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256 "" 0 "" {TEXT -1 31 "Alexander Rabe, IKG, 09. 02.2000" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 263 27 "Untersuchung von Vorzeichen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 23 "1) f(x)=1/4*x^4-1/2*x^2" }}{PARA 0 "" 0 "" {TEXT -1 29 "2) g(x)=1/4*x^4-4/3*x^3+2*x^2" }}{PARA 0 "" 0 "" {TEXT -1 15 "3) h(x)=1/4*x^4" }}{PARA 0 "" 0 "" {TEXT -1 20 "4) y=1/4* x^4+1/2*x^2" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "1) f(x)=1/4*x^4-1 /2*x^2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 292 "" 0 "" {TEXT -1 25 " Definition der Funktion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:=x->1/4*x^4-1/2*x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"%\"\"\"#\"\" \"F0*$)F/\"\"#F1#!\"\"F6F(F(F(" }}}{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 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 23 "Schaubild der Funktion :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "fplot:=plot(f(x),x=-5..5,y=-5..5):fplot:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 259 "" 0 " " {TEXT -1 25 "Nullstellen der Funktion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "nullf:=solve(f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&nullfG6&\"\"!F&*$-%%sqrtG6#\"\"#\"\"\",$F'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "nullf=0 ; nullf=-1.41 und nullf=1.41" }}{PARA 0 "" 0 "" {TEXT -1 55 "Nullstellen der Funktion: (0/0), (-1.41/0) und \+ (1.41/0)" }}}{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 260 "" 0 "" {TEXT -1 13 "1. Ableitun g:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "fs:=D(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fsGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$ \"\"$\"\"\"\"\"\"F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "fs(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"$\"\"\" \"\"\"F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 261 "" 0 "" {TEXT -1 27 "Schaubild der 1. Ableitung:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 54 "fsplot:=plot(fs(x),x=-5..5,y=-5..5,color=blue):fspl ot:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 29 "Nullstellen der 1. Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "xextr:=solve(fs(x));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&xextrG6%\"\"!\"\"\"!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 29 "xextr=0, xextr=-1 und xextr=1" }}{PARA 0 "" 0 "" {TEXT -1 53 "Nullstellen der 1. Ableitung: (0/0), (-1/0) und (1/0)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 263 "" 0 " " {TEXT -1 8 "Extrema:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f (xextr[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(xextr[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(xextr[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Extremstellen: (0/0), (1 / -1/4) und (-1 / -1/4)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 264 "" 0 " " {TEXT -1 13 "2. Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "fss:=D(fs);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fssGR6#%\"xG6 \"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"\"\"$!\"\"\"\"\"F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "fss(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\"\"\"\"$!\"\"\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 27 "Schaubild der 2. Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "fssplot:=plot(fss( x),x=-5..5,y=-5..5,color=green):fssplot:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 29 "Nullstellen der 2. Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "nullf ss:=evalf(solve(fss(x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(nullfs sG6$$\"+$p-Nx&!#5$!+$p-Nx&F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "n ullfss=0.58 und nullfss=-0.58" }}{PARA 0 "" 0 "" {TEXT -1 52 "Nullstel len der 2. Ableitung: (0.58/0) und (-0.58/0)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 12 "Wendep unkte:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f(nullfss[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+!*)))))Q\"!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f(nullfss[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+!*)))))Q \"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Wendepunkte: (0.58/-0.14 ) und (-0.58/0.14)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "WP1:=plot([[0.58,-0.14]],sty le=point,symbol=circle,color=magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "WP2:=plot([[-0.58,-0.14]],style=point,symbol=circle,c olor=magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 268 "" 0 " " {TEXT -1 10 "Schaubild:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "display(fplot,fsplot,fssplot ,WP1,WP2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "2) g(x)=1/4*x^4-4/3*x^3+2*x^2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 256 0 "" }{TEXT 257 0 "" }{TEXT 258 24 "Definition der Funktion:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g:=x->1/4*x^4-4/3*x^3+2*x^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&ar rowGF(,(*$)9$\"\"%\"\"\"#\"\"\"F0*$)F/\"\"$F1#!\"%F6*$)F/\"\"#F1F;F(F( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"%\"\"\"#\"\"\"F'*$)F&\"\"$F(#!\"%F-* $)F&\"\"#F(F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 269 "" 0 "" {TEXT -1 23 "Schaubild der Funktion:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "gplot:=plot(g(x),x=-5..5,y=-5..5):gplot:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 270 "" 0 " " {TEXT -1 25 "Nullstellen der Funktion:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nullg:=fsolve(g(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&nullgG6$\"\"!F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "nullg=0 \+ " }}{PARA 0 "" 0 "" {TEXT -1 31 "Nullstellen der Funktion: (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 "" }}} {EXCHG {PARA 271 "" 0 "" {TEXT -1 13 "1. Ableitung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "gs:=D(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gsGR6#%\"xG6\"6$%)operatorG%&arrowGF(,(*$)9$\"\"$\"\"\"\"\"\"*$) F/\"\"#F1!\"%F/\"\"%F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "gs(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"$\"\"\"\"\" \"*$)F&\"\"#F(!\"%F&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 272 "" 0 "" {TEXT -1 27 "Schaubild der 1. Ableitung :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 54 "gsplot:=plot(gs(x),x=-5..5,y=-5..5,color=blue) :gsplot:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 273 "" 0 "" {TEXT -1 29 "Nullstellen der 1. Ableitung:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "xextr:=solve(gs(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xextrG6%\"\"!\"\"#F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "xextr=0 und xextr=2" }}{PARA 0 "" 0 "" {TEXT -1 45 "Nullstellen der 1. 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