Lösungen

1

rem;  Remove["Global`*"]

M = {{2, -2, 3}, {-3, 1, 3}, {-2, -1, 
            2}};
MatrixForm[M]

MM = M + M;
MatrixForm[MM]

a

Det[M]

Det[MM]

b

Det[M] ungleich 0, Inverse existiert

c

{Inverse[M] // MatrixForm,  Inverse[M] // N // 
        MatrixForm}

d

{Inverse[Transpose[M]] // MatrixForm, Inverse[Transpose[M]] // N // 
        MatrixForm}

{Transpose[Inverse[M]] // MatrixForm, Transpose[Inverse[M]] // N // 
        MatrixForm}

{Inverse[M] // MatrixForm,  
      Inverse[Transpose[M]] // MatrixForm}

Die Transponierte der Inversen ist die Inverse der Transponierten.

e

Inverse[Transpose[M]] == Transpose[Inverse[M]]

f

IM = Inverse[M];
IM7 = 
      IM . IM . IM . IM . IM . IM . 
        IM;  IM7//MatrixForm

IM7 // N//MatrixForm

Det[IM]

Det[IM]^7

Det[IM7]

g

OP0 = {{2}, {5}, {8}}; OP0 //  MatrixForm

OP1 = M . OP0;  OP1  //  MatrixForm

{OP2 = Inverse[M] . OP0;  OP2  //  MatrixForm, 
      OP2 = Inverse[M] . OP0;  OP2  // N //  MatrixForm}

{Norm[OP2 - OP1],  Norm[OP2 - OP1] // N}

h

M2 = M . M;  M2  //  MatrixForm

M2 . OP2 == OP1

i

Eigenvalues[M] // N

Det[M - λ IdentityMatrix[3]] == 0

Solve[
      Det[M - λ IdentityMatrix[3]] == 
        0, {λ}]

Eigenvectors[M] // N

Aufgabe kann auch anders interpretiert werden:

Eigenvalues[M2] // N

Eigenvectors[M2] // N

2

rem;  Remove["Global`*"]

f[x_] := (2 E^(-(x - 4)^2) - 
            1)/(x (x + 5));   f[x]

a

Plot[f[x], {x, -7, 7}];

Plot[f[x], {x, -14, 10}, 
        PlotRange -> {-0.5, 0.5}];

b

f'[x]

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

steig = (f'[x] /. x -> -1) // N

ArcTan[steig]

ArcTan[steig]/Degree  // N

g[x_] :=  f[-1] + steig (x - (-1));  g[x]

g[x] // N

Plot[{f[x], g[x]}, {x, -7, 7}, 
        Epilog -> {PointSize[0.03], 
            Point[{-1, f[-1]}]}];

c

c1

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

solv1 = NSolve[f[x] == 0, {x}]

{x1 = x /. solv1[[1]], x2 = x /. solv1[[2]]}

c2

solv2 = Solve[1/f[x] == 0, {x}]

{x3 = x /. solv2[[1]], x4 = x /. solv2[[2]]}

Plot[{f[x], g[x]}, {x, -7, 7}, 
        Epilog -> {PointSize[0.03], Point[{-1, f[-1]}], 
            Point[{x1, f[x1]}], Point[{x2, f[x2]}], Point[{x3, 0}], 
            Point[{x4, 0}]}];

d

f'[x]

solv3 = FindRoot[f'[x] == 0, {x, -3}]

x5 = x /. solv3

solv4 = FindRoot[f'[x] == 0, {x, 4}]

x6 = x /. solv4

solv5 = FindRoot[f'[x] == 0, {x, 5}]

x7 = x /. solv5

Plot[{f[x], g[x]}, {x, -7, 7}, 
        Epilog -> {PointSize[0.03], Point[{-1, f[-1]}], 
            Point[{x1, f[x1]}], Point[{x2, f[x2]}], Point[{x3, 0}], 
            Point[{x4, 0}], Point[{x5, f[x5]}], Point[{x6, f[x6]}], 
            Point[{x7, f[x7]}]}];

e

solv6 = FindRoot[Evaluate[f''[x] == 0], {x, 2}]

solv7 = FindRoot[Evaluate[f''[x] == 0], {x, 3}]

solv8 = FindRoot[Evaluate[f''[x] == 0], {x, 4}]

solv9 = FindRoot[Evaluate[f''[x] == 0], {x, 7}]

x8 = x /. solv6

x9 = x /. solv7

x10 = x /. solv8

x11 = x /. solv9

Plot[{f[x], g[x]}, {x, -7, 7}, 
        Epilog -> {PointSize[0.03], Point[{-1, f[-1]}], 
            Point[{x1, f[x1]}], Point[{x2, f[x2]}], Point[{x3, 0}], 
            Point[{x4, 0}], Point[{x5, f[x5]}], Point[{x6, f[x6]}], 
            Point[{x7, f[x7]}], Point[{x8, f[x8]}], Point[{x9, f[x9]}], 
            Point[{x10, f[x10]}], Point[{x11, f[x11]}]}];

Plot[{f[x], g[x]}, {x, 2, 6}, 
        Epilog -> {PointSize[0.03], Point[{-1, f[-1]}], 
            Point[{x1, f[x1]}], Point[{x2, f[x2]}], Point[{x3, 0}], 
            Point[{x4, 0}], Point[{x5, f[x5]}], Point[{x6, f[x6]}], 
            Point[{x7, f[x7]}], Point[{x8, f[x8]}], Point[{x9, f[x9]}], 
            Point[{x10, f[x10]}], Point[{x11, f[x11]}]}];

f

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

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

g

ser = Series[f[x], {x, 4, 3}]

N[%]

serN = Series[f[x], {x, 4, 3}] // Normal

N[%]

ExpandAll[Evaluate[serN]]

N[%]

h

Integrate[-(13196/6561) + ((23419 x)/17496) - ((9953 (x^2))/34992) + ((32435 (x^3))/1679616), {x, 3, 5}]

int1 = 
      Integrate[-2.0112787684804148` + 1.3385345221764975` x - 
          0.284436442615455` (x^2) + 0.019310961553116904` (x^3), {x, 3, 
          5}]

i

int2 = NIntegrate[f[x], {x, 3, 5}]

Abs[int1 - int2]

3

rem;  Remove["Global`*"]

x1 = 0;  x2 = Pi;  y1 = 0;  y2 = 3  Pi/2;
f1[x_, y_] := Cos[x + y];
f2[x_, y_] := Cos[x] + Sin[y];
f3[x_, y_] := Cos[x]  Sin[y];

a

Plot3D[f1[x, y], {x, x1, x2}, {y, y1, y2}];

Plot3D[f2[x, y], {x, x1, x2}, {y, y1, y2}];

Plot3D[f3[x, y], {x, x1, x2}, {y, y1, y2}];

b

A1[y_] := Integrate[f1[x, y], {x, x1, x2}];   A1[y]

A2[y_] := Integrate[f2[x, y], {x, x1, x2}];   A2[y]

A3[y_] := Integrate[f3[x, y], {x, x1, x2}];   A3[y]

c

V1 = Integrate[A1[y], {y, y1, y2}]

V2 = Integrate[A2[y], {y, y1, y2}]

V3 = Integrate[A3[y], {y, y1, y2}]

d

D[f2[x, y], x] == 0

Plot[Sin[x], {x, x1, x2}];

D[f2[x, y], y] == 0

Plot[Cos[y], {y, y1, y2}];

Cos[Pi/2]

Cos[3  Pi/2]

f2[0, Pi/2]

f2[0, 3  Pi/2]

f2[Pi, Pi/2]

f2[Pi, 3  Pi/2]

Minimalstelle bei {Pi,  3Pi/2}

4

rem;  Remove["Global`*"]

{d/2 == x4 Sqrt[2]/2,   h^2 == 4^2 - (d/2)^2}

solv1 = Solve[d/2 == x4 Sqrt[2]/2, {d}] // Flatten

d[x4_] = d /. solv1;   d[k]

d[x4]

\(h\^2 == 16 - d[x4]\^2\/4\)

solv2 = Solve[(h^2) == 16 - ((x4^2)/2), {h}] // Flatten

h[x4_] = h /. solv2[[2]]

h[k]

a

V4[x4_] := x4^2 h[x4]/3;   V4[x4]

V4'[x4]

solv3 = Solve[Evaluate[V4'[x4] == 0], {x4}] // Flatten

N[%]

xMax = x4 /. solv3[[3]]

V4[k]

Plot[V4[x4], {x4, 0, 2 4/Sqrt[2]}];

x0   ->   4.6188

b

V4[x4]

V4[x4] /. x4 -> xMax

N[%]

c

rem;  Remove["Global`*"]

{d^2 ==  x3^2 - (x3/2)^2,  
      s^2 + (x3/2)^2 == 4^2}

solv1 = 
      Solve[d^2 ==  x3^2 - (x3/2)^2, {d}] // Flatten

d[x3_] = d /. solv1[[2]];   d[k]

d[x3]

solv2 = Solve[s^2 + (x3/2)^2 == 4^2, {s}] // Flatten

s[x3_] = s /. solv2[[2]];   s[k]

s[x3]

h[x3_] := Sqrt[s[x3]^2 - (d[x3]/3)^2];  h[k]

h[x3]

V3[x3_] := x3 d[x3]/2   h[x3]/3;  V3[k]

V3[x3]

V3'[x3]

solv3 = Solve[Evaluate[V3'[x3] == 0], {x3}] // Flatten

N[%]

xMax = x3 /. solv3[[3]]

V3[k]

Plot[V3[x3], {x3, 0, 2  Sqrt[4^2 - 2^2]}];

d

V3[xMax]

N[%]

5

rem;  Remove["Global`*"]

f[x_] := 1/2 (x - 2) (x + 3) E^x;   f[y]

h[x_] :=  f[x]  E^x;  h[x]

1 + Evaluate[h'[y]]^2  //  Simplify

(1 + Evaluate[h'[u]]^2) /. u -> 2

Plot[(1 + Evaluate[h'[u]]^2) /. u -> t, {t, 0, 
          2}];

a

dist[x_] := Sqrt[(x - 2)^2 + (f[x] - 2)^2];  
    dist[x]

Plot[dist[x], {x, -2, 4}];

Plot[dist[x], {x, 2, 2.2}];

dist'[x] // Simplify

solv = 
      FindRoot[Evaluate[D[dist[k], k] /. k -> x], {x, 2.1}]

x1 = x /. solv

dist[x1]

b

Sqrt[1 + Evaluate[h'[x]]^2]

laenge1 = 
      NIntegrate[Sqrt[1 + Evaluate[h'[x]]^2], {x, 0, 2}]

c

s[t_] := {t, t^2, Sqrt[1 + t]};   s[t]

norm[v_] := Sqrt[v . v];   D[s[t], t] . D[s[t], t]

len = 
      NIntegrate[
        Evaluate[D[s[t], t] . D[s[t], t]] /. t -> u, {u, 0, 
          2}]

d

v1 = {-1, 2, -3};  
    a1 = {0, 1, -2};
v2 = {1, -2, 2};  a2 = {1, 0, 1};

V = 
      Det[{v2 - v1, a1, a2}]/Norm[Cross[a1, a2]]

N[%]

6

rem;  Remove["Global`*"]

a = {-1, 2, -3};  b = {0, 1, -2};  
    c = Cross[a, b]

a

X = Transpose[{a, b, c}];  X // MatrixForm

Eλ = {{2, 0, 0}, {0, -1, 0}, {0, 0, 3}};   Eλ // 
      MatrixForm

M = X . Eλ . Inverse[X];   M // MatrixForm

M // N // MatrixForm

b

M . M - Transpose[M] . Y . M + M == 
        IdentityMatrix[3];
M . M - IdentityMatrix[3] + M == 
        Transpose[M] . Y . M;
Y == (Inverse[
              Transpose[M]] . (M . M - IdentityMatrix[3] + M) . 
            Inverse[M] // MatrixForm)

Inverse[Transpose[M]] . (M . M - IdentityMatrix[3] + M) . 
          Inverse[M] // N // MatrixForm

c

detQ =  (1/Det[M])^50

detQ // N

detQ 10^39

detQ 10^39   // N

d

M5 = M . M . M . M . M;
M5  //  MatrixForm

M5  // N //  MatrixForm

v = 
      Transpose[{b +  x b - x^2 b - x^3 b}] //  MatrixForm

M5 . v // Simplify  // MatrixForm

M5 . v // Expand // MatrixForm

Kommetar: Eigenvektoren ändern nur die Länge!

e

Dφ[φ_] := {{Cos[φ], -Sin[φ], 0}, {Sin[φ], Cos[φ], 0}, {0, 0, 
            1}};
MatrixForm[Dφ[φ]]

f

U = Inverse[M] . Dφ[φ] . M // Simplify;  
    MatrixForm[U]

Eigenvalues[U]

Reell für  Sin[φ]=0, also für φ= n π, n = ganze Zahl.

g

X = Transpose[{a, b, c}];  X // MatrixForm

EX = {{1, 0, 0}, {0, 1, 0}, {0, 0, -1}}; ; 
    MatrixForm[EX]

S = X . EX . Inverse[X];   S // MatrixForm

S // N // MatrixForm

e1 = {1, 0, 0};  e2 = {0, 1, 0};  
    e3 = {0, 0, 1};
OA = Transpose [{e1 + e2 + e3}];   OA // MatrixForm

S . OA // MatrixForm

S . OA // N // MatrixForm

7

rem;  Remove["Global`*"]

f[x_] := x Cos[(x + x^2) Pi]  -  x x^x;   f[x]

h[x_] := 2 x Sin[x^2 + 1] + x E^(2 x^2) + Cos[x] - x Sin[x];  
    h[x]

a

f'[x]

f'[x] // Simplify

b

steig = (f'[x] /. x -> 1) // Simplify

ArcTan[steig]

ArcTan[steig] // N

ArcTan[steig]/Degree  // N

c

Integrate[h[x], x]