{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 2 1 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 2 1 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 12 0 0 0 0 0 2 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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 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 "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Stephan Simon, LK12 Mathe, 6.10.1997" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 13 "Informationen" }} {PARA 3 "" 0 "" {TEXT 261 10 "Dateiname:" }{TEXT 262 1 " " }{TEXT 263 13 "folgenmw.mws\n" }{TEXT 264 250 "Dateigr\366\337e: 12.6 KB\nName: S tephan Simon\nSchule: Isolde-Kurz-Gymnasium\nKlasse: LK12 Mathe\nDatum : 6.10.97\nFach: Mathematik\nThema: Analysis: Folgen und Grenzwerte\nS tichw\366rter: Folgen\nKurzbeschreibung: Untersuchen von Folgen und ih ren (m\366gl.) Grenzwerten" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 100 " Mathe: Folgen und Grenzwerte / Untersuchen von Folgen / Beweisen von ( nicht) vorhandenen Grenzwerten" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "with(plots,contourplot):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 38 " Bestimmen der zu untersuchenden Folge:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:= n-> 1/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"nG6\"6$%)operatorG%&arrowGF(*$9$!\"\"F(F(" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 19 "Zeichnen der Folge:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(f( n),n,color=red);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 45 "Untersuchen der Folge in \"Zahlendarstellung\":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(f(n),n=1..25);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6;\"\"\"#F#\"\"##F#\"\"$#F#\"\"%#F#\"\"&#F#\"\"'#F# \"\"(#F#\"\")#F#\"\"*#F#\"#5#F#\"#6#F#\"#7#F#\"#8#F#\"#9#F#\"#:#F#\"#; #F#\"#<#F#\"#=#F#\"#>#F#\"#?#F#\"#@#F#\"#A#F#\"#B#F#\"#C#F#\"#D" }}} {EXCHG {PARA 258 "" 0 "" {TEXT -1 44 "Bestimmen des (nicht) m\366glich en Grenzwertes " }{TEXT 257 1 "g" }{TEXT -1 1 ":" }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Grenzwert:=limit(f(n),n=in finity);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "g:= Grenzwert;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*GrenzwertG\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"gG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "if (g = 0) then print(nullfolge) fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*nullfolgeG" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 14 " Bestimmen des " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 12 "-Intervalls:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "epsilon:=1/100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% (epsilonG#\"\"\"\"$+\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Zeigen, da\337 es ein " }{XPPEDIT 18 0 "n[0]" "&%\"nG6#\"\"!" }{TEXT -1 33 " \+ gibt, so da\337 f\374r alle folgenden " }{TEXT 258 1 "n" }{TEXT -1 7 " gilt: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "abs(x[n]) " 0 "" {MPLTEXT 1 0 11 "abs(f(n)); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%$absG6#%\"nG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 14 "Berechnen der " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 46 "-We rte f\374r die die Folgenglieder kleiner als " }{XPPEDIT 18 0 "epsilo n " "I(epsilonG6\"" }{TEXT -1 29 " sind (Intervall-Bestimmung):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(abs(f(n)) " 0 "" {MPLTEXT 1 0 29 "solve(abs(f(n)) " 0 "" {MPLTEXT 1 0 23 "Istart:=10; Iend:=1000; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'IstartG\"#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%IendG\"%+5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DrawFolge:= plot(\{f(n),-epsilon,epsilon\},n=Istart..Iend,-epsilon*5..epsilon*5,c olor=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "display(\{D rawFolge\});" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 111 "Zeichnen der F olge im Konturenfeld zur besseren Veranschaulichung der (nicht) m\366g lichen Grenzwerteigenschaften:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 92 "Konturen:= contourplot(f(n),n=Istart..Iend, \+ e=-epsilon..epsilon, grid=[10,10],\nfilled=true):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display(\{DrawFolge,Konturen\});" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 37 "N\344here Untersuchung der Folge ergibt:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "abs(f(n)-'g') " 0 "" {MPLTEXT 1 0 46 "n[0]:= expand(solve(abs(f( n)-g) = epsilon,n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2-%$absG6#,&*$ %\"nG!\"\"\"\"\"%\"gGF*#F+\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %\"nG6#\"\"!\"$+\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Hier wird a uf eine Ungleichung verzichtet, um das Wesentliche nicht zu \"beeinflu ssen\"." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 16 "E inzeichnen von " }{XPPEDIT 18 0 "n[0]" "&%\"nG6#\"\"!" }{TEXT -1 18 " \+ in die Folge mit " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 10 "-Streifen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Punkt:= p lot([[n[0],epsilon]],style=point,symbol=circle,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display(\{DrawFolge,Punkt\});" }}} {EXCHG {PARA 268 "" 0 "" {TEXT -1 88 "Untersuchen der Folge zur Beobac htung von Monotonieeigenschaften (M\366glichkeit 1 oder 2):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "1.) Zur Bere chnung der Extrema verwendet man z.B. wie in der Kurvendiskussion die \+ Ableitung: Gibt es mindestens ein Extremum, so ist die zu untersuchend e Folge nicht monoton." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f 1:= n-> diff(f(n),n$1): f1(n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "s olve(f1(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$%\"nG!\"#!\"\"" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "2.) Welche Eigenschaften hat die \+ Folge ? Ist sie monoton zunehmend, so ist die folgende Differenz " } {XPPEDIT 18 0 "f[n+1]-f[n]" ",&&%\"fG6#,&%\"nG\"\"\"\"\"\"F(F(&F$6#F'! \"\"" }{TEXT -1 68 " positiv. Wenn sie monoton abnehmend ist, so ist D ifferenz negativ. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dwert :=solve(f(n+1)-f(n)>0,n);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "dwert: =solve(f(n+1)-f(n)>0,\{n\}); about(dwert); AnzahlWerte:= nops(dwert); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dwertG-%*RealRangeG6$-%%OpenG6# !\"\"-F)6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dwertG<$2%\"nG\" \"!2!\"\"F'" }}{PARA 6 "" 1 "" {TEXT -1 50 "\{n < 0, -1 < n\}:\n noth ing known about this object" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,Anza hlWerteG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(f (n+1)-f(n)>0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2\"\"!,$*&,&%\"nG\" \"\"F)F)!\"\"F(F*F*" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 50 "Ausgeben aller m\366glichen Ergebnisse der Differenz:" }{MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dwerte:= op(dwert):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "for i to AnzahlWerte do dwerte[i] od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#2%\"nG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#2!\"\"%\"nG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Ist die Folge monoton zunehmend oder monoton abnehmend ?" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "if (dwerte[1] = 'n<0') or ( dwerte[2] = 'n<0') then print(monoton_abnehmend) else print(monoton_zu nehmend) fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2monoton_abnehmendG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "?op" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 24 "Bere chnen der Schranken:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Obere Schranke:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "minlim:= minimize(n,n,0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'minlimG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Untere Schranke:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "maxlim:= maximize(n,n,0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'maxlimG%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "if (minlim = infinity) then print(keine_untere_Schran ke) else print(untere_Schranke) fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "if (maxlim = infinity) then print(keine_obere_Schranke) else print (obere_Schranke) fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0untere_Schra nkeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5keine_obere_SchrankeG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(1-f(n) " 0 "" {MPLTEXT 1 0 64 "min limline:= plot(minlim,n=Istart..Iend,color=blue,thickness=3):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "maxlimline:= plot(maxlim,n=Istart.. Iend,color=blue,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display(DrawFolge,minlimline,maxlimline,Punkt);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Stephan Simon, 6.10.1997" } {MPLTEXT 1 0 0 "" }}}}{MARK "1 1 3" 16 }{VIEWOPTS 1 1 0 1 1 1803 }