restart;Differentialquotient, Tangentengleichung und Fehleranalyse

Die Funktion wird eingegeben:f:=x->(3+x)/(3-x);

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Maple leitet die Funktion ab:f1:=diff(f(x),x);

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subs(x=2,f1);

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Im Folgenden zwei M\366glichkeiten, den Grenzwert des Differenzenquotienten zu berechnen:limit( (f(x)-f(2))/(x-2), x=2);

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limit( (f(2+h)-f(2))/h, h=0);

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Die Funktion wird eingegeben:f:=x->2*x^3;

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Die Tangente wird definiert:t:=x->f(x0)+D(f)(x0)*(x-x0);

NiM+SSJ0RzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYtSSJmR0YlNiNJI3gwR0YlIiIiKiYtLUkiREdGJTYjRi5GL0YxLCY5JEYxRjAhIiJGMUYxRiVGJUYl

Die Tangente wird an der Stelle 2 berechnet:x0:=2;t(x);

NiM+SSN4MEc2IiIiIw==NiMsJiEjSyIiIkkieEc2IiIjQw==

Funktion V(q) wird definiert:V:=q->Pi/6*q^3;

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Die Ableitung wird berechnet:V1:=diff(V(q),q);

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Der relative Fehler dV/V wird durch V'(q)*dq/V(q) abgesch\344tzt: V1*dq/V(q);

NiMsJComSSJxRzYiISIiSSNkcUdGJiIiIiIiJA==

Ableitungstechniken, L'Hospital und Kurvendiskussion

Die Funktion wird eingegeben:f:=x->x^3/(sqrt(x)-x^2);

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Die Ableitung wird berechnet und vereinfacht:diff(f(x),x);

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f1:=normal(%);

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expand(numer(f1))/denom(f1);

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Es wird untersucht, wann der Nenner gleich 0 ist:solve(denom(f(x))=0,x);

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Ditto f\374r die zweite Funktion:f:=x->x^3*ln( (exp(x)-x)^2 );

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diff(f(x),x);

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simplify(%,ln);

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Die Funktion wird definiert:f:=x->(c*x-d)^m;

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Im Folgenden werden Ableitungen gebildet:assume(m,integer);assume(m>1);Die erste Ableitung, etc.:f1:=simplify(diff(f(x),x));

NiM+SSNmMUc2IiooKSwmKiZJImNHRiUiIiJJInhHRiVGK0YrSSJkR0YlISIiLCZJI218aXJHRiVGK0YuRitGK0YwRitGKkYr

f2:=simplify(diff(f1,x));

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f3:=simplify(diff(f2,x)); m:='m';

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Die Funktion wird eingegeben:f:=x->exp(-x)*cos(x);

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Die Ableitungen werden gebildet:(Beachte: Die vierte Ableitung gleicht der urspr\374nglichen Funktion bis auf einen Faktor, ab hier wiederholen sich die Ableitungen.)f1:=diff(f(x),x);

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f2:=diff(f1,x);

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f3:=diff(f2,x);

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f4:=diff(f3,x);

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f:=x->arcsin(x);

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Maple kann die Ableitung der Funktion arcsin(x) direkt angeben:diff(f(x),x);

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Maple kann aber auch die Formel f\374r die Ableitung der Umkehrfunktion einer Funktion auswerten:f^(-1) ' (x) = 1 / (f ' (f^(-1) (x)).(Beachte: f= sin, f^(-1) = arcsin.)f:=x->sin(x); fx:=diff(f(x),x); subs(x=arcsin(x),fx); 1/simplify(%);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkc2luRzYkSSpwcm90ZWN0ZWRHRi9JKF9zeXNsaWJHRiU2IzkkRiVGJUYlNiM+SSNmeEc2Ii1JJGNvc0c2JEkqcHJvdGVjdGVkR0YpSShfc3lzbGliR0YlNiNJInhHRiU=NiMtSSRjb3NHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjLUknYXJjc2luR0YlNiNJInhHRig=NiMqJCwmIiIiRiUqJEkieEc2IiIiIyEiIiNGKkYp

limit((exp(x)-exp(-x)-2*x)/(x-sin(x)),x=0);

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assume(m,integer); assume(m>1), assume(a>1); limit(x^(-m)*a^x,x=infinity);

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m:='m': a:='a':

Die Funktion wird definiert:f:=x->exp(-2*x^2);

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Die Funktion wird gezeichnet:plot(f(x),x=-2..2);

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Die Funktion ist gerade, wie man auch der Zeichnung entnimmt:simplify(f(x)-f(-x));

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Die Funktion hat keine Nullstellen:solve(f(x)=0,x);Nun werden Extrema ermittelt: F\374r x=0 liegt ein lokales Maximum vor, da die erste Ableitung 0 und die zweite negativ ist.f1:=diff(f(x),x);

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kandidat:=solve(f1=0,x);

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f2:=diff(f1,x);

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subs(x=kandidat,f2);

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evalf(%);

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Nun werden Wendepunkte ermittelt:x=1/2 und x=-1/2 sind Wendepunkte.kandidat:=[solve(f2=0,x)];

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f3:=diff(f2,x);

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subs(x=kandidat[1],f3);

NiMsJC1JJGV4cEc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiMjISIiIiIjIiM7

evalf(%);

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subs(x=kandidat[2],f3);

NiMsJC1JJGV4cEc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiMjISIiIiIjISM7

evalf(%);

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Der Grenzwert f\374r x gegen plus/minus unendlich ist jeweils 0:limit(f(x),x=-infinity);

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limit(f(x),x=infinity);

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Newtonverfahren und Taylorentwicklung

Die Funktion, deren Nullstelle mit Hilfe des Newtonverfahrens bestimmt werden soll, wird eingegeben:f:=x->3*cos(x)-x;

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Eine Nullstelle mu\337 im Intervall [1,2] liegen, da die Funktion stetig ist und an den Intervallgrenzen 1 und 2 verschiedene Vorzeichen annimmt:evalf(f(1)); evalf(f(2));

NiMkIio9cCE0aSEiKg==NiMkISs1MFdbSyEiKg==

Maple kann die Gleichung f(x)=0 numerisch l\366sen:(Beachte: Hier wurde das Intervall [1,2] f\374r die Suche einer L\366sung vorgegeben.)fsolve(f(x)=0,x,1..2);

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Nun wird die Iterationsformel f\374r das Newtonverfahren berechnet:f1:=diff(f(x),x);

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newton:=x-f(x)/f1;

NiM+SSduZXd0b25HNiIsJkkieEdGJSIiIiomLCYtSSRjb3NHNiRJKnByb3RlY3RlZEdGLkkoX3N5c2xpYkdGJTYjRiciIiRGJyEiIkYoLCYtSSRzaW5HRi1GMCEiJEYyRihGMkYy

Es werden einige Iterationen mit dem Startwert 1 durchgef\374hrt:Digits:=15; x[0]:=1;x[1]:=evalf(subs(x=x[0],newton));x[2]:=evalf(subs(x=x[1],newton));x[3]:=evalf(subs(x=x[2],newton));x[4]:=evalf(subs(x=x[3],newton));

NiM+SSdEaWdpdHNHNiIiIzo=NiM+JkkieEc2IjYjIiIhIiIiNiM+JkkieEc2IjYjIiIiJCIwSiMqZTl0aDwiISM5NiM+JkkieEc2IjYjIiIjJCIwXHU1ZUUsPCIhIzk=NiM+JkkieEc2IjYjIiIkJCIwYzIrJjQ3cTYhIzk=NiM+JkkieEc2IjYjIiIlJCIwai0rJjQ3cTYhIzk=

Der Test ergibt, dass hier wirklich eine Nullstelle vorliegt:f(1.17012); Digits:=10:x:='x':

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Das Polynom wird definiert:p:=x->2*x^3+3*x^2+5;

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Die Taylorentwicklung an der Stelle x=0 bis zur dritten Ordnung wird ausgef\374hrt:taylor(p(x),x=0,4);

NiMrKUkieEc2IiIiJiIiISIiJCIiI0YpRig=

Die Taylorentwicklung an der Stelle x=1 bis zur dritten Ordnung wird ausgef\374hrt:taylor(p(x),x=1,4);

NiMrKywmSSJ4RzYiIiIiISIiRiciIzUiIiEiIzdGJyIiKiIiI0YtIiIk

Der folgende Schritt ist in Maple n\366tig, um die obige Partialsumme in ein Polynom umzuwandeln:convert(%,polynom);

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Wie man nun sieht, stimmt die Taylorentwicklung an x=1 mit dem Polynom \374berein:p(x)-simplify(%);

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f:=x->sin(x);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkc2luRzYkSSpwcm90ZWN0ZWRHRi9JKF9zeXNsaWJHRiU2IzkkRiVGJUYl

taylor(f(x),x=0,9);

NiMrLUkieEc2IiIiIkYmIyEiIiIiJyIiJCNGJiIkPyIiIiYjRigiJVNdIiIoLUkiT0dJKnByb3RlY3RlZEdGMzYjRiYiIio=

p:=convert(%,polynom);

NiM+SSJwRzYiLCpJInhHRiUiIiIqJEYnIiIkIyEiIiIiJyokRiciIiYjRigiJD8iKiRGJyIiKCNGLCIlU10=

Funktionen mehrerer Ver\344nderlicher

Eine Funktion in zwei Ver\344nderlichen wird definiert:f:=(x,y)->3*x^2+4*y^2-6;

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJDkkIiIjIiIkKiQ5JUYwIiIlISInIiIiRiVGJUYl

Es wird nun y=1 gesetzt, nach x abgeleitet und diese Ableitung an der Stelle x=1 ausgewertet:z:=f(x,1);

NiM+SSJ6RzYiLCYqJEkieEdGJSIiIyIiJCEiIyIiIg==

zx:=diff(z,x);

NiM+SSN6eEc2IiwkSSJ4R0YlIiIn

subs(x=1,zx);

NiMiIic=

Es wird nun x=1 gesetzt, nach y abgeleitet und diese Ableitung an der Stelle y=1 ausgewertet:z:=f(1,y);

NiM+SSJ6RzYiLCYhIiQiIiIqJEkieUdGJSIiIyIiJQ==

zy:=diff(z,y);

NiM+SSN6eUc2IiwkSSJ5R0YlIiIp

subs(y=1,zy);

NiMiIik=

f:=(x,y)->x^2/y+y^2/x;

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJjkkIiIjOSUhIiIiIiIqJkYxRjBGL0YyRjNGJUYlRiU=

diff(f(x,y),x);

NiMsJiomSSJ4RzYiIiIiSSJ5R0YmISIiIiIjKiZGKEYqRiUhIiNGKQ==

diff(f(x,y),y);

NiMsJiomSSJ4RzYiIiIjSSJ5R0YmISIjISIiKiZGKCIiIkYlRipGJw==

f:=(x,y)->ln(tan(x/y));

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkjbG5HNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJTYjLUkkdGFuR0YvNiMqJjkkIiIiOSUhIiJGJUYlRiU=

diff(f(x,y),x);

NiMqKCwmIiIiRiUqJC1JJHRhbkc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliRzYiNiMqJkkieEdGLEYlSSJ5R0YsISIiIiIjRiVGJUYwRjFGJ0Yx

diff(f(x,y),y);

NiMsJCoqLCYiIiJGJiokLUkkdGFuRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHNiI2IyomSSJ4R0YtRiZJInlHRi0hIiIiIiNGJkYmRjBGJkYxISIjRihGMkYy

simplify(%,trig);

NiMsJCoqLCYiIiJGJiokLUkkdGFuRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHNiI2IyomSSJ4R0YtRiZJInlHRi0hIiIiIiNGJkYmRjBGJkYxISIjRihGMkYy

f:=(x,y)->x^y;

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKTkkOSVGJUYlRiU=

diff(f(x,y),x);

NiMqKClJInhHNiJJInlHRiYiIiJGJ0YoRiUhIiI=

diff(f(x,y),y);

NiMqJilJInhHNiJJInlHRiYiIiItSSNsbkc2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliR0YmNiNGJUYo

Die Funktion in zwei Ver\344nderlichen wird definiert:f:=(x,y)->sqrt(x-y);

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklc3FydEc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiMsJjkkIiIiOSUhIiJGJUYlRiU=

Die Formel f\374r die Tangentialebene wird angegeben:t:=(x,y)->f(x0,y0)+D[1](f)(x0,y0)*(x-x0)+D[2](f)(x0,y0)*(y-y0);

NiM+SSJ0RzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgtSSJmR0YlNiRJI3gwR0YlSSN5MEdGJSIiIiomLS0mSSJERzYkSSpwcm90ZWN0ZWRHRjpJKF9zeXNsaWJHRiU2I0YzNiNGL0YwRjMsJjkkRjNGMSEiIkYzRjMqJi0tJkY4NiMiIiNGPUYwRjMsJjklRjNGMkZARjNGM0YlRiVGJQ==

Die Tangentialebene an der Stelle x0=1, y0=-3 wird berechnet:x0:=1; y0:=-3;

NiM+SSN4MEc2IiIiIg==NiM+SSN5MEc2IiEiJA==

t(x,y);

NiMsKCIiIkYkSSJ4RzYiI0YkIiIlSSJ5R0YmIyEiIkYo

Funktion und Tangentialebene werden gezeichnet:p1:=plot3d(f(x,y),x=-149..151,y=-153..147,axes=boxed):p2:=plot3d(t(x,y),x=-149..151,y=-153..147,axes=boxed):with(plots,display):Zun\344chst wird die Funktion gezeichnet:(Beachte: Die Funktion ist nur f\374r x>=y definiert.)display([p1]);

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

Nun wird die Tangentialebene gezeichnet:display([p2]);

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2D0000000000004026C00000000000402080000000000040148000000000004000000000000000-%+AXESLABELSG6%Q"xF4Q"yF4Q!F4-%*AXESSTYLEG6#%$BOXG

Nun werden Funktion und Tangentialebene gemeinsam gezeichnet:display([p1,p2]);

-%'PLOT3DG6&-%%GRIDG6%;$!$\"""!$"$^"F+;$!$`"F+$"$Z"F+7;7;$""#F+I*undefinedGI*protectedGF8F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+-#>?1%!"*$"+++++?F<F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+2[;&Q&F<F:F=F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+j$\?W'F<F@F:F=F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+G#p%[tF<FCF@F:F=F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+:Kva")F<FFFCF@F:F=F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+<W>))))F<FIFFFCF@F:F=F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+NKcl&*F<FLFIFFFCF@F:F=F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+.R!)>5!")FOFLFIFFFCF@F:F=F7F7F7F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+d;Nz5FTFRFOFLFIFFFCF@F:F=F7F7F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+p;yN6FTFVFRFOFLFIFFFCF@F:F=F7F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+Dx`*="FTFYFVFRFOFLFIFFFCF@F:F=F7F7F7F7F7F7F7F7F7F7F7F7F77;$"+lt'4C"FTFfnFYFVFRFOFLFIFFFCF@F:F=F7F7F7F7F7F7F7F7F7F7F7F77;$"+!z[.H"FTFinFfnFYFVFRFOFLFIFFFCF@F:F=F7F7F7F7F7F7F7F7F7F7F77;$"+;)3zL"FTF\oFinFfnFYFVFRFOFLFIFFFCF@F:F=F7F7F7F7F7F7F7F7F7F77;$"+]_$QQ"FTF_oF\oFinFfnFYFVFRFOFLFIFFFCF@F:F=F7F7F7F7F7F7F7F7F77;$"+'o&GG9FTFboF_oF\oFinFfnFYFVFRFOFLFIFFFCF@F:F=F7F7F7F7F7F7F7F77;$"+(*QRr9FTFeoFboF_oF\oFinFfnFYFVFRFOFLFIFFFCF@F:F=F7F7F7F7F7F7F77;$"+&fuK^"FTFhoFeoFboF_oF\oFinFfnFYFVFRFOFLFIFFFCF@F:F=F7F7F7F7F7F77;$"+Fq-a:FTF[pFhoFeoFboF_oF\oFinFfnFYFVFRFOFLFIFFFCF@F:F=F7F7F7F7F77;$"+Xxt$f"FTF^pF[pFhoFeoFboF_oF\oFinFfnFYFVFRFOFLFIFFFCF@F:F=F7F7F7F77;$"+sF[K;FTFapF^pF[pFhoFeoFboF_oF\oFinFfnFYFVFRFOFLFIFFFCF@F:F=F7F7F77;$"+4$H.n"FTFdpFapF^pF[pFhoFeoFboF_oF\oFinFfnFYFVFRFOFLFIFFFCF@F:F=F7F77;$"+3rL2<FTFgpFdpFapF^pF[pFhoFeoFboF_oF\oFinFfnFYFVFRFOFLFIFFFCF@F:F=F77;$"+x&fNu"FTFjpFgpFdpFapF^pF[pFhoFeoFboF_oF\oFinFfnFYFVFRFOFLFIFFFCF@F:F=-F&6%F(F.X,6"FbqFbq[gl'!%"!!#\bm":":4000000000000000BFF2000000000000C011000000000000C01D800000000000C025000000000000C02B400000000000C030C00000000000C033E00000000000C037000000000000C03A200000000000C03D400000000000C040300000000000C041C00000000000C043500000000000C044E00000000000C046700000000000C048000000000000C049900000000000C04B200000000000C04CB00000000000C04E400000000000C04FD00000000000C050B00000000000C051780000000000C05240000000000040148000000000004000000000000000BFF2000000000000C011000000000000C01D800000000000C025000000000000C02B400000000000C030C00000000000C033E00000000000C037000000000000C03A200000000000C03D400000000000C040300000000000C041C00000000000C043500000000000C044E00000000000C046700000000000C048000000000000C049900000000000C04B200000000000C04CB00000000000C04E400000000000C04FD00000000000C050B00000000000C051780000000000402080000000000040148000000000004000000000000000BFF2000000000000C011000000000000C01D800000000000C025000000000000C02B400000000000C030C00000000000C033E00000000000C037000000000000C03A200000000000C03D400000000000C040300000000000C041C00000000000C043500000000000C044E00000000000C046700000000000C048000000000000C049900000000000C04B200000000000C04CB00000000000C04E400000000000C04FD00000000000C050B000000000004026C00000000000402080000000000040148000000000004000000000000000BFF2000000000000C011000000000000C01D800000000000C025000000000000C02B400000000000C030C00000000000C033E00000000000C037000000000000C03A200000000000C03D400000000000C040300000000000C041C00000000000C043500000000000C044E00000000000C046700000000000C048000000000000C049900000000000C04B200000000000C04CB00000000000C04E400000000000C04FD00000000000402D0000000000004026C00000000000402080000000000040148000000000004000000000000000BFF2000000000000C011000000000000C01D800000000000C025000000000000C02B400000000000C030C00000000000C033E00000000000C037000000000000C03A200000000000C03D400000000000C040300000000000C041C00000000000C043500000000000C044E00000000000C046700000000000C048000000000000C049900000000000C04B200000000000C04CB00000000000C04E4000000000004031A00000000000402D0000000000004026C00000000000402080000000000040148000000000004000000000000000BFF2000000000000C011000000000000C01D800000000000C025000000000000C02B400000000000C030C00000000000C033E00000000000C037000000000000C03A200000000000C03D400000000000C040300000000000C041C00000000000C043500000000000C044E00000000000C046700000000000C048000000000000C049900000000000C04B200000000000C04CB000000000004034C000000000004031A00000000000402D0000000000004026C00000000000402080000000000040148000000000004000000000000000BFF2000000000000C011000000000000C01D800000000000C025000000000000C02B400000000000C030C00000000000C033E00000000000C037000000000000C03A200000000000C03D400000000000C040300000000000C041C00000000000C043500000000000C044E00000000000C046700000000000C048000000000000C049900000000000C04B2000000000004037E000000000004034C000000000004031A00000000000402D0000000000004026C00000000000402080000000000040148000000000004000000000000000BFF2000000000000C011000000000000C01D800000000000C025000000000000C02B400000000000C030C00000000000C033E00000000000C037000000000000C03A200000000000C03D400000000000C040300000000000C041C00000000000C043500000000000C044E00000000000C046700000000000C048000000000000C049900000000000403B0000000000004037E000000000004034C000000000004031A00000000000402D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Die Funktion in zwei Ver\344nderlichen wird definiert:P:=(U,R)->U^2/R;

NiM+SSJQRzYiZio2JEkiVUdGJUkiUkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiIzklISIiRiVGJUYl

Die partiellen Ableitungen werden berechnet:PU:=D[1](P);

NiM+SSNQVUc2ImYqNiRJIlVHRiVJIlJHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwkKiY5JCIiIjklISIiIiIjRiVGJUYl

PR:=D[2](P);

NiM+SSNQUkc2ImYqNiRJIlVHRiVJIlJHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwkKiY5JCIiIzklISIjISIiRiVGJUYl

Die Formel f\374r die Fehlerrechnung wird eingegeben:dP:=PU*dU + PR*dR;

NiM+SSNkUEc2IiwmKiZJI1BVR0YlIiIiSSNkVUdGJUYpRikqJkkjUFJHRiVGKUkjZFJHRiVGKUYp

Die Formel f\374r die Fehlerrechnung wird ausgewertet:dU:=-10; dR:=-0.5;

NiM+SSNkVUc2IiEjNQ==NiM+SSNkUkc2IiQhIiYhIiI=

dP(220,10);

NiMkISUhKT4hIiI=

Die Funktion in zwei Ver\344nderlichen wird angegeben:f:=(x,y)->y*exp(x);

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JSIiIi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YzSShfc3lzbGliR0YlNiM5JEYvRiVGJUYl

with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

Der Gradient wird berechnet:grad(f(x,y),[x,y]);

NiMtSSd2ZWN0b3JHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjNyQqJkkieUdGKCIiIi1JJGV4cEdGJTYjSSJ4R0YoRi1GLg==

Der Gradient an der Stelle (0,5) wird berechnet:subs({x=0,y=5},%);

NiMtSSd2ZWN0b3JHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjNyQsJC1JJGV4cEdGJTYjIiIhIiImRiw=

gradient:=simplify(%);

NiM+SSlncmFkaWVudEc2Ii1JJ3ZlY3Rvckc2JEkqcHJvdGVjdGVkR0YpSShfc3lzbGliR0YlNiM3JCIiJiIiIg==

Das Skalarprodukt mit dem Richtungsvektor wird berechnet:v:=[1/sqrt(2),1/sqrt(2)];

NiM+SSJ2RzYiNyQsJCokIiIjIyIiIkYpRipGJw==

dotprod(gradient,v);

NiMsJCokIiIjIyIiIkYlIiIk

Die Funktion wird eingegeben:f:=(x,y)->ln(x^2+y^2+1);

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkjbG5HNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJTYjLCgqJDkkIiIjIiIiKiQ5JUY2RjdGN0Y3RiVGJUYl

Die ersten partiellen Ableitungen werden berechnet:fx:=diff(f(x,y),x);

NiM+SSNmeEc2IiwkKiZJInhHRiUiIiIsKCokRigiIiNGKSokSSJ5R0YlRixGKUYpRikhIiJGLA==

fy:=diff(f(x,y),y);

NiM+SSNmeUc2IiwkKiZJInlHRiUiIiIsKCokSSJ4R0YlIiIjRikqJEYoRi1GKUYpRikhIiJGLQ==

Die ersten partiellen Ableitungen werden gleich 0 gesetzt:solve({fx=0,fy=0},{x,y});

NiM8JC9JInhHNiIiIiEvSSJ5R0YmRic=

Die zweiten partiellen Ableitungen werden berechnet:fxx:=diff(fx,x);

NiM+SSRmeHhHNiIsJiokLCgqJEkieEdGJSIiIyIiIiokSSJ5R0YlRitGLEYsRiwhIiJGKyomRipGK0YoISIjISIl

fxy:=diff(fx,y);

NiM+SSRmeHlHNiIsJCooSSJ4R0YlIiIiLCgqJEYoIiIjRikqJEkieUdGJUYsRilGKUYpISIjRi5GKSEiJQ==

fyy:=diff(fy,y);

NiM+SSRmeXlHNiIsJiokLCgqJEkieEdGJSIiIyIiIiokSSJ5R0YlRitGLEYsRiwhIiJGKyomRi5GK0YoISIjISIl

Die Determinante der Hesse-Matrix wird an der gefundenen Stelle ausgewertet:subs({x=0,y=0},fxx*fyy-fxy^2);

NiMiIiU=

Analog wird fxx an der gefundenen Stelle ausgewertet:Es liegt ein lokales Minimum vor.subs({x=0,y=0},fxx);

NiMiIiM=

Maple ermittelt Minimum und Maximum (sowie die Stelle, an der sie angenommen werden) auch automatisch:(Beachte: Ein Maximum liegt nicht vor, ein Minimum wird an (0,0) angenommen.)maximize(f(x,y),location);

NiRJKWluZmluaXR5R0kqcHJvdGVjdGVkR0YkPCM3JEklRkFJTEdGJEYj

minimize(f(x,y),location);

NiQiIiE8IzckSSVGQUlMR0kqcHJvdGVjdGVkR0YnRiM=

Die Funktion wird eingegeben:f:=(x,y)->x^5*y+x*y^5+x*y;

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJjkkIiImOSUiIiJGMiomRi9GMkYxRjBGMiomRi9GMkYxRjJGMkYlRiVGJQ==

Die ersten partiellen Ableitungen werden berechnet:fx:=diff(f(x,y),x);

NiM+SSNmeEc2IiwoKiZJInhHRiUiIiVJInlHRiUiIiIiIiYqJEYqRixGK0YqRis=

fy:=diff(f(x,y),y);

NiM+SSNmeUc2IiwoKiRJInhHRiUiIiYiIiIqJkYoRipJInlHRiUiIiVGKUYoRio=

Die ersten partiellen Ableitungen werden gleich 0 gesetzt: fsolve({fx=0,fy=0},{x,y});

NiM8JC9JInlHNiIkIiIhRigvSSJ4R0YmRic=

Die zweiten partiellen Ableitungen werden berechnet:fxx:=diff(fx,x); fxy:=diff(fx,y); fyy:=diff(fy,y);

NiM+SSRmeHhHNiIsJComSSJ4R0YlIiIkSSJ5R0YlIiIiIiM/NiM+SSRmeHlHNiIsKCokSSJ4R0YlIiIlIiImKiRJInlHRiVGKUYqIiIiRi0=NiM+SSRmeXlHNiIsJComSSJ4R0YlIiIiSSJ5R0YlIiIkIiM/

Die Determinante der Hesse-Matrix wird an der ermittelten Stelle ausgewertet:Es liegt ein Sattelpunkt vor.subs({x=0,y=0},fxx*fyy-fxy^2);

NiMhIiI=

Die Funktionen maximize und minimize liefern hier kein Ergebnis, da ja kein Minimum oder Maximum vorliegt:maximize(f(x,y),location);

NiRJKWluZmluaXR5R0kqcHJvdGVjdGVkR0YkPCc3JDwkL0kieUc2IkYjL0kieEdGKkYjRiM3JDwkL0YsIiIhRihGIzckPCQvRilGMEYrRiM3JDwkL0YpLUkpSU5URVJWQUxHRio2IzssJEYjISIiRjAvRixGO0YjNyQ8JC9GKUY7L0YsRjdGIw==

minimize(f(x,y),location);

NiQsJEkpaW5maW5pdHlHSSpwcm90ZWN0ZWRHRiUhIiI8KDckPCQvSSJ5RzYiRiQvSSJ4R0YsLUkpSU5URVJWQUxHRiw2IztGIyIiIUYjNyQ8JC9GK0YzL0YuRiNGIzckPCQvRi5GJC9GK0YvRiM3JDwkRipGN0YjNyQ8JEY6L0YrRiNGIzckPCQvRi5GM0ZARiM=