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UNIVERSIDAD AUTNOMA DE GUERRERO
UNIDAD ACADMICA DE MATEMTICASACAPULCO
TESIS PARA OBTENER EL GRADO DEMAESTRO EN CIENCIAS: AREAMATEMTICA EDUCATIVA
LA HEURSTICA EN LA SOLUCIN DEECUACIONES DIFERENCIALES
POR: LIC. SERGIO LEE MARTNEZDIRECTOR: DR. JOS MARA SIGARRETA
ALMIRA
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Introduccin 2
Captulo I: Bases Tericas de la Investigacin 8
CAPITULO II. le!entos tericos so"re la #eurstica $ la solucin de
Pro"le!as %&2.1 Antecedentes histricos de la utilizacin de la Heurstica en la enseanza 16
2.2 El concepto de problema 27
2.3 Clasicaciones del concepto de problema 31
2.! Estrate"ia para la solucin de problemas !2
Captulo III. strategia !etodolgica para la solucin de pro"le!as decuaciones 'i(erenciales )*
CO+CLU,IO+, - CO/+'ACIO+, %%0
BIBLIO1A3A %%8
Ane4o I. 5aloracin del stado del pro"le!a %26
Ane4o II. 5alidacin de 4pertos %2&
Ane4o III Pro"le!as %28
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Introduccin
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)*t%)5t#,o!. R*27n' (o+ "* ,&*" %8%+/* !%+ &n* (*+t% (+#$#"%#**
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+%!o"&,#7n % (+o8"%)*! 3 %n (*+t#,&"*+ %" &!o % "o! +%,&+!o!
%&+/!t#,o! %n %" (+o,%!o % %n!%1*n2*>*(+%n#2*9% % "* M*t%)5t#,*?M""%+?@' Po"3*?@@' Ro+/&%2?@@' B%+)&o?'
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,o)("%)%nt*+#* o ,o)o $/* *(+%n#2*9% %
"*! M*t%)5t#,*!.
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E" #*n7!t#,o +%*"#2*o * %!t*nt%!' -&% *n +%,#8#o &n ,&+!o
% E,&*,#on%! D#4%+%n,#*"%!' 3 K (+o4%!o+%!' -&% *n #)(*+t#o "*
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% G&%++%+o' *++o97 -&% %" % "o! %!t*nt%! %$#%n,#7 ?$%+
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% (+o8"%)*!.
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,"5!#,*! ?$*+#*8"%! !%(*+*8"%! 3 #4%+%n,#*" tot*".
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%n!%1*n2*>*(+%n#2*9% % +%!o"&,#7n % "o! (+o8"%)*!' t*"%! ,o)o:
No !% (+o(#,#* %" t+*8*9o #n%(%n#%nt% 3 ,on!#%nt% % "o!
%!t*nt%!.
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%!t*,* n# !% *,% ,on!,#%nt% "o %!%n,#*" %n "* 4o+)* % t+*8*9o
3 t0,n#,*! %&+/!t#,*! &t#"#2**!.
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&t#"#2**! %n "* !o"&,#7n %" (+o8"%)* t*"%! ,o)o: $*+#*,#7n %
,on#,#on%!' 8
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En 4&n,#7n % "o! %"%)%nto! t%7+#,o! %6(&%!to! 3 %" #*n7!t#,o %"
%!t*o %" (+o8"%)* +%*"#2*o' no! (+o(&!#)o! +%!o"$%+ %" !#%nt%
(+o8"%)* ,#%nt/=,o:
#Cmo utilizar las potencialidades $ue brinda la heurstica para
%a&orecer proceso de solucin de problemas relacionados con las
Ecuaciones 'i%erenciales en el ni&el (uperior)
E" o89%to % "* #n$%!t#*,#7n %!: El proceso de enseanza*aprendiza+e
de la resolucin las Ecuaciones 'i%erenciales en el ni&el (uperior.
A" !%+ "* +%!o"&,#7n % (+o8"%)*! &n* % "*! *,t#$#*%! %n "* -&% "o!
(+o,%!o! %" (%n!*)#%nto !% )*n#=%!t*n % )*n%+* )5! %$#%nt%'
%n n%,%!*+#o $#n,&"*+"o #+%,t*)%nt% ,on %" (+o,%!o %n!%1*n2*>
*(+%n#2*9% 3 o8t%n%+ too! "o! 8%n%=,#o! (o!#8"%! -&% !&8!*n%n "*!
%=,#%n,#*! % "o! %!t*nt%!. E" ,*)(o % *,,#7n % "* (+%!%nt%
#n$%!t#*,#7n !% %n)*+,* %n:
El proceso de resolucin de problemas relacionados con las
Ecuaciones 'i%erenciales en el ni&el (uperior a tra&,s de la utilizacinde recursos heursticos.
En %" "#8+o A(+%n%+ * +%!o"$%+ (+o8"%)*! % "o! Do,to+%! L!
C*)(#!t+o&! 3 C%"#* R#2o *(&nt*n:
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"o! *"&)no! (*+* +%!o"$%+ (+o8"%)*! %n %n%+*"...Po+ "o %n%+*" "o!
(+o,%#)#%nto! )%too"7#,o! -&% !% *n %!t5n #+##o! * *,,#on%!-&% %8% +%*"#2*+ %" )*%!t+o' %! %,#+' %! &n* )%too"o/* %
%n!%1*n2* 3 no %!t5 #+##* * "* 8
*,t&*,#7n (*+* %" *"&)no.
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+%!o"&,#7n % (+o8"%)*! *n*"#2** ,o)o &n (+o,%!o t#%n% &n*
*4%,t*,#7n #+%,t* %n %" (+o,%!o % %n!%1*n2*>*(+%n#2*9% % "*
M*t%)5t#,* 3 %n 4&n,#7n % &t#"#2*+ "*! (ot%n,#*"#*%! % "*
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%&+/!t#,* %n %" (+o,%!o % %n!%1*n2*>*(+%n#2*9% E,&*,#on%!
D#4%+%n,#*"%! no! ("*nt%*)o! %" !#%nt% o89%t#$o:
-a elaboracin de una estrate"ia metodol"ica de carcter heurstico
para %a&orecer los procesos de solucin de problemas relacionados
con Ecuaciones 'i%erenciales en el ni&el (uperior.
En +%"*,#7n ,on %" o89%t#$o !% 4o+)&"*n "*! !#%nt%! (+%&nt*!
,#%nt/=,*!:
@. C&5"%! !on "*! 8*!%! t%7+#,*! -&% (%+)#t#+5n 4&n*)%nt*+ "*
%!t+*t%#* )%too"7#,*
. C&5" %! %" (*(%" % "* %&+/!t#,* %n "o! (+o,%!o! % !o"&,#7n
% (+o8"%)*! %n %n%+*" 3 % (+o8"%)*! +%"*,#on*o! ,on "*!
E,&*,#on%! D#4%+%n,#*"%! %n (*+t#,&"*+
. &0 %!t+*t%#* )%too"7#,*' !&!t%nt** %n "o! +%,&+!o!
%&+/!t#,o!' (%+)#t#+5 4*$o+%,%+ "o! (+o,%!o! % !o"&,#7n %
(+o8"%)*! +%"*,#on*o! ,on "*! E,&*,#on%! D#4%+%n,#*"%!
P*+* %!*++o""*+ "* #n$%!t#*,#7n !% +%*"#2*+on "*! !#%nt%! t*+%*!:
E!t*+ "* %$o"&,#7n #!t7+#,* % "* %&+/!t#,* ,o)o )%#o (*+*
4*$o+%,%+ "o! (+o,%!o! % !o"&,#7n % (+o8"%)*!.
F&n*)%nt*+ (%*7#,*' (!#,o"7#,* 3 %(#!t%)o"7#,*)%nt%
"*! (ot%n,#*"#*%! % )0too! %&+/!t#,o! %n "* +%!o"&,#7n %
E,&*,#7n D#4%+%n,#*".
An*"#2*+ 3 $*"o+*+ %" &!o % "* %&+/!t#,* %n "* %!,&%"* *!/
,ono,%+ "*! ,*+*,t%+/!t#,*! % !& #n!t+&)%nt*,#7n.
E"*8o+*+ &n* %!t+*t%#* )%too"7#,* !&!t%nt** %n "*
%&+/!t#,* -&% ,ont+#8&3* * 4*$o+%,%+ "o! (+o,%!o! % !o"&,#7n
% E,&*,#on%! D#4%+%n,#*"%!.
P*+* %" "o+o %" o89%t#$o !% &t#"#2*+on "o! )0too! t%7+#,o! !#%nt%!:
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An5"#!#! % 4&%nt%!' (*+* ,on!t*t*+ %" %!t*o %" (+o8"%)* 3
!%nt*+ "*! 8*!%! (*+* "* #n$%!t#*,#7n.
E" )0too #!t7+#,oQ"7#,o' (*+* %!t*+ %" %!*++o""o -&% *
t%n#o %" (+o8"%)* 3 !& ,on,*t%n*,#7n ,on "o! #4%+%nt%!
%"%)%nto! %" (+o,%!o o,%nt%Q%&,*t#$o.
E" )0too %" %n4o-&% !#!t0)#,o' (*+* *n*"#2*+ "* +%"*,#7n
!#!t0)#,* % "o! ,o)(on%nt%! %" )o%"o.
E" *n5"#!#! 3 "* !/nt%!#!' (*+* %!t*+ "o! #4%+%nt%! *!(%,to!
-&% ,o)(on%n "o! (+o,%!o! % +%!o"&,#7n % (+o8"%)*! 3
4o+)*,#7n % "* (%+!on*"#*.
E" )0too #(ot0t#,o>%&,t#$o' (*+* %&,#+ % "*! t%o+/*!
="o!7=,*!' (!#,o"7#,*!' (%*7#,*! 3 !o8+% "* +%!o"&,#7n %
(+o8"%)*! %n 8*!% * "* %&+/!t#,*' "o! 4&n*)%nto! t%7+#,o>
(+5,t#,o! %" )o%"o #5,t#,o (*+* "* 4*$o+%,%+ %" (+o,%!o
%n!%1*n2*>*(+%n#2*9% % "*! E,&*,#on%! D#4%+%n,#*"%!.
L* )o%"*,#7n' (*+* (+o(on%+ "* %!t+*t%#* )%too"7#,*.
E" *(o+t% t%7+#,o %!t5 %n &n* %!t+*t%#* )%too"7#,* (*+* 4*$o+%,%+
"o! (+o,%!o! % !o"&,#7n % (+o8"%)*! +%"*,#on*o! ,on "*!
E,&*,#on%! D#4%+%n,#*"%! * t+*$0! %" &!o % +%,&+!o! %&+/!t#,o!.
A%)5!' ,o)o %"%)%nto 4&n,#on*" % "* %!t+*t%#* )%too"7#,* !%
(+o(on% &n* ,"*!#=,*,#7n % "*! %,&*,#on%! #4%+%n,#*"%! %n 4&n,#7n
% "o! +%,&+!o! %&+/!t#,o!. A%)5!' !% ("*nt%*n "*! ,*+*,t%+/!t#,*!
% "*! %,&*,#on%! #4%+%n,#*"%! 3 &n* %!t+*t%#* %&+/!t#,* (*+*
%n4+%nt*+ !& !o"&,#7n.
En "* (+5,t#,*' %!t* #n$%!t#*,#7n t#%n% ,o)o )%t* (+o$%%+ *" (+o4%!o+
% %++*)#%nt*! n%,%!*+#*!' )*t%)5t#,*! 3 %&+/!t#,*!' -&%
(%+)#t*n "* *("#,*,#7n %n %" *&"* % "* %!t+*t%#* )%too"7#,*
(+o(&%!t* ,o)o )%#o (+#$#"%#*o (*+* 4*$o+%,%+ "o! (+o,%!o! %!o"&,#7n % (+o8"%)*! +%"*,#on*o! ,on "*! E,&*,#on%! D#4%+%n,#*"%!
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%n %" n#$%" !&(%+#o+ * t+*$0! %" &!o % "o! +%,&+!o! %&+/!t#,o!. E!t*!
%++*)#%nt*! (%+)#t#+5n ,*)8#*+ "* 4o+)* % ,+%*+ !#n#=,*o! % "o!
o89%to! )*t%)5t#,o!' %!t*no o+#%nt*o! *" %!,&8+#)#%nto ,+%*t#$o 3
o+##n*" %" ,on,%(to.
En %" (+#)%+ ,*(/t&"o !% %6(on%n "o! 4&n*)%nto! ="o!7=,o!'
(!#,o"7#,o! 3 (%*7#,o! -&% no! (%+)#t%n %!t*8"%,%+ %" $/n,&"o
%nt+% "* +%!o"&,#7n % (+o8"%)*!' %" (+o,%!o %n!%1*n2*>*(+%n#2*9% 3
"* %&+/!t#,*. En %" !%&no ,*(/t&"o' !% +%*"#2* &n %!to % "o!
%"%)%nto! 4&n*)%nt*"%! % "* %&+/!t#,*' 3 "*! ,*+*,t%+/!t#,*! -&%
%8% % t%n%+ (*+* !& #)("%)%nt*,#7n %n "* +%!o"&,#7n % (+o8"%)*!;
*%)5! %!t*8"%,%+%)o! ,7)o %" (+o,%!o %n!%1*n2*>*(+%n#2*9% %"*! E,&*,#on%! D#4%+%n,#*"%! %! 4*$o+%,#o (o+ "* +%!o"&,#7n %
(+o8"%)*!. E" t%+,%+ ,*(/t&"o %6(on+5 &n* %!t+*t%#* )%too"7#,*
(*+* 4*$o+%,%+ %" (+o,%!o % +%!o"&,#7n % (+o8"%)*! +%"*,#on*o!
,on "*! %,&*,#on%! #4%+%n,#*"%! * t+*$0! %" &!o % "o! +%,&+!o!
%&+/!t#,o!' on% %!t*+5n %!t*8"%,#*! "*! *,,#on%! 3 o(%+*,#on%!
-&% %8% % t%n%+ %n ,&%nt* %" (+o4%!o+ * "* o+* % !& *("#,*,#7n. A"
=n*" %!t*+5n "*! ,on,"&!#on%! 3 8#8"#o+*4/*.
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Captulo I: Bases Tericas de la Investigacin
En "* !o,#%* *,t&*" !% * %,o #n,*(#0 %n %" #n#$#&*"#!)o ,o)o
n%,%!*+#o 7 (+o(7!#to % $#*' %" %!t*,*+ %nt+o % #,* (o!#,#7n#)("#,* &t#"#2*+ "o! )%#o! -&% (o!%%! (*+* ,+%,%+
#n%(%n#%nt%)%nt% %" )%#o -&% t% +o%* ?!o,#%*; !#%no
(+*)5t#,o! !% *8"* % &n* &t#"#* %n %" ,ont%6to % "o! o89%t#$o!
(%+!on*"%!.
A" ,on,%8#+ %" #n#$#&o no ,o)o &n %nt% *" -&% !% "% t#%n%n -&%
#nt+o&,#+ #%*!' !#no *" ,ont+*+#o' &n !%+ !o,#*" -&% %8% % ,on!t++
"*! #%*! -&% "o 4o+)*+5n ,o)o #n#$#&o #n%(%n#%nt% 3
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N&%!t+* *("#,*,#7n +*#,* %n -&% "* *,t#$#* 7 (+5,t#,* %! "*
+%!o"&,#7n % (+o8"%)*!' on% "* %!t+*t%#* %!*++o""** (*+* "*
+%!o"&,#7n % (+o8"%)*! o+#%nt** * 4*$o+%,%+ %" (+o,%!o %n!%1*n2*
*(+%n#2*9% % "*! E,&*,#on%! D#4%+%n,#*"%! %)(#%2* ,on "* *,,#7n
4&n*)%nt*" % (%+,%(,#7n ,o)o +%%9o !%n!#8"% % "* +%*"#*. E!
%,#+' "o -&% !% (%+,#8% *",*n2* %" ,ono,#)#%nto %n %" (+o,%!o %"
%!*++o""o #!t7+#,o %" o)8+%.
D%nt+o %" ,*)(o -&% no! ,on,#%+n% %!t*8"%,%)o! "* +%"*,#7n !&9%to>
o89%to' on% %" o89%to %! %" (+o8"%)* )*t%)5t#,o' -&% no %! )5!
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% "* +%*"#*' 3 %" !&9%to %! *-&%" -&% +%*"#2* "* *,t#$#* %n %!t%,*!o %" %!t*nt%. Don% "* *,t#$#* %" !&9%to %! &n* **(t*,#7n
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!#t&*,#7n #!t7+#,o>,&"t&+*" "% 0 !%nt#o *" )#!)o.
En %!t* t%!#! !% *!&)% "* (o!#,#7n % -&% %" %!t*nt% %! !&!,%(t#8"%
*" %nto+no -&% "o +o%*' 3 %n 8*!% * !& !#t&*,#7n #!t7+#,*>,&"t&+*",+%*+5 #)5%n%! )%nt*"%! % "o! o89%to!' +%"*,#on%! 3 (+o,%!o! -&%
%!t5n #n$o"&,+*o! %n %" *&"*. T*"%! #)5%n%! !on (+o&,to % "*
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%n!%1*n2*>*(+%n#2*9%.
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Lo! 4&n*)%nto! (!#,o"7#,o! -&% no! (%+)#t%n $#n,&"*+ "* H%&+/!t#,*
,on %" (+o,%!o %n!%1*n2*>*(+%n#2*9% % "* )*t%)5t#,* !on "o! *o!
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%n &n (+o,%!o ,ont#n&o. D%!% %" (&nto % $#!t* % "* +%!o"&,#7n %
(+o8"%)*! no! %!t*)o! !o)%t#%no * &n (+o,%!o % **(t*,#7n %n %"
-&% o+#%nt*)o! n&%!t+o! ,ono,#)#%nto! *,#* "* !o"&,#7n !*t#!4*,to+#*
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*(+%n%+ !% %6t#%n% %n )
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8*9o "o! (+o,%!o! % (%n!*)#%nto %! -&% !% *(+%n%' !#no -&% %!t5
"#*o *" ,ont%6to ,&"t&+*" on% !% %!%n$&%"$% %" %!t*nt%. L*
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)#!)o t#%)(o !% #n,#% #+%,t*)%nt% %n "o! (+o,%!o! %
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#nt%+*,,#7n #n$#t*n *" %!t*nt% * %nt+*+ %n &n *)8#%nt% % ,ont+o"
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%!t+&,t&+* % (%n!*)#%nto -&% %!t5n #n)%+!o! %n "* !o"&,#7n %"
(+o8"%)*.
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%!t*nt% )#%nt+*! $* *(+%n#%no $* %!,&8+#%no n&%$o!
12
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,ono,#)#%nto! 3 +%%!,&8+#%no "o! *nt#&o! %n,ont+5no"% n&%$o!
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)#!)o t#%)(o "% %!t5 *o !%nt#o %n !& (%+!on* 3 +%*"#* ,&*no
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(+o8"%)*!. Too %!to %n (+o,%!o !#)&"t*n%o -&% $* % "o
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3 %!to "o ,ono,%)o! ,o)o "* "%3 % o8"% 4o+)*,#7n -&% ,on!t#t&3%&n 4&n*)%nto 85!#,o %" En4o-&% H#!t7+#,o>C&"t&+*".
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%nto+no %" %!t*nt%' 3* -&% ,o)o "o *8/*)o! %,"*+*o
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!o,#*" %6t%+no 4&n*)%nt*" %n "* #+%,,#7n %" )*n%9o % "*
+%!o"&,#7n % (+o8"%)*! %!% "* %&+/!t#,*' *-&/ (o%)o! *=+)*+
-&% %" )%9o+*)#%nto ,ont#n&o %" %!t*nt% t%n+5 ,o)o &n o89%t#$o
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&n %!*++o""o % "* +% !o,#*" % %!t*nt%' %! %,#+' t+*n!#t*+ % "*
%(%n%n,#* * "* #n%(%n%n,#*. D%" )#!)o )oo (*!* ,on "o!
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"* *&to++%&"*,#7n.
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"o %! (*+* *"%n (&%% no !%+"o (*+* ot+o. A!/ (&%!' ,&*no &n
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%n &n* 4*"t* % +%,&+!o! 3o ,ono,#)#%nto! (*+* %n,ont+*+ % 4o+)*
#n)%#*t* "* !o"&,#7n. D%!(&0! %!t* n%,%!#* "o *+5 -&%+%+
%n,ont+*+ &n* #n4o+)*,#7n !#n#=,*t#$* (*+* 0"' o !%*' %n,ont+*+"%
!%nt#o %n !& ,ont%6to #!t7+#,o>!o,#*". A ,ont#n&*,#7n' 8&!,*+5 "o!
)%#o! -&% "% (%+)#t*n o8t%n%+ &n* !o"&,#7n !*t#!4*,to+#* 3
=n*")%nt% $*"o+*+5 "* !o"&,#7n.
Enton,%!' &n (+o8"%)* %8% t%n%+ &n* %!t+&,t&+* -&% (%+)#t*
#%nt#=,*+ "*! ,*+*,t%+/!t#,*! *nt%+#o+%!' %n (+#)%+ "&*+ %" (+o8"%)*
%8% %!t*+ o+#%nt*o * "*! ,*+*,t%+/!t#,*! #n#$#&*"%! % "o!%!t*nt%! 3 *!/ %n,&%nt+%n &n* n%,%!#* (+#)*+#* % +%!o"$%+"o.
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no! *+5 &n ,o)(on%nt% !o,#*" 4*$o+*8"%. In#!,&t#8"%)%nt% %"
(+o,%!o % !o"&,#7n %! &no % "o! *!(%,to! )5! #)(o+t*nt%! %n %"
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%!t*8"%,%+ ,o)o (+#)o+#*" %n "* +%"*,#7n !&9%to>o89%to *" %!t*nt%>
(+o,%!o % !o"&,#7n 3 no *" %!t*nt%>!o"&,#7n.
E" %!*++o""o %" %!t*nt% %! %nt%n#o * t+*$0! % !&! ,*)8#o!
,&*"#t*t#$o! 3 no !#)("%)%nt% ,&*nt#t*t#$o!. Lo -&% no! ""%$* *"
*(+%n#2*9% +%*" ,o)o %" "o+o % %!t+&,t&+*! % (%n!*)#%nto -&%
(%+)#t*n *" %!t*nt% *("#,*+ "o! ,ono,#)#%nto! %n #!t#nt*!
!#t&*,#on%! 7 (+o8"%)*! % #!t#nt*! n*t&+*"%2*' %nton,%! %!t*+%)o!
*8"*no % &n %!*++o""o. no &n %!*++o""o -&% $#%n% % "* !#)("%
*("#,*,#7n % %!t+*t%#*! (*+t#,&"*+%! * (+o8"%)*! t/(#,o! o !#)#"*+%!'
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t/(#,o 3* !7"o !% +%)#t% * "* *("#,*,#7n #n)%#*t* % &n
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%!t*nt% 3 %9* % !%+ &n (+o8"%)* +%*".
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A!/' "* #+%,,#7n 7 "*8o+ %" (+o4%!o+ %! % $#t*" #)(o+t*n,#* %n "*
%n%+*,#7n % ,ono,#)#%nto!' on% %" %!t*nt% "o+*+5 ,&)("#+ "o!
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t+*n!#,#7n !% (o+5 #%nt#=,*+ &n %!*++o""o +%*" % 0!t%. En %"
%n4o-&% #!t7+#,o>,&"t&+*" !% ,ono,%n ,o)o D%!*++o""o (ot%n,#*" 3
D%!*++o""o +%*"' 3 * "* #!t*n,#* %nt+% %!to! o! n#$%"%! 4&% ""*)**
(o+ V#ot!3 Zon* % D%!*++o""o P+76#)o. S# %" o,%nt% t#%n% %n
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(%+)#t#+5 "* t+*n!#,#7n % &no * ot+o 3 %nton,%! t+*&,#+"o %n
%!*++o""o +%*" %!,o"*+.
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!o,#*"#2*+ "o! ,ono,#)#%nto!' %! *8!o+8#o (o+ %" +&(o 3 (o+ "*
n%,%!#* ,o)
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E" (+o4%!o+ %8% #n,&",*+ *" %!t*nt% -&% %" ,ono,#)#%nto no %!
*,*8*o n# #n)&t*8"%' 3* -&% %!t5 %n ,on!t*nt% ,*)8#o 3 n&%!t+*
+%*"#* #!t7+#,*>,&"t&+*" %!t5 +%no$5no!%. Lo )5! #)(o+t*nt% %!
%!t*,*+ "* #)(o+t*n,#* %" &!o % "o! ,ono,#)#%nto! ,o)o
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*("#,*+"o! %n !#t&*,#on%! (*+t#,&"*+%!.
E! #)(o+t*nt% %n,ont+*+ n&%$o! !#n#=,*o! % "*! !#t&*,#on%!
("*nt%**! (*+* %" %!t*nt%' t*" !#t&*,#7n (%+)#t#+5 "* 8
%9%,&,#7n' ,o)o #,% !& no)8+%' *,t&*)o! %n 8*!% * "* o+#%nt*,#7n
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,on!,#%nt%! %" ,ont+o" (*!* *" %!t*nt% )#!)o.
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)%#*nt% "*! o(%+*,#on%! -&% ,on!t#t&3%n "o! )%#o! o
(+o,%#)#%nto! ,on "o! -&% !% +%*"#2* "* *,,#7n %n "o! ,&*"%! %!t5*o %" o89%t#$o' *,,#7n -&% t#%n% ,o)o ,*+*,t%+/!t#,*!
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% ,on,#%n,#*' % #n%(%n%n,#* 3 % (%+)*n%n,#* ?o !o"#%2.
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%!t*nt% (&%* *n*"#2*+ "* 4o+)* % "o! o89%to!' *,%+
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16
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,+/t#,*)%nt% !& t+*8*9o 3 %" % ot+o! ,o)(*1%+o!' (*+t#,#(*+
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*,t#$#*%!.
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!#%nt%! %"%)%nto! ,o)o %!%n,#*"%! %n n&%!t+o t+*8*9o:
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3 % %nt+%n*+!% ,7)o *,%+"o.
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(+o,%!o! "7#,o! % (%n!*)#%nto 3 *" *",*n,% %" n#$%" t%7+#,o'%n "* )%#* -&% !% (+o&,% "* *(+o(#*,#7n % "o!
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CAPITULO II. le!entos tericos so"re la #eurstica $ lasolucin de Pro"le!as
2.% Antecedentes #istricos de la utili7acin de la eursticaen la ense9an7a
L* +%!o"&,#7n % (+o8"%)*! )*t%)5t#,o! !#%)(+% * !#o %" ,o+*27n
% "* *,t#$#* )*t%)5t#,*. S& %$o"&,#7n #!t7+#,* +%$%"* "* ("%n*
+%"*,#7n -&% * t%n#o %!t* *,t#$#* ,on "* %n!%1*n2* 3 %"
*(+%n#2*9% % "* (+o(#* M*t%)5t#,*. D%!% "* Ant#%* !% * #o
t+*n!)#t#%no too %" ,*&*" % ,ono,#)#%nto! *,&)&"*o! (o+ "*
&)*n#* &+*nt% )#"%n#o!; n&%!t+* ,#%n,#* no * !#o *9%n* * %!t*
t+*n!4%+%n,#*' 3 !% * )*t#2*o (o+ "* #)("%)%nt*,#7n % #4%+%nt%!
)0too! * "* o+* % +%*"#2*+ t*" *,,#7n.
L* *(*+#,#7n % "* %!,&%"* !% +%)ont* * "* )#!)* 0(o,* % "*
#n$%n,#7n % "* %!,+#t&+*. In$%!t#*,#on%! #!t7+#,*! %)&%!t+*n -&%
"* %n!%1*n2* % "* *+#t)0t#,* !% #n#,#*8* %n &n* 4*!% t%)(+*n* %n "*
$#* %!,o"*+' *" )#!)o t#%)(o -&% "* "%,t&+* 3 "* %!,+#t&+*; !% %8%
*,"*+*+ -&% "*! M*t%)5t#,*! %+*n ,on!#%+**! %"%)%nto!#)(o+t*nt%! %n "* 4o+)*,#7n % "o! %!,+#8*! 3 -&% "* %!,&%"*
+%!(on/* * "*! n%,%!#*%! % %!* !o,#%*' %! %,#+' * "*!
n%,%!#*%! %" )o)%nto #!t7+#,o ,on,+%to' *&n-&% %!t*8* #+##*
* +&(o! )&3 +%!t+#n#o!.
D% "* "%,t&+* % &n o,&)%nto #!t7+#,o %n -&% !% "o* * &no % "o!
+%3%! %" t%+,%+ #)(%+#o % U+ %n M%!o(ot*)#* !% (&%% #n4%+#+ -&% "*
=n*"#* 4&n*)%nt*" % "o! (+o8"%)*! )*t%)5t#,o! (+o(&%!to! %+*
(+%(*+*+ *" o)8+% (*+* %" ,5",&"o. E" !o8%+*no (+o,"*)*8* )&3
o+&""o!o S0 !&)*+ 3 +%!t*+ * "* (%+4%,,#7n' !o3 #%!t+o %n ,5",&"o 3
%n ,ont*8#"#*W ?G#)0n%2' J. @' (. . M&,o! *&to+%! ,o#n,#%n
%n ("*nt%*+ -&% 4&% %" )*t%)5t#,o +#%o H%+7n' -%n $#$#7 %n
A"%9*n+/* *(+o6#)**)%nt% %nt+% "o! !#"o! II 3 I *.n.%' %" (+#)%+o %n
#n,"+ %9%+,#,#o! ,on t%6to %n !&! t+*8*9o!; !#n %)8*+o' !% ,ono,%n'
% %,o' *"&no! t%6to! )*t%)5t#,o! %!,o"*+%! )5! *nt#&o!. E!to!
18
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t%6to! !on % o! t#(o!: % t*8"*! 3 % (+o8"%)*!; %!to!
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En "* G+%,#* !% ,ono,% -&%' *&n-&% %" ,5",&"o !% %n!%1*8* %n "*
%!,&%"* %"%)%nt*"' "* !o,#%* +#%* !% )o!t+*8*' +%*")%nt%' (o,o
#nt%+%!** (o+ "* 4o+)*,#7n #nt%"%,t&*" 3 t0,n#,* % "o! n#1o! 3
97$%n%!. A" #&*" -&% "o! t%6to! 8*8#"on#o! o %#(,#o!' "o! (+o8"%)*!
("*nt%*o! !% +%=%+%n %6("/,#t*)%nt% * &n* !#t&*,#7n ,on,+%t*'
#n,"&!o 0!t* %! )&,*! $%,%! &n *+t#=,#o ,on =n%! (%*7#,o!. S%
(&%% ("*nt%*+ -&% "* *(*+#,#7n % %!,&%"*!' *"&n*! % "*! ,&*"%!
""%*+on * !%+ 4*)o!*!' !% %8#7 * #n#,#*t#$*! #n#$#&*"%!; *!/' "o! o!
+*n%! ="7!o4o! *t%n#%n!%! % =n%! %" !#"o V 3 (+#)%+* )#t* %"
IV *nt%! % n&%!t+* %+*' S7,+*t%! ? K> *.n.% 3 P"*t7n ?K>K
*.n.%' 4&n*n !&! (+o(#*! %!,&%"*!. E" (+#)%+o' % +%t7+#,*' -&% %+* %"
*+t% % (%+!&*#+ *" ot+o (o+ )%#o % &n #!,&+!o )&3 *o+n*o 3
%"*8o+*o 3 %" !%&no' %n %" *1o ' 4&n7 %n At%n*! !& %!,&%"* %
F#"o!o4/* "* A,*%)#*W' #n!t#t&,#7n * )%n&o ,on!#%+** ,o)o "*
(+#)%+* &n#$%+!#* %&+o(%*' "* ,&*" o4+%,/* &n *)("#o ("*n %
%!to! -&% #n,"&/* t%)*! % A!t+ono)/*' B#o"o/*' M*t%)5t#,*'
T%o+/* Po"/t#,* 3 F#"o!o4/*.
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%" %9%+,#,#o % "*! M*t%)5t#,*!: (o+ &n "*o' "o! *+#)%n!o+%!'
#n%n#%+o! 3 ,ont*8"%! 3' (o+ ot+o' "o! )0#,o! 3 *!t+7"oo!' -&%
o2*8*n % &n* !#t&*,#7n !o,#*" !&(%+#o+; "o! %" (+#)%+ +&(o' %nt+o
% !&! %n!%1*n2*!' %n4*t#2*8*n %n "* +%!o"&,#7n % (+o8"%)*!
(+5,t#,o!' o4+%,#%no %t%+)#n*o! )o%"o! (*+* *"&n*! !#t&*,#on%!
%!(%,/=,*!.
En %" !#"o XIV' %n E&+o(*' "o! ,*)8#o! %,on7)#,o! *!/ ,o)o %"
%!*++o""o % "*! ,#&*%! 3 %" ,o)%+,#o $*n * 4*$o+%,%+ %" *!,%n!o
!o,#*" % "o! )*t%)5t#,o! (+5,t#,o!. Lo! #nt%+,*)8#o! ,o)%+,#*"%!
,** $%2 )5! ,o)("%9o! %6#/*n t0,n#,*! #7n%*! % ,5",&"o 3,ont*8#"#*. E6#!t/*n %n %!o! )o)%nto! t+*t*o! on% !% %6(on/*n
+%"*! (*+* "* !o"&,#7n % (+o8"%)*! %!(%,/=,o! +%"*,#on*o!
(+#n,#(*")%nt% ,on "*! t*!*! % #nt%+0!' "o! ,*)8#o!' "* ,#+,&"*,#7n 3
%" (%!o % "*! )on%*!' o "* +%(*+t#,#7n % "o! 8%n%=,#o!. En "o!
t+*t*o! %!to! )0too! !o"/*n (+%!%nt*+!% %n 4o+)* % ,*!o!
,on,+%to!' #nt%+5no!% %n &n ,ont%6to tot*")%nt% (+5,t#,o.
En "* (o,* Mo%+n*' ,on %" %!*++o""o %" ,*(#t*"#!)o' #)(%+* %"
%!(/+#t& &t#"#t*+#o 3 %!% %!% (&nto % $#!t* 4&% (&%!t* %n (+5,t#,*
to* "* %n!%1*n2*. E!t% (+o,%!o !% #n#,#* ,on %" &)*n#!)o
+%n*,%nt#!t*' #n,"&30no!% %n %!t* %no)#n*,#7n *-&%""o! -&% !%
*(*!#on*+on (o+ "*! "%t+*! 3 "*! *+t%! ,"5!#,*!. E! *t#n*o *,"*+*+ -&%
%n %" !#"o XVII ,o)#%n2* "* %,*%n,#* % "* %n!%1*n2* &)*n/!t#,*
3 "* !o,#%* (#% * "* %!,&%"* -&% (+o$%* * !&! #9o! % ,on&,t*! 3
,ono,#)#%nto! t%7+#,o>(+5,t#,o!' -&% "%! (%+)#t*n *,t&*+ 3
%!*++o""*+!% %n %""*.
E" #to 4&n*)%nt*"' %n %!t* 0(o,* 3 %n %" !%nt#o % "* &t#"#2*,#7n %
"* %&+/!t#,* %n "* *,t#$#* % +%!o"&,#7n % (+o8"%)*!' 4&% )*+,*o
(o+ %" ="7!o4o 3 )*t%)5t#,o R. D%!,*+t%! ?@>@. D%!,*+t%!
*!#n7 %nt+o %" (+o,%!o % ,ono,#)#%nto &n (*(%" %6t+*o+#n*+#o *
"* %&+/!t#,* 8*!**' %n "o 4&n*)%nt*"' %n "* %&,,#7n' %n "o!
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(+#n,#(#o! 85!#,o! ?*6#o)*! *",*n2*8"%! (o+ $/* #ntt#$*. P*+*
o8t%n%+ %" ,ono,#)#%nto' 0" ,+%/* n%,%!*+#o (on%+"o too %n &*'
!*"$o "* ,ono!,#8#"#* )#!)*; %!t% (+#n,#(#o !% )*n#=%!t* %n !&
)56#)*: (#%n!o' "&%o %6#!toW.
En %" 5)8#to % "* %&+/!t#,* 3 "* +%!o"&,#7n % (+o8"%)*!' "*
t+*!,%n%n,#* )5! %!(%,#*" !% ,%nt+* %n o! % !&! (+#n,#(*"%!
t+*t*o!: D#!,o&+! % "* M0to%W ?D#!,&+!o %" M0too' (&8"#,*o
(o+ (+#)%+* $%2 %n L%3%n' %n @ 3 R%&"*% * D#+%,t#on%)
In%n##W ?R%"*! (*+* "* D#+%,,#7n %" E!(/+#t&' (&8"#,*o (o!t )o+t%)
%n O8+*! P7!t&)*!W' A)!t%+*)' @@. En @ ,o)%n27 *
+%*,t*+ !&! R%"*!W %n t+%! to)o!' ,on &n* o,%n* % %""*! ,**&no; (%+o %!(&0! % *++#8*+ * "* )#t* %" !%&no $o"&)%n' !7"o
*",*n27 * (on%+ %" t/t&"o % t+%! +%"*! )5!' 3* -&% "* )&%+t% $#no *
!o+(+%n%+"o %n 4%8+%+o % @. En %!t* o8+* %" +*n (%n!*o+
%6("#,* * "o! )o+t*"%! ,o++#%nt%!W ,7)o %""o! (o+/*n (%n!*+ ,o)o
0"' 3 ,7)o' !#%no !& )0too' (o+/*n +%!o"$%+ (+o8"%)*! t*" 3
,o)o 0" "o #2o.
Con!#%+* Po"3* -&% "*! (*"*8+*! !#%nt%! % D%!,*+t%! %!,+#8%n %"
o+#%n % "*! R%"*!W: C&*no' %n )# 9&$%nt&' o/ *8"*+ %
#n$%n,#on%! #n%n#o!*!' t+*t*8* % !*8%+ !# no (o+/* #n$%nt*+"*! 3o
)#!)o' !#n #n,"&!o "%%+ *" *&to+' *!/ *$%+t/ -&% )% ,on4o+)*8* *
,#%+t*! +%"*!.W ?Po"3*' G. @K' (. @.
S& (+o3%,to %!,*n!*8* !o8+% &n ("*n %&+/!t#,o )&3 !#)("%: F*!% I:
R%&,#+ ,&*"-%+ (+o8"%)* *"%8+*#,o * "* +%!o"&,#7n % &n*%,&*,#7n !#)("%. F*!% II: R%&,#+ ,&*"-%+ (+o8"%)* )*t%)5t#,o * &n
(+o8"%)* *"%8+*#,o. F*!% III: R%&,#+ ,&*"-%+ (+o8"%)* * &n
(+o8"%)* )*t%)5t#,o. E" (+#)%+ "#8+o ,&")#n* ,on "*! +%"*! IXQXII'
-&% *3&*n * ,on!o"#*+ %" ,ono,#)#%nto. En4*t#2* "* n%,%!#* %
(+o4&n#2*+ %n "*! ,&%!t#on%! )5! !#)("%!; %n "* #)(o+t*n,#* % "*
%9%+,#t*,#7n; %n "* 8
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"* #)*#n*,#7n' "o! !%nt#o! 3 "* )%)o+#*. R%!(%,to * "*! 4*,&"t*%!
%)("%**! %n %" ,ono,#)#%nto' D%!,*+t%! %!t*,* -&% !7"o "*
#nt%"#%n,#* (&%% (%+,#8#+ "* $%+*' (%+o no %8% %9*+ % *3&*+!%
%" +%!to % "*! 4*,&"t*%! !%1*"**!.
En %" !%&no "#8+o !% %6*)#n*n ,&%!t#on%! )5! ,o)("#,**!.
V%*)o! "*! +%"*! )5! !#n#=,*t#$*!: R%"* XIII: C&*no !%
,o)(+%n% (%+4%,t*)%nt% &n* ,&%!t#7n %! n%,%!*+#o *8!t+*%+"* %
to* ,on,%(,#7n !&(%+&*' +%&,#+"* * !&! )5! !#)("%! %"%)%nto! 3
!&8#$##+"* %n t*nt*! (*+t%! ,o)o !%* (o!#8"% (o+ )%#o % "*
%n&)%+*,#7n. R%"* XIV: L* )#!)* +%"* %8% !%+ *("#,** * "*
%6t%n!#7n +%*" % "o! ,&%+(o! 3 %! n%,%!*+#o +%(+%!%nt*+"* ,o)("%t* *"* #)*#n*,#7n (o+ )%#o % =&+*! ,"*+*!; % %!t% )oo !%+/* )&,o
)%9o+ ,o)(+%n#* (o+ "* #nt%"#%n,#*. R%"* XV: E! % +*n &t#"#*
t+*2*+ %!t*! =&+*! 3 +%(+%!%nt*+"*! * "o! !%nt#o! %6t%+no! * =n %
,on!%+$*+ "* *t%n,#7n %" %!(/+#t&. Co)o !% (&%% *(+%,#*+' %!t*!
+%"*! !on )&3 *%,&**! (*+* %)(+%n%+ "* !o"&,#7n % &n
(+o8"%)*. En %" (+#)%+ ,*!o !% #n,#t* * %!,o)(on%+ %" (+o8"%)* %n
ot+o! )5! !%n,#""o!' (on#0no!% *" %!,&8#%+to "o! (+o,%!o! %*n5"#!#! 3 !/nt%!#!; %n %" +%!to !% !%+% "* ,on!t+&,,#7n % &n* =&+*
% *n5"#!#!' ,on 0n4*!#! %n "* $#!&*"#2*,#7n % "o! %"%)%nto! -&%
#nt%+$#%n% %n %" (+o8"%)*.
En %" !#"o XVIII +%!&"t* n%,%!*+#o %!t*,*+ *" !o L. E&"%+ ?@Q
@. E!t% %)#n%nt% ,#%nt/=,o no ""%7 * ("*nt%*+ %6("/,#t*)%nt%'
,o)o D%!,*+t%!' &n ,on9&nto % +%"*! %&+/!t#,*! (*+* *8o+*+ "o!
(+o8"%)*!. E" )0+#to 4&n*)%nt*" +*#,* %n "* %&,*,#7n %&+/!t#,*
)*n#4%!t** %n !& (+*6#! (%*7#,*. S%
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En "* o8+* %&"%+#*n*' no !o"*)%nt% "o! %!,&8+#)#%nto! %&+/!t#,o!
!&!t%nt*o! %n *n*"o/*! !on #no! % )%n,#on*+' %! #)(o+t*nt%
%,#+ -&% !& ,*(*,#* % *n5"#!#! %+* !o+(+%n%nt%' (%+o
4&n*)%nt*")%nt% !% #!t#n ,o)o: %" )*t%)5t#,o )5! 58#"
(*+* "* ,+%*,#7n % *"o+#t)o! 3 %!t+*t%#*! %n%+*"%! (*+* "*
!o"&,#7n % (+o8"%)*!' -&% 9*)5! *3* %6#!t#oW ?C*!t+o' I. @'
(.. T*)8#0n t#%n% &n "&*+ (+%(on%+*nt% "* ,+%*,#7n % n&%$*!
t%o+/*! 8*!**! %n "o! )0too! ,on "o! -&% +%!o"$#7 +*n%!
(+o8"%)*! )*t%)5t#,o!; &n %9%)("o % !& (+o##o!* ,*(*,#* (*+*
"* +%!o"&,#7n % (+o8"%)*! 3 !& 4*,#"#* (*+* "* %n%+*"#2*,#7n %
)0too! % !o"&,#7n' !% (on% % )*n#=%!to %n &no % "o! (+o8"%)*!
)5! (o(&"*+%! -&% +%!o"$#7 %!t% %n#o' %" % "o! !#%t% (&%nt%! %
Y]n#!8%+. L* %!t+*t%#* -&% %)("%7 (*+* !& !o"&,#7n !#+$#7 (*+*
%!*++o""*+ &n* +*)* % "* )*t%)5t#,* ""*)** To(o"o/*
,o)8#n*to+#*.
No !% %8% (*!*+ (o+ *"to' %n %" *n5"#!#! % "* 0(o,*' *" not*8"%
)*t%)5t#,o B. Bo"2*no ?@@>@K' -&% t*)8#0n #n,&+!#on7 !o8+% "*
4o+)* % *8o+*+ *-&%""o! (+o8"%)*! (*+* "o! ,&*"%! no !% (o!%/* &n(+o,%#)#%nto % +%!o"&,#7n; %n !& "#8+o ^#!!%n!,*4t!"%8+% #+##o *
"* L7#,*' %#,7 &n* %6t%n!* (*+t% * "* %&+/!t#,*. Mo%!t*)%nt%
+%"*t*:
No (+%t%no %n "o *8!o"&to (+%!%nt*+ *-&/ n#n
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E" $%+*%+o $*"o+ % !& o8+* %!t5 %n (+o(on%+!% *,%+ &n*
+%,o(#"*,#7n 3 #$&"*+ %!o! )oo! % *,t&*,#7n % "o! o)8+%! %
t*"%nto. E" t+*8*9o t#%n% &n $*"o+ %&,*,#on*" %6,%(,#on*"' (o+ ,&*nto
% too! %! ,ono,#o -&% )&,o! ,#%nt/=,o! % #n$%!t#*o+%!
%!t*,*o! 8*!*n !& *,t#$#* %n +%,&+!o! o8t%n#o! % &n* "*+*
%6(%+#%n,#* o * $%,%! #n!t+&)%nt*o! )*#!t+*")%nt% (o+ %""o!
)#!)o!' (%+o -&% (o,*! $%,%! !on %6("#,#t*o!' !*"#%no * "* "&2 !7"o
"* %6(o!#,#7n 4o+)*" 3 +#&+o!* %" +%!&"t*o o8t%n#o.
En "* *"8o+** %" !#"o XX *(*+%,%n "o! *(o+t%! % H. Po#n,*+0
?@KQ@@' )*t%)5t#,o 4+*n,0! -&% !% o,&(7 !o8+%)*n%+* % "*
)%too"o/* %n%+*" % "* ,#%n,#*. Po#n,*+0 ,on!#%+*8* -&% "*! "%3%!% "* ,#%n,#* no (%+t%n%,%n *" )&no +%*"' !#no -&% ,on!t#t&3%n
*,&%+o! ,on$%n,#on*"%! (*+* *,%+ )5! ,7)o* 3
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%!-&%)* *"o )5! %6*&!t#$o (*+* %6("#,*+ %" (+o,%!o % ,+%*,#7n
)*t%)5t#,*. S&! 4*!%! !on "*! !#%nt%!: Do,&)%nt*,#7n
?#n4o+)*+!%' "%%+ (+%$#*)%nt%' %!,&,*+' #!,&t#+; P+%(*+*,#7n
?+%*"#2*+ &n (+o,%!o % %n!*3oQ%++o+ !o8+% #4%+%nt%! $/*! % #(7t%!#!'
,on!#%+*no &n ,*)8#o %$%nt&*" % *,t#$#* %n ,*!o % no o8t%n%+
n#n
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En +%"*,#7n * "*! +%"*! M%"2* ?@ #%nt#=,* ,#n,o +%"*!
%n%+*"%! % t+*8*9o -&% !% &t#"#2*n (*+* +%!o"$%+ &n* )&"t#("#,#*
% (+o8"%)*! )*t%)5t#,o!. Ent+% %""o! !% )%n,#on* (o+ %9%)("o %"
(+#n,#(#o %" %!$/o. A-&/' %" (+o8"%)* * +%!o"$%+ %! ,on!#%+*o %n
ot+o o)#n#o "o ,&*" 4*,#"#t* !& !o"&,#7n 3 (o!t%+#o+)%nt% !% ""%$* *
!&! ,on#,#on%! #n#,#*"%!.
S,o%n4%" ?@ !%1*"* -&% %n Ho_ to So"$% ItW Po"3* no !%
,ont%nt* ,on %!t% !#)("% *4o+#!)o' *!/ -&% +%*"#2* &n %!to
#nt+o!(%,t#$o %" )0too ,*+t%!#*no. A&n-&% !& *",*n,% !% $#o
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*(o+t% 4&n*)%nt*": %" *#!"*)#%nto % ,&*t+o 4*!%! ,"*+*)%nt%
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("*n; 3 V#!#7n +%t+o!(%,t#$*. En ,** &n* Po"3* (+o(on% &n* !%+#% %
+%"*! %&+/!t#,*! 8*!t*nt% !&%+%nt%!' (%+o "o )5! noto+#o' %n (+#)%+
"&*+' ,on!#!t% %n -&% "* )*3o+/* % %""*! $*n #+##*! * "* !%&n*
4*!%.
Po"3* *n*"#2* "* #4%+%n,#* %nt+% %&+/!t#,*W 3 %&+/!t#,* )o%+n*W 3
%6(on%' %n "o 4&n*)%nt*"' -&% %n "* !%&n* !% t+*t* %:
,o)(+%n%+ %" )0too -&% ,on&,% * "* !o"&,#7n %" (+o8"%)*' %n
(*+t#,&"*+ "*! o(%+*,#on%! )%nt*"%! t/(#,*)%nt% @@K.
A (%!*+ % -&% Ho_ to So"$% ItW )*+,7 &n #to %n %" ,*)(o % "*
D#5,t#,* % "* M*t%)5t#,*' %n !& 4%,* % *(*+#,#7n no ,*&!7 +*n
#)(*,to' 3* -&% "o! ,&++/,&"o! %!,o"*+%! %!t*8*n 4&%+t%)%nt%
#n&%n,#*o! (o+ "o! *!o,#*,#on#!t*!' "o! ,&*"%! (+o(&n*8*n &n
*(+%n#2*9% (o+ +%(%t#,#7n. A
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o,%nt* -&% !% to)*n %n ,&%nt*' %n "o! EE.UU.' (*+* !&
#n!t+&)%nt*,#7n %n %" ,ont%6to %" *&"* "*! #%*! % Po"3*' !o8+% too
"o ,on,%+n#%nt% * "*! %t*(*! %n %" (+o,%!o % +%!o"&,#7n %
(+o8"%)*!.
T*)8#0n %! (+%,#!o *,"*+*+ "o! *(o+t%! +%*"#2*o! %n %" ,*)(o % "*
+%!o"&,#7n % (+o8"%)*! (o+ "* %!,&%"* % D#5,t#,* % "* M*t%)5t#,*
% "* %!*(*+%,#* R.D.A.' %n +%"*,#7n ,on "* #n!t+&,,#7n %&+/!t#,* %n
%" ,ont%6to % "*! M*t%)5t#,*! %!,o"*+%!. L* %!,&%"* *"%)*n*
,on,%8/* &n !#!t%)* % (+o,%#)#%nto! %&+/!t#,o!' ,"*!#=,*o! %n
(+#n,#(#o!' +%"*! 3 %!t+*t%#*! ?%n%+*"%! 3 (*+t#,&"*+%! -&% %8/*
!%+ o89%to % %n!%1*n2* * "o! %!t*nt%!' &+*nt% %" (+o,%!o %+%!o"&,#7n % (+o8"%)*!. Lo! t+*8*9o! +%*"#2*o! (o+ "* %!,&%"*
*"%)*n* !% (+o(on/*n 4o+)&"*+ &n P+o+*)* G%n%+*" H%&+/!t#,o
?PGH' -&% *8*+,*+* too %" (+o,%!o % +%!o"&,#7n % %9%+,#,#o! 3
(+o8"%)*! 3' *%)5!' -&% %!t&$#%+*n (+%!%nt%! too! "o! %)5!
(+o+*)*! ,o)o !&8(+o+*)*! o %n 4o+)* % ,*!o! %!(%,#*"%!.
L*! #%*! ,%nt+*"%! % "o! (+#n,#(*"%! )o%"o!' !on "*! !#%nt%!:
Pol$a ,#oen(eld /ller ;ung< > Co)(+%n%+ %"
(+o8"%)*> Con,%8#+ %"
("*n> E9%,&t*+ %"
("*n> V#!t*
+%t+o!(%,t#$*
> An5"#!#! 3,o)(+%n!#7n%" (+o8"%)*
> D#!%1*+ 3("*n#=,*+ "*!o"&,#7n
> E6("o+*+!o"&,#on%!
> V%+#=,*+!o"&,#on%!
> O+#%nt*,#7n> E"*8o+*,#7n> R%*"#2*,#7n> E$*"&*,#7n
> O+#%nt*,#7n*,#* %"(+o8"%)*
> T+*8*9o %n %"(+o8"%)*
> So"&,#7n %"(+o8"%)*
> E$*"&*,#7n %"* !o"&,#7n
E! #)(o+t*nt% +%!*"t*+ -&% %n %" P+o+*)* H%&+/!t#,o (+o(&%!to (o+
Po"3* "o! %"%)%nto! % !&! 4*!%! no *(*+%,%n %t*""*o!; (o+ "o
t*nto' !& *("#,*,#7n % )*n%+* #+%,t* * "* %!,&%"* +%!&"t* #4/,#". L*
%!t+*t%#* %!*++o""** (o+ S,o%n4%"' *&n-&% #+##* * %!t*nt%!
t*"%nto!' %! )5! %6("/,#t* 3 *("#,*t#$* 3 (&%% *("#,*+!% (*+,#*")%nt%'
,on **(t*,#on%!' * "o! %!t*nt%! % n&%!t+*! *&"*!. E" % M""%+ 3
29
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%" % J&n !on !#)#"*+%! 3 )5! ,o)("%to! -&% "o! *nt%+#o+%!; %!to!
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2.2 l concepto de pro"le!a
En %!t% %(/+*4% !% %n,*+* %" %!to % #4%+%nt%! %=n#,#on%! %
(+o8"%)*! &t#"#2**! %n %" ,ont%6to #5,t#,o % "*! M*t%)5t#,*!' ,on
%" =n % ""%*+ * &n* ,*+*,t%+#2*,#7n % "o! (+o8"%)*! %!,o"*+%!' (*+*
!& %4%,t#$* &t#"#2*,#7n' ,o)o &n +%,&+!o %n "* %n!%1*n2* 3 %"
*(+%n#2*9% % "* )*t%)5t#,*.
31
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E! n%,%!*+#o 8&!,*+ &n* %=n#,#7n -&% *,"*+% %" !#n#=,*o % "*
%6(+%!#7n (+o8"%)*' (&%!to -&% * (*+t#+ % !& &!o %n%+*"#2*o %!
,&*no ,o)#%n2*n * !&+#+ ,ont+*#,,#on%! *,%+,* % "o -&%
#4%+%nt%! *&to+%! -%+%n !#n#=,*+ ,&*no "* &!*n. P*+t*)o! (*+* %"
*n5"#!#! %" !#n#=,*o %" t0+)#no (+o8"%)*' % !& &!o %n %" "06#,o
,o)
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D%!% "o! (o!t&"*o! % "* P!#,o"o/* !% %!t*+on "*! %=n#,#on%!
t+*!,%n%nt*"%! %n %!t% ,*)(o. E! "/,#to )%n,#on*+ "*! **! (o+
R&8#n!t%#n ?@ 3 L%$ont#%$ ?@. D%" *n5"#!#! % "*! %=n#,#on%!
%!t*8"%,#*! (o+ %!to! (!#,7"oo! %n,*+n*n o! ,*+*,t%+/!t#,*!
,o)&n%!: %n too $%+*%+o (+o8"%)* %" !&9%to %!,ono,% "* $/* %
!o"&,#7n 3 *" (o!#,#on*+!% 4+%nt% *" (+o8"%)* )#!)o *o(t* &n
,*+5,t%+ *,t#$o. P*+* M*9)&to$ ?@ "o! %"%)%nto! 4&n*)%nt*"%!
% &n (+o8"%)* !on "o ,ono,#o 3 "o %!,ono,#o ?*""*+ %" n%6o' "*!
+%"*,#on%! %nt+% "o ,ono,#o 3 "o %!,ono,#o.
Bo+*!# ?@' +%"*,#on* (+o8"%)*! ,on t%6to * "o! t%6to! 4o+)&"*o!
,on (+%,#!#7n' on% *(*+%,%n too! "o! *to! n%,%!*+#o! (*+*o8t%n%+ "* !o"&,#7n. T*)8#0n t+*8*9* "o! (+o8"%)*! (*+* %"
%nt+%t%n#)#%nto 3 "*! (+&%8*! % ,on9%t&+*!' +%=+#0no!% * "*
%)o!t+*,#7n % t%o+%)*! o % &n* ,#%+t* (+o(#%*. Uno % "o!
(+o8"%)*! )5! !%+#o! %! -&% no -&%* ,"*+* "* 8*!% (*+* "* #$#!#7n
% "o! ,on,%(to!.
En "o! t+*8*9o! % Gon25"%2 ?@K' *(*+%,% &n ,on,%(to % (+o8"%)*
-&% %n4*t#2* 4&n*)%nt*")%nt% %n !& (*+t% ,&*nt#t*t#$*' *"
(&nt&*"#2*+ %n -&% (+o8"%)* %! to* (+o(o!#,#7n ?%n%+*")%nt% %
,*+5,t%+ (+5,t#,o %n -&% !% (#% "* %t%+)#n*,#7n % ,#%+t*!
,*nt#*%! ?n&)0+#,*!' %o)0t+#,*!' 4/!#,*!' %t,0t%+* )%#*nt% "*!
+%"*,#on%! -&% %6#!t%n %nt+% %""*! 3 ot+*! ,ono,#*!W ?Gon25"%2' M.
@K' (. .
P*+* Y*nto_!# ?@@ &n (+o8"%)* %! &n* !#t&*,#7n -&% #=%+% %&n %9%+,#,#o %n -&% %" +%!o"&to+ % (+o8"%)*! no t#%n% &n (+o,%!o
*"o+/t)#,o -&% "o ,on&,#+5 ,on ,%+t%2* * "* !o"&,#7n.W ?Y*nto_!#'
M. @@' (. @@@.
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Ot+* %=n#,#7n -&% *(*+%,% ,o)o ,on!t*nt% %n &n ,on9&nto %
#n$%!t#*,#on%! !o8+% %" ,*)(o % "* +%!o"&,#7n % (+o8"%)*!' %! "*
** (o+ P*"*,#o! 3 Z*)8+*no ?@ -&% (+%,#!*: E" (+o8"%)*
(&%% !%+ %=n#o ,o)o ,&*"-%+ !#t&*,#7n' -&% (+o&,% (o+ &n "*o
&n ,#%+to +*o % #n,%+t#&)8+% 3' (o+ ot+o "*o' &n* ,on&,t*
t%n%nt% * "* 8
@' (. .
A&n-&% %n "* %=n#,#7n *nt%+#o+ 3 %n "* ** (o+ C*)(#!t+o&! 3 R#2o
?@' !% o8!%+$* &n* ,#%+t* +%"*,#7n %n %" !#n#=,*o -&% !% "%
*t+#8&3% * "o! t0+)#no! &t#"#2*o!' !% %nt#%n% -&% "* ** (o+ %!to!
*&to+%! %! )5! *,*8**' (&%! %6("#,#t* % &n* )*n%+* )5! #+%,t*"o! %"%)%nto! %!%n,#*"%! % "* %=n#,#7n. En t*" !%nt#o %=n%n
(+o8"%)* ,o)o to* !#t&*,#7n %n "* -&% *3 &n ("*nt%*)#%nto #n#,#*"
3 &n* %6#%n,#* -&% o8"#* * t+*n!4o+)*+"*. L* $/* % (*!*+ % "*
!#t&*,#7n o ("*nt%*)#%nto #n#,#*" * "* n&%$* !#t&*,#7n %6##* t#%n%
-&% !%+ %!,ono,#* 3 "* (%+!on* %8% -&%+%+ +%*"#2*+ "*
t+*n!4o+)*,#7nW ?C*)(#!t+o&!' L. 3 R#2o' C. @' (. IX.
En %!t% )#!)o !%nt#o L*8*++%+% * !%1*"*o -&% ... &n (+o8"%)* %!
%t%+)#n** !#t&*,#7n %n "* ,&*" %6#!t%n n%6o!' +%"*,#on%!' ,&*"#*%!
% 3 %nt+% o89%to! -&% no !on *,,%!#8"%! % 4o+)* #+%,t* o
#n#+%,t*)%nt% * "* (%+!on*; ?... %! to* +%"*,#7n %n "* ,&*" *3 *"o
o,&"to (*+* %" !&9%to' -&% %!t% !% %!4&%+2* (o+ *""*+W ?L*8*++%+%' A. F.
@' (. .
A" *n*"#2*+ %!t*! %=n#,#on%! %n,ont+*)o! %"%)%nto! -&% !on %!&)* #)(o+t*n,#* (*+* *,%+ &n* (+%!%nt*,#7n t+*n!(*+%nt% %
(+o8"%)* %!,o"*+' "o -&% (%+)#t% &n *,,%!o "#n/!t#,oQ,on,%(t&*" %
)*3o+ (+%,#!#7n %n "* %"*8o+*,#7n % "o! (+o8"%)*! 3 -&% "o!
(+o4%!o+%! +%,ono2,*n ,&5no %!t5n +%*")%nt% %n (+%!%n,#* % %""o!.
E!to! %"%)%nto! !on:
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L* $/* % (*!*+ % "* !#t&*,#7n #n#,#*" * "* n&%$* !#t&*,#7n %8%
% !%+ %!,ono,#*; %!t*8"%,#%no #4%+%n,#*! %!%n,#*"%! %nt+%
%9%+,#,#o 3 (+o8"%)*'
L* (%+!on* -%+% +%*"#2*+ %!* t+*n!4o+)*,#7n' t%n#%no 8#%n
%n ,"*+o -&% "o -&% ,on!t#t&3% &n (+o8"%)* (*+* &no (&%% no
!%+"o (*+* ot+o.
A )oo % ,on,"&!#7n % %!t* (*+t%' (&%% +%!&)#+!% -&% *&n-&%
%6#!t% &n* +*n #$%+!#* % ,+#t%+#o!' "o! *&to+%! % )*n%+* %n%+*"
no !% ,ont+*#,%n; %n t*" !%nt#o' "o! +*!o! 4&n*)%nt*"%!
*n*"#2*o! !on:
E6#!t#+5 &n* !#t&*,#7n #n#,#*" o $*+#*! !#t&*,#on%! #n#,#*"%! 3 &n*
o $*+#*! !#t&*,#on%! =n*"%!.
L* $/* % (*!*+ % "* !#t&*,#7n #n#,#*" 7 !#t&*,#on%! #n#,#*"%! * "*
!#t&*,#7n =n*" o !#t&*,#on%! =n*"%! %8% % !%+ %!,ono,#* o
-&% no !% (&%* *,,%%+ * %""*! % 4o+)* #n)%#*t*.
D%8% %6#!t#+ %" %!t*nt% -&% -%+* +%!o"$%+"o.
E" %!t*nt% #!(on% % "o! %"%)%nto! n%,%!*+#o! (*+* 8&!,*+
"*! +%"*,#on%! -&% "% (%+)#t*n t+*n!4o+)*+ "* !#t&*,#7n o
!#t&*,#on%! ("*nt%**!.
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A%)5!' !% t#%n% (+%!%nt%' %n (+#)%+ "&*+' %" ,*+5,t%+ *,t#$o %"
%!t*nt% 4+%nt% *" (+o8"%)* 3 !& ,*+5,t%+ +%"*t#$o. E!to! o!
*!(%,to! !on )&3 #)(o+t*nt%! (*+* "* =n*"#* -&% !% (%+!#&%'
3* -&% %!t*8"%,%n "* n%,%!#* % t%n%+ %n ,&%nt* "o!
,ono,#)#%nto! 3 "* n*t&+*"%2* % "* *,t#$#* -&% +%*"#2* %"
%!t*nt%. E! 8&%no *,"*+*+ -&% (*+* (+%!%nt*+ &n (+o8"%)* -&%
+%!&"t% !#n#=,*t#$o (*+* %" %!t*nt%' %8%)o! ,%+,#o+*+no! %
-&% %!t0 * !& *",*n,% %n +%"*,#7n ,on %" n#$%" % ,ono,#)#%nto! 3
*8#"#*%! -&% 0!t% (o!%% 3 -&% !% +%"*,#on% +%*")%nt% ,on !&!
#nt%+%!%!.
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2.= Clasi>caciones del concepto de pro"le!a
En %!t% %(/+*4% !% +%*"#2* &n %!to % "*! (o!#,#on%! *!&)#*! (o+
"o! #4%+%nt%! *&to+%! * "* o+* % %!t*8"%,%+ "*! ,"*!#=,*,#on%! % "o!
(+o8"%)*! )*t%)5t#,o!; (*+* %""o !% %!*++o""* &n* ,"*!#=,*,#7n
!o8+% "* 8*!% % "o! +%,&+!o! %&+/!t#,o! -&% !% -%+*n %!*++o""*+.
Too! "o! *&to+%! (*+%,%n %!t*+ % *,&%+o %n -&% &n %"%)%nto
4&n*)%nt*" (*+* #+##+ %" (+o,%!o % %n!%1*n2*Q*(+%n#2*9% % "*
+%!o"&,#7n % (+o8"%)*!' %! -&% t*nto %" (+o4%!o+ ,o)o %" %!t*nt%
!%(*n %n (+%!%n,#* % -&0 ,"*!% % (+o8"%)*! !% %n,&%nt+*n; %n t*"
#+%,,#7n *n *(*+%,#o %n "o!
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"o! #!t#n&% %n (*+t#,&"*+%! 3 %n%+*"%!. M#%nt+*! -&% %n "o! t+*8*9o!
% B%+to"#* ?@ *(*+%,% &n* ,"*!#=,*,#7n -&% %!t5 )5! *,*8**'
3* -&% !% %n4*t#2* no !7"o %n %" (+o,%!o % !o"&,#7n' !#no -&% *%)5!
(on% *" %!,&8#%+to "* &t#"#2*,#7n % "* "7#,* %nt+o %" (+o,%!o;
("*nt%*no -&%:
P+o8"%)*! C%++*o!: L* !o"&,#7n !% %&,% % 4o+)* "7#,* * (*+t#+
% "* #n4o+)*,#7n -&% *(*+%,% %n %" ("*nt%*)#%nto %" (+o8"%)* 3
-&% +%!&"t* !&=,#%nt% (*+* %n,ont+*+ "* +%!(&%!t* ,o++%,t*. E"
+%!o"&to+ #!(on% % to* "* #n4o+)*,#7n' !7"o n%,%!#t* #nt%+*+"*
*("#,*no "o! +%,&+!o! % "* "7#,*; (o+ %""o !&%"%n ""*)*+!%
(+o8"%)*! % #n4%+%n,#*! "7#,*!W.
P+o8"%)*! *8#%+to!: E" +%!o"&to+ n%,%!#t* #+ )5! *""5 % "* #n4o+)*,#7n
+%,#8#*' &t#"#25no"* % )*n%+* 3o )o#=,*no "o! !#n#=,*o!
*t+#8o! * "o! %"%)%nto! %" (+o8"%)*W ?B%+to"#*' L. @' (. @@@.
J&n ?@ %"*8o+7 &n* ,"*!#=,*,#7n % "o! %9%+,#,#o!' to)*no
,o)o 8*!% %" +*o % *8!t+*,,#7n %n %" +%%9o % "o! %"%)%nto! 3
+%"*,#on%!' *!/ ,o)o %" t#(o % +%%9o -&% !% +%*"#2*. Co)o ,on,%(to
!&(%+#o+ to)7 "o! %9%+,#,#o! )*t%)5t#,o! (+o(&%!to! * "o!
%!t*nt%!' "o! ,&*"%! !% !&8#$#%n %n o! ,on,%(to! !&8o+#n*o!:
%9%+,#,#o! % *("#,*,#7n ?"o! -&% t#%n%n !& o+#%n %n "* (+5,t#,* 3
%9%+,#,#o! ,on!t+o!W ?*-&%""o! -&% !% ,on,#8%n ,on =n%!
#5,t#,o!' o !%*' (*+* %9%+,#t*+' (+o4&n#2*+' *("#,*+' *!%&+*+ "*!
,on#,#on%! (+%$#*!' %nt+% ot+o!. A !& $%2 "o! %9%+,#,#o! ,on!t+o!
!% %!,#n%n %n &n* (*+%9*; (o+ &n* (*+t% *(*+%,%n "o! %9%+,#,#o!4o+)*"%!' %nt+% "o! -&% !% #n,"&3%n: +%!o"$%+ &n* %,&*,#7n' +%!o"$%+
&n !#!t%)* % %,&*,#on%!' %t,0t%+*. Po+ ot+* (*+t% !% #!(on%n % "o!
%9%+,#,#o! ,on t%6to! ,on4o+)*o! (o+ *-&%""o! ,&3o #!,&+!o %!
(&+*)%nt% )*t%)5t#,o' o 8#%n !% +%"*,#on* ,on "* (+5,t#,*.
En +%"*,#7n ,on %!t* (+o(o!#,#7n %" (+o(#o *&to+ !%1*"* -&% "*!
4+ont%+*! %6#!t%nt%! %nt+% "o! #!t#nto! +&(o! !on )o$#8"%!; %n %!t%
!%nt#o' "o! %9%+,#,#o! ,on t%6to! )*t%)5t#,o! 3 "o! % t%6to!
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+%"*,#on*o! ,on "* (+5,t#,* no !on ,on,%(to! %6,"&3%nt%!' 3* -&% "o!
(+#)%+o! !on "* 8*!% % "o! !%&no!' 3 %n *)8o! ,*!o! !% n%,%!#t*
%n,ont+*+ %" )o%"o )*t%)5t#,o (*+* *8o+*+ !& !o"&,#7n. No !%
%8% *!&)#+ % 4o+)* *8!o"&t* "* #%nt#* %nt+% "o! %9%+,#,#o! ,on
t%6to 3 "o! % *("#,*,#7n ,o)o (+o8"%)*!' (&%!to -&% *(*+%,%n %n "*
8#8"#o+*4/* %9%+,#,#o! ,on t%6to ,&3o! o89%t#$o! !on %!*++o""*+ &n*
%t%+)#n** *8#"#* o %" %!*++o""o % &n %t%+)#n*o *"o+#t)o.
E!t5 %n 8o*' ,"*!#=,*+ "o! (+o8"%)*! %nt+o % t+%! +*n%! ,*)(o!'
(o+ %9%)("o "o! *n*"#2*o! (o+ P*"*,#o! 3 Z*)8+*no ?@:
S%
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%)("%o % &n ,#%+to *"o+#t)o (o+ (*+t% %" !o"&,#on*o+. Lo!
(+o8"%)*! *8#%+to!' (o+ %" ,ont+*+#o' #)("#,*n "* %6#!t%n,#* % &n* o
$*+#*! %t*(*! %n !& +%!o"&,#7n' -&% %8%n !%+ *(o+t**! (o+ %"
!o"&,#on*o+ )%#*nt% "* *,,#7n %" (%n!*)#%nto (+o&,t#$o. B*9o
%!t% ,+#t%+#o' "o! (+o8"%)*! ,&*"#t*t#$o! (&%%n !%+ ,on!#%+*o! %n "*
)*3o+/* % "o! ,*!o! ,o)o (+o8"%)*! *8#%+to! 3 "o! ,&*nt#t*t#$o!
,o)o ,%++*o!W ?P*"*,#o!' C. 3 Z*)8+*no' E. @' (. K.
A" *(+%,#*+ %!t* ,"*!#=,*,#7n !% %$*"
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A" no *n*"#2*+!% "*! E,&*,#on%! D#4%+%n,#*"%! %n !& +%"*,#7n ,on "*
4o+)*,#7n o %!*++o""o % +%,&+!o! %&+/!t#,o!' !&,%% -&% %nt+o %
"*! ,"*!#=,*,#on%! %!t**!' no *(*+%,%n "*! t#(o"o/*! %n %!t*
#+%,,#7n' *&n-&% !% %8% +*t#=,*+ -&% !% *n %,o %!4&%+2o! %n %!*
o+#%nt*,#7n %n %" ,*)(o % "* %n!%1*n2* % "* M*t%)5t#,* %n %n%+*"
3 "* +%!o"&,#7n % (+o8"%)*! %n (*+t#,&"*+. S#n %)8*+o' !on )/n#)*!
"*! o,*!#on%! %n -&% "o! (+o4%!o+%! !#t*(+%n#2*9% % "*
)*t%)5t#,*' %n (*+t#,&"*+ "o! +%"*,#on*o! ,on "* %&+/!t#,*' *3 -&%t%n%+ (+%!%nt% -&% "o! ,ont%n#o! * t+*t*+ %8%n $%+!% %n o!
#+%,,#on%!: ,o)o o89%to % *(+o(#*,#7n ?*(+%n%+ E,&*,#on%!
D#4%+%n,#*"%! 3 ,o)o 8*!% (*+* %" %!*++o""o % "* %&+/!t#,*. Lo!
%&,*o+%! )*t%)5t#,o! *n (&%!to *" %!,&8#%+to -&% "* +%!o"&,#7n
% (+o8"%)*! %8% !%+ %" o89%t#$o #n!t+&,t#$o 4&n*)%nt*" %n "*
%n!%1*n2* % "*! )*t%)5t#,*!' *&n-&% no "* ' !% *n
(*&t*! (*+* t+*t*+ %t%+)#n*o! (+o8"%)*! ,on %" o89%t#$o %
%!*++o""*+ #4%+%nt%! *!(%,to! +%"*,#on*o! ,on "* %&+/!t#,*.
L*! #%*! %!*++o""**! !#+$#%+on % 8*!% o !&!t%nto (*+* ,"*!#=,*+ "*!
#4%+%nt%! t#(o! % E,&*,#on%! D#4%+%n,#*"%! 3 %"#)#t*+ !&!
,*+*,t%+/!t#,*!. E!t* ,"*!#=,*,#7n %n,#%++* t+%! +*n%! +&(o! %
(+o8"%)*! 3 %nt+% %""o! 4o+)*n &n !#!t%)* (*+* %" %!*++o""o % "*
%&+/!t#,*' *!&)#%no ,o)o &n !#!t%)* &n ,on9&nto % %"%)%nto!
,&3* #nt%+*,,#7n (+o&,% "* *(*+#,#7n % n&%$*! ,&*"#*%!
41
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#nt%+*t#$*!' no #n%+%nt%! * "o! ,o)(on%nt%! *#!"*o! -&%
,on!t#t&3%n %" )#!)o.W ?A4*n*!#%$' V. @' (.@. Con %!t* (o!#,#7n
!% t#%n%n %n ,&%nt* t+%! *!(%,to! -&% !on #n#!(%n!*8"%! *" ,on,%8#+
&n !#!t%)*: %"%)%nto! -&% "o #nt%+*n' +%"*,#7n -&% !% %!t*8"%,%
%nt+% "o! %"%)%nto! %" !#!t%)* 3 +%!&"t*o -&% !% %n%+* % %!t*
+%"*,#7n.
E" o89%t#$o % "* ,"*!#=,*,#7n %!t5 *o' %n "o 4&n*)%nt*"' %n
o+*n#2*+ "* *,t#$#* 3 %t%+)#n*+ %" %4%,to -&% (+o$o,* %" !#!t%)*
% (+o8"%)*! %n %" %!*++o""o % &n %t%+)#n*o (+o,%#)#%nto
%&+/!t#,o 3 $#,%$%+!*. E!t* #$#!#7n (%+)#t#+5 +%$%"*+ "* %!t+&,t&+*
#nt%+n* % "o! (+o8"%)*! 3' *%)5!' *3&*+5 * "o! (+o4%!o+%! *%!t+&,t&+*+"o! %n 4&n,#7n % ,** &no % "o! %"%)%nto! * %!*++o""*+'
% 4o+)* t*" -&% !% *",*n,% &n (+o,%!o 7(t#)o % %n!%1*n2*Q
*(+%n#2*9% %n "* +%!o"&,#7n % (+o8"%)*! (*+* 4*$o+%,%+ "* 4o+)*,#7n
% +%,&+!o %&+/!t#,o %n "o! %!t*nt%!.
A!/ (&%!' "* ,"*!#=,*,#7n -&% no!ot+o! n%,%!#t*)o! %8% % %!t*+
%n,*)#n** *" +%,&+!o %&+/!t#,o -&% !% -%+* &t#"#2*+' *%)5! %8%
% ,ont%n%+ %n !& tot*"#* to*! "*! (o!#8"%! !#t&*,#on%! (+o8"%)*
-&% !% (&%*n (+%!%nt*+ %n &n ,&+!o % E,&*,#on%! D#4%+%n,#*"%!. L*
,"*!#=,*,#7n -&% (+o(on%)o! %! "* !#%nt%:
@. P+o8"%)*! %!t+&,t&+*"%!. E$#%n,#*o! %n S#*++%t*'R&%!* 3 Ro+/&%2 ? !on *-&%""o! ,&3* !#)#"#t& %,&%+(o 3 4o+)*' *!/ ,o)o % !o"&,#7n no! (%+)#t*n*("#,*+ *"&n* % "*! *n*"o/*! !#%nt%!:
* An*"o/* % %!t+&,t&+*. Son *-&%""o! (+o8"%)*!on% !% %$#%n,#* !#)#"#t& % ,&%+(o 3 4o+)*' 3 *"*("#,*+ &n* *n*"o/* (o%)o! ""%$*+ n&%!t+o!(+o8"%)*! * !#t&*,#on%! on% !% (&%*n%$#%n,#*+ &n* !o"&,#7n %n 8*!% * &n (+o8"%)* -&%3* !% ,ono,%.
8 An*"o/* % !o"&,#7n. En %!t% +&8+o' !# &n (+o8"%)*t#%n% !%)%9*n2*! not*8"%! %n !& (+o,%!o %!o"&,#7n %nton,%! !% %!t*+/* %n (o!#,#7n % %"*8o+*+&n* %!t+*t%#* -&% t% ""%$% * "* !o"&,#7n %!%**.
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. P+o8"%)*! R%&,t#$o!. Son *-&%""o! -&% t#%n%n !o"&,#7n%n (+o8"%)*! (o!#8"%)%nt% % #!t#nto ,*+5,t%+ o#)%n!#7n. En on% *("#,*)o! "o! !#%nt%! (+#n,#(#o!:
* P+#n,#(#o % T+*n!4o+)*,#7n. A-&%""o! (+o8"%)*!
-&% !% (&%%n ""%$*+ * ot+* #)%n!#7n )%#*nt% "*t+*n!4o+)*,#7n % *"&no o too! !&! ,o)(on%nt%!'t%n#%no "* !%&+#* -&% 5no"% +%!(&%!t* *"n&%$o (+o8"%)* !% "% *+5 t*)8#0n *" *nt%+#o+.
8 P+#n,#(#o % +%&,,#7n. Son *-&%""o! (+o8"%)*! -&%!% ""%$*n * &n +&8+o #!t#nto on% %" *,%+ &n**("#,*,#7n +%,&+!#$* o #n)%#*t* !% "% %n,ont+*+5 "*!o"&,#7n *" (+o8"%)* o+##n*".
L*! ,*+*,t%+/!t#,*! % "o! (+o8"%)*! (o+ *n*"o/* %!t+&,t&+*"
&t#"#2*o! !on:
P+o8"%)*! ,on *to! #n!&=,#%nt%!. Don% %" %!t*nt% !% $%
o8"#*o * 8&!,*+ (+o8"%)*! !%)%9*nt%! on% !% %$#%n,#%
,&*" %! %" *to 4*"t*nt%.
P+o8"%)*! ,on *to! !&(%+&o!. A-&/ %" %!t*nt% t%n+5 -&%
*(+%,#*+ ot+o! (+o8"%)*! (*+* *,"*+*+ "o! *to! -&% "% !on
*o!.
P+o8"%)*! on% *(*+%2,*n *to! ,ont+*#,to+#o!. D% #&*"
4o+)*' *" no (o%+ %n,ont+*+ $/*! % !o"&,#7n o(t*+5 (o+ 8&!,*+
(+o8"%)*! !%)%9*nt%! % ,&%+(o 3 4o+)*' 3 *!/ %)#t#+ 9,#o! %
$*"o+ !o8+% "o! *to! %" (+o8"%)*. In,"&!o (o+/* *n*"#2*+
,&5"%! %8%+/*n !%+ "*! ,*+*,t%+/!t#,*! % "o! *to! (*+* -&%(&%* t%n%+ &n* !o"&,#7n.
P+%!%nt*,#7n % !o"&,#on%! % (+o8"%)*! ,on %++o+%! !&t#"%!W o
#4/,#"%! % %t%,t*+ * (+#o+#. E!to ""%$*+5 %$%nt&*")%nt% *"
%!t*nt% * %,*+ $#!t*2o! * %9%+,#,#o! !#)#"*+%!.
L*! ,*+*,t%+/!t#,*! % "o! (+o8"%)*! (o+ *n*"o/* % !o"&,#7n
&t#"#2*o! !on:
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P+o8"%)*! %n "o! -&% *o &n ,on9&nto % (+%)#!*! !% (#*
o8t%n%+ &n ,on9&nto % t%!#!. S# %" %!t*nt% n%,%!#t* +%*"#2*+
,on,"&!#on%! % &n (+o8"%)* *" *,%+ ot+o (&%% *("#,*+ %"
)#!)o (+o,%!o -&% &t#"#27 (*+* +%!o"$%+ %" (+#)%+o.
P+%!%nt*,#7n % !o"&,#on%! % (+o8"%)*!' ,on &n* $/* %
!o"&,#7n ,o++%,t* 3 ot+* #n,o++%,t*. E" *n*"#2*+ -&0
(+o,%#)#%nto! (&%%n ""%$*+ o no * "* !o"&,#7n %" (+o8"%)*
,+%*+5 &n* 8*!% % #,o! (+o,%#)#%nto! (*+* !& 4&t&+*
#)("%)%nt*,#7n %n #$%+!o! %9%+,#,#o!.
P+%!%nt*,#7n % !o"&,#on%! #4%+%nt%! * &n )#!)o (+o8"%)*. E"(+%!%nt*+ #$%+!*! !o"&,#on%! *t+*%+5 *" %!t*nt% * #%nt#=,*+
(+o,%#)#%nto! )5! !%n,#""o! o )5! ,o)("#,*o! 3 *!/ %,##+
!& %8#* *("#,*,#7n %n ot+o! (+o8"%)*!.
L*! ,*+*,t%+/!t#,*! % "o! (+o8"%)*! (o+ (+#n,#(#o % t+*n!4o+)*,#7n
&t#"#2*o! !on:
P+o8"%)*! -&% no %6#9*n ,5",&"o (*+* %n,ont+*+ !& !o"&,#7n'(%+o +%-%+*n % &n* !7"#* 4&n*)%nt*,#7n t%7+#,*. E" t%n%+
&n* 4&n*)%nt*,#7n t%7+#,* % "o! *to! n%,%!*+#o! (*+*
+%!o"$%+ %" (+o8"%)* ""%$*+5 *" %!t*nt% * 8&!,*+ !& !o"&,#7n
%n !#t&*,#on%! % %-$*"%n,#*.
P+o8"%)*! ,&3* +%!(&%!t* +%-%+* &n *n5"#!#! ,o)("%)%nt*+#o.
E" t%n%+ -&% %4%,t&*+ *n5"#!#! % "o! *to! 8&!,*o! 3 "o!
%n,ont+*o! %$#%n,#*+5 "*! +%"*,#on%! -&% %6#!t%n %nt+% "o!
*to! 3 *-&%""o! -&% !% &t#"#2*+on (*+* !& !o"&,#7n' *!/ (&%!'
o8t%n+5 &n )*+,o % ,ono,#)#%nto! -&% !% (&%%n &t#"#2*+
(*+* "* !o"&,#7n % ot+o! (+o8"%)*!.
L*! ,*+*,t%+/!t#,*! % "o! (+o8"%)*! (o+ (+#n,#(#o % +%&,,#7n
&t#"#2*o! !on:
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P+o8"%)*! %n "o! ,&*"%! !& +%!o"&,#7n +%-%+* % "* +%!o"&,#7n
% !&8(+o8"%)*!. E" &!*+ !&8>(+o8"%)*! ,o)o !o"&,#7n %"
o+##n*" %! ,"*+*)%nt% "* *("#,*,#7n % &n* +%&,,#7n %"
(+o8"%)* * ot+o! 3* ,ono,#o!.
P+o8"%)*! -&% %6#%n (*+* !& !o"&,#7n "* 8
#n4o+)*,#7n -&% 0" no (o!%%. E" 8&!,*+ #n4o+)*,#7n -&% no !%
t#%n% % 4*,to (&%% ""%$*+ * %n,ont+*+ "* !o"&,#7n %n "*
+%&,,#7n * ,ono,#)#%nto! n&%$o!' o * ,ono,#)#%nto! -&% no
%+*n %$#%nt%! %n &n (+#n,#(#o.
P+o8"%)*! -&%' *o %" n#$%" % ,ono,#)#%nto %" %!t*nt%' "%!%* ,7)oo o8t%n%+ )
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H*,%)o! y=ux ' %nton,%! y'=u'x+u . S&!t#t)o! %n "* %,&*,#7n
o+##n*" y % y ' .
x (u'x+u)=x2(ux )2+ux
u'x
2+ux=x 1u2+ux
du
dxx=1u2
du
1u2=dx
x
Enton,%! arcsen (u )=Cx
S&!t#t)o! u=y
x ' 3 no! -&%*arcsen(yx )=Cx ' (o+ "o t*nto
y=x sin(|Cx|)
En %" n&%$o (+o8"%)*' !# t+*t*)o! % %n,ont+*+"% &n* %!t+&,t&+*
!#)#"*+ *,#%no "o !#%nt%:
(x y 'y )2=y2x2
x y'y=y2x2
x y'=y2x2+y
T#%n% !%)%9*n2* % %!t+&,t&+* 3 (o%)o! +%!o"$%+"o! ,o)o !%
+%!o"$#7 %" *nt%+#o+ N7t%!% -&% !% %!t5 *("#,*no &n +%,&+!o %&+/!t#,o
% +%&,,#7n * &n (+o8"%)* 3* ,ono,#o' !#n %)8*+o' "* 8
%!t5 #+##* *,#* &no -&% t%n* !%)%9*n2* % %!t+&,t&+*. A-&/ "*
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*n*n,#* (+#n,#(*" no %! "* &t#"#2*,#7n % &n )0too (*+* +%!o"$%+
%,&*,#on%!' !#no "* 8
.>T+*8*9*no %nt+o %" t%)* % E,&*,#on%! % B%+no&""#' &n
(+o8"%)* (o+ *n*"o/* % !o"&,#7n (o+/* !%+ %" !#%nt%:
x y'+y=x2 sinx
y'+
y
x=xsinx
H*,%)o!
y=u (x ) v (x)
Enton,%! y'=u '(x ) v (x )+u (x) v '(x)
!&!t#t)o! %n "* %,&*,#7n.
u'v+uv '+
uv
x=xsinx
F*,to+#2*)o! $.
v (u '+u
x)+uv '=xsinx
B&!,*)o! "* !o"&,#7n (*+t#,&"*+
(*+*:
u'+
u
x=0
du
dx=ux
u=1
x
x y'+y=x2 sinx
x y'+y=x2 sinx
H*,%)o! y=u (x ) v (x)
Enton,%! y'=u '(x ) v (x )+u (x) v '(x)
!&!t#t)o! %n "* %,&*,#7n.
u'v+uv '+
uv
x=xsinx
F*,to+#2*)o! &.
u'v+u(v '+
v
x)=xsinx
B&!,*)o! "* !o"&,#7n (*+t#,&"*+
(*+*:
v '+ vx=0
dv
dx=vx
v=1
x
S&!t#t)o! $ %n %" +%!to % "*
47
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S&!t#t)o! & %n %" +%!to % "*
%,&*,#7n:
uv'
=x
sinx
1
xv
'=xsinx
v'=x2 sinx
dv=x2 sinx dx
v=x2 cosx+1
2sin2x
2x
x2 cosx+1
2sin
y=uv=
y=uv=xcosx+ 1
2xsin 2x
%,&*,#7n:
u ' v=xsinx
1
xu
'=xsinx
u'=x2sinx
du=x2 sinxdx
u=x2cosx+1
2sin2x
2x
x2 cosx+1
2sin
y=uv=
y=uv=xcosx+ 1
2xsin 2x
D% %!t* 4o+)* -&%* ,"*+o ,7)o !% +%!&%"$% %" (+o8"%)* % #!t#nto!
5n&"o! 3 !% (&%% *n*"#2*+ * )*3o+ (+o4&n#* %" )0too %
!o"&,#7n.
R%!&"t* *t#n*o ("*nt%*+ -&% %" !#!t%)* % (+o8"%)*!' ,&*"%!-%+*
!%*n "*! ,*+*,t%+/!t#,*! -&% (o!%* (*+* #n,##+ %n ,&*"-%+
#)%n!#7n' %8% %!t*+ 8#%n o+*n#2*o 3 (%+)#t#+"% *" %!t*nt%
t%n%+ %n &n (+#)%+ n#$%" (o!#8#"#*%! % 06#to. L* &t#"#2*,#7n %"
,ont%n#o % "* #n4o+)*,#7n -&% !% t+*t% %n %" t%6to %" (+o8"%)* %!
% !&)* #)(o+t*n,#*; (%n!*)o! -&% %n %!t% !%nt#o "* (+%!%nt*,#7n
3 %!t+&,t&+* %" (+o8"%)* !on "*! $/*! 4&n*)%nt*"%! (*+* "o+*+ %"
48
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%nt%n#)#%nto % "o! (+o,%!o! %&+/!t#,o! #)("#,*o! %n "* !o"&,#7n
% "o! (+o8"%)*!.
49
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2.6 strategia para la solucin de pro"le!as
T%n#%no %n ,&%nt* %" #)(*,to -&% (+o&9o %n %" )&no % "*
%n!%1*n2* % "* )*t%)5t#,* "o! t+*8*9o! %" (+o4%!o+ Po"3*' %n
%!(%,#*" * "* %n!%1*n2*Q*(+%n#2*9% % "* +%!o"&,#7n % (+o8"%)*!'
%! (o,o ,on=*8"% &n* #n$%!t#*,#7n !o8+% %" t%)* -&% %"&* !&!
*(o+t%! * %!t* )*t%+#*. Po"3* ?@ %!t*8"%,%: ... !% %nt%n%+5 -&%
+%!o"$%+ &n (+o8"%)* %! %n,ont+*+ &n ,*)#no *""/ on% no !% ,ono,/*
,*)#no *"&no' %n,ont+*+ "* 4o+)* % !*"#+ % &n* #=,&"t*' %
!o+t%*+ &n o8!t5,&"o' ,on!%+ %" =n %!%*o -&% no %! ,on!%"%
% 4o+)* #n)%#*t* &t#"#2*no "o! )%#o! *%,&*o!.W ?Po"3*' G.
@@' (. @
Un* % "*! t%n%n,#*! *,t&*"%! (*+* *n*"#2*+ "* +%!o"&,#7n %
(+o8"%)*!' %! "* -&% (%+)#t% %!t*+ "* +%!o"&,#7n % (+o8"%)*!
,o)o &n (+o,%!o; %nt+o % %!t* (o!#,#7n *(*+%,%n "o! t+*8*9o! %
C*+"%! 3 L%!t%+ ?@' %n "o! -&% !% %6(+%!* -&% %! %" (+o,%!o %
,oo+#n*,#7n % "* %6(%+#%n,#* (+%$#*' ,ono,#)#%nto! % #nt,#7n' 3 &n
#nt%nto % %t%+)#n*+ &n )0too (*+* +%!o"$%+ &n* !#t&*,#7n ,&3o+%!&"t*o no! %! %!,ono,#o.W?C*+"%!' R. *n L%!t%+' F. @' (..
T*)8#0n %n "o! t+*8*9o! % C*++%+*! ?@' S#*++%t* 3 P*"*,#o ?
!% %!t* "* +%!o"&,#7n % (+o8"%)*! ,o)o &n (+o,%!o ,on!t#to
(o+ too! "o! %"%)%nto! -&% &t#"#2* %" +%!o"&to+ (*+* o8t%n%+ !&
!o"&,#7n. En %!t% ,*!o !% ,o)(+%n% "* +%!o"&,#7n % (+o8"%)*! ,o)o
&n (+o,%!o -&% ,o)#%n2* (*+* %" %!t*nt% %!% %" )o)%nto %n
-&% !% "% (+%!%nt* %" (+o8"%)* 3 -&% ""%$* ,on!#o *" ,on9&nto %
*,,#on%! 3 o(%+*,#on%! -&% !% %!*++o""*n *!t* -&% "o !o"&,#on*' 3
$*"o+* "* +%!(&%!t* %n,ont+**. S# *n*"#2*)o! "* ,*+*,t%+#2*,#7n %
(+o8"%)*! *!&)#* (o+ "* #n$%!t#*,#7n' +%!&"t* n*t&+*" *,%(t*+ -&%
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L* !o"&,#7n % &n (+o8"%)* no %8% $%+!% ,o)o &n )o)%nto =n*"'
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ACCIONES
A(+o6#)*,#7n