierendes Vorzeichen

?

152

sin x( ) = x ! 13! x

3 + 15! x

5 ! 17! x

7 ± " "" = !1( )k 12k+1( )! x

2k+1

k=0

#$

Nur ungerade ExponentenFakultäten von ungeraden Zahlen

153

sin x( ) = x ! 13! x

3 + 15! x

5 ! 17! x

7 ± " "" = !1( )k 12k+1( )! x

2k+1

k=0

#$

Nur ungerade ExponentenSinus ist eine ungerade Funktion

154

sin x( ) ! x " 13! x

3 + 15! x

5 " 17! x

7 + 19! x

9 ! # # #

! 2!!!!2!

155

sin x( ) ! x " 13! x

3 + 15! x

5 " 17! x

7 + 19! x

9 ! # # #

! 2!!!!2!

156

sin x( ) ! x " 13! x

3 + 15! x

5 " 17! x

7 + 19! x

9 ! # # #

! 2!!!!2!

157

sin x( ) ! x " 13! x

3 + 15! x

5 " 17! x

7 + 19! x

9 ! # # #

! 2!!!!2!

158

sin x( ) ! x " 13! x

3 + 15! x

5 " 17! x

7 + 19! x

9 ! # # #

! 2!!!!2!

159

sin x( ) ! x " 13! x

3 + 15! x

5 " 17! x

7 + 19! x

9 ! # # #

! 2!!!!2!

160

Entwicklung bis zum Grad 39

sin x( ) ! x " 13! x

3 + 15! x

5 " 17! x

7 + 19! x

9 ! # # #

! 2!!!!2!

161

Entwicklung bis zum Grad 15

CAS

sin x( ) ! x " 13! x

3 + 15! x

5 " 17! x

7 + 19! x

9 ! # # #

162


f x( ) = cos x( )Taylorreihe an der Stelle x0 = 0

Analog sin(x)

163

cos x( ) = 1! 12 x2 + 1

4! x4 ! 1

6! x6 ± " " " = !1( )k 1

2k( )! x2k

k=0

#$

164

cos x( ) = 1! 12 x2 + 1

4! x4 ! 1

6! x6 ± " " " = !1( )k 1

2k( )! x2k

k=0

#$

Alternierendes Vorzeichen

165

cos x( ) = 1! 12 x2 + 1

4! x4 ! 1

6! x6 ± " " " = !1( )k 1

2k( )! x2k

k=0

#$

Nur gerade ExponentenKosinus ist eine gerade Funktion

1 = x0

166

cos x( ) ! 1" 12 x2 + 1

4! x4 " 1

6! x6 + 1

8! x8 ! # # #

! 2!!!!2!

167

cos x( ) ! 1" 12 x2 + 1

4! x4 " 1

6! x6 + 1

8! x8 ! # # #

! 2!!!!2!

168

cos x( ) ! 1" 12 x2 + 1

4! x4 " 1

6! x6 + 1

8! x8 ! # # #

! 2!!!!2!

169

cos x( ) ! 1" 12 x2 + 1

4! x4 " 1

6! x6 + 1

8! x8 ! # # #

! 2!!!!2!

170

cos x( ) ! 1" 12 x2 + 1

4! x4 " 1

6! x6 + 1

8! x8 ! # # #

! 2!!!!2!

171

cos x( ) ! 1" 12 x2 + 1

4! x4 " 1

6! x6 + 1

8! x8 ! # # #

! 2!!!!2!

172

Vergleich

ex = 1 + x + 12 x2 + 1

3! x3 + 1

4! x4 + 1

5! x5 + 1

6! x6 + ! ! !

cos x( ) = 1 " 12 x2 + 1

4! x4 " 1

6! x6 ± ! ! !

sin x( ) = x " 13! x

3 + 15! x

5 ! ! ! !

Was ist der Link?

Problem mit Minuszeichen

173

!Erinnerung

174

Erinnerung

i so, dass i2 = !1

Menge der komplexen Zahlen

! = a + ib a,b "", i2 = !1{ }

175

i0 = 1

i1 = i

i2 = !1

i3 = !i

i4 = 1

i5 = i

Potenzen von i

176

i0 = 1

i1 = i

i2 = !1

i3 = !i

i4 = 1

i5 = i

Potenzen von i

Gemäß Definition

177

i0 = 1

i1 = i

i2 = !1

i3 = !i

i4 = 1

i5 = i

Potenzen von i

Gemäß Definition

178

i0 = 1

i1 = i

i2 = !1

i3 = !i

i4 = 1

i5 = i

Potenzen von i

Periodisches VerhaltenPeriodenlänge 4

179

Potenzen von i

i0 = 1

i1 = i

i2 = !1

i3 = !i

i4 = 1

i5 = i

180

Vergleich

eix = 1 + ix + 12 ix( )2 + 1

3! ix( )3 + 14! ix( )4 + 1

5! ix( )5 + 16! ix( )6 + ! ! !

eix = 1 + ix " 12 x2 " i 13! x

3 + 14! x

4 + i 15! x5 " 1

6! x6 !

cos x( ) = 1 " 12 x2 + 1

4! x4 " 1

6! x6 ± ! ! !

i sin x( ) = ix " i 13! x3 + i 15! x

5 ! ! ! !

181

Vergleich

eix = 1 + ix + 12 ix( )2 + 1

3! ix( )3 + 14! ix( )4 + 1

5! ix( )5 + 16! ix( )6 + ! ! !

eix = 1 + ix " 12 x2 " i 13! x

3 + 14! x

4 + i 15! x5 " 1

6! x6 !

cos x( ) = 1 " 12 x2 + 1

4! x4 " 1

6! x6 ± ! ! !

i sin x( ) = ix " i 13! x3 + i 15! x

5 ! ! ! !

182

Vergleich

eix = 1 + ix + 12 ix( )2 + 1

3! ix( )3 + 14! ix( )4 + 1

5! ix( )5 + 16! ix( )6 + ! ! !

eix = 1 + ix " 12 x2 " i 13! x

3 + 14! x

4 + i 15! x5 " 1

6! x6 !

cos x( ) = 1 " 12 x2 + 1

4! x4 " 1

6! x6 ± ! ! !

i sin x( ) = ix " i 13! x3 + i 15! x

5 ! ! ! !

Hurra hurradie Vorzeichen sind da!

183

Faktor idazu tun

Faktor idazu tun

Faktor idazu tun

Faktor idazu tun

Vergleich

eix = 1 + ix + 12 ix( )2 + 1

3! ix( )3 + 14! ix( )4 + 1

5! ix( )5 + 16! ix( )6 + ! ! !

eix = 1 + ix " 12 x2 " i 13! x

3 + 14! x

4 + i 15! x5 " 1

6! x6 !

cos x( ) = 1 " 12 x2 + 1

4! x4 " 1

6! x6 ± ! ! !

i sin x( ) = ix " i 13! x3 + i 15! x

5 ! ! ! !

184

eix = cos x( ) + i sin x( )

Formel von Euler

Vergleich

eix = 1 + ix + 12 ix( )2 + 1

3! ix( )3 + 14! ix( )4 + 1

5! ix( )5 + 16! ix( )6 + ! ! !

eix = 1 + ix " 12 x2 " i 13! x

3 + 14! x

4 + i 15! x5 " 1

6! x6 !

cos x( ) = 1 " 12 x2 + 1

4! x4 " 1

6! x6 ± ! ! !

i sin x( ) = ix " i 13! x3 + i 15! x

5 ! ! ! !

185

Sonderfall: x = 2!

e2!i = cos 2!( )1

!"# $# + i sin 2!( )0

!"# $#

e2!i = 1


Formel von Euler

186


Formel von Euler

Sonderfall: x = 2!

e2!i = cos 2!( )1

!"# $# + i sin 2!( )0

!"# $#

e2!i = 1

187


Formel von Euler

Sonderfall: x = 2!

e2!i = cos 2!( )1

!"# $# + i sin 2!( )0

!"# $#

e2!i = 1

188

e2! i = 1

Die schönste Formel

Eulersche Zahl

Bezeichnung von EulerBezeichnung von Eulerpopularisiert.

Sachverhalt von Euler

189

Die Geometrie der Sache

190

Realteil

Imaginärteil

|z| (Betra

g)

φ = arg(z)

z = 4 + 3i

191

tan arg z( )( ) = tan !( ) = Im z( )Re z( )

z = Re z( )( )2 + Im z( )( )2 = z z

Argument und Betrag„Polarkoordinaten“

Re z( ) = z cos !( )Im z( ) = z sin !( )

192

Re z( ) = z cos !( ) Im z( ) = z sin !( )

z = Re z( ) + i Im z( ) = z cos !( ) + i sin !( )( )

Euler: ei! = cos !( ) + i sin !( )

z = Re z( ) + i Im z( ) = z cos !( ) + i sin !( )( ) = z ei!

z = z ei!

193

Re z( ) = z cos !( ) Im z( ) = z sin !( )




z = z ei!

194

Re z( ) = z cos !( ) Im z( ) = z sin !( )




z = z ei!

195

Re z( ) = z cos !( ) Im z( ) = z sin !( )




z = z ei!

196

Re z( ) = z cos !( ) Im z( ) = z sin !( )




z = z ei!

197

Realteil

Imaginärteil

|z| (Betra

g)

φ = arg(z)

z = 4 + 3i z = z ei!

198

Realteil

Imaginärteil

|z| (Betra

g)

φ = arg(z)

z = z ei!

Polarkoordinaten

199

Multiplikation

z = z ei! w = w ei"

zw = z ei! w ei" = z w ei !+"( )

Andererseits:

zw = zw ei arg zw( )

Vergleich:zw = z w

arg zw( ) = arg z( ) + arg w( )

200

z = z ei! w = w ei"

zw = z ei! w ei" = z w ei !+"( )

Andererseits:


Vergleich:zw = z w


Multiplikation

201

z = z ei! w = w ei"

zw = z ei! w ei" = z w ei !+"( )

Andererseits:


Vergleich:zw = z w


Multiplikation

202

z = z ei! w = w ei"

zw = z ei! w ei" = z w ei !+"( )

Andererseits:


Vergleich:zw = z w


Multiplikation

203

Multiplikation

zw = z w


Der Betrag des Produktes ist gleich dem Produkt der Beträge der Faktoren.

Das Argument des Produktes ist gleich der Summe der Argumente der Faktoren

Division

zw = z

w

arg zw( ) = arg z( ) ! arg w( )

204

Multiplikation

zw = z w




Beispiel: !1= ei!

!1( ) !1( ) = ei!ei! = ei!+i! = e2!i = 1

205

Multiplikation

zw = z w




Beispiel: !1= ei!

!1( ) !1( ) = ei!ei! = ei!+i! = e2!i = 1

206

z = 3+ 2i w = 2 + 5i! zw = "4 +19i

z = 13, w = 29

und zw = 377 = 13 29

arg z( ) = arctan 23( ) # 0.588 # 33.69°

arg w( ) = arctan 52( ) #1.190 # 68.20°

und

arg zw( ) = arctan 19"4( ) + $ #1.778 #101.89°

zw = – 4 + 19i

w = 2 + 5i

z = 3 + 2i

Beispiel:

207

z = 3+ 2i w = 2 + 5i! zw = "4 +19i

z = 13, w = 29

und zw = 377 = 13 29

arg z( ) = arctan 23( ) # 0.588 # 33.69°

arg w( ) = arctan 52( ) #1.190 # 68.20°

und

arg zw( ) = arctan 19"4( ) + $ #1.778 #101.89°

zw = – 4 + 19i

w = 2 + 5i

z = 3 + 2i

Beispiel:

208

z = 3+ 2i w = 2 + 5i! zw = "4 +19i

z = 13, w = 29

und zw = 377 = 13 29

arg z( ) = arctan 23( ) # 0.588 # 33.69°

arg w( ) = arctan 52( ) #1.190 # 68.20°

und

arg zw( ) = arctan 19"4( ) + $ #1.778 #101.89°

zw = – 4 + 19i

w = 2 + 5i

z = 3 + 2i

Beispiel:

209

zw = – 4 + 19i

w = 2 + 5i

z = 3 + 2i

z = 3+ 2i w = 2 + 5i! zw = "4 +19i

z = 13, w = 29

und zw = 377 = 13 29

arg z( ) = arctan 23( ) # 0.588 # 33.69°

arg w( ) = arctan 52( ) #1.190 # 68.20°

und

arg zw( ) = arctan 19"4( ) + $ #1.778 #101.89°

Beispiel:

210

Drehstreckung mit

w = 2 + 5i

zw = – 4 + 19i

w = 2 + 5i

z = 3 + 2i

Faktor:

w = 29 ! 5.39

Drehwinkel:

arg w( ) = arctan 52( ) ! 1.190 ! 68.20°

Beispiel:

211

Drehstreckung mit

w = 2 + 5i

210-1-2-3-4-5

7

6

5

4

3

2

1

0

Beispiel:

212

Potenzen und Wurzeln

zn = z n

arg zn( ) = narg z( )

zn = zn

arg zn( ) = 1n arg z( )

213

Einheitswurzeln

zn =1 ! zn "1= 0

Lösungen?

214

Einheitswurzeln

zn =1 ! zn "1= 0

Lösungen?

Reelle Lösungen :

±1 falls n gerade+1 falls n ungerade

215

Einheitswurzeln

z2 !1= 0 Lösungen !1,1{ }

z4 !1= 0 Lösungen i,!1,!i, 1{ }

216

Einheitswurzeln

1

z2 !1= 0 Lösungen !1,1{ }

z4 !1= 0 Lösungen i,!1,!i, 1{ }

217

Einheitswurzeln

Lösungsmenge = ! 12 + i 3

2 ,! 12 ! i 3

2 , 1{ }

z3 !1 = 0

z3 !1 = z !1( ) z2 + z +1( ) = 0 " z3 = 1

z2 + z +1 = 0 " z1,2 =!1± !3

2 = ! 12 ± i32

Nummer 3

218

Einheitswurzeln


2 ,! 12 ! i 3

2 , 1{ }

z3 !1 = 0

z3 !1 = z !1( ) z2 + z +1( ) = 0 " z3 = 1

z2 + z +1 = 0 " z1,2 =!1± !3

2 = ! 12 ± i32

219

Einheitswurzeln


2 , ! 12 ! i 3

2 , 1{ }

z3 !1 = 0

z3 !1 = z !1( ) z2 + z +1( ) = 0 " z3 = 1

z2 + z +1 = 0 " z1,2 =!1± !3

2 = ! 12 ± i32

220

Einheitswurzeln

z3 !1 = 0 Lösungsmenge = ! 12 + i 3

2 , ! 12 ! i 3

2 , 1{ }

1

1

–1

–1

221

Einheitswurzeln

Argumente: 2!3 ,

4!3 , 0{ }

1

1

–1

–1

1

1

–1

–1

z1

z2

z3

z3 !1 = 0 Lösungsmenge = ! 12 + i 3

2 , ! 12 ! i 3

2 , 1{ }

222

Einheitswurzeln

zn = 1

Lösungen:

zk = eik 2!n = cos k 2!n( ) + isin k 2!n( ), k " 1,2, ... ,n{ }

223

Einheitswurzeln

Kontrolle:

zkn = eik

2!n"

#$%&'n= ei2k! = ei2!( )k = 1k = 1

zn = 1

Lösungen:


224

Einheitswurzeln

Regelmäßiges n-EckEine Ecke bei 1

zn = 1

Lösungen:


225

z5 = 32

Lösungen:

zk = 2eik 2!5 = 2 cos k 2!5( ) + isin k 2!5( )( ), k " 1,2, ... ,5{ }

Beispiel

226

Beispiel

z5 = 32

Lösungen:


227

Beispiel

Regelmäßiges 5-EckEine Ecke bei 2

z5 = 32

Lösungen:


Lösungsmenge =

= z1 = 2ei2!5 , z2 = 2e

i2 2!5 , z3 = 2ei32!5 , z4 = 2e

i4 2!5 , z5 = 2ei5 2!5"

#$

%&'

228

Beispiel

Regelmäßiges 5-EckEine Ecke bei 2

z5 = 32

Lösungen:

zk = 2eik 2!5 = 2 cos k 2!5( ) + isin k 2!5( )( ), k " 1,2, ... ,5{ }x

y

1

1

z1

z2

z3

z4

z5

229

Beispiel

Eine reelle Lösung, zwei Paare konjugiert komplexer Lösungen

z5 = 32

Lösungen:

zk = 2eik 2!5 = 2 cos k 2!5( ) + isin k 2!5( )( ), k " 1,2, ... ,5{ }x

y

1

1

z1

z2

z3

z4

z5

105 ppt taylor - walser-h-m.ch€¦ · 14 sichere extremalstelle: stationäre stelle mit...

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