Lösungen - Solutions

1 - 2

a

f1[x_] := -x^4/(3 - x^4)

Plot[f1[x], {x, -7, 7}];

f2[x_] := x^3/(3 + x^4 - Sin[x]); 
    Plot[f2[x], {x, -7, 7}];

f3[x_] := (-x^4 E^x)/(3 + x^4 + Abs[Tan[x]]); 
    Plot[f3[x], {x, -7, 7}];

f4[x_] := x^3/(3 + x^3 - Sin[x]); 
    Plot[f4[x], {x, -7, 7}];

f4[x_] := -x^3/(-3 - x^3 + Sin[x]); 
    Plot[f4[x], {x, -7, 7}];

x^3/(3 + x^n - Sin[x]) // TeXForm

fn[x_, n_] := 
        x^3/(3 + x^n - Sin[x]);
Table[
        Plot[fn[x, n], {x, -7, 7}, Axes -> None], {n, 1, 
          6}];

Show[
        GraphicsArray[{Table[
              Plot[fn[x, n], {x, -7, 7}, Axes -> None], {n, 1, 
                6}]}]];

Others

f1[x_] := x^3/(3 + x^4 - Sin[x])

Plot[f1[x], {x, -7, 7}];

f1a[x_] := x^4/(3 + x^4 - Sin[x]); 
    Plot[f1a[x], {x, -7, 7}];

f1a[x_] := x^3/(3 + x^3 - Sin[x]); 
    Plot[f1a[x], {x, -7, 7}];

f1a[x_] := x/(1 + x^6 + Cos[x]); 
    Plot[f1a[x], {x, -7, 7}];

x^3/(3 + x^n - Sin[x]) // TeXForm

fn[x_, n_] := 
        x^3/(3 + x^n - Sin[x]);
Table[
        Plot[fn[x, n], {x, -7, 7}, Axes -> None], {n, 1, 
          6}];

Show[
        GraphicsArray[{Table[
              Plot[fn[x, n], {x, -7, 7}, Axes -> None], {n, 1, 
                6}]}]];

2 - 1

x/(3 + x^4 - Tan[x]) // TeXForm

f0[x_] := x/(3 + x^4 - Tan[x]);

Plot[{f0[x]}, {x, -7, 7}, PlotPoints -> 400];

Plot[{f0[x]}, {x, -20, 20}];

Plot[{f0[x]}, {x, -100, 100}];

Plot[{f0[x]}, {x, 50, 100}, PlotPoints -> 200];

Plot[3 + x^4 - Tan[x], {x, 5, 15}, 
        PlotPoints -> 200];

Floor[Abs[f0[600]]] // N

Floor[Abs[f0[600 + 2  Pi]]] // N

Floor[Abs[f0[700]]] // N

Floor[Abs[f0[700 + 2  Pi]]] // N

f0[x_] := x/(3 + x^4 - Tan[x]);

Plot[-Tan[x], {x, 5, 20}, PlotPoints -> 200];

Plot[-Tan[x] + 3, {x, 5, 20}, 
        PlotPoints -> 200];

Plot[-Tan[x] + 3 + x, {x, 5, 20}, 
        PlotPoints -> 200];

Plot[-Tan[x] + 3 + x^2, {x, 5, 20}, 
        PlotPoints -> 200];

Plot[-Tan[x] + 3 + x^3, {x, 5, 20}, 
        PlotPoints -> 200];

f0[x_] := x/(3 + x^4 - Tan[x]); 
    Plot[-Tan[x] + 3 + x^4, {x, 5, 20}, 
      PlotPoints -> 200];

f0[x_] := x/(3 + x^4 - Tan[x]); 
    Plot[{f0[x]}, {x, -7, 7}];

f0[x_] := x/(3 + x^4 - Tan[x]); 
    Plot[{f0[x], 0.2}, {x, 0, 1.3}];

FindRoot[f0[x] == 0.2, {x, 0.2}]

f0[x^2]

f0[(-x)^2]

3

f[x_, n_] := 
        Ceiling[2  Sin[n Pi/2 x]]^2 - x;
g[x_, n_] := 
        Floor[2  Sin[n Pi/2 x]]^2 - x;
t = 
        Table[f[x, n] - g[x, n], {n, 1, 100}] /. 
          x -> 0.5;
Table[t[[k]] - t[[k + 8]], {k, 20, 27}]

t = Table[f[x, n] - g[x, n], {n, 1, 100}] /. x -> 0.5

Table[n, {n, 20, 27}]

Table[1, {n, 20, 27}]

3 - 4

4 - 3

Remove[f, g, h]

a,b

g[x_] := -2 + x;h[x_] := -2 + 
        x^2;

k = 1; f[k] = g[h[x]]; {f[k] // Expand, f[k]}

k = 2; f[k] = h[g[x]]; {f[k] // Expand, f[k]}

k = 3; f[k] = h[g[g[x]]]; {f[k] // Expand, f[k]}

k = 4; f[k] = g[h[h[x]]]; {f[k] // Expand, f[k]}

k = 5; f[k] = h[g[h[x]]]; {f[k] // Expand, f[k]}

k = 6; f[k] = g[h[g[x]]]; {f[k] // Expand, f[k]}

4

5

Remove[f1, f2, f3, g, h]

a,b

f1[
          x_] := (+x^2 - 2) (-1 + 
              x^2) ;
f2[x_] := Sqrt[f1[x]];
f3[x_] := f2[x]^4;

({{Plot[{f1[x]}, {x, 0, 2}, PlotRange -> {-3, 3}];}, {Plot[{f2[x]}, {x, 0, 2}, 
              PlotRange -> {-3, 4}];

          Plot[{f3[x]}, {x, 0, 2}, PlotRange -> {-3, 7}];}})

Solve[f1[x] == 0, {x}]

Solve[f2[x] == 0, {x}]

Solve[f3[x] == 0, {x}]

c,d

f1[
          x_] := (1 - x^2) (-x^2 + 
              2) ;
f2[x_] := Sqrt[f1[x]];
f3[x_] := f2[x]^6;

({{Plot[{f1[x]}, {x, 0, 2}, PlotRange -> {-3, 3}];}, {Plot[{f2[x]}, {x, 0, 2}, 
              PlotRange -> {-3, 4}];

          Plot[{f3[x]}, {x, 0, 2}, PlotRange -> {-3, 7}];}})

Solve[f1[x] == 0, {x}]

Solve[f2[x] == 0, {x}]

Solve[f3[x] == 0, {x}]

6

r[
          x_] := (Cos[2  x] - Sin[5  x])/(x + 
              2);
ParametricPlot[r[x] {Cos[x], Sin[x]}, {x, 0, 2  Pi}, 
        AspectRatio -> Automatic];

r[x_] := (Cos[2  x] - Sin[5  x]);
ParametricPlot[r[x] {Cos[x], Sin[x]}, {x, 0, 2  Pi}, 
        AspectRatio -> Automatic];

r[x_] := (Cos[2  x] + Sin[5  x]);
ParametricPlot[r[x] {Cos[x], Sin[x]}, {x, 0, 2  Pi}, 
        AspectRatio -> Automatic];

r[
          x_] := (2 + Cos[2  x] + 
            Sin[5  x]);
ParametricPlot[r[x] {Cos[x], Sin[x]}, {x, 0, 2  Pi}, 
        AspectRatio -> Automatic];

r[
          x_] := (3 + Sin[2  x] - 
            Sin[5  x]/2);
ParametricPlot[r[x] {Cos[x], Sin[x]}, {x, 0, 2  Pi}, 
        AspectRatio -> Automatic];

6

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

Solve[x == 4 + 4/(4 + x), {x}]

Solve[(Sqrt[6]) == x + 2 (Sqrt[5]), {x}]

7

f5a[x_] := (x^2 + 1) (x);
f5b[x_] := (x^2 - 1) (x + 3)

Expand[f5a[x]]

Expand[f5b[x]]

Expand[f5a[f5b[x]]]

Solve[f5b[x] - 3 == 0, {x}]

Solve[f5b[x] == 0, {x}]

Plot[f5a[f5b[x]], {x, -4, 2}];

Plot[f5a[x]/f5b[x], {x, -4, 2}];

Apart[f5a[x]/f5b[x]]

?Nest

8

r[x_] := 2 Sin[x/4] + 2 + Cos[2  x]; 
    ParametricPlot[r[x] {Cos[x], Sin[x]}, {x, 0, 8  Pi}];

r[x_] := 2 Sin[x/2] + 2 + Cos[4  x]; 
    ParametricPlot[r[x] {Cos[x], Sin[x]}, {x, 0, 8  Pi}];

r[x_] := 2 Sin[x/4] + 2 + Cos[4  x]; 
    ParametricPlot[r[x] {Cos[x], Sin[x]}, {x, 0, 8  Pi}];

r[x_] := 2 Sin[x/2] + 2 + Cos[2  x]; 
    ParametricPlot[r[x] {Cos[x], Sin[x]}, {x, 0, 8  Pi}];

r[x_] := Sin[x/3] + 2 + 4 Cos[3  x]; 
    ParametricPlot[r[x] {Cos[x], Sin[x]}, {x, 0, 12  Pi}];