Lösungen/ Solutions

1a

Plot[{x^2 + Sin[x], x^2}, {x, -2, 5}];

Plot[{x^2 + Sin[x], x^2}, {x, -4, 10}];

1b

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

1c

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

2a

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

% // N

x  > 4   ⇒   x ≈ 4.828....,  x ∉ Q

2b

Solve[x - 2 == 3 + 3/(x - 2), {x}]

% // N

x  > 5   ⇒   x ≈ 5.791....,  x ∉ Q

3a

f[x_] :=  ArcCos[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}];

3b

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

h[1]

(Subscript[D, h]) = [-1,1],  (Subscript[W, h]) = [0, h(-1)]

3c

{h[0],  h[0] // N}

h[1]

4a

Remove[f, g, h]

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

h[x]

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

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

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

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

4b

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

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

Polstelle/ Place du pôle

Solve[1/u[x] == 0, {x}]

5a

Remove[h]

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

Apart[h[x]]

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

5b

Show[Plot[h[x], {x, -100, -5}], 
        Plot[h[x], {x, 5, 100}]];

Plot[x + 1, {x, -100, 100}];

6a

Remove[f, g]

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

Plot[{f[x], g[x]}, {x, -2, 2}];

6b

Plot[{f[x], g[x]}, {x, 0.650, 0.655}];

6c

m = 0; M = 1;

Plot[{f[x], m, M}, {x, -4, 4}];

6d

{f[0], f[Log[E]], f[1]}

% // N

6d

{f[0], f[Log[E]], f[1]}

% // N

7

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

Solve[expr, {x}]

% // N

expr1 =  
      Simplify[3^x (4/9)^(-x) ==  E^3]

Solve[x  Log[27/4] == 3, {x}]

% // N