{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 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 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 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 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 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 " " 0 256 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 "" 0 257 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 }} {SECT 0 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "Quelle" }}{PARA 0 "" 0 "" {TEXT -1 289 "Dateiname: referat2.mws\nDateigr\366\337e: 7 KB\nName: C hristoph Schill\nSchule: Isolde-Kurz-Gymnasium\nKlasse: 11 D\nDatum: 0 3.07.97\nFach: Mathematik\nThema: Kurvenfit\nStichw\366rter: Naeherung einer Geraden\nKurzbeschreibung: Naeherung einer Geraden an 3 gegegeb ene Punkte, die verbunden werden sollten\n" }}}{EXCHG {PARA 256 "" 0 " " {TEXT 256 30 "Referat Nr. 2:Christoph Schill" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 17 "Thema 7:Kurvenfit" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Mathe-Referat" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Problemstellung" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 218 "Bei einer Me\337ung bekommt man verschiedene Me\337werte. Man nimmt an, da\337 sich diese Punkte durc h eine Funktion ersetzen lassen. Man mu\337 nun eine Funktion finden, \+ bei denen die Abweichung zur Funktion m\366glichst gering ist." }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT 258 19 "Beliebige Me\337werte " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 263 9 "Achtung: " }{TEXT -1 34 "Bei \304nderung der Me\337punkte auf die" }{TEXT 264 10 " Bereiche " }{TEXT -1 82 "der Plots achten!Da man weiter unten noc hmals die Punkte ge\344ndert werden m\374ssen ->" }{HYPERLNK 17 "Punkt e" 1 "" "Punkte" }}}{EXCHG {PARA 0 "> " 0 "Zur\374ck" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "x1:=-2: y1:=1: x2:=0: y2: =2: x3:=1: y3:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Mw1:=[ x1,y1]: Mw2:=[x2,y2]: Mw3:=[x3,y3]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot(\{Mw1,Mw2,Mw3\},style=point,symbol=box,color=gre en,title=`Me\337punkte`,thickness=10);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 259 27 "Geraden mit verschiedenem m" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "y:=m*x+b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG,&* &%\"mG\"\"\"%\"xGF(F(%\"bGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "b:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Pkte:=plot(\{ Mw1,Mw2,Mw3\},style=point,symbol=box,color=green,thickness=10):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Gerade1:=plot(\{seq(y,m=-5.. 5)\},x=-15..15,-15..15):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "display(\{Pkte,Gerade1\});" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 260 27 "Geraden mit verschiedenem b" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "b:='b':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "m:=1.5:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Gerade2:=plot(\{seq(y,b=-15. .15)\},x=-15..15,-15..15):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "display(\{Pkte,Gerade2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "Durch diese beiden Se quenzen kann man nun erkennen, da\337 einer dieser Geraden die gesucht e ist, d.h. die Gerade mit dem genausten N\344herungswert." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 261 16 "Genaueste Gerade" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Um nun die genaueste Gerade zu bekommen ben\374tzt m an eines der sogenannten " }{TEXT 262 19 "Iterationsverfahren" }{TEXT -1 81 ". Man nimmt die einzelnen Me\337werte und f\374gt sie dann in d ie folgende Formel ein: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 9 "Achtung: " }{TEXT -1 31 "Hier die Punkte \+ einsetzen, aber" }{TEXT 266 6 " ohne " }{TEXT -1 17 "eckigen Klammern! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "b:='b': m:='m': " }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "jetzt wieder die \+ Abst\344nde summieren," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 " Naeherungswert:=sum('(deltay(i))^2','i'=1..3);\ndeltay(1):=(y1-(m*x1+b ));\ndeltay(2):=(y2-(m*x2+b));\ndeltay(3):=(y3-(m*x3+b));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/NaeherungswertG,(*$-%'deltayG6#\"\"\"\"\" #F**$-F(6#F+F+F**$-F(6#\"\"$F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>- %'deltayG6#\"\"\",(F'F'%\"mG\"\"#%\"bG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%'deltayG6#\"\"#,&F'\"\"\"%\"bG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%'deltayG6#\"\"$,(\"\"%\"\"\"%\"mG!\"\"%\"bGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "expand(Naeherungswert);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.\"#@\"\"\"%\"mG!\"%%\"bG!#9*$F&\" \"#\"\"&*&F&F%F(F%!\"#*$F(F+\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Para:=p lot3d(Naeherungswert,b=-50..50,m=-50..50): " }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 217 "Hier habe ich nun die Verschiedenen Geraden nochmals z usammen in einem 3D-Plot gezeigt. Die genaueste Gerade bekomme ich, we nn ich \"summe\" einmal ableite und gleich 0 setze. Ab jetzt k\374rze \+ ich N\344herungswert mit Nw ab." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Das ist die 1. Ableitu ng \374ber m:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Nw1:=diff( Naeherungswert,m$1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Nw1G,(!\"% \"\"\"%\"mG\"#5%\"bG!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(Nw1=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"bG,&!\"#\"\"\" %\"mG\"\"&/F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Ger1:=pl ot3d(Nw1,b=-50..50,m=-50..50): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Das ist die 1 Ableitun g \374ber b:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Nw2:=diff(N aeherungswert,b$1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Nw2G,(!#9\" \"\"%\"mG!\"#%\"bG\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " solve(Nw2=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"mG,&!\"(\"\"\"% \"bG\"\"$/F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Ger2:=plo t3d(Nw2,b=-50..50,m=-50..50,axes=frame): " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "display(\{Para,Ger1,Ger2\});" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Die Sc hnittgerade ist die gesuchte Gerade. Diese rechne ich im folgenden aus !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Nw3:=(m,b)->sum((y[i]-f(x[i]))^2,i=1..n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Nw3G:6$%\"mG%\"bG6\"6$%)operatorG%& arrowGF)-%$sumG6$*$,&&%\"yG6#%\"iG\"\"\"-%\"fG6#&%\"xGF4!\"\"\"\"#/F5; F6%\"nGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=x->m*x+b; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%&ar rowGF(,&*&%\"mG\"\"\"9$F/F/%\"bGF/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Nw3(m,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"nG \"\"\"%\"bG\"\"#F&-%$sumG6$,,*$&%\"yG6#%\"iGF(F&*(F.F&%\"mGF&&%\"xGF0F &!\"#*&F.F&F'F&F6*&F3F(F4F(F&*(F3F&F4F&F'F&F(/F1;F&F%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "nw3y:=\{diff(Nw3(m,b),m)=0,diff(Nw3 (m,b),b)=0\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nw3yG<$/-%$sumG6$, (*&&%\"yG6#%\"iG\"\"\"&%\"xGF.F0!\"#*&%\"mGF0F1\"\"#F6*&F1F0%\"bGF0F6/ F/;F0%\"nG\"\"!/,&*&F;F0F8F0F6-F(6$,&F,F3*&F5F0F1F0F6F9F0F<" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "n:=3;nw3y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$/,4 *&&%\"yG6#\"\"\"F*&%\"xGF)F*!\"#*&%\"mGF*F+\"\"#F0*&F+F*%\"bGF*F0*&&F( 6#F0F*&F,F5F*F-*&F/F*F6F0F0*&F6F*F2F*F0*&&F(6#\"\"$F*&F,F;F*F-*&F/F*F= F0F0*&F=F*F2F*F0\"\"!/,0F2\"\"'F'F-*&F/F*F+F*F0F4F-*&F/F*F6F*F0F:F-*&F /F*F=F*F0F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "solve(nw3y, \{m,b\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$/%\"mG,$*&,4*&&%\"yG6# \"\"\"F-&%\"xGF,F-\"\"#*&F.F-&F+6#F0F-!\"\"*&F.F-&F+6#\"\"$F-F4*&F2F-& F/F3F-F0*&F:F-F*F-F4*&F:F-F6F-F4*&F6F-&F/F7F-F0*&F>F-F*F-F4*&F>F-F2F-F 4F-,.*$F.F0F-*&F.F-F:F-F4*&F.F-F>F-F4*$F:F0F-*&F:F-F>F-F4*$F>F0F-F4#F- F0/%\"bG,$*&,:*&F:F0F6F-F4*&F2F-F>F0F4*(F.F-F2F-F:F-F-*&F.F0F6F-F4*&F* F-F:F0F4*&F*F-F>F0F4*&F.F0F2F-F4*(F*F-F.F-F>F-F-*(F*F-F.F-F:F-F-*(F:F- F6F-F>F-F-*(F2F-F:F-F>F-F-*(F.F-F6F-F>F-F-F-FAF4#F4F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(\");" }}}{EXCHG {PARA 0 "" 0 "Punkte" {TEXT -1 17 "Aber jetzt wieder" }{TEXT 267 5 " mit " } {TEXT -1 19 "eckigen Klammern!->" }{HYPERLNK 17 "Zur\374ck" 1 "" "Zur \374ck" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "x[1]:=-2: y[1]:=1 : x[2]:=0: y[2]:=2: x[3]:=1: y[3]:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Nw3(m,b) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"*\"#9" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 85 "Gerade:=plot(f(x),x=-15..15,-15..15,color=gree n,thickness=2,title=`Kuerzester Weg`): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Punkte:=plot(\{[x[1],y[1]],[x[2],y[2]],[x[3],y[3]]\}, style=point,symbol=box,thickness=10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "display(\{Punkte,Gerade\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Das ist \+ die gesuchte Gerade in einem normalen Plot." }}}}}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 }