{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 "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }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 1 }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 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helveti ca" 1 10 0 0 255 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Norma l" -1 258 1 {CSTYLE "" -1 -1 "Helvetica" 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 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 256 25 "Moderne Physik mit Mapl e " }}{PARA 258 "" 0 "" {TEXT 257 9 "PDF-Buch " }{URLLINK 17 "Moderne \+ Physik mit Maple" 4 "http://mikomma.de/fh/modphys.pdf" "" }}{PARA 258 "" 0 "" {TEXT -1 19 "Update auf Maple 10" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 13 "Kapitel 4.1.2" }}{PARA 258 " " 0 "" {TEXT -1 22 "Worksheet wirf3_10.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "c International Thomson Publishing Bonn 1995 filename: wirf3.ms" } }{PARA 0 "" 0 "" {TEXT -1 103 "Autor: Komma \+ Datum: 28.3.94" }} {PARA 0 "" 0 "" {TEXT -1 54 "Thema: Wirkungsfunktion des harmonischen Oszillators:" }}{PARA 0 "" 0 "" {TEXT -1 75 " N\344herung sl\366sung durch schwaches Extremum der Wirkungsfunktion," }}{PARA 0 " " 0 "" {TEXT -1 75 " wenn die Ortsfunktion als Polynom n-t en Grades angesetzt wird." }}{PARA 0 "" 0 "" {TEXT -1 68 " \+ Vergleich mit der Reihenentwicklung der exakten L\366sung." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 194 "Die Bewegung des ha rmonischen Oszillators kann ebenfalls \374ber das schwache Minimum der Wirkungsfunktion n\344herungsweise bestimmt werden, wenn wir zum Bei spiel ein Polynom n-ten Grades ansetzen. " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "T:=m/2*v^2: v:=diff(x(t),t): L:=T-V: S:=int(L,t=t0..t1): t0:=0:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 131 "Das Polynom n-ten Grades durch die zwei Punkte (0|0) u nd (t1|x1) bauen wir zur Abwechslung mit Hilfe des Punkt-Operators (ca t) auf:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "xx:=proc(t) local xx,i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "xx:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "for i to \+ n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "xx:=xx+a||i*t^i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "RETURN(xx);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxGf*6#% \"tG6$F$%\"iG6\"F*C%>8$\"\"!?(8%\"\"\"F1%\"nG%%trueG>F-,&F-F1*&(%\"aGF 0F1)9$F0F1F1-%'RETURNG6#F-F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Ein Koeffizi ent (z.B. a1) l\344\337t sich durch die Bedingung x(t1)=x1 und die and eren Koeff. ausdr\374cken: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "as1:=solve(xx(t1)=x1,a1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ as1G,$*&,6*&%#a5G\"\"\")%#t1G\"\"&F*F**&%#a2GF*)F,\"\"#F*F**&%#a3GF*)F ,\"\"$F*F**&%#a4GF*)F,\"\"%F*F**&%#a9GF*)F,\"\"*F*F**&%#a6GF*)F,\"\"'F *F**&%#a7GF*)F,\"\"(F*F**&%#a8GF*)F,\"\")F*F**&%$a10GF*)F,\"#5F*F*%#x1 G!\"\"F*F,FOFO" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Unser CAS liefert uns auf Knopfdru ck die Weg-Zeit-Funktion zu den aufgestellten Bedingungen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x:=t->subs(a1=as1,xx(t)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6*(,6*&%#a5G\"\"\")%#t1G\"\"&F(F(*&%#a2GF()F*\"\"#F(F( *&%#a3GF()F*\"\"$F(F(*&%#a4GF()F*\"\"%F(F(*&%#a9GF()F*\"\"*F(F(*&%#a6G F()F*\"\"'F(F(*&%#a7GF()F*\"\"(F(F(*&%#a8GF()F*\"\")F(F(*&%$a10GF()F* \"#5F(F(%#x1G!\"\"F(F*FM%\"tGF(FM*&F-F()FNF/F(F(*&F1F()FNF3F(F(*&F5F() FNF7F(F(*&F'F()FNF+F(F(*&F=F()FNF?F(F(*&FAF()FNFCF(F(*&FEF()FNFGF(F(*& F9F()FNF;F(F(*&FIF()FNFKF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Und mit dem qudratischen Potential" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k:='k':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "V:=1/2*k*x(t)^2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "erhalten wir f\374r die Wirkung einen etw as l\344ngeren Ausdruck ..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Ss:=simplify(S);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#SsG,$*(\"*g DzK#!\"\"%#t1GF(,fw**\")g()zQ\"\"\")F)\"\"%F-%\"mGF-)%#a2G\"\"#F-F-** \"*[yA!QF-)F)\"#;F-F0F-)%#a8GF3F-F-**\"(w)zQF-)F)\"\"'F-%\"kGF-F1F-F(* *\"*+q`k#F-)F)\"#7F-F0F-)%#a6GF3F-F-**\"()Go))F-)F)\"\")F-F>F-)%#a3GF3 F-F(**\")g?%\\#F-F>F-)%$a10GF3F-)F)\"#AF-F(**\")sEp?F-F6F-F>F-)%#a7GF3 F-F(**\"*?d@'\\F-)F)\"#?F-F0F-FMF-F-**\")?H$H\"F-)F)\"#5F-F>F-)%#a4GF3 F-F(**\")]Kl=F-)F)\"#9F-F>F-FCF-F(**\")Cq6$*F-FQ% F-)F)\"#=F-F0F-)%#a9GF3F-F-**\"*g\"GBKF-F[oF-F0F-FSF-F-**\"*gBl\\\"F-F GF-F0F-FgnF-F-**\")gT7;F-FAF-F>F-)%#a5GF3F-F(**\"*?n#p?F-FenF-F0F-F[pF -F-**F,F-)F)F3F-F>F-)%#x1GF3F-F(**\")?>wBF-FWF-F>F-FcoF-F(**\")WjOAF-F aoF-F>F-F8F-F(*,\")3[n[F-F>F-FdoF-FNF-)F)\"#@F-F(*,\")C6F-FN F-F9F-F(*,\"*S-F-F0F-FdoF-FNF-F-*,\")C#)3YF-F_qF-F>F-FdoF- F9F-F(*,\")?_EXF-F_qF-F>F-FNF-FTF-F(*,\"*S=qi)F-FaoF-F0F-FNF-F9F-F-*, \")]FzUF-FaoF-F>F-FNF-FDF-F(*,\")+CEWF-FaoF-F>F-FdoF-FTF-F(*,\"*gRx9)F -)F)\"#F-FdoF-FDF-F(*,\")f>+VF-F]rF-F>F-F9F-FTF-F(*,\")&f\"\\RF-F]rF-F> F-FNF-F\\pF-F(*,\"))pQ2%F-F6F-F>F-F9F-FDF-F(*,\")%)zpQF-F6F-F>F-FdoF-F \\pF-F(*,\")%))=\\$F-F6F-F>F-FNF-FhnF-F(*,\"*!ox$)pF-F6F-F0F-FNF-FDF-F -*,\"*#>O\\uF-F6F-F0F-FdoF-FTF-F-*,\"*+;7l'F-)F)\"#:F-F0F-FdoF-FDF-F-* ,F^sF-FcsF-F0F-F9F-FTF-F-*,\"*S%4')fF-FcsF-F0F-FNF-F\\pF-F-*,\")C-pPF- FcsF-F>F-F9F-F\\pF-F(*,\")owEGF-FcsF-F>F-FNF-FJF-F(*,\")]1ERF-FcsF-F>F -FTF-FDF-F(*,\")SQEMF-FcsF-F>F-FdoF-FhnF-F(*,\"*+#\\niF-F[oF-F0F-F9F-F DF-F-*,\"*Sy-t&F-F[oF-F0F-FdoF-F\\pF-F-*,\"*SA\\$[F-F[oF-F0F-FNF-FhnF- F-*,\")?r!z\"F-F[oF-F>F-FNF-F2F-F(*,\")oJyFF-F[oF-F>F-FdoF-FJF-F(*,\") ?FQOF-F[oF-F>F-F\\pF-FTF-F(*,\")CmULF-F[oF-F>F-F9F-FhnF-F(*,\"*SE=V&F- )F)\"#8F-F0F-F9F-F\\pF-F-*,\"*S))=\\$F-F`uF-F0F-FNF-FJF-F-*,\"*+9)>eF- F`uF-F0F-FTF-FDF-F-*,\"*?^el%F-F`uF-F0F-FdoF-FhnF-F-*,\")+ej F-FdoF-F2F-F(*,\")]F-F\\pF-FDF-F(*,\")K\"fr#F-F`uF-F>F-F9F -FJF-F(*,\")S\")>eF-FAF-F>F-FNF-FbpF-F-*,\")+BLKF-F`uF-F>F-FhnF-FTF-F( *,\")%3$GF-F9F-F2F-F(*,\")]E'3$F-FAF-F>F-FhnF-FDF-F(*,\")GhLE F-FAF-F>F-FTF-FJF-F(*,\")gXVcF-)F)\"#6F-F>F-FdoF-FbpF-F-*,\"*SmY!>F-FA F-F0F-FNF-F2F-F-*,\"*gtgQ$F-FAF-F0F-FdoF-FJF-F-*,\"*S5\"z]F-FAF-F0F-F \\pF-FTF-F-*,\"*g@UW%F-FAF-F0F-F9F-FhnF-F-*,\")k#=V&F-FenF-F>F-F9F-Fbp F-F-*,\"*[SB'=F-FjvF-F0F-FdoF-F2F-F-*,FguF-FjvF-F0F-F\\pF-FDF-F-*,\"*% e4fKF-FjvF-F0F-F9F-FJF-F-*,\"*3m->%F-FjvF-F0F-FhnF-FTF-F-*,\")'z7o\"F- FjvF-F>F-FTF-F2F-F(*,\")%>>_#F-FjvF-F>F-FDF-FJF-F(*,\")O>#)GF-FjvF-F>F -F\\pF-FhnF-F(*,\")!oJ<&F-)F)\"\"*F-F>F-FTF-FbpF-F-*,\"*!)31\"=F-FenF- F0F-F9F-F2F-F-*,\"*+w)zQF-FenF-F0F-FhnF-FDF-F-*,\"*!3!R5$F-FenF-F0F-FT F-FJF-F-*,\")o([O#F-FenF-F>F-F\\pF-FJF-F(*,\")]h;;F-FenF-F>F-FDF-F2F-F (*,\")]%)\\[F-FGF-F>F-FDF-FbpF-F-*,\"*?Wfu\"F-FexF-F0F-FTF-F2F-F-*,\"* +2*4HF-FexF-F0F-FDF-FJF-F-*,FcuF-FexF-F0F-F\\pF-FhnF-F-*,\")=$R8#F-Fex F-F>F-FhnF-FJF-F(*,\")qBC:F-FexF-F>F-F\\pF-F2F-F(*,\")S9MWF-)F)\"\"(F- F>F-F\\pF-FbpF-F-*,\")+n&Q\"F-FGF-F>F-FhnF-F2F-F(*,\"*S'[gEF-FGF-F0F-F \\pF-FJF-F-*,\"*+/Gm\"F-FGF-F0F-FDF-F2F-F-*,F'F-F^zF-F0F-FhnF-FJF-F-*, \"*S]>b\"F-F^zF-F0F-F\\pF-F2F-F-*,F,F-FF-FhnF-FbpF-F-*,\")G'R;\"F -F^zF-F>F-FJF-F2F-F(*,\"*ObnR\"F-FF-FJF-FbpF-F-*,\"*!G'R;\"F-F`[lF-F0F-FJF-F2F-F-*,\")!Q*R>F-F.F- F>F-F2F-FbpF-F-*(Fc[lF-F0F-FapF-F-F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Wir stellen wieder unser Gleichungssystem auf und lassen \+ es l\366sen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#x(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sys:=se q(diff(Ss,a||j),j=2..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " #sys;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "#a.(2..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "# Probleme in R4!erst Werte einsetzen!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "# R5 packt es wieder, aber langsame r als R3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# schnelle L \366sung in R6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sol:=solv e(\{sys\},\{a||(2..n)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "#sol;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 413 "Wenn Sie sich die L\366 sung xs im Worksheet ausgeben lassen, werden Sie sicher meine Begeiste rung f\374r dieses CAS teilen: Probleme, die von der Struktur her einf ach sind, jedoch einen immensen Rechenaufwand bedeuten w\374rden, woll te man sie von Hand l\366sen, sind ein gefundenes Fressen f\374r Maple . Hier kann man zuschauen, wie Quantit\344t in Qualit\344t umschl\344g t, und - was ebenso wichtig ist - man kann damit weiterarbeiten." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "xs:=subs(sol,x(t)):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Nun k\366nnen wir Zahlen f\374r di e Federkonstante, die Masse und den Endpunkt einsetzen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "k:=2: m:=1/4: t1:=2: x1:=3:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "und f\374r eine vergleichende Dars tellung die exakte L\366sung der Newton-DGL parat stellen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "exakt:=rhs(dsolve(\{diff(y(t),t$2)= -k/m*y(t),y(0)=0,y(t1)=3\},y(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&exaktG,$*(\"\"$\"\"\"-%$sinG6#,$*&\"\"%F(\"\"##F(F/F(!\"\"-F*6#,$*( F/F(F/F0%\"tGF(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "si mplify(exakt,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"$\"\"\" -%$sinG6#,$*&\"\"%F&\"\"##F&F-F&!\"\"-F(6#,$*(F-F&F-F.%\"tGF&F&F&F&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "#plot(\{exakt\},t=-1..t1+1,-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{exakt,xs\},t=-1..t1+1,-10..1 0);" }}{PARA 13 "" 1 "" {GLPLOT2D 681 681 681 {PLOTDATA 2 "6&-%'CURVES G6$7dq7$$!\"\"\"\"!$\"37AqnF]ow:!#<7$$!3ommm;p0k&*!#=$\"3[/R%))pdN;#F- 7$$!3PLLLLQ6G\"*F1$\"3edFk@zd^eXq[F-7$$!3Bnm\" z%4&QW'F1$\"3M)4v6TCl&\\F-7$$!3'*****\\7(p,B'F1$\"3yk?'*>M\\C]F-7$$!3z LL3x%)[;gF1$\"3)=5=xn9T2&F-7$$!3hnmmTs!G!eF1$\"3r'Q)Q%)p?0^F-7$$!3nMLL $Q(p/cF1$\"3iz;'3q!Q<^F-7$$!3i+++Dve1aF1$\"3N?8^i5\\8^F-7$$!3cmmmmwZ3_ F1$\"3=[O$)y-b$4&F-7$$!3iLLL3yO5]F1$\"3Q4laV4id]F-7$$!37nmmT_5+YF1$\"3 `s?'\\Z?H$\\F-7$$!3i+++vE%)*=%F1$\"3#=6D!)fr=u%F-7$$!3)RLL$3WDTLF1$\"3 /15bvF?[TF-7$$!3Znmm\"*4K=HF1$\"3%**4#4j;$3w$F-7$$!3'4++]d(Q&\\#F1$\"3 P\"=nT*yq>LF-7$$!3cLL$3d[.1#F1$\"3uoYuL:_;GF-7$$!3:mmmm&4`i\"F1$\"3[hH 6-[uqAF-7$$!3p*****\\(p7U7F1$\"3Kk;!)\\zHh$\"3KwN !fe#=J7F-7$$!3+qmm;arvUF_s$\"35GA&4)**HuhF17$$\"3HI#*******H,Q!#@$!3VR )yAvFE]&!#?7$$\"3%[*******RXpVF_s$!3vn[+V(z*3jF17$$\"3Q(*******\\*3q)F _s$!3EI-+P8v8FF-7$$\"3s****\\([Wdb*F1$!3% >@S\"=kXu@F-7$$\"3Immm;kD!)**F1$!3!yTIYP]Qg\"F-7$$\"3[mmT+07U5F-$!3e;: >Kv8o)*F17$$\"3Mmm;f`@'3\"F-$!3v+9))fFZWNF17$$\"32LLLj+gC6F-$\"3dpYi^q g3?F17$$\"3y****\\nZ)H;\"F-$\"3T&fu#*[M!QvF17$$\"37LLe*HTW?\"F-$\"3/E? TsL$3M\"F-7$$\"3YmmmJy*eC\"F-$\"3/MuJ'3\\%4>F-7$$\"3;LL$e[E()G\"F-$\"3 Y#*fXjxDpCF-7$$\"3')******R^bJ8F-$\"33Q<**yf(G*HF-7$$\"3s****\\AYXt8F- $\"3#*[wBXqtiMF-7$$\"3f*****\\5a`T\"F-$\"3!z.[F-7$$\"3k*** **\\@fke\"F-$\"3'G7\"[`'z'))\\F-7$$\"3+LLe\\@o1;F-$\"3Io$*37g&e/&F-7$$ \"3Nmm;%30pi\"F-$\"3/_u4f\"Gl3&F-7$$\"3q***\\(=!Grk\"F-$\"3+MsAiIc5^F- 7$$\"3/LLL`4Nn;F-$\"3!e@Uq5#)y6&F-7$$\"3m****\\_K%*)o\"F-$\"3Vl'o2w=s5 &F-7$$\"3\\mmm^b`5$=F-$\"3!eBWGhCZc%F-7$$\"3$*******pfa<>F-$\"3#HYu2Y eq(QF-7$$\"3Umm\"zy*zd>F-$\"3c5y'R.-CZ$F-7$$\"3#HLLeg`!)*>F-$\"3n\"3th g\"yAIF-7$$\"3cmm;W/8S?F-$\"3,)f4^0e5^#F-7$$\"3w****\\#G2A3#F-$\"3#))* )fNj5Q'>F-7$$\"3Ymm\"H3XL7#F-$\"37&*y&RZf=S\"F-7$$\"3;LLL$)G[k@F-$\"3> (*zTSd]4#)F17$$\"3\\mm\"zM]v?#F-$\"3x\\dOdFJ5?F17$$\"3#)****\\7yh]AF-$ !35U.a(Gt'=UF17$$\"31LLe**o4#H#F-$!3^\"H$ob*))f,\"F-7$$\"3xmmm')fdLBF- $!3=P^2CC9'f\"F-7$$\"3mmm;WV*fP#F-$!3#***)))\\Axm;#F-7$$\"3bmmm,FT=CF- $!3#*>%*\\0611FF-7$$\"39++v8)z/Y#F-$!3#)*)yOrZd-KF-7$$\"3FLL$e#pa-DF-$ !3C=\">qh-Ql$F-7$$\"3!*******Rv&)zDF-$!3Bd:N\"o+UM%F-7$$\"3#ommT)3;CEF -$!3%pgfGY.$[YF-7$$\"3ILLLGUYoEF-$!3qZkF\"p7&z[F-7$$\"3\"*****\\n'*33F F-$!3`dhZQ[c@]F-7$$\"3_mmm1^rZFF-$!3U6r9&3015&F-7$$\"31LeR_tFeFF-$!3&= D>:u\"*36&F-7$$\"3-+]7)fR)oFF-$!3:Q&ehKr)=$F-7$FI$\"32]41H<;2QF-7$ FN$\"3e4]Ng<7vUF-7$FX$\"34DrH,5h1YF-7$F\\o$\"3*f,T$>_m;[F-7$Fao$\"31r0 =46$e([F-7$Ffo$\"3$QJ65)\\67\\F-7$$!3.LL$efKvI&F1$\"3^@tiDK&>#\\F-7$F[ p$\"3D!o@g](RE\\F-7$$!34++]PFU4^F1$\"3'RBWK(4bD\\F-7$F`p$\"3hbm[;X^>\\ F-7$$!3l+++DlB0[F1$\"3)\\Sc&od$4*[F-7$Fep$\"3_xI+brPT[F-7$$!3eLLLeR(\\ R%F1$\"39bQcZ'p;x%F-7$Fjp$\"3#))o#e)3FEo%F-7$$!3InmmT&[bw$F1$\"3bG6!4o X0W%F-7$F_q$\"3#>=J8[^i7%F-7$Fdq$\"3a%ei\\C7#[PF-7$Fiq$\"3lLL75S\"GJ$F -7$F^r$\"3;IWYlP18GF-7$Fcr$\"3=^&zS!\\;pAF-7$Fhr$\"3k(3BWn)eg(othF17$Fhs$!3C#Q(GuvU-bF]t7$F_t$!3RNw 4G[$*3jF17$Fdt$!3OiuI%HSoC\"F-7$Fit$!3k6^#GCMG#=F-7$F^u$!3oZiclHWtBF-7 $Fcu$!3w0a2ZF-7$Fgv$!3u,Kde_/A\\F-7$F\\w$!3E=nk`-(*e]F-7$Faw$!339l/&**)>%4&F -7$Ffw$!3]0\\rlIo8^F-7$F[x$!3$R!=5%>it6&F-7$F`x$!3LEOC*)\\A0^F-7$Fex$! 3=]s82&eP2&F-7$Fjx$!3%pKydQa\\F-7$Fdy$!3ax]S=P#p' [F-7$Fiy$!3Y)=[Zm/1l%F-7$F^z$!3Ai?r%>'erVF-7$Fcz$!3tfNa1&\\[g$F-7$Fhz$ !3E)fDEBp*yJF-7$F][l$!3K>jh!))RPr#F-7$Fb[l$!3%fR:!)f\\W<#F-7$Fg[l$!31B P#*Q&\\Qg\"F-7$F\\\\l$!3q`rc3J>o)*F17$Fa\\l$!3WZ-=&y!eWNF17$Ff\\l$\"3; Q6Ka:Z3?F17$F[]l$\"3q!pn9U$*y`(F17$F`]l$\"3nba@_7#3M\"F-7$Fe]l$\"3iW'4 sHT%4>F-7$Fj]l$\"3%Q\\K[/c#pCF-7$F_^l$\"3o$G8$)o!)G*HF-7$Fd^l$\"3UL>+A quiMF-7$Fi^l$\"31Z\"Q'3b.%)QF-7$F^_l$\"3YF'\\8gl*RXF-7$Fc_l$\"3I7qAXq> .[F-7$Fh_l$\"31z9,?on))\\F-7$F]`l$\"3=1BW;,&e/&F-7$Fb`l$\"3/S]!)G'>l3& F-7$Fg`l$\"3f36.ODb5^F-7$F\\al$\"3Eyb://(y6&F-7$Faal$\"3/[&zK&o?2^F-7$ Ffal$\"3w)\\k&z8]x]F-7$F[bl$\"3:%z5_rk)G]F-7$F`bl$\"3Ga\\_e!y9'\\F-7$F ebl$\"3y7/7c(HCz%F-7$Fjbl$\"3a\">aN0KZc%F-7$F_cl$\"3]'4],Uiq(QF-7$Fdcl $\"31brgjqRsMF-7$Ficl$\"3OI,-M0yAIF-7$F^dl$\"35sndA'=6^#F-7$Fcdl$\"3Ia !)fF-7$Fhdl$\"3\"pPWe:]ES\"F-7$F]el$\"33_x)G6U$G#)F17$Fbel$\"3oj \\R(zL90#F17$Fgel$!3(e:oOZ*oOTF17$F\\fl$!3x&pq$e//,5F-7$Fafl$!3+&4yLt% >q:F-7$Fffl$!3ca)*o+3#H7#F-7$F[gl$!3l'f\">ET![j#F-7$F`gl$!3i?kDhAx$F-7$Fjgl$!3qxa! zZaa)RF-7$$\"3OLL37#4?g#F-$!3g]$p#y$Qw1%F-7$F_hl$!3(*H-#oL`t6%F-7$$\"3 LL$3-sO_j#F-$!3#)oQ0\\<8HTF-7$$\"3%)***\\ib7jk#F-$!398iGs)z;8%F-7$$\"3 !om\"H#R)QdEF-$!3;k(eXe%fCTF-7$Fdhl$!36N$**\\dbu5%F-7$Fihl$!3)*H#fXqSp &RF-7$F^il$!3C/csLpf\\OF-7$Fbjl$!3^P()G**[`>JF-7$F\\[m$!3+=Ho0pUVBF-7$ $\"35++D1>V_GF-$!3)))oq&y\"))>(=F-7$Fa[m$!3*Q_cj13#G8F-7$$\"33++vt&pG* GF-$!3'*Q5h&*y`eqF17$Ff[m$\"3'[qOH#oW')=F]t7$$\"31+]i0j\"[$HF-$\"3_un3 1y]h')F17$$\"3/++v.UacHF-$\"3Wq-+#\\^x%=F-7$$\"3E+D\"G:3u'HF-$\"3/j\\4 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 273 "Der Vergleich unseres N\344herungspolynoms mit der exakt en L\366sung ist aber nicht \"nur\" graphisch m\366glich. Man kann mit Hilfe der Reihenentwicklung der exakten L\366sung auch untersuchen, w elche Koeffizienten eine Abweichung verursachen bzw. was eine Erh\366h ung der Ordnung bewirkt." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(series(exakt,t,10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"tG $!+\"*\\cZ9!\")\"\"\"$\"+bm3I>F'\"\"$$!+@mM?x!\"*\"\"&$\"+8Aaq9F.\"\"( $!+$zNRj\"!#5\"\"*-%\"OG6#F(\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(xs);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6*&$\" +&y6vW\"!\")\"\"\"%\"tGF(!\"\"*&$\"+6kf\\9!#6F()F)\"\"#F(F**&$\"+\\?\" H%>F'F()F)\"\"$F(F(*&$\"+Pf]5a!#5F()F)\"\"%F(F**&$\"+cl*zW'!\"*F()F)\" \"&F(F**&$\"+)>****y\"F?F()F)\"\"'F(F**&$\"+#\\'4-IF?F()F)\"\"(F(F(*&$ \"+NYJ\"f(F9F()F)\"\")F(F**&$\"+F&H^9)!#7F()F)\"\"*F(F(*&$\"+R?]u6F.F( )F)\"#5F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "komma@oe.uni-tuebingen.de" }}}}{MARK "0 4 0" 13 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }