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Z:^:f?IJJ?nEcLifd:^;QR:IJ:^YrI>:C:Uk<^:>XIJ@>:C;V;wldnEiL@f; IJ:^YbJiAC:Uk=^:>XEJ@>:C;f;wlenEmD:SjA@j=>:sAL>;J:sAHNv?:C:Uk=^:>XEJ@>:C;V:sA:Uk<^:>XIJ@>:C;f:sAtmqKjFne;VFwLlW^:f?IJJGn ECMinFSJHvZ:>:sAX=^:f?EJJHnEGMi>g:^;wR:IJ:^YRGB:C:Uk=^:>XEJ@ >:C;n=wT:KMiFGSjIv:;Jx[mJ:^:f?EJ:sAXEJ@>:C;F>wlonEUM@V^;v:;Jx[h:C:Uk<^:>XIJ@>:C;V>wlpnEYE:SjL@j=>:sAl <IJ:^YbD>:C:Uk<^:>XIJ@>:C;n>wLrnE_E:SJN @j=>:sAXL^?Z:^:f?IJJNnE_Mi^h:^;OS:IJ:^YRCJXEJ@>:C;^?wLunEiE:SjP@:;Jx[[^Q^:f?EJjPnEiMiFISjQ@j=>:sAtKV?:C:Uk=^:>XEJ@>:C;v?wlvnEqE:SjR@j=>:sAl;;J:sAdkR=J:^:f?IJJSnEsMini:^;cS:IJ :^YB@JJwL ;_;kk=>:sAH;;JXEJ@>:C;FA wl;oEEF:SjW@j=>:sAx:;J:sAp:;B:C:Uk=^ :>XEJ@>:C;fAwl=oEKN@nAIJ:^YrJYnEKNiNKSJZ@j=>:sA\\: ^:f?IJJZnEONi^k:^;?T:IJ:^Yb;SN:^:f?EJJ[nESNink:^;C T:IJ:^YB;M><:Uk=^:>XEJ@>:C;^BwLAoEYF:Sj\\v:;Jx[j\\n EYNiFl:^;IT:IJ:^YB:?N:^:f?IJj]nE]NiVl:^;MT:IJ:^Yb:G>C:Uk<^:> XIJ@>:C;FCwlCoEcN@NCIJ:^YB;O><:Uk=^:>XEJ@>:C;NCwLDoEgN@^CIJ:^Yr;W>C:Uk <^:>XIJ@>:C;^CwLEoEiF:Sj`@j=>:sA`:^:f?IJj`nEiNiFMSjav:;Jx[En L;Jd;v:;Jx[GNM;JJcnEsNifMSjcv:;Jx[L:C:Uk=^:>XEJ@>:C;VDwlH oEyF:Sjd@j=>:sALKMK:sATKOKNwLK_;kT:IJ:^YR@>:C:Uk<^:>XIJ@>:C;>EwLKoEAG:Sjf@j=>:sAh;;J:sApKU?:C:Uk<^:>XIJ@>:C;VEwlLoEGG:SJh@j=>:sAxK W?:C:Uk=^:>XEJ@>:C;^EwLMoEKG:SJiv:;Jx[\\:C:Uk<^:>XIJ@>:C;nEwLNoEMO@ve; v:;Jx[]Z:^:f?IJjinEMOiNo:^;;U:IJ:^YRCC@C:Uk<^:>XIJ@>:C;>FwLO oESO@NFIJ:^YBDJXIJ@>:C;V FwlPoEYG:Sjl@j=>:sAl<X IJ@>Z:^>JmnE[OiNp:^;KU:IJ:^YU^:f?IJ:sa:r F>:C:Uk<^:>XIJ@JL:MMiVPwlT_;Qm=>:sAP=:sAd=^:f?IJJqnEkOiFq :^;YU:IJ:^YBISQ:^:f?EJjqnEmOiVq:^;]U:IJ:^YbI[a:^:f?IJjrnEqOi^q:^;_U:IJ:^YRJcAC:Uk<^:>XIJ@>:C;NHwLXoEwG:SJt@j=>:sAH>^:f?I JJtnEwOivQSjtv:;Jx[?_Y;J:sAX>; J<:GkAn:yayIZ:>;NYB::C:UkJM:ry=JH> :qi:[B:IB:>:sg:>Z:Z " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "1.) Beispiel mit der Funktion f(x)=" }{XPPEDIT 18 0 "1/4*x^4-1/ 2*x^2;" "6#,&*(\"\"\"\"\"\"\"\"%!\"\"%\"xG\"\"%F&*(\"\"\"F&\"\"#F(F)\" \"#F(" }{TEXT -1 23 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Pa rabelfunktion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=x->a*x ^4+b*x^3+c*x^2+d*x+e;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"x G6\"6$%)operatorG%&arrowGF(,,*&%\"aG\"\"\")9$\"\"%\"\"\"F/*&%\"bGF/)F1 \"\"$F3F/*&%\"cGF/)F1\"\"#F3F/*&%\"dGF/F1F/F/%\"eGF/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Einsetzen der Werte:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "a:=1/4; b:=0; c:=-1/2; d:=0; e:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG#! \"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"eG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Kontollausgabe:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"%\"\"\"#\"\"\"F'*$)F &\"\"#F(#!\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Zeichnen der \+ Parabelfunktion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p1:=plo t(f(x),x=-2..2,y=-2..2): p1;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Die Nullstellen der Parabel:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "null:=solve(f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nullG6&\"\"!F&*$-%%sqrtG6#\"\"#\"\"\",$F' !\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!F#$\"+iN@99!\"*$!+iN@99F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Die Nullstellen als Punkte:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n1:=[null[1],0];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#n1G7$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n2:=[null[2],0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#n2G7$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n3:=[null [3],0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n3G7$*$-%%sqrtG6#\"\"#\" \"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n4:=[null[4],0 ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n4G7$,$*$-%%sqrtG6#\"\"#\"\" \"!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Zeichnung der Nul lstellen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "p8:=plot([n1,n 2,n3,n4],style=point,color=blue):p8;" }}{PARA 13 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Bilden der Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "af:=D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#afGR6# %\"xG6\"6$%)operatorG%&arrowGF(,**&%\"aG\"\"\")9$\"\"$\"\"\"\"\"%*&%\" bGF/)F1\"\"#F3F2*&%\"cGF/F1F/F8%\"dGF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "af(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)% \"xG\"\"$\"\"\"\"\"\"F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Z eichnen der Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "p 2:=plot(af(x),x=-2..2,y=-2..2,color=blue): p2;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Nullstellen der A bleitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nulla:=solve(a f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&nullaG6%\"\"!\"\"\"!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%\"\"!$\"\"\"F#$!\"\"F#" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 81 "Extrema der Parabel durch Einsetzen der Nullstellen der Ableitung in die Parabel:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(nulla[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(nulla[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(nulla[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Die Extrema als Punkte:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e1:=[nulla[1],f(nulla[1])]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G7$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e2:=[nulla[2],f(nulla[2])];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#e2G7$\"\"\"#!\"\"\"\"%" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "e3:=[nulla[3],f(nulla[3])];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#e3G7$!\"\"#F&\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Zeichnen der Extrema:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "p5:=plot([e1,e2,e3],style=point,color=navy,symbol=cir cle):p5;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Bilden der zweiten Ableitung:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "aaf:=D(af);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a afGR6#%\"xG6\"6$%)operatorG%&arrowGF(,(*&%\"aG\"\"\")9$\"\"#\"\"\"\"#7 *&%\"bGF/F1F/\"\"'%\"cGF2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "aaf(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\" \"\"\"$!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Zeichnen de r zweiten Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "p3: =plot(aaf(x),x=-2..2,y=-2..2,color=navy): p3;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Nullstellen der z weiten Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "nullaa :=solve(aaf(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'nullaaG6$,$*$-% %sqrtG6#\"\"$\"\"\"#\"\"\"F+,$F'#!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+$p-Nx &!#5$!+$p-Nx&F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Extrema der Ab leitung durch Einsetzen der Nullstellen der zweiten Ableitung in die A bleitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "af(nullaa[1]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#\"\"$\"\"\"#!\"#\"\" *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!+&z,!\\Q!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "af(nullaa[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* $-%%sqrtG6#\"\"$\"\"\"#\"\"#\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&z,! \\Q!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Die Extrema als Punkte: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ea1:=[nullaa[1],af(null aa[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ea1G7$,$*$-%%sqrtG6#\" \"$\"\"\"#\"\"\"F+,$F'#!\"#\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ea2:=[nullaa[2],af(nullaa[2])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ea2G7$,$*$-%%sqrtG6#\"\"$\"\"\"#!\"\"F+,$F'#\"\"#\" \"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Zeichnen der Extrema:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "p6:=plot([ea1,ea2],style=poi nt,symbol=circle,color=red):p6;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Bilden der dritten Ableitung:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "aaaf:=D(aaf);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%aaafGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&% \"aG\"\"\"9$F/\"#C%\"bG\"\"'F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "aaaf(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"xG\" \"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Zeichnen der dritten Ablei tung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "p4:=plot(aaaf(x),x =-2..2,y=-2..2,color=black): p4;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Nullstelle der dritten Ableitung: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "nullaaa:=solve(aaaf(x)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(nullaaaG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Einsetzen der Nullstelle der dritten Abl eitung in die zweite Ableitung, um den Scheitel zu bestimmen:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "aaf(nullaaa);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Sc heitelpunkt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eaa1:=[null aaa[1],aaf(nullaaa[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eaa1G7$ \"\"!!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Zeichnen des Scheit elpunktes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "p7:=plot([eaa 1],style=point,symbol=circle,color=magenta):p7;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Parabel, Ableitun g und Extrema in einem Schaubild:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "with(plots): display(p1,p2,p3,p4,p5,p6,p7,p8);" }} {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 " Ableitungen und Parabel in einem Schaubild:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(\{f(x),af(x),aaf(x),aaaf(x)\},x=-2..2,y=-1.5. .2);" }}{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 35 "2.) Beispiel mit der Funktion f(x )=" }{XPPEDIT 18 0 "1/4*x^4+4/3*x^3+2*x^2;" "6#,(*(\"\"\"\"\"\"\"\"%! \"\"%\"xG\"\"%F&*(\"\"%F&\"\"$F(F)\"\"$F&*&\"\"#F&*$F)\"\"#F&F&" } {TEXT -1 21 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Parab elfunktion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=x->a*x^4+ b*x^3+c*x^2+d*x+e;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6 \"6$%)operatorG%&arrowGF(,,*&%\"aG\"\"\")9$\"\"%\"\"\"F/*&%\"bGF/)F1\" \"$F3F/*&%\"cGF/)F1\"\"#F3F/*&%\"dGF/F1F/F/%\"eGF/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Einsetzen der Werte:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "a:=1/4; b:=4/3; c:=2; d:=0; e:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG#\"\"%\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"cG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"eG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Kontollausgabe:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"%\"\"\"#\"\"\"F'*$)F &\"\"$F(#F'F-*$)F&\"\"#F(F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Ze ichnen der Parabelfunktion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "p1:=plot(f(x),x=-3..1,y=-1.5..2): p1;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Die Nullstellen der Par abel:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "null:=fsolve(f(x)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nullG6$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Die Nul lstellen als Punkte:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n1: =[null[1],0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n1G7$\"\"!F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n2:=[null[2],0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n2G7$\"\"!F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Zeichnung der Nullstellen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p8:=plot([n1,n2],style=point,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Bilden der Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "af:=D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#afGR6#%\"xG6\"6$% )operatorG%&arrowGF(,**&%\"aG\"\"\")9$\"\"$\"\"\"\"\"%*&%\"bGF/)F1\"\" #F3F2*&%\"cGF/F1F/F8%\"dGF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "af(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\" \"$\"\"\"\"\"\"*$)F&\"\"#F(\"\"%F&F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Zeichnen der Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "p2:=plot(af(x),x=-3..1,y=-1.5..2,color=blue): p2;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Nullstell en der Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nulla: =solve(af(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&nullaG6%\"\"!!\"# F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!$!\"#F#F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Extrema der Parabel durch Einsetzen der Nullstellen der A bleitung in die Parabel:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(nulla[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(nulla[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(nulla[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"\"$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Die Extrema als Punkte:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e1:=[nulla[1],f(nulla[1])]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G7$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e2:=[nulla[2],f(nulla[2])];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#e2G7$!\"##\"\"%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e3:=[nulla[3],f(nulla[3])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G7$!\"##\"\"%\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Zeichnen der Extrema:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "p5:=plot([e1,e2,e3],style=point,color=navy,symbol=cir cle):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Bilden der zweiten Ablei tung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "aaf:=D(af);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$aafGR6#%\"xG6\"6$%)operatorG%&arrow GF(,(*&%\"aG\"\"\")9$\"\"#\"\"\"\"#7*&%\"bGF/F1F/\"\"'%\"cGF2F(F(F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "aaf(x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"\"\"$F&\"\")\"\"%\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Zeichnen der zweiten Ableitung:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "p3:=plot(aaf(x),x=-3..1,y= -1.5..2,color=navy): p3;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Nullstellen der zweiten Ableitung:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "nullaa:=solve(aaf(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'nullaaG6$!\"##F&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!\"#\"\"!$!+nmmmm!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Extrema der Ableitung durch Einsetzen der Nullstellen der zweit en Ableitung in die Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "af(nullaa[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a f(nullaa[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!#K\"#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+&=&=&=\"!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Die Extrema als Punkte:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ea1:=[nullaa[1],af(nullaa[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $ea1G7$!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ea2:=[n ullaa[2],af(nullaa[2])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ea2G7$# !\"#\"\"$#!#K\"#F" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Zeichnen der Extrema:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "p6:=plot([ea1, ea2],style=point,symbol=circle,color=red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Bilden der dritten Ableitung:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "aaaf:=D(aaf);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %aaafGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&%\"aG\"\"\"9$F/\"#C%\"bG\" \"'F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "aaaf(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"'\"\")\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Zeichnen der dritten Ableitung:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "p4:=plot(aaaf(x),x=-3..1,y=- 1.5..2,color=black): p4;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Nullstelle der dritten Ableitung:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "nullaaa:=solve(aaaf(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(nullaaaG#!\"%\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Einsetzen der Nullstelle der dritten Abl eitung in die zweite Ableitung, um den Scheitel zu bestimmen:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "aaf(nullaaa);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6##!\"%\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Scheitelpunkt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eaa1: =[nullaaa[1],aaf(nullaaa[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e aa1G7$&#!\"%\"\"$6#\"\"\",(*$)F&\"\"#\"\"\"F)F&\"\")\"\"%F+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Zeichnen des Scheitelpunktes:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "p7:=plot([eaa1],style=point, symbol=circle,color=magenta):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 " Parabel, Ableitung und Extrema in einem Schaubild:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "with(plots): display(p1,p2,p3,p4,p5,p6,p7); " }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Ableitungen und Parabel in einem Schaubild:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "plot(\{f(x),af(x),aaf(x),aaaf(x)\},x=-3..1,y=- 1.5..2);" }}{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 35 "3.) Beispiel mit der Funktion f(x )=" }{XPPEDIT 18 0 "1/4*x^4;" "6#*(\"\"\"\"\"\"\"\"%!\"\"%\"xG\"\"%" } {TEXT -1 35 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Parabelfunktion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:= x->a*x^4+b*x^3+c*x^2+d*x+e;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR 6#%\"xG6\"6$%)operatorG%&arrowGF(,,*&%\"aG\"\"\")9$\"\"%\"\"\"F/*&%\"b GF/)F1\"\"$F3F/*&%\"cGF/)F1\"\"#F3F/*&%\"dGF/F1F/F/%\"eGF/F(F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Einsetzen der Werte:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a:=1/4; b:=0; c:=0; d:=0; e:=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"\"\"\"\"%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"bG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 " Kontollausgabe:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"xG\"\"%\"\"\"#\"\"\"F'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Zeichnen der Parabelfunktion:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "p1:=plot(f(x),x=-1.5..1.5,y= -2..3): p1;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 28 "Die Nullstellen der Parabel:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "null:=solve(f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%nullG6&\"\"!F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"!F#F#F#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Die Nullstellen als Punkte:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n1:=[null[1],0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n1G7$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "n2:=[null[2],0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#n2G7$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n3:=[nu ll[3],0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n3G7$\"\"!F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n4:=[null[4],0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n4G7$\"\"!F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Zeichnung der Nullstellen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "p8:=plot([n1,n2,n3,n4],style=point,color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Bilden der Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "af:=D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#afGR6# %\"xG6\"6$%)operatorG%&arrowGF(,**&%\"aG\"\"\")9$\"\"$\"\"\"\"\"%*&%\" bGF/)F1\"\"#F3F2*&%\"cGF/F1F/F8%\"dGF/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "af(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"x G\"\"$\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Zeichnen der Able itung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "p2:=plot(af(x),x= -1.5..1.5,y=-2..3,color=blue): p2;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Nullstellen der Ableitung:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nulla:=solve(af(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&nullaG6%\"\"!F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!F#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Extrema der Parab el durch Einsetzen der Nullstellen der Ableitung in die Parabel:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(nulla[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(nulla[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(nulla[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Die Extre ma als Punkte:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e1:=[null a[1],f(nulla[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G7$\"\"!F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e2:=[nulla[2],f(nulla[2 ])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G7$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e3:=[nulla[3],f(nulla[3])];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G7$\"\"!F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Zeichnen der Extrema:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "p5:=plot([e1,e2,e3],style=point,color=navy,symbol=cir cle):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Bilden der zweiten Ablei tung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "aaf:=D(af);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$aafGR6#%\"xG6\"6$%)operatorG%&arrow GF(,(*&%\"aG\"\"\")9$\"\"#\"\"\"\"#7*&%\"bGF/F1F/\"\"'%\"cGF2F(F(F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "aaf(x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*$)%\"xG\"\"#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 31 "Zeichnen der zweiten Ableitung:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "p3:=plot(aaf(x),x=-1.5..1.5,y=-2..3,color=navy ): p3;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Nullstellen der zweiten Ableitung:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "nullaa:=solve(aaf(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'nullaaG6$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Extrema der Ableitung durch Ein setzen der Nullstellen der zweiten Ableitung in die Ableitung:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "af(nullaa[1]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "af(nullaa[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eval f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Die Extrema als Punkte:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "ea1:=[nullaa[1],af(nullaa[1])];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$ea1G7$\"\"!F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ea2:=[nullaa[2],af(nullaa[2])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ea2G7$\"\"!F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Zeichnen der Extrema:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "p6:=plot([ea1,ea2],style=point,symbol=circle,color=red):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Bilden der dritten Ableitung:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "aaaf:=D(aaf);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%aaafGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&% \"aG\"\"\"9$F/\"#C%\"bG\"\"'F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "aaaf(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"xG\" \"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Zeichnen der dritten Ablei tung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "p4:=plot(aaaf(x),x =-1.5..1.5,y=-2..3,color=black): p4;" }}{PARA 13 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Nullstelle der dritten Ableit ung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "nullaaa:=solve(aaaf (x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(nullaaaG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Einsetzen der Nullstelle der dritten Abl eitung in die zweite Ableitung, um den Scheitel zu bestimmen:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "aaf(nullaaa[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Scheitelpunkt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eaa1: =[nullaaa[1],aaf(nullaaa[1])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e aa1G7$\"\"!F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Zeichnen des Sch eitelpunktes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "p7:=plot([ eaa1],style=point,symbol=circle,color=magenta):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Parabel, Ableitung und Extrema in einem Schaubild:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "with(plots): display(p1,p 2,p3,p4,p5,p6,p7);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Ableitungen und Parabel in einem Schaubild:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot(\{f(x),af(x),aaf(x),aa af(x)\},x=-1.5..1.5,y=-2..3);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 0 0" 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