Lösungen

1

Remove["Global`*"]

a

f1[x_] := x^2/(1 + x^2 + Cos[x]^2);
Plot[f1[x], {x, -6, 6}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}, 
        PlotRange -> {0, 1}];

b

Plot[{0.8, f1[x]}, {x, -4, 4}, 
        PlotRange -> {0.7, 0.9}];

FindRoot[0.8 == f1[x], {x, -2.6}]

FindRoot[0.8 == f1[x], {x, -2.1}]

FindRoot[0.8 == f1[x], {x, -1.5}]

FindRoot[0.8 == f1[x], {x, 2.6}]

FindRoot[0.8 == f1[x], {x, 2.1}]

FindRoot[0.8 == f1[x], {x, 1.5}]

c

f2[x_] := 2  Cos[Floor[x] - x];
Plot[f2[x], {x, -6, 6}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

d

f3[x_] := 2  Cos[Abs[Log[x]]];
Plot[f3[x], {x, 0.5, 3.5}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

e

Remove[φ, v, t]

φ[t_] := 
        1/(2 + t^2 + (t - 5)^4);
v[
          t_] := φ[t] {Cos[t], 
            Sin[t]};
ParametricPlot[Evaluate [v[t]], {t, -,2  Pi}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}, 
        AspectRatio -> Automatic];

ParametricPlot[Evaluate [v[t]], {t, -2 Pi, 2  Pi}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}, 
        AspectRatio -> Automatic];

ParametricPlot[Evaluate [v[t]], {t, -2 Pi, 21}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}, 
        AspectRatio -> Automatic];

ParametricPlot[Evaluate [v[t]], {t, -2 Pi, 30}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}, 
        AspectRatio -> Automatic];

Plot[φ[t], {t, -2 Pi, 10}]

φ[t_] := 
        1/(2 + t^2 + (t - 5)^4);
v[t_] := φ[t] {Cos[t], Sin[t]}; 
    ParametricPlot[Evaluate [v[t]], {t, -5, 10}, 
      PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}, 
      AspectRatio -> Automatic];

φ[t_] := 
        1/(2 + t^2 + (t - 5)^4);
v[t_] := φ[t] {Cos[t], Sin[t]}; 
    ParametricPlot[Evaluate [v[t]], {t, -2 Pi, 500}, 
      PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}, 
      AspectRatio -> Automatic];

2

Remove[x]

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

Solve[x == 2 + 2/x, {x}] // N

x not in Q !

3

Remove[f, p]

f[x_] := Sin[x];  p[x_] := Cos[x]

a

h[x_] := p[f[x]];  h[x]

Plot[h[x], {x, -6, 6}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

b

h[x_] := f[p[x]];  h[x]

Plot[h[x], {x, -6, 6}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

c

h[x_] := f[p[x]*f[x]];  h[x]

Plot[h[x], {x, -6, 6}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

4

Remove[f, p]

f[x_] := (x^2 - x) (16 - x^2);  
    p[x_] := (2 + x)^2*(4 + x);
{Expand[f[x]], Expand[p[x]]}

a

h[x_] := f[x] p[x];  h[x]

h[x] // Expand

Solve[h[x] == 0, x]

Plot[h[x], {x, -6, 6}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

Plot[h[x], {x, -4.1, 1.1}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

b

u[x_] := f[x]/p[x]; u[x]

u[x] // Simplify

u[x] // ExpandAll // Together

Solve[Denominator[u[x]] == 0, x]

Solve[u[x] == 0, x]

u[x] // Apart

Plot[u[x], {x, -6, 6}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

Plot[u[x], {x, -0.3, 4.1}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

Plot[u[x], {x, -0.1, 1.1}, 
        PlotStyle -> {{Thickness[0.01], GrayLevel[0.1]}}];

5

Beispiel

Remove[p]

p[x_] := 0; p[x_ /; PrimeQ[x]] := x
tabP = Union[Table[p[x], {x, 0, 10000}]]; 
    Union[Table[
        If[tabP[[k + 1]] - tabP[[k]] == 2, {tabP[[k]], 
            tabP[[k + 1]]}, {}], {k, 1, Length[tabP] - 1}]]

p[x_] := 0; p[x_ /; PrimeQ[x]] := x
tabP = Union[Table[p[x], {x, 0, 10000}]]; 
    Union[Table[
        If[tabP[[k + 2]] == 
            tabP[[k]] + tabP[[k + 1]] + 1, {tabP[[k]], 
            tabP[[k + 1]], tabP[[k + 2]]}, {}], {k, 1, 
          Length[tabP] - 2}]]

p[x_] := 0; p[x_ /; PrimeQ[x]] := x
tabP = Union[Table[p[x], {x, 0, 10000}]]; 
    Union[Table[
        If[tabP[[k + 2]] == 
            tabP[[k]]*tabP[[k + 1]] + 2, {tabP[[k]], 
            tabP[[k + 1]], tabP[[k + 2]]}, {}], {k, 1, 
          Length[tabP] - 2}]]