{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 21 "L\366sen von Gleichungen " }}{PARA 19 "" 0 "" {TEXT -1 53 "von Baumann angepasst f\374r Braille nutzer von D.Stephan" }}{PARA 19 "" 0 "" {TEXT -1 25 "06 Gleichungen L \366sung.mws" }}{PARA 0 "" 0 "" {TEXT -1 73 "Wir kommen nun zu dem (f \374r uns) vielleicht m\344chtigsten Befehl von Maple: " }{TEXT 256 5 "solve" }}{PARA 0 "" 0 "" {TEXT -1 103 "Maple l\366st (fast) alle Glei chungen, auch Lineare Gleichungssysteme oder Ungleichungen mit einem R eturn." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Grundlegendes zu " }{TEXT 260 5 " solve" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 44 "Definition einer Gleichu ng, L\366sung und Probe" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g l1:=3*x-5=8;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(gl1);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(x=%,gl1);" }{TEXT -1 19 " Gleich die Probe !" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 "Varianten im Umgang mit solve" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Was rechnet Maple aus ?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(3*x-5);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(3*x-5=8);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " solve(3*x-5-8);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 72 "Was berechnet Maple, wenn dem Befehl solve nur ein Term \374bergeben wird ?" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 40 "Ei ne Gleichung enth\344lt mehrere Variablen" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gl2:= 4*x-a=9;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Wir versuchen die L\366sung wie vo rher zu erhalten. Seltsames Ergebnis ?? Oder doch nicht ?" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 11 "solve(gl2);" }}{PARA 11 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(gl2,x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " solve(gl2,a);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Fazit: Was ist also zu beachten ?" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "Quadratische Gleichungen" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 46 "Erwartete Ergebnisse: Quadratische Gleich ungen" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gl3:=x^2+5*x-6=0;so lve(gl3);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "gl4:=4*x^2-8*x-1= 0;solve(gl4);evalf(%,3);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gl5:=x^2-2*x+1=0;solve(gl5);" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Ist die L\366sungsangabe von gl5 n icht komisch ?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "factor(gl5);" } {TEXT -1 12 " Nun klar ? " }{TEXT 257 8 "Nebenbei" }{TEXT -1 60 ": Der Befehl factor funktioniert auch bei einer Gleichung !!" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gl6: =x^2+6*x+t=0;solve(gl6,x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " factor(gl6);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "Was macht Maple nun ?" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Achtung: Jetzt wird es abenteuerlich. Berechne die L\366 sungsmenge mit Papier und Bleistift. Was stellst du fest ?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gl7:=2*x^2+3*x+5=0;solve(gl7);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fac tor(gl7);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 32 "Hoffentlich etwas ganz Bekanntes" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "QGL:=a*x^2+b*x+c=0;Mitternachtsformel:=solve(Q GL,x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "Weitere Gleichungen" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "Grad ist h\366her als 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "x^4-13*x^2+36=0;solve(%);factor(%%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "x^3+8*x^2-9*x=0;solv e(%);factor(%%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "2*x^6-22*x^4+36*x^2=0;solve(%);factor(%%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "x^5-10*x^4+39*x^3-74*x^2+68*x-24=0;solve(%);factor(%%);" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "(x^2+2)^2+3*(2*x+1)=(3*x+1)^ 2;solve(%);factor(%%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 26 "Bruchgle ichungen mit Probe" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "gl8:=1 /x^2+1/(2*x)=3;solve(gl8);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " subs(x=-1/2,gl8);subs(x=2/3,gl8);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "gl9 := (x+1 1)/(2*x+1)=(x+3)/(5+x);solve(gl9);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "subs(x=-4,gl9);subs(x=13,g l9);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "gl10:=x/a-a/x=3/2;solve (gl10,x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "subs(x=-1/2*a,gl10);subs(x=2*a,gl10);" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "Wurzelgleichungen mit Probe" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "gl11:=sqrt(x-2)+14=x;solve(gl11);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(x=18,gl11);evalf(%);" } }{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 21 "Kleine Zusatzrechnung" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "lhs(gl11)-14;LS:=%^2;rhs(gl11)-14;%^2;RS:=expand(%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 " " {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "gl12:=LS=RS ;solve(%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(x=11,gl11);e valf(%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 173 "Maple hat die Probe offensichtlich schon gemacht; wir h \344tten mit Papier und Bleistift auch x = 11 als L\366sung gefunden. \+ Sie h\344tte aber auch bei uns nicht die Probe bestanden." }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Maple kann sch\366ne Wurzeln schreiben:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "gl13:=sqrt(x+sqrt(x))=30;solve(gl 13);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(x=%,gl13);evalf(%) ;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" } }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 28 "Trigonometrische Gleichungen" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(sin(x) = 0.75);" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 " Sp\344ter mehr." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "Numerische \+ L\366sungen mit " }{TEXT 258 6 "fsolve" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Mit fsolve (floating) kann man Maple anweisen, gleich num erische L\366sungen zu suchen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "gl7;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 11 "Vergleiche:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve (gl7); " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve (gl7); " }{TEXT -1 28 "Es gibt keine reelle L \366sung." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "3*x^2-5*x-4;fs olve(%);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Fsolve gibt also nur " }{TEXT 259 6 "reelle" }{TEXT -1 13 " L\366sungen an." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "Lineare Gleichungssyteme (LGS)" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "Eindeu tige L\366sungen" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "Die Gleichung en und die Variablen, nach denen aufgel\366st werden soll, m\374ssen s olve in einer Mengenklammer (Alt GR+7 bzw 0) \374bergeben werden." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "lg1:=5*x-2*y=24;lg2:=x+3*y=-2;" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(\{lg1,lg2\},\{x,y\});" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "lg3:=x+y+z=6;lg4:=-x+2*y-3* z=-7;lg5:=-x-4*y+2*z=-3;solve(\{lg3,lg4,lg5\},\{x,y,z\});" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 35 "Keine oder unendlich viele L\366sungen" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Es folgt eine unl\366sbares LGS. \+ Maple verf\344hrt nach dem Motto: \"Keine Antwort ist auch eine Antwor t\"." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(\{x-2*y=-2,x-2*y=2\}, \{x,y\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Beispiel f\374r unendlich viele L\366sung en:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(\{x-2*y=-2,-x+2*y=2\}, \{x,y\});" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Interpretiere diese L\366sungsmenge. Gib 3 verschiedene L \366sungspare an." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 "Ungleichu ngen" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ugl1:=x-2<3;" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(ugl1);" }}{PARA 0 "" 0 "" {TEXT -1 93 "Schau dir diese L \366sungsdarstellung von Maple an: Alle reellen Zahlen, die kleiner al s 5 sind." }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Es gibt auc h eine f\374r uns leichter lesbare Darstellung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(ugl1,\{x\});" }{TEXT -1 59 "Die L\366sungsmenge als Menge (also mit geschweiften Klammern)" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ugl2:=x^2+x-2>0;solve(ugl2,\{x\});" } {TEXT -1 39 "Wo liegt die Parabel \374ber der x-Achse ?" }}{PARA 0 "" 0 "" {TEXT -1 36 "Die beiden L\366sungsmengen m\374ssen mit " }{TEXT 261 4 "oder" }{TEXT -1 91 " verkn\374pft werden. Skizziere zum bessere n Verst\344ndnis die Parabel in ein Koordinatensystem." }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Wir nehmen diesselbe Parabel und \+ wollen wissen, f\374r welche x-Werte das Schaubild unter der x-Achse l iegt." }}{PARA 0 "" 0 "" {TEXT -1 64 "Betrachte vor allem die neue Sch reibweise der L\366sungsmenge: Nur " }{TEXT 262 4 "eine" }{TEXT -1 51 " Klammer. Verkn\374pfung der beiden Ungleichungen mit " }{TEXT 263 3 "und" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ugl3:=x^2+x -2<0;solve(ugl3,\{x\});" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "7 6 2 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }