diese Funktion linearisiert eine Funktion f(x,y) im Punkt x0, y0der Rueckgabewert der Funktion ist eine Funktion im Stile: z := (x,y) -> blabla

restrestart:linearisieren := proc( f, x0, y0 ) local fx, fy, z: printf("\t\t\tEs wird folgende Gleichung berechnet :"): print( z = fx*dx + fy*dy + z0 ): fx := D[1](f): fy := D[2](f): printf("\t\t\tEs ergeben sich folgende partielle Ableitungen : "): print(fx = fx(x,y)): print(fy = fy(x,y)): printf("\t\t\tlinearisiert wird im Punkt:"): print(P = (x0, y0, f(x0,y0))): return (x,y) -> D[1](f)(x0,y0)*(x-x0) + D[2](f)(x0,y0)*(y-y0) + f(x0,y0): end proc:

Beispiel 1 ( Aus der Theorie )f := (x,y) -> 5*x^2*sqrt(y);

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqJjkkIiIjLUklc3FydEdGJTYjOSUiIiIiIiZGJUYlRiU=

z := linearisieren( f, 2, 1);

Es wird folgende Gleichung berechnet :NiMvSSJ6RzYiLCgqJkkjZnhHRiUiIiJJI2R4R0YlRilGKSomSSNmeUdGJUYpSSNkeUdGJUYpRilJI3owR0YlRik= Es ergeben sich folgende partielle Ableitungen : NiMvSSNmeEc2IiwkKiZJInhHRiUiIiJJInlHRiUjRikiIiMiIzU=NiMvSSNmeUc2IiwkKiZJInhHRiUiIiNJInlHRiUjISIiRikjIiImRik= linearisiert wird im Punkt:NiMvSSJQRzYiNiUiIiMiIiIiIz8=NiM+SSJ6RzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJi0tJkkiREc2JEkqcHJvdGVjdGVkR0Y0SShfc3lzbGliR0YlNiMiIiI2I1QoNiRUJFQmRjcsJjkkRjdGOyEiIkY3RjcqJi0tJkYyNiMiIiNGOEY6RjcsJjklRjdGPEY/RjdGNy1GOUY6RjdGJUYlNihJI3gwR0YlRkVJI3kwR0YlRjdJImZHRiVGTA==

z(x,y);

NiMsKEkieEc2IiIjPyEjSSIiIkkieUdGJSIjNQ==

diese Funktion gibt fuer messwerte von x0+dx und y0+dy zur Funktion f(x,y), den Wert f(x,y) sowie deren maximalen Fehler

restart:messfehler := proc( f, x0, dx, y0, dy) local z, fx, fy, dz: printf("\t\t\tBerechne folgende Gleichung:"); print(dz = abs(fx(x0,y0)*dx) + abs(fy(x0,y0)*dy)): dz := abs(D[1](f)(x0,y0)*dx) + abs(D[2](f)(x0,y0)*dy); z := f(x0,y0): fx := D[1](f): fy := D[2](f): printf("\n\t\t\tDie partiellen Ableitungen sind:"); print(fx = fx(x,y)); print(fy = fy(x,y)); printf("\n\t\t\tDer Wert ist z = %10f\n", z): printf("\t\t\tbzw. z = %10e\n\n", z): printf("\t\t\tMessfehler betraegt dz = %10f\n", dz); printf("\t\t\tbzw. dz = %10e\n\n", dz): end proc:

f := (x,y) -> x*sin(y):messfehler(f, 32, 0.5, 8, 0.2);

Berechne folgende Gleichung:NiMvSSNkekc2IiwmLUkkYWJzR0kqcHJvdGVjdGVkR0YpNiMtSSNmeEdGJTYkIiNLIiIpJCIiJiEiIi1GKDYjLUkjZnlHRiVGLSQiIiNGMg==

Der Wert ist z = 31.659464 bzw. z = 3.165946e+01 Messfehler betraegt dz = 1.425879 bzw. dz = 1.425879e+00

f := (x,y) -> 2*Pi*(x*y)^(1/2);

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqJkkjUGlHSSpwcm90ZWN0ZWRHRjAiIiIqJjkkRjE5JUYxI0YxIiIjRjZGJUYlRiU=

messfehler(f, 7*10^(-6), 0.3*10^(-6), 0.22, 0.01);

Berechne folgende Gleichung:NiMvSSNkekc2IiwmLUkkYWJzR0kqcHJvdGVjdGVkR0YpNiMtSSNmeEdGJTYkIyIiKCIoKysrIiQiI0EhIiMkIisrKysrSSEjOy1GKDYjLUkjZnlHRiVGLSQiIiJGMw== Die partiellen Ableitungen sind:NiMvSSNmeEc2IiooSSNQaUdJKnByb3RlY3RlZEdGKCIiIiomSSJ4R0YlRilJInlHRiVGKSMhIiIiIiNGLEYpNiMvSSNmeUc2IiooSSNQaUdJKnByb3RlY3RlZEdGKCIiIiomSSJ4R0YlRilJInlHRiVGKSMhIiIiIiNGK0Yp Der Wert ist z = 0.007797 bzw. z = 7.797228e-03 Messfehler betraegt dz = 0.000344 bzw. dz = 3.442932e-04

g:=(L,T)->L/((T/(2*Pi))^2);

NiM+SSJnRzYiZio2JEkiTEdGJUkiVEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqKDkkIiIiOSUhIiNJI1BpR0kqcHJvdGVjdGVkR0Y0IiIjIiIlRiVGJUYl

messfehler(g,0.992,0.001,2.0,0.003);

Berechne folgende Gleichung:NiMvSSNkekc2IiwmLUkkYWJzR0kqcHJvdGVjdGVkR0YpNiMtSSNmeEdGJTYkJCIkIyoqISIkJCIjPyEiIiQiIiJGMC1GKDYjLUkjZnlHRiVGLSQiIiRGMA== Die partiellen Ableitungen sind:NiMvSSNmeEc2IiwkKiZJInlHRiUhIiNJI1BpR0kqcHJvdGVjdGVkR0YrIiIjIiIlNiMvSSNmeUc2IiwkKihJInhHRiUiIiJJInlHRiUhIiRJI1BpR0kqcHJvdGVjdGVkR0YtIiIjISIp Der Wert ist z = 9.790648 bzw. z = 9.790648e+00 Messfehler betraegt dz = 0.039242 bzw. dz = 3.924155e-02

Diese Funktion berechnet die Loesungen fuer die Funktion 'f(x0,x1, . . , xn)' unter der Nebenbedingung 'gleichung' Parameter: 1. : f( x0, x1, . . . ,xn ) Dieser Parameter ist die eigentliche Funktion, welche ihr maximieren wollt. 2. : params params ist die Auflistung aller Parameter, welche eure Funktion verlangt. 3. : g( x0, x1, . . . , xn ) Dieser Parameter ist die Nebenbedingung Achtung: Die Funktion verlangt die beiden Funktionen f() und g() in der normalen Form,dass heisst nicht in der Form f := (x,y) -> x*y; Solltet ihr eure Funktionen dennoch so definiert haben, dann muesst ihr sicherstellen,dass ihr sie in der Form f(x,y) an die laGrange-Funktion uebergibt. lagrange(f(x,y), [x,y], g(x,y));

restart:lagrange := proc(f, params, g) local F, fs, i: F := f + lambda*g; printf("\t\t\tLaGrange Funktion:"); print(F); printf("\t\t\tLoese somit folgendes Gleichungssystem:"); for i from 1 to nops(params) do print(diff(F, op(i,params)) = 0); end do: print(g = 0); printf("\t\t\tDie Loesungen sind: "); fs := solve( {seq( diff(F, op(i, params)) = 0 , i=1..nops(params)) , g = 0}, {seq(op(ii, params), ii=1..nops(params)),lambda}): print(fs); printf("\t\t\tbzw.:"): print(evalf(fs)); end proc:

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

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiIjklRi9GJUYlRiU=

g := (x,y) -> x + y -20;

NiM+SSJnRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCg5JCIiIjklRi8hIz9GL0YlRiVGJQ==

evalf(lagrange(f(x,y), [x,y], g(x,y)));

LaGrange Funktion:NiMsJiomSSJ4RzYiIiIiSSJ5R0YmRidGJyomSSdsYW1iZGFHRiZGJywoRiVGJ0YoRichIz9GJ0YnRic= Loese somit folgendes Gleichungssystem:NiMvLCZJInlHNiIiIiJJJ2xhbWJkYUdGJkYnIiIhNiMvLCZJInhHNiIiIiJJJ2xhbWJkYUdGJkYnIiIhNiMvLChJInhHNiIiIiJJInlHRiZGJyEjP0YnIiIh Die Loesungen sind: NiM8JS9JInlHNiIkIiM1IiIhL0kieEdGJkYnL0knbGFtYmRhR0YmJCEjNUYp

f := (x,y) -> sqrt(x+2*y);

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklc3FydEdGJTYjLCY5JCIiIjklIiIjRiVGJUYl

g := (x,y) -> 2*x^2+y^2-4;

NiM+SSJnRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJDkkIiIjRjAqJDklRjAiIiIhIiVGM0YlRiVGJQ==

lagrange(f(x,y), [x,y], g(x,y));

LaGrange Funktion:NiMsJiokLCZJInhHNiIiIiJJInlHRiciIiMjRihGKkYoKiZJJ2xhbWJkYUdGJ0YoLCgqJEYmRipGKiokRilGKkYoISIlRihGKEYo Loese somit folgendes Gleichungssystem:NiMvLCYqJCwmSSJ4RzYiIiIiSSJ5R0YoIiIjIyEiIkYrI0YpRisqJkknbGFtYmRhR0YoRilGJ0YpIiIlIiIhNiMvLCYqJCwmSSJ4RzYiIiIiSSJ5R0YoIiIjIyEiIkYrRikqJkknbGFtYmRhR0YoRilGKkYpRisiIiE=NiMvLCgqJEkieEc2IiIiI0YoKiRJInlHRidGKCIiIiEiJUYrIiIh Die Loesungen sind: NiQ8JS9JJ2xhbWJkYUc2IiwkLUknUm9vdE9mRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2JCwmKiRJI19aR0YmIiIlIiIiISM9RjJeIyQiK1dyd2Y/ISIqIyEiIiIjOy9JInhHRiYsJCokRigiIiMjRjIiIiovSSJ5R0YmLCRGPiNGMUZBPCUvRkMsJCokLUYpNiRGLkY1Rj9GRS9GPCwkRklGQC9GJSwkRkpGOA== bzw.:NiQ8JS9JJ2xhbWJkYUc2Il4jJCErbFdOKEciISM1L0kieEdGJiQhKzNfLzlaRiovSSJ5R0YmJCErJDM9Yyk9ISIqPCUvRiVGKC9GMCQiKyQzPWMpPUYzL0YsJCIrM18vOVpGKg==

evalf(sqrt(3));

NiMkIiszMzBLPCEiKg==

restart;

Die Funktion Rechnet Kartesische Koordinaten in Polarkoordinaten um.Es k\366nnen auch negative Werte eingegeben werden, die Werte werden automatisch richtig berechnet.Ausgabewerte:- Radius exakt und approximiert- Winkel im Radiantmass und im Winkelmass- Alle Formeln f\374r die Umwandlung

kart_to_polar := proc (x, y) local r, phi; r:=sqrt(x^2+y^2); lprint("Formel f\374r den Radius auszurechnen r = sqrt(x^2 + y^2)"); printf("Radius exakt:"); print(r); printf("Radius approximiert:"); print(evalf(r)); if x>0 then if y>=0 then lprint("Formel um den Winkel auszurechnen: tan(phi) = y/x"); printf("Winkel in Radiant:"); phi:=evalf(arctan(y/x)); print(phi); lprint("Winkel von Radiant in Grad umwandeln: (rad*180)/Pi"); print(evalf((phi*180)/Pi)); elif y<0 then lprint("Formel um den Winkel auszurechnen: tan(phi) = x/y"); printf("Winkel in Radiant:"); phi:=evalf(arctan(abs(x)/abs(y))+3*Pi/2); print(phi); lprint("Winkel von Radiant in Grad umwandeln: (rad*180)/Pi"); print(evalf((phi*180)/Pi)); end if: elif x<0 then if y>=0 then lprint("Formel um den Winkel auszurechnen: tan(phi) = y/x"); printf("Winkel in Radiant:"); phi:=evalf(arctan(abs(x)/abs(y))+Pi/2); print(phi); lprint("Winkel von Radiant in Grad umwandeln: (rad*180)/Pi"); print(evalf((phi*180)/Pi)); elif y<0 then lprint("Formel um den Winkel auszurechnen: tan(phi) = x/y"); printf("Winkel in Radiant:"); phi:=evalf(arctan(abs(y)/abs(x))+Pi); print(phi); lprint("Winkel von Radiant in Grad umwandeln: (rad*180)/Pi"); print(evalf((phi*180)/Pi)); end if: end if: end proc:;

kart_to_polar(-1,1);

"Formel f\374r den Radius auszurechnen r = sqrt(x^2 + y^2)" Radius exakt:NiMqJCIiIyMiIiJGJA==Radius approximiert:NiMkIitpTkA5OSEiKg=="Formel um den Winkel auszurechnen: tan(phi) = y/x" Winkel in Radiant:

NiMkIishXCU+Y0IhIio="Winkel von Radiant in Grad umwandeln: (rad*180)/Pi" NiMkIisrKytdOCEiKA==

restart:

Die Funktion Rechnet Polarkoordinaten in Kartesische Koordinaten um.Es k\366nnen auch negative Werte eingegeben werden, die Werte werden automatisch richtig berechnet.Ausgabewerte:- x-Koordinate- y-Koordinate- Formeln um dies auszurechnen

polar_to_kart := proc (r, phi) local x, y, phirad; if phi<90 then phirad:=(Pi*phi)/180: lprint("Formel um die x-Koordinate auszurechnen x = r*cos(phi)"); x:=r*cos(phirad): printf("x-Koordinate:"); print(evalf(x)); lprint("Formel um die y-Koordinate auszurechnen y = r*sin(phi)"); y:=r*sin(phirad): printf("y-Koordinate:"); print(evalf(y)) elif phi<180 then phirad:=(Pi*(phi-90))/180: lprint("Formel um die x-Koordinate auszurechnen x = (-1)*r*sin(phi-90)"); x:=(-1)*r*sin(phirad): printf("x-Koordinate:"); print(evalf(x)); lprint("Formel um die y-Koordinate auszurechnen y = r*cos(phi-90)"); y:=r*cos(phirad): printf("y-Koordinate:"); print(evalf(y)) elif phi<270 then phirad:=(Pi*(phi-180))/180: lprint("Formel um die x-Koordinate auszurechnen x = (-1)*r*cos(phi-180)"); x:=(-1)*r*cos(phirad): printf("x-Koordinate:"); print(evalf(x)); lprint("Formel um die y-Koordinate auszurechnen y = (-1)*r*sin(phi-180)"); y:=(-1)*r*sin(phirad): printf("y-Koordinate:"); print(evalf(y)) elif phi<360 then phirad:=(Pi*(phi-270))/180: lprint("Formel um die x-Koordinate auszurechnen x = r*sin(phi-270)"); x:=r*sin(phirad): printf("x-Koordinate:"); print(evalf(x)); lprint("Formel um die y-Koordinate auszurechnen y = (-1)*r*cos(phi-270)"); y:=(-1)*r*cos(phirad): printf("y-Koordinate:"); print(evalf(y)) end if: lprintf("Zus\344tzlich ben\366tigte Formeln:"); lprint("Gradmass in Radiantmass:"); lprint("Pi*phi/180"); lprint(); end proc:;

polar_to_kart(23,150);

"Formel um die x-Koordinate auszurechnen x = (-1)*r*sin(phi-90)" x-Koordinate:NiMkIStIJWU9Kj4hIik="Formel um die y-Koordinate auszurechnen y = r*cos(phi-90)" y-Koordinate:

NiMkIisrKytdNiEiKQ=="Gradmass in Radiantmass:" "Pi*phi/180"

restart;

Erstellt die implitze ableitung: die leichung y= x+5 wird alles auf eine seite genommen und eine Gleichung erzeugt: f(x,y)=x+5-y dnach kann sie der funktion gef\374ttert werden. verfahren nach Implizitzer ableitung muss je nach fall vorgegangen werden. meist wird dnach jeduch nach y aufgel\366st und somit eine funktioin f(x) erzeugt welche y abbildet.Das Resultat stellt die Gleichung fuer die Steigung an die uebergebene Kurve dar.

implizit := proc (f) local Fx,Fy,F,l; print(Fx=D[1](f)(x,y)); print(Fy=D[2](f)(x,y)); print(F=-Fx/Fy); Fx:=D[1](f)(x,y): Fy:=D[1](f)(x,y): return -D[1](f)(x,y)/D[2](f)(x,y); end proc:

f:=(x,y)->x**2-y**2-4;

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJDkkIiIjIiIiKiQ5JUYwISIiISIlRjFGJUYlRiU=

l:=implizit(f);

NiMvSSNGeEc2IiwkSSJ4R0YlIiIjNiMvSSNGeUc2IiwkSSJ5R0YlISIjNiMvSSJGRzYiLCQqJkkjRnhHRiUiIiJJI0Z5R0YlISIiRis=NiM+SSJsRzYiKiZJInhHRiUiIiJJInlHRiUhIiI=

l;

NiMqJkkieEc2IiIiIkkieUdGJSEiIg==

Folgendes Beispiel ist aus der UEbung 5 die Nummer 5f := (x,y) -> x^3 - 3*x^2 + 4*y^4 -4;

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCoqJDkkIiIkIiIiKiRGLyIiIyEiJCokOSUiIiVGNyEiJUYxRiVGJUYl

l := implizit(f);

NiMvSSNGeEc2IiwmKiRJInhHRiUiIiMiIiRGKCEiJw==NiMvSSNGeUc2IiwkKiRJInlHRiUiIiQiIzs=NiMvSSJGRzYiLCQqJkkjRnhHRiUiIiJJI0Z5R0YlISIiRis=NiM+SSJsRzYiLCQqJiwmKiRJInhHRiUiIiMiIiRGKiEiJyIiIkkieUdGJSEiJCMhIiIiIzs=

restart:

Mit Hilfe dieser Funktion laesst sich die Tangentialebene an eine gegebene 2-D-Kurve berechnen.als Parameter wird folgendes \374bergeben: 1. Parameter: f Die Funktion abhaenig von zwei Variablen (x,y). Die Definition der Funktion muss wie folgt lauten f := (x,y) -> x^2+y^2; ACHTUNG ! : Es wird im Funktionsaufruf nur f und NICHT f(x,y) uebergeben!!! ACHTUNG ! : Die Variablen in der Funktion muessen x und y heissen!!! 2. Parameter: x0 x-Wert, an welchem die Tangentialebene gebildet werden soll. 3. Parameter: y0 y-Wert, an welchem die Tangentialebene gebildet werden soll. 4. Parameter: X X-Wert fuer die Plottaufbereitung ( x = -X .. X ). 5. Parameter: Y Y-Wert fuer die Plottaufbereitung ( y = -Y .. Y ).

tangential_Ebene := proc( f, x0, y0, X, Y ) local t, p1, p2, fx, fy: fx := D[1] (f); fy := D[2] (f); t := (x,y) -> f(x0,y0) + fx(x0,y0) * (x-x0) + fy(x0,y0) * (y-y0); p1 := plot3d (f(x,y), x=-X..X, y=-Y..Y, axes=boxed): p2 := plot3d (t(x,y), x=-X..X, y=-Y..Y, style=patchnogrid): with (plots, display): printf("\n\n\n\t\t\tDie Ableitung nach x lautet:"); print(fx(x,y)); printf("\n\n\n\t\t\tDie Ableitung nach y lautet:"); print(fy(x,y)); printf("\n\n\n\t\t\tDie Tangentialebenengleichung lautet:\n"); print (z = t(x,y)); printf("\n\n\n\t\t\tDie Tangentialebenengleichung genau lautet:\n"); print (z = evalf(t(x,y))); printf("\n\n\n\t\t\tDie Funktion und die Ebene geplottet sehen wie folgt aus:\n"); print (display ([p1,p2])); end proc:

f:= (x,y) -> exp(-(x^2+y^2)):tangential_Ebene ( f, 0.15, 0.15, 2, 2 );

Die Ableitung nach x lautet:NiMsJComSSJ4RzYiIiIiLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2IywmKiRGJSIiIyEiIiokSSJ5R0YmRjBGMUYnISIj Die Ableitung nach y lautet:NiMsJComSSJ5RzYiIiIiLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2IywmKiRJInhHRiYiIiMhIiIqJEYlRjFGMkYnISIj Die Tangentialebenengleichung lautet: NiMvSSJ6RzYiLCgkIiticy5VNSEiKiIiIkkieEdGJSQhK1lDKnonRyEjNUkieUdGJUYs Die Tangentialebenengleichung genau lautet: NiMvSSJ6RzYiLCgkIiticy5VNSEiKiIiIkkieEdGJSQhK1lDKnonRyEjNUkieUdGJUYs Die Funktion und die Ebene geplottet sehen wie folgt aus:

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z := (x,y) -> sin(x)*cos(y);

NiM+SSJ6RzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYtSSRzaW5HNiRJKnByb3RlY3RlZEdGMUkoX3N5c2xpYkdGJTYjOSQiIiItSSRjb3NHRjA2IzklRjVGJUYlRiU=

tangential_Ebene(z, 0.9*Pi, 1.1*Pi, 5, 5);

Die Ableitung nach x lautet:NiMqJi1JJGNvc0c2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiNJInhHRikiIiItRiU2I0kieUdGKUYs Die Ableitung nach y lautet:NiMsJComLUkkc2luRzYkSSpwcm90ZWN0ZWRHRihJKF9zeXNsaWJHNiI2I0kieEdGKiIiIi1GJjYjSSJ5R0YqRi0hIiI= Die Tangentialebenengleichung lautet: NiMvSSJ6RzYiLCgqJi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YrSShfc3lzbGliR0YlNiMsJEkjUGlHRiskIiIqISIiIiIiLUkkY29zR0YqNiMsJEYvJCIjNkYyRjNGMyooLUY1Ri1GM0Y0RjMsJkkieEdGJUYzRi8kISIqRjJGM0YzKihGKEYzLUYpRjZGMywmSSJ5R0YlRjNGLyQhIzZGMkYzRjI= Die Tangentialebenengleichung genau lautet: NiMvSSJ6RzYiLCgkISslNEQ4PSQhIioiIiJJInhHRiUkIit1XDNYISohIzVJInlHRiUkIitlLTpcJiohIzY= Die Funktion und die Ebene geplottet sehen wie folgt aus:

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f:=(x,y)->(x^2+5*y^2)*exp(-y^2);

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plot3d(f(x,y),x=-100..100, y=-100..100);

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tangential_Ebene ( f, 0.0, 0.0, 2, 2 );

Die Ableitung nach x lautet:NiMsJComSSJ4RzYiIiIiLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2IywkKiRJInlHRiYiIiMhIiJGJ0Yx Die Ableitung nach y lautet:NiMsJiomSSJ5RzYiIiIiLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2IywkKiRGJSIiIyEiIkYnIiM1KigsJiokSSJ4R0YmRjBGJ0YvIiImRidGJUYnRihGJyEiIw== Die Tangentialebenengleichung lautet: NiMvSSJ6RzYiJCIiIUYn Die Tangentialebenengleichung genau lautet: NiMvSSJ6RzYiJCIiIUYn Die Funktion und die Ebene geplottet sehen wie folgt aus:

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f:=(x,y)->(sin(x) * cos(y));

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYtSSRzaW5HRiU2IzkkIiIiLUkkY29zR0YlNiM5JUYyRiVGJUYl

plot3d(f(x,y),x=-10..10, y=-10..10);

-%'PLOT3DG6%-%%GRIDG6%;$!#5""!$"#5F+F(X,I)anythingGI*protectedGF06"F1[gl'!%"!!#\bm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34598DBFE024ED23A07AB4BFC82349F307A3B03FD00ED2BA72A79E3FE0D4FD734B25E53FDD36D8F55D3CE1-%+AXESLABELSG6%Q"xF1Q"yF1Q!F1-%%FONTG6$%*HELVETICAGF-

tangential_Ebene ( f, 0.9*Pi, 1.1*Pi, 10, 10 );

Die Ableitung nach x lautet:NiMqJi1JJGNvc0c2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiNJInhHRikiIiItRiU2I0kieUdGKUYs Die Ableitung nach y lautet:NiMsJComLUkkc2luRzYkSSpwcm90ZWN0ZWRHRihJKF9zeXNsaWJHNiI2I0kieEdGKiIiIi1GJjYjSSJ5R0YqRi0hIiI= Die Tangentialebenengleichung lautet: NiMvSSJ6RzYiLCgqJi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YrSShfc3lzbGliR0YlNiMsJEkjUGlHRiskIiIqISIiIiIiLUkkY29zR0YqNiMsJEYvJCIjNkYyRjNGMyooLUY1Ri1GM0Y0RjMsJkkieEdGJUYzRi8kISIqRjJGM0YzKihGKEYzLUYpRjZGMywmSSJ5R0YlRjNGLyQhIzZGMkYzRjI= Die Tangentialebenengleichung genau lautet: NiMvSSJ6RzYiLCgkISslNEQ4PSQhIioiIiJJInhHRiUkIit1XDNYISohIzVJInlHRiUkIitlLTpcJiohIzY= Die Funktion und die Ebene geplottet sehen wie folgt aus:

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restart:

Mit Hilfe dieser Funktion laesst sich das totale Differential einer Funktion berechnen.als Parameter wird folgendes \374bergeben: 1. Parameter: f Die Funktion abhaenig von mehreren Variablen. - Wird die Funktion f := (x,y) -> x^2+y^2; auf diese Weise definiert, wird im Funktionsaufruf f(x,y) uebergeben. - Wird die Funktion f := x^2+y^2: auf diese Weise definiert, wird im Funktionsaufruf f uebergeben. 2. Parameter: VAR Ist eine Liste von Variablen, die in der Funktion vorkommen. ACHTUNG ! : Definition mit eckigen Klammern ( [] ) !!!

totales_Differential := proc ( f, VAR ) local n, i, var, df: n := nops (VAR): for i from 1 to n do var[i] := args[2][i]: od: df := sum ('diff(f, var[i]) * cat(d, var[i])', 'i' = 1..n); end proc:

f(x,y):=x*ln(x+y);

NiM+LUkiZkc2IjYkSSJ4R0YmSSJ5R0YmKiZGKCIiIi1JI2xuRzYkSSpwcm90ZWN0ZWRHRi9JKF9zeXNsaWJHRiY2IywmRihGK0YpRitGKw==

df:=totales_Differential(f(x,y), [x,y]);

NiM+SSNkZkc2IiwmKiYsJi1JI2xuRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiU2IywmSSJ4R0YlIiIiSSJ5R0YlRjFGMSomRjBGMUYvISIiRjFGMUkjZHhHRiVGMUYxKihGMEYxRi9GNEkjZHlHRiVGMUYx

u:=(p,r,t)->r**2*exp(-t**2)*(1-(p/(1+p)**2));

NiM+SSJ1RzYiZio2JUkicEdGJUkickdGJUkidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKig5JSIiIy1JJGV4cEdGJTYjLCQqJDkmRjAhIiIiIiIsJkY4RjgqJjkkRjgsJkY4RjhGO0Y4ISIjRjdGOEYlRiVGJQ==

df:=totales_Differential(u(p,r,t), [p,r,t]);

NiM+SSNkZkc2IiwoKipJInJHRiUiIiMtSSRleHBHNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJTYjLCQqJEkidEdGJUYpISIiIiIiLCYqJCwmRjRGNEkicEdGJUY0ISIjRjMqJkY4RjRGNyEiJEYpRjRJI2RwR0YlRjRGNCoqRihGNEYqRjQsJkY0RjQqJkY4RjRGN0Y5RjNGNEkjZHJHRiVGNEYpKixGKEYpRjJGNEYqRjRGPkY0SSNkdEdGJUY0Rjk=

x:=(r,t)->r**2*(1+cos(t)**2);

NiM+SSJ4RzYiZio2JEkickdGJUkidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiIywmIiIiRjEqJC1JJGNvc0dGJTYjOSVGL0YxRjFGJUYlRiU=

y:=(r,t)->r**(-1)*(1-sin(t)**2);

NiM+SSJ5RzYiZio2JEkickdGJUkidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCEiIiwmIiIiRjEqJC1JJHNpbkdGJTYjOSUiIiNGL0YxRiVGJUYl

z:=x(r,t)+y(r,t);

NiM+SSJ6RzYiLCYqJkkickdGJSIiIywmIiIiRisqJC1JJGNvc0c2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiNJInRHRiVGKUYrRitGKyomRighIiIsJkYrRisqJC1JJHNpbkdGL0YyRilGNUYrRis=

df:=totales_Differential(z,[r,t]);

NiM+SSNkZkc2IiwmKiYsJiomSSJyR0YlIiIiLCZGK0YrKiQtSSRjb3NHNiRJKnByb3RlY3RlZEdGMUkoX3N5c2xpYkdGJTYjSSJ0R0YlIiIjRitGK0Y1KiZGKiEiIywmRitGKyokLUkkc2luR0YwRjNGNSEiIkYrRjxGK0kjZHJHRiVGK0YrKiYsJiooRipGNUYuRitGOkYrRjcqKEYqRjxGLkYrRjpGK0Y3RitJI2R0R0YlRitGKw==

restart:

Mit Hilfe dieser Funktion lassen sich die Extremalwerte einer Funktion berechnen.als Parameter wird folgendes \374bergeben: 1. Parameter: f Die Funktion abhaenig von zwei Variablen. Die Definition der Funktion muss wie folgt lauten f := (x,y) -> x^2+y^2; ACHTUNG ! : Es wird im Funktionsaufruf nur f und NICHT f(x,y) uebergeben!!! 2. Parameter: VAR Ist eine Liste von Variablen, die in der Funktion vorkommen. ACHTUNG ! : Definition mit eckigen Klammern ( [] ) !!!

extremalWerte := proc ( f , VAR ) local fx, fy, stellen, i, Delta, X, Y: fx := D[1](f): fy := D[2](f): stellen := sort (simplify (remove (has,[evalf(solve ( {fx(VAR[1],VAR[2]) = 0, fy(VAR[1],VAR[2]) = 0}, {VAR[1],VAR[2]} ))],I))); printf("\t\t\tEs gibt folgende moeglichen Extremwerte: %a\n\n\n\n", stellen); Delta:=(x0,y0)->D[1](fx)(x0,y0)*D[2](fy)(x0,y0)-D[2](fx)(x0,y0)^2; for i from 1 to nops(stellen) do X := 1: Y := 2: if op(1,stellen[i][1]) <> VAR[1] then X := 2: Y := 1: end if; #printf("X = %a: ", X); #printf("Y = %a: ", Y); printf("\t\t\tIm folgenden wird der Punkt [ %a , %a ] betrachtet\n", op(2,stellen[i][X]), op(2,stellen[i][Y])); printf("\t\t\tDas Delta an diesem Punkt lautet: %a\n", Delta(op(2,stellen[i][X]),op(2,stellen[i][Y]))); #printf("XXX = %a: ", op(2,stellen[i][1])); #printf("YYY = %a: \n\n\n", op(2,stellen[i][2])); if Delta(op(2,stellen[i][X]),op(2,stellen[i][Y])) > 0 then printf("\t\t\tDie zweite Ableitung nach der ersten Variable lautet: %a\n", D[1](fx)(op(2,stellen[i][X]),op(2,stellen[i][Y]))); if D[1](fx)(op(2,stellen[i][X]),op(2,stellen[i][Y])) > 0 then printf("\t\t\tDer Punkt [ %a , %a ] ist ein lokales Minimum\n\n\n", op(2,stellen[i][X]), op(2,stellen[i][Y])); end if; if D[1](fx)(op(2,stellen[i][X]),op(2,stellen[i][Y])) < 0 then printf("\t\t\tDer Punkt [ %a , %a ] ist ein lokales Maximum\n\n\n", op(2,stellen[i][X]), op(2,stellen[i][Y])); end if: print(plot3d([f(x,y)],x=op(2,stellen[i][X])-10..op(2,stellen[i][X])+10, y=op(2,stellen[i][Y])-10..10, axes=boxed)); end if; if Delta(op(2,stellen[i][X]),op(2,stellen[i][Y])) < 0 then printf("\t\t\tDer Punkt [ %a , %a ] ist kein Extremalwert\n\n\n", op(2,stellen[i][X]), op(2,stellen[i][Y])); end if; if Delta(op(2,stellen[i][X]),op(2,stellen[i][Y])) = 0 then printf("\t\t\tUeber den Punkt [ %a , %a ] kann mit diesem Verfahren keine Aussage gemacht werden\n\n\n", op(2,stellen[i][X]), op(2,stellen[i][Y])); end if; end do; end proc:

f:=(x,y)->3*x*y**2+4*x**3-3*y**2-12*x**2+1;

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCwqJjkkIiIiOSUiIiMiIiQqJEYvRjMiIiUqJEYxRjIhIiQqJEYvRjIhIzdGMEYwRiVGJUYl

extremalWerte(f,[x,y]);

Es gibt folgende moeglichen Extremwerte: [{y = 0., x = 0.}, {y = 0., x = 2.}, {x = 1., y = 2.}, {y = -2., x = 1.}] Im folgenden wird der Punkt [ 0. , 0. ] betrachtet Das Delta an diesem Punkt lautet: 144. Die zweite Ableitung nach der ersten Variable lautet: -24. Der Punkt [ 0. , 0. ] ist ein lokales Maximum

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Im folgenden wird der Punkt [ 2. , 0. ] betrachtet Das Delta an diesem Punkt lautet: 144. Die zweite Ableitung nach der ersten Variable lautet: 24. Der Punkt [ 2. , 0. ] ist ein lokales Minimum

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Im folgenden wird der Punkt [ 1. , 2. ] betrachtet Das Delta an diesem Punkt lautet: -144. Der Punkt [ 1. , 2. ] ist kein Extremalwert Im folgenden wird der Punkt [ 1. , -2. ] betrachtet Das Delta an diesem Punkt lautet: -144. Der Punkt [ 1. , -2. ] ist kein Extremalwert

f:=(x,y)->1+x^2+y^2;

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgiIiJGLiokOSQiIiNGLiokOSVGMUYuRiVGJUYl

extremalWerte(f,[x,y]);

Es gibt folgende moeglichen Extremwerte: [{y = 0., x = 0.}] Im folgenden wird der Punkt [ 0. , 0. ] betrachtet Das Delta an diesem Punkt lautet: 4 Die zweite Ableitung nach der ersten Variable lautet: 2 Der Punkt [ 0. , 0. ] ist ein lokales Minimum

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

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extremalWerte(f,[x,y]);

Es gibt folgende moeglichen Extremwerte: [{y = 0., x = 0.}] Im folgenden wird der Punkt [ 0. , 0. ] betrachtet Das Delta an diesem Punkt lautet: 4. Die zweite Ableitung nach der ersten Variable lautet: -2. Der Punkt [ 0. , 0. ] ist ein lokales Maximum

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

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDkkIiIjIiIiKiQ5JUYwISIiRiVGJUYl

extremalWerte(f,[x,y]);

Es gibt folgende moeglichen Extremwerte: [{y = 0., x = 0.}] Im folgenden wird der Punkt [ 0. , 0. ] betrachtet Das Delta an diesem Punkt lautet: -4 Der Punkt [ 0. , 0. ] ist kein Extremalwert

f:=(x,y)->3*x*y-x^3-y^3;

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJjkkIiIiOSVGMCIiJCokRi9GMiEiIiokRjFGMkY0RiVGJUYl

extremalWerte(f,[x,y]);

Es gibt folgende moeglichen Extremwerte: [{y = 0., x = 0.}, {y = 1., x = 1.}] Im folgenden wird der Punkt [ 0. , 0. ] betrachtet Das Delta an diesem Punkt lautet: -9. Der Punkt [ 0. , 0. ] ist kein Extremalwert Im folgenden wird der Punkt [ 1. , 1. ] betrachtet Das Delta an diesem Punkt lautet: 27. Die zweite Ableitung nach der ersten Variable lautet: -6. Der Punkt [ 1. , 1. ] ist ein lokales Maximum

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u:=(s,t)->s*t-27*(1/s-1/t);

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extremalWerte(u,[s,t]);

Es gibt folgende moeglichen Extremwerte: [{t = -3., s = 3.}] Im folgenden wird der Punkt [ 3. , -3. ] betrachtet Das Delta an diesem Punkt lautet: 3.000000000 Die zweite Ableitung nach der ersten Variable lautet: -2.000000000 Der Punkt [ 3. , -3. ] ist ein lokales Maximum

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h:=(x,y)->(x^2+y^2)*exp(-x);

NiM+SSJoRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYsJiokOSQiIiMiIiIqJDklRjFGMkYyLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRjhJKF9zeXNsaWJHRiU2IywkRjAhIiJGMkYlRiVGJQ==

extremalWerte(h,[x,y]);

Es gibt folgende moeglichen Extremwerte: [{x = 2., y = 0.}, {y = 0., x = 0.}] Im folgenden wird der Punkt [ 2. , 0. ] betrachtet Das Delta an diesem Punkt lautet: -.7326255562e-1 Der Punkt [ 2. , 0. ] ist kein Extremalwert Im folgenden wird der Punkt [ 0. , 0. ] betrachtet Das Delta an diesem Punkt lautet: 4. Die zweite Ableitung nach der ersten Variable lautet: 2. Der Punkt [ 0. , 0. ] ist ein lokales Minimum

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f:=(x,y)->(x^2+5*y^2)*exp(-y^2);

NiM+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYsJiokOSQiIiMiIiIqJDklRjEiIiZGMi1JJGV4cEdGJTYjLCRGMyEiIkYyRiVGJUYl

fx:=D[1](f);

NiM+SSNmeEc2ImYqNiRJInhHRiVJInlHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwkKiY5JCIiIi1JJGV4cEc2JEkqcHJvdGVjdGVkR0Y0SShfc3lzbGliR0YlNiMsJCokOSUiIiMhIiJGMEY6RiVGJUYl

fy:=D[2](f);

NiM+SSNmeUc2ImYqNiRJInhHRiVJInlHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmKiY5JSIiIi1JJGV4cEc2JEkqcHJvdGVjdGVkR0Y0SShfc3lzbGliR0YlNiMsJCokRi8iIiMhIiJGMCIjNSooLCYqJDkkRjlGMEY4IiImRjBGL0YwRjFGMCEiI0YlRiVGJQ==

extremalWerte(f,[x,y]);

Es gibt folgende moeglichen Extremwerte: [{y = 0., x = 0.}, {x = 0., y = 1.}, {y = -1., x = 0.}] Im folgenden wird der Punkt [ 0. , 0. ] betrachtet Das Delta an diesem Punkt lautet: 20. Die zweite Ableitung nach der ersten Variable lautet: 2. Der Punkt [ 0. , 0. ] ist ein lokales Minimum

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Im folgenden wird der Punkt [ 0. , 1. ] betrachtet Das Delta an diesem Punkt lautet: -5.413411332 Der Punkt [ 0. , 1. ] ist kein Extremalwert Im folgenden wird der Punkt [ 0. , -1. ] betrachtet Das Delta an diesem Punkt lautet: -5.413411332 Der Punkt [ 0. , -1. ] ist kein Extremalwert