Lösungen

Aufgabe 1

f1[x_] := E^Sin[x];  f1[x]

?Series

Series[f1[x], {x, 0, 5}]

f1e[x] := Normal[Series[E^x, {x, 0, 5}]]; f1e[
      x]

f1s[x] := Normal[Series[Sin[x], {x, 0, 5}]]; f1s[
      x]

f1e[x]  /.  x -> f1s[x]

f1b[x_] := f1e[x]  /.  x -> f1s[x]  //  Expand ;  
    f1b[x]

Integrate[f1b[x], {x, -0.1, 0.1}]

NIntegrate[f1[x], {x, -0.1, 0.1}]

Aufgabe 2

f2[x_] := 1/(x^2 - 5  x + 6);  f2[x]

Apart[f2[x]]

s1 = Series[-3/(x - 3), {x, 0, 10}]

s2 = Series[-2/(x - 2), {x, 0, 10}]

s1/(-3) - s2/(-2)

Series[f2[x], {x, 0, 10}]

a[n_] := (1/3)^n /(-3) -  (1/2)^n /(-2);  
    a[n]

?Limit

Limit[a[n]/a[n + 1], n -> Infinity]

1/Limit[a[n + 1]/a[n], n -> Infinity]

Aufgabe 3

f3[x_, y_] := x^2 y /(x^2 + y^2);  f3[x, y]

Plot3D[f3[x, y], {x, -1, 1}, {y, -1, 1}];

f3a[r_, α_] := 
      x^2 y /(x^2 + y^2) /. {x -> r Cos[α],  
          y -> r Sin[α]};  
    f3a[r, α] // Simplify

Limit[f3a[r, α], r -> 0]

Aufgabe 4

f4[x_, y_] := x^7 Sin[E^(x y)]

grad[f_] := {{D[f, x], D[f, y]}} // Transpose;  
    grad[f4[x, y]] // MatrixForm

res4 = 
      Flatten[grad[f4[x, y]]] . {Cos[α], Sin[α]}

res4 /. {x -> 0, y -> 0, α -> Pi/3}

Plot3D[f4[x, y], {x, -1, 1}, {y, -1, 1}];

Aufgabe 5

f5[x_, y_] := -2 x^2  +  (2  x^2 - 1) y^2;  
    f5[x, y]

D[f5[x, y], x]

D[f5[x, y], y]

s5 = 
      Solve[{D[f5[x, y], x] == 0, D[f5[x, y], y] == 0}, {x, 
          y}]

hess = {{D[f5[x, y], x, x], D[f5[x, y], y, x]}, {D[f5[x, y], x, y], 
          D[f5[x, y], y, y]}};  hess // MatrixForm

h = Det[hess]

Table[h /. s5[[n]], {n, 1, 5}]

{?,Sattel,Sattel,Sattel,Sattel}

D[f5[x, y], x, x] /. s5[[1]]

{Max.,Sattel,Sattel,Sattel,Sattel}

Plot3D[
        f5[x, y], {x, -1.5, 1.5}, {y, -1.5, 1.5}];

Aufgabe 6

f6[x_, y_] := -x  +  2  y;  f6[x, y]

g6[x_, y_] := x^2 - y^2 - 1;  g6[x, y]

grad[f_] := {{D[f, x], D[f, y]}} // Transpose;

MatrixForm[
        grad[f6[x, y]]] ==  λ  MatrixForm[
          grad[g6[x, y]]]

Solve[{grad[f6[x, y]] ==  λ grad[g6[x, y]],  
        g6[x, y] == 0}, {λ, x, y}]

Keine reellen Lösungen!

<< Graphics`ImplicitPlot`

ImplicitPlot[g6[x, y] == 0, {x, -10, 10}];

x1[t_] := Sinh[t] - Cosh[t]; y1[t_] := Sinh[t] + Cosh[t];

x1[u_, v_] := u Cos[Pi/4] + v Sin[Pi/4]; 
    y1[u_, v_] := u Sin[Pi/4] - v Cos[Pi/4];

g6[x1[u, v], y1[u, v]] // Expand

Solve[g6[x1[u, v], y1[u, v]] == 0, {v}]

Hyperbel!!!!

v = 1/(2  u)

v1[u_] := 1/(2  u)

Senkrechte Fläche durch Kurve g6[x,y]==0, Parametrisierung:

{x1[u, v], y1[u, v], z}

p1 = 
        ParametricPlot3D[{x1[u, v], y1[u, v], 
            z}, {u, -3, -0.0001}, {z, -7, 7}, 
          DisplayFunction :> Identity];

p2 = 
        ParametricPlot3D[{x1[u, v], y1[u, v], z}, {u, 0.0001, 3}, {z, -7, 
            7}, DisplayFunction :> Identity];

p3 = 
        ParametricPlot3D[{x, y, f6[x, y]}, {x, -3, 3}, {y, -3, 3}, 
          DisplayFunction :> Identity];

Show[p1, p2, p3, 
        DisplayFunction :> $DisplayFunction];

ParametricPlot3D[{x1[u, v1[u]], y1[u, v1[u]], 
          f6[x1[u, v1[u]], y1[u, v1[u]]]}, {u, -5, 5}];

Keine Extrema!

Aufgabe 7

V[x_, y_] := 1/3 x y^2  -  y

V[0.5, 0.5]

Δx = 0.01;  Δy =  0.01;

ΔV =  
      Abs[D[V[x, y], x] Δx] +  
        Abs[D[V[x, y], y] Δy]

ΔV  /.  {x -> 0.5,  y -> 0.5}