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0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 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 Out put" -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 Plot" -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 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 257 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" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 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 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 256 34 "Stetigkeit und Differenzierbarkeit" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 257 74 "Beispiele zur Stetigkeit und Differenzierbarkeit\nganzrationaler Funktionen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 14 "Autor: D.Komma" }} {PARA 258 "" 0 "" {TEXT -1 16 "Datum: M\344rz 2003" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(student):\nwith(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Stetig und differenzierbar" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=x->x^2-x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f GR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"F1F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "fPlot:=plot(f(x),x=-2..2, f=-2..5,color=black,discont=true):\ndisplay(fPlot);" }}{PARA 13 "" 1 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Ableitungsfunkt ion von f" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eva lf(diff(f(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG$\"\"#\"\" !$\"\"\"F'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Steigung der S ekante durch die Punkte " }{TEXT 263 8 "(a|f(a))" }{TEXT -1 5 " und " }{TEXT 266 10 "(x0|f(x0))" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "d:=(a,x0)->(f(a)-f(x0))/(a-x0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGR6$%\"aG%#x0G6\"6$%)operatorG%&arrowGF)*&,&-%\"fG 6#9$\"\"\"-F06#9%!\"\"F3,&F2F3F6F7F7F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Sekante durch die Punkte " }{TEXT 267 8 "(a|f(a))" } {TEXT -1 5 " und " }{TEXT 268 10 "(x0|f(x0))" }{MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "g:=(x,a,x0)->f(a)+d(a,x0)*(x-a);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6%%\"xG%\"aG%#x0G6\"6$%)operat orG%&arrowGF*,&-%\"fG6#9%\"\"\"*&-%\"dG6$F29&F3,&9$F3F2!\"\"F3F3F*F*F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N:=100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Darstellung d er Sekante durch einen Kurvenpunkt an der Stelle " }{TEXT 261 1 "x" } {TEXT -1 29 " und den Punkt an der Stelle " }{TEXT 259 2 "x0" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "sekantPlot:=seq(display (\n plottools[point]([-2.05+(4.0/N)*i,f(-2.05+(4.0/N)*i)],\n \+ symbol=diamond,color=red,symbolsize=5),\n plot(g(x,-2.05+(4.0/N)*i, 0),\n x=-2.05..2.0,g=-2..5,color = blue),\n fPlot),i=0..N):\n display(sekantPlot,insequence=true);" }}{PARA 13 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Darstellung der " }{TEXT 262 7 "Sekante" }{TEXT -1 1 " " }{TEXT 269 6 "(blau)" }{TEXT -1 24 " durch die Kurvenpunkte " } {TEXT 270 8 "(a|f(a))" }{TEXT -1 5 " und " }{TEXT 271 10 "(x0|f(x0))" }{TEXT -1 11 ", sowie der" }{TEXT 264 23 " Sekantensteigung (rot)" } {TEXT -1 21 " in Abh\344ngigkeit von " }{TEXT 265 2 "x " }{TEXT -1 1 " :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 299 "seqPlot:=seq(display(\n poi nt([-2.05+(4.0/N)*i,f(-2.05+(4.0/N)*i)],\n symbol=diamond,colo r=red),\n plot(d(x,0),x=-2.05..-2.05+(4.0/N)*i,color=red,discont=tru e),\n plot(g(x,-2.05+(4.0/N)*i,0),\n x=-2.05..2.0,g=-3..6,col or = blue),\n fPlot),i=0..N):\ndisplay(seqPlot, insequence=true);" } }{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "Stetig und nicht differenzierbar?" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f:=x->piecewise(x<-0.5,x+2,1 -x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG %&arrowGF(-%*piecewiseG6%29$$!\"&!\"\",&F0\"\"\"\"\"#F5,&F5F5F0F3F(F(F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "fPlot:=plot(f(x),x=-2. .2,f=-2..5,color=black,discont=true):\ndisplay(fPlot);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Ableitungsfu nktion von f" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " evalf(diff(f(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6 %7$$\"\"\"\"\"!2%\"xG$!\"&!\"\"7$$F(%*undefinedG/F+F,7$$F.F)2F,F+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Steigung der Sekante durch die Pun kte " }{TEXT 275 8 "(a|f(a))" }{TEXT -1 5 " und " }{TEXT 278 10 "(x0|f (x0))" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "d:=(a,x 0)->(f(a)-f(x0))/(a-x0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGR6$% \"aG%#x0G6\"6$%)operatorG%&arrowGF)*&,&-%\"fG6#9$\"\"\"-F06#9%!\"\"F3, &F2F3F6F7F7F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Sekante durc h die Punkte " }{TEXT 279 8 "(a|f(a))" }{TEXT -1 5 " und " }{TEXT 280 10 "(x0|f(x0))" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "g:=(x,a,x0)->f(a)+d(a,x0)*(x-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"gGR6%%\"xG%\"aG%#x0G6\"6$%)operatorG%&arrowGF*,&-%\"fG6#9%\"\"\" *&-%\"dG6$F29&F3,&9$F3F2!\"\"F3F3F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N:=100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"$+ \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 62 "Darstellung der Sekante durch einen Kurvenpunkt an der Stelle " }{TEXT 273 1 "x" }{TEXT -1 29 " und den Punkt an der Ste lle " }{TEXT 272 2 "x0" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "sekantPlot:=seq(display(\n plottools[point]([-2.05+(4.0/N)* i,f(-2.05+(4.0/N)*i)],\n symbol=diamond,color=red,symbolsize=5 ),\n plot(g(x,-2.05+(4.0/N)*i,-0.5),\n x=-2.05..2.0,g=-2..5,c olor = blue),\n fPlot),i=0..N):\ndisplay(sekantPlot,insequence=true) ;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Darstellung d er " }{TEXT 274 7 "Sekante" }{TEXT -1 1 " " }{TEXT 281 6 "(blau)" } {TEXT -1 24 " durch die Kurvenpunkte " }{TEXT 282 8 "(a|f(a))" }{TEXT -1 5 " und " }{TEXT 283 10 "(x0|f(x0))" }{TEXT -1 11 ", sowie der" } {TEXT 276 23 " Sekantensteigung (rot)" }{TEXT -1 21 " in Abh\344ngigke it von " }{TEXT 277 2 "x " }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 305 "seqPlot:=seq(display(\n point([-2.05+(4.0/N)*i,f(- 2.05+(4.0/N)*i)],\n symbol=diamond,color=red),\n plot(d(x,-0 .5),x=-2.05..-2.05+(4.0/N)*i,color=red,discont=true),\n plot(g(x,-2. 05+(4.0/N)*i,-0.5),\n x=-2.05..2.0,g=-3..6,color = blue),\n f Plot),i=0..N):\ndisplay(seqPlot, insequence=true);" }}{PARA 13 "" 1 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Unstetig u nd" }{TEXT 260 1 " " }{TEXT -1 5 "nicht" }{TEXT 258 1 " " }{TEXT -1 15 "differenzierbar" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:=x->piecewise(x<1,x,x+1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&ar rowGF(-%*piecewiseG6%29$\"\"\"F0,&F0F1F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "fPlot:=plot(f(x),x=-2..2,f=-2..5,color=black, discont=true):\ndisplay(fPlot);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Ableitungsfunktion von f" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(diff(f(x) ,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$$\"\"\"%*und efinedG/%\"xG$F(\"\"!7$F,%*otherwiseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Steigung der Sekante durch die Punkte " }{TEXT 287 8 "(a| f(a))" }{TEXT -1 5 " und " }{TEXT 290 10 "(x0|f(x0))" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "d:=(a,x0)->(f(a)-f(x0))/(a-x0) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGR6$%\"aG%#x0G6\"6$%)operato rG%&arrowGF)*&,&-%\"fG6#9$\"\"\"-F06#9%!\"\"F3,&F2F3F6F7F7F)F)F)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Sekante durch die Punkte " }{TEXT 291 8 "(a|f(a))" }{TEXT -1 5 " und " }{TEXT 292 10 "(x0|f(x0))" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "g:=(x,a,x0)->f( a)+d(a,x0)*(x-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6%%\"xG%\" aG%#x0G6\"6$%)operatorG%&arrowGF*,&-%\"fG6#9%\"\"\"*&-%\"dG6$F29&F3,&9 $F3F2!\"\"F3F3F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N:=1 00;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "D arstellung der Sekante durch einen Kurvenpunkt an der Stelle " }{TEXT 285 1 "x" }{TEXT -1 29 " und den Punkt an der Stelle " }{TEXT 284 2 "x 0" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 263 "sekantPlot:=s eq(display(\n plottools[point]([-2.05+(4.0/N)*i,f(-2.05+(4.0/N)*i)], \n symbol=diamond,color=red,symbolsize=5),\n plot(g(x,-2.05+ (4.0/N)*i,1),\n x=-2.05..2.0,g=-2..5,color = blue),\n fPlot), i=0..N):\ndisplay(sekantPlot,insequence=true);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Darstellung der " }{TEXT 286 7 "Sekante" }{TEXT -1 1 " " }{TEXT 293 6 "(blau)" }{TEXT -1 24 " durch die Kurvenp unkte " }{TEXT 294 8 "(a|f(a))" }{TEXT -1 5 " und " }{TEXT 295 10 "(x0 |f(x0))" }{TEXT -1 11 ", sowie der" }{TEXT 288 23 " Sekantensteigung ( rot)" }{TEXT -1 21 " in Abh\344ngigkeit von " }{TEXT 289 2 "x " } {TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "seqPlot:=seq(disp lay(\n point([-2.05+(4.0/N)*i,f(-2.05+(4.0/N)*i)],\n symbol= diamond,color=red),\n plot(d(x,1),x=-2.05..-2.05+(4.0/N)*i,color=red ,discont=true),\n plot(g(x,-2.05+(4.0/N)*i,1),\n x=-2.05..2.0 ,g=-3..15,color = blue),\n fPlot),i=0..N):\ndisplay(seqPlot, inseque nce=true);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 43 "Differenzierbar aber trotzdem nicht stetig?" }}{EXCHG {PARA 0 "" 0 "" {TEXT 296 13 "Voraussetzung" }{TEXT -1 83 ": f ist dif ferenzierbar auf dem ganzen Defintionsbereich (hier alle reellen Zahle n)" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f:=x->f(x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Damit hat f erstmal zumindest keinen Knick, wie beispi elsweise die Betragsfunktion:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(abs(x),x=-2..2);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "W\344re noch m\366glich, das s f als Unstetigkeit einen Sprung hat, wie etwa" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "f:=x->piecewise(x<1,x,x+1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG 6%29$\"\"\"F0,&F0F1F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "fPlot:=plot(f(x),x =-2..2,f=-2..5,color=black,discont=true):\ndisplay(fPlot);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 288 "Was in diesem Beispiel mit d em Differenzenquotient geschieht hat man oben gesehen: er divergiert. \+ Also existiert im Widerspruch zur Voraussetzung (f diff'bar) an dieser Stelle kein reeller Ableitungswert. (Anschaulich gesprochen h\344tte \+ damit auch die Ableitungsfunktion hier einen Sprung)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "display(seqPlot, insequence=true);" }}{PARA 13 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f:=' f': x0:='x0': x:='x': f:=x->f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"fGF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 331 "Also folgt aus der Di fferenzierbarkeit einer Funktion auch die Stetigkeit auf dem entsprech enden Bereich (Intervall). Rechnerisch argumentiert man \344hnlich: De r Grenzwert des Differenzenquotienten soll existeren, also endlich ble iben. F\374r differenzierbares f existiert also der folgende Grenzwert auf dem gesamten Definitionsbereich" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "'limit((f(x)-f(x0))/(x-x0),x=x0)';" }{TEXT -1 0 "" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limitG6$* &,&-%\"fG6#%\"xG\"\"\"-F)6#%#x0G!\"\"F,,&F+F,F/F0F0/F+F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Wir " }{TEXT 301 7 "nehmen " }{TEXT -1 7 " zuerst " }{TEXT 302 19 "an, dass f unstetig" }{TEXT -1 10 " ist mit \+ " }{TEXT 299 12 "f(x) > f(x0)" }{TEXT -1 25 ", dann bleibt der Nenner \+ " }{TEXT 298 17 "echt gr\366\337er als 0" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "assume('f(x)'>'f(x0)');" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "und f\374r den Grenzwert erhalten wir" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "'limit((f(x)-f( x0))/(x-x0),x=x0)'=limit(('f(x)'-'f(x0)')/(x-x0),x=x0);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%&limitG6$*&,&-%\"fG6#%\"xG\"\"\"-F*6#%#x0G!\" \"F-,&F,F-F0F1F1/F,F0%*undefinedG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Entsprechendes gilt f\374r " }{TEXT 297 12 "f(x) < f(x0)" }{TEXT -1 25 ", dann bleibt der Nenner " }{TEXT 300 18 "echt kleiner als 0" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "assume('f(x)'<' f(x0)');" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "und f\374r den Grenzw ert erhalten wir wieder" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "'limit(( f(x)-f(x0))/(x-x0),x=x0)'=limit(('f(x)'-'f(x0)')/(x-x0),x=x0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&limitG6$*&,&-%\"fG6#%\"xG\"\"\"-F* 6#%#x0G!\"\"F-,&F,F-F0F1F1/F,F0-F%6$*&F(F-,&%#x|irGF-%$x0|irGF1F1/F8F9 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Das bedeutet beides Mal, dass die Ableitung an der Stelle" }{TEXT 303 4 " x0 " }{TEXT -1 26 "gar ni cht existiert, wenn " }{TEXT 304 2 "f " }{TEXT -1 63 "unstetig sein so ll. Das ist ein Widerspruch zur Voraussetzung, " }{TEXT 307 22 "als mu ss die Funktion " }{TEXT 305 1 "f" }{TEXT 308 17 " an jeder Stelle " } {TEXT 306 2 "x0" }{TEXT 309 50 " stetig sein, an der sie auch differen zierbar ist!" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "x:='x':f:='f':" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Weitere Beispiele" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "Flossenprofil" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f1:= x->3-x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1GR6#%\"xG6\"6$%)oper atorG%&arrowGF(,&\"\"$\"\"\"*$)9$\"\"#F.!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "f2:=x->2/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2GR6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F.9$!\"\"\"\"# F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot([f1(x),f2(x) ],x=-4..4,y=-4..4);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=x->piecewise(x<1,f1(x),f2(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6%29$\"\"\" -%#f1G6#F0-%#f2GF4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$,&\"\"$\"\" \"*$)%\"xG\"\"#F)!\"\"2F,F)7$,$*&F)F)F,F.F-%*otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot([f(x),f1(x),f2(x)], x=-4..6, y =-4..4, color=[blue,red,green]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fPlot:=plot(f(x),x=-4..4,col or=black):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Ableitungsfunktion \+ von f" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(d iff(f(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$,$% \"xG$!\"#\"\"!1F($\"\"\"F+7$,$*&F.F.*$)F(\"\"#F.!\"\"F)2F-F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Steigung der Sekante durch die Pun kte " }{TEXT 311 8 "(a|f(a))" }{TEXT -1 5 " und " }{TEXT 314 10 "(x0|f (x0))" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "d:=(a,x 0)->(f(a)-f(x0))/(a-x0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGR6$% \"aG%#x0G6\"6$%)operatorG%&arrowGF)*&,&-%\"fG6#9$\"\"\"-F06#9%!\"\"F3, &F2F3F6F7F7F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Sekante durc h die Punkte " }{TEXT 315 8 "(a|f(a))" }{TEXT -1 5 " und " }{TEXT 316 10 "(x0|f(x0))" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "g:=(x,a,x0)->f(a)+d(a,x0)*(x-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"gGR6%%\"xG%\"aG%#x0G6\"6$%)operatorG%&arrowGF*,&-%\"fG6#9%\"\"\" *&-%\"dG6$F29&F3,&9$F3F2!\"\"F3F3F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N:=100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"$+ \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 16 "Darstellung der " }{TEXT 310 7 "Sekante" }{TEXT -1 1 " " }{TEXT 317 6 "(blau)" }{TEXT -1 24 " durch die Kurvenpunkte \+ " }{TEXT 318 8 "(a|f(a))" }{TEXT -1 5 " und " }{TEXT 319 10 "(x0|f(x0) )" }{TEXT -1 11 ", sowie der" }{TEXT 312 23 " Sekantensteigung (rot)" }{TEXT -1 21 " in Abh\344ngigkeit von " }{TEXT 313 2 "x " }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 299 "seqPlot:=seq(display(\n po int([-4.05+(8.0/N)*i,f(-4.05+(8.0/N)*i)],\n symbol=diamond,col or=red),\n plot(d(x,1),x=-4.05..-4.05+(8.0/N)*i,color=red,discont=tr ue),\n plot(g(x,-4.05+(8.0/N)*i,1),\n x=-4.05..4.0,g=-3..5,co lor = blue),\n fPlot),i=0..N):\ndisplay(seqPlot, insequence=true);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 28 "Kinematik au f einer Kartbahn" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 664 "v:=t->piecewise(t>0 and t<= 6, 2.3*t, #Start\n t>6 and t<=9, v(6)+1.5*(6-t), #Kurve B\n \+ t>9 and t<=13, v(9)+1.3*(-9+t),\n t>13 and t<=17, v(13)+3.3*(13-t), \+ #Kurve C\n t>17 and t<=21, v(17)+1.7*(-17+t),\n t>21 and t<=29, v (21)+0.9*(21-t), #Kurve D\n t>29 and t<=38, v(29)+0.6*(-29+t),\n \+ t>38 and t<=46, v(38)+0.5*(38-t), #Kurve E,F\n t>46 and t<=53, v(46 )+0.7*(-46+t), \n t>53 and t<=61, v(53)+0.5*(53-t), #Kurve G\n t> 61 and t<=68, v(61)+1.2*(-61+t),\n t>68 and t<=73, v(68)+1.4*(68-t), #Kurve H\n t>73 and t<=78, v(73)+1.4*(-73+t),\n t>78 and t<=82, \+ v(78)+1.3*(78-t), #Kurve I\n t>82 and t<=90, v(82)+2.1*(-82+t) \+ #Ziel\n):\nv(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWIS EG617$,$*&$\"#B!\"\"\"\"\"%\"tGF,F,32,$F-F+\"\"!1,&F-F,\"\"'F+F17$,&$ \"$G#F+F,*&$\"#:F+F,F-F,F+32F0!\"'1,&F-F,\"\"*F+F17$,&$\"#CF+F+*&$\"#8 F+F,F-F,F,32F0!\"*1,&F-F,FHF+F17$,&$\"$u&F+F,*&$\"#LF+F,F-F,F+32F0!#81 ,&F-F,\"# " 0 "" {MPLTEXT 1 0 40 "vPlot:=plot(10*v(t),t=0..50,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sPlot:=plot(int(v(t),t),t=0..50,col or=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "aPlot:=plot(5 0*diff(v(t),t),t=0..50):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "display(sPlot,vPlot,aPl ot);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "13 0 0" 14 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }