restart;kernzerfall := proc( n ) local i, res, roll, hallo; res := n; roll := rand(1..6); for i from 1 to n do if (roll() = 6) then res := res-1; end if; end do; printf("\t\t\t\t\tZerfallen: %a\n", n-res); printf("\t\t\t\t\tUebrige: %a", res); end proc:kernzerfall(100);

Zerfallen: 17 Uebrige: 83

restart;kernzerfall := proc( n, roll ) local i, res; res := n; for i from 1 to n do if (roll() = 6) then res := res-1; end if; end do; return res; end proc:zerfallsreihe_schritte := proc( time, n ) local i, liste, ret, roll; ret := n; liste := []; roll := rand(1..6); for i from 0 to time do ret := kernzerfall(ret,roll); liste := [seq(op(j,liste),j=1..nops(liste)), ret]; end do; return liste; end proc:zerfallsreihe := proc( n ) local i, liste, ret, roll; ret := n; liste := [n]; roll := rand(1..6); while op(nops(liste),liste) > 0 do ret := kernzerfall(ret,roll); liste := [seq(op(j,liste),j=1..nops(liste)), ret]; end do; return liste; end proc:zerfallsreihe_schritte(10,1000);

NiM3LSIkUikiJDooIiQlZiIkJ1wiJD8lIiRjJCIkJEgiJFIjIiQjPiIkYSIiJEUi

zerfallsreihe(1000);

NiM3TCIlKzUiJEwpIiQncCIkcyYiJG0lIiQhUiIkMyQiJGAjIiQ3IyIkJj0iJF8iIiRBIiIjKioiIyMpIiNxIiNoIiNeIiNWIiNRIiNLIiNJIiNIIiNCIiNAIiM9IiM5IiM4IiM1IiIpIiIoIiInIiIkRkMiIiMiIiJGRUZFRkVGRUZFRkUiIiE=

restart;kernzerfall := proc( n, roll ) local i, res; res := n; for i from 1 to n do if (roll() = 6) then res := res-1; end if; end do; return res; end proc:halbwaertszeit := proc( n ) local ret, time, roll; ret := n; time := 0; roll := rand(1..6); while (ret > n/2) do ret := kernzerfall(ret,roll); time := time+1; end do; return time; end proc:halbwaertszeit(1000);

NiMiIiU=

efunc := proc( n , t , halb ) local res; res := n * exp((-t*ln(2))/halb); return evalf(res); end proc:efunc(1000,5,4);

NiMkIit5P1svVSEiKA==

Wenn man den Bereich der Zufallszahlen vergroessert!!! => mehr Zahlen die keinen Zerfall bewirken

restart;f3:=unapply(mtaylor(exp(x), x=0, 4),x);

NiM+SSNmM0c2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwqIiIiRi05JEYtKiRGLiIiIyNGLUYwKiRGLiIiJCNGLSIiJ0YlRiVGJQ==

f6:=unapply(mtaylor(exp(x), x=0, 7),x);

NiM+SSNmNkc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwwIiIiRi05JEYtKiRGLiIiIyNGLUYwKiRGLiIiJCNGLSIiJyokRi4iIiUjRi0iI0MqJEYuIiImI0YtIiQ/IiokRi5GNSNGLSIkPyhGJUYlRiU=

f9:=unapply(mtaylor(exp(x), x=0, 10),x);

NiM+SSNmOUc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSw2IiIiRi05JEYtKiRGLiIiIyNGLUYwKiRGLiIiJCNGLSIiJyokRi4iIiUjRi0iI0MqJEYuIiImI0YtIiQ/IiokRi5GNSNGLSIkPygqJEYuIiIoI0YtIiVTXSokRi4iIikjRi0iJj8uJSokRi4iIiojRi0iJyEpR09GJUYlRiU=

evalf(f3(2)-exp(2));

NiMkISttRnNiNSEiKg==

evalf(f6(2)-exp(2));

NiMkISlWMF1MISIq

evalf(f9(2)-exp(2));

NiMkISd4Tk0hIio=

evalf(f3(5)-exp(5));

NiMkIStlIyl6ITQiISIo

evalf(f6(5)-exp(5));

NiMkISpONSZITiEiKA==

evalf(f9(5)-exp(5));

NiMkISlEcUJaISIo

restart;f3:=unapply(mtaylor(1/x, x=1, 4),x);

NiM+SSNmM0c2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwqIiIjIiIiOSQhIiIqJCwmRi9GLkYwRi5GLUYuKiRGMiIiJEYwRiVGJUYl

f6:=unapply(mtaylor(1/x, x=1, 7),x);

NiM+SSNmNkc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwwIiIjIiIiOSQhIiIqJCwmRi9GLkYwRi5GLUYuKiRGMiIiJEYwKiRGMiIiJUYuKiRGMiIiJkYwKiRGMiIiJ0YuRiVGJUYl

f9:=unapply(mtaylor(1/x, x=1, 10),x);

NiM+SSNmOUc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSw2IiIjIiIiOSQhIiIqJCwmRi9GLkYwRi5GLUYuKiRGMiIiJEYwKiRGMiIiJUYuKiRGMiIiJkYwKiRGMiIiJ0YuKiRGMiIiKEYwKiRGMiIiKUYuKiRGMiIiKkYwRiVGJUYl

evalf(f3(2)-1/3);

NiMkIStMTExMTCEjNQ==

evalf(f6(2)-1/3);

NiMkIitubW1tbSEjNQ==

evalf(f9(2)-1/3);

NiMkIStMTExMTCEjNQ==

evalf(f3(5)-1/7);

NiMkISs5ZEc5XiEiKQ==

evalf(f6(5)-1/7);

NiMkIitWciZvRiQhIic=

evalf(f9(5)-1/7);

NiMkIStIOTooNCMhIiU=

restart;lagrange := proc( liste1 , liste2) local res, zaehler, nenner, laenge, i, n; Digits:=20; res := 0; if nops(liste1) = nops(liste2) then laenge := nops(liste1); for n from 1 to laenge do zaehler := 1; nenner := 1; for i from 1 to laenge do if i <> n then zaehler := zaehler * (x - op(i,liste1)); nenner := nenner * (op(n,liste1)-op(i,liste1)); end if; end do; res := res + ((zaehler/nenner)*op(n,liste2)); end do; else printf("Die Listen sind nicht gleich lang"); end if; res := sort(expand(res)); return res; end proc:

liste1 := [-Pi/4,0,Pi/4];

NiM+SSdsaXN0ZTFHNiI3JSwkSSNQaUdJKnByb3RlY3RlZEdGKSMhIiIiIiUiIiEsJEYoIyIiIkYs

liste2 := map(cos,liste1);

NiM+SSdsaXN0ZTJHNiI3JSwkKiQiIiMjIiIiRilGKkYrRic=

lagrange(liste1,liste2);

NiMsKCooSSNQaUdJKnByb3RlY3RlZEdGJiEiIyIiIyMiIiJGKEkieEc2IkYoIiIpKiZGJUYnRitGKCEjO0YqRio=

f := unapply(evalf(%),x);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDkkIiIjJCEqLTEjW1ohIiokIiIiIiIhRjRGJUYlRiU=

plot(f(x),x=-Pi/4..Pi/4);

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

liste1 := [8.0,8.2,8.5,8.6,8.8,9.0];

NiM+SSdsaXN0ZTFHNiI3KCQiIyEpISIiJCIjIylGKSQiIyYpRikkIiMnKUYpJCIjKSlGKSQiIyEqRik=

liste2 := map(x->x*ln(x),liste1);

NiM+SSdsaXN0ZTJHNiI3KCQiK01LYmo7ISIpJCIrMStSRDxGKSQiK1JpMD49RikkIismXDowJj1GKSQiKzk6eTg+RikkIis+QF14PkYp

#liste2 := [op(1,liste1)*ln(op(1,liste1)),op(2,liste1)*ln(op(2,liste1)),op(3,liste1)*ln(op(3,liste1)),op(4,liste1)*ln(op(4,liste1)),op(5,liste1)*ln(op(5,liste1)),op(6,liste1)*ln(op(6,liste1))];lagrange(liste1,liste2);

NiMsLiokSSJ4RzYiIiImJCEsKysrRCJ5ISM7KiRGJSIiJSQiLUBMZSpbbyUhIzoqJEYlIiIkJCEtKipcaTpmNyEjOCokRiUiIiMkIi08L14nekMjISM3RiUkIi40XXM1MzUiRjkkIS4rK1kkKVJ4IkY5IiIi

f := unapply(%,x);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLC4qJDkkIiImJCEsKysrRCJ5ISM7KiRGLiIiJSQiLUBMZSpbbyUhIzoqJEYuIiIkJCEtKipcaTpmNyEjOCokRi4iIiMkIi08L14nekMjISM3Ri4kIi40XXM1MzUiRkEkIS4rK1kkKVJ4IkZBIiIiRiVGJUYl

restart;Digits:=20;

NiM+SSdEaWdpdHNHNiIiIz8=

neuville := proc( liste1 , liste2 , stelle , grad ) local laenge, res, zaehler1, zaehler2, nenner; Digits:=20; laenge := nops(liste1); if grad = 0 then res := op(stelle+1,liste2); else zaehler1 := (x-op(stelle-grad+1,liste1))*neuville(liste1,liste2,stelle,grad-1); zaehler2 := (x-op(stelle+1,liste1))*neuville(liste1,liste2,stelle-1,grad-1); nenner := (op(stelle+1,liste1)-op(stelle-grad+1,liste1)); res := (zaehler1 - zaehler2) / nenner; end if; res := sort(expand(res)); return res; end proc:

liste1 := [8.0,8.2,8.5,8.6,8.8,9.0];

NiM+SSdsaXN0ZTFHNiI3KCQiIyEpISIiJCIjIylGKSQiIyYpRikkIiMnKUYpJCIjKSlGKSQiIyEqRik=

liste2 := map(x->x*ln(x),liste1);

NiM+SSdsaXN0ZTJHNiI3KCQiNUV1b1FNTEtiajshIz0kIjU/NHE6XTErUkQ8RikkIjVfOkk9KCpRaTA+PUYpJCI1QE9QSiFbXDowJj1GKSQiNVBcaGchXF4ieTg+RikkIjVYVyhmLSc+QF14PkYp

neuville(liste1,liste2,5,5);

NiMsLiokSSJ4RzYiIiImJCEzU08keXIkUkREJiohI0IqJEYlIiIlJCI0UT01KlJJbHQyYSEjQSokRiUiIiQkITQ7eS8weXM4NlEiISM/KiRGJSIiIyQiNSEpZUU3MC01el1CRjRGJSQiNDMzdztDNChbZDUhIz0kITR4blQqXCUqKTQ1cSJGOyIiIg==

g := unapply(%,x);

NiM+SSJnRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLC4qJDkkIiImJCEzU08keXIkUkREJiohI0IqJEYuIiIlJCI0UT01KlJJbHQyYSEjQSokRi4iIiQkITQ7eS8weXM4NlEiISM/KiRGLiIiIyQiNSEpZUU3MC01el1CRjxGLiQiNDMzdztDNChbZDUhIz0kITR4blQqXCUqKTQ1cSJGQyIiIkYlRiVGJQ==

g(8.4);

NiMkIjU9L2lXKUdqOXh5IiEjPQ==

gen := 8.4*ln(8.4):printf("\t\t\tgenauer Wert : %a", gen);

genauer Wert : 17.877146329133850623

printf("\t\t\tinterpolierter Wert : %a", g(8.4));

interpolierter Wert : 17.877146328844620418

printf("\t\t\tDifferenz (genau - interpoliert) : %a", gen - g(8.4));

Differenz (genau - interpoliert) : .289230205e-9

liste1 := [8.0,8.1,8.2,8.3,8.5,8.6,8.8,9.0];

NiM+SSdsaXN0ZTFHNiI3KiQiIyEpISIiJCIjIilGKSQiIyMpRikkIiMkKUYpJCIjJilGKSQiIycpRikkIiMpKUYpJCIjISpGKQ==

liste2 := map(x->x*ln(x),liste1);

NiM+SSdsaXN0ZTJHNiI3KiQiNUV1b1FNTEtiajshIz0kIjVoUilcZioqKSk0V3AiRikkIjU/NHE6XTErUkQ8RikkIjUpZSQ9aEd4P1xjPEYpJCI1XzpJPSgqUWkwPj1GKSQiNUBPUEohW1w6MCY9RikkIjVQXGhnIVxeInk4PkYpJCI1WFcoZi0nPkBdeD5GKQ==

neuville(liste1,liste2,7,7);

NiMsMiokSSJ4RzYiIiIoJCE0K3VgJil5bHE+aSchI0UqJEYlIiInJCI0djhKbSdSKik0JHAlISNDKiRGJSIiJiQhND9GZE8hZWRgJVsiISNBKiRGJSIiJSQiNDFbVT9KYHdAeSMhI0AqJEYlIiIkJCE0YW5HTFY3Zyg9TiEjPyokRiUiIiMkIjQ7OyMpcFQlZV9mTiEjPkYlJCI0XTF2XU88VVkib0ZDJCE0QHcpemw8I2VQPyIhIz0iIiI=

g := unapply(%,x);

NiM+SSJnRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLDIqJDkkIiIoJCE0K3VgJil5bHE+aSchI0UqJEYuIiInJCI0djhKbSdSKik0JHAlISNDKiRGLiIiJiQhND9GZE8hZWRgJVsiISNBKiRGLiIiJSQiNDFbVT9KYHdAeSMhI0AqJEYuIiIkJCE0YW5HTFY3Zyg9TiEjPyokRi4iIiMkIjQ7OyMpcFQlZV9mTiEjPkYuJCI0XTF2XU88VVkib0ZLJCE0QHcpemw8I2VQPyIhIz0iIiJGJUYlRiU=

g(8.4);

NiMkIjUiZSN5TCJIajl4eSIhIz0=

printf("\t\t\tDifferenz (genau - interpoliert) : %a", gen - g(8.4));

Differenz (genau - interpoliert) : .68042e-13