Abgabe Papier

Lösungen

1

Ÿ a

Abgabe Papier

Ÿ b

Abgabe elektronisch

2

Remove @ "Global`*" D

Ÿ a

LaplaceTransform @ Sin @ 3 t + Pi D , t, s D

- 3

€€€€€€€€€€€€€€€

9 + s

LaplaceTransform @ Cos @ 3 t D , t, s D

€€€€€€€€€€€€€€€ s 9 + s

LaplaceTransform @ Sin @ 3 t + Pi D + Cos @ 3 t D + t E ^ -H 3 t L , t, s D

€€€€€€€€€€€€€€€€€€€€ 1 H 3 + s L

- 3

€€€€€€€€€€€€€€€

9 + s

+ s

€€€€€€€€€€€€€€€

9 + s

% •• Together -18 - 9 s + 4 s

+ s

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€€€€€€

H 3 + s L

H 9 + s

L

Ÿ b

LaplaceTransform @ Sin @ 3 t + Pi D + Cos @ 3 t D + t E ^ -H 3 t L , t, s D • . s ® 3

€€€€€€€ 1

36

Ÿ c

LaplaceTransform @ Sin @ 3 t + Pi D + Cos @ 3 t D + t E ^ - H 3 t L , t, s D • . s ® -3 ComplexInfinity

Kommentar: s < 0 Š> s ausserhalb des Definitionsbereiches von F(s)! Die Polstelle von F(s) ist bedeutungslos.

3

Remove @ "Global`*" D f @ t_ D := Sin @ t D ; f @ t_ • ; t < 1 D := t;

f @ t_ • ; 1 £ t && t < Pi • 2 D := 1;

Plot @ f @ t D , 8 t, -1, 10 <D ;

2 4 6 8 10

-1 -0.5 0.5 1

Integrate @ E ^ H-s t L t, 8 t, 0, 1 <D •• ExpandAll

€€€€€€€ 1 s

- €€€€€€€€€ ã

s

- €€€€€€€€€ ã

s

Integrate @ E ^ H -s t L 1, 8 t, 1, Pi • 2 <D •• ExpandAll ã

€€€€€€€€€

s - €€€€€€€€€€€€€ ã

s

Integrate @ E ^ H-s t L Sin @ t D , 8 t, Pi • 2, Infinity <D •• ExpandAll If A Re @ s D > 0, ã

s

€€€€€€€€€€€€€€€€€

1 + s

, Integrate Aã

Sin @ t D , 9 t, €€€€ p

2 , ¥= , Assumptions ® Re @ s D £ 0 EE Integrate @ E ^ H-s t L t, 8 t, 0, 1 <D + Integrate @ E ^ H-s t L 1, 8 t, 1, Pi • 2 <D +

Integrate @ E ^ H -s t L Sin @ t D , 8 t, Pi • 2, Infinity <D •• ExpandAll

€€€€€€€ 1 s

- ã

€€€€€€€€€

s

- ã

€€€€€€€€€€€€€

s + If A Re @ s D > 0, ã

s

€€€€€€€€€€€€€€€€€

1 + s

, Integrate A ã

Sin @ t D , 9 t, €€€€ p

2 , ¥ = , Assumptions ® Re @ s D £ 0 EE

€€€€€€€ 1

s

- €€€€€€€€€€€€€ · @ s D

s

- €€€€€€€€€€€€€€€€€€€ · @ s D

s + €€€€€€€€€€€€€€€€€€€€€€€ s · @ s D

1 + s

4

Ÿ a

x @ 0 D = 0; y @ 0 D = 1;

H LaplaceTransform @ x ' @ t D - 2 y @ t D , t, s D == LaplaceTransform @ DiracDelta @ t D , t, s DL • . 8 LaplaceTransform @ x @ t D , t, s D ® Xs, LaplaceTransform @ y @ t D , t, s D ® Ys <

s Xs - 2 Ys Š 1

H LaplaceTransform @ x @ t D + y ' @ t D , t, s D == LaplaceTransform @-Sin @ t D , t, s DL • . 8 LaplaceTransform @ x @ t D , t, s D ® Xs, LaplaceTransform @ y @ t D , t, s D ® Ys <

-1 + Xs + s Ys Š - 1

€€€€€€€€€€€€€€€

1 + s

Ÿ b

solv = Solve A9 s Xs - 2 Ys Š 1, -1 + Xs + s Ys Š - 1

€€€€€€€€€€€€€€€

1 + s

= , 8 Xs, Ys <E •• Flatten 9 Xs ® €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ s H 1 + 2 s + s

L

2 + 3 s

+ s

, Ys ® - 1 + s

- s

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

2 + 3 s

+ s

=

X @ s_ D := Xs • . solv; X @ s D s H 1 + 2 s + s

L

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

2 + 3 s

+ s

Apart @ X @ s DD

- 2

€€€€€€€€€€€€€€€

1 + s

+ 4 + s

€€€€€€€€€€€€€€€

2 + s

Y @ s_ D := Ys • . solv; Y @ s D - 1 + s

- s

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

2 + 3 s

+ s

Apart @ Y @ s DD

- s

€€€€€€€€€€€€€€€

1 + s

+ -1 + 2 s

€€€€€€€€€€€€€€€€€€€€

2 + s

Ÿ c

InverseLaplaceTransform @ X @ s D , s, t D •• Expand Cos A•!!!! 2 t E - 2 Sin @ t D + 2 •!!!! 2 Sin A•!!!! 2 t E

InverseLaplaceTransform @ Y @ s D , s, t D •• Expand -Cos @ t D + 2 Cos A•!!!! 2 t E - Sin A•!!!! 2 t E

€€€€€€€€€€€€€€€€€€€€€€€€€€€€ •!!!! 2

5

Remove @ "Global`*" D

Ÿ a

y @ 0 D = 0; y ' @ 0 D = 1;

u = H LaplaceTransform @ y '' @ t D - y ' @ t D + 2 y @ t D , t, s D == LaplaceTransform @ 1, t, s DL • . 8 LaplaceTransform @ y @ t D , t, s D ® Ys <

-1 + 2 Ys - s Ys + s

Ys Š 1

€€€€ s

solv = Solve @ u, 8 Ys <D •• Flatten

9 Ys ® 1 + s

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

s H 2 - s + s

L =

H Y @ s_ D := Ys • . solv L ; H Y @ s DL •• Apart

€€€€€€€€€ 1

2 s + 3 - s

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

2 H 2 - s + s

L

Ÿ b

InverseLaplaceTransform @ Y @ s D , s, t D •• Expand

€€€€ 1 2 - 1

€€€€ 2 ã

Cos A •!!!! 7 t

€€€€€€€€€€€€€€

2 E + 5 ã

Sin A €€€€€€€€€€

E

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ €€€€€€€€€€

2 •!!!! 7

Plot A 1

€€€€€€€

14 i

k jjjjjj 7 + ã

i

k jjjjjj -7 Cos A •!!!! 7 t

€€€€€€€€€€€€€€

2 E + 5 •!!!!

7 Sin A •!!!! 7 t

€€€€€€€€€€€€€€

2 E y { zzzzzz y

{ zzzzzz , 8 t, 0, 7 <E ;

1 2 3 4 5 6 7

-5 5 10 15 20 25

Ÿ d

ã

wächst über alle Grenzen. Cosinus und Sinus sorgen für ein Hin - und Herpendeln. Der Limes für t gegen unendlich ist nicht definiert.

6

Remove @ "Global`*" D

<<Statistics`StatisticsPlots`

M1 = Round @ 100 * Table @ 6.2 + 0.4 Random @D , 8 t, 1, 20 <DD • 100 •• N 8 6.34, 6.5, 6.44, 6.51, 6.26, 6.34, 6.33, 6.33, 6.58,

6.4, 6.6, 6.58, 6.57, 6.36, 6.5, 6.55, 6.48, 6.26, 6.42, 6.21 <

H* M1=8 6.36`,6.56`,6.42`,6.27`,6.36`,6.47`,6.29`,6.54`,6.3`,

6.55`,6.46`,6.49`,6.38`,6.4`,6.35`,6.41`,6.56`,6.25`,6.34`,6.52` < *L M1 = 8 6.36`, 6.56`, 6.42`, 6.27`, 6.36`, 6.47`, 6.29`, 6.54`,

6.3`, 6.55`, 6.46`, 6.49`, 6.38`, 6.4`, 6.35`, 6.41`, 6.56`, 6.25`, 6.34` <

8 6.36, 6.56, 6.42, 6.27, 6.36, 6.47, 6.29, 6.54,

6.3, 6.55, 6.46, 6.49, 6.38, 6.4, 6.35, 6.41, 6.56, 6.25, 6.34 <

M2 = Round @ 100 * Table @ 6.3 + 0.3 Random @D , 8 t, 1, 20 <DD • 100 •• N 8 6.46, 6.49, 6.43, 6.44, 6.35, 6.57, 6.56, 6.5, 6.31,

6.47, 6.46, 6.4, 6.32, 6.32, 6.46, 6.42, 6.34, 6.5, 6.53, 6.46 <

H* M2=8 6.55`,6.48`,6.51`,6.51`,6.43`,6.51`,6.35`,6.46`,6.31`,

6.31`,6.58`,6.51`,6.54`,6.35`,6.39`,6.59`,6.4`,6.5`,6.57`,6.44` < *L

M2 = 8 6.55`, 6.48`, 6.51`, 6.51`, 6.43`, 6.51`, 6.35`, 6.46`,

6.31`, 6.31`, 6.58`, 6.51`, 6.54`, 6.35`, 6.39`, 6.59`, 6.4`, 6.5`, 6.57` <

8 6.55, 6.48, 6.51, 6.51, 6.43, 6.51, 6.35, 6.46,

6.31, 6.31, 6.58, 6.51, 6.54, 6.35, 6.39, 6.59, 6.4, 6.5, 6.57 <

MM = 8 M1, M2 < •• Transpose

88 6.36, 6.55 < , 8 6.56, 6.48 < , 8 6.42, 6.51 < , 8 6.27, 6.51 < ,

8 6.36, 6.43 < , 8 6.47, 6.51 < , 8 6.29, 6.35 < , 8 6.54, 6.46 < , 8 6.3, 6.31 < , 8 6.55, 6.31 < , 8 6.46, 6.58 < , 8 6.49, 6.51 < , 8 6.38, 6.54 < , 8 6.4, 6.35 < , 8 6.35, 6.39 < , 8 6.41, 6.59 < , 8 6.56, 6.4 < , 8 6.25, 6.5 < , 8 6.34, 6.57 <<

Ÿ a

<< Statistics`DescriptiveStatistics`

minmax @ t_ D := 8 Max @ t D , Min @ t D , Max @ t D - Mean @ t D , Mean @ t D - Min @ t D , Max @ t D - Min @ t D<

minmax @ M1 D

8 6.56, 6.25, 0.151579, 0.158421, 0.31 <

minmax @ M2 D

8 6.59, 6.31, 0.124211, 0.155789, 0.28 <

LocationReport @ M1 D

8 Mean ® 6.40842, HarmonicMean ® 6.40694, Median ® 6.4 <

LocationReport @ M2 D

8 Mean ® 6.46579, HarmonicMean ® 6.46458, Median ® 6.5 <

Ÿ b

DispersionReport @ M1 D

8 Variance ® 0.010014, StandardDeviation ® 0.10007, SampleRange ® 0.31,

MeanDeviation ® 0.0825485, MedianDeviation ® 0.07, QuartileDeviation ® 0.07125 <

DispersionReport @ M2 D

8 Variance ® 0.00821462, StandardDeviation ® 0.0906345, SampleRange ® 0.28, MeanDeviation ® 0.0764543, MedianDeviation ® 0.07, QuartileDeviation ® 0.07 <

q @ M_ D := Quartiles @ M D q @ M1 D

8 6.3425, 6.4, 6.485 <

q @ M1 D@@ 3 DD - q @ M1 D@@ 1 DD H* Quartilsdifferenz *L

0.1425

q @ M2 D@@ 3 DD - q @ M2 D@@ 1 DD H* Quartilsdifferenz *L 0.14

Ÿ c

Abs @ Mean @ M1 D - Min @ M1 DD > 2 StandardDeviation @ M1 D False

Abs @ Mean @ M2 D - Min @ M2 DD > 2 StandardDeviation @ M2 D False

Keine schwachen Ausreisser!

Ÿ d

BoxWhiskerPlot @ MM D ;

1 2

6.25 6.3 6.35 6.4 6.45 6.5 6.55

7

Ÿ a

14 Š> 13 nummerierte Zwischenstellen. 2 daraus auswählen.

Binomial @ 13, 2 D

78

Ÿ b

5 auswählen, aus dem Rest nochmals 5 und dann die bleibenden 4 Binomial @ 14 - 5 2, 4 D

1

Binomial @ 14, 5 D Binomial @ 14 - 5, 5 D Binomial @ 14 - 5 2, 4 D 252252

Ÿ c

Binomial @ 14, 5 D 5 Binomial @ 14 - 5, 5 D 5 Binomial @ 14 - 5 2, 4 D 4 25225200

8

1000 Leute:

Nicht infisziert, Test richtig (richtigerweise nicht behandlet) 1000 0.9 0.7

630.

Nicht infisziert, Test falsch (fälschlicherweise behandlet) 1000 0.9 0.3

270.

Infisziert, Test richtig (richtigerweise behandlet) 1000 0.1 0.7

70.

Infisziert, Test falsch (fälschlicherweise nicht behandlet) 1000 0.1 0.3

30.

270 Menschen werden hier fälschlicherweise behandlet mit grossen Folgen und 30 fälschlicherweise nicht behandlet mit

noch grösseren Folgen