{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 "" 1 16 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 16 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 1 16 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 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 "Maple Output" 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 "" 1 14 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 {SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "Quelle" }}{PARA 0 "" 0 "" {TEXT -1 356 "Dateiname: kurvdisk.mws\nDateigr\366\337e: 15 KB\nName: \+ Kerstin M\374ller\nSchule: Isolde-Kurz-Gymnasium\nKlasse: 11d\nDatum: \+ 25.03.97\nFach: Mathematik\nThema: Kurvendiskussion\nStichw\366rter: B erechnung der charakteristischen Punkte\nKurzbeschreibung: erste und z weite Ableitung einer Funktion\nBerechnung von Nullstellen, Extrema, W endestellen\nAufbau eines Tangentenger\374sts\n" }}}{EXCHG {PARA 256 " " 0 "" {TEXT 256 14 "Kerstin M\374ller" }{TEXT 257 24 " \+ " }{TEXT 258 31 "Worksheet zu Kurvendiskussionen" }{TEXT 259 15 " " }{TEXT 260 10 "09.03.1997" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Die Funktion und ihre ersten beiden " }{TEXT 261 11 "Ableitungen" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with (plots):" }}}{EXCHG {PARA 0 "> " 0 "x^4" {MPLTEXT 1 0 20 "f:=x->x^4+x^ 3-2*x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)opera torG%&arrowGF(,(*$9$\"\"%\"\"\"*$F.\"\"$F0*$F.\"\"#!\"#F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "fs:=D(f);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#fsG:6#%\"xG6\"6$%)operatorG%&arrowGF(,(*$9$\"\"$\" \"%*$F.\"\"#F/F.!\"%F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "fss:=D(fs);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fssG:6#%\"xG6\"6$%) operatorG%&arrowGF(,(*$9$\"\"#\"#7F.\"\"'!\"%\"\"\"F(F(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Berechnun g der " }{TEXT 262 11 "Nullstellen" }}{PARA 0 "" 0 "" {TEXT -1 117 "Um die Nullstellen zu berechnen, mu\337 man die Funktion gleich null set zen, weil der y-Wert an diesen Punkten null ist." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "Nullstellen:=solve(f(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,NullstellenG6&\"\"!F&!\"#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Berechnung \+ der " }{TEXT 269 29 "Schnittpunkte mit der y-Achse" }}{PARA 0 "" 0 "" {TEXT -1 103 "Um diese Punkte zu berechnen, mu\337 man die Funktion vo n null berechnen, der x-Wert ist also gleich null." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "Schnittpunktey:=f(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/SchnittpunkteyG\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Berechnung der " }{TEXT 263 7 "Extrema" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Zur Berechnung d er Extrempunkte braucht man die erste Ableitung, die gleich null geset zt wird." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Extremstellen:= fsolve(fs(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.Extremstellen G6%\"\"!$!+o/+V9!\"*$\"+#o/+$p!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Das sind die x-Werte der Extrempunkte. Die y-Werte erh\344lt man, indem man diese Werte jetzt als x-Wert in die Funktion einsetzt. " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f(Extremstellen[1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f(Extremstellen[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+4CULG!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f(Extrem stellen[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+4MYqR!#5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Das sind jetzt also die y-Werte de r Extrempunkte." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 15 "Berechnung der " }{TEXT 264 11 "Wendepunkte" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Um die Wendepunkte berechnen zu k \366nnen, braucht man die zweite Ableitung, die dazu gleich null geset zt wird." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Wendestellen:=f solve(fss(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-WendestellenG 6$$!+'pG:z)!#5$\"+'pG:z$F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Da s ist der x-Wert der Wendepunktes. F\374r den y-Wert mu\337 man diesen jetzt wieder wie bei den Extrempunkten als Wert f\374r x einsetzen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f(Wendestellen[1]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$!+ne$zi\"!\"*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "f(Wendestellen[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+0\">M7#!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "\334berpr\374fung der " }{TEXT 265 15 "Achsensymmetrie" }{TEXT -1 12 " zur y-Achse" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Falls bei dieser Funktion eine Achsensymmetrie he rrscht, so mu\337 f(-x)-f(x) gleich null sein." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "f(-x)-f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*$%\"xG\"\"$!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "\334berpr\374fung der " }{TEXT 266 14 "Pu nktsymmetrie" }{TEXT -1 13 " zum Ursprung" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Wenn das Schaubild dieser Funktion punktsymmetrisch zum \+ Ursprung ist, dann mu\337 f(-x)-(-f(x)) gleich null sein." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f(-x)-(-f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$%\"xG\"\"%\"\"#*$F%F'!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 267 9 "Schaubild" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Jetzt zeichne ich die Funktion mit ihren ersten beiden Ableitungen in ein Schaubild." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "xmin:=min(Nullstellen,Extremstellen,Wendest ellen)-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xminG!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "xmax:=max(Nullstellen,Extremstellen ,Wendestellen)+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xmaxG\"\"#" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A:=plot(f,xmin..xmax,color= green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "B:=plot(fs,xmin. .xmax,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "C:=pl ot(fss,xmin..xmax,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(A,B,C);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 268 15 "Tangentenger\374st" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "Nun werde ich versuchen, ein Tangentenger\374st auf zubauen. Dazu berechne und zeichne ich die Tangenten an den charakteri stischen Punkten, also an den Nullstellen, den Extrema und den Wendepu nkten." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m1:=diff(f(x1),x1 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m1G,(*$%#x1G\"\"$\"\"%*$F'\" \"#F(F'!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PSF1:=(y-y1) /(x-x1)=m1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PSF1G/*&,&%\"yG\"\" \"%#y1G!\"\"F),&%\"xGF)%#x1GF+F+,(*$F.\"\"$\"\"%*$F.\"\"#F1F.!\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x1:=Nullstellen[1]:y1:=f(x1) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "T1:=solve(PSF1,y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T1G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Tang1:=plot(T1,x=-2..2,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m2:=diff(f(x2),x2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m2G,(*$%#x2G\"\"$\"\"%*$F'\"\"#F(F'!\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PSF2:=(y-y2)/(x-x2)=m2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PSF2G/*&,&%\"yG\"\"\"%#y2G!\"\"F),& %\"xGF)%#x2GF+F+,(*$F.\"\"$\"\"%*$F.\"\"#F1F.!\"%" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 29 "x2:=Nullstellen[2]:y2:=f(x2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "T2:=solve(PSF2,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T2G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Tang2:=plot(T2,x=-2..2,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m3:=diff(f(x3),x3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m3G,(*$%#x3G\"\"$\"\"%*$F'\"\"#F(F'!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PSF3:=(y-y3)/(x-x3)=m3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PSF3G/*&,&%\"yG\"\"\"%#y3G!\"\"F),&%\"xGF )%#x3GF+F+,(*$F.\"\"$\"\"%*$F.\"\"#F1F.!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x3:=Nullstellen[3]:y3:=f(x3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "T3:=solve(PSF3,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T3G,&%\"x G!#7!#C\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Tang3:=plo t(T3,x=-2.3..-1.7,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m4:=diff(f(x4),x4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m4G, (*$%#x4G\"\"$\"\"%*$F'\"\"#F(F'!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PSF4:=(y-y4)/(x-x4)=m4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PSF4G/*&,&%\"yG\"\"\"%#y4G!\"\"F),&%\"xGF)%#x4GF+F+,(*$F.\"\" $\"\"%*$F.\"\"#F1F.!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " x4:=Nullstellen[4]:y4:=f(x4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y4 G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "T4:=solve(PSF4,y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T4G,&%\"xG\"\"$!\"$\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Tang4:=plot(T4,x=0.7..1.3,co lor=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m5:=diff(f(x 5),x5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m5G,(*$%#x5G\"\"$\"\"%*$ F'\"\"#F(F'!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PSF5:=(y -y5)/(x-x5)=m5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PSF5G/*&,&%\"yG \"\"\"%#y5G!\"\"F),&%\"xGF)%#x5GF+F+,(*$F.\"\"$\"\"%*$F.\"\"#F1F.!\"% " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "x5:=Schnittpunktey[1]:y 5:=f(x5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y5G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "T5:=solve(PSF5,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T5G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Tang5:=plot(T5,x=-2..2,color=grey):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m6:=diff(f(x6),x6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m6G,(*$%#x6G\"\"$\"\"%*$F'\"\"#F(F'!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PSF6:=(y-y6)/(x-x6)=m6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PSF6G/*&,&%\"yG\"\"\"%#y6G!\"\"F),&%\"xGF)%#x6GF+F+, (*$F.\"\"$\"\"%*$F.\"\"#F1F.!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x6:=Extrema[1]:y6:=f(x6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y6G,(*$&%(ExtremaG6#\"\"\"\"\"%F**$F'\"\"$F**$F'\"\" #!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "T6:=solve(PSF6,y); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#T6G,$*&&%(ExtremaG6#\"\"\"F*,.* $F'\"\"$F-*$F'\"\"#F/F'!\"#*&F'F/%\"xGF*!\"%*&F'F*F2F*!\"$F2\"\"%F*!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Tang6:=plot(T6,x=-2.. 2,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m7:=diff( f(x7),x7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m7G,(*$%#x7G\"\"$\"\" %*$F'\"\"#F(F'!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PSF7: =(y-y7)/(x-x7)=m7;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PSF7G/*&,&%\" yG\"\"\"%#y7G!\"\"F),&%\"xGF)%#x7GF+F+,(*$F.\"\"$\"\"%*$F.\"\"#F1F.!\" %" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "x7:=Extremstellen[2]:y 7:=f(x7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y7G$!+4CULG!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "T7:=solve(PSF7,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T7G,&$!+-CULG!\"*\"\"\"%\"xG$\"+++++]!#= " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Tang7:=plot(T7,x=-2..-1 ,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m8:=diff(f (x8),x8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m8G,(*$%#x8G\"\"$\"\"% *$F'\"\"#F(F'!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PSF8:= (y-y8)/(x-x8)=m8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PSF8G/*&,&%\"y G\"\"\"%#y8G!\"\"F),&%\"xGF)%#x8GF+F+,(*$F.\"\"$\"\"%*$F.\"\"#F1F.!\"% " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "x8:=Extremstellen[3]:y8 :=f(x8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y8G$!+4MYqR!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "T8:=solve(PSF8,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T8G$!+4MYqR!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Tang8:=plot(T8,x=0.4..1,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "m9:=diff(f(x9),x9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m9G,(*$%#x9G\"\"$\"\"%*$F'\"\"#F(F'!\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "PSF9:=(y-y9)/(x-x9)=m9;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%PSF9G/*&,&%\"yG\"\"\"%#y9G!\"\"F),& %\"xGF)%#x9GF+F+,(*$F.\"\"$\"\"%*$F.\"\"#F1F.!\"%" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "x9:=Wendestellen[1]:y9:=f(x9);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#y9G$!+ne$zi\"!\"*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "T9:=solve(PSF9,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T9G,&$\"+)oiE6\"!\"*\"\"\"%\"xG$\"+?uJ " 0 "" {MPLTEXT 1 0 39 "Tang9:=plot(T9,x=-1.4..-0.2,color=red):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "m10:=diff(f(x10),x10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$m10G,(*$%$x10G\"\"$\"\"%*$F'\"\"#F( F'!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "PSF10:=(y-y10)/(x -x10)=m10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&PSF10G/*&,&%\"yG\"\" \"%$y10G!\"\"F),&%\"xGF)%$x10GF+F+,(*$F.\"\"$\"\"%*$F.\"\"#F1F.!\"%" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "x10:=Wendestellen[2]:y10:= f(x10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$y10G$!+0\">M7#!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "T10:=solve(PSF10,y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$T10G,&$\"+#yR];\"!#5\"\"\"%\"xG$!+, U " 0 "" {MPLTEXT 1 0 40 "Tang10:=plot(T10 ,x=-0.5..0.9,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Funktion:=plot(f(x),x=-2..1.5,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "display(Tang1,Tang2,Tang3,Tang4,Tang5,Tang6,Tang 7,Tang8,Tang9,Tang10,Funktion);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 210 "So, in diesem Schaubi ld sieht man jetzt die Funktion und ihr Tangentenger\374st, das durch \+ Tangenten durch die Nullstellen, die Schnittpunkte mit der y-Achse, di e Extrema und die Wendepunkte zustande gekommen ist." }}{PARA 0 "" 0 " " {TEXT -1 199 "Wenn man jetzt oben eine andere Funktion eingibt, dann mu\337 man die Bereichsangaben \344ndern und eventuell noch neue Tang enten hinzuf\374gen (je nachdem, wieviele Nullstellen usw. diese Funkt ion dann hat)." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 3 "" 0 "" {HYPERLNK 17 "Neue Funktion eingeben" 1 "" "x^4" }}}{MARK "0" 0 } {VIEWOPTS 1 1 0 1 1 1803 }