lee - tesis.docx

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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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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

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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:

34


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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%**!.

35


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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%+%!%!.

36


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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!

38


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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%


41/158

%)("%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 !% %$*"


42/158

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 %!%**.

42


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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:

43


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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:

44


46/158

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%+ )


47/158

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-&/ "*

46


48/158

*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


49/158

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


50/158

%nt%n#)#%nto % "o! (+o,%!o! %&+/!t#,o! #)("#,*o! %n "* !o"&,#7n

% "o! (+o8"%)*!.

49


51/158

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*+ -&%

+%!o"$%+ &n (+o8"%)* %! *+"% !o"&,#7n * "* ,ont+*#,,#7n %6#!t%nt%

%nt+% %" %!t*o *,t&*" 3 %" %!%*o %" o89%to.

En %!t* )#!)* #+%,,#7n L*8*++%+% ?@ *!%$%+* -&%:

50


52/158

L* !o"&,#7n % &n (+o8"%)* no %8% $%+!% ,o)o &n )o)%nto =n*"'

!#no ,o)o too &n ,o)("%9o (+o,%!o % 8

*$*n,%! 3 +%t+o,%!o! %n %" t+*8*9o )%nt*". E!t% ,o)("%9o (+o,%!o %

t+*8*9o )%nt*" !% )*t%+#*"#2* %n %" *n5"#!#! % "* !#t&*,#7n *nt% "* ,&*"

&no !% *""*: %n "* %"*8o+*,#7n % #(7t%!#! 3 "* 4o+)&"*,#7n %

,on9%t&+*!; %n %" %!,&8+#)#%nto 3 !%"%,,#7n % (o!#8#"#*%!; %n "*

(+%$#!#7n 3 (&%!t* %n (+5,t#,* % (+o,%#)#%nto! % !o"&,#7n.W

?L*8*++%+%' A. F. @' (. .

Un %"%)%nto #)(o+t*nt% * t%n%+ %n ,&%nt* %! no %nt%n%+ "* !o"&,#7n

% &n (+o8"%)* ,o)o &n +%!&"t*o. L* %!t+*t%#* t#%n% -&% #+##+!%

*,#* %" *(+o$%,*)#%nto %" (ot%n,#*" -&% 8+#n* %!t% (+o,%!o' (*+*-&% %n !& ,&+!o !% (&%* #n,##+ %n "o! (+o,%!o! )%t*,on#t#$o! !#n

)*+#n*+ %" %!*++o""o %" (%n!*)#%nto "7#,o %" %!t*nt%. D% %!to

!% %&,% -&%' *" *n*"#2*+ "*! (o!#,#on%! %6(&%!t*!' (+%t%n% %n "o

4&n*)%nt*" Q%nt+o %" -&%*,%+ #5,t#,o >' ,*)8#*+ "* (%+,%(,#7n

3 *,t&*,#7n %" %!t*nt%; *%)5! !&(on% )o#=,*+ "o! !#n#=,*o!

*t+#8o! (+%$#*)%nt% * "o! %"%)%nto! %" (+o8"%)*' 3o &t#"#2*+ !&!

,ono,#)#%nto % )*n%+* #4%+%nt%' o % 4o+)* -&% no %!t5*,o!t&)8+*o' %!' % 4o+)* #n*8#t&*" (*+* 0"; * "o -&% !% *+%*'

(o+


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ACCIONES

A(+o6#)*,#7n

lee - tesis.docx

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