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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 35 "Successioni definite per \+
ricorrenza" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Preliminari" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Una successione e' definita per ri
correnza se abbiamo una funzione " }{TEXT 256 4 "f(x)" }{TEXT -1 10 " \+
tale che " }{XPPEDIT 18 0 "x[n+1] = f(x[n]);" "6#/&%\"xG6#,&%\"nG\"\"
\"\"\"\"F)-%\"fG6#&F%6#F(" }{TEXT -1 83 ". Naturalmente abbiamo bisogn
o di un punto di partenza, per cui dobbiamo assegnare " }{XPPEDIT 18 
0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 4 " (o " }{XPPEDIT 18 0 "x[0];
" "6#&%\"xG6#\"\"!" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 55 "Ne
i modelli di popolazione le funzioni piu' usate sono " }{TEXT 257 15 "
f(x) = ax(1-x) " }{TEXT -1 2 "e " }{TEXT 258 11 "f(x) = a x " }
{XPPEDIT 18 0 "exp(-b*x);" "6#-%$expG6#,$*&%\"bG\"\"\"%\"xGF)!\"\"" }
{TEXT -1 30 " e si suppone che i valori di " }{XPPEDIT 18 0 "x[n];" "6
#&%\"xG6#%\"nG" }{TEXT -1 27 " siano sempre nonnegativi. " }}{PARA 0 "
" 0 "" {TEXT -1 24 "Nel caso della funzione " }{TEXT 259 15 "f(x) = ax
(1-x) " }{TEXT -1 9 "se fosse " }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"n
G" }{TEXT -1 0 "" }{TEXT 260 4 " > 1" }{TEXT -1 12 " otterremmo " }
{XPPEDIT 18 0 "x[n+1] = f(x[n]);" "6#/&%\"xG6#,&%\"nG\"\"\"\"\"\"F)-%
\"fG6#&F%6#F(" }{TEXT -1 1 " " }{TEXT 261 5 "< 0; " }{TEXT -1 33 "i va
lori accettabili sono quindi " }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG
" }{TEXT -1 1 " " }{TEXT 262 2 "e " }{TEXT -1 121 "[0,1] (notare che 1
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 dire che c'e' 1 individuo). Se " }{TEXT 263 6 "a <= 4" }{TEXT -1 4 ",
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264 2 "e " }{TEXT -1 14 "[0,1] implica " }{XPPEDIT 18 0 "x[n+1] = f(x[
n]);" "6#/&%\"xG6#,&%\"nG\"\"\"\"\"\"F)-%\"fG6#&%\"xG6#%\"nG" }{TEXT 
-1 1 " " }{TEXT 265 2 "e " }{TEXT -1 51 "[0,1] ; altrimenti, cio' non \+
e' vero. La funzione  " }{TEXT 266 14 "f(x) = ax(1-x)" }{TEXT -1 39 " \+
sara' quindi considerata soltanto con " }{TEXT 267 2 "a " }{TEXT 268 
2 "e " }{TEXT -1 7 "[0,4] ." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "
Costruzione della successione" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "
Definizioni" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Prima di tutto deci
diamo il numero di termini della successione che vogliamo costruire" }
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"" {XPPMATH 20 "6#>%\"nG\"$+\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 
"Creiamo il vettore per mettere la successione" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "x := array (1..n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"xG-%&arrayG6$;\"\"\"\"$+\"7\"" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 21 "Definiamo la funzione" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "f := t -> a*t*(1-t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fGR6#%\"tG6\"6$%)operatorG%&arrowGF(*(%\"aG\"\"\"9$F.,&F.F.F
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ci" }}{PARA 0 "" 0 "" {TEXT -1 120 "Potremmo costruire la successione \+
in modo simbolico, ma le espressioni sarebbero molto complicate e in s
ostanza inutili." }}{PARA 0 "" 0 "" {TEXT -1 46 "Conviene invece dare \+
dei valori alla costante " }{TEXT 269 1 "a" }{TEXT -1 20 " e al dato i
niziale " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 1 "." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "a := 3.1;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"aG$\"#J!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "x[1] := 0.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"
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XwF,7$$\"#lF)$\"+#eT,e&F,7$$\"#hF)$\"++A9!e&F,7$$\"#fF)$\"+eG9!e&F,7$$
\"#gF)$\"+AYmXwF,7$$\"#aF)$\"+#Rick(F,7$$\"#bF)$\"+ee9!e&F,7$$\"#cF)$
\"+VNmXwF,7$$\"#dF)$\"+pR9!e&F,7$$\"#[F)$\"+C:lXwF,7$$\"#\\F)$\"+$oj,e
&F,7$$\"\"&F)$\"+cS[VhF,7$$\"\"'F)$\"+0xlWtF,7$$\"\"(F)$\"+y)*zXgF,7$$
\"#_F)$\"+T/mXwF,7$$\"#`F)$\"+e!\\,e&F,7$$\"#eF)$\"+@UmXwF,7$$\"#]F)$
\"+KrlXwF,7$$\"#^F)$\"+'[a,e&F,7$$\"#PF)$\"+NDn!e&F,7$$\"#YF)$\"+A?kXw
F,7$$\"#ZF)$\"+r#z,e&F,7$$\"#WF)$\"+<fiXwF,7$$\"#XF)$\"+(o0-e&F,7$$\"#
UF)$\"+E')fXwF,7$$\"#VF)$\"+_/D!e&F,7$$\"#SF)$\"+vBbXwF,7$$\"#TF)$\"+:
jK!e&F,7$$\"#QF)$\"+<SZXwF,7$$\"#RF)$\"+R[X!e&F,7$$\"#NF)$\"+!4T5e&F,7
$$\"#OF)$\"+)GT`k(F,7$$\"#MF)$\"+ll6XwF,7$$\"#IF)$\"+GR4WwF,7$$\"#JF)$
\"+kyr#e&F,7$$\"#KF)$\"+njtWwF,7$$\"#LF)$\"+hXm\"e&F,7$$\"#GF)$\"+Y1,V
wF,7$$\"#HF)$\"+kL\\%e&F,7$$\"#EF)$\"+I/>TwF,7$$\"#FF)$\"+k]Z(e&F,7$$
\"#CF)$\"+'z\\\"QwF,7$$\"#DF)$\"+`8X#f&F,7$$\"#=F)$\"+\">&)=h(F,7$$\"#
>F)$\"+#G(>NcF,7$$\"#9F)$\"+i6jivF,7$$\"#:F)$\"+tc?9dF,7$$\"#;F)$\"+\"
=s=f(F,7$$\"#<F)$\"+l:[ncF,7$$\"#7F)$\"+ao^AvF,7$$\"#8F)$\"+/<WxdF,7$$
\"\")F)$\"+5Q&4T(F,7$$\"\"*F)$\"+NN1[fF,7$$\"#5F)$\"+;WOruF,7$$\"#6F)$
\"+c4jceF,7$$\"\"$F)$\"(HfB'!\"(7$$\"\"%F)$\"+Z'okF(F,7$$\"\"#F)$\"$z#
!\"$7$$\"\"\"F)$Fghl!\"\"" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 205 "Questi sono i val
ori della successione che (da 20 in poi) sembrano avere due possibili \+
valori. Per vedere meglio, riusiamo il comando di prima coll'opzione \+
\"connect=true\" (e una scala precisa per l'asse y)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 52 "pointplot(points,connect=true,view = [1..
100,0..1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 290 290 290 {PLOTDATA 2 "6$-
%'CURVESG6#7`q7$$\"#?\"\"!$\"+mD#\\i(!#57$$\"#@F*$\"+rA.9cF-7$$\"#AF*$
\"+b*=Jj(F-7$$\"#BF*$\"+;An+cF-7$$\"#(*F*$\"+\\79!e&F-7$$\"$+\"F*$\"+,
_mXwF-7$$\"#)*F*$\"++_mXwF-7$$\"#**F*$\"+^79!e&F-7$$\"#\"*F*FO7$$\"##*
F*FE7$$\"#&*F*FO7$$\"#'*F*FE7$$\"#$*F*F@7$$\"#%*F*FJ7$$\"#%)F*FE7$$\"#
&)F*F@7$$\"#')F*FJ7$$\"##)F*FJ7$$\"#$)F*FO7$$\"#*)F*F@7$$\"#!*F*FJ7$$
\"#uF*$\"+%=lck(F-7$$\"#vF*$\"+x79!e&F-7$$\"#!)F*$\"+)>lck(F-7$$\"#\")
F*$\"+a79!e&F-7$$\"#yF*$\"+$>lck(F-7$$\"#zF*$\"+j79!e&F-7$$\"#()F*FO7$
$\"#))F*FE7$$\"#wF*$\"+\">lck(F-7$$\"#xF*$\"+m79!e&F-7$$\"#rF*$\"+?89!
e&F-7$$\"#sF*$\"+u^mXwF-7$$\"#tF*$\"+%HT,e&F-7$$\"#pF*$\"+q89!e&F-7$$
\"#qF*$\"+e^mXwF-7$$\"#mF*$\"+z]mXwF-7$$\"#nF*$\"+\\99!e&F-7$$\"#oF*$
\"+G^mXwF-7$$\"#iF*$\"+f[mXwF-7$$\"#jF*$\"+5=9!e&F-7$$\"#kF*$\"+)*\\mX
wF-7$$\"#lF*$\"+#eT,e&F-7$$\"#hF*$\"++A9!e&F-7$$\"#fF*$\"+eG9!e&F-7$$
\"#gF*$\"+AYmXwF-7$$\"#aF*$\"+#Rick(F-7$$\"#bF*$\"+ee9!e&F-7$$\"#cF*$
\"+VNmXwF-7$$\"#dF*$\"+pR9!e&F-7$$\"#[F*$\"+C:lXwF-7$$\"#\\F*$\"+$oj,e
&F-7$$\"\"&F*$\"+cS[VhF-7$$\"\"'F*$\"+0xlWtF-7$$\"\"(F*$\"+y)*zXgF-7$$
\"#_F*$\"+T/mXwF-7$$\"#`F*$\"+e!\\,e&F-7$$\"#eF*$\"+@UmXwF-7$$\"#]F*$
\"+KrlXwF-7$$\"#^F*$\"+'[a,e&F-7$$\"#PF*$\"+NDn!e&F-7$$\"#YF*$\"+A?kXw
F-7$$\"#ZF*$\"+r#z,e&F-7$$\"#WF*$\"+<fiXwF-7$$\"#XF*$\"+(o0-e&F-7$$\"#
UF*$\"+E')fXwF-7$$\"#VF*$\"+_/D!e&F-7$$\"#SF*$\"+vBbXwF-7$$\"#TF*$\"+:
jK!e&F-7$$\"#QF*$\"+<SZXwF-7$$\"#RF*$\"+R[X!e&F-7$$\"#NF*$\"+!4T5e&F-7
$$\"#OF*$\"+)GT`k(F-7$$\"#MF*$\"+ll6XwF-7$$\"#IF*$\"+GR4WwF-7$$\"#JF*$
\"+kyr#e&F-7$$\"#KF*$\"+njtWwF-7$$\"#LF*$\"+hXm\"e&F-7$$\"#GF*$\"+Y1,V
wF-7$$\"#HF*$\"+kL\\%e&F-7$$\"#EF*$\"+I/>TwF-7$$\"#FF*$\"+k]Z(e&F-7$$
\"#CF*$\"+'z\\\"QwF-7$$\"#DF*$\"+`8X#f&F-7$$\"#=F*$\"+\">&)=h(F-7$$\"#
>F*$\"+#G(>NcF-7$$\"#9F*$\"+i6jivF-7$$\"#:F*$\"+tc?9dF-7$$\"#;F*$\"+\"
=s=f(F-7$$\"#<F*$\"+l:[ncF-7$$\"#7F*$\"+ao^AvF-7$$\"#8F*$\"+/<WxdF-7$$
\"\")F*$\"+5Q&4T(F-7$$\"\"*F*$\"+NN1[fF-7$$\"#5F*$\"+;WOruF-7$$\"#6F*$
\"+c4jceF-7$$\"\"$F*$\"(HfB'!\"(7$$\"\"%F*$\"+Z'okF(F-7$$\"\"#F*$\"$z#
!\"$7$$\"\"\"F*$Fhhl!\"\"-%%VIEWG6$;FghlFC;F*Fghl" 1 2 0 1 0 2 9 1 4 
2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
134 "E' venuto uno schifo. Il problema e' che points e' un insieme (NO
N ordinato) e quindi possono essere disposti in un modo arbitrario...
" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Primitive di grafica" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Per riuscire a fare dei grafici d
ecenti (e PREVEDIBILI) mi sembra necessario usare le primitive grafich
e di Maple." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "PLOT (tutto in MA
IUSCOLO) e' la primitiva dentro cui mettiamo tutti is egmenti (CURVES)
 e i punti (POINTS) da tracciare. Ad esempio:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 72 "PLOT(CURVES([[0,0],[0.1,1],[0.5,0.5]]),POINTS([0
,0],[0.1,1],[0.5,0.5]));" }}{PARA 13 "" 1 "" {GLPLOT2D 349 262 262 
{PLOTDATA 2 "6$-%'CURVESG6#7%7$\"\"!F(7$$\"\"\"!\"\"F+7$$\"\"&F,F.-%'P
OINTSG6%F'F)F-" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Ora costruiamo \"pointsb\" che no
n e' piu' un insieme, ma un array (almeno credo)" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 30 "pointsb:=seq([i,x[i]],i=1..n):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 72 "e con la seguente sintassi, facciamo il p
lot dei punti e dei segmenti..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 72 "plx_n.1 := PLOT(POINTS(pointsb),CURVES([pointsb]),VIEW(1 .. n+
1,0 ..1)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Abbiamo costruito i
l plot, ma non lo abbiamo mostrato. Per vederlo:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "plx_n.1;" }}{PARA 13 "" 1 "" {GLPLOT2D 349 
262 262 {PLOTDATA 2 "6%-%'POINTSG6`q7$\"\"\"$F'!\"\"7$\"\"#$\"$z#!\"$7
$\"\"$$\"(HfB'!\"(7$\"\"%$\"+Z'okF(!#57$\"\"&$\"+cS[VhF87$\"\"'$\"+0xl
WtF87$\"\"($\"+y)*zXgF87$\"\")$\"+5Q&4T(F87$\"\"*$\"+NN1[fF87$\"#5$\"+
;WOruF87$\"#6$\"+c4jceF87$\"#7$\"+ao^AvF87$\"#8$\"+/<WxdF87$\"#9$\"+i6
jivF87$\"#:$\"+tc?9dF87$\"#;$\"+\"=s=f(F87$\"#<$\"+l:[ncF87$\"#=$\"+\"
>&)=h(F87$\"#>$\"+#G(>NcF87$\"#?$\"+mD#\\i(F87$\"#@$\"+rA.9cF87$\"#A$
\"+b*=Jj(F87$\"#B$\"+;An+cF87$\"#C$\"+'z\\\"QwF87$\"#D$\"+`8X#f&F87$\"
#E$\"+I/>TwF87$\"#F$\"+k]Z(e&F87$\"#G$\"+Y1,VwF87$\"#H$\"+kL\\%e&F87$
\"#I$\"+GR4WwF87$\"#J$\"+kyr#e&F87$\"#K$\"+njtWwF87$\"#L$\"+hXm\"e&F87
$\"#M$\"+ll6XwF87$\"#N$\"+!4T5e&F87$\"#O$\"+)GT`k(F87$\"#P$\"+NDn!e&F8
7$\"#Q$\"+<SZXwF87$\"#R$\"+R[X!e&F87$\"#S$\"+vBbXwF87$\"#T$\"+:jK!e&F8
7$\"#U$\"+E')fXwF87$\"#V$\"+_/D!e&F87$\"#W$\"+<fiXwF87$\"#X$\"+(o0-e&F
87$\"#Y$\"+A?kXwF87$\"#Z$\"+r#z,e&F87$\"#[$\"+C:lXwF87$\"#\\$\"+$oj,e&
F87$\"#]$\"+KrlXwF87$\"#^$\"+'[a,e&F87$\"#_$\"+T/mXwF87$\"#`$\"+e!\\,e
&F87$\"#a$\"+#Rick(F87$\"#b$\"+ee9!e&F87$\"#c$\"+VNmXwF87$\"#d$\"+pR9!
e&F87$\"#e$\"+@UmXwF87$\"#f$\"+eG9!e&F87$\"#g$\"+AYmXwF87$\"#h$\"++A9!
e&F87$\"#i$\"+f[mXwF87$\"#j$\"+5=9!e&F87$\"#k$\"+)*\\mXwF87$\"#l$\"+#e
T,e&F87$\"#m$\"+z]mXwF87$\"#n$\"+\\99!e&F87$\"#o$\"+G^mXwF87$\"#p$\"+q
89!e&F87$\"#q$\"+e^mXwF87$\"#r$\"+?89!e&F87$\"#s$\"+u^mXwF87$\"#t$\"+%
HT,e&F87$\"#u$\"+%=lck(F87$\"#v$\"+x79!e&F87$\"#w$\"+\">lck(F87$\"#x$
\"+m79!e&F87$\"#y$\"+$>lck(F87$\"#z$\"+j79!e&F87$\"#!)$\"+)>lck(F87$\"
#\")$\"+a79!e&F87$\"##)$\"++_mXwF87$\"#$)$\"+^79!e&F87$\"#%)$\"+,_mXwF
87$\"#&)$\"+\\79!e&F87$\"#')Fi_l7$\"#()F]`l7$\"#))Fa`l7$\"#*)Fe`l7$\"#
!*Fi_l7$\"#\"*F]`l7$\"##*Fa`l7$\"#$*Fe`l7$\"#%*Fi_l7$\"#&*F]`l7$\"#'*F
a`l7$\"#(*Fe`l7$\"#)*Fi_l7$\"#**F]`l7$\"$+\"Fa`l-%'CURVESG6#7`qF&F*F/F
4F9F=FAFEFIFMFQFUFYFgnF[oF_oFcoFgoF[pF_pFcpFgpF[qF_qFcqFgqF[rF_rFcrFgr
F[sF_sFcsFgsF[tF_tFctFgtF[uF_uFcuFguF[vF_vFcvFgvF[wF_wFcwFgwF[xF_xFcxF
gxF[yF_yFcyFgyF[zF_zFczFgzF[[lF_[lFc[lFg[lF[\\lF_\\lFc\\lFg\\lF[]lF_]l
Fc]lFg]lF[^lF_^lFc^lFg^lF[_lF__lFc_lFg_lF[`lF_`lFc`lFg`lFi`lF[alF]alF_
alFaalFcalFealFgalFialF[blF]blF_blFablFcbl-%%VIEWG6$;F'\"$,\";\"\"!F'
" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 51 "Effettivamente la successione oscilla fra 2 val
ori." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 11 "Esercizio
: " }{TEXT -1 37 "rifate lo stesso con altri valori di " }{TEXT 271 1 
"a" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Qua invece rifacciamo lo st
esso con un valore diverso di " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"
\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "x[1] \+
:= 0.02;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"$\"\"#!\"#
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "for i to n-1 do x[i+1] \+
:= f(x[i]) od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "pointsb:=
seq([i,x[i]],i=1..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "pl
x_n.2 := PLOT(POINTS(pointsb),CURVES([pointsb]),VIEW(1 .. n+1,0 ..1),C
OLOR(RGB,1,0,0)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Abbiamo cost
ruito il plot della nuova successione in ROSSO (questo dice l'opzione \+
\"COLOR(RGB...)\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "plx_n.2;" }}{PARA 13 "" 1 "" 
{GLPLOT2D 349 262 262 {PLOTDATA 2 "6&-%'POINTSG6`q7$\"\"\"$\"\"#!\"#7$
F)$\"%wg!\"&7$\"\"$$\"+%*[6p<!#57$\"\"%$\"+W#GS^%F37$\"\"&$\"+_xywwF37
$\"\"'$\"+s-zGbF37$\"\"($\"+i!=Lm(F37$\"\")$\"+R%)3^bF37$\"\"*$\"+vM&e
l(F37$\"#5$\"+=rRjbF37$\"#6$\"+X4g^wF37$\"#7$\"+`QRqbF37$\"#8$\"+i<9\\
wF37$\"#9$\"+c[VubF37$\"#:$\"+BwqZwF37$\"#;$\"+Y(*ywbF37$\"#<$\"+6s'ok
(F37$\"#=$\"+H\"p\"ybF37$\"#>$\"+ULPYwF37$\"#?$\"+A&z*ybF37$\"#@$\"+UE
3YwF37$\"#A$\"+ikXzbF37$\"#B$\"+k8\"fk(F37$\"#C$\"+ZutzbF37$\"#D$\"+&R
5ek(F37$\"#E$\"+zI!*zbF37$\"#F$\"+[3vXwF37$\"#G$\"+g2+!e&F37$\"#H$\"+G
drXwF37$\"#I$\"+q$e+e&F37$\"#J$\"+5]pXwF37$\"#K$\"+`B4!e&F37$\"#L$\"+&
y#oXwF37$\"#M$\"+1C6!e&F37$\"#N$\"+wbnXwF37$\"#O$\"+KU7!e&F37$\"#P$\"+
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\"+damXwF37$\"#]$\"+J39!e&F37$\"#^$\"+``mXwF37$\"#_$\"+**49!e&F37$\"#`
$\"+#Hlck(F37$\"#a$\"+-69!e&F37$\"#b$\"+b_mXwF37$\"#c$\"+h69!e&F37$\"#
d$\"+L_mXwF37$\"#e$\"+(>T,e&F37$\"#f$\"+>_mXwF37$\"#g$\"+?79!e&F37$\"#
h$\"+5_mXwF37$\"#i$\"+N79!e&F37$\"#j$\"+1_mXwF37$\"#k$\"+U79!e&F37$\"#
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lFh^lFj^lF\\_lF^_lF`_lFb_lFd_lFf_lFh_lFj_lF\\`l-%%VIEWG6$;F'\"$,\";\"
\"!F'-%&COLORG6&%$RGBGF'Fh`lFh`l" 1 2 0 1 0 2 6 1 4 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "e queste
 sono le due successioni insieme..." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "display([plx_n.1,plx_n.2]);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 349 262 262 {PLOTDATA 2 "6'-%'POINTSG6`q7$\"\"\"$F'!\"\"7$\"
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7$\"\"'$\"+0xlWtF87$\"\"($\"+y)*zXgF87$\"\")$\"+5Q&4T(F87$\"\"*$\"+NN1
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F87$\"#9$\"+i6jivF87$\"#:$\"+tc?9dF87$\"#;$\"+\"=s=f(F87$\"#<$\"+l:[nc
F87$\"#=$\"+\">&)=h(F87$\"#>$\"+#G(>NcF87$\"#?$\"+mD#\\i(F87$\"#@$\"+r
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#X$\"+(o0-e&F87$\"#Y$\"+A?kXwF87$\"#Z$\"+r#z,e&F87$\"#[$\"+C:lXwF87$\"
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45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Non si capisce ni
ente perche' quest'altra successione oscilla fra gli stessi valori ma \+
sfasata di una unita'..." }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Ana
lisi" }}{PARA 0 "" 0 "" {TEXT -1 26 "I due valori (chiamiamoli " }
{XPPEDIT 18 0 "x[a];" "6#&%\"xG6#%\"aG" }{TEXT -1 3 " e " }{XPPEDIT 
18 0 "x[b];" "6#&%\"xG6#%\"bG" }{TEXT -1 25 ") devono essere tali che \+
" }{XPPEDIT 18 0 "f(x[a]) = x[b];" "6#/-%\"fG6#&%\"xG6#%\"aG&F(6#%\"bG
" }{TEXT -1 3 " e " }{XPPEDIT 18 0 "f(x[b]) = x[a];" "6#/-%\"fG6#&%\"x
G6#%\"bG&F(6#%\"aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "Per
 altri valori di " }{TEXT 272 1 "a" }{TEXT -1 39 " si trovera' invece \+
che la successione " }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT 
-1 41 " tende a un valore costante (chiamiamolo " }{TEXT 273 2 "x*" }
{TEXT -1 40 "). E' abbastanza facile convincersi che " }{TEXT 274 3 "x
* " }{TEXT -1 15 "deve soddisfare" }{TEXT 276 13 " f(x*) = x*. " }
{TEXT -1 51 "Un punto che soddisfa tale identita' viene detto un" }
{TEXT 277 13 " punto fisso " }{TEXT -1 2 "di" }{TEXT 278 3 " f." }}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Punti fissi di f" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 56 "Per cominciare guardiamo graficamente i punti f
issi di f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot([f(t),t],
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"Si vede chiaramente che ci sono 2 punti fissi. E' facilissimo trovarl
i analiticamente; sono 0 e " }{TEXT 279 3 "x*=" }{XPPEDIT 18 0 "1-1/a;
" "6#,&\"\"\"\"\"\"*&\"\"\"F%%\"aG!\"\"F)" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 65 "Quest'ultimo e' positivo (il caso che ci interessa
) se e solo se " }{TEXT 280 6 "a > 1." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 36 "Controlliamo che sia un punto fisso:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 9 "f(1-1/a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
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{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[N>un!#5" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 96 "errore di arrotondamento... (rifacendo il conto in modo
 simbolico, sarebbero esattamente uguali)" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 18 "Cicli di periodo 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
4 "Se  " }{XPPEDIT 18 0 "x[a];" "6#&%\"xG6#%\"aG" }{TEXT -1 3 " e " }
{XPPEDIT 18 0 "x[b];" "6#&%\"xG6#%\"bG" }{TEXT -1 15 " sono tali che \+
" }{XPPEDIT 18 0 "f(x[a]) = x[b];" "6#/-%\"fG6#&%\"xG6#%\"aG&F(6#%\"bG
" }{TEXT -1 3 " e " }{XPPEDIT 18 0 "f(x[b]) = x[a];" "6#/-%\"fG6#&%\"x
G6#%\"bG&F(6#%\"aG" }{TEXT -1 18 ", necessariamente " }{TEXT 281 36 "f
(f(x[a]))=x[a] e f(f(x[b]))=x[b] , " }{TEXT -1 6 "ossia " }{XPPEDIT 
18 0 "x[a];" "6#&%\"xG6#%\"aG" }{TEXT -1 3 " e " }{XPPEDIT 18 0 "x[b];
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{TEXT -1 48 "Rifacciamo quindi lo stesso plot con la composta" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " f2 := t -> f(f(t));  " }}
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{TEXT -1 36 "Il grafico non si vede bene perche' " }{TEXT 284 2 "f2" }
{TEXT -1 73 " e la bisettrice sono molto vicine per un tratto; riguard
iamone una parte" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot([f
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45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Adesso s
i vedono abbastanza chiaramente 3 punti fissi di " }{TEXT 285 2 "f2" }
{TEXT -1 39 " piu' lo 0. Fra questi c'e' certamente " }{TEXT 286 2 "x*
" }{TEXT -1 34 " : infatti tutti i punti fissi di " }{TEXT 287 1 "f" }
{TEXT -1 27 " sono anche punti fissi di " }{TEXT 288 4 "f2; " }{TEXT 
-1 0 "" }{TEXT 289 8 "perche'?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 
"Per trovarli, possiamo usare il comando \"solve\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "fixed:=solve(f2(x)-x,x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&fixedG6&\"\"!$\"+[N>un!#5$\"+_79!e&F)$\"++_mXwF)" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "per vedere quali corrispondono \+
effettivamente a cicli (non costanti) di periodo 2, calcoliamo " }
{TEXT 290 1 "f" }{TEXT -1 17 " in questi punti:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "f(fixed[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#$\"+\\N>un!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "questo e' " }
{TEXT 291 2 "x*" }{TEXT -1 20 ", il punto fisso di " }{TEXT 292 1 "f" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(fixed[3]);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#$\"+*>lck(!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "f(fixed[3])-fixed[4];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$!\"\"!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "errore di arro
tondamento..." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 293 10 "
Importante" }{TEXT -1 14 ": nel caso di " }{TEXT 294 1 "f" }{TEXT -1 
241 " non polinomiale (come il secondo esempio), il comando \"solve\" \+
non fornirebbe in genere soluzioni esplicite. Bisogna invece usare \"f
solve\" (risoluzione numerica) e si trovano le radici una alla volta, \+
assegnando l'intervallo in cui cercarle." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 295 10 "Esercizio:" }{TEXT -1 47 " se ripetete qu
esta procedura con un valore di " }{TEXT 296 1 "a" }{TEXT -1 44 " piu'
 piccolo, non troverete punti fissi di " }{TEXT 297 2 "f2" }{TEXT -1 
36 " che non siano anche punti fissi di " }{TEXT 298 1 "f" }}}}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 27 "Diagramma a \"tela di ragno\"" }}{PARA 
0 "" 0 "" {TEXT -1 149 "La successione si puo' costruire graficamente \+
a mano tracciando (a partire dal punto di partenza) segmenti verticali
 fino a incontrare il grafico di " }{TEXT 299 1 "f" }{TEXT -1 53 " e s
egmenti orizzontali fino a toccare la bisettrice." }}{PARA 0 "" 0 "" 
{TEXT -1 33 "Cio' corrisponde a rappresentare " }{XPPEDIT 18 0 "x[n];
" "6#&%\"xG6#%\"nG" }{TEXT -1 27 " sull'asse delle ascisse e " }
{XPPEDIT 18 0 "x[n+1];" "6#&%\"xG6#,&%\"nG\"\"\"\"\"\"F(" }{TEXT -1 
60 " sull'asse delle ordinate e fare cio' per tutti i valori di " }
{TEXT 300 1 "n" }{TEXT -1 2 ". " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
26 "Costruzione degli elementi" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 68 "pp := seq ([[x[i],x[i]],[x[i],x[i+1]],[x[i+1],x[i+1]]], i = 1..n
-1):" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Grafico " }}{EXCHG 
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