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}{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{SECT 1 {PARA 258 "" 0 "" {TEXT 262 12 "Ebene Kurven" }{TEXT 262 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 259 "" 0 " " {TEXT 264 52 "Kurvenplots ebener Kurven erh\344lt man mit dem Befehl " }{TEXT 346 4 "plot" }{TEXT 264 44 ". Zum Beispiel erh\344lt man den Kreis mit der " }{TEXT 264 0 "" }{TEXT 264 22 "\nParameterdarstellung " }{XPPEDIT 344 0 "x(t) = (cos(t), sin(t));" "6#/-%\"xG6#%\"tG6$-%$co sGF&-%$sinGF&" }{TEXT 347 1 " " }{TEXT 264 1 "," }{TEXT 346 4 " t " } {XPPEDIT 342 0 "epsilon;" "6#%(epsilonG" }{TEXT 347 1 " " }{TEXT 346 6 " [0; 2" }{XPPEDIT 337 0 "Pi;" "6#%#PiG" }{TEXT 347 1 " " }{TEXT 346 1 "]" }{TEXT 264 7 ", durch" }{TEXT 316 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 32 "plot([c os(t),sin(t),t=0..2*Pi]);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 107 "Man beachte in der B efehlssyntax die Stellung der eckigen Klammern! Durch klicken mit der \+ linken Maustaste " }{TEXT 264 0 "" }{TEXT 264 109 "\nauf die Graphik v er\344ndern sich die Men\374leiste und die Symbolleisten und er\366ffn en M\366glichkeiten zur \304nderung " }{TEXT 264 0 "" }{TEXT 264 20 " \nder Plot-Optionen. " }{TEXT 264 0 "" }}{PARA 259 "" 0 "" {TEXT 264 116 "Der obige Plot \344hnelt mehr einer Ellipse als einem Kreis, da M aple standardm\344\337ig die Koordinatenachsen so skaliert, " }{TEXT 264 0 "" }{TEXT 264 98 "\ndass sie dem Fenster angepasst sind. Die Ska lierung kann man mit Hilfe der Men\374s oder der Option " }{TEXT 346 7 "scaling" }{TEXT 264 8 " \344ndern:" }{TEXT 316 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 52 "pl ot([cos(t),sin(t),t=0..2*Pi],scaling=constrained);" }{MPLTEXT 1 259 0 "" }}}{PARA 261 "" 0 "" {TEXT 348 0 "" }}{PARA 262 "" 0 "" {TEXT 258 8 "Aufgabe:" }{TEXT 258 0 "" }}{PARA 263 "" 0 "" {TEXT 257 109 "Erstel le (durch Ab\344nderung obiger Befehlszeile) einen Kurvenplot der Arch imedischen Spirale, die gegeben ist " }{TEXT 257 0 "" }}{PARA 264 "" 0 "" {TEXT 349 31 "durch die Parameterdarstellung " }{XPPEDIT 18 0 "x(t ) = (t*cos(t), t*sin(t));" "6#/-%\"xG6#%\"tG6$*&F'\"\"\"-%$cosGF&F**&F 'F*-%$sinGF&F*" }{TEXT 350 1 " " }{TEXT 349 3 ", " }{TEXT 252 1 "t" } {TEXT 349 1 " " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT 350 1 " " }{TEXT 349 1 " " }{TEXT 252 2 "[0" }{TEXT 349 2 "; " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT 350 1 " " }{TEXT 252 4 "[ . " } {TEXT 349 45 "Variiere f\374r den Plot das Parameterintervall." } {TEXT 315 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 265 "" 0 "" {TEXT 261 7 "L\366sung:" }{TEXT 261 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 56 "plot([t*cos(t ),t*sin(t),t=0..4*Pi],scaling=constrained);" }{MPLTEXT 1 259 0 "" }}} {PARA 266 "" 0 "" {TEXT 351 0 "" }}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}}{SECT 0 {PARA 267 "" 0 "" {TEXT 251 10 "Raumkurven" } {TEXT 251 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 268 "" 0 "" {TEXT 352 3 "F\374r" }{TEXT 353 1 " " }{TEXT 352 56 "Kurvenplots von R aumkurven ist zun\344chst das Maple-Paket " }{TEXT 354 6 "plots " } {TEXT 352 9 "zu laden:" }{TEXT 352 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 12 "with(plots):" } {MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 259 "" 0 "" {TEXT 264 79 "F\374r die Darstellung einer Raumkurve gibt es zwe i M\366glichkeiten: Mit dem Befehl " }{TEXT 346 10 "spacecurve" } {TEXT 264 19 " wird die Kurve in " }{TEXT 264 0 "" }{TEXT 264 38 "\n\3 74blicher Weise als Linie dargestellt:" }{TEXT 316 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 62 "s pacecurve([cos(t),sin(t),t/5],t=0..4*Pi,scaling=constrained);" } {MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 110 "Durch Anklicken des Plots und Ziehen mit der link en Maustaste kann das Schaubild gedreht werden.(Teste dies!) " }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 107 "Weitere Einstellungen k\366nnen in den Men\374- und Symbolleisten ver\344ndert werden, z. B. das Einzeichnen eines " } {TEXT 264 0 "" }{TEXT 264 29 "\nAchsenkreuzes. (Teste dies!)" }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 259 "" 0 "" {TEXT 264 80 "Einen noch besseren r\344umlichen Eindruck von der Kurve erh\3 44lt man mit dem Befehl " }{TEXT 346 8 "tubeplot" }{TEXT 264 30 ", der nicht die Kurve selbst, " }{TEXT 264 0 "" }{TEXT 264 108 "\nsondern e ine R\366hre um diese Kurve zeichnet, wobei der Radius dieser R\366hre als Option angegeben werden kann:" }{TEXT 316 0 "" }}{PARA 254 "" 0 " " {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 43 "tubep lot([2*cos(t),2*sin(t),t/5],t=0..4*Pi," }{MPLTEXT 1 259 0 "" } {MPLTEXT 1 259 35 "\n scaling=constrained,radius=0.2);" }{MPLTEXT 1 259 0 "" }}}{PARA 269 "" 0 "" {TEXT 355 0 "" }}{PARA 270 "" 0 "" {TEXT 250 9 "Aufgaben:" }{TEXT 250 0 "" }}{PARA 259 "" 0 "" {TEXT 264 104 "1. Zeichne durch \304nderung des obigen Befehls eine Windung eine r Schraubenlinie mit Zylinderradius 3 und " }{TEXT 264 0 "" }{TEXT 264 32 "\nGangh\366he 4 (Beachte: Gangh\366he=2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 356 1 " " }{TEXT 264 5 "b) . " }{TEXT 316 0 "" }} {PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 265 "" 0 "" {TEXT 261 7 "L\36 6sung:" }{TEXT 261 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 46 "tubeplot([3*cos(t),3*sin(t),2/P i*t],t=0..2*Pi," }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 35 "\n scaling= constrained,radius=0.2);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 101 "2. \304ndere nun den Befehl so ab, dass nicht die x3-Achse, sondern die x2-Achse die Schra ubenachse ist." }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }} {EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}{PARA 271 "" 0 "" {TEXT 248 7 "L\366sung:" }{TEXT 248 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 46 "tubeplot([2*cos(t),2/Pi*t,3*sin(t)],t=0..2*Pi," } {MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 35 "\n scaling=constrained,radius =0.2);" }{MPLTEXT 1 259 0 "" }}}{PARA 261 "" 0 "" {TEXT 348 0 "" }} {PARA 256 "" 0 "" {TEXT 264 105 "Durch Verwendung geschweifter Klammer n k\366nnen auch mehrere Kurven gleichzeitig dargestellt werden, z. B. :" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 71 "tubeplot(\{[cos(t),sin(t),0],[0,sin(t)-1,cos(t)]\}, t=0..2*Pi,radius=1/6);" }{MPLTEXT 1 259 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}}} {SECT 1 {PARA 257 "" 0 "" {TEXT 263 33 " Berechnung der L\344nge einer Kurve" }{TEXT 263 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{SECT 0 {PARA 267 "" 0 "" {TEXT 251 25 "Berechnung mit Integralen" }{TEXT 251 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 272 "" 0 "" {TEXT 247 32 "Beispiel: Ein Bogen der Zykloide" }{TEXT 247 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 8 "rest art;" }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 98 "\nZun\344chst wird die Parameterdarstellung dieser Kurve mit d em Funktionskonzept von Maple definiert:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 20 "x1 := t -> t-si n(t);" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 21 "\nx2 := t -> 1-cos(t); " }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 47 "Es interessiert die L\344nge des folgenden B ogens:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 50 "plot([x1(t),x2(t),t=0..2*Pi],scaling=constrained); " }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 115 "Nun soll die L\344nge dieser Kurve mit der \+ Integralformel exakt berechnet werden. Die Integrandenfunktion f in di eser " }{TEXT 264 0 "" }{TEXT 264 90 "\nIntegralformel ergibt sich dab ei als Wurzel der Summe der Quadrate der Ableitungen, also:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 48 "f := t -> sqrt(diff(x1(t),t)^2+diff(x2(t),t)^2);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 37 "Diese Funktion ist also gegeben durch" }{TEXT 264 0 "" } {TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 5 "f(t);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 16 "oder vereinfacht" }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 12 "s implify(%);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 96 "Bestimmt f\344llt es Dir schwer, hie rvon mit Papier und Bleistift eine Stammfunktion zu bestimmen. " } {TEXT 264 0 "" }{TEXT 264 40 "\nF\374r Maple ist dies jedoch kein Prob lem:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 12 "int(f(t),t);" }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 83 "\nDas bestimmte Integral, aus welch em sich die L\344nge des Zykloidenbogens ergibt, ist" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 20 "int( f(t),t=0..2*Pi);" }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 71 "\nMaple muss nun noch veranlasst werden dieses Erg ebnis zu vereinfachen:" }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 12 "simplify(%);" } {MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 265 "" 0 "" {TEXT 261 9 "Aufgaben:" } {TEXT 261 0 "" }}{PARA 259 "" 0 "" {TEXT 264 81 "1. Bestimme mit Maple die L\344nge des Parabelsegments mit der Parameterdarstellung " } {XPPEDIT 244 0 "x(t) = (t, t^2),t*epsilon;" "6$/-%\"xG6#%\"tG6$F'*$F' \"\"#*&F'\"\"\"%(epsilonGF," }{TEXT 347 1 " " }{TEXT 346 8 " [-1; 1]" }{TEXT 264 2 ". " }{TEXT 264 0 "" }{TEXT 264 55 "\n Hierzu kannst D u mit obiger Befehlsfolge arbeiten." }{TEXT 264 0 "" }{TEXT 316 1 "\n" }}{SECT 0 {PARA 273 "" 0 "" {TEXT 246 7 "L\366sung:" }{TEXT 246 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 13 "x1 := t -> t;" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 16 "\nx2 := t -> t^2;" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 47 "Es interessiert die L \344nge des folgenden Bogens:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 48 "plot([x1(t),x2(t),t=-1.. 1],scaling=constrained);" }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 109 "\nNun soll die L\344nge dieser Kurve mi t der Integralformel exakt berechnet werden. Die Integrandenfunktion f in " }{TEXT 264 0 "" }{TEXT 264 97 "\ndieser Integralformel ergibt si ch dabei als Wurzel der Summe der Quadrate der Ableitungen, also:" } {TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 48 "f := t -> sqrt(diff(x1(t),t)^2+diff(x2(t),t)^2);" } {MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 38 "\nDiese Funktion ist also gegeben durch" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 5 "f(t);" } {MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 82 "Das bestimmte Integral, aus welchem sich die L\344 nge des Zykloidenbogens ergibt, ist" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 18 "int(f(t),t=-1..1);" } {MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 32 "\nEinen numerischen Wert liefert:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 9 "evalf(%);" }{MPLTEXT 1 259 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 259 "" 0 "" {TEXT 264 71 "2. Bestimme mit Maple die L\344nge der Kurve mit der Parameterdarstellung " } {XPPEDIT 18 0 "x(t) = (t, t^2, 2*t^3/3),t*epsilon;" "6$/-%\"xG6#%\"tG6 %F'*$F'\"\"#*(F*\"\"\"*$F'\"\"$F,F.!\"\"*&F'F,%(epsilonGF," }{TEXT 356 1 " " }{TEXT 346 5 "[0;1]" }{TEXT 264 1 "." }{TEXT 316 0 "" }} {SECT 0 {PARA 274 "" 0 "" {TEXT 245 7 "L\366sung:" }{TEXT 245 0 "" }} {PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 9 "x1:=t->t;" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 12 " \nx2:=t->t^2;" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 16 "\nx3:=t->2*t^3 /3;" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 108 "Nun soll die L\344nge dieser Kurve mit der \+ Integralformel exakt berechnet werden. Die Integrandenfunktion f in " }{TEXT 264 0 "" }{TEXT 264 97 "\ndieser Integralformel ergibt sich dab ei als Wurzel der Summe der Quadrate der Ableitungen, also:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 46 "f := t -> sqrt(diff(x1(t),t)^2+diff(x2(t),t)^2" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 30 "\n +diff(x3(t),t)^2);" } {MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 37 "Diese Funktion ist also gegeben durch" }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 5 "f(t);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 82 "Das bestimmte Integra l, aus welchem sich die L\344nge des Zykloidenbogens ergibt, ist" } {TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 17 "int(f(t),t=0..1);" }{MPLTEXT 1 259 0 "" }}}}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 53 " Maple kann auch Integrale mit Parametern berechnen: " }{TEXT 264 0 "" }}{PARA 256 "" 0 "" {TEXT 264 67 "F\374r die L\344nge eines Bogens de r Zykloide mit Radius r>0 ergibt sich:" }{TEXT 264 0 "" }{TEXT 264 1 " \n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 25 "x1 := t -> r*(t-si n(t)); " }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 25 "\nx2 := t -> r*(1-co s(t));" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 54 "\nint(sqrt(diff(x1(t) ,t)^2+diff(x2(t),t)^2),t=0..2*Pi);" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 1 "\n" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 259 "" 0 "" {TEXT 264 4 "Mit " }{TEXT 346 11 "assume(r>0)" }{TEXT 264 34 " kann Ma ple die Einschr\344nkung von " }{TEXT 357 1 "r" }{TEXT 264 53 " mitget eilt und dann der Ausdruck vereinfacht werden:" }{TEXT 264 0 "" } {TEXT 316 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 25 "assum e(r>0); simplify(%);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 62 "Die Tilde im Ergebnis deute t dabei die Einschr\344nkung f\374r r an." }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 265 "" 0 "" {TEXT 261 8 "Aufgabe:" }{TEXT 261 0 "" }}{PARA 259 "" 0 "" {TEXT 264 104 "\304ndere obige Bef ehlsfolge so ab, dass die L\344nge einer Windung einer Schraubenlinie \+ mit Radius r und b=1 " }{TEXT 264 0 "" }{TEXT 264 17 "\n(also Gangh\36 6he 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 356 1 " " }{TEXT 264 17 " ) berechnet wird." }{TEXT 264 0 "" }{TEXT 316 1 "\n" }}{SECT 0 {PARA 273 "" 0 "" {TEXT 246 7 "L\366sung:" }{TEXT 246 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 21 "x1 : = t -> r*cos(t); " }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 21 "\nx2 := t \+ -> r*sin(t);" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 14 "\nx3 := t -> t; " }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 41 "\nint(sqrt(diff(x1(t),t)^2+ diff(x2(t),t)^2" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 39 "\n + diff(x3(t),t)^2),t=0..2*Pi);" }{MPLTEXT 1 259 0 "" }}}}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}}{SECT 1 {PARA 267 "" 0 "" {TEXT 251 37 "Berechnung mit N\344herungsstreckenz\374gen" }{TEXT 251 0 "" }} {PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 265 "" 0 "" {TEXT 261 32 "Bei spiel: Ein Bogen der Zykloide" }{TEXT 261 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 8 "restart; " }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 101 "\nEs soll jetzt die Berechnung der L\344nge eines Zykloidenbogens mit N\344herungsstreckenz\374gen, wie sie im " }{TEXT 264 0 "" } {TEXT 264 71 "\nUnterricht vorgestellt wurde, mit Maple nochmals nachv ollzogen werden." }{TEXT 264 0 "" }}{PARA 256 "" 0 "" {TEXT 264 49 "Zu n\344chst definieren wir die Parameterdarstellung:" }{TEXT 264 0 "" } {TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 20 "x1 := t -> t-sin(t);" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 21 "\nx2 := t -> 1-cos(t);" }{MPLTEXT 1 259 0 "" }}}{PARA 259 "" 0 "" {TEXT 264 0 "" } {TEXT 264 69 "\nWir bilden eine gleichm\344\337ige Zerlegung des Param eterintervalls [0; 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 356 1 " " }{TEXT 264 22 "] in 6 gleiche Teile: " }{TEXT 264 0 "" }{TEXT 316 1 " \n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 19 "t := i -> i*2*Pi/6 ;" }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 26 "\nHier ist diese Zerlegung:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 35 "t(0),t(1),t(2),t(3),t (4),t(5),t(6);" }{MPLTEXT 1 259 0 "" }}}{PARA 259 "" 0 "" {TEXT 264 0 "" }{TEXT 264 23 "\nOder als Maple-Folge (" }{TEXT 346 5 "seq) " } {TEXT 264 12 "dargestellt:" }{TEXT 264 0 "" }{TEXT 316 1 "\n" }} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 17 "seq(t(i),i=0..6);" } {MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 103 "\nNun wollen wir den N\344herungsstreckenzug zu dieser Zerlegung \+ bilden. Dies erreichen wir in Maple durch " }{TEXT 264 0 "" }{TEXT 264 53 "\nBildung der Folge der Eckpunkte dieses Streckenzugs:" } {TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 46 "polygon6 := [seq([x1(t(i)),x2(t(i))],i=0..6)];" } {MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 37 "Dieser Streckenzug sieht dann so aus:" }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 35 "plot(polygon6,scaling=constrained);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 49 "Zusammen mit dem Zykloidenbogen sieht das so aus:" } {TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 78 "plot(\{[x1(t),x2(t),t=0..2*Pi],polygon6\},color=[re d,blue],scaling=constrained);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 62 "Wir berechnen die \+ L\344ngen der Sehnen (der i-ten Sehne/Strecke):" }{TEXT 264 0 "" } {TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 69 "lsehn e := i -> sqrt((x1(t(i))-x1(t(i-1)))^2+(x2(t(i))-x2(t(i-1)))^2);" } {MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 73 "\nDie Summe der 6 Sehnenl\344ngen ergibt die L\344nge des N\344herung sstreckenzugs:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 " > " 0 "" {MPLTEXT 1 259 22 "sum(lsehne(i),i=1..6);" }{MPLTEXT 1 259 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 9 "evalf(%);" } {MPLTEXT 1 259 0 "" }}}{PARA 275 "" 0 "" {TEXT 233 0 "" }{TEXT 233 9 " \nAufgabe:" }{TEXT 233 0 "" }}{PARA 264 "" 0 "" {TEXT 349 111 "\304nde re die Befehlsfolge so ab, dass die L\344nge des N\344herungsstreckenz ugs eines Bogens der verl\344ngerten Zykloide " }{TEXT 349 0 "" } {TEXT 349 60 "\nmit r=1 und d=2 bei Zerlegung des Parameterintervalls \+ [0; 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 350 1 " " }{TEXT 349 37 " ] in 8 gleiche Teile berechnet wird!" }{TEXT 230 0 "" }{TEXT 315 1 " \n" }}{SECT 0 {PARA 273 "" 0 "" {TEXT 246 7 "L\366sung:" }{TEXT 246 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 49 "Zun\344chst definieren wir die Parameterdarstellung:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 22 "x1 := t -> t-2*sin(t);" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 23 "\nx2 := t -> 1-2*cos(t);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 259 "" 0 "" {TEXT 264 68 "Wir bilden eine gleichm\344 \337ige Zerlegung des Parameterintervalls [0; 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 356 1 " " }{TEXT 264 22 "] in 8 gleiche Teile: " } {TEXT 264 0 "" }{TEXT 316 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 15 "t:=i->i*2*Pi/8;" }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 26 "\nHier ist diese Zerlegung:" } {TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 17 "seq(t(i),i=0..8);" }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 103 "\nNun wollen wir den N\344h erungsstreckenzug zu dieser Zerlegung bilden. Dies erreichen wir in Ma ple durch " }{TEXT 264 0 "" }{TEXT 264 53 "\nBildung der Folge der Eck punkte dieses Streckenzugs:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 46 "polygon8 := [seq([x1(t(i )),x2(t(i))],i=0..8)];" }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 70 "\nDieser Streckenzug sieht dann zusammen mit dem Zykloidenbogen so aus:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 78 "plot(\{[x1(t),x2(t),t=0. .2*Pi],polygon8\},color=[red,blue],scaling=constrained);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 62 "Wir berechnen die L\344ngen der Sehnen (der i-ten Sehne/ Strecke):" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 69 "lsehne := i -> sqrt((x1(t(i))-x1(t(i-1)))^2+(x2 (t(i))-x2(t(i-1)))^2);" }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 73 "\nDie Summe der 8 Sehnenl\344ngen ergibt die L\344nge des N\344herungsstreckenzugs:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 22 "sum(lsehne( i),i=1..8);" }{MPLTEXT 1 259 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 9 "evalf(%);" }{MPLTEXT 1 259 0 "" }}}}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }} {PARA 276 "" 0 "" {TEXT 229 83 "Jetzt soll die L\344ngenberechnung mit N\344herungsstreckenz\374gen verallgemeinert werden: " }{TEXT 229 0 " " }{TEXT 229 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 8 "res tart;" }{MPLTEXT 1 259 0 "" }}}{PARA 264 "" 0 "" {TEXT 349 0 "" } {TEXT 349 89 "\nDas Parameterintervall [a; b] soll in n gleiche Teile \+ zerlegt werden (vgl. Unterricht): " }{TEXT 227 0 "" }{TEXT 315 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 26 "t := (i,n) -> a+i*(b-a )/n;" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }} {PARA 256 "" 0 "" {TEXT 264 56 "Dann bilden wir den (allgemeinen) N\34 4herungsstreckenzug: " }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 54 "polygon := n -> [seq([x1(t(i,n) ),x2(t(i,n))],i=0..n)]:" }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 64 "\n(An der ausgegebenen Warnung brauchst \+ du dich nicht zu st\366ren!)" }{TEXT 264 0 "" }{TEXT 264 80 "\nWir ber echnen die L\344nge der i-ten Sehne bei einer Unterteilung in n Interv alle:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 81 "lsehne := (i,n) -> sqrt((x1(t(i,n))-x1(t(i-1,n)))^2 +(x2(t(i,n))-x2(t(i-1,n)))^2);" }{MPLTEXT 1 259 0 "" }}}{PARA 277 "" 0 "" {TEXT 226 0 "" }{TEXT 226 113 "\nWir erstellen einen Kurvenplot am Beispiel der verl\344ngerten Zykloide einschlie\337lich eines N\344he rungsstreckenzugs:" }{TEXT 226 0 "" }{TEXT 226 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 22 "x1 := t -> t-2*sin(t);" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 23 "\nx2 := t -> 1-2*cos(t);" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 19 "\na := 0; b := 2*Pi;" }{MPLTEXT 1 259 0 " " }{MPLTEXT 1 259 43 "\nplot([[x1(t),x2(t),t=0..2*Pi],polygon(6)]," } {MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 41 "\n color=[red,blue],scaling=c onstrained);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 55 "Die L\344nge eines N\344herungsstre ckenzugs berechnen wir mit" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 31 "evalf(sum(lsehne(i,6),i= 1..6));" }{MPLTEXT 1 259 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 33 "evalf(sum(lsehne(i,60),i=1..60));" }{MPLTEXT 1 259 0 "" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 35 "evalf(sum(lsehne(i,600), i=1..600));" }{MPLTEXT 1 259 0 "" }}}{PARA 278 "" 0 "" {TEXT 358 0 "" }}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}{PARA 279 "" 0 "" {TEXT 359 0 "" }}{PARA 280 "" 0 "" {TEXT 216 8 "Aufgabe:" }{TEXT 216 0 "" }}{PARA 264 "" 0 "" {TEXT 349 117 "\304ndere obige Befehlssequenz dahingehend ab, dass die L\344nge eines Bogens einer verk\374rzten Zy kloide (mit r=1und d=1/2) " }{TEXT 349 0 "" }{TEXT 349 31 "\nn\344heru ngsweise berechnet wird!" }{TEXT 214 0 "" }{TEXT 315 1 "\n" }}{SECT 0 {PARA 273 "" 0 "" {TEXT 246 7 "L\366sung:" }{TEXT 246 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 55 "Es muss n ur die Parameterdarstellung abge\344ndert werden:" }{TEXT 264 0 "" } {TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 24 "x1 := t -> t-1/2*sin(t);" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 25 "\nx2 := \+ t -> 1-1/2*cos(t);" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 19 "\na := 0; b := 2*Pi;" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 43 "\nplot([[x1(t),x 2(t),t=0..2*Pi],polygon(6)]," }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 41 "\n color=[red,blue],scaling=constrained);" }{MPLTEXT 1 259 0 "" }}} {PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 55 "Die L\344nge eines N\344herungsstreckenzugs berechnen wir mit" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 31 "evalf(sum(lsehne(i,6),i=1..6));" }{MPLTEXT 1 259 0 "" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 33 "evalf(sum(lsehne(i,60),i =1..60));" }{MPLTEXT 1 259 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 35 "evalf(sum(lsehne(i,600),i=1..600));" }{MPLTEXT 1 259 0 "" }}}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}}}{SECT 1 {PARA 257 "" 0 "" {TEXT 263 29 " Weit ere Themen und Beispiele" }{TEXT 263 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{SECT 0 {PARA 281 "" 0 "" {TEXT 360 57 "Die Astroide (Stern kurve) als Beispiel einer ebenen Kurve" }{TEXT 213 0 "" }{TEXT 313 1 " \n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 8 "restart;" } {MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 56 "Die Astroide ist gegeben durch die Parameterdarste llung " }{TEXT 264 0 "" }}{PARA 259 "" 0 "" {XPPEDIT 239 0 "x(t) = (r* cos(t)^3, r*sin(t)^3),t*epsilon;" "6$/-%\"xG6#%\"tG6$*&%\"rG\"\"\"*$-% $cosGF&\"\"$F+*&F*F+*$-%$sinGF&F/F+*&F'F+%(epsilonGF+" }{TEXT 347 1 " \+ " }{TEXT 346 6 " [0; 2" }{XPPEDIT 237 0 "Pi;" "6#%#PiG" }{TEXT 347 1 " " }{TEXT 346 1 "]" }{TEXT 264 1 "." }{TEXT 316 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 265 "" 0 "" {TEXT 261 9 "Aufgaben:" }{TEXT 261 0 "" }}{PARA 256 "" 0 "" {TEXT 264 54 "1. Erstelle einen Kurvenplo t f\374r die Astroide mit r=1." }{TEXT 264 0 "" }{TEXT 264 1 "\n" }} {PARA 265 "" 0 "" {TEXT 261 7 "L\366sung:" }{TEXT 261 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 56 "plot([cos(t)^3,sin(t)^3,t=0..2*Pi],scaling=constrained);" } {MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 52 "\n2. Berechne die L\344nge der Kurve f\374r beliebiges r>0." }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 282 "" 0 "" {TEXT 212 7 "L\366sung:" }{TEXT 212 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 23 "x1 := t -> r*cos(t)^3; " }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 23 "\nx2 := t -> r*sin(t)^3;" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 54 "\nint(sqrt(diff(x1(t),t)^2+di ff(x2(t),t)^2),t=0..2*Pi);" }{MPLTEXT 1 259 0 "" }}}{EXCHG {PARA 260 " > " 0 "" {MPLTEXT 1 259 25 "assume(r>0); simplify(%);" }{MPLTEXT 1 259 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}}{SECT 0 {PARA 267 "" 0 "" {TEXT 251 37 "Ein weiteres Beispiel einer Raumkurve" }{TEXT 251 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 " " {TEXT 264 61 "Eine Kurve im Raum sei gegeben durch die Parameterdars tellung" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 " " {MPLTEXT 1 259 8 "restart;" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 22 "\nx1 := t -> 1+cos(t); " }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 20 "\nx 2 := t -> sin(t); " }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 23 "\nx3 := t -> 2*sin(t/2);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 259 "" 0 "" {TEXT 264 28 "mit dem Parameterintervall [" } {XPPEDIT 18 0 "-2*Pi;" "6#,$*&\"\"#\"\"\"%#PiGF&!\"\"" }{TEXT 356 1 " \+ " }{TEXT 264 3 "; 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 356 1 " " } {TEXT 264 2 "]." }{TEXT 316 0 "" }}{PARA 256 "" 0 "" {TEXT 264 117 "Er stelle einen Kurvenplot f\374r diese Kurve und gewinne einen r\344umli chen Eindruck von dieser Kurve (mit Achsenkreuz!). " }{TEXT 264 0 "" } {TEXT 264 1 "\n" }}{PARA 265 "" 0 "" {TEXT 261 7 "L\366sung:" }{TEXT 261 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 12 "with(plots):" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 47 "\nspacecurve([x1(t),x2(t),x3(t)],t=-2*Pi..2*Pi);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 101 "Diese Kurve (auch als Viviani-Kurve bekannt) liegt auf \+ einer Kugel und einem Zylinder und stellt die " }{TEXT 264 0 "" } {TEXT 264 102 "\nSchnittkurve dieser beiden Fl\344chen dar. Versuche, \+ dies am Kurvenplot zu erkennen. Dass sie auf einem " }{TEXT 264 0 "" } {TEXT 264 56 "\nZylinder mit Radius 1 liegt, best\344tigt man mit (war um?)" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 20 "(x1(t)-1)^2+x2(t)^2;" }{MPLTEXT 1 259 0 "" }}} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 12 "simplify(%);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 86 "Best\344tige nun selbst, dass die Kurve auf der Kugel um den Ursprung mit Radius 2 liegt:" }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 265 "" 0 "" {TEXT 261 7 "L\366sung:" }{TEXT 261 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 24 "x1(t)^2+x2(t)^2+x3(t)^2;" }{MPLTEXT 1 259 0 "" } }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 12 "simplify(%);" } {MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 77 "Berechne die L\344nge dieser Kurve mit der Integra lformel (exakt und numerisch):" }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 265 "" 0 "" {TEXT 261 7 "L\366sung:" }{TEXT 261 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 40 "int(sqrt(diff(x1(t),t)^2+diff(x2(t),t)^2" } {MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 45 "\n +diff(x3(t),t)^2) ,t=-2*Pi..2*Pi);" }{MPLTEXT 1 259 0 "" }}}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 9 "evalf(%);" }{MPLTEXT 1 259 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}}{SECT 0 {PARA 267 "" 0 "" {TEXT 251 42 "Z usatzthema: Eine verknotete Kurve im Raum" }{TEXT 251 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 283 "" 0 "" {TEXT 211 46 "Zeichne d ie Kurve mit der Parameterdarstellung" }{TEXT 211 0 "" }{TEXT 211 1 " \n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 45 "x1 := t -> -10*cos (t)-2*cos(5*t)+15*sin(2*t);" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 46 " \nx2 := t -> -15*cos(2*t)+10*sin(t)-2*sin(5*t);" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 24 "\nx3 := t -> 10*cos(3*t);" }{MPLTEXT 1 259 0 "" }} }{PARA 264 "" 0 "" {TEXT 349 0 "" }{TEXT 349 29 "\nmit dem Parameterin tervall " }{TEXT 209 5 "[0; 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 350 1 " " }{TEXT 209 1 "]" }{TEXT 349 68 " , und verschaffe Dir einen \+ r\344umlichen Eindruck von ihr. Kann diese " }{TEXT 349 0 "" }{TEXT 349 37 "\ngeschlossene Kurve entknotet werden?" }{TEXT 209 0 "" } {TEXT 315 1 "\n" }}{PARA 259 "" 0 "" {TEXT 346 7 "L\366sung:" }{TEXT 264 0 "" }{TEXT 316 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 12 "with(plots):" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 52 "\ntubep lot([x1(t),x2(t),x3(t)],t=0..2*Pi,radius=0.5);" }{MPLTEXT 1 259 0 "" } }}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}}{SECT 0 {PARA 267 "" 0 "" {TEXT 251 41 "Zusatzthema: Eine Kurve unendlicher L\344nge" } {TEXT 251 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 281 "" 0 "" {TEXT 361 49 "Eine Kurve (unendlicher L\344nge!) ist gegeben durch" } {TEXT 213 0 "" }{TEXT 313 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 8 "restart;" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 14 " \nx1 := t -> t;" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 41 "\nx2 := t -> piecewise(t=0,0,t*cos(Pi/t));" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 259 "" 0 "" {TEXT 264 87 "mit dem Parameter intervall [0; 1]. Beachte, dass die Funktion x2 mit Hilfe des Befehls \+ " }{TEXT 346 9 "piecewise" }{TEXT 264 12 " st\374ckwiese " }{TEXT 264 0 "" }{TEXT 264 28 "\ndefiniert ist. (Den Befehl " }{TEXT 346 9 "piece wise" }{TEXT 264 45 " kannst Du mit der Online-Hilfe analysieren.)" } {TEXT 316 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 82 "Wir definieren nun einen N\344herungsstreckenzug als Fol ge von Punkten auf der Kurve:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 50 "polygon:=n->[seq([x1(1/i ),x2(1/i)],i=1..n),[0,0]]:" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 46 "Ein Kurvenplot mit ei nem N\344herungsstreckenzug:" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 58 "plot([[x1(t),x2(t),t=0.. 1],polygon(10)],color=[red,blue]);" }{MPLTEXT 1 259 0 "" }}}{PARA 256 "" 0 "" {TEXT 264 0 "" }{TEXT 264 57 "\nDer N\344herungsstreckenzug mi t 10 Teilstrecken besteht aus" }{TEXT 264 0 "" }{TEXT 264 1 "\n" }} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 12 "polygon(10);" }{MPLTEXT 1 259 0 "" }}}{PARA 259 "" 0 "" {TEXT 264 71 "Mache Dir klar, dass die ser Streckenzug eine L\344nge hat, die gr\366\337er als " }{XPPEDIT 18 0 "1+1/2+1/3;" "6#,(\"\"\"F$*&F$F$\"\"#!\"\"F$*&F$F$\"\"$F'F$" } {TEXT 356 1 " " }{TEXT 264 5 "+...+" }{XPPEDIT 18 0 "1/10;" "6#*&\"\" \"F$\"#5!\"\"" }{TEXT 356 1 " " }{TEXT 264 5 " ist." }{TEXT 316 0 "" } }{PARA 259 "" 0 "" {TEXT 264 91 "Eine Verallgemeinerung dieser \334ber legung f\374hrt darauf, dass die L\344nge der Kurve gr\366\337er als " }{XPPEDIT 18 0 "1+1/2+1/3;" "6#,(\"\"\"F$*&F$F$\"\"#!\"\"F$*&F$F$\"\" $F'F$" }{TEXT 356 1 " " }{TEXT 264 5 "+...+" }{XPPEDIT 18 0 "1/n;" "6# *&\"\"\"F$%\"nG!\"\"" }{TEXT 356 1 " " }{TEXT 264 28 " f\374r jede na t\374rliche Zahl " }{TEXT 357 2 "n " }{TEXT 264 4 "ist." }{TEXT 316 0 "" }}{PARA 284 "" 0 "" {TEXT 341 67 "Wir untersuchen deshalb das Ver halten dieser Summen f\374r wachsendes " }{TEXT 357 2 "n:" }{TEXT 341 2 " " }{TEXT 341 0 "" }{TEXT 316 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 23 "sum(1/i,i=1..infinity);" }{MPLTEXT 1 259 0 "" }}} {PARA 259 "" 0 "" {TEXT 264 0 "" }{TEXT 264 41 "\nHieraus folgt, dass \+ die L\344nge der Kurve " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" } {TEXT 356 1 " " }{TEXT 264 5 " ist." }{TEXT 264 0 "" }{TEXT 316 1 "\n" }}}{SECT 0 {PARA 267 "" 0 "" {TEXT 251 57 "Zusatzthema: Einzelne Punk te in einen Kurvenplot zeichnen" }{TEXT 251 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 259 "" 0 "" {TEXT 264 90 "Mit etwas mehr Aufwan d k\366nnen auch einzelne Punkte in das Schaubild eingezeichnet werden , " }{TEXT 264 0 "" }{TEXT 264 24 "\nz. B. der Anfangspunkt " } {XPPEDIT 18 0 "x(0);" "6#-%\"xG6#\"\"!" }{TEXT 356 1 " " }{TEXT 264 1 ":" }{TEXT 316 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 21 "restart; with(plots):" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 62 "\ndisplay(\{plot([cos(t),sin(t),t=0..2*Pi ],scaling=constrained)," }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 67 "\npl ot([[cos(0),sin(0)]],style=POINT,symbol=CIRCLE,color=magenta)\});" } {MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 84 "\304ndere die Befehlssequenz so ab, dass andere Pu nkte der Kurve eingezeichnet werden. " }{TEXT 264 0 "" }{TEXT 264 1 " \n" }}}{SECT 0 {PARA 267 "" 0 "" {TEXT 251 33 "Zusatzthema: Weitere Pl otoptionen" }{TEXT 251 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }} {PARA 285 "" 0 "" {TEXT 339 8 "Aufgabe:" }{TEXT 339 0 "" }}{PARA 259 " " 0 "" {TEXT 264 50 "Informiere Dich in der Online-Hilfe (Topic Search " }{TEXT 346 4 "plot" }{TEXT 264 53 ") \374ber weitere Plotoptionen u nd experimentiere damit!" }{TEXT 264 0 "" }{TEXT 316 1 "\n" }}}{SECT 1 {PARA 267 "" 0 "" {TEXT 251 34 "Zusatzthema: Animationen mit Maple" }{TEXT 251 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 265 "" 0 "" {TEXT 261 29 "Eine Animation f\374r Zykloiden:" }{TEXT 261 0 "" }} {PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 259 "" 0 "" {TEXT 346 5 "zykl o" }{TEXT 264 37 " liefert das Schaubild der Zykloiden;" }{TEXT 316 0 "" }}{PARA 259 "" 0 "" {TEXT 346 6 "fkreis" }{TEXT 264 67 " liefert da s Schaubild des Kreises in der Endposition dieser Kurve;" }{TEXT 316 0 "" }}{PARA 259 "" 0 "" {TEXT 346 6 "radius" }{TEXT 264 64 " liefert \+ den Kreisradius zu dem die Zykloide erzeugenden Punkt; " }{TEXT 316 0 "" }}{PARA 259 "" 0 "" {TEXT 346 5 "punkt" }{TEXT 264 53 " zeichnet de n Punkt auf dem Umfang als kleinen Kreis;" }{TEXT 264 0 "" }{TEXT 316 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 8 "restart;" } {MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 50 "\nwith(plots): with(plottools) : T := 4*Pi: N := 20:" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 66 "\nzykl o := plot([t-sin(t),1-cos(t),t=0..T],color=red,thickness=2):" } {MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 63 "\nfkreis := x -> plot([x+cos(t ),1+sin(t),t=0..2*Pi],color=blue):" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 76 "\nradius := x -> plot([[x,1],[x-sin(x),1-cos(x)]],color=magent a,thickness=2):" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 59 "\npunkt := \+ x -> disk([x-sin(x),1-cos(x)],0.15,color=black):" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 62 "\nzbild := x -> display(\{zyklo,fkreis(x),radius (x),punkt(x)\}):" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 23 "Eine Animation liefert:" } {TEXT 264 0 "" }{TEXT 264 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 72 "display([seq(zbild(T/N*i),i=0..N)],scaling=constrai ned,insequence=true);" }{MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 75 "Starte die Animation \+ durch Anklicken der Grafik und Dr\374cken der Starttaste!" }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 256 "" 0 "" {TEXT 264 114 "Man kann dies alles auf die verk\374rzte oder verl\344 ngerte Zykloide verallgemeinern, indem man den Zeichenstift statt " } {TEXT 264 0 "" }{TEXT 264 117 "\nauf der Kreislinie auf einer festen S tange anbringt, die sich mit dem Kreis dreht. Man ben\366tigt dazu ein en weiteren " }{TEXT 264 0 "" }{TEXT 264 93 "\nParameter f\374r die St angenl\344nge r und davon abh\344ngig einige kleine \304nderungen in d en Formeln:" }{TEXT 264 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }} {EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 8 "restart;" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 50 "\nwith(plots): with(plottools): T := 4*Pi : N := 20:" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 76 "\nfkreis := x -> plot([x+cos(t),1+sin(t),t=0..2*Pi],color=blue,thickness=2):" } {MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 76 "\nfzyklo1 := r -> plot([t-r*si n(t),1-r*cos(t),t=0..T],color=red,thickness=2):" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 69 "\npunkt1 := (x,r) -> disk([x-r*sin(x),1-r*cos(x)] ,0.15,color=black): " }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 85 "\nradiu s1 := (x,r) -> plot([[x,1],[x-r*sin(x),1-r*cos(x)]],color=magenta,thic kness=2):" }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 78 "\nzbild1 := (x,r) -> display(\{fzyklo1(r),fkreis(x),radius1(x,r),punkt1(x,r)\}):" } {MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 55 "\nanim := r -> display([seq (zbild1(T/N*i,r),i=0..N)]," }{MPLTEXT 1 259 0 "" }{MPLTEXT 1 259 62 " \n scaling=CONSTRAINED,insequence=true):" } {MPLTEXT 1 259 0 "" }}}{PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 259 "" 0 "" {TEXT 264 13 "Die Funktion " }{TEXT 346 5 "anim " }{TEXT 264 44 "liefert nun f\374r jeden eingegebenen Wert von " }{TEXT 346 2 "r " } {TEXT 264 26 "eine zugeh\366rige Animation:" }{TEXT 264 0 "" }{TEXT 316 1 "\n" }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 259 8 "anim(4);" } {MPLTEXT 1 259 0 "" }}}{EXCHG {PARA 254 "> " 0 "" {TEXT 345 0 "" }}}}} {PARA 254 "" 0 "" {TEXT 345 0 "" }}{PARA 254 "" 0 "" {TEXT 345 0 "" }} {PARA 286 "" 0 "" {TEXT 362 0 "" }}{PARA 287 "" 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 }