Lösungen/ Solutions

(Beispiele von analytischen, nicht-geometrischen Lösunge/ 
        Exemples de solution analytiques, non-géométriques)

Remove["Global`*"]

1

l = 2 (1/2^0+1/2^1 +1/2^2  +1/2^3+  ...)

l = 2  Sum[1/2^n, {n, 0, Infinity}]

l = 2  (1/(1 - 1/2))

A = 2 * 1/2 + 2/2 * 1/2/2 + 2/2^2*1/2/2^2+....
    =  2 * 1/2 *(1+1/2*1/2+1/2^2*1/2^2+1/2^3*1/2^3+...)
    =  2 * 1/2 *(1+1/4+1/4^2+1/4^3+...)

A = 2 * 1/2   Sum[1/4^n, {n, 0, Infinity}]

A = 2 * 1/2   1/(1 - 1/4)

2

Limit[(n Cos[2  n] - Sin[n^2] + 8  n^2 - 4  n + 5)/(2  n^2 + 
            4  n - 5 Sin[n]), n -> Infinity]

3

f[x_] := x^2;  f'[x]

tg[t_] := (f'[x])  t  +  b;  
    tg[x] == f[x]

solv1 = Solve[tg[x] == f[x], {b}] // Flatten

tg[t_] := (f'[x])  t  +  b  /.  
          solv1;
tg[t]

y1 = tg[0]

(-y1 + f[x]) / x  ==  
      x / (y2 - f[x])

solv2 = Solve[(-y1 + f[x]) / x  ==  
              x / (y2 - f[x]), {y2}] // 
          Flatten//Simplify

y2 = y2  /.  solv2

A[x_] := x * (-y1  +  y2) / 2;    A[x]

A[x]  /.  x -> 1

Remove["Global`*"]

4

x1 = -2;  x3 = 0;  x5 = 2;  x6 = 4;
f[x_] := a (x - x1) (x - x3) (x - x5);  f[x]

f[x6] == 48

solv3 = Solve[f[x6] == 48, {a}] // Flatten

f[x_] := a (x - x1) (x - x3) (x - x5) /. solv3 // Expand; 
    f[x]

f'[x]

Solve[f'[x] == 0, {x}] //  Flatten

Solve[f'[x] == 0, {x}] //  Flatten // N

5

Remove["Global`*"]

f[x_] := x^3 - 2  x^2 + x;   f[x]

f'[x]

f'[x] /.  x -> 0

Solve[Evaluate[1 - 4 x + 3 (x^2) == 1], {x}] // 
      Flatten

ArcTan[1]

ArcTan[1]/ Degree // N

tg[t_] := 1 * t  +  b;  tg[t]

tg[(4/3)] == f[(4/3)]

solv4 = Solve[tg[(4/3)] == f[(4/3)], {b}] // Flatten

% // N

tg[t_] := 1 * t  +  b  /.  solv4;  tg[t]

Solve[tg[t] == 0, {t}] // Flatten

% // N

6

Remove["Global`*"]

f[x_] := -x^2 + 4;  f[x]

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

f'[x]

f'[x] /.  x -> -2

ArcTan[4]

ArcTan[4] // N

alpha = %/Degree

tg[t_] := 4 t + b;  tg[t]

tg[-2] == 0

solv5 = 
      Solve[tg[-2] == 0, {b}] // Flatten

tg[t_] := 4 t + b /. solv5;  tg[t]

tg[t] == f[0]

solv6 = Solve[ tg[t] == f[0], {t}] // Flatten

{-1, tg[-1]}

A = (4 + 2)/2 * 4