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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Kurvendiskussion mit Schar
parameter" }}{PARA 0 "" 0 "" {TEXT -1 31 "Gegeben ist die Funktion f m
it " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:=x->x^4-x^2;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$
\"\"%\"\"\"F1*$)F/\"\"#F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "1. Die ersten drei Ableitungen:
" }}{PARA 0 "" 0 "" {TEXT -1 24 "    Die erste Ableitung:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 28 "f1:=unapply(diff(f(x),x),x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#f1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"\"%
\"\"\")9$\"\"$F/F/*&\"\"#F/F1F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 25 "    Die zweite Ableitung:" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "f2:=unapply(diff(f1(x),x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#f2Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"#7\"\"\")9$\"\"#F/F/
F2!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "    Die dritte A
bleitung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f3:=unapply(diff(f2(x)
,x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f3Gf*6#%\"xG6\"6$%)opera
torG%&arrowGF(,$*&\"#C\"\"\"9$F/F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "2. Nullstellen:  Bed. f(x
) = 0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "nst:=\{solve(f(x)=0,x)\};
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nstG<%!\"\"\"\"!\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "3.
 Extrempunkte:" }}{PARA 0 "" 0 "" {TEXT -1 40 "    a) Notwendige Bedin
gung: f ' (x) = 0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "extr:=\{solve(
f1(x)=0,x)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%extrG<%\"\"!,$*&\"
\"#!\"\"F)#\"\"\"F)F*,$*&F)F*F)F+F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 61 "    b) Hinreichende Bedingung: f ' (x) = 0  und f '' (x) <> 0" 
}}{PARA 0 "" 0 "" {TEXT -1 18 "         f '' (xE)" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 69 "for i from 1 to nops(extr) do f2(extr[i]):=subs(x=e
xtr[i],f2(x)); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#f2G6#\"\"!!
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#f2G6#,$*&\"\"#!\"\"F)#\"\"
\"F)F*\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#f2G6#,$*&\"\"#!\"\"
F)#\"\"\"F)F,\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 57 "for i from 1 to nops(extr) do f(extr[i]):=f
(extr[i]); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#\"\"!F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#,$*&\"\"#!\"\"F)#\"\"\"F)F*#F
*\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#,$*&\"\"#!\"\"F)#\"
\"\"F)F,#F*\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 15 "4. Wendepunkte:" }}{PARA 0 "" 0 "" {TEXT -1 41 "  \+
  a) Notwendige Bedingung: f ''(x) = 0 " }}{PARA 0 "" 0 "" {TEXT -1 4 
"    " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "wende:=\{solve(f2(x)=0,x)
\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&wendeG<$,$*&\"\"'!\"\"F(#\"
\"\"\"\"#F+,$*&F(F)F(F*F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 61 "    b) Hinreichende Bedingung: f ''(x) = \+
0  und f '''(x) <> 0" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "for i from \+
1 to nops(wende) do f3(wende[i]):=simplify(factor(subs(x=wende[i],f3(x
))));od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#f3G6#,$*&\"\"'!\"\"F)#
\"\"\"\"\"#F,,$*&\"\"%F,F)F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#
f3G6#,$*&\"\"'!\"\"F)#\"\"\"\"\"#F*,$*&\"\"%F,F)F+F*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 60 "for i from 1 to nops(wende) do f(wende[i]
):=f(wende[i]); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#,$*&\"
\"'!\"\"F)#\"\"\"\"\"#F,#!\"&\"#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
-%\"fG6#,$*&\"\"'!\"\"F)#\"\"\"\"\"#F*#!\"&\"#O" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 13 "5. Grenzwerte" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "
Limit(subs(t=t,f(x)),x=infinity)=limit(subs(t=1,f(x)),x=infinity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$,&*$)%\"xG\"\"%\"\"\"F,*$)
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1 0 68 "Limit(subs(t=t,f(x)),x=-infinity)=limit(subs(t=1,f(x)),x=-infi
nity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$,&*$)%\"xG\"\"%
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"" {TEXT -1 8 "6. Graph" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot(\{s
ubs(t=1,f(x)),subs(t=-1,f(x)),subs(t=1/2,f(x))\},x=-4..4,-8..8);" }}
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