Lösungen

1

Remove["Global`*"]

a

f[x_] := (x^2 - 1)/(E^(2  x) - 1); 
    Plot[f[x], {x, -5, 5}];

Plot[f[x], {x, 0.5, 5}, 
        PlotRange -> {-0.1, 0.1}];

b

(Evaluate[f'[y] // Simplify]) /. y -> -1

N[%]

ArcTan[%]

%/Degree

c

FindRoot[f[x] == 0, {x, 1}]

f[1]

FindRoot[f[x] == 0, {x, -1}]

f[-1]

FindRoot[1/f[x] == 0, {x, 0}]

Limit[1/f[x], x -> 0]

==>  Pol

d

Evaluate[f'[x] // Simplify]

Plot[Evaluate[f'[x] // Simplify], {x, -5, 5}, 
        PlotRange -> {-5, 5}];

FindRoot[(Evaluate[f'[y] // Simplify] == 0) /. 
        y -> x, {x, 2}]

f[1.5879776025618713`]

e

Plot[Evaluate[f''[x] // Simplify], {x, -5, 5}, 
        PlotRange -> {-5, 5}];

Plot[Evaluate[f''[x] // Simplify], {x, 2, 5}, 
        PlotRange -> {-0.1, 0.1}];

fr1 = 
      FindRoot[(Evaluate[f''[y] // Simplify] == 0) /. 
          y -> x, {x, -1}]

x1 = x /. fr1

f[x1]

fr2 = 
      FindRoot[(Evaluate[f''[y] // Simplify] == 0) /. 
          y -> x, {x, 2}]

x2 = x /. fr2

f[x2]

Plot[Evaluate[f''[x] // Simplify], {x, -5, 5}, 
        PlotRange -> {-1, 1}];

f

Limit[f[x], x -> Infinity]

Limit[f[x], x -> -Infinity]

g

Normal[Series[f[x], {x, -3, 3}]] // N

% // Expand

h

A1 = 
      Integrate[
        N[Normal[
            Series[f[x], {x, -3, 3}]]], {x, -4, -2}]

i

A2 = Integrate[N[f[x]], {x, -4, -2}] // Chop

Abs[A2 - A1]

2

Remove["Global`*"]

f1[x_, y_] := 1 + Sin[x + y];
f2[x_, y_] := 1 + Sin[x y];
f3[x_, y_] := 1 + Sin[x] + Sin[y];
f4[x_, y_] := 1 + Sin[x] Sin[y];

a

Plot3D[
        1 + Sin[x + y], {x, -2 Pi, 2  Pi}, {y, -2 Pi, 
          2  Pi}];

Wellblech

Plot3D[
        1 + Sin[x y], {x, -Pi, Pi}, {y, -Pi, Pi}];

Keine der genannten Formen

Plot3D[
        1 + Sin[x] + Sin[y], {x, -2 Pi, 2  Pi}, {y, -2 Pi, 
          2  Pi}];

Eierschachtel

Plot3D[
        1 + Sin[x] Sin[y], {x, -2 Pi, 2  Pi}, {y, -2 Pi, 
          2  Pi}];

Eierschachtel

b

V1 = 
      Integrate[
        f3[x, y], {x, -2 Pi, 2  Pi}, {y, -2 Pi, 2  Pi}]

N[%]

V2 = 
      Integrate[
        f4[x, y], {x, -2 Pi, 2  Pi}, {y, -2 Pi, 2  Pi}]

N[%]

c

s1 = 
      Integrate[(Evaluate[Sqrt[1 + D[f3[0, t], t]^2]] /. 
            t -> y), {y, -2 Pi, 2  Pi}]

N[%]

s2 = 
      Integrate[(Evaluate[Sqrt[1 + D[f4[0, t], t]^2]] /. 
            t -> y), {y, -2 Pi, 2  Pi}]

N[%]

s1 > s2

3

Remove["Global`*"]

a

s[x_, r_] := Sqrt[10^2 - x^2] - r;
Plot[{s[x, 8], -s[x, 8]}, {x, -6, 6}, 
        AspectRatio -> Automatic];

Plot[{s[x, 7], -s[x, 7]}, {x, -6, 6}, 
        AspectRatio -> Automatic];

b

A[r_] := s[6, r]^2 Pi;  A[r]

s[x, r]^2 Pi // Expand

V[r_] := Integrate[s[x, r]^2 Pi, {x, -6, 6}];  V[r]

N[%]

q[r_] := V[r]/ A[r]  // Simplify;   q[r]

N[%]

Plot[Evaluate[ q[r]], {r, 3, 7.5}];

Plot[Evaluate[ q[r]], {r, 3, 10}];

V[7]

N[%]

V[8]

N[%]

q[7]

N[%]

q[8]

c

solv = Solve[V[r]/ A[r]  == 25, {r}] // Flatten

solv // N

Nur  r = 4.84166  liegt im geometrisch möglichen Bereich

d

Integrate[(Evaluate[2 s[t, 7] Pi Sqrt[1 + D[s[t, 7], t]^2]] /. 
          t -> x), {x, -6, 6}]

N[%]

12 Pi 2 2 // 
      N

4

Remove["Global`*"]

Daten

v1 = {2, -8, 0}; v2 = {-2, 6, 2};

    OP = {-1, 2, 3};

    v3 = Cross[v1, v2]

a

mX = Transpose[{v1, v2, v3}];  mX // MatrixForm

Dλ = {{1, 0, 0}, {0, 1, 0}, {0, 0, -1}};  
    Dλ // MatrixForm

mS = mX . Dλ . Inverse[mX]; mS // MatrixForm

N[%] // MatrixForm

9*mS  // MatrixForm

b

OP1 = mS . OP

N[%]

c

Dreh[φ_] := {{Cos[φ], -Sin[φ], 
          0}, {Sin[φ], Cos[φ], 0}, {0, 0, 1}}; 
    Dreh[Pi/5] // MatrixForm

N[%] // MatrixForm

d

OP2 = Dreh[Pi/5] . OP1

N[%]

e

OP3 = mS . OP2 // N

f

Dλ1 = {{1, 0, 0}, {0, 1, 0}, {0, 0, 0}};  
    Dλ1 // MatrixForm

mP = mX . Dλ1 . Inverse[mX]; mP  // MatrixForm

N[mP]  // MatrixForm

18*mP  // MatrixForm

g

mP . OP3 // N

5

Remove["Global`*"]

Vektornormierung

nor[m_] := 
      Table[m[[k]]/Norm[m[[k]]], {k, 1, Length[m]}]

Daten

a1 = {2, 2, 2};  a2 = {2, -2, 2};  
    a3 = {2, 2, -2};
b1 = 2 a1;
OQ  =  {0, -2, 2};

A = Transpose[{a1, a2, a3}];  A // MatrixForm

B = Transpose[{b1, a2, a3}];  B // MatrixForm

a

ew1 = Eigenvalues[A]

% // N

b

ew2 = Eigenvalues[B]

% // N

c

Der Eigenwert  -4  ist gemeinsam

d

ev1 = Eigenvectors[A]

Transpose[ev1] // MatrixForm

ev11 = nor[ev1] // N

Transpose[ev11] // MatrixForm

ev2 = Eigenvectors[B]

Transpose[ev2] // MatrixForm

ev21 = nor[ev2] // N

Transpose[ev21] // MatrixForm

Der Eigenvektor  {0,-1,1}  zum Eigenwert -4 ist gemeinsam

e

ew3 = Eigenvalues[A . B]

% // N

ew3 = Eigenvalues[B . A]

% // N

Gleiche Eigenwerte

f

Apply[Plus, Eigenvalues[A]]

Apply[Plus, Eigenvalues[B]]

% // Simplify

Apply[Plus, Eigenvalues[A . B]] // Simplify

Apply[Plus, Eigenvalues[B . A]] // Simplify

Gleiche Eigenwerte von A.B und B.A ==> gleiche Summen

g

Apply[Times, Eigenvalues[A]]

Apply[Times, Eigenvalues[B]] // Simplify

Apply[Times, Eigenvalues[A . B]] // Simplify

Apply[Times, Eigenvalues[B . A]] // Simplify

Gleiche Eigenwerte von A mal Eigenwerte von B gleich Eigenwerte von B.A oder A.B

h

Det[A]

Det[B]

Det[A . B]

Det[B . A]

Die Determinante ist das Produkt der Eigenwerte: Resultate der vorangehenden Teilaufgabe

i

OP1  = A . OQ

OP2  = B . OQ

OQ ist Eigenvektor von A und von B !!!!

6

a

Remove["Global`*"]

A.(A+X).A+A+(A^(-1)) = A.(A^T)+E

==>   (A^(-1)).A.(A+X).A.(A^(-1))+(A^(-1)).A.(A^(-1))+(A^(-1)).(A^(-1)).(A^(-1)) = (A^(-1)).A.(A^T).(A^(-1))+(A^(-1)).E.(A^(-1))

==>  A+X+(A^(-1))+(A^(-3)) = (A^T).(A^(-1))+(A^(-2))

==>  X  = (A^T).(A^(-1))+(A^(-2))-A-(A^(-1))-(A^(-3)) =  (A^T).(A^(-1)) - A - (A^(-1)) + (A^(-2))  - (A^(-3))

\(\(\(
\)\(
\)\(B . \((B + X)\) . B + 
        B + B\^\(-1\)\  = B . B\^T + E


    \(\(==\)\(>\)\(B\^\(-1\) . B . \((B + X)\) . B . B\^\(-1\) + 
          B\^\(-1\) . B . B\^\(-1\) + 
          B\^\(-1\) . B\^\(-1\) . B\^\(-1\)\)\)\  = 
      B\^\(-1\) . B . B\^T . B\^\(-1\) + 
        B\^\(-1\) . E . B\^\(-1\)


    \(\(==\)\(>\)\(B + X + B\^\(-1\) + B\^\(-3\)\)\)\  = 
      B\^T . B\^\(-1\) + B\^\(-2\)


    \(\(==\)\(>\)\(X\)\) = \(B\^T . B\^\(-1\) + B\^\(-2\) - B - 
          B\^\(-1\) - B\^\(-3\)\  = 
        B\^T . B\^\(-1\) - B - B\^\(-1\) + B\^\(-2\) - 
          B\^\(-3\)\)\)\)\)

b

i

U1 = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {0, 0, 0, 0}};  
    U1 // MatrixForm

(U1 . U1) // MatrixForm

(U1 . U1 . U1) // MatrixForm

(U1 . U1 . U1 . U1) // MatrixForm

ii

(U1 . U1) . (U1 . U1) // MatrixForm

(U1) . (U1 . U1 . U1) // MatrixForm

(U1) . (U1 . U1 . U1 . U1) // MatrixForm

c

Remove["Global`*"]

\(U2 = {{1, \ 2, \ 3, \ 4, \ 5\ } + {1, \ 1, \ 2, \ 1, \ 2}, 
{3, \ 2, \ 
          1, \ 5, \ 4}, 
{7, \ 2, \(-3\), \ 7, \ 
          2\ }, 
{1, \ 1, \ 2, \ 1, \ 2} - {1, \ 2, \ 3, \ 4, \ 5\ }}; 
    U2 // MatrixForm\)

b1 = {4 + 2, 3, 0, 2 - 4}

b2 = {4 + 2, 3, 1, 2 - 4}

i

x = {x1, x2, x3, x4, x5};

Solve[U2 . x == b1, x]

Solve[U2 . x == b2, x]

ii

Ordnung = 5

iii

Fall für b2: Dim Lösungsraum = 2

iiii

Rang = Ordnung - Dimension = 5 - 2 = 3

d

Remove["Global`*"]

v1[t1_] := {-3, 1, -1} + 
          t1 {-1, 2, 1};
v2[t2_] := {1, 4, 2} + t2 {-2, 4, 2};

Solve[k  {-1, 2, 1} == {-2, 4, 2}, {k}]

==>  Geraden parallel  ==> Abstand:

a = 
      Norm[Cross[(v1[0] - v2[0]),  {-1, 2, 1}]]/
        Norm[ {-1, 2, 1}]

N[%]

7

Remove["Global`*"]

a

OP1 = {2, 0, 4};

    a = {3, 1, 4};

    b = {1, 0, 0};

    c = Cross[a, b]

d=Cross[a, c]

ea = a/Norm[a]

ea // N

ec = c/Norm[c]

ec // N

ed = d/Norm[d]

ed // N

b

M = Transpose[{ea, ec, ed}];  M // MatrixForm

M // N // MatrixForm

c

Minv = Inverse[M]

Minv // MatrixForm

Minv // N // MatrixForm

OP1s = Minv . OP1

OP1s // N

d

Dreh[φ_] := {{1, 0, 0}, {0, 
          Cos[φ], -Sin[φ]}, {0, Sin[φ], 
          Cos[φ]}}

Dreh[2  Pi/3] // MatrixForm

Dreh[2  Pi/3] // N // MatrixForm

Dreh[4  Pi/3] // MatrixForm

Dreh[4  Pi/3] // N // MatrixForm

OP2s = Dreh[2  Pi/3] . OP1s

OP2s // N

OP3s = Dreh[4  Pi/3] . OP1s

OP3s // N

OP3as = Dreh[2  Pi/3] . OP2s

OP3as // N

e

OP2 = M . OP2s;  OP2 // N

OP3 = M . OP3s;  OP3 // N

OP3a = M . OP3as;  OP3a // N

f

V = Det[{OP1, OP2, OP3}]/6

V // N

Va = Det[{OP1, OP2, OP3a}]/6

Va // N