{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 "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 }1 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 "Helvetica" 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 "R 3 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 "Normal" -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 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "Helvetica " 0 1 0 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 "" 0 260 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 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 "" 0 261 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 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 259 "" 0 "" {TEXT 256 25 "Moderne Physik mit Mapl e " }}{PARA 261 "" 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 260 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 23 "Kapitel 2.1.5 bis 2.1.7" }} {PARA 258 "" 0 "" {TEXT -1 22 "Worksheet kino2_10.mws" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "c Intern ational Thomson Publishing Bonn filename: kino2. ms" }}{PARA 0 "" 0 "" {TEXT -1 116 "Autor: Komma \+ Datum: 14.6.94 \+ " }}{PARA 0 "" 0 "" {TEXT -1 40 "Thema: Kinematik, beschle unigte Bewegung" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Beschleunigte Bewegunge n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 470 "Der Momentanwert der \304nderung einer Gr\366\337e ist die Ableitung dies er Gr\366\337e nach der Zeit. Stellvertretend f\374r den Momentanwert \+ einer \304nderung wird hier die Geschwindigkeit behandelt. Sie kann mi t dem Befehl " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "v:=t->diff(x(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vGf*6#%\"tG6\"6$%)operatorG%&arrowGF(-%%diffG6$-%\" xG6#9$F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Ortsfunktion:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "x:=t->t^3/5-t+3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,(*&#\"\"\"\" \"&F/*$)9$\"\"$F/F/F/F3!\"\"F4F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Anhand der g raphischen Darstellung l\344\337t sich der Zusammenhang von Funktion u nd Ableitung leicht \374berpr\374fen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(\{x(t),v(t)\},t=-2..4);" }}{PARA 13 "" 1 "" {GLPLOT2D 542 542 542 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$!\"#\"\"!$\"3!** ************R\"!#<7$$!3!******\\2<#p=F-$\"3=LH3%[$Q'4\"F-7$$!31++D^NUb zl,'!#>7$$!3$****\\7Sw%G6F-$!3u%4J42Y#fBF87$$!3***** \\7;)=,5F-$!3Akqj%fLd)RF87$$!3M++]i83V()F8$!3`3BK(p6NT&F87$$!3B****** \\V'zV(F8$!3EYxoz6g!o'F87$$!3%)*****\\d;%)G'F8$!3k:fv=!\\ti(F87$$!3#*) *****\\!)H%*\\F8$!3*H(fE>#>M])F87$$!3Q+++]d'[p$F8$!3?;'o`-y3=*F87$$!3/ ******\\>iUCF8$!35#pi0)e,U'*F87$$!3B++]7YY08F8$!3k81tGdu(*)*F87$$\"3%z -+++XDn%!#?$!3'yLRR+p)****F87$$\"3C++++y?#>\"F8$!3X\\rpL%=Z\"**F87$$\" 3h****\\(3wY_#F8$!3$=M#>R1c<'*F87$$\"3F)******HOTq$F8$!3CR8EcCww\"*F87 $$\"3I,+](3\">)*\\F8$!3_L,<^^3,&)F87$$\"3_,+]isVIiF8$!3')*3$3\"4**3n(F 87$$\"3&=++](o:;vF8$!3'GTB)\\JW5mF87$$\"3#>++v$)[op)F8$!3k\"\\gy@*)=Y& F87$$\"3W*****\\i%Qq**F8$!3)z\"QxDe[NSF87$$\"3&****\\(QIKH6F-$!3_'z!*[ %oxZBF87$$\"3#****\\7:xWC\"F-$!3C+6+,sfwqFM7$$\"37++]Zn%)o8F-$\"34tz%) 3&[CC\"F87$$\"3y******4FL(\\\"F-$\"3Ifchm9._MF87$$\"3#)****\\d6.B;F-$ \"34NZHH3Q0eF87$$\"3(****\\(o3lW6F-7$$\"3e******Hk-,?F-$\"3DE^8kSY-9F-7$$\"36+++D-eI @F-$\"3cIm4dKiBw7$F-$\"3#))fODc+#p[F-7$$\"3O++v)Q?QD$F-$\"3KArMt#3CN&F-7$$\"3G+++5 jypLF-$\"30]!Q]'eF8eF-7$$\"3<++]Ujp-NF-$\"3yExl+!H8O'F-7$$\"3++++gEd@O F-$\"3Y&o&**=JZpoF-7$$\"39++v3'>$[PF-$\"3>^QgL*R*HuF-7$$\"37++D6EjpQF- $\"3u'pHw#RV%)zF-7$$\"\"%F*$\"3k*************f)F--%'COLOURG6&%$RGBG$\" #5!\"\"$F*F*Fb[l-F$6$7U7$F($\"3!**************R$F-7$F/$\"3y'R,OZ=Ic$F- 7$F4$\"3?.zEe=btOF-7$F:$\"3oV\")R5)e`w$F-7$F?$\"39`SDryZDQF-7$FD$\"3S) o8%QhnbQF-7$FI$\"3/eE`$Qq%fQF-7$FO$\"3(3DE!=F1TQF-7$FT$\"3`M#GrTu/!QF- 7$FY$\"3`\"y@\\LT1u$F-7$Fhn$\"3W^p.oy\\hOF-7$F]o$\"3u:&4$)f2\"zNF-7$Fb o$\"35v.vO_^uMF-7$Fgo$\"3)p?O#*3)RfLF-7$F\\p$\"3ROq5iuMTKF-7$Fap$\"3$f $z!o\\,,8$F-7$Ffp$\"3sKGSvuK&*HF-7$F\\q$\"3O3=;J\"=6)GF-7$Faq$\"3Kmr%H %3v]FF-7$Ffq$\"31%GkN)4vREF-7$F[r$\"3G4yPnP:DDF-7$F`r$\"3SG,sItKDCF-7$ Fer$\"3zEbd#y0LL#F-7$Fjr$\"3%Rl$=hE(=E#F-7$F_s$\"3gkwAs)*=,AF-7$Fds$\" 3?ot![-Q(e@F-7$Fis$\"3&==\\2-$*49#F-7$F^t$\"31fdn@i7W@F-7$Fct$\"3u'4(p ZG2u@F-7$Fht$\"3s^*p0Oc?B#F-7$F]u$\"3c`B$e6DuJ#F-7$Fbu$\"3)QBm7z&e[CF- 7$Fgu$\"34Ml[m#Q9g#F-7$F\\v$\"3p!Rt*e!>P!GF-7$Fav$\"3X32>T]+CIF-7$Ffv$ \"38%HxxBWuI$F-7$F[w$\"3vsZ50R)ph$F-7$F`w$\"3w(Q1\">%)G()RF-7$Few$\"3W M\\B(*\\!yR%F-7$Fjw$\"3'o8`O309)[F-7$F_x$\"3\")QiZ]@d$Qy))F-7$F]z$\"3;Kt[UMQ%y*F-7$$\"39+++5h(*3QF-$\"3epP DvwLC5!#;7$Fbz$\"3_$*3a*o=>2\"Fhdl7$$\"31+]i0j\"[$RF-$\"3c@A+Bi&\\7\"F hdl7$Fgz$\"33++++++!=\"Fhdl-F\\[l6&F^[lFb[lF_[lFb[l-%+AXESLABELSG6$Q\" t6\"Q!Fjel-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 234 "Wir k\366 nnen diesen Zusammenhang geometrisch veranschaulichen, indem wir zun \344chst die Gleichung der Tangente (Punkt-Steigungsform) aufstellen. \+ (Um flexibel zu bleiben, wird hier nicht der fertige Befehl " 0 "" {MPLTEXT 1 0 19 " t0:='t0'; xt:='xt':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t0GF$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "xt:=simplify(solve((xt-x(t0) )/(t-t0)=v(t0),xt));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xtG,**&#\" \"#\"\"&\"\"\"*$)%#t0G\"\"$F*F*!\"\"F.F**&#F.F)F**&)F-F(F*%\"tGF*F*F*F 4F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 150 "Diese Tangente kann nun im Punkt (t0|x(t0)) mit e ingezeichnet werden. Dann mu\337 ihre Steigung gleich v(t0) sein, was \+ sich mit der Maus ausmessen l\344\337t. 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Ein e ander Ortsfunktion kann wegen der Formulierung von xt als Ausdruck n icht im nachhinein angegeben werden, bzw. wenn dies erw\374nscht ist, \+ mu\337 nach dem L\366schen von xt das Assignment von xt neu abgearbeit et werden." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "# z.B. x:=t-> t^2; xt:='xt': #zurueck zu xt:=...;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 314 "Die Veranschaulichung gelingt besser, wenn wir ein Steigungsdr eieck mit der Abszisse 1 einzeichnen (bei der Betrachtung auf die ggf. verschiedenen Ma\337st\344be von Abszisse und Ordinate achten, bzw. 1 :1 w\344hlen). 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Es wird die Palette der Maple-Befehle vorgestellt . Aber der rote Faden ist die Infinitesimalrechnung, die hier \"in ers ter N\344herung\" behandelt wird, d.h. \"ersetze Kurve durch Gerade\": die lineare Approximation mit Grenz\374bergang ist das A und O in der mathematischen Physik von Newton und Leibniz." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 659 "Wie kommt man (wie kamen Newton und Leibniz) auf Momentanwerte? Man geht von Mittelwerten aus \+ und l\344\337t das Zeitintervall gegen Null gehen. Ob die Natur es tat s\344chlich auch so macht, wie Leibniz das meinte (\"natura non facit \+ saltus\") wei\337 man bis heute noch nicht, es erscheint aber nach der Quantenphysik und nun auch nach der Chaostheorie immer unwahrscheinli cher. Abgesehen davon: der Grenzwert bedeutet in der Regel eine Zahl m it unendlich vielen Stellen und wer kann schon mit dieser Genauigkeit \+ rechnen (Rauschen des Kontinuums). Dennoch hat sich diese Mathematik g l\344nzend bew\344hrt, ja sie ist geradezu die Basis unserer heutigen \+ technischen Gesellschaft." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 335 "Wir ersetzen zun \344chst die Ortsfunktion x(t) im Intervall [t , t+dt] durch eine glei chf\366rmige Bewegung. Diese lineare Approximation ist in vielen folge nden \334berlegungen und Beispielen ein elementarer Baustein und f\374 hrt hier auf die mittlere Geschwindigkeit vq (wie im vorangehenden Abs chnitt bei der st\374ckweise gleichf\366rmigen Bewegung)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "unassign('vq','x','dt','n','t1','t0 ','i');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "vq:=t->(x(t+dt)- x(t))/dt;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vqGf*6#%\"tG6\"6$%)ope ratorG%&arrowGF(*&,&-%\"xG6#,&9$\"\"\"%#dtGF3F3-F/6#F2!\"\"F3F4F7F(F(F (" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 229 "Neugierige wollen sicher gl eich wissen, wie sich aus diesem Differenzenquotient der Differentialq uotient machen l\344\337t. Daf\374r gibt es in Maple den Befehl " 0 "" {MPLTEXT 1 0 22 "DQ:=limit(vq(t),dt=0); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DQG--%\"DG6#%\"xG6#%\"tG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Testfunktion, Differenzenquotient \+ und Differetialquotient:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "x:=t->t^5+t^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGf*6#%\"tG6\"6 $%)operatorG%&arrowGF(,&*$)9$\"\"&\"\"\"F1*$)F/\"\"#F1F1F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "vq(t);DQ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&,**$),&%\"tG\"\"\"%#dtGF)\"\"&F)F)*$)F'\"\"#F)F)*$) F(F+F)!\"\"*$)F(F.F)F1F)F*F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\" \"&\"\"\")%\"tG\"\"%F&F&*&\"\"#F&F(F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 217 "Und wieder k \366nnen Sie die letzten Befehle in Ihrem Worksheet \374berschreiben u nd damit testen, ob Maple auch tats\344chlich die Regeln zur Berechnun g von Grenzwerten beherrscht. Oder sind es die Regeln zur Differentiat ion?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 217 " Wir ersetzen nun die beschleunigte Bewegung durch eine st\374ckweise g leichf\366rmige, indem wir n Kurvenpunkte durch Sekanten verbinden. Da zu gibt es in Maple (mindestens) zwei M\366glichkeiten. Die erste ist \+ der " 0 "" {MPLTEXT 1 0 27 "x:=t->1/5*t^3-t+3; #sin(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,(*&#\"\"\" \"\"&F/*$)9$\"\"$F/F/F/F3!\"\"F4F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=5: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x liste:=[seq([i*dt,x(i*dt)],i=0..n)]; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'xlisteG7(7$\"\"!\"\"$7$%#dtG,(*&#\"\"\"\"\"&F.*$)F*F(F.F.F.F*!\" \"F(F.7$,$*&\"\"#F.F*F.F.,(*&#\"\")F/F.F0F.F.*&F6F.F*F.F2F(F.7$,$*&F(F .F*F.F.,(*&#\"#FF/F.F0F.F.*&F(F.F*F.F2F(F.7$,$*&\"\"%F.F*F.F.,(*&#\"#k F/F.F0F.F.*&FGF.F*F.F2F(F.7$,$*&F/F.F*F.F.,(*&\"#DF.F1F.F.*&F/F.F*F.F2 F(F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "Der " 0 "" {MPLTEXT 1 0 21 "n:='n': tt:=t0+j*dt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "xliste:=[[tt,x(tt)] $ j=0..n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xlisteG7#-%\"$G6$7$,&%#t0G\"\"\"*&%\"jGF,%#dtGF,F,,* *&\"\"&!\"\"F*\"\"$F,F+F3F-F3F4F,/F.;\"\"!%\"nG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "unassign('dt','t1','t0','n'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dt:=(t1-t0)/n: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "vtitle:=`lin. Approx.`:pspl(`p3kino2.ps`):" }} {PARA 2 "" 0 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "t1:=4: t0:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:= 3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(\{x(t),xliste\}, t=-2..t1);" }}{PARA 13 "" 1 "" {GLPLOT2D 542 542 542 {PLOTDATA 2 "6&-% 'CURVESG6$7&7$$\"\"!F)$\"\"$F)7$$\"3ELLLLLLL8!#<$\"3$3uS2uS29#F/7$$\"3 _mmmmmmmEF/$\"3yDf#f#f#f7%F/7$$\"\"%F)$\"33++++++!=\"!#;-%'COLOURG6&%$ RGBG$\"#5!\"\"F(F(-F$6$7U7$$!\"#F)$\"3!**************R$F/7$$!3!****** \\2<#p=F/$\"3y'R,OZ=Ic$F/7$$!31++D^NUbiUCFap$\"3ROq 5iuMTKF/7$$!3B++]7YY08Fap$\"3$f$z!o\\,,8$F/7$$\"3%z-+++XDn%!#?$\"3sKGS vuK&*HF/7$$\"3C++++y?#>\"Fap$\"3O3=;J\"=6)GF/7$$\"3h****\\(3wY_#Fap$\" 3Kmr%H%3v]FF/7$$\"3F)******HOTq$Fap$\"31%GkN)4vREF/7$$\"3I,+](3\">)*\\ Fap$\"3G4yPnP:DDF/7$$\"3_,+]isVIiFap$\"3SG,sItKDCF/7$$\"3&=++](o:;vFap $\"3zEbd#y0LL#F/7$$\"3#>++v$)[op)Fap$\"3%Rl$=hE(=E#F/7$$\"3W*****\\i%Q q**Fap$\"3gkwAs)*=,AF/7$$\"3&****\\(QIKH6F/$\"3?ot![-Q(e@F/7$$\"3#**** \\7:xWC\"F/$\"3&==\\2-$*49#F/7$$\"37++]Zn%)o8F/$\"31fdn@i7W@F/7$$\"3y* *****4FL(\\\"F/$\"3u'4(pZG2u@F/7$$\"3#)****\\d6.B;F/$\"3s^*p0Oc?B#F/7$ $\"3(****\\(o3lWP! 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(Alter native Formulierungen werden als Kommentare mitgef\374hrt und k\366nne n zum Experimentieren mit der Maple-Syntax verwendet werden.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "stufe:=(a,b,c)->[[a,c(a)],[b ,c(a)],[b,c(b)]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&stufeGf*6%%\"a G%\"bG%\"cG6\"6$%)operatorG%&arrowGF*7%7$9$-9&6#F07$9%F17$F5-F26#F5F*F *F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "#treppe:=seq(stufe(t 0+i*dt,t0+(i+1)*dt,vq),i=0..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "treppe:=s tufe(tt,tt+dt,vq) $ j=0..n;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'trep peG-%\"$G6$7%7$,$*(\"\"%\"\"\"%\"jGF-%\"nG!\"\"F-,$*(F,F0,(*&\"\"&F0,& *(F,F-F.F-F/F0F-*&F,F-F/F0F-\"\"$F-*&F,F-F/F0F0**\"#kF-F5F0F.F9F/!\"$F 0F-F/F-F-7$F6F17$F6,$*(F,F0,(*&F5F0,&*(F,F-F.F-F/F0F-*&\"\")F-F/F0F-F9 F-*&F,F-F/F0F0*&F5F0F6F9F0F-F/F-F-/F.;\"\"!F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "vtitle:= Differenzieren:xtick:=2:pspl(`p4kino2.ps`):" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=5:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "plots[display](\{plot(\{x(t ),diff(x(t),t)\},t=t0..t1),plot(\{treppe\},t0..t1,color=red),plot(\{xl iste\},t=0..t1,color=blue)\});" }}{PARA 13 "" 1 "" {GLPLOT2D 542 542 542 {PLOTDATA 2 "6--%'CURVESG6$7S7$$\"\"!F)$\"\"$F)7$$\"3Hmmmm;')=()!# >$\"3[&[vB%R%H\"H!#<7$$\"3RLLLe'40j\"!#=$\"3$eY(G'*f\"y$GF27$$\"3mmmm; 6m$[#F6$\"3OLr3 (>%F6$\"3qobK'*)\\F6$\"3H#zffe\")e_#F27$$\"3P*** **\\Kd,\"eF6$\"3mUL>U?@eCF27$$\"3-mmm\"fX(emF6$\"3gV-IAP<$R#F27$$\"3.* ****\\U7Y](F6$\"3Ml%)fA&pSL#F27$$\"3'QLLLV!pu$)F6$\"3gEN5wM+!G#F27$$\" 3xmmm;c0T\"*F6$\"3W()4sWxlQAF27$$\"3#*******H,Q+5F2$\"3yv-rM![)*>#F27$ $\"3)*******\\*3q3\"F2$\"3+txn&\\q)p@F27$$\"3)*******p=\\q6F2$\"3+S(oB *[B]@F27$$\"3mmm;fBIY7F2$\"3n[;&oMm39#F27$$\"3GLLLj$[kL\"F2$\"3=7A=yh& 49#F27$$\"3?LLL`Q\"GT\"F2$\"3onOk#e$>^@F27$$\"3!*****\\s]k,:F2$\"3/xD! >@yb<#F27$$\"39LLL`dF!e\"F2$\"3E'[HJv***3AF27$$\"33++]sgam;F2$\"3m]2px )y\"fAF27$$\"3/++]F2$\"3'4YS*)\\)H([#F27$$\"3immmTc-)*>F2$\"3w\"3. 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Bei der Geschwindigkeit mu\337 man schon zu h \366heren n-Werten greifen, wenn die Ableitungsfunktion wie eine steti ge Kurve aussehen soll: das Differenzieren bringt die N\344herung an d en Tag." }}{PARA 0 "" 0 "" {TEXT -1 770 "Der letzte Plot zeigt den Zus ammenhang von Funktion und Ableitung, bzw. in umgekehrter Richtung ge lesen den Zusammenhang von Funktion und Stammfunktion ... wenn man sic h jeweils den Grenz\374bergang dazudenkt. L\344\337t man den Grenz\374 bergang aber bewu\337t weg, bleibt man also bei der deutlich sichtbare n linearen Approximation, und damit bei der st\374ckweise gleichf\366r migen Bewegung, so ist die geometrische Bedeutung der Kurven unmittel bar klar: Die Steigung der Ortskurvenst\374cke ist der Funktionswert d er Geschwindigkeitskurvenst\374cke. Umgekehrt mu\337 dann die Fl\344ch e unter den Geschwindigkeitskurvenst\374cken den Funktionswert der Ort skurve ergeben ... bis auf eine additive Konstante. Summiert man also \+ diese Fl\344chen alle auf, so mu\337 man den insgesamt zur\374ckgelegt en Weg erhalten." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "i:='i': n:='n':t0:='t0':t1:='t1':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "xq:=sum(vq(t0+i*dt),i=0..n-1)*dt;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xqG*(,***\"\"&!\"\",&%#t1G\"\"\"%#t0GF)F)%\"nGF,F-\"\"$F)*(F*F)F .F,F+F,F)*(F*F)F.F,F-F,F,**F(F)F*F)F.F,F+F/F,F,F*F,F.F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Der insgesamt zur\374ckgelegte Weg darf aber bei dem gemachten Ansatz nicht von der Anzahl der Wegst\374cke abh\344ngen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"&!\"\",&%#t1GF&%#t0G\"\"\"F*,**$)F)\"\"#F*F**&F(F*F)F*F* F%F&*$)F(F.F*F*F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "(x(t 1)-x(t0))-xq;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"\"&!\"\"%#t1G\" \"$\"\"\"F'F&*&F%F&%#t0GF(F&F+F)*(,***F%F&,&F'F)F+F&F&%\"nGF)F+F(F&*(F /F&F0F)F'F)F&*(F/F&F0F)F+F)F)**F%F&F/F&F0F)F'F(F)F)F/F)F0F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Oder zum Experimentieren mit konkreten Zahlen n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x(t) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"&F&*$)%\"tG\"\"$F& F&F&F*!\"\"F+F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "n:=5: xq; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"&!\"\",**&,&%#t1G\"\"\"%#t 0GF&F&F,\"\"$F&*(F%F+F)F&F*F+F&*(F%F+F)F&F,F+F+*&F)F&F*F-F+F+F)F+F+" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*(\"\"&!\"\",&%#t1GF&%#t0G\"\"\"F*,**$)F)\"\"# F*F**&F(F*F)F*F*F%F&*$)F(F.F*F*F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 254 "Anstatt vers chiedene n einzugeben, k\366nnen wir Maple auch den Grenzwert berechne n lassen. Das ist bei dem gemachten Ansatz zwar nicht besonders sinnvo ll, weil sich die Terme mit n herausheben, die Befehle werden aber der Vollst\344ndigkeit halber aufgef\374hrt:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "n:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "lxq:=limit(xq,n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$l xqG,$*(\"\"&!\"\",&%#t1GF(%#t0G\"\"\"F,,**$)F+\"\"#F,F,*&F*F,F+F,F,F'F (*$)F*F0F,F,F,F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "x(t1)-x (t0)-lxq;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"\"&!\"\"%#t1G\"\"$ \"\"\"F'F&*&F%F&%#t0GF(F&F+F)*(F%F&,&F'F&F+F)F),**$)F+\"\"#F)F)*&F'F)F +F)F)F%F&*$)F'F1F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "#simplify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "Wenn wir umgekehrt die Funktion f \374r die Momentangeschwindigkeit vorgeben wollen, um von ihr auf die \+ Ortsfunktion zu schlie\337en, k\366nnen wir so vorgehen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "unassign('v','x','dt','n','t1','t0' ,'i');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Wie oben, nur jetzt mit allgemeinem v(t)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "xq:=sum(v(t0+i*dt),i=0..n-1) *dt;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xqG*&-%$sumG6$-%\"vG6#,&%#t 0G\"\"\"*&%\"iGF.%#dtGF.F./F0;\"\"!,&%\"nGF.F.!\"\"F.F1F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Bildung des Grenzwertes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dt:=(t1-t0)/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dtG*&,&%# t1G\"\"\"%#t0G!\"\"F(%\"nGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "lxq:=limit(xq,n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $lxqG-%&limitG6$*(-%$sumG6$-%\"vG6#,&%#t0G\"\"\"*(%\"iGF1,&%#t1GF1F0! \"\"F1%\"nGF6F1/F3;\"\"!,&F7F1F1F6F1F4F1F7F6/F7%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Angabe einer Funktion (wenn Sie sin(t) zur Funktion hinzuf\374g en, wird die Ausgabe etwas l\344nger):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "v:=t->3/5*t^2-1; # +sin(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&#\"\"$\"\" &\"\"\"*$)9$\"\"#F1F1F1F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "xq;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,4*(\"\"&!\" \"%#t0G\"\"#%\"nG\"\"\"F+F*F'**F&F'F(F+F*F+%#t1GF+F+*(\"\"$F+\"#5F'F(F )F+*(F&F'F*F+F-F)F+*(F/F+F0F'F-F)F'*(F0F'F*F'F-F)F+**F&F'F*F'F-F+F(F+F '*(F0F'F*F'F(F)F+F+,&F-F+F(F'F+F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "lxq;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"&!\"\" ,&%#t1GF&%#t0G\"\"\"F*,**$)F)\"\"#F*F**&F(F*F)F*F*F%F&*$)F(F.F*F*F*F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "expand(lxq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%#t1G!\"\"*&#\"\"\"\"\"&F(*$)F$\"\"$F(F(F( *&#F(F)F(*$)%#t0GF,F(F(F%F1F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "So berechnet man also oh ne einen Integrationsbefehl ein bestimmtes Integral. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 "Die Graphische Darste llung von Momentangeschwindigkeit und mittlerer Geschwindigkeit k\366n nen wir im wesentlichen von oben \374bernehmen:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "v:='v':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " tt:=t0+j*dt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "st ufe:=(a,b,c)->[[a,c(a)],[b,c(a)],[b,c(b)]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&stufeGf*6%%\"aG%\"bG%\"cG6\"6$%)operatorG%&arrowGF*7 %7$9$-9&6#F07$9%F17$F5-F26#F5F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "#treppe:=seq(stufe(t0+i*dt,t0+(i+1)*dt,vq),i=0..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "treppe:=stufe(tt,tt+dt,v) $ j=0..n;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%'treppeG-%\"$G6$7%7$,&%#t0G\"\"\"*(% \"jGF,,&%#t1GF,F+!\"\"F,%\"nGF1F,-%\"vG6#F*7$,(F+F,F-F,*&F/F,F2F1F,F37 $F7-F46#F7/F.;\"\"!F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Zur Abwechslung einmal eine etwa s seltenere Funktion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "v: =t->2^t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vGf*6#%\"tG6\"6$%)oper atorG%&arrowGF()\"\"#9$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 " vtitle:=`momentanes Mittel`:vylab:=`v `:ytick:=3:pspl(`p5kino2.ps`): " }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "t0:=0: t1:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=50:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plots[disp lay](\{plot(\{treppe\},t=t0..t1,color=red),plot(v(t),t=t0..t1,color=bl ue)\});" }}{PARA 13 "" 1 "" {GLPLOT2D 542 542 542 {PLOTDATA 2 "6X-%'CU RVESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"3Hmmmm;')=()!#>$\"3[J3Kh!)Hi5!#<7$$\" 3RLLLe'40j\"!#=$\"3o%)*>6F27$$\"3mmmm;6m$[#F6$\"3u\\VYq5'y=\"F27$ $\"3fmmm;yYULF6$\"3c1UrE!>2E\"F27$$\"3%HLL$eF>(>%F6$\"378b$>CnwL\"F27$ $\"3Qmmm\">K'*)\\F6$\"3&*pJh9w>89F27$$\"3P*****\\Kd,\"eF6$\"3-6G*42-f \\\"F27$$\"3-mmm\"fX(emF6$\"3s\"puKtHle\"F27$$\"3.*****\\U7Y](F6$\"3k6 $o1gIBo\"F27$$\"3'QLLLV!pu$)F6$\"31InDkD\"py\"F27$$\"3xmmm;c0T\"*F6$\" 3#G/9RU+W)=F27$$\"3#*******H,Q+5F2$\"3?#[1:/F0+#F27$$\"3)*******\\*3q3 \"F2$\"3wvff^:LC@F27$$\"3)*******p=\\q6F2$\"3)QuI7D%)3D#F27$$\"3mmm;fB IY7F2$\"3_(yAu8EBP#F27$$\"3GLLLj$[kL\"F2$\"3(Q\\>gv)GDDF27$$\"3?LLL`Q \"GT\"F2$\"3QzT7i'fDm#F27$$\"3!*****\\s]k,:F2$\"3wS6obTlJGF27$$\"39LLL `dF!e\"F2$\"3y#>F2$\"3Wjbi5cBmPF27$$\"3immmTc-)*>F2$\"3O4AMg'HX*RF27$$\"3Mmm;f`@'3# F2$\"3ghtWhmKYUF27$$\"3y****\\nZ)H;#F2$\"35(*HupUSyWF27$$\"3YmmmJy*eC# F2$\"3!QjEY2ALu%F27$$\"3')******R^bJBF2$\"3]AA([OwM.&F27$$\"3f*****\\5 a`T#F2$\"3jQ\"R])Q]M`F27$$\"3o****\\7RV'\\#F2$\"3!Gi_9x))Gk&F27$$\"3k* ****\\@fke#F2$\"3nvun*)yA1gF27$$\"3/LLL`4NnEF2$\"3/z#>$GF2$\"3,e$Q1\"3 5!#;7$$\"3bmmm,FT=MF2$\"3a\"Fax7$$\"3ILLLGUYoOF2$\"3[n%)4< T]r7Fax7$$\"3_mmm1^rZPF2$\"3%Ht:g40LM\"Fax7$$\"34++]sI@KQF2$\"35z30=:L C9Fax7$$\"34++]2%)38RF2$\"3v$4&GbyX1:Fax7$$\"\"%F)$\"#;F)-%'COLOURG6&% $RGBGF(F($\"*++++\"!\")-F$6$7%F'7$$\"3=+++++++!)F/F*7$Fe[l$\"3E!Qh0/=q 0\"F2-F[[l6&F][lF^[lF(F(-F$6$7%Fg[l7$$\"3-+++++++;F6Fh[l7$F`\\l$\"3-?A 2QrG<6F2Fj[l-F$6$7%Fb\\l7$$\"3!**************R#F6Fc\\l7$Fi\\l$\"3II&H9 m#*4=\"F2Fj[l-F$6$7%F[]l7$$\"31+++++++KF6F\\]l7$Fb]l$\"3)=h,*[0L[7F2Fj [l-F$6$7%Fd]l7$$\"3A+++++++SF6Fe]l7$F[^l$\"3?%*Gx5z]>8F2Fj[l-F$6$7%F]^ l7$$\"3#)*************z%F6F^^l7$Fd^l$\"3O0/NmOu%R\"F2Fj[l-F$6$7%Ff^l7$ $\"3a+++++++cF6Fg^l7$F]_l$\"3;,6H<#pUZ\"F2Fj[l-F$6$7%F__l7$$\"39++++++ +kF6F`_l7$Ff_l$\"3!)**4Kf\"H$e:F2Fj[l-F$6$7%Fh_l7$$\"3u*************>( F6Fi_l7$F_`l$\"3AY^`M?=Z;F2Fj[l-F$6$7%Fa`l7$$\"3U+++++++!)F6Fb`l7$Fh`l $\"3D[AfE65TF 2Fj[l-F$6$7%F\\bl7$$\"3/++++++S5F2F]bl7$Fcbl$\"33L@J`wAc?F2Fj[l-F$6$7% Febl7$$\"36++++++?6F2Ffbl7$F\\cl$\"3+;@0D(pM<#F2Fj[l-F$6$7%F^cl7$$\"3% **************>\"F2F_cl7$Fecl$\"3?qS**4nR(H#F2Fj[l-F$6$7%Fgcl7$$\"3-++ ++++!G\"F2Fhcl7$F^dl$\"3%R4!zo(*QGCF2Fj[l-F$6$7%F`dl7$$\"35++++++g8F2F adl7$Fgdl$\"3_3e7&z^oc#F2Fj[l-F$6$7%Fidl7$$\"3%*************R9F2Fjdl7$ F`el$\"3dV`*[l3Kr#F2Fj[l-F$6$7%Fbel7$$\"3-++++++?:F2Fcel7$Fiel$\"3ga;. 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0 "> " 0 "" {MPLTEXT 1 0 29 "xliste:=[[tt,x[j]] $ \+ j=0..n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xlisteG7#-%\"$G6$7$,$*( \"\"%\"\"\"%\"jGF-%\"nG!\"\"F-&%\"xG6#F./F.;\"\"!F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Und berechnen deren Elemente in einer " 0 "" {MPLTEXT 1 0 13 "t0:=0: t1:=4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x0:=10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "x[0]:=x0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:=50: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " for i to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x[i]:=x[i-1]+v(t0+ i*dt)*dt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Schauen Sie sich die Liste ruhig an ..." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "#xliste;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 45 "Und ver\344ndern Sie in der vorigen Region n ..." }}{PARA 0 "" 0 "" {TEXT -1 40 "vtitle:=Genauigkeit: pspl(`p6kino2.ps`) :" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 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"" 0 "" {TEXT -1 79 "Im Plot-Befeh l ist schon die exakte L\366sung eingebaut, die Stammfunktion lautet: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "int(v(tau),tau=0..t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"!\"\")\"\"#%\"tGF%F%-%#lnG 6#F(F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Noch einmal der Verglei ch mit dem Grenzwert:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "i: ='i': n:='n': t1:='t': dt:='dt':t0:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "xq:=Sum(v(t0+i*dt),i=0..n-1)*dt;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xqG*&-%$SumG6$)\"\"#*&%\"iG\"\"\"%#dtGF-/F,;\"\"!,&% \"nGF-F-!\"\"F-F.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "xq:= value(xq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xqG*&,&*&))\"\"#%#dtG %\"nG\"\"\",&F)F-F-!\"\"F/F-*&F-F-F.F/F/F-F+F-" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "dt:=(t1-t0)/n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "lxq:=limit(xq,n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$lxqG*&,&\"\"\"!\"\")\"\"#%\"tGF'F'-%#lnG6#F*F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 183 "Anmerkung: xq wird hier in zwei S chritten \"berechnet\": " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Statistik-Befehle :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "F \374r die Behandlung der st\374ckweise gleichf\366rmigen Bewegung (ode r allgemeiner von diskreten Funktionen) kann das " 0 " " {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "with(stats): with(describe): with(statplots): with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Mittelwerte" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 380 "Die Geschwindigke it v wird als zeitliche Ableitung einer noch unbekannten Ortsfunktion \+ vordefiniert. 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