Lösungen/Solutions/F2S1Ue/Ex16

L = Lektion/ Leçon   -- Sprechstunde / heures de consultation
W = Work / Arbeit / Travail
A=Anhang/Annexe/Appendice

Ueb/Ex 1      W

h = p*q; s = p + q;
solv = Solve[ s*h/2 == 1, {q}]

((-(p^(3/2)) + (Sqrt[8 + (p^3)]))/(2 (Sqrt[p])))  / .0 -> 
      0

Plot[((-(p^(3/2)) + (Sqrt[8 + (p^3)]))/(2 (Sqrt[p]))), {p, 0, 
          10}];

h*s /. solv[[2]] // Simplify

p*(p*q) + q*(p*q) /. solv[[2]] // Simplify

f1[x_, p_] := 
        p x + (p*((-(p^(3/2)) + (Sqrt[8 + (p^3)]))/(2 (Sqrt[p])))) // 
          Simplify;
f2[x_, 
          p_] := -((-(p^(3/2)) + (Sqrt[8 + (p^3)]))/(2 (Sqrt[p]))) x + (p*((-(p^(3/2)) + (Sqrt[8 + (p^3)]))/(2 (Sqrt[p])))) // 
          Simplify;

p x + (p*q) /. solv[[2]]

f1[x, 2]

f2[x, 2]

Plot[{f1[x, 2], f2[x, 2]}, {x, -1, 2}, 
        AspectRatio -> Automatic];

Remove[f]

f[x_, p_] := 
      0 /; x ≤ -((-(p^(3/2)) + (Sqrt[8 + (p^3)]))/(2 (Sqrt[p]))); f[x_, p_] := 
      f1[x, p] /; -((-(p^(3/2)) + (Sqrt[8 + (p^3)]))/(2 (Sqrt[p]))) < 
            x  &&  x ≤ 0 ;
f[x_, p_] := 
        f2[x, p] /; 
          0 < x  &&  x  ≤ p;
f[x_, p_] := 0 /; p < x

f[-1, 2]

Plot[{f[x, 2]}, {x, -0.7, 2.5}, 
        AspectRatio -> Automatic];

Plot[{f[x, 1]}, {x, -1, 1}, 
        AspectRatio -> Automatic];

e[n_, p_] := 
        NIntegrate[
          x^n f[x, p], {x, -Infinity, 
            Infinity}];
e[3, 2]

e[n_, p_] := 
        Integrate[
          x^n f[x, p], {x, ((-(p^(3/2)) + (Sqrt[8 + (p^3)]))/(2 (Sqrt[p]))), 
            p}];
e[3, 2]

eNew[n_, p_] := 
        Integrate[
            x^n f1[x, 
                p], {x, -((-(p^(3/2)) + (Sqrt[8 + (p^3)]))/(2 (Sqrt[p]))), 0}] + Integrate[x^n f2[x, p], {x, 0, p}];

eNew[3, 2]

{eNew[3, 1], eNew[2, 1], eNew[1, 1], eNew[0, 1]}

μ[p_] := eNew[1, p];  σ[p_] := 
      Sqrt[eNew[2, p] - μ[p]^2];
γ[p_] := 
      1/σ[p]^3 * (eNew[3, p] - 3 μ[p] eNew[2, p] +  
            2  μ[p]^3 )

{μ[1], σ[1], γ[1]}

Plot[γ[p], {p, 0.8, 5}, 
        PlotRange -> {-0.3, 0.7}];