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0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle27" -1 226 1 {CSTYLE "" -1 -1 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle28" -1 227 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle14" -1 228 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle29" -1 228 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 220 "" 0 "" {TEXT 215 24 "Maple-Kurs 10. 11. 200 6" }{TEXT 215 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 24 "Stand: 11. November 2006" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{SECT 1 {PARA 222 "" 0 "" {TEXT 213 1 " \+ " }{TEXT 226 34 "Funktionen in zwei Ver\344nderlichen " }{TEXT 201 0 " " }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 " " }}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 223 "" 0 "" {TEXT 214 20 "Die Funktionsschar " }{XPPEDIT 18 0 "f := x^3-a*x" "6#>%\"fG, &*$%\"xG\"\"$\"\"\"*&%\"aGF)F'F)!\"\"" }{TEXT 214 7 " mit " } {XPPEDIT 18 0 "a=0..5" "6#/%\"aG;\"\"!\"\"&" }{TEXT 214 51 " des 1. T reffens benutzen wir nun als \"Aufh\344nger\"," }{TEXT 200 0 "" }} {PARA 221 "" 0 "" {TEXT 214 59 "uns mit Funktionen von zwei Ver\344nde rlichen zu besch\344ftigen." }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 96 "Wir definieren die Funktion sschar als Funktion der beiden Ver\344nderlichen x und a mittels d er" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 94 "\"Pfeilnotation \" und zeichnen sie mit Hilfe des Maple-Befehls 'plot3d' in dem bekann ten Bereich" }{TEXT 214 0 "" }}{PARA 223 "" 0 "" {XPPEDIT 18 0 "x=-2.2 5..2.25" "6#/%\"xG;,$-%&FloatG6$\"$D#!\"#!\"\"F'" }{TEXT 214 2 ", " } {XPPEDIT 18 0 "a=0..5" "6#/%\"aG;\"\"!\"\"&" }{TEXT 214 2 " :" }{TEXT 200 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 44 "f := (x,a) -> x^3-a*x; #x=-2.25..2.25,a=0..5" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 36 "plot3 d(f(x,a),x=-2.25..2.25,a=0..5);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 101 "Wir stel len fest: Maple zeichnet dreidimensionale Graphiken grunds\344tzlich O HNE Achsenkreuz. Wenn wir " }{TEXT 214 0 "" }{TEXT 214 105 "\nein solc hes w\374nschen, m\374ssen wir dies explizit \"sagen\". Dabei haben wi r folgende drei Wahlm\366glichkeiten:" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 48 "plo t3d(f(x,a),x=-2.25..2.25,a=0..5,axes=normal);" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 61 "plot3d(f(x,a),x=-2.25..2.2 5,a=0..5,axes=frame); # axes=framed" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 58 "plot3d(f(x,a),x=-2.25..2.25,a=0.. 5,axes=box); # axes=boxed" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 99 "Standardm\344\337 ig wir die Fl\344che durch ein achsenparalleles Liniengitter \374berzo gen. Wir erkennen ferner" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 95 "die \"Parameterachse\" (mit a beschriftet) sowie die x-Achse . Mittels einer Animation --- deren" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 92 "Einzelheiten Sie noch nicht zu verstehen brauchen --- versuchen wir, die Kurvenschar auf der" }{TEXT 214 0 "" }}{PARA 221 " " 0 "" {TEXT 214 24 "Fl\344che wiederzuerkennen:" }{TEXT 214 0 "" }} {PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 61 "Animation := seq(display(plot3d(f(x,a),x=-2.25..2.25, a=0..5)," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 68 "\n plot3d([x,0.5*k,z], x=-2.25..2.25,z=-10..10),axes=frame),k=0..10):" }{MPLTEXT 1 0 0 "" } {MPLTEXT 1 0 36 "\ndisplay(Animation,insequence=true);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 96 "Im Hinblick auf Extremwertaufgaben f\374r Funktionen von zwei Ver\344nderlichen ist es zweckm\344\337ig, die" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 96 "Fl\344che mit H\366henlinien (an Ste lle des Liniengitters) zu versehen. Dies geht mit Hilfe der Option" } {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 21 "'style=patchcontour': " }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 64 "plot3d(f(x,a),x=-2.25..2.25,a=0..5,styl e=patchcontour,axes=box);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 106 "Will man die sta ndardm\344\337ig gezeichnete Anzahl von H\366henlinien auf 30 ver\34 4ndern, so kann man dies mittels" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 37 "der Option 'contours=30' erreichen:" }{TEXT 214 0 "" } }{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 76 "plot3d(f(x,a),x=-2.25..2.25,a=0..5,style=patchcontour ,contours=30,axes=box);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 " " {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 94 "Will man die Blickr ichtung (dauerhaft) \344ndern, so geht dies mittels der Option 'orien tation':" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}} {EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 75 "plot3d(f(x,a),x=-2.25..2.2 5,a=0..5,style=patchcontour,contours=30,axes=box," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 25 "\n orientation=[135,45]);" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 101 "Mittels der Option 'orientation=[-90,0]' k\366nnen wir die Fl\3 44che aus der Vogelperspektive betrachten:" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 75 "plot3d(f(x,a),x=-2.25..2.25,a=0..5,style=patchcontour,contours=30, axes=box," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 24 "\n orientation=[-90,0 ]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }} {PARA 221 "" 0 "" {TEXT 214 46 "Eine \344hnliche Wirkung hat der folge nde Befehl:" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}} {EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 53 "contourplot(f(x,a),x=-2.25 ..2.25,a=0..5,contours=30);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 87 "Will man die Be reiche zwischen den H\366henlinien f\344rben, mu\337 man 'filled=true ' eingeben:" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}} {EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 65 "contourplot(f(x,a),x=-2.25 ..2.25,a=0..5,contours=30,filled=true);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 100 "Kl icken Sie mal die mit 'contourplot' erzeugten Graphiken an. Stellen S ie (bis auf die Gr\366\337e) einen" }{TEXT 214 0 "" }}{PARA 221 "" 0 " " {TEXT 214 64 "Unterschied fest zu der mit 'plot3d' erzeugten Graphik ? Welchen?" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }} {PARA 221 "" 0 "" {TEXT 214 106 "Wir kehren zu 'plot3d' zur\374ck un d versuchen, die F\344rbung der Fl\344che zu ver\344ndern. Nat\374rlic h verf\374gen wir" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 36 "a uch hier \374ber die Option 'color' :" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 77 "pl ot3d(f(x,a),x=-2.25..2.25,a=0..5,color=cyan,axes=box,orientation=[135, 45]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" } }{PARA 221 "" 0 "" {TEXT 214 72 "Oder gef\344llt Ihnen die folgende F \344rbung mittels 'shading=zhue' besser?" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 79 "plot3d(f(x,a),x=-2.25..2.25,a=0..5,shading=zhue,axes=box,orientati on=[135,45]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 56 "Jetzt bringen wir \"unsere \" H\366henlinien wieder ins Spiel:" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 57 "plot3 d(f(x,a),x=-2.25..2.25,a=0..5,shading=zhue,axes=box," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 65 "\n style=patchcontour,contours=30,orientation=[1 35,45],axes=box);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 104 "Falls Sie f\374r die Ausgabe einer Graphik nur \374ber einen Schwarz-Wei\337-Drucker verf \374gen, empfiehlt sich die" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 98 "Option 'shading=zgrayscale' , welche die Fl\344che in A bh\344ngigkeit vom z-Wert durch Grau-Abstufungen" }{TEXT 214 0 "" }} {PARA 221 "" 0 "" {TEXT 214 6 "f\344rbt:" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 63 " plot3d(f(x,a),x=-2.25..2.25,a=0..5,shading=zgrayscale,axes=box," } {MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 56 "\n style=patchcontour,contours=20 ,orientation=[135,45]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 " " {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 107 "Manchmal m\366chte man eine Graphik wie ein Drahtmodell \"transparent\" gestalten --- et wa um dahinter liegende " }{TEXT 214 0 "" }{TEXT 214 77 "\nObjekte seh en zu k\366nnen. Dann empfiehlt sich die Option 'style=wireframe' :" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 62 "plot3d(f(x,a),x=-2.25..2.25,a=0..5,styl e=wireframe,color=gray," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 34 "\n orie ntation=[135,45],axes=box);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 223 "" 0 "" {TEXT 227 14 "\334bungsaufgab e:" }{TEXT 214 69 " F\374hren Sie die oben gemachten Experimente mit der Funktionsschar " }{XPPEDIT 18 0 "fa=x^3-(3+a)*x^2+4*a*x" "6#/%#f aG,(*$%\"xG\"\"$\"\"\"*&,&F(F)%\"aGF)F)*$F'\"\"#F)!\"\"*(\"\"%F)F,F)F' F)F)" }{TEXT 214 7 " durch" }{TEXT 200 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 222 "" 0 "" {TEXT 213 1 " " }{TEXT 226 18 "Extremwertauf gaben" }{TEXT 201 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }} {PARA 223 "" 0 "" {TEXT 214 106 "Aus einem rechteckigen Karton soll du rch Schneiden und Falten eine oben offene Schachtel mit einem Volumen" }{TEXT 214 0 "" }{TEXT 214 9 "\nvon 1 " }{XPPEDIT 18 0 "dm^3" "6#*$ %#dmG\"\"$" }{TEXT 214 104 " hergestellt werden. Sind x und y (in dm) die Grundkanten der Schachtel und h (in dm) ihre H\366he," } {TEXT 200 0 "" }}{PARA 221 "" 0 "" {TEXT 214 56 "so gilt f\374r den Fl \344cheninhalt A des ben\366tigten Kartons" }{TEXT 214 0 "" }}{PARA 223 "" 0 "" {TEXT 214 10 " " }{XPPEDIT 18 0 "A = (x+2*h)*(y+ 2*h) " "6#/%\"AG*&,&%\"xG\"\"\"*&\"\"#F(%\"hGF(F(F(,&%\"yGF(F)F(F(" } {TEXT 214 11 " " }{TEXT 200 0 "" }}{PARA 221 "" 0 "" {TEXT 214 50 "und f\374r das Volumen V der entstehenden Schachtel" }{TEXT 214 0 "" }}{PARA 223 "" 0 "" {TEXT 214 11 " " }{XPPEDIT 18 0 "V = x*y*h" "6#/%\"VG*(%\"xG\"\"\"%\"yGF'%\"hGF'" }{TEXT 214 2 " ." }{TEXT 200 0 "" }}{PARA 223 "" 0 "" {TEXT 214 11 "Setzt man " } {XPPEDIT 18 0 "h =V/(x*y)" "6#/%\"hG*&%\"VG\"\"\"*&%\"xGF'%\"yGF'!\"\" " }{TEXT 214 38 " in A ein (und ber\374cksichtigt, da\337 " } {XPPEDIT 18 0 "V=1" "6#/%\"VG\"\"\"" }{TEXT 214 21 " ist), so ergibt s ich" }{TEXT 200 0 "" }}{PARA 223 "" 0 "" {TEXT 214 11 " " } {XPPEDIT 18 0 "A = (x+2/(x*y))*(y+2/(x*y))" "6#/%\"AG*&,&%\"xG\"\"\"*& \"\"#F(*&F'F(%\"yGF(!\"\"F(F(,&F,F(F)F(F(" }{TEXT 214 2 " ." }{TEXT 200 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 29 "A := (x+2/(x*y))*(y+2/(x*y ));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 38 "f := (x,y) -> (x+2/(x*y))*(y+2/(x*y));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 7 "f(1,2);" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 52 "plot3d(f(x,y),x=0..6,y=0.. 6,view=0..16,shading=zhue," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 53 "\n s tyle=patchcontour,orientation=[-15,30],axes=box);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 53 "#plot3d(f(x,y),x=0..6,y =0..6,view=0..16,shading=zhue," }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 67 " \n# style=patchcontour,orientation=[-15,30],grid=[50,50],axes=box);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 96 "Wir schneiden die Fl\344che f\374r x=1 par allel zur yz-Ebene bzw. f\374r y=2 parallel zur xz-Ebene:" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 28 "plot(f(1,y),y=0..5,z=0..16);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 28 "plot(f(x,2),x=0..5,z=0. .16);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" } }{PARA 221 "" 0 "" {TEXT 214 72 "Solche Schnitte k\366nnen wir uns auc h mit Hilfe einer Animation anschauen:" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 48 "an imate(f(x,y),y=0..6,x=0..6,view=[0..6,0..16]);" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 48 "animate(f(x,y),x=0..6,y=0. .6,view=[0..6,0..16]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 81 "Und nun versuchen wi r, mit 'contourplot' das Funktionsverhalten zu analysieren:" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 46 "contourplot(f(x,y),x=0..6,y=0..6,contours=20);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 223 "" 0 "" {TEXT 214 81 "Dieses Bild ist wenig aussagekr\344ftig. Wir werden sehen, da\337 es nicht reicht, die " }{TEXT 227 6 "Anzahl" } {TEXT 214 27 " der H\366henlinien anzugeben." }{TEXT 200 0 "" }}{PARA 221 "" 0 "" {TEXT 214 109 "Stattdessen werden wir nun spezielle H\366h enlinien zeichnen, die durch Angabe der entsprechenden H\366hen in ein er" }{TEXT 214 0 "" }{TEXT 214 26 "\nListe beschrieben werden:" } {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 69 "contourplot(f(x,y),x=0..6,y=0..6,contours=[ 5.7,5.8,6.0,6.5,7.0,8.0]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 126 "Das ist schon w esentlich besser. Vielleicht st\366rt es Sie jedoch, da\337 die H\366h enlinien \"etwas verwackelt\" sind. Bevor wir einen " }{TEXT 214 0 "" }{TEXT 214 118 "\nweiteren Versuch unternehmen, machen wir uns klar, d a\337 die Graphik durch Auswerten der Funktion an den Punkten eines " }{TEXT 214 0 "" }{TEXT 214 114 "\nGitters entsteht. Wom\366glich war d ieses Gitter zu grob. Mit der Option 'grid' kann man die Voreinstellu ng \344ndern:" }{TEXT 214 0 "" }{TEXT 214 1 "\n" }}}{EXCHG {PARA 224 " > " 0 "" {MPLTEXT 1 0 82 "contourplot(f(x,y),x=0..6,y=0..6,contours=[5 .7,5.8,6.0,6.5,7.0,8.0],grid=[60,60]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 100 "Wi r ermitteln aus obiger Graphik eine N\344herung f\374r die Minimumstel le: Nach Anklicken k\366nnen wir etwa" }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 33 "den Wert [1.55, 1.66] ablesen. " }{TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 0 "" }}{PARA 221 "" 0 "" {TEXT 214 46 "U nd nun berechnen wir die Minimumstelle exakt:" }{TEXT 214 0 "" }} {PARA 221 "" 0 "" {TEXT 214 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 21 "fx := diff(f(x,y),x);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 22 "\nfy := diff(f(x,y),y);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 33 "solve(\{fx=0,fy=0,x>0,y>0\},\{x,y\});" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 33 "x := 4^(1/3): `x = y` = evalf(x);" }{MPLTEXT 1 0 0 "" }{MPLTEXT 1 0 26 "\n 'f'(x,x) = evalf(f(x,x));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{PARA 225 "" 0 "" {TEXT 208 0 "" }}{PARA 226 "" 0 "" {TEXT 218 0 "" }}{PARA 227 "" 0 "" {TEXT 228 0 "" }}{PARA 228 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }