\334bung 4 -- Differenzieren, Integrieren, Taylor-Reihe Aufgabe 1 Gegeben sei die Funktion NiMvLUkiZkc2IjYjSSJ4R0YmKiZGKCIiJC1JJGV4cEdGJjYjLCQqJEYoIiIjISIiIiIi . (a) Berechnen Sie alle Extrema und Wendepunkte der Funktion. Stellen Sie bei jedem Extremum fest, ob es sich um ein Minimum oder ein Maximum handelt. restart; f := x^3 * exp(-x^2); NiM+SSJmRzYiKiZJInhHRiUiIiQtSSRleHBHNiRJKnByb3RlY3RlZEdGLEkoX3N5c2xpYkdGJTYjLCQqJEYnIiIjISIiIiIi plot(f, x=-5..5); 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 Diff(f, x);
d1f := simplify(value(%)); NiMtSSVEaWZmRzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiI2JComSSJ4R0YoIiIkLUkkZXhwR0YlNiMsJCokRisiIiMhIiIiIiJGKw== NiM+SSRkMWZHNiIsJCooSSJ4R0YlIiIjLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRi1JKF9zeXNsaWJHRiU2IywkKiRGKEYpISIiIiIiLCYhIiRGM0YxRilGM0Yy Diff(f, x$2);
d2f := simplify(value(%)); NiMtSSVEaWZmRzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiI2JComSSJ4R0YoIiIkLUkkZXhwR0YlNiMsJCokRisiIiMhIiIiIiItSSIkR0YmNiRGK0Yy NiM+SSRkMmZHNiIsJCooSSJ4R0YlIiIiLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRi1JKF9zeXNsaWJHRiU2IywkKiRGKCIiIyEiIkYpLCgiIiRGKUYxISIoKiRGKCIiJUYyRilGMg== Diff(f, x$3);
d3f := simplify(value(%)); NiMtSSVEaWZmRzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiI2JComSSJ4R0YoIiIkLUkkZXhwR0YlNiMsJCokRisiIiMhIiIiIiItSSIkR0YmNiRGK0Ys NiM+SSRkM2ZHNiIsJComLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiU2IywkKiRJInhHRiUiIiMhIiIiIiIsKiEiJEYzRi8iI0YqJEYwIiIlISNDKiRGMCIiJ0Y4RjMhIiM= ext := solve(d1f=0, x); NiM+SSRleHRHNiI2JiIiIUYnLCQqJCIiJyMiIiIiIiNGKywkRikjISIiRi0= ext_unique := { ext }; NiM+SStleHRfdW5pcXVlRzYiPCUiIiEsJCokIiInIyIiIiIiI0YrLCRGKSMhIiJGLQ== subs(x=ext_unique[1], d2f);
signum(%); NiMiIiE= NiMiIiE= subs(x=ext_unique[2], d2f);
signum(%); NiMsJComIiInIyIiIiIiIy1JJGV4cEc2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliRzYiNiMjISIkRihGJ0Yx NiMhIiI= subs(x=ext_unique[3], d2f);
signum(%); NiMsJComIiInIyIiIiIiIy1JJGV4cEc2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliRzYiNiMjISIkRihGJyIiJA== NiMiIiI= minmax := [ ext_unique[2], ext_unique[3] ]; NiM+SSdtaW5tYXhHNiI3JCwkKiQiIicjIiIiIiIjRiosJEYoIyEiIkYs solve(d2f, x);
inflex := { % }; NiciIiEsJCokIiIjIyIiIkYmRicsJEYlIyEiIkYmKiQiIiRGJywkRixGKw== NiM+SSdpbmZsZXhHNiI8JyIiISwkKiQiIiMjIiIiRipGKyokIiIkRissJEYpIyEiIkYqLCRGLUYx seq(signum(subs(x=inflex[i], d3f)), i=1..nops(inflex)); Alle Punkte in inflex sind Wendepunkte. NiciIiIhIiJGI0YkRiM= (b) Zeichnen Sie die Funktion, und markieren Sie die Extrema und die Wendepunkte mit Punkten jeweils unterschiedlicher Farbe. with(plots): Warning, the name changecoords has been redefined pl1 := plot(f, x=-5..5, y=-0.5..0.5):
display(%); 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 points_ext := seq([ minmax[i], subs(x=minmax[i], f)], i=1..nops(minmax)); NiM+SStwb2ludHNfZXh0RzYiNiQ3JCwkKiQiIicjIiIiIiIjRissJComRipGKy1JJGV4cEc2JEkqcHJvdGVjdGVkR0YzSShfc3lzbGliR0YlNiMjISIkRi1GLCMiIiQiIiU3JCwkRikjISIiRi0sJEYvI0Y3Rjo= pl2 := plot([ points_ext ], x=-5..5, y=-0.5..0.5, style=point, symbol=CIRCLE, symbolsize=10, color=green):
display(%); LSUlUExPVEc2JS0lJ0NVUlZFU0c2JjckNyQkIjMlKillIlJyW3VDNyEjPCQiMy0rJz0lKnlpIio0JSEjPTckJCEzJSopZSJSclt1QzdGLCQhMy0rJz0lKnlpIio0JUYvLSUnU1lNQk9MRzYkJSdDSVJDTEVHIiM1LSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJEY5RkBGPi0lJlNUWUxFRzYjJSZQT0lOVEctJStBWEVTTEFCRUxTRzYkUSJ4NiJRInlGSi0lJVZJRVdHNiQ7JCEjXUZAJCIjXUZAOyQhIiZGQCRGUyEiIw== points_inflex := seq([ inflex[i], subs(x=inflex[i], f) ], i=1..nops(inflex)); NiM+SS5wb2ludHNfaW5mbGV4RzYiNic3JCIiIUYoNyQsJCokIiIjIyIiIkYsRi0sJComRixGLS1JJGV4cEc2JEkqcHJvdGVjdGVkR0Y0SShfc3lzbGliR0YlNiMjISIiRixGLiNGLiIiJTckKiQiIiRGLSwkKiZGPUYtLUYyNiMhIiRGLkY9NyQsJEYrRjcsJEYwI0Y4Rjo3JCwkRjxGOCwkRj9GQg== pl3 := plot([ points_inflex ], x=-5..5, y=-0.5..0.5, style=point, symbol=CIRCLE, symbolsize=10, color=blue):
display(%); LSUlUExPVEc2JS0lJ0NVUlZFU0c2JjcnNyQkIiIhRitGKjckJCIzdHZhJz0ieTFycSEjPSQiMz9uPFM3KDRXOSNGLzckJCIzP3gpb3YhMzBLPCEjPCQiMyUpcDg+Zj4sKGUjRi83JCQhM3R2YSc9InkxcnFGLyQhMz9uPFM3KDRXOSNGLzckJCEzP3gpb3YhMzBLPEY1JCEzJSlwOD5mPiwoZSNGLy0lJ1NZTUJPTEc2JCUnQ0lSQ0xFRyIjNS0lJkNPTE9SRzYmJSRSR0JHJEYrISIiRkskRkZGTC0lJlNUWUxFRzYjJSZQT0lOVEctJStBWEVTTEFCRUxTRzYkUSJ4NiJRInlGVi0lJVZJRVdHNiQ7JCEjXUZMJCIjXUZMOyQhIiZGTCRGaW4hIiM= display({pl1, pl2, pl3}); 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 Aufgabe 2 Das Tr\344gheitsmoment eines K\366rpers wird mit der Formel NiMtSSRpbnRHNiI2JComSSJyR0YlIiIjSSRyaG9HRiUiIiJJIlZHRiU= berechnet, wobei NiMtSSJyRzYiNiVJInhHRiVJInlHRiVJInpHRiU= der Abstand eines Punktes von der Drehachse ist, und NiMtSSRyaG9HNiI2JUkieEdGJUkieUdGJUkiekdGJQ== die Dichte des K\366rpers. Integriert wird \374ber das Volumen des K\366rpers. (a) Berechnen Sie das Tr\344gheitsmoment eines Quaders mit den Kantenl\344ngen NiMvSSJhRzYiIiIk , NiMvSSJiRzYiIiM1 , c=8 bez\374glich der drei Hauptachsen (Drehachsen entlang der NiNJInhHNiI= -, NiNJInlHNiI= - und NiNJInpHNiI= -Achse. Die Masse des Quaders soll NiNJIk1HNiI= =5 betragen, mit der gleichm\344\337igen Massenverteilung NiMvSSRyaG9HNiIqJkkiTUdGJSIiIiooSSJhR0YlRihJImJHRiVGKEkiY0dGJUYoISIi .restart; Int(Int(Int(M/(a*b*c)*(y^2+z^2), x=-a/2..a/2), y=-b/2..b/2), z=-c/2..c/2);
mom_x := simplify(value(%)); NiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUYkNiQtRiQ2JCosSSJNR0YoIiIiSSJhR0YoISIiSSJiR0YoRjJJImNHRihGMiwmKiRJInlHRigiIiNGMCokSSJ6R0YoRjhGMEYwL0kieEdGKDssJEYxI0YyRjgsJEYxI0YwRjgvRjc7LCRGM0Y/LCRGM0ZBL0Y6OywkRjRGPywkRjRGQQ== NiM+SSZtb21feEc2IiwmKiZJIk1HRiUiIiJJImJHRiUiIiMjRikiIzcqJkYoRilJImNHRiVGK0Ys Int(Int(Int(M/(a*b*c)*(x^2+z^2), x=-a/2..a/2), y=-b/2..b/2), z=-c/2..c/2);
mom_y := simplify(value(%)); NiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUYkNiQtRiQ2JCosSSJNR0YoIiIiSSJhR0YoISIiSSJiR0YoRjJJImNHRihGMiwmKiRJInhHRigiIiNGMCokSSJ6R0YoRjhGMEYwL0Y3OywkRjEjRjJGOCwkRjEjRjBGOC9JInlHRig7LCRGM0Y+LCRGM0ZAL0Y6OywkRjRGPiwkRjRGQA== NiM+SSZtb21feUc2IiwmKiZJIk1HRiUiIiJJImFHRiUiIiMjRikiIzcqJkYoRilJImNHRiVGK0Ys Int(Int(Int(M/(a*b*c)*(x^2+y^2), x=-a/2..a/2), y=-b/2..b/2), z=-c/2..c/2);
mom_z := simplify(value(%)); NiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUYkNiQtRiQ2JCosSSJNR0YoIiIiSSJhR0YoISIiSSJiR0YoRjJJImNHRihGMiwmKiRJInhHRigiIiNGMCokSSJ5R0YoRjhGMEYwL0Y3OywkRjEjRjJGOCwkRjEjRjBGOC9GOjssJEYzRj4sJEYzRkAvSSJ6R0YoOywkRjRGPiwkRjRGQA== NiM+SSZtb21fekc2IiwmKiZJIk1HRiUiIiJJImFHRiUiIiMjRikiIzcqJkYoRilJImJHRiVGK0Ys (b) Wie ver\344ndert sich das Tr\344gheitsmoment, wenn der Quader um eine der Kanten rotiert wird? Int(Int(Int(M/(a*b*c)*(y^2+z^2), x=0..a), y=0..b), z=0..c);
mom_x2 := simplify(value(%)); NiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUYkNiQtRiQ2JCosSSJNR0YoIiIiSSJhR0YoISIiSSJiR0YoRjJJImNHRihGMiwmKiRJInlHRigiIiNGMCokSSJ6R0YoRjhGMEYwL0kieEdGKDsiIiFGMS9GNztGPkYzL0Y6O0Y+RjQ= NiM+SSdtb21feDJHNiIsJiomSSJNR0YlIiIiSSJiR0YlIiIjI0YpIiIkKiZGKEYpSSJjR0YlRitGLA== Int(Int(Int(M/(a*b*c)*(x^2+z^2), x=0..a), y=0..b), z=0..c);
mom_y2 := simplify(value(%)); NiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUYkNiQtRiQ2JCosSSJNR0YoIiIiSSJhR0YoISIiSSJiR0YoRjJJImNHRihGMiwmKiRJInhHRigiIiNGMCokSSJ6R0YoRjhGMEYwL0Y3OyIiIUYxL0kieUdGKDtGPUYzL0Y6O0Y9RjQ= NiM+SSdtb21feTJHNiIsJiomSSJNR0YlIiIiSSJhR0YlIiIjI0YpIiIkKiZGKEYpSSJjR0YlRitGLA== Int(Int(Int(M/(a*b*c)*(x^2+y^2), x=0..a), y=0..b), z=0..c);
mom_z2 := simplify(value(%)); NiMtSSRJbnRHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYkLUYkNiQtRiQ2JCosSSJNR0YoIiIiSSJhR0YoISIiSSJiR0YoRjJJImNHRihGMiwmKiRJInhHRigiIiNGMCokSSJ5R0YoRjhGMEYwL0Y3OyIiIUYxL0Y6O0Y9RjMvSSJ6R0YoO0Y9RjQ= NiM+SSdtb21fejJHNiIsJiomSSJNR0YlIiIiSSJhR0YlIiIjI0YpIiIkKiZGKEYpSSJiR0YlRitGLA== Aufgabe 3 In Einsteins Relavitit\344tstheorie ist die Energie eines Teilchens der Masse NiNJIm1HNiI= , das sich mit der Geschwindigkeit NiNJInZHNiI= bewegt, als NiMvSSJFRzYiKihJIm1HRiUiIiIqJEkiY0dGJSIiI0YoLUklc3FydEc2JEkqcHJvdGVjdGVkR0YvSShfc3lzbGliR0YlNiMsJkYoRigqJkkidkdGJUYrRikhIiJGNUY1 gegeben. Zeigen Sie mittels Taylorreihenentwicklung bei NiNJInZHNiI= =0, dass diese Formel f\374r kleine Geschwindigkeiten in die klassische kinetische Energie NiMvSSJFRzYiKihJIm1HRiUiIiIqJEkidkdGJSIiI0YoRishIiI= \374bergeht. Wissen Sie was der konstante Term ist?
Stellen Sie das Ergebnis graphisch dar, indem Sie die relativistische Energie und die durch Reihenentwicklung gewonnene in einer Abbildung darstellen. (Setzen Sie dabei NiNJIm1HNiI= =1 und NiNJImNHNiI= =1.)restart; Erel := m*c^2 / sqrt(1-v^2/c^2); NiM+SSVFcmVsRzYiKihJIm1HRiUiIiJJImNHRiUiIiMsJkYoRigqJkkidkdGJUYqRikhIiMhIiIjRi9GKg== taylor(Erel, v=0, 3);
Erel_apprx2 := convert(%, polynom); NiMrKUkidkc2IiomSSJtR0YlIiIiSSJjR0YlIiIjIiIhLCRGJyNGKEYqRiotSSJPR0kqcHJvdGVjdGVkR0YwNiNGKCIiJQ== NiM+SSxFcmVsX2FwcHJ4Mkc2IiwmKiZJIm1HRiUiIiJJImNHRiUiIiNGKSomRihGKUkidkdGJUYrI0YpRis= values := {c= 1, m= 1}; NiM+SSd2YWx1ZXNHNiI8JC9JImNHRiUiIiIvSSJtR0YlRik= plot([subs(values, Erel), subs(values, Erel_apprx2)], v=0..0.9, color=[red, blue], legend=["relativistic", "classic"], labels=["v/c", "E/mc^2"]); 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