Lösungen / Solutions

1

a

f[x_] := x^2/(x^2 + Sin[x]^2);
Plot[f[x], {x, -6, 6}];

f[0] = x^2/(x^2 + Sin[x]^2)

===>    Definitionslücke / Ne pas difini pour x=0

b

Solve[f[x] == 1, {x}]

(1 == x^2/(x^2 + Sin[x]^2))

Equal[1, x^2/(x^2 + Sin[x]^2)]

eq[x_, y_, z_] := Equal[ x z,  y z];
eq[1, x^2/(x^2 + Sin[x]^2), (x^2 + Sin[x]^2)]

Reduce[(x^2) + Sin[x]^2 == (x^2)]

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

FindRoot[f[x] == 1, {x, 3}]

c

f[x_] := Sin[Floor[x]] - 1;
Plot[f[x], {x, -6, 6}];

d

f[x_] := Cos[E^Abs[x]];
Plot[f[x], {x, -2.5, 2.5}];

2

solv = 
      Solve[x + 1 == 5 + 5/(x + 1), {x}] // Flatten

N[%]

solv[[1]][[2]]

solv[[1]][[2]] ∈ Rationals

solv[[2]][[2]] ∈ Rationals

3

Ln[x_^2  _] := 2 Ln[Abs[x]];
Ln[e^x_] := x;
exp[Ln[x]] := x;
exp[x_/2] := Sqrt[e^x];
exp[x_] := E^x;
Ln[x_] := Log[x];

f[x_] := E^x;
p[x_] := 1/2 Ln[x^2];
p[f[x]] // Simplify

p[f[x]]

Plot[Evaluate[p[f[x]]], {x, -2.5, 2.5}];

Evaluate[f[p[x]]]

Plot[f[p[x]], {x, -2.5, 2.5}];

f[p[x]^2] // Simplify

Plot[Evaluate[f[p[x]^2]], {x, -2.5, 2.5}];

Plot[Evaluate[f[p[x]^2]], {x, -2.5, 2.5}, 
        PlotRange -> {0, 10}];

4

f[x_] := (x - 1) (x^2 - 1)
p[x_] := (x^3 - 6  x^2 + 12  x - 8);
f[x] p[x]

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

Plot[f[x] p[x], {x, -1.05, 2.8}];

Plot[f[x] p[x], {x, 1, 2}];

Plot[f[x] /p[x], {x, -1.05, 2.8}];

Plot[f[x] /p[x]  - x^2, {x, -0, 3}];

Plot[f[x] /p[x]  - x^2, {x, -0, 20}, 
        PlotRange -> {-20, 30}];

5

f[x_] := E^-(Sin[x]^2);
g[x_] := -0.5 x;

Plot[{f[x], g[x], E^0, E^(-1)} , {x, -6, 
          6}];

FindRoot[f[x] == g[x], {x, -1}]

6

Plot[
        4^(2  x) - 3^x* Pi^5 * 4^(-3 x) , {x, -2, 
          3}];

Plot[
        4^(2  x) - 3^x* Pi^5 * 4^(-3 x) , {x, -2, 3}, 
        PlotRange -> {-20, 20}];

FindRoot[
      4^(2  x) == 
        3^x* Pi^5 * 4^(-3 x), {x, -1}]

4^(2  x)*3^(-x) * 4^(+3 x) ==  
      Pi^5

u = 
      4^(2  x)*3^(-x) * 4^(+3 x) ==  
          Pi^5 /. {4 -> e^Ln[4], 3 -> e^Ln[3], 
          Pi -> e^Ln[Pi]}

Solve[-x Log[3] + 5 x Log[4] == 
        5  Log[Pi], {x}]

7

Remove["Global`*"]

Geometrie / Géométrie   ===>  α=90 °, β=30 °, γ=60 °

30 Degree

R = 1* Sin[30 Degree]

Proportionen / proportions

a = 2; b = 1; c = Sqrt[2^2 - 1^2]

Remove[r]

g = (r[2]/
              r[1] == ((Sqrt[3]) - 
                  1)/(Sqrt[3])); 
solve2 = Solve[g, {r[2]}] // Flatten

solve2 = solve2 /. r[1] -> 1

u = r[2] /. solve2 // Apart

r[1] = R;
r[n_] := r[n - 1] *u

Table[r[n], {n, 1, 10}]

r[n_] := (1/2) (1 - (1/(Sqrt[3])))^(n - 1)

r[n]

Table[r[n], {n, 1, 10}]

Table[r[n]^2 Pi /2, {n, 1, 10}]

Pi /2  Sum[r[n]^2, {n, 1, Infinity}]

N[%]

8

Limit[(4  n^3 - 3  n^2 + n^2 Cos[n] - E^(-n)) 
        n/(Sin[-n^2] + 4  n^2  + 3  n^4), 
      n -> Infinity]

N[%]

9

s1 = 1/2 Sum[1/(3^k), {k, 0, Infinity}]

s2 = 1/3 Sum[1/(4^k), {k, 0, Infinity}]

s1 + s2 - 7/6

Sum[1/(2*3^k) + 1/(3*4^k), {k, 0, Infinity}] - 7/6

N[%]

10

Limit[((n + 2  Log[n])/(3  n - 4  Log[n] + 5  Tan[1/n])), 
      n -> Infinity]

N[%]