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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 16 "Tangenten-Lineal" }}
{PARA 259 "" 0 "" {TEXT -1 77 "Nach Vorgabe der Funktionsgleichung mit
 passenden Zeichenparametern berechnet" }}{PARA 259 "" 0 "" {TEXT -1 
93 "das Arbeitsblatt einen Ausschnitt der lokalen Kurventangente mit d
er L\344nge 1 und l\344sst dieses" }}{PARA 259 "" 0 "" {TEXT -1 95 "Li
neal im Zeichenintervall den Funktionsgraphen entlang wandern. Die Lin
eal-L\344nge ist variabel." }}{PARA 259 "" 0 "" {TEXT -1 19 "Das Vorze
ichen der " }{TEXT 270 17 "Tangentensteigung" }{TEXT -1 6 " wird " }
{TEXT 269 23 "farblich gekennzeichnet" }{TEXT -1 7 ":durch:" }}{PARA 
259 "" 0 "" {TEXT 271 53 " f\264(x)>0 in gr\374n, f\264(x)<0 in rot un
d f\264(x)=0 in blau." }}{PARA 0 "" 0 "" {TEXT 272 99 "Bei dieser Vers
ion werden vorab die Extremstellen berechnet und in die Partition eins
ortiert, damit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 273 96 "das Tan
genten-Lineal auf jeden Fall die Extremstelle trifft und dort horizont
al (blau) verl\344uft." }}{PARA 259 "" 0 "" {TEXT -1 78 "Bisher getest
et an Polynomfunktionen und einigen trigonometrischen Funktionen." }}
{PARA 259 "" 0 "" {TEXT -1 87 "Aktivieren Sie nur das jeweilige Beispi
el, indem Sie die Raute # setzen bzw. entfernen." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 6 "Autor:" }{TEXT -1 10 " Re
ggentin" }}{PARA 0 "" 0 "" {TEXT 266 20 "letzte Modifikation:" }{TEXT 
-1 11 " 08.03.2006" }}{PARA 0 "" 0 "" {TEXT 267 8 "Version:" }{TEXT 
-1 6 " 1.4.4" }}{PARA 0 "" 0 "" {TEXT 268 4 "CAS:" }{TEXT -1 8 " Maple
 8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):
" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has be
en redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 6 "Bsp. 1" }{TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "# f:=x->1/80*x^3-
x^2/10-x/2:          xmin:=-8:xmax:=14:      ymin:=-10:   ymax:=10:" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 6 "Bsp. 2" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 84 "# f:=x->1/10*x^4-x^2/2+3:             xmin:=-4:x
max:=4:       ymin:=0:     ymax:=10:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
262 6 "Bsp. 3" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 83 "# f:=x->1/1000*x^6+1/80*x^4-x^2/5+2:  xmin:=-5:xmax:=5:       ym
in:=0:     ymax:=6:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 6 "Bsp. 4" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "# f:=x->1/1000*x^6-1/40*x^4
+x^2/10+1: xmin:=-5:xmax:=5:       ymin:=0:     ymax:=5:" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT -1 6 "Bsp. 5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 85 "# f:=x->sin(x):                       xmin:=-2*Pi:xma
x:=2*Pi: ymin:=-1.1:  ymax:=1.1:" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 
-1 6 "Bsp. 6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "# f:=x->x*s
in(x):                     xmin:=-3*Pi:xmax:=3*Pi: ymin:=-8:    ymax:=
8:" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 6 "Bsp. 7" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 87 "# f:=x->(x+7)*(x+4)*(2*x-15)*(x-3)*(5-x)/10
00: xmin:=-10:xmax:=10: ymin:=-10: ymax:=10:" }}}{EXCHG {PARA 257 "" 
0 "" {TEXT -1 6 "Bsp. 8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "
# f:=x->sum((-2/5)^(k)*x^(k)/(10^(1/2*k)),k=0..4):  xmin:=-15: xmax:=1
5: ymin:=-10:  ymax:=10:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 290 6 "Bsp. \+
9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "# f:=x->(x^4-6*x^2)/5:
                      xmin:=-3: xmax:=3: ymin:=-2:  ymax:=4:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 291 7 "Bsp. 10" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 89 "f:=x->-x^4/4+x^3/3+x^2/2+x/4:                     \+
 xmin:=-3: xmax:=3: ymin:=-2:  ymax:=4:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&#\"\"\"\"
\"%F&*$)%\"xGF'F&F&!\"\"*&#F&\"\"$F&*$)F*F.F&F&F&*&#F&\"\"#F&*$)F*F3F&
F&F&*&#F&F'F&F*F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 37 "Ben\366ti
gte Prozeduren (Unterprogramme)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
52 "Filtert aus einer Liste L die komplexen Daten heraus" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ReFilter:=proc(L::list)" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 147 "  local R::list, nr::integer,i::integer:\n
  R:=\{\}:                              # leere Hilfsmenge, wird sp
\344ter zur rein reellen Liste              " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "  nr:=nops(L):                        # Anzahl der El
emente der \374bergebenen Liste L" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
79 "  for i from 1 to nr do if (Im(L[i])=0) then R:=R union \{L[i]\}: \+
end if: end do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "  R:=[op(R)]:   \+
                      # In der obigen Schleife wird mengenweise verein
igt," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "end proc:                  \+
           # dann wird R wieder zur Liste umgewandelt" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 9 "Sortiert " }{TEXT 276 6 "exakte" }{TEXT 
-1 121 " Listenwerte aufsteigend anstatt mit gerundeten Werten zu arbe
iten, wird im Befehl sort() als Sortier-Relation verwendet." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "sortiere:=proc(z1,z2)       \+
          # z1,z2 beinhalten exakte Werte" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 88 "if evalf(z1)<=evalf(z2) then true else false end if: \+
# gerundete Werte werden verglichen" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "end proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Es wird ein Farbs
chalter aufgebaut, der bei " }{TEXT 278 9 "positiver" }{TEXT -1 1 " " 
}{TEXT 279 8 "Steigung" }{TEXT -1 12 " das Lineal " }{TEXT 280 4 "gr
\374n" }{TEXT -1 19 " einf\344rbt bzw. bei " }{TEXT 281 9 "negativer" 
}{TEXT -1 1 " " }{TEXT 282 8 "Steigung" }{TEXT -1 12 " das Lineal " }
{TEXT 283 3 "rot" }{TEXT -1 29 " darstellt. Ist die Steigung " }{TEXT 
284 5 "exakt" }{TEXT -1 1 " " }{TEXT 285 1 "0" }{TEXT -1 24 ", so wird
 die Farbe auf " }{TEXT 286 4 "blau" }{TEXT -1 114 " eingestellt. F
\374r die Farbe Blau m\374ssen die Partitionsstellen so gew\344hlt sei
n, dass die Stelle mit der Steigung 0 " }{TEXT 274 5 "genau" }{TEXT 
-1 121 " getroffen wird. Die Partiton muss also die exakte Extremstell
e (oder: Sattelpunktstelle) bereits enthalten (siehe oben)." }}{PARA 
0 "" 0 "" {TEXT 287 68 "Kann keine exakte Nullstelle berechnet werden,
 so wird die Tangente " }{TEXT 288 11 "nicht blau " }{TEXT 289 24 "gef
\344rbt (s. Beispiel 6)!" }}{PARA 0 "" 0 "" {TEXT -1 45 "Es werden exa
kte Vergleiche mit der Funktion " }{TEXT 277 4 "is()" }{TEXT -1 11 " v
erwendet." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Farbe:=proc (w
1,w2)                   # exakter Vergleich" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 93 " if is(w1<w2) then color=green else if is(w1>w2) then
 color=red else color=blue end if end if" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 16 "Zei
chenparameter" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ZB:=x=xmin
..xmax,y=ymin..ymax:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "PSt
il:=style=point,symbol=circle,symbolsize=24:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "LStil:=style=line,thickness=3:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 67 "Anzahl:=50:                           # A
nzahl der Intervallstellen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "L\344nge:=(xmax-xmin)/Anzahl:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
257 26 "Intervallstellenberechnung" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "Part:=\{seq(xmin+i*(xmax-xmin)/Anzahl,i=1..Anzahl)\}:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Lsg:=[solve(D(f)(x)=0,x
)]:            # Stellen mit Horizontaltangenten berechnen und .." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Lsg:=\{op(ReFilter(Lsg))\}: \+
            # .. komplexe L\366sungen herausfiltern und .." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "Partition:=[op(Part union Lsg)]:   \+
   # ..in die Partitonsliste einf\374gen, dann sortieren." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "Partition:=sort(Partition,sortiere)
:  # Fertige sortierte Liste mit exakten Extremstellen" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 258 15 "Lokale Tangente" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 57 "t:=unapply(D(f)(xmitte)*(x-xmitte)+f(xmitte),[x,
xmitte]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 29 "Linealpunkte links \+
und rechts" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "LTL:=1:      \+
                                    # L\344nge des Tangenten-Lineals" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "right:=(x0,f)->x0+LTL/sqr
t(1+(D(f)(x0))^2):      # hier werden die entscheidenden Daten erzeugt
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "left:=(x0,f)->x0-LTL/sq
rt(1+(D(f)(x0))^2):       # dto." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "Links:=x0->[left(x0,f),t(left(x0,f),x0)]:        # li
nker Linealpunkt" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Rechts:
=x0->[right(x0,f),t(right(x0,f),x0)]:     # rechter Linealpunkt" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Mitte:=x0->[x0,f(x0)]:      \+
                     # Punkt in der Linealmitte" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 260 19 "Zeichnung erstellen" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "Graf:=plot(f(x),ZB,LStil,color=brown):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "Punkte:=seq(plot([Links(Partition[
i]),Mitte(Partition[i]),Rechts(Partition[i])],ZB,PStil,color=[navy,red
,navy]),i=1..Anzahl):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "L
ineal:=seq(plot([Links(Partition[i]),Rechts(Partition[i])],ZB,thicknes
s=5,Farbe(Links(Partition[i])[2],Rechts(Partition[i])[2])),i=1..Anzahl
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Film1:=display(Punkte
,insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Film2
:=display(Lineal,insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "Hintergrund:=display(Graf):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "display(Hintergrund,Film1,Film2,scaling=constrai
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