Lösungen/ Solutions

1a

Plot[{x^3 + 20  Sin[x], x^2}, {x, -6, 6}];

1b

Plot[Floor[-x^2] + 1, {x, -2, 4}];

1c

Plot[{Cos[Floor[x + 1]] - 1, Cos[x + 1] - 1}, {x, -7, 
          14}];

2a

Solve[x == 5 + 6/x, {x}]

2b

Solve[x == 5 + 7/x, {x}]

% // N

x ∉ Q

3a

f[x_] :=  ArcTan[x];  
    p[y_] :=  y^2 + 2  y + 2;
h[x_] := -p[f[x]];       h[x]

Plot[f[x], {x, -1, 1}];

Plot[h[x], {x, -1, 1}];

Plot[{f[x], h[x]}, {x, -1, 1}];

3b

{h[-1],  h[-1] // N}

h[1]

h[-1] -  h[-1]

Sin[h[-1] -  h[-1]]

4

Remove[f, g, h, u, s, x]

f[x_] := x^2 - x + 1/4;  g[x_] := x^2 + x;  h[x_] := f[x] g[x]; 
    h[x]

u[x_] := 1/(f[x] - g[x]); u[x]

s[x_] := f[x]/g[x] + x;
s[x] // Simplify

% // Together

% // Apart

4a

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

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

4b

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

4c

Plot[h[x], {x, -1, 2}];

Plot[u[x], {x, -1, 2}];

Denominator[u[x]]

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

4e

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

Solve[s[x] == 0, {x}] // N

Solve[Denominator[Together[s[x]]] == 0, {x}] // N

Plot[s[x], {x, -3, 2}];

Polstelle/ Place du pôle

4f

s[x] // Apart

ta = 
      Table[Apart[s[x]][[k]], {k, 1, Length[Apart[s[x]]]}]

Table[PolynomialQ[ta[[k]], x], {k, 1, Length[ta]}]

sel = Select[ta, PolynomialQ]

as[x_] := Sum[sel[[k]], {k, 1, Length[sel]}];  as[x]

Plot[{s[x], as[x]}, {x, -6, 6}];

5a

Remove[f, g]

f[x_] := 2^(-x^2);  g[x_] := 1/2 x

Plot[{f[x], g[x]}, {x, -3, 3}];

5b

Plot[{f[x], g[x]}, {x, 0.5, 1.5}];

Plot[{f[x], g[x]}, {x, 0.9, 1.1}];

5c

Plot[{0, f[x], 1}, {x, -3, 3}];

6

expr =  
      Simplify[2^x *3^(-x) * 2^(2  x)* 
          E^(-3)]

Solve[expr == 1, {x}]

% // N

expr1 =  
      Simplify[2^x* 3^(-x)*  2^(2  x) ==  
          E^3]

Solve[x  Log[8/3] == 3, {x}]

% // N