Restfunktion

#0000ff

#ff0000

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Schranke Haushalte

Schranke SH

\text{    Beschränktes Wachstum}\ 
 \scalebox{0.8}\text{Formeltransformation ( Monate ↔ Jahre)}

\scalebox{1.1}{\mathbf{Aufgabe:}}\
\text{Eine Firma bringt in einer Stadt mit 40 000  Haushalten  einen neuen Haushaltartikel auf den Markt. }\
\text{Die Firma geht davon aus, dass drei Viertel der Haushalte den Artikel kaufen werden und sich die Zahl}\
\text{der verkauften Artikel mit einem beschränkten Wachstum beschreiben lässt.} \
\text{Im ersten Monat werden 2400 Stück verkauft. Kann man auf Grund dieser Erfahrung davon ausgehen, }\
\text{dass im ersten Jahr 20 000 Stück verkauft werden ?}\
\mathit{(Quelle: Lambacher \space  Schweizer)}\ \space \
\mathbf{Lösungshinweis:}\
\text{ Bei korrekter Dateneingabe bzw. Einstellung der Regler lässt sich der gesuchte Vergleichswert direkt} \ 
\text{aus der Tabelle ablesen.}

Werner Knobloch, 2015

S_H :

a(1) :

#00ff00

#7f00ff

Beschränktes Wachstum

explizit

Eingaben korrekt!

Tab4Z1: "\textcolor{" + (LaTeX[ColorBlue]) + " }{R}(\textcolor{ " + (LaTeX[ColorGreen]) + " }{t}) =(\textcolor{ " + (LaTeX[ColorRed]) + "  }{S}− \textcolor{" + (LaTeX[ColorBlue]) + "   }{B}(\textcolor{ " + (LaTeX[ColorGreen]) + "  }{0}))· ℯ ^{-\lambda \space ·\textcolor{" + (LaTeX[ColorGreen]) + "   }{t}}"
B_0 = 1501 (slider from 0 to S, step 0.1)
S = 75000.40000000001 (slider from 1 to 120000, step 0.1)
k = 0.1 (slider from 0 to 1, step 0.1)
q: 1 - k
λ: (-log(q))
Tab4Z2: "\textcolor{" + (LaTeX[ColorBlue]) + " }{R}(\textcolor{ " + (LaTeX[ColorGreen]) + " }{t}) = (\textcolor{ " + (LaTeX[ColorRed]) + "  }{" + (LaTeX[S]) + "}− \textcolor{" + (LaTeX[ColorBlue]) + "   }{" + (LaTeX[B_0]) + "})  ·ℯ ^{-" + (LaTeX[λ]) + " \space · \space \textcolor{" + (LaTeX[ColorGreen]) + "   }{t}}"
Tab4 = TableText({{Tab4Z1, Tab4Z2}}, "v")
t = 1 (slider from 0 to h, step 0.1)
Tab5b = LaTeX(t, true, true)
a_1Monat = 2400
S_H = 45000 (slider from 1 to 60000, step 1)
a_0 = 1290 (slider from 0 to a_1Monat, step 0.1)
k_A: (a_1Monat - a_0) / (S_H - a_0)
λ_A: (-log(1 - k_A))
Tabelle2_2: "\textcolor{" + (LaTeX[ColorBlue]) + " }{a}(\textcolor{ " + (LaTeX[ColorGreen]) + " }{t}) = \textcolor{ " + (LaTeX[ColorRed]) + " }{" + (LaTeX[S_H]) + " }− (\textcolor{ " + (LaTeX[ColorRed]) + "  }{" + (LaTeX[S_H]) + "}− \textcolor{" + (LaTeX[ColorBlue]) + "   }{" + (LaTeX[a_0]) + "})  ·ℯ ^{-" + (LaTeX[λ_A]) + " \space ·\space \textcolor{" + (LaTeX[ColorGreen]) + "   }{\space t}}"
λ'_A: (12 * λ_A)
Tabelle2_3: "\textcolor{" + (LaTeX[ColorBlue]) + " }{b}(\textcolor{ " + (LaTeX[ColorGreen]) + " }{t}) = \textcolor{ " + (LaTeX[ColorRed]) + " }{" + (LaTeX[S_H]) + " }− (\textcolor{ " + (LaTeX[ColorRed]) + "  }{" + (LaTeX[S_H]) + "}− \textcolor{" + (LaTeX[ColorBlue]) + "   }{" + (LaTeX[a_0]) + "})  ·ℯ ^{-" + (LaTeX[λ'_A]) + " \space ·\space \textcolor{" + (LaTeX[ColorGreen]) + "   }{\space t}}"
Tab3Z2: "\textcolor{" + (LaTeX[ColorBlue]) + " }{B}(\textcolor{ " + (LaTeX[ColorGreen]) + " }{t}) = \textcolor{ " + (LaTeX[ColorRed]) + " }{" + (LaTeX[S]) + " }− (\textcolor{ " + (LaTeX[ColorRed]) + "  }{" + (LaTeX[S]) + "}− \textcolor{" + (LaTeX[ColorBlue]) + "   }{" + (LaTeX[B_0]) + "})  ·ℯ ^{-" + (LaTeX[λ]) + " \space · \space \textcolor{" + (LaTeX[ColorGreen]) + "   }{t}}"
Tab3Z1: "\textcolor{" + (LaTeX[ColorBlue]) + " }{B}(\textcolor{ " + (LaTeX[ColorGreen]) + " }{t}) = \textcolor{ " + (LaTeX[ColorRed]) + " }{S }− (\textcolor{ " + (LaTeX[ColorRed]) + "  }{S}− \textcolor{" + (LaTeX[ColorBlue]) + "   }{B}(\textcolor{ " + (LaTeX[ColorGreen]) + "  }{0}))· ℯ ^{-\lambda \space ·\textcolor{" + (LaTeX[ColorGreen]) + "   }{t}}"
Tab3 = TableText({{Tab3Z1, Tab3Z2}}, "v")
Tab2Z1: "\textcolor{" + (LaTeX[ColorBlue]) + " }{B}(\textcolor{ " + (LaTeX[ColorGreen]) + " }{t}) = \textcolor{ " + (LaTeX[ColorRed]) + " }{S }− (\textcolor{ " + (LaTeX[ColorRed]) + "  }{S}− \textcolor{" + (LaTeX[ColorBlue]) + "   }{B}(\textcolor{ " + (LaTeX[ColorGreen]) + "  }{0}))· (1−\textcolor{" + (LaTeX[ColorRed]) + "   }{k})^\textcolor{" + (LaTeX[ColorGreen]) + "   }{ \space t}"
Tab2Z2: "\textcolor{" + (LaTeX[ColorBlue]) + " }{B}(\textcolor{ " + (LaTeX[ColorGreen]) + " }{t}) = \textcolor{ " + (LaTeX[ColorRed]) + " }{" + (LaTeX[S]) + " }− (\textcolor{ " + (LaTeX[ColorRed]) + "  }{" + (LaTeX[S]) + "}− \textcolor{" + (LaTeX[ColorBlue]) + "   }{0" + (LaTeX[B_0]) + "})· (1−\textcolor{" + (LaTeX[ColorRed]) + "   }{" + (LaTeX[k]) + "})^\textcolor{" + (LaTeX[ColorGreen]) + "   }{ \space t}"
Tab2 = TableText({{Tab2Z1, Tab2Z2}}, "v")
B_k(t) = S - ((S - B_0) * (1 - k)^(t))
Tab7b = LaTeX(k, true, true)
Tab7: "\text{  }λ = ln(q) = ln(1−\textcolor{" + (LaTeX[ColorRed]) + "}{k}) =" + (LaTeX[λ]) + ""
Rekursionslänge = 100 (slider from 0 to 100, step 1)
Rekursion(x) = x + (k * (S - x))
a(t) = S_H - ((S_H - a_0) * ℯ^(((-λ_A) * t)))
a_{12}: round((10 * a(12)) + 0) / 10
b(t) = S_H - ((S_H - a_0) * ℯ^(((-λ'_A) * t)))
b_1: round((10 * b(1)) + 0) / 10
e: y = S_H
Tabelle2_9: "b(1)=" + (LaTeX[a_{12}]) + "          "
Tabelle2_8: "a(12)=" + (LaTeX[a_{12}]) + ""
Text1 = LaTeX(λ'_A, false, true)
Tabelle2_6 = LaTeX(λ_A, true, true)
Tabelle2_7 = LaTeX(λ'_A, true, true)
Liste09a: {Tabelle2_9, "t = Jahre"}
Liste07a: {Tabelle2_7, "für t = Jahre"}
Liste06a: {Tabelle2_6, "für t = Monate"}
Liste02a: {Tabelle2_2, "t = Monate"}
Liste03a: {Tabelle2_3, "t = Jahre"}
a_{1}: a(1)
Textfeld1 = Textfield(S_H)
Textfeld2 = Textfield(a_1Monat)
Seite = 3 (slider from 1 to 4, step 1)
Tab1Z1: "  \textcolor{" + (LaTeX[ColorBlue]) + "  }{B(\textcolor{ " + (LaTeX[ColorGrün]) + "}{t}+1)=\textcolor{" + (LaTeX[ColorBlue]) + "  }{B}(\textcolor{" + (LaTeX[ColorGrün]) + "  }{t}) +\textcolor{" + (LaTeX[ColorRed]) + "  }{ k}}·(\textcolor{  " + (LaTeX[ColorRed]) + "}{S} - \textcolor{" + (LaTeX[ColorBlue]) + "  }{B}(\textcolor{ " + (LaTeX[ColorGrün]) + " }{t})) "
Liste01a: {Tab1Z1, "Rekursion"}
Tabelle2_4: "k=" + k_A + ""
Liste04a: {Tabelle2_4, "Monate"}
k'_A: 1 - (1 - k_A)^(12)
λ'Test: (-log(1 - k'_A))
a_1Jahr: a_0 + (k'_A * (S_H - a_0))
Tabelle2_5: "k'=" + k'_A + ""
Liste08a: {Tabelle2_8, "t = Monate"}
Liste05a: {Tabelle2_5, "Jahre"}
Text11: "\text{ }Bestimmungsgleichung:\\
\text{  }a(1)  = a(0) + k (S−a(0)) \\
 \text{       }        =" + (LaTeX[a_0]) + " + k·( S−" + (LaTeX[a_0]) + ")\\
\text{ }" + (LaTeX[a_1Monat]) + "  = " + (LaTeX[a_0]) + "+k·(" + (LaTeX[S_H]) + "−" + (LaTeX[a_0]) + ") \\

\text{ }k = \frac{" + (LaTeX[a_1Monat]) + "−" + (LaTeX[a_0]) + "}{" + (LaTeX[S_H]) + "−" + (LaTeX[a_0]) + "} = " + (LaTeX[k_A]) + "\\
\text{ (alles in Monatsangaben)}\\ "
Text12: "\scalebox{1.2}{ \mathbf{ \text{Anhang: Herleitung der expliziten Formel} }  }\\  \space \\

\textcolor{" + (LaTeX[ColorBlue]) + " }{B(\textcolor{" + (LaTeX[ColorGreen]) + "}{t}+1)=\textcolor{" + (LaTeX[ColorBlue]) + " }{B}(\textcolor{" + (LaTeX[ColorGreen]) + " }{t}) +\textcolor{" + (LaTeX[ColorRed]) + " }{ k}}·(\textcolor{ " + (LaTeX[ColorViolett]) + "}{S} - \textcolor{" + (LaTeX[ColorBlue]) + " }{B}(\textcolor{ " + (LaTeX[ColorGreen]) + "}{t})) \\
⇒   B(1) = \textcolor{ " + (LaTeX[ColorRed]) + "}{B(0) + k ·(S - B(0))} \\
  \text{          }             =S - S + k·S + B(0) - k·B(0) \\
    \text{          }                        = S - [S - k·S - B(0) + k·B(0)] \\
   \text{          }                         = S - (S - B(0))·(1 - k) \\
 ⇒    \textcolor{" + (LaTeX[ColorGreen]) + "  }{B(1) = S - (S - B(0))·(1 - k)^1} \\  \space \\
   \text{    }                  B(2) = B(1) + k ·(S - B(1))  \\
 ⇒   B(2 ) = \textcolor{" + (LaTeX[ColorRed]) + "}{B(0) + k ·(S - B(0))} + k (S - (\textcolor{" + (LaTeX[ColorRed]) + "}{ B(0) + k ·(S - B(0))})) \\
      \text{          }              = B(0) + k·(S - B(0)) + k·S - k·(B(0)+k·(S - B(0))) \\
      \text{          }               = B(0) + k·(S -B(0)) + k·S - k·B(0) - k^2·(S - B(0))) \\
     \text{          }                = (B(0) - \textcolor{" + (LaTeX[ColorViolett]) + " }{S})+k·(S - B(0))+k·(S - B(0)) - k^2·(S - B(0)))+\textcolor{" + (LaTeX[ColorViolett]) + " }{S} \\
    \text{          }                  =\textcolor{ " + (LaTeX[ColorViolett]) + " }{S} - (\textcolor{" + (LaTeX[ColorViolett]) + "}{S} - B(0))+k·(S - B(0))+k·(S - B(0)) - k^2·(S - B(0))) \\
      \text{          }                = S - (S - B(0))+ 2 k·(S - B(0)) - k^2·(S - B(0))) \\
      \text{          }                = S - (S - B(0))·(1 - 2·k+k^2) \\ 
 ⇒   \textcolor{" + (LaTeX[ColorGreen]) + "   }{B(2)  = S - (S - B(0))·(1 - k)^2}  \\ 
     \text{           }            . . . . . \\ \space  \\
     \text{           }                . . . . . \\ \space  \\
    \text{           }              . . . . . \\ \space  \\
  ⇒   \textcolor{ " + (LaTeX[ColorGreen]) + "   }{B(t)  = S - (S - B(0))·(1 - k)^t}  \\ "
B(t) = S - ((S - B_0) * ℯ^(((-λ) * t)))
Schranke: y = S
R(t) = S - B(t)
f(t) = If[0 ≤ t ≤ 100000000,R(t)]
C: (t, B(t))
d = Segment((t, 0), C)
g = Segment((0, B(t)), C)
Restfunktion = true
h: round(x(Corner[2]))
Tab5: "\text{  }B(" + t + ")=" + (B(t)) + ""
Tab6: "\text{  }R(" + t + ")=" + (R(t)) + ""
Rekursionsliste = IterationList(Rekursion, B_0, Rekursionslänge)
Liste06: {Tab6, Tab5b}
Liste02: {Tab2, Tab2b}
Liste07: {Tab7, Tab7b}
Rekursionspunkte = Sequence((i, Element[Rekursionsliste, i + 1]), i, 0, Rekursionslänge)
Tab1Z2: "  \textcolor{" + (LaTeX[ColorBlue]) + "  }{B(\textcolor{ " + (LaTeX[ColorGrün]) + "}{t}+1)=\textcolor{" + (LaTeX[ColorBlue]) + "  }{B}(\textcolor{" + (LaTeX[ColorGrün]) + "  }{t}) +\textcolor{" + (LaTeX[ColorRed]) + "  }{ " + (LaTeX[k]) + "}}·(\textcolor{  " + (LaTeX[ColorRed]) + "}{" + (LaTeX[S]) + "} - \textcolor{" + (LaTeX[ColorBlue]) + "  }{B}(\textcolor{ " + (LaTeX[ColorGrün]) + " }{t})) "
Tab1 = TableText({{Tab1Z1, Tab1Z2}}, "v")
Liste03: {Tab3, "t = Zeiteinheit"}
Liste05: {Tab4, Tab4b}
Liste01: {Tab1, "Rekursion"}
Text10a: {Text1, "Transformation"}
Text8 = TableText({Liste01a, Liste02a, Liste03a, Liste04a, Liste05a, Liste06a, Liste07a, Liste08a, Liste09a, Text10a}, "|_")
Liste04: {Tab5, Tab5b}
Text9 = TableText({Liste01, Liste02, Liste03, Liste04, Liste05, Liste06, Liste07}, "|_")
Eingabe = If(a_{12} ≟ 18970, true, false)
D: (0, a_0)
Text16: " \text{    } \textcolor{" + (LaTeX[ColorBlue]) + " }{B}(\textcolor{" + (LaTeX[ColorGreen]) + "}{t}+1)=\textcolor{" + (LaTeX[ColorBlue]) + " }{B}(\textcolor{" + (LaTeX[ColorGreen]) + " }{t}) +\textcolor{" + (LaTeX[ColorRed]) + "}{ k}·(\textcolor{ " + (LaTeX[ColorViolett]) + "}{S}−\textcolor{" + (LaTeX[ColorBlue]) + "}{B}(\textcolor{ " + (LaTeX[ColorGreen]) + "}{t})) \\
  ⇒ \textcolor{" + (LaTeX[ColorBlue]) + "}{B}(\textcolor{" + (LaTeX[ColorGrün]) + " }{t}+1)−\textcolor{ " + (LaTeX[ColorBlue]) + " }{B}(\textcolor{" + (LaTeX[ColorGrün]) + "  }{t}) =\textcolor{ " + (LaTeX[ColorRed]) + " }{ k}·(\textcolor{" + (LaTeX[ColorViolett]) + "  }{S} −\textcolor{ " + (LaTeX[ColorBlue]) + " }{B}(\textcolor{" + (LaTeX[ColorGreen]) + "  }{t})) \\ 
 ⇒\text{Der Zuwachs ist mit dem Faktor }\textcolor{" + (LaTeX[ColorRed]) + "  }{k} \mathbf{ \space proportional}\text{  zum }\mathbf{Sättigungsmanko  \space } (\textcolor{ " + (LaTeX[ColorRed]) + "  }{S}−\textcolor{" + (LaTeX[ColorBlue]) + "   }{B}(\textcolor{" + (LaTeX[ColorGreen]) + "   }{t}))\\
     \text{    Durch rekursive Substitution läßt sich für t= 1, 2, 3 , ... zeigen: }\\ 
 \text{    }   \textcolor{" + (LaTeX[ColorBlue]) + " }{B}(\textcolor{" + (LaTeX[ColorGrün]) + " }{t}) =\textcolor{" + (LaTeX[ColorRed]) + " }{S} − (\textcolor{ " + (LaTeX[ColorRed]) + "   }{S}−\textcolor{ " + (LaTeX[ColorBlue]) + "   }{B}(\textcolor{  " + (LaTeX[ColorGreen]) + "  }{0}))·(1−\textcolor{ " + (LaTeX[ColorRed]) + "   }{k})^\textcolor{ " + (LaTeX[ColorGrün]) + "   }{t} \\  
 \text{    }    \textcolor{" + (LaTeX[ColorBlue]) + " }{B}(\textcolor{" + (LaTeX[ColorGrün]) + "  }{t}) =\textcolor{" + (LaTeX[ColorRed]) + " }{S}−(\textcolor{" + (LaTeX[ColorRed]) + "}{S}− \textcolor{" + (LaTeX[ColorBlue]) + "}{B}(\textcolor{ " + (LaTeX[ColorGreen]) + "}{0}))· q^{\textcolor{" + (LaTeX[ColorGreen]) + " }{  \space t} }  \space          mit \space   q= (1−\textcolor{" + (LaTeX[ColorRed]) + "     }{k})                \\  
 \text{    }    \textcolor{ " + (LaTeX[ColorBlue]) + " }{B}(\textcolor{" + (LaTeX[ColorGrün]) + "   }{t}) =\textcolor{" + (LaTeX[ColorRed]) + "}{S}−(\textcolor{" + (LaTeX[ColorRed]) + "}{S}−\textcolor{" + (LaTeX[ColorBlue]) + "}{B}(\textcolor{" + (LaTeX[ColorGreen]) + "}{0}))· e^{−λ \space  · \textcolor{" + (LaTeX[ColorGrün]) + "}{ \space t} }  \space       mit \space    λ =−ln((1−\textcolor{   " + (LaTeX[ColorRed]) + "   }{k})  \space   als \space  Wachstumskonstante             \\  \space  \\ 
  \text{    Die Restfunktion R(t)=S−B(t) folgt übrigens den Gesetzen der exponentiellen Abnahme (Zerfall):} \\
  \text{    }    R(\textcolor{ " + (LaTeX[ColorGreen]) + " }{t}) =  \textcolor{ " + (LaTeX[ColorRed]) + "}{S} − \textcolor{" + (LaTeX[ColorBlue]) + "   }{B}(\textcolor{  " + (LaTeX[ColorGreen]) + "  }{t}) = (\textcolor{ " + (LaTeX[ColorRed]) + "  }{S}−\textcolor{  " + (LaTeX[ColorBlue]) + " }{B}(\textcolor{" + (LaTeX[ColorGreen]) + "   }{0}))· e^{−λ \space  ·  \space \textcolor{ " + (LaTeX[ColorGreen]) + "   }{t} }     \text{          mit    }   λ = −ln(1−\textcolor{" + (LaTeX[ColorRed]) + "       }{k})       \\
 \text{    }    R(\textcolor{" + (LaTeX[ColorGreen]) + "   }{t}) =  \textcolor{  " + (LaTeX[ColorRed]) + "}{S} − \textcolor{ " + (LaTeX[ColorBlue]) + "   }{B}(\textcolor{  " + (LaTeX[ColorGreen]) + "   }{t}) = (\textcolor{" + (LaTeX[ColorRed]) + "    }{S}−\textcolor{" + (LaTeX[ColorBlue]) + "    }{B}(\textcolor{" + (LaTeX[ColorGreen]) + "    }{0}))·q^{ \textcolor{" + (LaTeX[ColorGreen]) + "     }{t} }     \text{               mit    }     q= (1−\textcolor{ " + (LaTeX[ColorRed]) + "       }{k})       \\
 \text{    }    R(\textcolor{ " + (LaTeX[ColorGreen]) + "  }{t}) =  \textcolor{" + (LaTeX[ColorRed]) + "  }{S} − \textcolor{" + (LaTeX[ColorBlue]) + "    }{B}(\textcolor{    " + (LaTeX[ColorGreen]) + " }{t}) = (\textcolor{ " + (LaTeX[ColorRed]) + "   }{S}−\textcolor{" + (LaTeX[ColorBlue]) + "    }{B}(\textcolor{  " + (LaTeX[ColorGreen]) + "  }{0}))· (1+\textcolor{" + (LaTeX[ColorRed]) + "}{k})^{   \textcolor{ " + (LaTeX[ColorGreen]) + "    }{t} }      \text{        mit    }R(\textcolor{" + (LaTeX[ColorGreen]) + "}0) = \textcolor{" + (LaTeX[ColorRed]) + "}{S} − \textcolor{" + (LaTeX[ColorBlue]) + "}{B}(\textcolor{" + (LaTeX[ColorGreen]) + "}0)           \\ 
  ⇒ \frac{R(t+1)}{R(t)} = \frac{q^{t+1}}{q^t} = \frac{e^{−λ (t+1)}}{e^{−λ t}} \\
              ⇒ \frac{R(t+1)}{R(t)} = q = e^{−λ }  =1−\textcolor{" + (LaTeX[ColorRed]) + " } k    = const. \text{                    } Zerfalls\textcolor{" + (LaTeX[ColorRed]) + "  }{faktor}\text{   } \scalebox{0.8}{(   \space bezogen \space  auf  \space \Delta \space  t =1 \space  Zeiteinheit)) }  \\
               ⇒  \frac{R(1)}{R(0)} = \frac{R(2)}{R(1)} = \frac{R(3)}{R(2)} =  . . .\frac{R(t+1)}{R(t)} = q = e^{-λ }  =1−\textcolor{" + (LaTeX[ColorRed]) + "   } k \\
              ⇒  k =  \frac{R(t)−R(t+1)}{R(t)} = 1−q = 1−e^{−λ }  = const. \text{       } Zerfalls\textcolor{ " + (LaTeX[ColorRed]) + "  }{rate} \text{   } \scalebox{0.8}{(   \space bezogen \space  auf  \space \Delta \space  t =1  \space Zeiteinheit) }  \\
              ⇒  k · 100  = Zerfalls\textcolor{" + (LaTeX[ColorRed]) + " }{rate} \  in \ %  \space  \\  \space  \\ 
\text{ } \textcolor{ " + (LaTeX[ColorBlue]) + "  }{\text{ Hinweis: Durch Vorgabe von Werten zu bestimmten Vielfachen von ausgewählten Zeiteinheiten B(1)= , B(2)= , B(3)= , . . .}\\
 \text{              werden gleichzeitig die Wachstumskonstante λ und die damit zusammenhängenden Konstanten q und k  an die } \\
  \text{              Verwendung dieser Zeiteinheit in den obigen Formeln gebunden.} \\ \space  \\ 
\text{              Beispiel:}\\
 \text{              Wurde die Konstante λ für Monate berechnet, dann gelten für die Verwendung der Zeitangaben in Jahren : }\\
 \text{              λ' = 12·λ  ;       q'=e^{−λ'} =e^{−12\space·\space λ}=(e^{−λ})^{12}=q^{12}  \space   und  \space k' =1−(1−k)^{12}  }    \space  \\ "