{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 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 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 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 0 }{CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 1 12 0 0 0 0 0 2 0 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 0 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 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 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 " Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Output" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 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 0 "" 0 "" {TEXT -1 82 " Claudia Loewenstein \+ Isolde-Kurz-Gymnasium" }{MPLTEXT 1 0 23 " " }{TEXT -1 12 "2. Juni 2002" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT 256 24 "L \344nge eines Kurvenst\374cks" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT 257 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 " Eine Funktion sei im Intervall [a;b] differenz ierbar und f \264 au\337erdem noch stetig." }}{PARA 0 "" 0 "" {TEXT -1 24 " F\374r die Ermittlung der " }{TEXT 259 15 "L\344nge der Kurve " }{TEXT -1 35 " wird das Kurvenst\374ck K durch einen" }{TEXT 260 12 " Streckenzug" }{TEXT -1 26 " angen\344hert, die einzelnen" }{TEXT 263 25 " L\344ngen der Teilstrecken " }{TEXT -1 6 "werden" }{TEXT 264 8 " addiert" }{TEXT -1 9 " und der " }{TEXT 262 27 "Grenz\374bergang z u unendlich " }{TEXT -1 25 "vielen Strecken vollzogen" }{TEXT 261 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "Dies entspricht das Vorgehen in der I ntegralrechnung." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "restart:with(plots):f:=x->x^2:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "p1:=plot(f(x),x=0..4,color=r ed,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "p2:=plo t(\{seq([[x,f(x)],[x+1,f(x)]],x=0..3)\},x=0..4,color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "p3:=plot(\{seq([[x+1,f(x)],[ x+1,f(x+1)]],x=0..3)\},x=0..4,color=blue):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "p4:=plot(\{seq([[x,f(x)],[x+1,f(x+1)]],x=0..3)\},x= 0..4,color=blue):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 " Vorgehen:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "displa y(p1,p2,p3,p4);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "61-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3Hmmmm;')=()!#>$\"3$Q%pCw[&=g(!#?7$$ \"3RLLLe'40j\"!#=$\"3m$G=fuh&eEF-7$$\"3mmmm;6m$[#F4$\"3c8>WUDdohF-7$$ \"3fmmm;yYULF4$\"3OV_a5\"4s6\"F47$$\"3%HLL$eF>(>%F4$\"33x010FkhK'*)\\F4$\"3'*H;\"3%Hk*[#F47$$\"3P*****\\Kd,\"eF4$\"3y9^79Gz vLF47$$\"3-mmm\"fX(emF4$\"3sDSX&G*)QV%F47$$\"3.*****\\U7Y](F4$\"3XOk% \\w?>j&F47$$\"3'QLLLV!pu$)F4$\"30'[;a)Ra8qF47$$\"3xmmm;c0T\"*F4$\"3Q@$ *py(*)eN)F47$$\"3#*******H,Q+5!#<$\"3w;))\\//w+5Fao7$$\"3)*******\\*3q 3\"Fao$\"3a-,Qd%)e\"=\"Fao7$$\"3)*******p=\\q6Fao$\"3%o4Ox@^+P\"Fao7$$ \"3mmm;fBIY7Fao$\"31*Rk/dpKb\"Fao7$$\"3GLLLj$[kL\"Fao$\"3?Xj&yA%4'y\"F ao7$$\"3?LLL`Q\"GT\"Fao$\"37\"eqT)H/'*>Fao7$$\"3!*****\\s]k,:Fao$\"33I NwBz$\\D#Fao7$$\"39LLL`dF!e\"Fao$\"3YLKdc9F(\\#Fao7$$\"33++]sgam;Fao$ \"3ov^w6ePxFFao7$$\"3/++]9iq$z0$Fao7$$\"3QLLLe/TM=Fao$ \"355FkH<1lLFao7$$\"3JLL$eDBJ\">Fao$\"3gL.,#fS+m$Fao7$$\"3immmTc-)*>Fa o$\"3'[\\dZY1@*RFao7$$\"3Mmm;f`@'3#Fao$\"3-.H#[_%H_VFao7$$\"3y****\\nZ )H;#Fao$\"3)z-PW5.&yYFao7$$\"3YmmmJy*eC#Fao$\"3DM]Gqq0W]Fao7$$\"3')*** ***R^bJBFao$\"3;>/'3P\\hV&Fao7$$\"3f*****\\5a`T#Fao$\"3i[.a_a$R$eFao7$ $\"3o****\\7RV'\\#Fao$\"3)e0![zA=KiFao7$$\"3k*****\\@fke#Fao$\"3E9%e3F r(*o'Fao7$$\"3/LLL`4NnEFao$\"3ES#[#36w9rFao7$$\"3#*******\\,s`FFao$\"3 '=-;XmuHe(Fao7$$\"3[mm;zM)>$GFao$\"3$p$HFE/8?!)Fao7$$\"3$*******pfa&R,5!#;7$$\"3# )****\\7yh]KFao$\"3QGUH;;lc5Ffw7$$\"3xmmm')fdLLFao$\"3,k!))e)GF66Ffw7$ $\"3bmmm,FT=MFao$\"3%)*f\"*)RXbo6Ffw7$$\"3FLL$e#pa-NFao$\"3S`kw'\\$yE7 Ffw7$$\"3!*******Rv&)zNFao$\"3-&[p1+Q:G\"Ffw7$$\"3ILLLGUYoOFao$\"3!y7c %zHwX8Ffw7$$\"3_mmm1^rZPFao$\"3OaP2_o`/9Ffw7$$\"34++]sI@KQFao$\"37*)RI .deo9Ffw7$$\"34++]2%)38RFao$\"3h)3\"\\)3E7`\"Ffw7$$\"\"%F)$\"#;F)-%'CO LOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-F$6$7$7$$\"\"$F)$ \"\"*F)7$FezFj[l-Fjz6&F\\[lF(F(F][l-F$6$7$7$$Fc[lF)Fez7$Fh[lFezF]\\l-F $6$7$7$$\"\"\"F)Fi\\l7$Fc\\lFi\\lF]\\l-F$6$7$F'7$Fi\\lF(F]\\l-F$6$7$F \\\\lFdzF]\\l-F$6$7$F[]lFb\\lF]\\l-F$6$7$Fd\\lFg[lF]\\l-F$6$7$F_]lFh\\ lF]\\l-F$6$7$F'Fh\\lF]\\l-F$6$7$Fh\\lFb\\lF]\\l-F$6$7$Fb\\lFg[lF]\\l-F $6$7$Fg[lFdzF]\\l-%+AXESLABELSG6%Q\"x6\"Q!F\\_l-%%FONTG6#%(DEFAULTG-%% VIEWG6$;F(FezFa_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 " 1. Das Intervall [a;b ] wird in n gleich lange Teilintervalle der L\344nge:" }}{PARA 256 "" 0 "" {TEXT -1 17 " " }{XPPEDIT 18 0 "Delta;" "6#%&Delt aG" }{TEXT -1 6 " x = " }{XPPEDIT 18 0 "(b-a)/n;" "6#*&,&%\"bG\"\"\"% \"aG!\"\"F&%\"nGF(" }{TEXT -1 27 " " } {MPLTEXT 1 0 15 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 " Die den x-Werten a = x0, x1, x2, ..., xn = b zugeordneten Punkt en auf dem Kurvenst\374ck bilden einen Sehnenzug P0 P1 P2 ...Pn" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 69 "F\374r die Gesamtl\344nge der Sehnen gilt dann nac h dem Satz des Pytagoras:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f:='f':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "sn:=sum(sqrt((D elta(x))^2+(Delta(y[i]))^2),i=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#snG-%$sumG6$*$-%%sqrtG6#,&*$)-%&DeltaG6#%\"xG\"\"#\"\"\"F4*$)-F06 #&%\"yG6#%\"iGF3F4F4F4/F<;F4%\"nG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "sn:=sum(sqrt((Delta(x))^2+(Delta(y[i]))^2),i=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#snG,**$-%%sqrtG6#,&*$)-%&DeltaG6# %\"xG\"\"#\"\"\"F2*$)-F.6#&%\"yG6#F2F1F2F2F2F2*$-F(6#,&F+F2*$)-F.6#&F8 6#F1F1F2F2F2F2*$-F(6#,&F+F2*$)-F.6#&F86#\"\"$F1F2F2F2F2*$-F(6#,&F+F2*$ )-F.6#&F86#\"\"%F1F2F2F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 " f \374r diesen Fall mit n = 4" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 " oder:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 63 "sn1:=Delta(x)*(sum(sqrt(1+(Delta(y[i])/(Delta(x)))^ 2),i=1..4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sn1G*&-%&DeltaG6#% \"xG\"\"\",**$-%%sqrtG6#,&F*F**&-F'6#&%\"yG6#F*\"\"#F&!\"#F*F*F**$-F.6 #,&F*F**&-F'6#&F56#F7F7F&F8F*F*F**$-F.6#,&F*F**&-F'6#&F56#\"\"$F7F&F8F *F*F**$-F.6#,&F*F**&-F'6#&F56#\"\"%F7F&F8F*F*F*F*" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 5 " Mit:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {XPPEDIT 18 0 "Delta(x) := x[k]-x[k-1] = (b-a)/n;" "6#>-%&DeltaG6#% \"xG/,&&F'6#%\"kG\"\"\"&F'6#,&F,F-F-!\"\"F1*&,&%\"bGF-%\"aGF1F-%\"nGF1 " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 257 "" 1 "" {TEXT -1 4 "Und " }}{PARA 258 "" 0 "" {TEXT -1 17 " " }{XPPEDIT 18 0 "Delta(y[k]) := f(x[k]) -f(x[k-1]);" "6#>-%&DeltaG6#&%\"yG6#%\"kG,&-%\"fG6#&%\"xG6#F*\"\"\"-F- 6#&F06#,&F*F2F2!\"\"F8" }{TEXT -1 26 " , k = 1, 2 , ... , n " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 " " {TEXT -1 13 " Dabei ist " }{XPPEDIT 18 0 "Delta(y[k])/Delta(x);" " 6#*&-%&DeltaG6#&%\"yG6#%\"kG\"\"\"-F%6#%\"xG!\"\"" }{TEXT -1 42 " di e \304nderungsrate von f im Intervall [ " }{XPPEDIT 18 0 "x[k-1]" "6#& %\"xG6#,&%\"kG\"\"\"F(!\"\"" }{TEXT -1 3 " ; " }{XPPEDIT 18 0 "x[k]" " 6#&%\"xG6#%\"kG" }{TEXT -1 2 "]." }{MPLTEXT 1 0 12 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 43 " Geht man in der Intervallteilung mit n -> " }{XPPEDIT 18 0 "infinity ;" "6#%)infinityG" }{TEXT -1 12 " , so gilt " }{XPPEDIT 18 0 "Delta(x );" "6#-%&DeltaG6#%\"xG" }{TEXT -1 6 " -> 0 " }{MPLTEXT 1 0 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 19 " Weiterhin strebt " }{XPPEDIT 18 0 "Delta(y[ k])/Delta(x);" "6#*&-%&DeltaG6#&%\"yG6#%\"kG\"\"\"-F%6#%\"xG!\"\"" } {TEXT -1 41 " gegen die momentane \304nderungsrate f \264( " } {XPPEDIT 18 0 "x[k]" "6#&%\"xG6#%\"kG" }{TEXT -1 2 " )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Limit(Delta(y[k] )/Delta(x),Delta(x) = 0) = `f\264`(x[k]);" "6#/-%&LimitG6$*&-%&DeltaG6 #&%\"yG6#%\"kG\"\"\"-F)6#%\"xG!\"\"/-F)6#F2\"\"!-%#f|_vG6#&F26#F." } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 53 " So geht man, nach Definiton des Integrals, \374ber in: " }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "s: =Int(sqrt(1+(f\264(x))^2),x=a..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"sG-%$IntG6$*$-%%sqrtG6#,&\"\"\"F-*$)-%#f|_vG6#%\"xG\"\"#F-F-F-/F3; %\"aG%\"bG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 " Das Schaubilb der Funktion f : = x -> f (x) , die im Intervall I = [a;b] differenzierba r ist, mit f \264 in I stetig, besitzt zwischen A ( a / f (a) ) und B \+ ( b / f (b) ) die Bogenl\344nge :" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " s := Int(sqrt(1+`f\264`(x)^2),x = a .. b);" "6#>%\"sG-%$IntG6$-%%sqrtG 6#,&\"\"\"F,*$-%#f|_vG6#%\"xG\"\"#F,/F1;%\"aG%\"bG" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 258 30 "Kurve in Parameterdarste llung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 130 " Ist die Kurve in Parameterdarstellung gegeben, so ist ihr Schaubild in Teilkurven zerlegbar, die Schaubilder von Funktionen sind." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 9 "Wegen " }{XPPEDIT 18 0 "`f\264`(x) := `v\264`(t)/`u\264`(t) = `y \264`(t)/`x\264`(t);" "6#>-%#f|_vG6#%\"xG/*&-%#v|_vG6#%\"tG\"\"\"-%#u| _vG6#F-!\"\"*&-%#y|_vG6#F-F.-%#x|_vG6#F-F2" }}{PARA 0 "" 0 "" {TEXT -1 59 " kann man das Integral auch mit den Variablen t schreiben:" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f\264(x):=v\264(t)/u\264(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#f|_vG6#%\"xG*&-%#v|_vG6#%\"tG\"\"\"-%#u|_vGF+! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "s;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$-%%sqrtG6#,&\"\"\"F+*&-%#v|_vG6#%\"tG\"\" #-%#u|_vGF/!\"#F+F+/%\"xG;%\"aG%\"bG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "s1:=Int(sqrt((u\264(t))^2+(v\264(t))^2)*1/(u\264(t)), x=a..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1G-%$IntG6$*&,&*$)-%#u |_vG6#%\"tG\"\"#\"\"\"F1*$)-%#v|_vGF.F0F1F1#F1F0F,!\"\"/%\"xG;%\"aG%\" bG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 " Wegen :" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "1/u\264(t)*dx=dt;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%#u|_vG6#%\"tG!\"\"%#dxG\"\"\"%#dt G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 " folgt:" }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "S:=Int(sqrt((u\264(t))^2+( v\264(t))^2),t=t1..t2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG-%$In tG6$*$-%%sqrtG6#,&*$)-%#u|_vG6#%\"tG\"\"#\"\"\"F4*$)-%#v|_vGF1F3F4F4F4 /F2;%#t1G%#t2G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 83 " Das Kurvenst\374ck K, mit der Parameterdarstellung \+ x = u ( t ) , y = v ( t ) mit t " }{XPPEDIT 18 0 "epsilon;" "6#%(eps ilonG" }{TEXT -1 87 " [t1 ; t2] , und mit u`, v` in [t1 ; t2] stetig, besitzt zwsichen x1 und x2 die L\344nge:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "S := Int(sqrt(`u\264`(t)^2+`v \264`(t)^2),t = t1 .. t2);" "6#>%\"SG-%$IntG6$-%%sqrtG6#,&*$-%#u|_vG6# %\"tG\"\"#\"\"\"*$-%#v|_vG6#F0F1F2/F0;%#t1G%#t2G" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 11 " Beispiele: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 268 12 "1. Astroide:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 " Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 " Die Parameterdarstellung einer Astroide ist: \+ " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x:=t ->(cos(t))^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGf*6#%\"tG6\"6$% )operatorG%&arrowGF(*$)-%$cosG6#9$\"\"$\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "y:=t->(sin(t))^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*$)-%$sinG6#9$ \"\"$\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 " Die jeweill ige Ableitungen x\264(t) und y\264(t) sind:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "xs:=D(x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#xsGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,$*&)-%$cosG 6#9$\"\"#\"\"\"-%$sinGF1F4!\"$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ys:=D(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ysGf*6 #%\"tG6\"6$%)operatorG%&arrowGF(,$*&)-%$sinG6#9$\"\"#\"\"\"-%$cosGF1F4 \"\"$F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 " mit t " } {XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 7 " [0 ; " } {XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 267 0 "" }{TEXT -1 1 ")" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "p1:=plot([x(t),y(t),t=0..2*Pi],color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "p2:=plot([x(t),y(t),t=0..Pi/2],thickness=3): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(p1,p2);" }} {PARA 13 "" 1 "" {GLPLOT2D 399 271 271 {PLOTDATA 2 "6&-%'CURVESG6$7in7 $$\"\"\"\"\"!$F*F*7$$\"3J$fMKJ42L+#Rd[a#!#?7$$\"3kF:eE2)H0* F/$\"3V2eH/;vD;!#>7$$\"3QM[NsM!4\"zF/$\"3l9Uye\\o+bF87$$\"3shc;7>/zkF/ $\"3wkOrl)Q$f7F/7$$\"3bI%HoOb$Q\\F/$\"3#oi)eZ7\\)H#F/7$$\"3[#e7=a@Gb$F /$\"33TrF,WF=NF/7$$\"3C^u.\\\"H#)G#F/$\"3!eopb.8;&\\F/7$$\"3g:Ap$3'4e7 F/$\"3u^d8Iy=\"['F/7$$\"3QfjhZ\"3\\d&F8$\"3o&R>nq+H*yF/7$$\"3xp@B^Ue5; F8$\"37^qi0Bye!*F/7$$\"3#)eM8Z7)QV#F2$\"3/bl%******!#<7$$!31%oQS-cc A$Fco$\"3'f0iH+K&H**F/7$$!3R'*Rlr_DHDF2$\"3d%o[SdUGs*F/7$$!3k')3gF3$H& =F8$\"3+rM\"3Sl#o*)F/7$$!3q=!HDJI:P&F8$\"3'[p:'pqXUzF/7$$!3[pRE+e+#G\" F/$\"3!p,v!Qq1SkF/7$$!3:VlCD_N5D)4tuB%F/$\"3csSsbhIxGF/7$$!3`7e&Qsl#Q\\F/$\"3M \"fv!>2c)H#F/7$$!3+VD%4MfI\\'F/$\"3%*3c!3rI7D\"F/7$$!3!*)Q^u#yplyF/$\" 37P)eElF/$!3'G*=8N4$>B\"F/7$$!3ly$4mjYr,&F/$!38\"p')41h yB#F/7$$!3=Wv)\\ClFH%F/$!3M)*)4c<\\*GGF/7$$!3E!frvl7^f$F/$!3UR\"3d&zGw MF/7$$!3[7@S[0\")pGF/$!3SXn7mw+YUF/7$$!3gTI*oS**G@#F/$!3E%eHc98*\\]F/7 $$!3?E&*yJR&=p\"F/$!3yI!*G\"GpFy&F/7$$!3!*)>]9JHIC\"F/$!3R'y576zs]'F/7 $$!3YDkvG>8q`F8$!3;?\"z'\\F/7$$\"3C?jqy?0yNF/$!3qMTX&\\&=$\\$F/7$$ \"3)y&y59SsI\\F/$!3/&)Hb/%4WI#F/7$$\"3a0r,9*Hp_'F/$!3Fb%>A#\\tJ7F/7$$ \"3kX\"G)*3N0&yF/$!3)HR&zU^5^dF87$$\"3OXxb#zE%***)F/$!3g\"G-6!Q8okKo?F3_#F27$F($\"3GMDF^'H?_&!#X-%'COLOURG6&%$RGBG F*F*F*-F$6%7SF'7$$\"3s%RHu\\FC)**F/$\"3^F$*)oHy9,%!#A7$$\"3%fE&Hb$\\'Q **F/$\"3!Q%GEesu>EFco7$$\"3;\\nS0*)4e)*F/$\"3gg'zoM5SB*Fco7$$\"3**=@-u F:W(*F/$\"3IAh^+D,UAF27$$\"33&y)pqe!*)f*F/$\"3FbBGc)[sT%F27$$\"3IstqoO &oV*F/$\"3uLebv+oztF27$$\"3\"o2mpatCC*F/$\"3')**Gb'oGs:\"F87$$\"3yS^H& )HY9!*F/$\"3C-M)G\\Fxs\"F87$$\"3oLMN?4fh()F/$\"3WwMPmOZ]CF87$$\"37$H>k `KoZ)F/$\"3ME4![+0\"pLF87$$\"3')>%ek\"o32#)F/$\"3U_:Bd1xNVF87$$\"3Odmc 0@M%)yF/$\"3G_du5-L5cF87$$\"3H7X_f#\\Ba(F/$\"3wG'z*yA*o4(F87$$\"3g,%oM HE#)>(F/$\"3QG@TJ_HJ()F87$$\"3g>%QBt9a(oF/$\"3!oUs\">l**Q5F/7$$\"3u!Q1 (3RU\"['F/$\"3=J#Q]RfzD\"F/7$$\"3(Qd***oPDThF/$\"3KJIRA'>=Y\"F/7$$\"3^ =X-\\?&3u&F/$\"3r1nu16z>-Jm>F/7$$\"3/ XzONe)R*\\F/$\"3=/M],UfbAF/7$$\"3Y7#QXb6\\i%F/$\"3i[!Q]^o$[DF/7$$\"3_^ _pJT9XUF/$\"35=\"[^Vo0(GF/7$$\"38:yKMs?.RF/$\"3s'*Q*zN!>!=$F/7$$\"3A]' *[\"pgPa$F/$\"3fo_O'\\8t_$F/7$$\"3+0.iKBl')\\!)>F/$\"3yYJ`*R(ok`F/7$$\" 3r[h6VIhD92\"Q8#)\\'F/7$$\"3-+E$=zW*Q5F/$\"3%3c^t%=^voF/7 $$\"3AA#4v\\Y*z')F8$\"3x8e)e;7'3sF/7$$\"3,q6a,$eM,(F8$\"3I?gGXmtgvF/7$ $\"3)[hj,4Y`j&F8$\"3WetsL0JyyF/7$$\"3mAx6=U9'Q%F8$\"3Uk.v]9p$>)F/7$$\" 3?l$GR#4XYLF8$\"3U%zIS5zM[)F/7$$\"3A0vBT[ASCF8$\"3m!>='y)p\\w)F/7$$\"3 zxf&Qx4>t\"F8$\"3Z!p2u'3!H,*F/7$$\"3EW,d!f(eg6F8$\"3v$o4jxF5C*F/7$$\"3 $Gk-t2GPJ(F2$\"3b&)[gL:=S%*F/7$$\"3!=Sc&>=]IWF2$\"3czM`.&4\")f*F/7$$\" 33B\"RA10#)=#F2$\"3OJD>A`C[(*F/7$$\"3gexgBwcw'*Fco$\"3/u+a<:h`)*F/7$$ \"3GY0AedPaGFco$\"3&*o\"3OyU]$**F/7$$\"3Y%*zS&QxL(RFd_l$\"3Ejvh5*QD)** F/7$$!3\"[L8,Lr\"G')!#ZF(-Fi^l6&F[_l$\"#5!\"\"F+F+-%*THICKNESSG6#\"\"$ -%+AXESLABELSG6%Q!6\"F[_m-%%FONTG6#%(DEFAULTG-%%VIEWG6$F`_mF`_m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "Gesucht ist die L\344nge eines einzelnen Bogens einer Astroide. Als Beispiel nehmen wir der rote Bog en im Schaubild." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 154 "Um die L\344nge berechnen zu \+ k\366nnen braucht man die zwei Punkten, wo der Bogen beginnt und endet , in diesem Fall sind es die Punkte P1 (1 / 0) und P2 (0 / 1)" }} {PARA 0 "" 0 "" {TEXT -1 77 "Dann muss man die zugeh\366rige Werte von t f\374r die Punkte P1 und P2 berechnen:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 " P1: " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solv e(x(t)=1,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!-%'arccosG6#,&#! \"\"\"\"#\"\"\"*&^##F+F*F+-%%sqrtG6#\"\"$F+F+,&%#PiGF+-F%6#,&F.F+F,F+F )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "t1:=solve(y(t)=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t1G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 " Bei t1 = 0 befindet sich die Kurve im Punkt P1" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 4 " P2:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "solve(y(t)=1,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %,$%#PiG#\"\"\"\"\"#-%'arcsinG6#,&#!\"\"F'F&*&^#F%F&-%%sqrtG6#\"\"$F&F &,$-F)6#,&F%F&F.F&F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "t2: =solve(x(t)=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t2G,$%#PiG#\" \"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 " Und bei t2 = " } {XPPEDIT 18 0 "1/2*Pi;" "6#*(\"\"\"F$\"\"#!\"\"%#PiGF$" }{TEXT -1 9 " \+ bei P2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 " Bogenl\344nge:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "S := Int(sqrt(`u \264`(t)^2+`v\264`(t)^2),t = t1 .. t2)" "6#>%\"SG-%$IntG6$-%%sqrtG6#,& *$-%#u|_vG6#%\"tG\"\"#\"\"\"*$-%#v|_vG6#F0F1F2/F0;%#t1G%#t2G" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "s:=int( sqrt(xs(t)^2+ys(t)^2),t=t1..t2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"sG,$*$-%%sqrtG6#\"\"%\"\"\"#\"\"$F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\" $\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 " Die L\344nge eines ein zelnen Bogens einer Astroide betr\344gt 3 / 2." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 71 " Die L\344nge der g anzen Kurve betr\344gt das vierfache der Bogenl\344nge, also:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "gesl:=4*s;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%geslG,$*$-%%sqrtG6#\"\"%\"\"\"\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Oder mit der Formel berechnet:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sges:=int(sqrt(xs(t)^2+ys(t) ^2),t=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sgesG,$*$-%%sqrt G6#\"\"%\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simp lify(sges);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Eine Astroide hat die L\344nge 6." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 260 "" 0 "" {TEXT -1 35 "2. Beispiel : Zykolide ; Bogenl \344nge" }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{TEXT 265 0 "" }{TEXT 266 0 "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 " Ei n Rad mit dem Radius a ( a > 0) rollt in x-Richtung ab.( siehe Animati on)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 149 " Gesucht ist die Parameterdarstellung de r Kurve, die ein Punkt P auf der Radoberfl\344che beschreibt, der zu B eginn der Rollbewegung in O (0 / 0) liegt." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the \+ name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 " Parameterdarstellung des Kreises, mit " }{TEXT 271 1 "a " }{TEXT -1 12 " als Radius:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "x:=t->a*cos(t)+b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&%\"aG\"\" \"-%$cosG6#9$F/F/%\"bGF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "y:=t->a*sin(t)+a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGf*6 #%\"tG6\"6$%)operatorG%&arrowGF(,&*&%\"aG\"\"\"-%$sinG6#9$F/F/F.F/F(F( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "a:=1:b:=1.2:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "p:=plot([x(t),y(t),t=-Pi..Pi ],color=black,scaling=constrained):p;" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6#7S7$$\"3a**************>!#=$\"3 w1-T++++5!#<7$$\"3Ag)yaiPO4#F*$\"3)G*f]@=sM')F*7$$\"358x8+z>EBF*$\"3aP LPW5rmuF*7$$\"3c.*y&>NT^FF*$\"3I)\\P*y'*)o>'F*7$$\"3m4q9z!HpM$F*$\"3! 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