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Simon\nSchule: Isolde-Kurz-Gymnasium\nKla sse: 12 \nDatum: 19.11.1997\nFach: Mathematik (LK)\nThema: Analysis, F olgen\nStichw\366rter: Folgen, Grenzwerte\nKurzbeschreibung: \334bungs worksheet zu Folgen und Grenzwerten" }{MPLTEXT 1 0 1 "\n" }{TEXT -1 44 "Update auf Maple 8: 07.05.04 David Ausl\344nder" }}}}{EXCHG {PARA 256 "" 0 "" {TEXT 256 59 "\334bungsworksheet zur Mathe-Klausur Nr.1 / \+ Folgen, Grenzwerte" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 24 "1. Folgen und Grenzwerte" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;\nwith(plots):\n" }}}{EXCHG {PARA 258 "" 0 " " {TEXT -1 14 "Folge angeben:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 694 "n : #bestimmt divergent, monoton ste igend\n-n : #bestimmt divergent, monoton fallend\n1/n : # konvergent, beschr\344nkt, Nullfolge\n1-1/n : #konvergent und besc hr\344nkt\n\nsin(n) : #oszilierend, H\344ufungspunkte (unendl. viel e Folgenglieder), beschr\344nkt\n(-1/n)^n: #alternierend, konvergent \n(-1)^n*n: #oszilierend/alternierend \"konvergent aber auch diverge nt\"\n(-n)^n : #unbestimmt divergent\n\n[sin(i)/i,cos(Pi*i/2)] \+ : #Folge mit H\344ufungspunkten\n[sin(Pi*i/3)+1/i,cos(Pi*i/2)+1/ i]: #Folge mit vielen H\344ufungspunkten, einer Grenzfigur (-->Lissajo us-Figuren)\nevalc(sin(n)) : #reelle Folge\n[rand() *10^(-10),rand()*10^(-10)]: #Folge aus Zufallszahlen\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "\nf:= n->1-1/n;\n" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 25 "Folge zeichnen (diskret):" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot([seq([n,f(n)],n=1. .100)],style=point);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 32 "Folge z eichnen (kontinuierlich):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot([seq([n,f(n)],n=1..100)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot([seq([0,f(n)],n=1..100)],style =point,symbol=circle);\n" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 16 "Fol ge animieren:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "display(seq(plot([seq([f(n),n],n=1..i)],0..1,style=po int),i=1..50),insequence=true);" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 31 "Folgen auf eine Ebene zeichnen:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "display(seq(plot([seq([sin(i)/i,co s(i)/i],i=1..n)],style=point,symbol=circle),n=1..50)); #Animation mit \+ \"insequence = true\"" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 43 "Punkte folgen aus zwei Algorithmen zeichnen:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "HPP:=(x,y)-> display(seq(plot([seq( [x(i),y(i)],i=1..n)]),n=1..50),insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "x:= i->f(i)^2: y:= i->f(i):\nHPP(x,y); #Gute s Beispiel: x:= i->f(i)^2: y:= i->f(i): f\374r f->sin(n).\n" }}} {EXCHG {PARA 261 "" 0 "" {TEXT -1 11 "Grenzwert \"" }{TEXT 257 1 "g" } {TEXT -1 12 "\" berechnen:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g:= limit(f(n),n=infinity);\n" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 32 "Folge auf Schranken untersuchen:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 16 "untere Schranke:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "`untere Schranke`:= minimize(f(n),n,n=1..infinity); US:= %;\n" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 15 "obere Schranke:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "`obere Schranke`:= maximize( f(n),n,n=1..infinity); OS:= %;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "\nIstart:= 1: Iend:= 1000:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 316 "DrawFolge:= plot(f(n),n=Istart..Iend,color=blac k,thickness=2):\nSchrankeUnten:= \+ plot(US,n=Istart..Iend,color=blue,thickness=3):\nSchrank eOben:= plot(OS,n=Istart..Iend,color=blue, thickness=3):\ndisplay(DrawFolge,SchrankeUnten,SchrankeOben);\n" }}} {EXCHG {PARA 270 "" 0 "" {TEXT -1 86 "Graphische Ann\344herung an die \+ Schranken (obere oder untere Schranke - falls vorhanden):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "drawb:= plot(\{s eq(f(-k),k=1..25)\},color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "drawf:= plot(\{f(n)\},n=1..10,-2..2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display(\{drawf,drawb\});" }}} {EXCHG {PARA 269 "" 0 "" {TEXT -1 26 "Untersuchen auf Monotonie:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Ist die Folge monoton steigend ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "solv e(f(n+1)-f(n)>0,n);\nsolve(f(n+1)-f(n)>0,\{n\});\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "\nsimplify(f(n+1)-f(n));\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Ist die Folge monoton fallend ?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "solve(f(n+1)-f(n)<0,n);\nsol ve(f(n+1)-f(n)<0,\{n\});\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 87 "Be stimmen der Extrema (falls welche vorhanden sind) von der Folge (Annah me: Kontinuum):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(di ff(f(n),n));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 272 10 "Bemerkung:" } {TEXT -1 64 " Erscheint kein Ergebnis (also keine Extrema), so ist die Folge " }{TEXT 270 7 "streng " }{TEXT 271 7 "monoton" }{TEXT -1 1 ". " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "\ndi ff(f(n),n$1);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 259 34 "1. Konvergenzkriterium (Anwendung )" }{TEXT 264 19 " Zeigen, da\337 es ein" }{TEXT -1 1 " " }{XPPEDIT 265 0 "n[0]" "6#&%\"nG6#\"\"!" }{TEXT -1 1 " " }{TEXT 261 31 "gibt, so da\337 f\374r alle folgenden" }{TEXT 266 1 " " }{TEXT 258 1 "n" } {TEXT -1 1 " " }{TEXT 267 5 "gilt:" }{TEXT -1 1 " " }{XPPEDIT 260 0 "a bs(x[n]-g) n[0]" "6#2&%\"nG6#\"\"!F%" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Epsilon-Streifen (Umgebun g). Aufzeigen der \"noch so kleinen Umgebung\" mit " }{TEXT 262 10 "fa st allen" }{TEXT -1 20 " Folgengliedern --> " }{TEXT 263 9 "Grenzwert " }{TEXT -1 1 "." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 19 "Epsilon definieren:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "epsilon:= 1/1000; g:= g;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "\nabs(x[n]-g) " 0 "" {MPLTEXT 1 0 30 "\n\nsolve(abs(f(n)-g) " 0 "" {MPLTEXT 1 0 32 "solve(abs(f(n)-g)< epsilon,\{n\});\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "\nn[0] := expand(solve(abs(f(n)-g) = epsilon,n));\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "(n[0]+1)*`ten`;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "...Folgenglied ist die Aussage erf\374llt." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Zeichnen von " } {XPPEDIT 268 0 "n[0]" "6#&%\"nG6#\"\"!" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "ZeichneN[0]:= plot([[n[0],epsilon] ],style=point,symbol=circle,color=red):\nText:= textplot([[n[0],epsilo n*1.1,`Beginn der Umgebung (epsilon-Streifen)`],[n[0],0,`n[0]`]],color =blue):\nMarkierung:= plot([seq([n[0],i],i=-(1/epsilon)..1/epsilon)],c olor=red,thickness=1):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "(Bewe is der Konvergenz, wenn Intervallangabe ohne \"infinity\".)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "\nIstart:= 100: Iend:= 1000 0:\nDrawFolge:= plot(\{f(n),-epsilon,epsilon\},n=Istart..Iend,-epsilon *5..epsilon*5,color=black,thickness=2):\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "\ndisplay(\{DrawFolge,Markierung,ZeichneN[0],Text\} );\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;\nwith(plots):\n" }}}{EXCHG {PARA 274 "" 0 "" {TEXT -1 14 "Folge angeben:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "\nRFolge:= f(n) = 1/2*(f(n-1)+f(n-2));\n" }}}{EXCHG {PARA 275 "" 0 "" {TEXT 283 8 "Rekursiv" }{TEXT -1 21 " definierte Fol ge in " }{TEXT 284 8 "explizit" }{TEXT -1 28 " definierte Folge umwand eln:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 " explFolge:= rsolve((RFolge), f(k));\n" }}}{EXCHG {PARA 276 "" 0 "" {TEXT -1 22 "Werte einf\374gen ergibt:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "f(0):= 0: f(1):= 1:\ny:= k->explFol ge: y(k);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "\nplot([seq( [k,y(k)],k=1..50)],thickness=2);\n" }}}{EXCHG {PARA 277 "" 0 "" {TEXT -1 43 "K\374rzerer Weg, um ein Schaubild zu erhalten:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "R:= rsolve(\{f2(n) = 1/2*(f2(n-1)+f2(n-2)),f2(0)=0,f2(1)=1\}, f2(n),makeproc):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plot(R);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 271 "" 0 "" {TEXT -1 38 "Arimthmetische und geometrische \+ Folgen" }}}{EXCHG {PARA 272 "" 0 "" {TEXT -1 27 "a.) Die arithmetische Folge" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;\nwith(plots):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 43 " Def. \"arithmetisches Mittel\" (aus a und b):" }{TEXT 278 1 " " } {XPPEDIT 279 0 "(a+b)/2" "6#*&,&%\"aG\"\"\"%\"bGF&F&\"\"#!\"\"" } {TEXT 280 1 " " }{TEXT -1 12 "(Allgemein: " }{XPPEDIT 18 0 "(x[1]+x[2] +x[n])/n" "6#*&,(&%\"xG6#\"\"\"F(&F&6#\"\"#F(&F&6#%\"nGF(F(F.!\"\"" } {TEXT -1 1 ")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f(n)=(f(n+ 1)+f(n-1))/2;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 273 "" 0 " " {TEXT -1 26 "b.) Die geometrische Folge" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;\nwith(plots):\n" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 274 42 "Def. \"geometrisches Mittel\" (a us a und b):" }{TEXT 275 1 " " }{XPPEDIT 276 0 "sqrt(a+b)" "6#-%%sqrtG 6#,&%\"aG\"\"\"%\"bGF(" }{TEXT 277 1 " " }{TEXT -1 12 "(Allgemein: " } {XPPEDIT 18 0 "(x[1]*x[2]*x[n])^(1/n)" "6#)*(&%\"xG6#\"\"\"F(&F&6#\"\" #F(&F&6#%\"nGF(*&F(F(F.!\"\"" }{TEXT -1 1 ")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f(n)=sqrt(f(n+1)*f(n-1));\n" }}}{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 "" {MPLTEXT 1 0 8 "\n\n\n\n\n\n\n\n" }}}{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 21 "Stephan J. Simon 1997" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "96" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }