Lösung zu Aufgabe 1

(1)

Lösung zu Aufgabe 1

à a

f[x_,y_]:= Sqrt[x^2+y^2] E^(-Sqrt[x^2+y^2]/2);

Plot3D[f[x,y],{x,-2,2},{y,-2,2}];

-2

-1

0

1 2 -2 -1

0 1

2 0.5

0.6 0.7

-2

-1

0

1 Plot3D[f[x,y],{x,-2,2},{y,-2,2},ViewPoint->{3.667, 2.785, -1.120},PlotRange->{0,1}];

-1 -2 1 0

2 -2 -1 0

1

2

0

0.2

0.4

0.6

0.8

1

(2)

Plot3D[f[x,y],{x,-20,20},{y,-20,20},ViewPoint->{3.667, 2.785, -1.120},PlotRange->{0,1}];

-10 -20 10 0

20 -20 -10 0 10

20 0 0.2 0.4 0.6 0.8 1

h[r_]:= r E^(-r/2);

ParametricPlot3D[{r Cos[t],r Sin[t], h[r]}, {r,0,3}, {t,0,2Pi}, ViewPoint->{3.667, 2.785, -1.120},

AspectRatio->1];

-2 0

2 -2

0

2

0

0.2

0.4

0.6

(3)

ParametricPlot3D[{r Cos[t],r Sin[t], h[r]}, {r,0,2}, {t,0,2Pi}, AspectRatio->1];

-2

-1

0

1 2 -2 -1

0 1

2 0 0.2 0.4 0.6

-2

-1

0

1 à b

D[f[x,y],x]/.x->u

- 1

€€€€ 2 ã

^-^€€€€¹² •!!!!!!!!!!!!!!!u²+y²

u + ã

^-^€€€€¹² •!!!!!!!!!!!!!!!u²+y²

u

€€€€€€€€€€€€€€€€€€€€€€€€€€€€ •!!!!!!!!!!!!!!!! u

²

+ y

²

(4)

Plot[Evaluate[D[f[x,0],x]/.x->u],{u,-3,3}];

-3 -2 -1 1 2 3

-1 -0.5 0.5 1

Limit[Evaluate[D[f[x,0],x]/.x->u],u->0, Direction ® 1]

-1

Limit[Evaluate[D[f[x,0],x]/.x->u],u->0, Direction ® -1]

1 à c

Plot[h[r],{r,0,3}];

0.5 1 1.5 2 2.5 3

0.55

0.6

0.65

0.7

(5)

D[h[r],r]/.r->2 0

hMax=h[2]

— General::spell1 :

Possible spelling error: new symbol

name "hMax" is similar to existing symbol "Max". Mehr…

€€€€ 2 ã

hMax=h[2]//N 0.735759

à d

<<Calculus`VectorAnalysis`

Drop[Grad[f[x,y], Cartesian[x, y, z]],{3}]

9 - 1

€€€€ 2 ã

^-^€€€€¹² •!!!!!!!!!!!!!!!x²+y²

x + ã

^-^€€€€¹²•!!!!!!!!!!!!!!!x²+y²

x

€€€€€€€€€€€€€€€€€€€€€€€€€€€€ •!!!!!!!!!!!!!!!! x

²

+ y

²

, - 1

€€€€ 2 ã

^-^€€€€¹² •!!!!!!!!!!!!!!!x²+y²

y + ã

^-^€€€€¹² •!!!!!!!!!!!!!!!x²+y²

y

€€€€€€€€€€€€€€€€€€€€€€€€€€€€ •!!!!!!!!!!!!!!!! x

²

+ y

²

=

Drop[Grad[f[x,y], Cartesian[x, y, z]],{3}] /. {x->1,y->1}

9 - 1

€€€€ 2 ã

^-^{€€€€€€€€}^•!!!!!¹²

+ €€€€€€€€€€€€€ ã •!!!!

^-^{€€€€€€€€}^•!!!!!

2

¹²

, - 1

€€€€ 2 ã

^-^{€€€€€€€€}^•!!!!!¹²

+ €€€€€€€€€€€€€ ã •!!!!

^-^{€€€€€€€€}^•!!!!!

2

¹²

=

N[%]

8 0.102118, 0.102118 <

(6)

à e

Oberfl=

Integrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],x]^2+1]/.{x-

>x1,y->y1}],{x1,1,2},{y1,1,2}]

á

1 2

á

1 2

&'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' 1 + ã

^-•!!!!!!!!!!!!!!!!!!!!!x1²+y1²

x1

²

I -2 + •!!!!!!!!!!!!!!!!!!!!! x1

²

+ y1

²

M

²

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

2 H x1

²

+ y1

²

L â y1 â x1

NIntegrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],x]^2+1]/.{x ->x1,y->y1}],{x1,1,2},{y1,1,2}]

1.00133

NIntegrate[Evaluate[Sqrt[D[f[x,y],x]^2+D[f[x,y],x]^2+1]/.{x ->x1,y->y1}],{x1,0,2},{y1,1,2}]

2.00319

**Remove["Global`*"]**

Lösung zu Aufgabe 2

à a

DSolve @8 y '' @ x D + 2 y @ x D Š Cos @ x D< , y @ x D , x D •• Simplify 88 y @ x D ® Cos @ x D + C @ 1 D Cos A•!!!! 2 x E + C @ 2 D Sin A•!!!! 2 x E<<

N[%]

88 y @ x D ® Cos @ x D + C @ 1 D Cos @ 1.41421 x D + C @ 2 D Sin @ 1.41421 x D<<

(7)

à b

DSolve[{y''[x]-y'[x] + 2 y[x] ŠCos[x]},y[x],x ]//Simplify

99 y @ x D ® 1

€€€€ 2 i

k jjj Cos @ x D + 2 ã

^x^•²

C @ 2 D Cos A •!!!! 7 x

€€€€€€€€€€€€€

2 E - Sin @ x D + 2 ã

^x^•²

C @ 1 D Sin A •!!!! 7 x

€€€€€€€€€€€€€

2 Ey { zzz==

N[%]

88 y @ x D ® 0.5 H Cos @ x D + 2. 2.71828

^{0.5 x}

C @ 2 D Cos @ 1.32288 x D - 1. Sin @ x D + 2. 2.71828

^{0.5 x}

C @ 1 D Sin @ 1.32288 x DL<<

à c

DSolve @8 y '' @ x D - y ' @ x D + 2 y @ x D Š Cos @ x D , y @ 0 D Š 0, y ' @ 0 D Š 0 < , y @ x D , x D •• Simplify

99 y @ x D ® 1

€€€€€€€

14 i

k jjj 7 Cos @ x D - 7 ã

^x^•²

Cos A •!!!! 7 x

€€€€€€€€€€€€€

2 E - 7 Sin @ x D + 3 •!!!! 7 ã

^x^•²

Sin A •!!!! 7 x

€€€€€€€€€€€€€

2 Ey { zzz==

N[%]

88 y @ x D ® 0.0714286 H 7. Cos @ x D - 7. 2.71828

^{0.5 x}

Cos @ 1.32288 x D -

7. Sin @ x D + 7.93725 2.71828

^{0.5 x}

Sin @ 1.32288 x DL<<

(8)

solv=DSolve[{y''[x]-y'[x]+2y[x]ŠCos[x],y[0]Š0,y'[0]Š0},y,x ]//Simplify//Flatten

9 y ®

Function A8 x < , 1

€€€€€€€

14 i

k jjjjj -7 ã

^x^•²

Cos A •!!!! 7 x

€€€€€€€€€€€€€

2 E + 7 Cos @ x D Cos A •!!!! 7 x

€€€€€€€€€€€€€

2 E

2

-

7 Cos A •!!!! 7 x

€€€€€€€€€€€€€

2 E

2

Sin @ x D + 3 •!!!! 7 ã

^x^•²

Sin A •!!!! 7 x

€€€€€€€€€€€€€

2 E + 7 Cos @ x D Sin A •!!!! 7 x

€€€€€€€€€€€€€

2 E

2

- 7 Sin @ x D Sin A •!!!! 7 x

€€€€€€€€€€€€€

2 E

2

y { zzzzzE=

N[%]

8 y ® Function @8 x < ,

0.0714286 H -7. 2.71828

^{0.5 x}

Cos @ 0.5 2.64575 x D + 7. Cos @ x D Cos @ 0.5 2.64575 x D

²

- 7. Cos @ 0.5 2.64575 x D

²

Sin @ x D + 3. 2.64575 2.71828

^{0.5 x}

Sin @ 0.5 2.64575 x D + 7. Cos @ x D

Sin @ 0.5 2.64575 x D

²

- 7. Sin @ x D Sin @ 0.5 2.64575 x D

²

LD<

y=y/.solv;

y[z]//Simplify

€€€€€€€ 1 14

i

k jjj 7 Cos @ z D - 7 ã

^z^•²

Cos A •!!!! 7 z

€€€€€€€€€€€€€

2 E - 7 Sin @ z D + 3 •!!!! 7 ã

^z^•²

Sin A •!!!! 7 z

€€€€€€€€€€€€€

2 Ey { zzz

N[%]

0.0714286 H 7. Cos @ z D - 7. 2.71828

^{0.5 z}

Cos @ 1.32288 z D -

7. Sin @ z D + 7.93725 2.71828

^{0.5 z}

Sin @ 1.32288 z DL

(9)

à d

pl=Plot[y[z],{z,0,12}];

2 4 6 8 10 12

-60 -40 -20 20 40 60

à e

FindRoot[y[x],{x,3}]

8 x ® 2.75558 <

y[2.75558]

1.56584 ´ 10

^-6

**Remove["Global`*"]**

(10)

Lösung zu Aufgabe 3

à a

s1=Normal[Series[Sin[x],{x,0,10}]]

x - x

³

€€€€€€€

6 + x

⁵

€€€€€€€€€€

120 - x

⁷

€€€€€€€€€€€€€

5040 + x

⁹

€€€€€€€€€€€€€€€€€€

362880

Plot[{s1,Sin[x]},{x,-10,10}];

-10 -5 5 10

-4 -2 2 4

s2=Normal[Series[Cos[x],{x,0,10}]]

1 - x

²

€€€€€€€

2 + x

⁴

€€€€€€€

24 - x

⁶

€€€€€€€€€€

720 + x

⁸

€€€€€€€€€€€€€€€€

40320 - x

¹⁰

€€€€€€€€€€€€€€€€€€€€€

3628800

(11)

à b

Plot[s1-Sin[x],{x,-4,4},PlotRange->{-0.01,0.01}]

-4 -2 2 4

-0.01 -0.0075 -0.005 -0.0025 0.0025 0.005 0.0075 0.01

… Graphics …

à c

D[s1,x]

1 - x

²

€€€€€€€

2 + x

⁴

€€€€€€€

24 - x

⁶

€€€€€€€€€€

720 + x

⁸

€€€€€€€€€€€€€€€€

40320

à d

s1+s2

1 + x - x

²

€€€€€€€

2 - x

³

€€€€€€€

6 + x

⁴

€€€€€€€

24 + x

⁵

€€€€€€€€€€

120 - x

⁶

€€€€€€€€€€

720 - x

⁷

€€€€€€€€€€€€€

5040 + x

⁸

€€€€€€€€€€€€€€€€

40320 + x

⁹

€€€€€€€€€€€€€€€€€€

362880 - x

¹⁰

€€€€€€€€€€€€€€€€€€€€€

3628800

(12)

Sqrt[2] Sin[x+Pi/4]//TrigExpand Cos @ x D + Sin @ x D

s3=Normal[Series[Sqrt[2] Sin[x+Pi/4],{x,0,10}]]

1 + x - x

²

€€€€€€€

2 - x

³

€€€€€€€

6 + x

⁴

€€€€€€€

24 + x

⁵

€€€€€€€€€€

120 - x

⁶

€€€€€€€€€€

720 - x

⁷

€€€€€€€€€€€€€

5040 + x

⁸

€€€€€€€€€€€€€€€€

40320 + x

⁹

€€€€€€€€€€€€€€€€€€

362880 - x

¹⁰

€€€€€€€€€€€€€€€€€€€€€

3628800

à e

Plot[{Sin[x]/x,s1/x},{x,-1,1}]

-1 -0.5 0.5 1

0.85 0.875 0.9 0.925 0.95 0.975

… Graphics …

à f

Integrate[s1/x,x]

x - x

³

€€€€€€€

18 + x

⁵

€€€€€€€€€€

600 - x

⁷

€€€€€€€€€€€€€€€€

35280 + x

⁹

€€€€€€€€€€€€€€€€€€€€€

3265920

(13)

NIntegrate[Simplify[s1/x],{x,-1,1}]

1.89217

NIntegrate[Sin[x]/x,{x,-1,1}]

1.89217

Lösung zu Aufgabe 4

**Remove["Global`*"]**

à a

tr[x_]:= (Pi+ (Pi-2 x))/2 Sin[x]; tr[x]//Simplify H p - x L Sin @ x D

Plot[Evaluate[D[tr[x],x]/.x->u],{u,0,Pi}];

0.5 1 1.5 2 2.5 3

-1 1 2 3

FindRoot[(Evaluate[D[tr[x],x]]==0) /.x->u,{u,1}]

8 u ® 1.11283 <

(14)

(Evaluate[D[tr[x],x]]) /.x->1.11283 0.0000130208

tr[x] /.x->1.11283 1.81971

Integrate[Sin[x],{x,0,Pi}]

2 Integrate[Sin[x],{x,0,Pi}] - (tr[x] /.x->1.11283) 0.180294

à b

h[a_] := 2.5 + 4 Tan[a];

g[a_] := 1/Tan[a] h[a];

laenge[a_] := Sqrt[h[a]^2+g[a]^2]

Plot[laenge[a],{a,0,Pi/2}];

0.25 0.5 0.75 1 1.25 1.5

100

200

300

400

(15)

D[laenge[a],a]

H 8 Csc @ a D

²

H 2.5 + 4 Tan @ a DL + 8 Sec @ a D

²

H 2.5 + 4 Tan @ a DL - 2 Cot @ a D Csc @ a D

²

H 2.5 + 4 Tan @ a DL

²

L ’

J 2 "############################################################################################################ H 2.5 + 4 Tan @ a DL

²

+ Cot @ a D

²

H 2.5 + 4 Tan @ a DL

²

N

fr=FindRoot[(Evaluate[D[laenge[a],a]==0] /.a->u),{u,0.75}]

8 u ® 0.707383 <

laenge[u] /. fr 9.10977

**Remove["Global`*"]**

Lösung zu Aufgabe 5,

differenzieren und integrieren

à a Resultat so kurz wie möglich halten

f[x_]:=x^2+ 0.5 Sin[x];

f[x]

x

²

+ 0.5 Sin @ x D

f'[x]

2 x + 0.5 Cos @ x D

f'[x]/.x->1

2.27015

(16)

ArcTan[f'[x]/.x->1]

1.15587

ArcTan[f'[x]/.x->1]/Degree 66.2265

f''[x]/.x->1 1.57926

à b Resultat so kurz wie möglich halten

f[x_]:=(((x^(3/4)-3)^3)^(1/4))^5

f[x]

IH -3 + x

³^•⁴

L

³

M

⁵^•⁴

D[f[x],x]//Simplify

45 H -3 + x

³^•⁴

L

²

IH -3 + x

³^•⁴

L

³

M

¹^•⁴

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

16 x

¹^•⁴

D[f[x],x]//Expand

405 IH -3 + x

³^•⁴

L

³

M

¹^•⁴

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

16 x

¹^•⁴

-

€€€€€€€€€€ 135

8 IH -3 + x

³^•⁴

L

³

M

¹^•⁴

•!!!! x + 45

€€€€€€€

16 IH -3 + x

³^•⁴

L

³

M

¹^•⁴

x

⁵^•⁴

(17)

à c Resultat so kurz wie möglich halten

f[x_]:= 4 x^2 Sin[3x^2-2]

f[x]

-4 x

²

Sin @ 2 - 3 x

²

D

D[f[x],x]//Simplify

24 x

³

Cos @ 2 - 3 x

²

D - 8 x Sin @ 2 - 3 x

²

D

D[f[x],x]//Expand

24 x

³

Cos @ 2 - 3 x

²

D - 8 x Sin @ 2 - 3 x

²

D

à d Resultat so kurz wie möglich halten

f[x_]:= Log[x/(1-x)] (x/(1-x))

f[x]

x Log @ €€€€€€€

_1-x^x

D

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

1 - x

D[f[x],x]//Simplify 1 + Log @ €€€€€€€

_1-x^x

D

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

H -1 + x L

²

D[f[x],x]//Expand

€€€€€€€€€€€€ 1

1 - x + x

€€€€€€€€€€€€€€€€€€€€

H 1 - x L

²

+ €€€€€€€€€€€€€€€€€€€€€€€€ Log @ €€€€€€€

₁^x_-_x

D

1 - x + €€€€€€€€€€€€€€€€€€€€€€€€€€€€€ x Log @ €€€€€€€

₁^x_-_x

D

H 1 - x L

²

(18)

à e Resultat so kurz wie möglich halten

f[x_]:= 4 E^(-x) Cos[2-E^x]

f[x]

4 ã

^-^x

Cos @ 2 - ã

^x

D

D[f[x],x]//Simplify

-4 ã

^-x

Cos @ 2 - ã

^x

D + 4 Sin @ 2 - ã

^x

D

D[f[x],x]//Expand

-4 ã

^-^x

Cos @ 2 - ã

^x

D + 4 Sin @ 2 - ã

^x

D

à f Resultat so kurz wie möglich halten

Remove[a]

f[x_]:= x^2+a Sin[x]+2/x

Integrate[f[x],{x,1,E^2}]

€€€€ 1

3 H 11 + ã

⁶

+ 3 a Cos @ 1 D - 3 a Cos @ ã

²

DL

Integrate[f[x],{x,1,E^2}]//Expand

€€€€€€€ 11 3 + ã

⁶

€€€€€€€

3 + a Cos @ 1 D - a Cos @ ã

²

D

(19)

N[%]//Expand

138.143 + 0.0919461 a

**Remove["Global`*"]**

Lösung zu Aufgabe 6, Kurven und Flächen

Gegeben sei die Kurvenschar y=x^a auf I=[0,1]

à a Plot

Plot[{x^5,x^(1/5)},{x,0,1}];

0.2 0.4 0.6 0.8 1

à b y=x^(a) soll die eingeschlossenen Fläche halbieren. Berechne a

r1=Evaluate[Integrate[x^5-x^(a),{x,0,1}]==Integrate[x^(a)-x

^(1/5),{x,0,1}]][[1]][[2]];

r2=Evaluate[Integrate[x^5-x^(a),{x,0,1}]==Integrate[x^(a)-x

^(1/5),{x,0,1}]][[2]][[2]];

h=(r1==r2)

€€€€ 1 6 - 1

€€€€€€€€€€€€

1 + a Š - 5

€€€€ 6 + 1

€€€€€€€€€€€€

1 + a

(20)

Solve[h,{a}]

88 a ® 1 <<

r1=Evaluate[Integrate[x^5-x^(a),{x,0,1}]==Integrate[x^(a)-x

^(1/5),{x,0,1}]][[1]][[2]];

r2=Evaluate[

Integrate[x^5-x^(a),{x,0,1}]==Integrate[x^(a)-x^(1/5),{x,0, 1}]][[2]][[2]];

h=(r1==3 r2)

€€€€ 1 6 - 1

€€€€€€€€€€€€

1 + a Š 3 J - 5

€€€€ 6 + 1

€€€€€€€€€€€€

1 + a N

Solve[h,{a}]

99 a ® 1

€€€€ 2 ==

r1=Evaluate[Integrate[x^5-x^(a),{x,0,1}]==Integrate[x^(a)-x

^(1/5),{x,0,1}]][[1]][[2]];

r2=Evaluate[

Integrate[x^5-x^(a),{x,0,1}]==Integrate[x^(a)-x^(1/5),{x,0, 1}]][[2]][[2]];

h=(r1==(Sqrt[5]+1)/2 r2)

€€€€ 1 6 - 1

€€€€€€€€€€€€

1 + a Š 1

€€€€ 2 I 1 + •!!!! 5 M J - 5

€€€€ 6 + 1

€€€€€€€€€€€€

1 + a N

solv1=Solve[h,{a}]//Flatten

9 a ® 11 + •!!!! 5

€€€€€€€€€€€€€€€€€€€€€

7 + 5 •!!!! 5 =

N[%]

8 a ® 0.728043 <

(21)

r3=Evaluate[Integrate[x^5-x^(a),{x,0,1}]==Integrate[x^(a)-x

^(1/5),{x,0,1}]][[1]][[2]];

r4=Evaluate[

Integrate[x^5-x^(a),{x,0,1}]==Integrate[x^(a)-x^(1/5),{x,0, 1}]][[2]][[2]];

h=(r3==(Sqrt[5]-1)/2 r2)

€€€€ 1 6 - 1

€€€€€€€€€€€€

1 + a Š 1

€€€€ 2 I -1 + •!!!! 5 M J - 5

€€€€ 6 + 1

€€€€€€€€€€€€

1 + a N

N[%]

0.166667 - 1.

€€€€€€€€€€€€€€€

1. + a Š 0.618034 J -0.833333 + 1

€€€€€€€€€€€€€€€

1. + a N

solv2=Solve[h,{a}]//Flatten

9 a ® 9 + •!!!! 5

€€€€€€€€€€€€€€€€€€€€€€€€

-3 + 5 •!!!! 5 =

N[%]

8 a ® 1.37355 <

Plot[{x^5,x^(1/5),x^1},{x,0,1}];

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

(22)

Plot[{x^5,x^(1/5),x^(1/2)},{x,0,1}];

0.2 0.4 0.6 0.8 1

à Plot Goldener Schnitt

Plot[Evaluate[{x^5,x^(1/5),x^(a)/.solv1}],{x,0,1},AspectRat io->Automatic];

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

(23)

Plot[Evaluate[{x^5,x^(1/5),x^(a)/.solv2}],{x,0,1},AspectRat io->Automatic];

0.2 0.4 0.6 0.8 1

**Remove["Global`*"]**

Lösung zu Aufgabe 7, Rotationskörper, Parabel

Funktion durch Parabel angenähert. Gleiche Ableitung bei -1 und Werte bei -1 und 1, Winkelabweichung bei 1?

f[x_]:=(x^2-1) E^(-x);

p[x_,a_,b_,c_]:=a x^2+b x + c

f'[x]/.x->-1

-2 ã

(24)

solv=Solve[Evaluate[{p[-1,a,b,c]==0, p[1,a,b,c]==0, (D[p[x,a,b,c],x]==f'[x]/.x->-1)}]]//Flatten

8 a ® ã, b ® 0, c ® -ã <

%//N

8 a ® 2.71828, b ® 0., c ® -2.71828 <

p[x_]:=a x^2+b x + c/.solv; p[x]

-ã + ã x

²

p'[x]/.x->1 2 ã

f'[x]/.x->1

€€€€ 2 ã

ArcTan[2 E]-ArcTan[2/ E] //N 0.754566

(ArcTan[2 E]-ArcTan[2/ E]) /Degree//N

43.2334

(25)

à a Plot

Plot[{f[x],p[x]},{x,-1,1}];

-1 -0.5 0.5 1

-2.5 -2 -1.5 -1 -0.5

à Flächenverhältnis

Integrate[(x^2-1) E^(-x),x]

-ã

^-x

H 1 + x L

²

Integrate[(x^2-1) E^(-x),{x,-1,1}]

- 4

€€€€ ã

%//N -1.47152

Integrate[p[x],x]

-ã x + ã x

³

€€€€€€€€€€€

3

(26)

Integrate[p[x],{x,-1,1}]

- 4 ã

€€€€€€€€

3 %//N -3.62438

int=Integrate[(f[x]-p[x]),{x,-1,1}]/Integrate[(p[x]),{x,-1, 1}]

- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 3 H - €€€

⁴_ã

+ €€€€€€

⁴₃^ã

L 4 ã

int//N -0.593994

Integrate[(f[x]),{x,-1,1}]/Integrate[(p[x]),{x,-1,1}]

€€€€€€€ 3 ã

²

%//N 0.406006

1/%//N

2.46302

(27)

à Wendepunkt

Evaluate[f''[x]]/.x->0 1

Solve[Evaluate[f''[x]==0],{x}]

— Solve::ifun :

Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. Mehr…

88 x ® 2 - •!!!! 3 < , 8 x ® 2 + •!!!! 3 <<

— Solve::ifun :

Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. Mehr…

88 x ® 2 - •!!!! 3 < , 8 x ® 2 + •!!!! 3 <<

Solve[Evaluate[f''[x]==0],{x}]//N

— Solve::ifun :

Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. Mehr…

88 x ® 0.267949 < , 8 x ® 3.73205 <<

à Rotationsvolunen bei Parabe, Vergleich mit Zylinder gleicher Länge, r =

? (Hat Volumen in Zylinder mit r=2 Platz?)

Pi Integrate[p[x]^2,{x,-1,1}]

16 ã

²

p

€€€€€€€€€€€€€€€€€

15 %//N

24.761

(28)

Sqrt[Integrate[p[x]^2,{x,-1,1}]/2]

2 $%%%%%%%%% 2

€€€€€€€

15 ã

%//N 1.98515

Plot[{f[x],p[x],-1.9851,1.9851},{x,-1,1}];

-1 -0.5 0.5 1

-2 -1 1 2

à Länge der Bögen:

{f[x],p[x]}

8 ã

^-x

H -1 + x

²

L , -ã + ã x

²

<

l1=NIntegrate[Evaluate[Sqrt[1+D[p[x],x]^2]],{x,-1,1}]

5.96824

l2=NIntegrate[Evaluate[Sqrt[1+D[f[x],x]^2]],{x,-1,1}]

3.37052

(29)

l3=2 2

l1/l2 1.77072

l1/l3 2.98412

l2/l3 1.68526

**Remove["Global`*"]**

Lösung zu Aufgabe 7,

Variante: Rotationskörper, Parabel mit Punkteverwechslung

Funktion durch Parabel angenähert. Gleiche Ableitung bei -1 und Werte bei -1 und 1, Winkelabweichung bei 1?

f[x_]:=(x^2-1) E^(-x);

p[x_,a_,b_,c_]:=a x^2+b x + c

f'[x]/.x->1

€€€€ 2 ã

solv=Solve[Evaluate[{p[-1,a,b,c]==0, p[1,a,b,c]==0, (D[p[x,a,b,c],x]==f'[x]/.x->1)}]]//Flatten

9 c ® - 1

€€€€ ã , a ® 1

€€€€ ã , b ® 0 =

(30)

%//N

8 c ® -0.367879, a ® 0.367879, b ® 0. <

p[x_]:=a x^2+b x + c/.solv; p[x]

- 1

€€€€ ã + x

²

€€€€€€€

ã

p'[x]/.x->1

€€€€ 2 ã

f'[x]/.x->1

€€€€ 2 ã

ArcTan[2 E]-ArcTan[2/ E] //N 0.754566

(ArcTan[2 E]-ArcTan[2/ E]) /Degree//N

43.2334

(31)

à a Plot

Plot[{f[x],p[x]},{x,-1,1}];

-1 -0.5 0.5 1

-1.2 -1 -0.8 -0.6 -0.4 -0.2

à Flächenverhältnis

Integrate[(x^2-1) E^(-x),x]

-ã

^-x

H 1 + x L

²

Integrate[(x^2-1) E^(-x),{x,-1,1}]

- 4

€€€€ ã

%//N -1.47152

Integrate[p[x],t]

t H -1 + x

²

L

€€€€€€€€€€€€€€€€€€€€€€€€€€€

ã

(32)

Integrate[p[x],{x,-1,1}]

- 4

€€€€€€€€

3 ã

%//N -0.490506

int=Integrate[(f[x]-p[x]),{x,-1,1}]/Integrate[(p[x]),{x,-1, 1}]

2 int//N 2.

Integrate[(f[x]),{x,-1,1}]/Integrate[(p[x]),{x,-1,1}]

3 %//N 3.

1/%//N 0.333333

à Wendepunkt

Evaluate[f''[x]]/.x->0

1

(33)

Solve[Evaluate[f''[x]==0],{x}]

— Solve::ifun :

Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. Mehr…

88 x ® 2 - •!!!! 3 < , 8 x ® 2 + •!!!! 3 <<

Solve[Evaluate[f''[x]==0],{x}]//N

— Solve::ifun :

Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. Mehr…

88 x ® 0.267949 < , 8 x ® 3.73205 <<

à Rotationsvolunen bei Parabe, Vergleich mit Zylinder gleicher Länge, r =

? (Hat Volumen in Zylinder mit r=2 Platz?)

Pi Integrate[p[x]^2,{x,-1,1}]

16 p

€€€€€€€€€€€€€

15 ã

²

%//N 0.453513

Sqrt[Integrate[p[x]^2,{x,-1,1}]/2]

2 "######## €€€€€

₁₅²

€€€€€€€€€€€€€€€€€€

ã

%//N

0.268661

(34)

Plot[{f[x],p[x],-1.9851,1.9851},{x,-1,1}];

-1 -0.5 0.5 1

-2 -1 1 2

à Länge der Bögen:

{f[x],p[x]}

9 ã

^-^x

H -1 + x

²

L , - 1

€€€€ ã + x

²

€€€€€€€

ã =

l1=NIntegrate[Evaluate[Sqrt[1+D[p[x],x]^2]],{x,-1,1}]

2.16805

l2=NIntegrate[Evaluate[Sqrt[1+D[f[x],x]^2]],{x,-1,1}]

3.37052

l3=2 2

l1/l2

0.643241

(35)

l1/l3 1.08403

l2/l3 1.68526

**Remove["Global`*"]**

Lösung zu Aufgabe 8,

Annäherung Hügel durch Potenzreihe

Funktion durch Parabel angenähert. Gleiche Ableitung bei -1 und Werte bei -1 und 1, Winkelabweichung bei 1?

à Test

**Remove["Global`*"]**

— Remove::rmnsm :

There are no symbols matching "Global`*".

Mehr…

q[x_,a_,b_,c_,d_,e_]:=(a x^2+b x + c)/(x^2+d x+e)

q[x,a,b,c,d,e]//TeXForm

\frac{a x^2+b x+c}{x^2+d x+e}

(36)

sys={

q[-2,a,b,c,d,e]==0, q[2,a,b,c,d,e]==0, q[0,a,b,c,d,e]==2,

(D[q[x,a,b,c,d,e],x]/.x->1)==-1, (D[q[x,a,b,c,d,e],x]/.x->-1)==1}

9 4 a - 2 b + c

€€€€€€€€€€€€€€€€€€€€€€€€€€€€

4 - 2 d + e Š 0, 4 a + 2 b + c

€€€€€€€€€€€€€€€€€€€€€€€€€€€€

4 + 2 d + e Š 0,

€€€€ c

e Š 2, - €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ H a + b + c L H 2 €€€€€€€€€€€ + d L

H 1 + d + e L

²

+ 2 a + b

€€€€€€€€€€€€€€€€€€€€

1 + d + e Š -1,

- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ H a - b + c L H -2 €€€€€€€€€€€€€€ + d L

H 1 - d + e L

²

+ -2 a + b

€€€€€€€€€€€€€€€€€€€€

1 - d + e Š 1 =

solv=Solve[Evaluate[sys,{a,b,c,d,e}]]

99 b ® 0, c ® 1, a ® - 1

€€€€ 4 , d ® 0, e ® 1

€€€€ 2 = , 8 b ® 0, c ® 8, a ® -2, d ® - •!!!! 7 , e ® 4 < , 8 b ® 0, c ® 8, a ® -2, d ® •!!!! 7 , e ® 4 <=

q[x_]:=(a x^2+b x + c)/(x^2+d x+e)/.solv ; q[x]

9 1 - €€€€€

^x₄²

€€€€€€€€€€€€€€€€

₁

€€€

₂

+ x

²

, 8 - 2 x

²

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

4 - •!!!! 7 x + x

²

, 8 - 2 x

²

€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

4 + •!!!! 7 x + x

²

=

à Symmetrie

**Remove["Global`*"]**

q[x_,a_,c_,e_]:=(a x^2+ c)/(x^2+e)

sys={

q[2,a,c,e]==0, q[0,a,c,e]==2,

(D[q[x,a,c,e],x]/.x->1)==-1}

9 4 a + c

€€€€€€€€€€€€€€€€

4 + e Š 0, c

€€€€ e Š 2, - €€€€€€€€€€€€€€€€€€€€€€ 2 H a + c L

H 1 + e L

²

+ 2 a

€€€€€€€€€€€€

1 + e Š -1 =

(37)

solv=Solve[Evaluate[sys,{a,c,e}]]//Flatten 9 c ® 1, a ® - 1

€€€€ 4 , e ® 1

€€€€ 2 =

q[x_]:=(a x^2+ c)/(x^2+e)/.solv ; q[x]

1 - €€€€€

^x₄²

€€€€€€€€€€€€€€€€

₁

€€€

₂

+ x

²

q[x]//Simplify 4 - x

²

€€€€€€€€€€€€€€€€€€

2 + 4 x

²

à Konvergenzbereich: Partialbruchzerlegung, Vergleich mit geometrischer Reihe! ==> r = 1

q1[x_]:= Evaluate[q[x]//Apart]; q1[x]

- 1

€€€€ 4 + 9

€€€€€€€€€€€€€€€€€€€€€€€€€€€€

4 H 1 + 2 x

²

L

Table[D[q1[x],{x,k}],{k,0,10}]//Together

9 4 - x

²

€€€€€€€€€€€€€€€€€€€€€€€€€€€€

2 H 1 + 2 x

²

L , - 9 x

€€€€€€€€€€€€€€€€€€€€€€€€€€

H 1 + 2 x

²

L

²

, €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 9 H -1 + 6 x

²

L H 1 + 2 x

²

L

³

, - €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 216 H -x + 2 x

³

L

H 1 + 2 x

²

L

⁴

, €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 216 H 1 - 20 x

²

€€€€€€€€€€€€€€€€€€€€€€ + 20 x

⁴

L H 1 + 2 x

²

L

⁵

, - €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 4320 H 3 x - 20 x €€€€€€€€€€€€€€€€€€€€€€€€€€€€€

³

+ 12 x

⁵

L

H 1 + 2 x

²

L

⁶

, €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 12960 H -1 + 42 x €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

²

- 140 x

⁴

+ €€€€€€€€€€€€€€€€€€ 56 x

⁶

L

H 1 + 2 x

²

L

⁷

,

- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 1451520 H -x + €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 14 x

³

- 28 x

⁵

€€€€€€€€€€€€€€€€€€ + 8 x

⁷

L

H 1 + 2 x

²

L

⁸

,

1451520 H 1 - 72 x

²

+ 504 x

⁴

- 672 x

⁶

+ 144 x

⁸

L

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

H 1 + 2 x

²

L

⁹

,

- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ 52254720 H 5 x €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ - 120 x

³

+ 504 x €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€

⁵

- 480 x

⁷

+ €€€€€€€€€€€€€€€€€€ 80 x

⁹

L

H 1 + 2 x

²

L

¹⁰

, 1

€€€€€€€€€€€€€€€€€€€€€€€€€€€€

H 1 + 2 x

²

L

¹¹

H 261273600 H -1 + 110 x

²

- 1320 x

⁴

+ 3696 x

⁶

- 2640 x

⁸

+ 352 x

¹⁰

LL=

(38)

Table[D[q1[x],{x,k}],{k,0,10}]/.x->1.

8 0.5, -1., 1.66667, -2.66667, 0.888889, 29.6296,

-254.815, 1548.64, -7005.76, 9734.32, 290555. <

q1[x_]:=Evaluate[q[x]]

Plot[q1[x],{x,-2,2}];

-2 -1 1 2

0.5 1 1.5 2

Plot[q1[x],{x,-8,8}];

-7.5 -5 -2.5 2.5 5 7.5

0.5

1

1.5

2

(39)

à Potenzreihe, Abweichung

q2[x_]:=Normal[Series[q1[z],{z,0,10}]]/.z->x; q2[x]

2 - 9 x

²

€€€€€€€€€€€

2 + 9 x

⁴

- 18 x

⁶

+ 36 x

⁸

- 72 x

¹⁰

q2[x_]:=Evaluate[Normal[Series[q1[x],{x,0,10}]]]; q2[x]

2 - 9 x

²

€€€€€€€€€€€

2 + 9 x

⁴

- 18 x

⁶

+ 36 x

⁸

- 72 x

¹⁰

{q1[x],q2[x]}

9 1 - €€€€€

^x₄²

€€€€€€€€€€€€€€€€

₁

€€€

₂

+ x

²

, 2 - 9 x

²

€€€€€€€€€€€

2 + 9 x

⁴

- 18 x

⁶

+ 36 x

⁸

- 72 x

¹⁰

=

{q1[5],q2[5]}//N

8 -0.205882, -6.89338 ´ 10

⁸

<

Plot[{q1[x],q2[x]},{x,-1,1}];

-1 -0.5 0.5 1

-5

-4

-3

-2

-1

1

2

(40)

q2[x_]:=Evaluate[Normal[Series[q1[x],{x,0,20}]]];

Plot[{q1[x],q2[x]},{x,-1,1}];

-1 -0.5 0.5 1

5 10 15 20 25 30 35

q2[x_]:=Evaluate[Normal[Series[q1[x],{x,0,150}]]];

Plot[{q1[x],q2[x]},{x,-1,1}];

-1 -0.5 0.5 1

-1.4´ 10

¹⁸

-1.2´ 10

¹⁸

-1´ 10

¹⁸

-8´ 10

¹⁷

-6´ 10

¹⁷

-4´ 10

¹⁷

-2´ 10

¹⁷

q2[x_]:=Evaluate[Normal[Series[q1[x],{x,0,10}]]];

d1=q2[x]/.x->0.5 1.22656

d2=q[x]/.x->0.5

1.25

(41)

d2-d1 0.0234375

q[x]-q2[x]/.x->0.9 15.5228

**Remove["Global`*"]**

Lösung zu Aufgabe 9,

Reihe, Maximum Geometrie

à a

Normal[Series[ArcTan[x],{x,0,10}]]

x - x

³

€€€€€€€

3 + x

⁵

€€€€€€€

5 - x

⁷

€€€€€€€

7 + x

⁹

€€€€€€€

9 4(Normal[Series[ArcTan[x],{x,0,10}]]/.x->1)//N 3.33968

4(Normal[Series[ArcTan[x],{x,0,2000}]]/.x->1)//N 3.14059

à b

10m : 2m Breite 10 m

Höhe: x : 2 = (x+10) : h, l = f[x]

(42)

Solve[ x / 2 == (x+10) / h,{h}]

99 h ® €€€€€€€€€€€€€€€€€€€€€€€€€ 2 H 10 + x L

x ==

f @ x_ D := Sqrt Ai

k jj 2 H 10 + x L

€€€€€€€€€€€€€€€€€€€€€€€€€

x y {

zz ^ 2 + H x + 10 L ^ 2 E ; f @ x D

$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H 10 + x L

²

+ €€€€€€€€€€€€€€€€€€€€€€€€€€€ 4 H 10 + x L

²

x

²

Plot[f[x],{x,0,20}];

5 10 15 20

20 40 60 80 100 120

f'[x]

2 H 10 + x L + €€€€€€€€€€€€€€€€

⁸^H¹⁰_x2⁺^x^L

- €€€€€€€€€€€€€€€€€€

⁸^H¹⁰_x3⁺^x^L²

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

2 "############################################## H 10 + x L

²

+ €€€€€€€€€€€€€€€€€€

⁴^H^10+x_x2 ^L²

Solve[Evaluate[f'[x]==0],{x}]

88 x ® -2 H -5 L

¹^•³

< , 8 x ® 2 5

¹^•³

< , 8 x ® 2 H -1 L

²^•³

5

¹^•³

<<

Solve[Evaluate[f'[x]==0],{x}]//N 88 x ® -1.70998 - 2.96177 ä < ,

8 x ® 3.41995 < , 8 x ® -1.70998 + 2.96177 ä <<

(43)

H -5 L

¹^•³

•• N

0.854988 + 1.48088 ä

H -1 L

²^•³

•• N

-0.5 + 0.866025 ä

HH -1 L

²^•³

L ^ 2 •• N

-0.5 - 0.866025 ä

HH -1 L

²^•³

L ^ 3 •• N

1. à c

tan1=49/10

€€€€€€€ 49 10

weite=2/tan1 -0.7 -0.291837

**Remove["Global`*"]**

(44)