{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 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 22 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 13 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 267 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 18 269 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 18 272 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 273 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 18 274 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 18 276 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 18 277 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 278 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 13 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 1 15 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 1 13 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 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 "Heading 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 "Maple Out put" 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 "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "M aple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 256 1 {CSTYLE "" -1 -1 " " 1 15 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 "" 4 257 1 {CSTYLE "" -1 -1 "" 1 15 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 "" 0 258 1 {CSTYLE "" -1 -1 "" 1 15 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 "" 4 259 1 {CSTYLE "" -1 -1 "" 1 15 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 56 " 27.11.1997 \+ " }{TEXT 256 16 "Kleines Referat " } {TEXT 257 1 " " }{TEXT 264 56 " \+ " }{TEXT -1 16 "Christoph Ankele" }}{PARA 0 "" 0 "" {TEXT -1 172 " \+ \+ LK-Mathe" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 7 "Referat" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 256 "" 0 "" {TEXT 260 5 "Thema" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 17 "Aufgabenstellung:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Untersuche die Fibonacci- Folge" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 261 19 "Wer war Fibonacci ?" }}{PARA 258 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 19 "Leonardo Fibonacci " } {TEXT -1 22 "( 1170 bis nach 1240 )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 116 "Leonardo Fibonacci ( eigentlich Leonard o von Pisa ) war ein italienischer Kaufmann und Mathematiker, dessen V ater in" }}{PARA 0 "" 0 "" {TEXT -1 115 "Nordamerika Handel getrieben \+ hatte. Im Jahre 1202 schrieb Fibonacci ein Rechenbuch ( \"Liber abacci \" ), das \374ber die" }}{PARA 0 "" 0 "" {TEXT -1 115 "schriftlichen R echenmethoden der indisch-arabischen Ziffern berichtet. Au\337erdem be fasst es sich mit arithmetischen " }}{PARA 0 "" 0 "" {TEXT -1 90 "Oper ationen, Vermischungsrechnung, Einzelaufgaben \374ber Preise und Waren , Tauschhandel etc." }}{PARA 0 "" 0 "" {TEXT -1 116 "Dar\374ber hinaus hat Leonardo Fibonacci, der als erster Fachmathematiker des europ\344 ischen Feudalismus bezeichnet wird," }}{PARA 0 "" 0 "" {TEXT -1 89 "im Anschlu\337 an antike und islamische Quellen Selbst\344ndiges zur Zah lentheorie beigetragen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 16 "Fibonacci-Folge:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Die Folge ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 277 0 "F [n]" "&%\"FG6#%\"nG" }{TEXT 279 1 " " }{TEXT 281 15 "( n Element N )" }{TEXT 282 1 " " }{TEXT 283 3 "mit" }{TEXT 278 1 " " }{XPPEDIT 267 0 " F[1]" "&%\"FG6#\"\"\"" }{TEXT 268 3 " = " }{XPPEDIT 269 0 "F[2]" "&%\" FG6#\"\"#" }{TEXT 270 5 " =1 " }{TEXT 266 3 "und" }{TEXT 271 2 " " } {XPPEDIT 272 0 "F[n+2" "&%\"FG6#,&%\"nG\"\"\"\"\"#F'" }{TEXT 273 3 " = " }{XPPEDIT 274 0 "F[n]" "&%\"FG6#%\"nG" }{TEXT 275 3 " + " } {XPPEDIT 276 0 "F[n+1" "&%\"FG6#,&%\"nG\"\"\"\"\"\"F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "wird als Fibonacci-Fol ge bezeichnet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Sie beginnt mit den Elementen 1, 1, 2, 3, 5, 8, 13, 21, 3 4, 55, 89, 144, 233..." }}{PARA 0 "" 0 "" {TEXT -1 89 "Es werden also \+ jeweils die beiden aufeinanderfolgenden Glieder der Folge zusammengez \344hlt." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 259 "" 0 "" {TEXT 262 29 "Die Fibonacci-Folge mit Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Nun geht`s los:" }}{PARA 0 "" 0 "" {TEXT -1 66 "Auch in Maple gibt es einen eigenen Befehl f\374r di e Fibonacci-Folge" }}{PARA 0 "" 0 "" {TEXT -1 10 "Er lautet:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart:with(combinat,fibona cci);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%*fibonacciG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Lassen wir uns einfach mal eine Zahl dies er Folge ausgeben" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "fibonacci(7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Also 13. Und wie sieht es mit einer " }{TEXT 263 9 "negativen" }{TEXT -1 10 " Zahl aus." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fibonacci(-7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Auch 13 ? Nochmal " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fibonacci(-8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#@" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Also ein negativer Wert." }}{PARA 0 "" 0 "" {TEXT -1 31 "Versuchen wir es nochmal mit -9" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fibonacci(-9);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#M" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Wieder positiv. " }}{PARA 0 "" 0 "" {TEXT -1 95 "Es sieht so a us, da\337 wir eine Abfolge von abwechselnd positiven und negativen Za hlen bekommen." }}{PARA 0 "" 0 "" {TEXT -1 49 "Eigentlich ja logisch, \+ denn - und - gibt eben +. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Am besten wird dies wohl durch eine Zahlenfolge mit se quence vereanschaulicht. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "seq(fibonacci(i),i=-25..0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6<\"&D](!&oj%\"&d'G!&6x\"\"&Y4\"!%ln\"%\"=%!%%e#\"%(f\"!$ ()*\"$5'!$x$\"$L#!$W\"\"#*)!#b\"#M!#@\"#8!\")\"\"&!\"$\"\"#!\"\"\"\"\" \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Sieht so aus, als liegen wir mit unserer Annahme richtig." }}{PARA 0 "" 0 "" {TEXT -1 48 "Scha uen wir uns nochmal die positiven Zahlen an." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fibonacci(13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$ L#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 " 233. Versuchen wir es noch mal mit 14." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fibonacci(14);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$x$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Ebenfalls positiv. Sollte sie auch sein, denn wir haben j a keine negativen Vorzeichen. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 66 "Schauen wir uns auch die positiven Zahlen nochmal \+ mit sequence an." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "seq(fibonacci(i ),i=0..25);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6<\"\"!\"\"\"F$\"\"#\"\"$ \"\"&\"\")\"#8\"#@\"#M\"#b\"#*)\"$W\"\"$L#\"$x$\"$5'\"$()*\"%(f\"\"%%e #\"%\"=%\"%ln\"&Y4\"\"&6x\"\"&d'G\"&oj%\"&D](" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Sieht eigentlich auch so aus, wie wir uns das vorgeste llt haben." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Au\337erdem ist zu bemerken, da\337 die Werte bei dieser Folge sehr s chnell extrem gro\337 werden." }}{PARA 0 "" 0 "" {TEXT -1 41 "Nun stel len wir noch eine Folge in einem " }{TEXT 284 4 "Plot" }{TEXT -1 5 " d ar:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "restart:with(combinat, fibonacci):" }}{PARA 0 "" 0 " " {TEXT -1 53 "Wir legen Punkte von 0 bis 30 auf der x-Achse fest..." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Punkte:=seq([i,fibonacci(i)],i=0. .30);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'PunkteG6A7$\"\"!F'7$\"\"\" F)7$\"\"#F)7$\"\"$F+7$\"\"%F-7$\"\"&F17$\"\"'\"\")7$\"\"(\"#87$F4\"#@7 $\"\"*\"#M7$\"#5\"#b7$\"#6\"#*)7$\"#7\"$W\"7$F7\"$L#7$\"#9\"$x$7$\"#: \"$5'7$\"#;\"$()*7$\"#<\"%(f\"7$\"#=\"%%e#7$\"#>\"%\"=%7$\"#?\"%ln7$F9 \"&Y4\"7$\"#A\"&6x\"7$\"#B\"&d'G7$\"#C\"&oj%7$\"#D\"&D](7$\"#E\"'$R@\" 7$\"#F\"'=k>7$\"#G\"'6yJ7$\"#H\"'HU^7$\"#I\"'S?$)" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 96 "und stellen sie mit den dazugeh\366rigen Fibonacci -Zahlen als Folge in einem Plot mit sequence dar." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(\{Punkte\},color=blue,style= point);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6#7A7$$\"\"&\"\"!F(7$$\"\"'F*$\"\")F*7$$\"\"(F*$ \"#8F*7$$\"#9F*$\"$x$F*7$$\"#;F*$\"$()*F*7$F3$\"$L#F*7$$\"#7F*$\"$W\"F *7$$\"#AF*$\"&6x\"F*7$$\"#BF*$\"&d'GF*7$$\"#EF*$\"'$R@\"F*7$$\"#=F*$\" %%e#F*7$$\"#>F*$\"%\"=%F*7$$\"#?F*$\"%lnF*7$$\"#F*7$$\"#GF*$\"'6yJF*7$ $\"#HF*$\"'HU^F*7$F.F_p7$$\"\"*F*$\"#MF*7$$\"#5F*$\"#bF*7$$\"#6F*$\"#* )F*7$$\"\"#F*Fjp7$$\"\"$F*Fas7$$\"\"%F*Fds-%'COLOURG6&%$RGBGF*F*$\"*++ ++\"!\")-%&STYLEG6#%&POINTG" 2 365 365 365 5 0 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 182 3 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Bei positiven Zahle n bekommen wir eine " }{TEXT 285 23 "exponentiell steigende " }{TEXT -1 6 "Kurve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Jetzt stellen wir noch eine neg ative Zahlenfolge,0 bis -30, in einem Plot mit sequence dar." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Punkte:=seq([i,fibonacci(i)] ,i=-30..0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'PunkteG6A7$!#I!'S?$)7$!#H\"'HU^7$!#G!'6yJ7$!#F\"'=k> 7$!#E!'$R@\"7$!#D\"&D](7$!#C!&oj%7$!#B\"&d'G7$!#A!&6x\"7$!#@\"&Y4\"7$! #?!%ln7$!#>\"%\"=%7$!#=!%%e#7$!#<\"%(f\"7$!#;!$()*7$!#:\"$5'7$!#9!$x$7 $!#8\"$L#7$!#7!$W\"7$!#6\"#*)7$!#5!#b7$!\"*\"#M7$!\")FB7$!\"(\"#87$!\" 'Fco7$!\"&\"\"&7$!\"%!\"$7$F^p\"\"#7$!\"#!\"\"7$Fcp\"\"\"7$\"\"!Fgp" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(\{Punkte\},color=gree n,style=point);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6#7A7$\"\"!F (7$$!#9F($!$x$F(7$$!#8F($\"$L#F(7$$!#7F($!$W\"F(7$$!#6F($\"#*)F(7$$!#5 F($!#bF(7$$!\"*F($\"#MF(7$$!#HF($\"'HU^F(7$$!#GF($!'6yJF(7$$!#FF($\"'= k>F(7$$!#EF($!'$R@\"F(7$$!#DF($\"&D](F(7$$!#CF($!&oj%F(7$$!#BF($\"&d'G F(7$$!#AF($!&6x\"F(7$$!#@F($\"&Y4\"F(7$$!#?F($!%lnF(7$$!#>F($\"%\"=%F( 7$$!#=F($!%%e#F(7$$!# " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Bei einer negativen Zahle nfolge bekommen wir durch die abwechselnd positiven und negativen Zahl en eine oszillierende Folge." }}}}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }