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Ableitung\n" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Daniel Weihing Gomaringen 19.02. 1997 \"Untersuchung einer Funktion\"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 18 "" 0 "Untersuchng einer Funktion" {TEXT -1 27 "Untersuchung einer Funktion" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 17 "Besondere Punkte:" }}{PARA 0 "" 0 "" {TEXT -1 35 "1.) Schnittpunkt e mit der x-Achse " }{XPPEDIT 18 0 "f(x)=0" "/-%\"fG6#%\"xG\"\"!" }} {PARA 0 "" 0 "" {TEXT -1 35 "2.) Schnittpunkte mit der y-Achse " } {XPPEDIT 18 0 "f(0)" "-%\"fG6#\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 53 "3. ) Extrempunkte f '(x) = 0" }}{PARA 0 "" 0 " " {TEXT -1 53 "4.) Wendepunkte f ''(x) = 0" }}{PARA 0 "" 0 "" {TEXT 256 22 "Globale Eigenschaften:" }}{PARA 0 "" 0 "" {TEXT -1 20 "1.) Symmetrie " }{XPPEDIT 18 0 "f(-x)=f(x)" "/ -%\"fG6#,$%\"xG!\"\"-F$6#F'" }{TEXT -1 31 " achsensymmetrisch zu y-A chse" }}{PARA 0 "" 0 "" {TEXT -1 28 " " } {XPPEDIT 18 0 "f(-x)=-f(x)" "/-%\"fG6#,$%\"xG!\"\",$-F$6#F'F(" }{TEXT -1 30 " punktsymmetrisch zum Ursprung" }}{PARA 0 "" 0 "" {TEXT -1 17 " 2.) Verhalten f\374r" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest art:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Berechnung am Beispiel vo n " }{XPPEDIT 18 0 "x->x^3-3*x" ":6#%\"xG7\"6$%)operatorG%&arrowG6\",& *$F$\"\"$\"\"\"*&\"\"$F-F$F-!\"\"F)F)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:=x->x^3-3*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$9$\"\"$\"\" \"F.!\"$F(F(" }}}{EXCHG {PARA 257 "" 0 "Besondere Punkte der Stammfunk tion" {TEXT 267 17 "Besondere Punkte:" }}{PARA 0 "" 0 "" {TEXT -1 4 "1 .) " }{TEXT 265 1 "S" }{TEXT -1 6 "chnitt" }{TEXT 266 1 "p" }{TEXT -1 14 "unkte mit der " }{TEXT 264 1 "x" }{TEXT -1 1 "-" }{TEXT 263 1 "A" }{TEXT -1 9 "chse " }{XPPEDIT 18 0 "f(x)=0" "/-%\"fG6#%\"xG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "spxa:=solve(f(x)=0,x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%spxaG6%\"\"!*$\"\"$#\"\"\"\"\"#,$F '!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "spxa1:=[spxa[1],f (spxa[1])];\nspxa2:=[spxa[2],f(spxa[2])];\nspxa3:=[spxa[3],f(spxa[3])] ;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&spxa1G7$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&spxa2G7$*$\"\"$#\"\"\"\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&spxa3G7$,$*$\"\"$#\"\"\"\"\"#!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "2.) " }{TEXT 259 1 "S" }{TEXT -1 6 "chnitt" }{TEXT 260 1 "p" }{TEXT -1 14 "unkte mit der " }{TEXT 261 1 "y" }{TEXT -1 1 "-" }{TEXT 262 1 "A" }{TEXT -1 8 "chse " } {XPPEDIT 18 0 "f(0)" "-%\"fG6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "spya:=f(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%spy aG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "spya1:=[spya[1], f(spya[1])];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&spya1G7$\"\"!F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "3.) " }{TEXT 257 2 "Ex" }{TEXT -1 4 "trem" }{TEXT 258 2 "pu" }{TEXT -1 22 "nkte f '(x) = 0" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "expu:=solve(diff(f(x),x)=0, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%expuG6$\"\"\"!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "expu1:=[expu[1],f(expu[1])]; \nexpu2:=[expu[2],f(expu[2])];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &expu1G7$\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expu2G7$!\" \"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "4.) " }{TEXT 269 2 "We " }{TEXT -1 3 "nde" }{TEXT 268 2 "pu" }{TEXT -1 18 "nkte f ''(x) = 0 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "wepu:=solve(diff(diff(f (x),x),x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%wepuG\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "wepu1:=[wepu[1],f(wepu[1])]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&wepu1G7$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "Globale Eigenschaften der St ammfunkeion" {TEXT 284 22 "Globale Eigenschaften:" }{TEXT -1 1 "\n" } {TEXT 270 19 "1.) Symmetrie " }{XPPEDIT 18 0 "f(-x)=f(x)" "/-%\"f G6#,$%\"xG!\"\"-F$6#F'" }{TEXT -1 32 " achsensymmetrisch zu y-Achse " }}{PARA 0 "" 0 "" {TEXT -1 27 " " } {XPPEDIT 18 0 "f(-x)=-f(x)" "/-%\"fG6#,$%\"xG!\"\",$-F$6#F'F(" }{TEXT -1 31 " punktsymmetrisch zum Ursprung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalb(f(-x)=f(x));\nevalb(f(-x)=-f(x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tr ueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Die Funktion ist, wie man \+ sehen kann, punktsymmetrisch zum Ursprung." }}{PARA 0 "" 0 "" {TEXT -1 17 "2.) Verhalten ..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Genug gerechnet, jetzt wird geplottet!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Stammfunktion mit Ableitungen..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "fp:=plot(f(x),x=-3..3,-6..6,color=red):\nf sp:=plot(diff(f(x),x),x=-3..3,color=blue):\nfssp:=plot(diff(diff(f(x), x),x),x=-3..3,color=green):\nA:=display(fp,fsp,fssp):A;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Besondere Punkte..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "punkte:=plot([spxa1,spxa2,spxa3,spya1,expu1,e xpu2,wepu1],x,style=point,symbol=circle,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "B:=display(punkte,axes=framed):B;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Zusammen..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(A,B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Das waren die besonderen Punkte der Stammfunktion," }}{PARA 0 "" 0 "" {TEXT -1 43 " aber es gibt ja auch noch die 1. Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "fs:=diff(f(x),x):\nfs:=unapply(fs,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fsG:6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$9$ \"\"#\"\"$!\"$\"\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "???unapply" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "Besondere \+ Punkte der 1. Ableitung" {TEXT 275 34 "Besondere Punkte der 1. Ableitu ng:" }}{PARA 0 "" 0 "" {TEXT -1 4 "1.) " }{TEXT 273 1 "S" }{TEXT -1 6 "chnitt" }{TEXT 274 1 "p" }{TEXT -1 14 "unkte mit der " }{TEXT 272 1 " x" }{TEXT -1 1 "-" }{TEXT 271 1 "A" }{TEXT -1 9 "chse " }{XPPEDIT 18 0 "fs(x)=0" "/-%#fsG6#%\"xG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "spxafs:=solve(fs(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'spxafsG6$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "spxafs1:=[spxafs[1],fs(spxafs[1])];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "spxafs2:=[spxafs[2],fs(spxafs[2])];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(spxafs1G7$\"\"\"\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(spxafs2G7$!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "2.) " }{TEXT 276 1 "S" }{TEXT -1 6 "chnitt" }{TEXT 277 1 " p" }{TEXT -1 14 "unkte mit der " }{TEXT 278 1 "y" }{TEXT -1 1 "-" } {TEXT 279 1 "A" }{TEXT -1 8 "chse " }{XPPEDIT 18 0 "fs(0)" "-%#fsG6 #\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "spyafs:=f(0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'spyafsG\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "spyafs1:=[spyafs[1],fs(spyafs[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(spyafs1G7$\"\"!!\"$" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 4 "3.) " }{TEXT 280 2 "Ex" }{TEXT -1 4 "trem" }{TEXT 281 2 "pu" }{TEXT -1 42 "nkte fs '(x) = 0\n oder Scheitel" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "expufs:=solve(diff(fs(x), x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'expufsG\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "expufs1:=[expufs[1],fs(expuf s[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(expufs1G7$\"\"!!\"$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "4.) Wendepunkte gibt es hier nicht , da nur ein Extrempunkt vorhanden ist." }}{PARA 0 "" 0 "" {TEXT -1 99 "Au\337erdem ist die 2. Ableitung von fs(x) (bzw. die 3. Ableitung \+ von f(x)) eine Parallel zur x-Achse." }}}{EXCHG {PARA 0 "" 0 "Globale \+ Eigenschaften der 1. Ableitung" {TEXT 283 39 "Globale Eigenschaften de r 1. Ableitung:" }{TEXT 282 20 "\n1.) Symmetrie " }{XPPEDIT 18 0 "fs(-x)=fs(x)" "/-%#fsG6#,$%\"xG!\"\"-F$6#F'" }{TEXT -1 32 " achsen symmetrisch zu y-Achse" }}{PARA 0 "" 0 "" {TEXT -1 27 " \+ " }{XPPEDIT 18 0 "fs(-x)=-fs(x)" "/-%#fsG6#,$%\"xG!\"\",$- F$6#F'F(" }{TEXT -1 31 " punktsymmetrisch zum Ursprung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalb(fs(-x)=fs(x));\nevalb(fs(-x)= -fs(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Aha, die 1.Ableitung ist nicht punktsymmetrisch wie die Stammfunktion, son dern achsensymmetrisch." }}{PARA 0 "" 0 "" {TEXT -1 24 "Erweitern wir \+ den Plot. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "punktefs:=pl ot(\{spxafs1,spxafs2,spyafs1,expufs1\},x,style=point,symbol=box,color= violet,axes=framed):C:=display(punktefs):C;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(A,B,C);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Wenn wir schon mal dabei sind, dann k\366nnen wir mit der 2. Ab leitung auch gleich weitermachen." }}}{EXCHG {PARA 0 "" 0 "Besondere P unkte der 2. Ableitung" {TEXT 287 34 "Besondere Punkte der 2. Ableitun g:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "fss:=diff(fs(x),x);\n fss:=unapply(fss,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fssG,$%\"xG \"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fssG:6#%\"xG6\"6$%)operato rG%&arrowGF(,$9$\"\"'F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "1.) \+ Schnittpunkt mit der x-Achse " }{XPPEDIT 18 0 "fss(x)=0" "/-%$fssG6# %\"xG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "spxafss:=solv e(fss(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(spxafssG\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "spxafss1:=[spxafss[1],fss(s pxafss[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)spxafss1G7$\"\"!F& " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "2.) Schnittpunkt mit der y-Ac hse " }{XPPEDIT 18 0 "fss(0)" "-%$fssG6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "spyafss:=fss(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(spyafssG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "spyafss1:=[spyafss[1],fss(spyafss[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)spyafss1G7$\"\"!F&" }}}{EXCHG {PARA 0 "" 0 "Globale \+ Eigenschaften der 2. Ableitung" {TEXT 286 39 "Globale Eigenschaften de r 2. Ableitung:" }{TEXT 285 20 "\n1.) Symmetrie " }{XPPEDIT 18 0 "fss(-x)=fss(x)" "/-%$fssG6#,$%\"xG!\"\"-F$6#F'" }{TEXT -1 32 " ach sensymmetrisch zu y-Achse" }}{PARA 0 "" 0 "" {TEXT -1 27 " \+ " }{XPPEDIT 18 0 "fss(-x)=-fss(x)" "/-%$fssG6#,$%\"xG! \"\",$-F$6#F'F(" }{TEXT -1 31 " punktsymmetrisch zum Ursprung" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalb(fss(-x)=fss(x));\neval b(fss(-x)=-fss(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Die 2. Ableitung der Stammfunktion ist punktsymmetrisch z um Ursprung." }}{PARA 0 "" 0 "" {TEXT -1 25 "Und jetzt noch plotten... " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "punktefss:=plot(\{spxaf ss1,spyafss1\},x,style=point,symbol=diamond):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "E:=display(punktefss):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(A,B,C,E);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Na ja, E kann man kaum erkennen, aber was soll's." }}{PARA 0 "" 0 "" {TEXT -1 32 "I ch glaub ich bin fertig. Hurra!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Untersuchng einer Fun ktion" 1 "" "Untersuchng einer Funktion" }{TEXT -1 1 "\n" }{HYPERLNK 17 "Besondere Punkte der Stammfunktion" 1 "" "Besondere Punkte der Sta mmfunktion" }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Globale Eigenschaf ten der Stammfunkeion" 1 "" "Globale Eigenschaften der Stammfunkeion" }{TEXT -1 1 "\n" }{HYPERLNK 17 "Besondere Punkte der 1. Ableitung" 1 " " "Besondere Punkte der 1. Ableitung" }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Globale Eigenschaften der 1. Ableitung" 1 "" "Globale Ei genschaften der 1. Ableitung" }{TEXT -1 1 "\n" }{HYPERLNK 17 "Besonder e Punkte der 2. Ableitung" 1 "" "Besondere Punkte der 2. Ableitung" }} }{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Globale Eigenschaften der 2. Abl eitung" 1 "" "Globale Eigenschaften der 2. Ableitung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }