Lösungen

Aufgabe 1

f[x_, a4_, a3_, a2_, a1_, a0_] := 
a4 x^4  +  
        a3 x^3  +  a2 x^2  +  a1 x  +  a0

s = 
      Solve[{f[-3, a4, a3, a2, a1, a0] == 1, 
            f[-2, a4, a3, a2, a1, a0] == 0, 
            f[0, a4, a3, a2, a1, a0] == 0, 
            f[1, a4, a3, a2, a1, a0] == 0, 
            f[2, a4, a3, a2, a1, a0] == 1}, {a4, a3, a2, a1, a0}] // 
        Flatten

p[x_] := f[x, a4, a3, a2, a1, a0] /. s;  p[x]

Plot[
        p[x], {x, -4.8, 
          4.8}, 
Prolog -> {PointSize[ .03], 
            Point[{-3, 1}], Point[{-2, 0}], Point[{0, 0}], 
            Point[{1, 0}], Point[{2, 1}]}, 
        PlotStyle -> {PointSize[ .03]}];

DisplayTogether[
        ListPlot[{{-3, 1}, {-2, 0}, {0, 0}, {1, 0}, {2, 1}}, 
          PlotStyle -> {PointSize[ .03]}], 
        Plot[p[x], {x, -5, 5}]];

Aufgabe 2

f2[x_] := Sin[x]^2 -  Abs[Floor[x]];  f2[x]

points = 
      Prepend[Map[Point, Table[{n, f2[n]}, {n, -3, 3}]], 
        PointSize[ .03]]

Plot[
        f2[x],  {x, -3, 3}, 
Prolog -> 
          points
];

Aufgabe 3

f31[x_] := 2 x^2  +  2 x - 4;  
    f32[x_] := -x^2  +  x  -  2;

Plot[{f31[x], f32[x]}, {x, -3, 2}];

s = Solve[f31[x] == f32[x], {x}] // Flatten

y = {f31[x] /. s[[1]], f31[x] /. s[[2]]}

f33[x_] := 1/(f31[x] - f32[x]) + 2

Plot[{f33[x], 2}, {x, -2, 2}];

Aufgabe 4

f4[x_] := Cosh[Sqrt[x + 1]] - 1

p41 = Plot[{f4[x], x}, {x, -1, 3}];

f4um[y_] := ArcCosh[y + 1]^2 - 1;  f4um[x]

p42 = Plot[f4um[x], {x, 0, 3}];

Show[p41, p42, AspectRatio -> 1];

Aufgabe 5

PolarPlot[f_, i_] := ParametricPlot[{Cos[f], Sin[f]}, i]

Plot[Abs[Cos[t]], {t, 0, 2  Pi}];

ParametricPlot[{Cos[t] Abs[Cos[t]], Sin[t] Abs[Cos[t]]}, {t, 0, 
          2  Pi}, AspectRatio -> 0.5];

Aufgabe 6

Sum[(1/11)^n, {n, 0, Infinity}]

Sum[(1/n)^n, {n, 0, Infinity}]

Aufgabe 7

Limit[((1 + n)/n^2 - 1) (3 - 6/n + Cos[n]/Sqrt[n]), 
      n -> Infinity]

Limit[x/Tan[x] (Sin[x] + Cos[x] + 2  x), x -> 0]

Aufgabe 8

Limit[1/n  Sin[n!] + 3^n /(n n!) Sqrt[n], 
      n -> Infinity]

Limit[Log[n]/Floor[n^2] + (1 + 1/n)^(2  n), 
      n -> Infinity]

Limit[(((1 + (1/n)))^(2  n)),n->∞]

Limit[Log[n]/n^2,n->∞]