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341 "" 0 1 0 0 4 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 344 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 } {PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Heading 4" 5 20 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 74 0 220 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "5. Sitzung" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Erinnerung an die 4.te Sitzung" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "a) Gehe zurueck zur 4. Sitzung, falls diese noch nicht ab geschlossen war." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "b) Es sind in den ersten 4 Sitzungen die folgenden Operatoren u nd Funktionen aufgetaucht:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 257 296 "$ , ` ` , ' ' , . , add , BesselJ , cat , convert , \+ diff , evalf , evaln , expand , factor , factorial , fsolve , help , i factor , int , isprime , lhs, mul , nops , op , plot , print , product , rhs , seq , simplify , sort , sqrt , subs , subsop , sum , taylor , type, writeto , whattype" }}{PARA 0 "" 0 "" {TEXT -1 58 "Was ist die \+ jeweilige Bedeutung? Benutze im zweifelsfalle " }{TEXT 256 4 "help" } {TEXT -1 35 " , um die Erinnerung aufzufrischen." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "help( );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "c) W as steckt hinter den Typenbezeichnungen" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 258 132 "`+` , `*` , `^` , algebraic , constant , float , fraction , list , name , numeric , polynom , posint , range , serie s , set , string" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "help(type[algeb raic]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "d) MAPLEs Integrator \+ " }{TEXT 259 3 "int" }{TEXT -1 20 " kann in der Form " }{TEXT 260 11 "int(f(x),x)" }{TEXT -1 25 " (Stammfunktion) oder " }{TEXT 261 16 "int(f(x),x=a..b)" }{TEXT -1 82 " (bestimmtes Integral) aufgerufen werden. Falls das 2.te Argument nicht vom Typ " }{TEXT 262 4 "name" }{TEXT -1 32 " oder vom Typ `=` (genauer: " }{TEXT 263 12 "name = \+ range" }{TEXT -1 14 " ) ist, muss " }{TEXT 264 3 "int" }{TEXT -1 61 " mit einer Fehlermeldung abbrechen. Schreibe eine Funktion " }{TEXT 265 16 "f := expr -> ..." }{TEXT -1 7 " , die " }{TEXT 266 4 "true" } {TEXT -1 15 " ergibt, falls " }{TEXT 267 4 "expr" }{TEXT -1 49 " entwe der ein Name oder eine Gleichung der Form " }{TEXT 268 12 "name = ran ge" }{TEXT -1 6 " ist !" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Probiere zunaechst eine eigene Loesung !" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f:= \+ expr -> type(expr,\{name,name=range\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f(x),f(x=1..2),f(x=y);" }}}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Mehr MAPLE" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Loese Gleich ungen: " }{TEXT 269 6 "solve " }{TEXT -1 3 "und" }{TEXT 288 7 " fsolve " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Lese zunaechst die Help-Seite zu " } {TEXT 270 5 "solve" }{TEXT -1 2 " :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "help(solve);" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 21 "Einzelne Gle ichungen:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Mehre Loesungen werde n als Folge ausgegeben:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(x**4+x^2+a=0,x); nops([\"]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "In der Regel gibt es keine Darstellungen fuer d ie Loesungen nichtlinearer Gleichungen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "loesung:=solve(x**6+x=1,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "MAPLEs " }{TEXT 272 6 "RootOf" }{TEXT -1 25 "-Funktion repraese ntiert " }{TEXT 271 4 "alle" }{TEXT -1 37 " Nullstellen seines Argumen ts (d.h., " }{TEXT 273 11 "RootOf(..) " }{TEXT -1 86 "hat die algebrai schen Eigenschaften, die allen Wurzeln gemeinsam sind). Die Funktion \+ " }{TEXT 274 8 "simplify" }{TEXT -1 99 " (der universelle Vereinfache r fuer Ausdruecke) verwendet diese algebraischen Eigenschaften, z.B.: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify( loesung^6 );" }} {PARA 0 "" 0 "" {TEXT -1 91 "Beachte, wie hierbei die Eigenschaft x^6 =1-x der Loesung zur Vereinfachung benutzt wurde:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "MAPLEs " }{TEXT 275 5 "alias" }{TEXT -1 91 "-Mecha nismus ist hilfreich, die haessliche RootOf-Darstellung eleganter auss ehen zu lassen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "help(alias);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Man kann natuerlich einfach mit \+ " }{TEXT 276 27 "alpha := RootOf(_Z^6+_Z-1);" }{TEXT -1 63 " die Eing abe vereinfachen, die Ausgabe wuerde aber immer noch " }{TEXT 277 7 " \+ RootOf" }{TEXT -1 21 " benutzen. Mittels " }{TEXT 278 6 "alias " } {TEXT -1 49 " werden sowohl Ein- als auch Ausgabe vereinfacht:" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "alias(alpha=loe sung): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve(x^6+x-1); simplify ( alpha^6);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "MAPLE's Funktion \+ " }{TEXT 279 9 "allvalues" }{TEXT -1 196 " versucht, alternative Dars tellungen aller in RootOf(..) repraesentierten Loesungen zu finden. Di es sind entweder Radikal-Darstellungen (Wurzeln von Wurzeln ...), oder numerische Approximationen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "all values(alpha);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Ein anderes Bei spiel:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(x^4+x^3+x^2-x-1,x); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "alias(alpha=\");" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "allvalues(alpha);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "Beachte, dass sich eine Folge mit 4 Elementen (die 4 komplexen Wurzeln) ergibt. Bilde daraus eine Liste und berechne numerische Approximationen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf([\"]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 7 " Achtung" }{TEXT -1 12 ": man kann " }{TEXT 283 6 "evalf " }{TEXT -1 16 " direkt auf den " }{TEXT 282 6 "RootOf" }{TEXT -1 74 "-Ausdruck an wenden. Der Floatingpoint-Evaluierer sucht dann aber nur nach " } {TEXT 280 5 "einer" }{TEXT -1 12 " Nullstelle:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(alpha);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Betrachte eine nicht-polynomiale Gleichung mit unendlich vielen Loesu ngen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve( tan(x)=1-x, x);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "alias(alpha=\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "In diesem Fall begnuegt sich " }{TEXT 284 9 "a llvalues" }{TEXT -1 6 " mit " }{TEXT 285 5 "einer" }{TEXT -1 21 " num erischen Loesung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "allvalues(alph a); evalf(alpha);" }}{PARA 0 "" 0 "" {TEXT -1 110 "Dies ist verstaendl ich, denn allvalues ware nicht in der Lage, alle (unendliche viele) Lo esungen zu bestimmen." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Man muss in so einem Fall mit " }{TEXT 286 6 "fsolve" }{TEXT -1 72 " und ein em Suchintervall arbeiten, um bestimmte Loesungen zu ermitteln:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(tan(x)-1+x,x=0..7,view=[1..6,- 2..5]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "fsolve(tan(x)=1- x,x= 2 .. 3 ); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fsolve(tan(x)= 1-x,x=4.5 .. 5.5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Falls statt einer Gleichung ein anderer Ausdruck " }{TEXT 291 5 "expr " }{TEXT -1 5 " an " }{TEXT 292 5 "solve" }{TEXT -1 8 " bzw. " }{TEXT 293 7 "fsolve " }{TEXT -1 40 " uebergeben wird, so wird die Gleichung " } {TEXT 294 7 " expr=0" }{TEXT -1 13 " angenommen:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "solve( x-Ausdruck(a,b,c) , x );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Statt Gleichungen koennen auch Ungleichungen gelo est werden:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve( x^2-x+1 <= 2 \+ , x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve( x^2-x+1 < 2 , x);" }}{PARA 0 "" 0 "" {TEXT -1 76 "Dies sind MAPLEs Darstellungen von abge schlossenen bzw. offenen Intervallen:" }{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Gleichungssysteme:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Betrachte mehrere Gleichungen fuer mehrere Unbekannte:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eq1:= x+y=A; eq2:= x-y= B;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "loesung:=solve(\{eq1,eq2\},\{ x,y\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Beachte, dass als Erg ebnis eine Menge von ``aufgeloesten Gleichungen'' geliefert wird, die \+ Loesungen sind " }{TEXT 338 5 "nicht" }{TEXT -1 19 " zugewiesen worden :" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "x,y;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Dieses Ausgabeformat ist aber so, \+ dass die Loesungen mittels des Substituierers " }{TEXT 289 5 "subs " }{TEXT -1 60 " unmittelbar in andere Ausdruecke eingesetzt werden koen nen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "subs(loesung,[x^2+y^2,eq1,e q2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Mittels " }{TEXT 287 6 " assign" }{TEXT -1 44 " koennen aufgeloeste Gleichungen der Form " } {TEXT 339 13 "Name=Ausdruck" }{TEXT -1 17 " in Zuweisungen " }{TEXT 341 17 " Name := Ausdruck" }{TEXT -1 21 " umgewandelt werden:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "loesung; assign(loesung); x,y;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Wenn keine Loesung gefunden wird, \+ dann ist das Ergebnis " }{TEXT 290 4 "NULL" }{TEXT -1 19 " (die leere \+ Folge):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "restart: solve( \{x+y=1, x+y=2\},\{x,y\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "Man kann er zwingen, dass Zusatzinformationen wie z.B. Warnungen von den System-Fu nktionen geliefert werden. Der voreingestellte Wert ist " }{TEXT 340 21 "infolevel[Funktion]=0" }{TEXT -1 2 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve( \{x+y=1,x+y=2\},\{x,y\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "help(infolevel);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Mehr Unbekannte als Gleichungen:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "infolevel[solve]:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x+y=0,\{x,y\});" }}{PARA 0 "" 0 "" {TEXT -1 26 "In diesem Fall verbleibt " }{TEXT 295 1 "y" }{TEXT -1 35 " als beli ebiger freier Parameter ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Ein \+ weiterer Fall mit vielen Loesungen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eq1:= x+y=1; eq2:=2*x+2*y=2; eq3:=3*x+3*y=3;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve( \{eq1,eq2,eq3\},\{x,y\});" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 70 "Bei nichtlinearen Gleichungssystemen mit mehreren \+ Loesungen wird eine " }{TEXT 304 16 "Folge von Mengen" }{TEXT -1 36 " \+ aufgeloester Gleichungen geliefert:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eq1:=x+2*y=1; eq2:=x**2-y**2=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "LoesungsFolge:=solve(\{eq1,eq2\},\{x,y\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Waehle das 2.te Element dieser Folge und weise die W erte zu:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "LoesungsFolge[2]; assig n(\"); x; y;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 296 9 "Aufgabe: " } {TEXT -1 4 "Sei " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "restart; n:=10; " }}{PARA 0 "" 0 "" {TEXT -1 48 "Finde die Loesung des linearen Gleich ungssystems" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 305 21 "[ 1 0 \+ 0 .. 0 " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT 306 8 " ] [ " }{XPPEDIT 18 0 "x[1]" "&%\"xG6#\"\"\"" }{TEXT 307 16 " ] \+ [ 1 ]" }}{PARA 0 "" 0 "" {TEXT 308 18 " [ 0 2 0 .. " } {XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT 309 12 " 0 ] [ " } {XPPEDIT 18 0 "x[2]" "&%\"xG6#\"\"#" }{TEXT 310 16 " ] [ 2 ]" }}{PARA 0 "" 0 "" {TEXT 311 31 " [ 0 0 3 .. 0 0 ] [ " } {XPPEDIT 18 0 "x[3]" "&%\"xG6#\"\"$" }{TEXT 312 16 " ] = [ 3 ]" }}{PARA 0 "" 0 "" {TEXT 313 49 " [.. .. .. .. .. .. ] [ .. ] \+ [ .. ]" }}{PARA 0 "" 0 "" {TEXT 314 7 " [ 0 " }{XPPEDIT 18 0 "e psilon" "I(epsilonG6\"" }{TEXT 315 22 " 0 .. n-1 0 ] [ " } {XPPEDIT 18 0 "x[n-1]" "&%\"xG6#,&%\"nG\"\"\"\"\"\"!\"\"" }{TEXT 316 14 " ] [ n-1]" }}{PARA 0 "" 0 "" {TEXT 317 3 " [ " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT 318 27 " 0 0 .. 0 n ] [ \+ " }{XPPEDIT 18 0 "x[n]" "&%\"xG6#%\"nG" }{TEXT 319 16 " ] [ n \+ ]" }}{PARA 0 "" 0 "" {TEXT -1 16 "fuer beliebiges " }{XPPEDIT 18 0 "ep silon" "I(epsilonG6\"" }{TEXT -1 91 " (erzeuge die Menge von Gleichun gen, die Menge von Unbekannten \{x[1],..,x[n]\} und benutze " }{TEXT 297 5 "solve" }{TEXT -1 53 " ; es werden keine Matrizen oder Vektoren \+ benoetigt)!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}} {SECT 1 {PARA 20 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Versuche zunaechst eine eigene Loesung !" }}} {SECT 1 {PARA 20 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "restart; n:=10;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Gleichungen:=\{seq(i*x[i]+epsilon*x[n+1-i]=i,i=1..n)\};" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Unbekannte :=\{seq(x[i],i=1..n)\}; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Loesung:=solve(Gleichungen,Unbe kannte);" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Die Loesung fuer " }{XPPEDIT 18 0 "epsilon = 0 " "/%(epsilonG\"\"!" }{TEXT -1 22 " ist o ffensichtlich " }{XPPEDIT 18 0 "x[1]*`=`*x[2]*`= ... =`*x[n]*`=`*0" " *0&%\"xG6#\"\"\"\"\"\"%\"=GF'&F$6#\"\"#F'%(=~...~=GF'&F$6#%\"nGF'F(F' \"\"!F'" }{TEXT -1 67 " . Welche der Loesungskomponenten waechst mit w achsendem (kleinen) " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 62 " am staerksten ? (Betrachte die Taylor-Entwicklung von x[i]). " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Probiere zunaechst eine eigene Loesung!" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "assi gn(Loesung);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "for i from 1 to n d o\n evaln(x[i])=taylor(x[i],epsilon=0,2);\nod;" }}}}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 9 "Aufgabe: " }{TEXT 298 12 "Finde fuer " } {TEXT 299 4 "n=20" }{TEXT 300 28 " alle Werte des Parameters " } {XPPEDIT 18 0 "alpha" "I&alphaG6\"" }{TEXT 301 52 " , fuer den wenigst ens eine der Loesungskomponenten " }{TEXT 302 4 "x[i]" }{TEXT 303 22 " des Gleichungssystems" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 320 24 " [ 2 -1 0 .. 0 0] [ " }{XPPEDIT 18 0 "x[1]" "&%\"xG6#\"\"\"" } {TEXT 321 11 " ] [ 1+" }{XPPEDIT 18 0 "alpha" "I&alphaG6\"" } {TEXT 322 2 " ]" }}{PARA 0 "" 0 "" {TEXT 323 25 " [-1 2 -1 .. 0 0] [ " }{XPPEDIT 18 0 "x[2]" "&%\"xG6#\"\"#" }{TEXT 324 11 " ] [ 2+ " }{XPPEDIT 18 0 "alpha" "I&alphaG6\"" }{TEXT 325 2 " ]" }}{PARA 0 "" 0 "" {TEXT 326 38 " [ 0 -1 2 .. 0 0] [ .. ] = [ 3+" }{XPPEDIT 18 0 "alpha" "I&alphaG6\"" }{TEXT 327 2 " ]" }}{PARA 0 "" 0 "" {TEXT 328 41 " [.. .. .. .. .. ..] [ .. ] [ .. ]" }}{PARA 0 "" 0 "" {TEXT 329 41 " [ 0 0 0 .. 2 -1] [ .. ] [ .. ]" }}{PARA 0 "" 0 "" {TEXT 330 25 " [ 0 0 0 .. -1 2] [ " }{XPPEDIT 18 0 "x[n]" "& %\"xG6#%\"nG" }{TEXT 331 11 " ] [ n+" }{XPPEDIT 18 0 "alpha" "I&al phaG6\"" }{TEXT 332 2 " ]" }}{PARA 0 "" 0 "" {TEXT -1 26 "verschwindet ! (Betrachte " }{XPPEDIT 18 0 "p(alpha):=x[1](alpha)*`...`*x[n](alpha )" ">-%\"pG6#%&alphaG*(-&%\"xG6#\"\"\"6#F&\"\"\"%$...GF.-&F*6#%\"nG6#F &F." }{TEXT -1 3 " )." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}} {SECT 1 {PARA 20 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Versuche zunaechst eine eigene Loesung!" }}} {SECT 1 {PARA 20 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "restart; n:=20;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Gleichungen:=\{2*x[1]-x[2] =1+alpha,\n seq(-x[i-1]+2*x[i]-x[i +1]=i+alpha,i=2..n-1),\n -x[n-1]+2*x[n] =n+alpha\};" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Unbekannte:=\{seq(x[i],i=1..n)\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Loesung:=solve(Gleichungen,Unbeka nnte);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(Loesung,prod uct(x[i],i=1..n));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\"=0,al pha); " }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT 333 8 "Aufgabe:" }{TEXT -1 47 " Finde alle lokalen Extrema /Sattelpunkte von " } {XPPEDIT 18 0 "f(x,y,z)=x^2+y^3-x*y+z*x-z " "/-%\"fG6%%\"xG%\"yG%\"zG, ,*$F&\"\"#\"\"\"*$F'\"\"$F,*&F&F,F'F,!\"\"*&F(F,F&F,F,F(F0" }{TEXT -1 49 " und berechne die entsprechenden Funktionswerte." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 8 "L oesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Versuche zunaechs t eine eigene Loesung !" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 8 "Loesu ng:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "restart; f:=x^2+y^3-x *y+z*x-z;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Gleichungen:=\{diff(f, x)=0,diff(f,y)=0,diff(f,z)=0\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " Loesungen:=solve(Gleichungen,\{x,y,z\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "alleLoesungen:=allvalues(Loesungen);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Anzahl_der_Loesungen=nops(\{alleLoesungen\});" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f1:=subs(alleLoesungen[1],f);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f2:=subs(alleLoesungen[2],f);" }}}} }{EXCHG {PARA 0 "" 0 "" {TEXT 344 8 "Aufgabe:" }{TEXT -1 46 " Finde al le lokalen Extrema/Sattelpunkte von " }{XPPEDIT 18 0 "f(x,y,z)=x*y*z \+ " "/-%\"fG6%%\"xG%\"yG%\"zG*(F&\"\"\"F'F*F(F*" }{TEXT -1 27 " unter d er Nebenbedingung " }{XPPEDIT 18 0 "x^2+y^2+z^2=1 " "/,(*$%\"xG\"\"#\" \"\"*$%\"yG\"\"#F'*$%\"zG\"\"#F'\"\"\"" }{TEXT -1 121 " (Lagrange-Mult iplikator)! Erzeuge eine Menge von Listen, welche die Koordinaten und \+ den Funktionswert von f enthalten!" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6 "Platz;" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 8 "Loesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Versuche zunaec hst eine eigene Loesung !" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 8 "Loe sung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "restart; f:=x*y*z; \+ Neben:=x^2+y^2+z^2-1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "Gleichung en:=\{diff(f,x)+lambda*diff(Neben,x)=0,\n diff(f,y)+lambd a*diff(Neben,y)=0,\n diff(f,z)+lambda*diff(Neben,z)=0,\n \+ Neben=0\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Loesunge n:=solve(Gleichungen,\{x,y,z,lambda\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Dies ist eine Folge von Mengen " }{TEXT 335 12 "Loesunge n[i]" }{TEXT -1 67 " , in denen jeweils RootOf-Ausdruecke auftauchen. \+ Diese werden mit " }{TEXT 334 10 " allvalues" }{TEXT -1 24 " in Wurze ln verwandelt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "FolgeAllerLoesung en:=\n seq(allvalues(Loesungen[i]),\n i=1..nops([Loesungen]));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Setze mit " }{TEXT 336 4 "subs " }{TEXT -1 32 " diese Loesungen in die Liste " }{TEXT 337 9 "[x,y,z ,f]" }{TEXT -1 53 " ein und fasse diese Listen zu einer Menge zusamme n:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "Ergebnis:=\{seq( \n \+ subs(FolgeAllerLoesungen[i],[x,y,z,f])\n ,i=1..nops([Fo lgeAllerLoesungen])\n )\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(Ergebnis);" }}}}}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }