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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 259 10 "PR\301CTICA V" }}{PARA 
274 "" 0 "" {TEXT -1 25 "TEMA 6 Y TEMA 7. C\301LCULO." }}{PARA 263 "" 
0 "" {TEXT -1 20 "MATEM\301TICA APLICADA." }}{PARA 275 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "OBJETIVOS" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 67 "Esta pr\341ctica tiene como objetivo apre
nder a utilizar MAPLE V para " }{TEXT 256 79 "evaluar polinomios inter
poladores de Lagrange y f\363rmulas de derivaci\363n num\351rica" }
{TEXT -1 1 "." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "FORMA DE DESAR
ROLLAR ESTA PR\301CTICA" }}{PARA 0 "" 0 "" {TEXT -1 247 "El desarrollo
 de la pr\341ctica consistir\341 en la realizaci\363n de ejemplos resu
eltos (ep\355grafe de EJEMPLOS) y tras ello se propone al alumno el de
sarrollo de algunos ejercicios (ep\355grafe de EJERCICIOS PROPUESTOS) \+
que deber\341 desarrollar individualmente." }}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 34 "DURACI\323N ESTIMADA DE ESTA PR\301CTICA" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 71 "El tiempo estimado para la realizaci\363n
 de esta pr\341ctica es de 2 horas. " }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 8 "EJEMPLOS" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Interpola
ci\363n polin\363mica de Lagrange" }}{EXCHG {PARA 259 "" 0 "" {TEXT 
-1 37 "Interpolaci\363n polin\363mica de Lagrange." }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 91 "Consideramos el soporte [2,3,4] y una funci\363n \+
que toma en el soporte los valores [8,27,64]." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "
Entrada de los datos conocidos." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 5 "n:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x[0]:=2; x[1]:=3; x[2]:=4; \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!\"\"#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"xG6#\"\"#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "y[0]:=8; y[1]:=27; y[2]:=64;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"yG6#\"\"!\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>&%\"yG6#\"\"\"\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"#
\"#k" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 46 "Utilizando los polinomi
os de base de Lagrange." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 46 "C\341
lculo de los polinomios de base de Lagrange." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }{XPPEDIT 18 0 "L[i] := Product((x-x[j])/(x[i]-x[j]),j
 = 0 .. n)" "6#>&%\"LG6#%\"iG-%(ProductG6$*&,&%\"xG\"\"\"&F-6#%\"jG!\"
\"F.,&&F-6#F'F.&F-6#F1F2F2/F1;\"\"!%\"nG" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{XPPEDIT 18 0 "j <> i" "6#0%\"jG%\"iG" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "L[i]:=Product(LL[
j]/DLL[j],j=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"LG6#%\"iG-%
(ProductG6$*&&%#LLG6#%\"jG\"\"\"&%$DLLGF.!\"\"/F/;\"\"!\"\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "En los c\341lculos siguientes, el \+
caso j=i no se considera gracias a la multiplicaci\363n por 1. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "i:=0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"iG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
263 "for j from 0 by 1 to n do                                        \+
                                if (j=i) then LL[j]:=1; fi;           \+
            if (j<>i) then LL[j]:=x-x[j]; fi;                         \+
                                                 LL[j] od;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&
%\"xG\"\"\"!\"$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"!\"%
F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "for j from 0 by 1 to
 n do                                                                 \+
       if (j=i) then DLL[j]:=1; fi;                       if (j<>i) th
en DLL[j]:=x[i]-x[j]; fi;                                             \+
                             DLL[j] od;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 
"L[i]:=product(LL[jk]/DLL[jk],jk=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%\"LG6#\"\"!*&,&%\"xG!\"\"\"\"$\"\"\"F-,&F*#F+\"\"#F0F-F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "i:=1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"iG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
263 "for j from 0 by 1 to n do                                        \+
                                if (j=i) then LL[j]:=1; fi;           \+
            if (j<>i) then LL[j]:=x-x[j]; fi;                         \+
                                                 LL[j] od;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"!\"#F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"
!\"%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "for j from 0 by \+
1 to n do                                                             \+
           if (j=i) then DLL[j]:=1; fi;                       if (j<>i
) then DLL[j]:=x[i]-x[j]; fi;                                         \+
                                 DLL[j] od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "L[i]:=product(LL[jk]/DLL[jk],jk=0..n);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"LG6#\"\"\"*&,&%\"xGF'!\"#F'F',&F*!\"\"\"\"%
F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "i:=2;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"iG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 263 "for j from 0 by 1 to n do                           \+
                                             if (j=i) then LL[j]:=1; f
i;                       if (j<>i) then LL[j]:=x-x[j]; fi;            \+
                                                              LL[j] od
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"!\"#F%" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"!\"$F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "f
or j from 0 by 1 to n do                                              \+
                          if (j=i) then DLL[j]:=1; fi;                \+
       if (j<>i) then DLL[j]:=x[i]-x[j]; fi;                          \+
                                                DLL[j] od;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 38 "L[i]:=product(LL[jk]/DLL[jk],jk=0..n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"LG6#\"\"#*&,&%\"xG#\"\"\"F'!\"\"F
,F,,&F*F,!\"$F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Se trata aho
ra de hacer lo mismo pero con un verdadero bucle sobre i." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 695 "for i from 0 by 1 to n do         \+
                for j from 0 by 1 to n do                             \+
                                           if (j=i) then LL[j]:=1; fi;
                       if (j<>i) then LL[j]:=x-x[j]; fi;              \+
                                                            LL[j] od; \+
                                       for j from 0 by 1 to n do      \+
                                                                  if (
j=i) then DLL[j]:=1; fi;                       if (j<>i) then DLL[j]:=
x[i]-x[j]; fi;                                                        \+
                  DLL[j] od;             L[i]:=product(LL[jk]/DLL[jk],
jk=0..n);                   od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
\"LG6#\"\"!*&,&%\"xG!\"\"\"\"$\"\"\"F-,&F*#F+\"\"#F0F-F-" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"LG6#\"\"\"*&,&%\"xGF'!\"#F'F',&F*!\"\"\"\"%
F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"LG6#\"\"#*&,&%\"xG#\"\"\"
F'!\"\"F,F,,&F*F,!\"$F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Repr
esentaci\363n gr\341fica de los polinomios de base de Lagrange: " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dibl0:=plot(L[0],x=x[0]..x[n
],color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "point0:=po
intplot([x[0],1],color=red,symbol=diamond):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 43 "dibl1:=plot(L[1],x=x[0]..x[n],color=green):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "point1:=pointplot([x[1],1],c
olor=green,symbol=diamond):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "dibl2:=plot(L[2],x=x[0]..x[n],color=blue):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 54 "point2:=pointplot([x[2],1],color=blue,symbol=d
iamond):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "display(dibl0,point0,dibl1
,point1,dibl2,point2,title=`Polinomios de base de Lagrange`);" }}
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[$*oT<\"F37$Fep$!1-H^BJ]36F37$Fjp$!12,7.4)p.\"F37$F_q$!1weXlFSa$*F]r7$
Fdq$!17z;E1+\"H)F]r7$Fiq$!13`VlTXYpF]r7$F_r$!1(3D2;#=$\\&F]r7$Fdr$!1h%
*zP)*)pz$F]r7$Fir$!1^Tq&*Rdx?F]r7$F^s$!1Bzz>K-J\\Fbs7$Fds$\"1C'>6e(H[A
F]r7$Fis$\"1XG'zhpmS%F]r7$F^t$\"1ki*oenK!pF]r7$Fct$\"1S_vN'))Hm*F]r7$F
ht$\"1Pa<`S.a7F37$F]u$\"1r+5oI9\\:F37$Fbu$\"1>I2^g1'*=F37$Fgu$\"1op(>+
u]A#F37$F\\v$\"1E]kbzT%f#F37$Fav$\"10X#\\L/_%HF37$Ffv$\"1^!HQ=GiM$F37$
F[w$\"1y%HZws-u$F37$F`w$\"1U+7fR[pTF37$Few$\"1CPI!pKig%F37$Fjw$\"1byUs
1g\"3&F37$F_x$\"1KT$p5rpb&F37$Fdx$\"1G$yW****31'F37$Fix$\"1Lt*\\RE%ylF
37$F^y$\"1k&o$e6epqF37$Fcy$\"1i#oBnw3l(F37$Fhy$\"1g_D%)GU(=)F37$F]z$\"
1k)\\+h)yw()F37$Fbz$\"14')Q(30wN*F37$FgzF+-Fjz6&F\\[lF*F*F][l-Fa[l6%F_
^mF`^mFc[l-%+AXESLABELSG6$Q\"x6\"%!G-%&TITLEG6#%?Polinomios~de~base~de
~LagrangeG-%%VIEWG6$;F(Fgz%(DEFAULTG" 2 564 615 615 2 0 1 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Podemos com
probar que se verifican las 2 propiedades de los polinomios de base de
 Lagrange: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "El polinomio de ba
se de Lagrange L0 toma el valor 1 en x=2 y 0 en x=3 y x=4 ; " }}{PARA 
0 "" 0 "" {TEXT -1 78 "El polinomio de base de Lagrange L1 toma el val
or 1 en x=3 y 0 en x=2 y x=4 ; " }}{PARA 0 "" 0 "" {TEXT -1 78 "El pol
inomio de base de Lagrange L2 toma el valor 1 en x=4 y 0 en x=2 y x=3 \+
; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "En cualquier punto x, la su
ma de los polinomios de base de Lagrange vale 1 ; por ejemplo en 2'75 \+
: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Sum(L[k],k=0..2)=subs
(x=2.75,L[0])+subs(x=2.75,L[1])+subs(x=2.75,L[2]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%$SumG6$&%\"LG6#%\"kG/F*;\"\"!\"\"#$\"+++++5!\"*" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "O en 3'28 : " }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 71 "Sum(L[k],k=0..2)=subs(x=3.28,L[0])+subs(x=3
.28,L[1])+subs(x=3.28,L[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$Su
mG6$&%\"LG6#%\"kG/F*;\"\"!\"\"#$\"+++++5!\"*" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 257 47 "C\341lculo del polinomio interpolador de Lagrange." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "p[n] := Sum(f[i]
*L[i],i = 0 .. n);" "6#>&%\"pG6#%\"nG-%$SumG6$*&&%\"fG6#%\"iG\"\"\"&%
\"LG6#F/F0/F/;\"\"!F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Sum(y[ik]*L[ik],ik=0..n);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"yG6#%#ikG\"\"\"&%\"LGF)F+/F*;
\"\"!\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "sum(y[ik]*L[i
k],ik=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&%\"xG!\"\"\"\"$
\"\"\"F),&F&#F'\"\"#F,F)F)\"\")*&,&F&F)!\"#F)F),&F&F'\"\"%F)F)\"#F*&,&
F&#F)F,F'F)F),&F&F)!\"$F)F)\"#k" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "p:=expand(sum(y[ik]*L[ik],ik=0..n));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"pG,(*$)%\"xG\"\"#\"\"\"\"\"*F(!#E\"#C\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Sabiendo que la funci\363n es f(x
)=x^3, calculamos el error real y la estimaci\363n del error de interp
olaci\363n." }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 28 "Error real de in
terpolaci\363n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "err:=f(x
)-p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$errG,*-%\"fG6#%\"xG\"\"\"*$
)F)\"\"#\"\"\"!\"*F)\"#E!#CF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "
Sabiendo que la funci\363n es: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "f:=x->x^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6
\"6$%)operatorG%&arrowGF(*$)9$\"\"$\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "err:=f(x)-p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$errG,**$)%\"xG\"\"$\"\"\"\"\"\"*$)F(\"\"#F*!\"*F(\"#E!#CF+" }}}
{EXCHG {PARA 261 "" 0 "" {TEXT -1 38 "Estimaci\363n del error de inter
polaci\363n." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "
E := Product(x-x[i],i = 0 .. n)*`@@`(D,n+1)(f)/(n+1)!;" "6#>%\"EG*(-%(
ProductG6$,&%\"xG\"\"\"&F*6#%\"iG!\"\"/F.;\"\"!%\"nGF+--%#@@G6$%\"DG,&
F3F+\"\"\"F+6#%\"fGF+-%*factorialG6#,&F3F+\"\"\"F+F/" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 53 "Calculamos la derivada n+1 de la funci\363n en \+
cuesti\363n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "der:=(D@@(n
+1))(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$derG\"\"'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 50 "Calculamos el producto que interviene en \+
el error." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "prod:=1:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "for ii from 0 by 1 to n do  \+
    prod:=prod*(x-x[ii])                             od;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%prodG,&%\"xG\"\"\"!\"#F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%prodG*&,&%\"xG\"\"\"!\"#F(F(,&F'F(!\"$F(F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%prodG*(,&%\"xG\"\"\"!\"#F(F(,&F'F(!\"$F(F
(,&F'F(!\"%F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Calculamos la \+
expresi\363n del error." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "
ER:=prod*der/((n+1)!);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ERG*(,&%
\"xG\"\"\"!\"#F(F(,&F'F(!\"$F(F(,&F'F(!\"%F(F(" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 52 "Evaluamos ahora su cota en el intervalo considerado.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "EVR:=abs(maximize(prod,
x,x[0]..x[n]))*abs(maximize(der,x,x[0]..x[n]))/((n+1)!);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$EVRG,$*$-%%sqrtG6#\"\"$\"\"\"#\"\"#\"\"*" }}}
{EXCHG {PARA 265 "" 0 "" {TEXT -1 32 "Utilizando la f\363rmula de Newt
on." }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 18 "F\363rmula de Newton." }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "p[n] := f[0]+Su
m(f[[x[0] .. x[i]]],i = 1 .. n)*Product(x-x[j],j = 0 .. i-1);" "6#>&%
\"pG6#%\"nG,&&%\"fG6#\"\"!\"\"\"*&-%$SumG6$&F*6#7#;&%\"xG6#F,&F76#%\"i
G/F;;\"\"\"F'F--%(ProductG6$,&F7F-&F76#%\"jG!\"\"/FE;F,,&F;F-\"\"\"FFF
-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Construcci\363n de la tabl
a de diferencias divididas." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "f[[x[0] .. x[n]]] := (f[[x[0] .. x[n-1]]]-f[[x[1] .. x[
n]]])/(x[0]-x[n])" "6#>&%\"fG6#7#;&%\"xG6#\"\"!&F*6#%\"nG*&,&&F%6#7#;&
F*6#F,&F*6#,&F/\"\"\"\"\"\"!\"\"F;&F%6#7#;&F*6#\"\"\"&F*6#F/F=F;,&&F*6
#F,F;&F*6#F/F=F=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Las columnas \+
de la tabla de diferencias divididas corresponden cada una a una etapa
 " }{TEXT 263 4 "etap" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 91 "for ll from 0 by 1 to n do             didiv[1,ll,ll]
:=y[ll]                            od;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%&didivG6%\"\"\"\"\"!F(\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>&%&didivG6%\"\"\"F'F'\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&di
divG6%\"\"\"\"\"#F(\"#k" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 326 
"for etap from 2 by 1 to (n+1) do                 for l from 0 by 1 to
 (n-(etap-1)) do             didiv[etap,l,l+etap-1]:=(didiv[etap-1,l,l
+etap-2]-didiv[etap-1,l+1,l+etap-1])/(x[l]-x[l+etap-1]):    print(etap
, l,l+etap-1,'Diferencia',didiv[etap,l,l+etap-1])                od;  \+
                                            od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'\"\"#\"\"!\"\"\"%+DiferenciaG\"#>" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'\"\"#\"\"\"F#%+DiferenciaG\"#P" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'\"\"$\"\"!\"\"#%+DiferenciaG\"\"*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 67 "Calculamos el polinomio interpolador mediante la f
\363rmula de Newton." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "pnw
:=didiv[1,0,0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pnwG\"\")" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "for etp from 2 by 1 to n+1 \+
do pnw:=pnw+didiv[etp,0,etp-1]*product(x-x[jjj],jjj=0..etp-2):        \+
                                print(didiv[etp,0,etp-1],product(x-x[j
jj],jjj=0..etp-2),pnw)                                            od;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pnwG,&!#I\"\"\"%\"xG\"#>" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#>,&%\"xG\"\"\"!\"#F&,&!#IF&F%F#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pnwG,(!#I\"\"\"%\"xG\"#>*&,&F(F'!\"
#F'F',&F(F'!\"$F'F'\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"**&,&%
\"xG\"\"\"!\"#F'F',&F&F'!\"$F'F',(!#IF'F&\"#>F$F#" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "expand(pnw);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,(*$)%\"xG\"\"#\"\"\"\"\"*F&!#E\"#C\"\"\"" }}}{EXCHG {PARA 266 "
" 0 "" {TEXT -1 37 "Resolviendo un sistema de ecuaciones." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "p[n] := Sum(a[j]*x[i]^j,
j = 0 .. n) = f[i]" "6#>&%\"pG6#%\"nG/-%$SumG6$*&&%\"aG6#%\"jG\"\"\")&
%\"xG6#%\"iGF0F1/F0;\"\"!F'&%\"fG6#F6" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{XPPEDIT 18 0 "0 <= i;" "6#1\"\"!%\"iG" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{XPPEDIT 18 0 "i <= n;" "6#1%\"iG%\"nG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 83 "for pti from 0 by 1 to n do eq[pti]:=sum(a[jj]*x
[pti]^jj,jj=0..n)=y[pti];       od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>&%#eqG6#\"\"!/,(&%\"aGF&\"\"\"&F+6#F,\"\"#&F+6#F/\"\"%\"\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#eqG6#\"\"\"/,(&%\"aG6#\"\"!F'&F+F&
\"\"$&F+6#\"\"#\"\"*\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#eqG6#
\"\"#/,(&%\"aG6#\"\"!\"\"\"&F+6#F.\"\"%&F+F&\"#;\"#k" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 133 "Ahora resolvemos este sistema de ecuaciones pa
ra obtener los coeficientes del polinomio interpolador de Lagrange med
iante el comando " }{TEXT 264 5 "solve" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 29 "El sistema de ecuaciones es: " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "equas:=\{eq[0],eq[1],eq[2]\};" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&equasG<%/,(&%\"aG6#\"\"!\"\"\"&F)6#
F,\"\"#&F)6#F/\"\"%\"\")/,(F(F,F-\"\"$F0\"\"*\"#F/,(F(F,F-F2F0\"#;\"#k
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Su resoluci\363n nos da los c
oeficientes: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "coef:=solv
e(equas);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coefG<%/&%\"aG6#\"\"!
\"#C/&F(6#\"\"#\"\"*/&F(6#\"\"\"!#E" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 57 "Comprobamos que efectivamente las soluciones son buenas: " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(coef,equas);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<%/\"\")F%/\"#FF'/\"#kF)" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 242 "Ahora asignamos valores a los coeficientes de la
 soluci\363n para poder utilizar los componentes de esa soluci\363n in
diferentemente del orden en el que han aparecido esos componentes dura
nte la resoluci\363n del sistema ; empleamos el comando assign:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "assign(coef);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a[0];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a[1];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#!#E" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "a[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "As\355 obtenemos el polinomio inte
rpolador de Lagrange: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p
rs:=sum(a[jj]*x^jj,jj=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$prs
G,(*$)%\"xG\"\"#\"\"\"\"\"*F(!#E\"#C\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 55 "Ahora dibujamos la funci\363n y el polinomio interpolador
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "with(plots):         \+
     dibujfun:=plot(f(x),x=2..4,color=blue):    dibujpol:=plot(prs,x=2
..4,color=green):   display(\{dibujfun,dibujpol\},title=`Funci\363n y \+
Polinomio interpolador`); " }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6
$7S7$$\"\"#\"\"!$\"\")F*7$$\"1LLL3VfV?!#:$\"18drJthM&)F07$$\"1nm\"H[D:
3#F0$\"1vU^sgs=!*F07$$\"1LL$e0$=C@F0$\"1bxgQ.k%e*F07$$\"1LL$3RBr;#F0$
\"1R_4&)Hx<5!#97$$\"1nm\"zjf)4AF0$\"1lkK]/=z5FB7$$\"1LLe4;[\\AF0$\"1yD
$fPv#Q6FB7$$\"1++Dmy]!H#F0$\"16-jn!)p,7FB7$$\"1LLezs$HL#F0$\"1#\\tBfB(
p7FB7$$\"1++D@1BvBF0$\"1^GzH(Q+M\"FB7$$\"1nmm@Xt=CF0$\"1^gC=m-:9FB7$$
\"1LL$3y_qX#F0$\"1=(H.R\\L[\"FB7$$\"1+++l+>+DF0$\"1:#p*)RcGc\"FB7$$\"1
+++vW]VDF0$\"1w='Rg)\\X;FB7$$\"1+++NfC&e#F0$\"1h%3tA[ys\"FB7$$\"1LLez6
:BEF0$\"1/e>&*)p\\!=FB7$$\"1nmm\"=C#oEF0$\"1tJ-O4i**=FB7$$\"1mmmEpS1FF
0$\"1\\TPC_M#)>FB7$$\"1++DOD#3v#F0$\"1!pRt=a:3#FB7$$\"1mmmwy8!z#F0$\"1
6*R'*)e3s@FB7$$\"1++DOIFLGF0$\"1)p^Y#=RuAFB7$$\"1++v3zMuGF0$\"1Bcqr0vu
BFB7$$\"1nm;H_?<HF0$\"1I)=)Roc#[#FB7$$\"1nm\"zihl&HF0$\"18\\(>S1We#FB7
$$\"1LL$3#G,**HF0$\"1MUA$\\Ntp#FB7$$\"1LLezw5VIF0$\"1<(*G!>r!=GFB7$$\"
1++v$Q#\\\"3$F0$\"1Qb.j/1EHFB7$$\"1LL$e\"*[H7$F0$\"11Z+?FvXIFB7$$\"1++
+qvxlJF0$\"1I?s9!*ysJFB7$$\"1++]_qn2KF0$\"1Kg<I0W+LFB7$$\"1++Dcp@[KF0$
\"1.UJzb;FMFB7$$\"1++]2'HKH$F0$\"1(o+lZE;d$FB7$$\"1nmmwanLLF0$\"1Zva)H
W[q$FB7$$\"1+++v+'oP$F0$\"1>,Jmcp]QFB7$$\"1LLeR<*fT$F0$\"1E0H717')RFB7
$$\"1+++&)HxeMF0$\"1v.wi$ox8%FB7$$\"1mm\"H!o-*\\$F0$\"1nZ4&\\CRG%FB7$$
\"1++DTO5TNF0$\"1tC)3\"oLSWFB7$$\"1nmmT9C#e$F0$\"1a.'3w%*of%FB7$$\"1++
D1*3`i$F0$\"1gHjKWpkZFB7$$\"1LLL$*zymOF0$\"1t3r'**=,$\\FB7$$\"1LL$3N1#
4PF0$\"1go0TY?.^FB7$$\"1nm\"HYt7v$F0$\"1\"[2ju6)y_FB7$$\"1+++q(G**y$F0
$\"1veJfpoVaFB7$$\"1mm;9@BMQF0$\"1`z#\\ULoj&FB7$$\"1LLL`v&Q(QF0$\"1))*
3%R)4M\"eFB7$$\"1++DOl5;RF0$\"1.?uk!)p0gFB7$$\"1++v.UacRF0$\"1RODf+n$>
'FB7$$\"\"%F*$\"#kF*-%'COLOURG6&%$RGBGF*F*$\"*++++\"!\")-F$6$7SF'7$F.$
\"1z[!oDZIX)F07$F4$\"1H)\\%oC2v))F07$F9$\"1!GQ0WB1Q*F07$F>$\"18g?efgA*
*F07$FD$\"1I0y)f'\\\\5FB7$FI$\"1%f^dd)\\06FB7$FN$\"1:yc/KYm6FB7$FS$\"1
\\g])y*pK7FB7$FX$\"1&Hh%Q)[>I\"FB7$Fgn$\"1Q&Q8ERlP\"FB7$F\\o$\"1nS&3Ig
]W\"FB7$Fao$\"1MA,c6OD:FB7$Ffo$\"1A0\"z(=O4;FB7$F[p$\"1A1**eu]$p\"FB7$
F`p$\"1By$H$ojs<FB7$Fep$\"1VV$HQ&4q=FB7$Fjp$\"1Y]5)f;b&>FB7$F_q$\"1&z'
4pN=e?FB7$Fdq$\"1XcRaR-_@FB7$Fiq$\"1lr*Qf#=eAFB7$F^r$\"1]>tZPQiBFB7$Fc
r$\"1wiL=TMuCFB7$Fhr$\"1\"**=6wq+e#FB7$F]s$\"1rPSvnB(p#FB7$Fbs$\"1([ow
%RPAGFB7$Fgs$\"1I$[AebT$HFB7$F\\t$\"1mukc<'y0$FB7$Fat$\"1&fV%y6\"*)=$F
B7$Fft$\"1z:KHDJ?LFB7$F[u$\"17I8pzX]MFB7$F`u$\"1WJWV\"G%)f$FB7$Feu$\"1
>D-En\\MPFB7$Fju$\"11h^u$HI)QFB7$F_v$\"1WZ\\&3@0-%FB7$Fdv$\"1G#*eW(*)R
<%FB7$Fiv$\"1tH>B+S@VFB7$F^w$\"12m,JSgyWFB7$Fcw$\"1Q8UB1QNYFB7$Fhw$\"1
:?\"H/vF![FB7$F]x$\"1x9e')>:n\\FB7$Fbx$\"1L%*elSXQ^FB7$Fgx$\"1(=T1LO:J
&FB7$F\\y$\"1Vj]r#*QtaFB7$Fay$\"1aH%z\"))>icFB7$Ffy$\"1!G*\\qa1MeFB7$F
[z$\"1vRooVU?gFB7$F`z$\"1[\\&G!H!=?'FBFdz-Fjz6&F\\[lF*F][lF*-%+AXESLAB
ELSG6$Q\"x6\"%!G-%&TITLEG6#%AFunci|^zn~y~Polinomio~interpoladorG-%%VIE
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0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Si ampliamos el inte
rvalo de dibujo, podemos ver que la interpolaci\363n no es correcta fu
era del soporte: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "with(
plots):              dibujfun:=plot(f(x),x=-2..8,color=blue):    dibuj
pol:=plot(prs,x=-2..8,color=green):   display(\{dibujfun,dibujpol\},ti
tle=`Funci\363n y Polinomio interpolador`);" }}{PARA 13 "" 1 "" 
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\"1Q:#R-Q$f;F`v7$Faw$\"1S[O/QJd=F`v7$Ffw$\"1?x?Lf^l?F`v7$F[x$\"1_=H\"z
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F`v7$F_y$\"1G]Q)o2lN$F`v7$Fdy$\"1Tl$)H$3yo$F`v7$Fiy$\"1%)G&H0&*>+%F`v7
$F^z$\"1\"GNHO8hN%F`v7$Fcz$\"1&*4i:A09ZF`v7$Fhz$\"$7&F*-F][l6&F_[lF*F*
F`[l-%+AXESLABELSG6$Q\"x6\"%!G-%&TITLEG6#%AFunci|^zn~y~Polinomio~inter
poladorG-%%VIEWG6$;F(Fhz%(DEFAULTG" 2 400 300 300 2 0 1 0 2 9 1 4 2 
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0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Derivaci\363n num\351rica" }}
{EXCHG {PARA 256 "" 0 "" {TEXT -1 20 "Derivaci\363n num\351rica." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "En este ejemplo, derivamos directa
mente el polinomio y el error de interpolaci\363n." }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 91 "Consideramos el soporte [2,3,4] y una funci\363n \+
que toma en el soporte los valores [8,27,64]." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "
Entrada de los datos conocidos." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 5 "n:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x[0]:=2; x[1]:=3; x[2]:=4; \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!\"\"#" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"xG6#\"\"#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "y[0]:=8; y[1]:=27; y[2]:=64;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"yG6#\"\"!\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>&%\"yG6#\"\"\"\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"#
\"#k" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Calculamos el polinomio i
nterpolador de Lagrange mediante la " }{TEXT 265 17 "f\363rmula de New
ton" }{TEXT -1 32 " (como en el ejemplo anterior): " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 91 "for ll from 0 by 1 to n do             di
div[1,ll,ll]:=y[ll]                            od;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>&%&didivG6%\"\"\"\"\"!F(\"\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&didivG6%\"\"\"F'F'\"#F" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&didivG6%\"\"\"\"\"#F(\"#k" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 326 "for etap from 2 by 1 to (n+1) do               \+
  for l from 0 by 1 to (n-(etap-1)) do             didiv[etap,l,l+etap
-1]:=(didiv[etap-1,l,l+etap-2]-didiv[etap-1,l+1,l+etap-1])/(x[l]-x[l+e
tap-1]):    print(etap, l,l+etap-1,'Diferencia',didiv[etap,l,l+etap-1]
)                od;                                              od;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#\"\"!\"\"\"%+DiferenciaG\"#>
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#\"\"\"F#%+DiferenciaG\"#P" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"$\"\"!\"\"#%+DiferenciaG\"\"*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Calculamos el polinomio interpolad
or mediante la f\363rmula de Newton." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "pnw:=didiv[1,0,0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$pnwG\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "for etp f
rom 2 by 1 to n+1 do pnw:=pnw+didiv[etp,0,etp-1]*product(x-x[jjj],jjj=
0..etp-2):                                        print(didiv[etp,0,et
p-1],product(x-x[jjj],jjj=0..etp-2),pnw)                              \+
              od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pnwG,&!#I\"\"
\"%\"xG\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#>,&%\"xG\"\"\"!\"#F&
,&!#IF&F%F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pnwG,(!#I\"\"\"%\"xG
\"#>*&,&F(F'!\"#F'F',&F(F'!\"$F'F'\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%\"\"**&,&%\"xG\"\"\"!\"#F'F',&F&F'!\"$F'F',(!#IF'F&\"#>F$F#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "expand(pnw);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(\"#C\"\"\"%\"xG!#E*$)F&\"\"#\"\"\"\"\"*" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT 262 38 "Estimaci\363n del error de inte
rpolaci\363n." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Sabiendo que la \+
funci\363n es f(x)=x^3: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 
"f:=x->x^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)op
eratorG%&arrowGF(*$)9$\"\"$\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 53 "Calculamos la derivada n+1 de la funci\363n en cuesti\363
n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "der:=(D@@(n+1))(f);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$derG\"\"'" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 50 "Calculamos el producto que interviene en el error." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "prod:=1:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 84 "for ii from 0 by 1 to n do      prod:=pro
d*(x-x[ii])                             od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%prodG,&%\"xG\"\"\"!\"#F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%prodG*&,&%\"xG\"\"\"!\"#F(F(,&F'F(!\"$F(F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%prodG*(,&%\"xG\"\"\"!\"#F(F(,&F'F(!\"$F(F
(,&F'F(!\"%F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Calculamos la \+
expresi\363n del error." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "
ER:=prod*der/(n+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ERG,$*(,&%\"
xG\"\"\"!\"#F)F),&F(F)!\"$F)F),&F(F)!\"%F)F)\"\"#" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 14 "Calculamos la " }{TEXT 260 63 "derivada de primer \+
orden del polinomio interpolador de Lagrange" }{TEXT -1 6 " y la " }
{TEXT 261 50 "derivada de primer orden de la expresi\363n del error" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "pprim:=dif
f(pnw,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&pprimG,&!#E\"\"\"%\"xG
\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "erprim:=diff(ER,x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'erprimG,(*&,&%\"xG\"\"\"!\"$F)F
),&F(F)!\"%F)F)\"\"#*&,&F(F)!\"#F)F)F+\"\"\"F-*&F/F1F'F1F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "expand(erprim);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"\"\"'F&!#O\"#_\"\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 15 "En el punto x0." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "subs(x=x[0],pprim);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs
(x=x[0],erprim);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 15 "En el punto x1." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "subs(x=x[1],pprim);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs
(x=x[1],erprim);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 15 "En el punto x2." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "subs(x=x[2],pprim);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#Y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs
(x=x[2],erprim);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 14 "Calculamos la " }{TEXT 256 64 "derivada de segundo orden del po
linomio interpolador de Lagrange" }{TEXT -1 6 " y la " }{TEXT 257 51 "
derivada de segundo orden de la expresi\363n del error" }{TEXT -1 1 ".
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "psec:=diff(pnw,x,x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%psecG\"#=" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "ersec:=diff(ER,x,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&ersecG,&%\"xG\"#7!#O\"\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "expand(ersec);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,&%\"xG\"#7!#O\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "En el p
unto x0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x=x[0],pse
c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#=" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 19 "subs(x=x[0],ersec);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#!#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "En el punto x1." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x=x[1],psec);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "subs(x=x[1],ersec);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "En el punto x2." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x=x[2],psec);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 
"subs(x=x[2],ersec);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 62 "Derivaci\363n num\351rica de primer orden con un soport
e de 2 puntos" }}{EXCHG {PARA 268 "" 0 "" {TEXT -1 50 "Aplicaci\363n d
e las f\363rmulas de derivaci\363n num\351rica." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 86 "Consideramos el soporte [2,4] y una funci\363n que t
oma en el soporte los valores [8,64]." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Entra
da de los datos conocidos." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
5 "n:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x[0]:=2; x[1]:=4; " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%\"xG6#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Pa
ra poder aplicar las f\363rmulas de derivaci\363n, la funci\363n y deb
e venir en funci\363n de x y no seg\372n el orden de las x en el vecto
r. As\355: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "y[x[0]]:=8; \+
y[x[1]]:=64;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"#\"\")" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"%\"#k" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 14 "El paso h es: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "h:=x[1]-x[0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
hG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Aplicamos la f\363rmul
a de derivaci\363n num\351rica en un soporte de 2 puntos  " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "fprim(xast) := (f[1]-f[0
])/(x[1]-x[0]);" "6#>-%&fprimG6#%%xastG*&,&&%\"fG6#\"\"\"\"\"\"&F+6#\"
\"!!\"\"F.,&&%\"xG6#\"\"\"F.&F56#F1F2F2" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 38 "a los 2 casos particulares siguientes." }}}{EXCHG {PARA 
267 "" 0 "" {TEXT -1 13 "Primer caso: " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 18 0 "xast := x[0];" "6#>%%xastG&%\"xG6#\"\"!
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[1] := x[0]
+h;" "6#>&%\"xG6#\"\"\",&&F%6#\"\"!\"\"\"%\"hGF," }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 59 "En este caso la f\363rmula de derivaci\363n num
\351rica se escribe: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "fp
rim[x[0]]:=(y[x[0]+h]-y[x[0]])/h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
&%&fprimG6#\"\"#\"#G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Aplicamos
 ahora la f\363rmula del error de derivaci\363n: " }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "rf := -1/2*h*`@@`(D,2)(f);" "6#>%
#rfG,$**\"\"\"\"\"\"\"\"#!\"\"%\"hGF(--%#@@G6$%\"DG\"\"#6#%\"fGF(F*" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "La funci\363n es x^3:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^3;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"$\"
\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dersec:=diff(
f(x),x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dersecG,$%\"xG\"\"'" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rf:=(-h/2)*dersec;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rfG,$%\"xG!\"'" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 51 "Una cota del error en el intervalo considerado es:
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "abs(maximize(rf,x,\{x=
x[0]..x[1]\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 14 "Segundo caso: " }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[0] := xast-1/2*h;" "6#>&%\"xG6#\"\"
!,&%%xastG\"\"\"*(\"\"\"F*\"\"#!\"\"%\"hGF*F." }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[1] := xast+1/2*h;" "6#>&%\"xG6#\"\"
\",&%%xastG\"\"\"*(\"\"\"F*\"\"#!\"\"%\"hGF*F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "xast:=x[0]+h/2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%xastG\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "En este ca
so la f\363rmula de derivaci\363n num\351rica se escribe: " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "fprim[xast]:=(y[xast+h/2]-y[xast-h/
2])/h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&fprimG6#\"\"$\"#G" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Aplicamos ahora la f\363rmula del \+
error de derivaci\363n: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "rf := -1/24*h^2*`@@`(D,3)(f);" "6#>%#rfG,$**\"\"\"\"\"
\"\"#C!\"\"%\"hG\"\"#--%#@@G6$%\"DG\"\"$6#%\"fGF(F*" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 18 "La funci\363n es x^3:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 10 "f:=x->x^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"$\"\"\"F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "derter:=diff(f(x),x,x,x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'derterG\"\"'" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 21 "rf:=(-h^2/24)*derter;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#rfG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Una
 cota del error en el intervalo considerado es: " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 35 "abs(maximize(rf,x,\{x=x[0]..x[1]\}));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "Derivaci
\363n num\351rica de primer orden con un soporte de 3 puntos" }}
{EXCHG {PARA 258 "" 0 "" {TEXT -1 50 "Aplicaci\363n de las f\363rmulas
 de derivaci\363n num\351rica." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 
"Consideramos el soporte [2,3,4] y una funci\363n que toma en el sopor
te los valores [8,27,64]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 
"restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Entrada de los datos
 conocidos." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"#" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 27 "x[0]:=2; x[1]:=3; x[2]:=4; " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"xG6#\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>&%\"xG6#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#
\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Para poder aplicar las \+
f\363rmulas de derivaci\363n, la funci\363n y debe venir en funci\363n
 de x y no seg\372n el orden de las x en el vector. As\355: " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "y[x[0]]:=8; y[x[1]]:=27; y[x
[2]]:=64;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"#\"\")" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"$\"#F" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>&%\"yG6#\"\"%\"#k" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 14 "El paso h es: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "h:
=x[1]-x[0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Aplicamos la f\363rmula de derivac
i\363n num\351rica en un soporte de 3 puntos  " }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "fprim(xast) := (f[1]-f[0])/(x[1]-x[
0])+((f[2]-f[1])/(x[2]-x[1])-(f[1]-f[0])/(x[1]-x[0]))*(2*xast-x[0]-x[1
])/(x[2]-x[0]);" "6#>-%&fprimG6#%%xastG,&*&,&&%\"fG6#\"\"\"\"\"\"&F,6#
\"\"!!\"\"F/,&&%\"xG6#\"\"\"F/&F66#F2F3F3F/*(,&*&,&&F,6#\"\"#F/&F,6#\"
\"\"F3F/,&&F66#\"\"#F/&F66#\"\"\"F3F3F/*&,&&F,6#\"\"\"F/&F,6#F2F3F/,&&
F66#\"\"\"F/&F66#F2F3F3F3F/,(*&\"\"#F/F'F/F/&F66#F2F3&F66#\"\"\"F3F/,&
&F66#\"\"#F/&F66#F2F3F3F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "a lo
s 2 casos particulares siguientes." }}}{EXCHG {PARA 257 "" 0 "" {TEXT 
-1 13 "Primer caso: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "xast := x[0];" "6#>%%xastG&%\"xG6#\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[1] := x[0]+h;" "6#>&%
\"xG6#\"\"\",&&F%6#\"\"!\"\"\"%\"hGF," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 18 0 "x[2] := x[0]+2*h;" "6#>&%\"xG6#\"\"#,&&F
%6#\"\"!\"\"\"*&\"\"#F,%\"hGF,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
59 "En este caso la f\363rmula de derivaci\363n num\351rica se escribe
: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "fprim[x[0]]:=(-3*y[x[
0]]+4*y[x[0]+h]-y[x[0]+2*h])/(2*h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>&%&fprimG6#\"\"#\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Aplicam
os ahora la f\363rmula del error de derivaci\363n: " }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "rf := -1/6*h^2*(2*`@@`(D,3)(f1
)-4*`@@`(D,3)(f2));" "6#>%#rfG,$**\"\"\"\"\"\"\"\"'!\"\"%\"hG\"\"#,&*&
\"\"#F(--%#@@G6$%\"DG\"\"$6#%#f1GF(F(*&\"\"%F(--F26$F4\"\"$6#%#f2GF(F*
F(F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "La funci\363n es x^3:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^3;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"$
\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "derter:=dif
f(f(x),x,x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'derterG\"\"'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "El valor m\341ximo de la derivada \+
de orden 3 en el intervalo considerado es: " }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 39 "abs(maximize(derter,x,\{x=x[0]..x[2]\}));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 32 "Una cota del error es entonces: " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "mrf:=(-h^2/6)*(2*derter-4*derter);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$mrfG\"\"#" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 
-1 14 "Segundo caso: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "x[0] := xast-h;" "6#>&%\"xG6#\"\"!,&%%xastG\"\"\"%\"hG!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[1] := \+
xast;" "6#>&%\"xG6#\"\"\"%%xastG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }{XPPEDIT 18 0 "x[2] := xast+h;" "6#>&%\"xG6#\"\"#,&%%xastG\"\"\"
%\"hGF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "En este caso la f\363r
mula de derivaci\363n num\351rica se escribe: " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "fprim[x[1]]:=(y[x[1]+h]-y[x[1]-h])/(2*h);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&fprimG6#\"\"$\"#G" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 52 "Aplicamos ahora la f\363rmula del error d
e derivaci\363n: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 
18 0 "rf := -1/12*h^2*`@@`(D,3)(f);" "6#>%#rfG,$**\"\"\"\"\"\"\"#7!\"
\"%\"hG\"\"#--%#@@G6$%\"DG\"\"$6#%\"fGF(F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 18 "La funci\363n es x^3:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "f:=x->x^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR
6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"$\"\"\"F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "derter:=diff(f(x),x,x,x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'derterG\"\"'" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "rf:=(-h^2/12)*derter;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#rfG#!\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Una co
ta del error en el intervalo considerado es: " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "abs(maximize(rf,x,\{x=x[0]..x[2]\}));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Derivaci
\363n num\351rica de segundo orden" }}{EXCHG {PARA 258 "" 0 "" {TEXT 
-1 48 "Aplicaci\363n de la f\363rmula de derivaci\363n num\351rica." }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Consideramos el soporte [2,3,4] \+
y una funci\363n que toma en el soporte los valores [8,27,64]." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 31 "Entrada de los datos conocidos." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 5 "n:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"nG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x[0]:=2; x[1]
:=3; x[2]:=4; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!\"\"#
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"\"\"$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 135 "Para poder aplicar las f\363rmulas de derivaci\363n, la \+
funci\363n y debe venir en funci\363n de x y no seg\372n el orden de l
as x en el vector. As\355: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "y[x[0]]:=8; y[x[1]]:=27; y[x[2]]:=64;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"yG6#\"\"#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>&%\"yG6#\"\"$\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"%\"
#k" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "El paso h es: " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "h:=x[1]-x[0];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"hG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Ap
licamos la f\363rmula de derivaci\363n num\351rica en un soporte de 3 \+
puntos  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "fsec
(xast) := 2*((f[2]-f[1])/(x[2]-x[1])-(f[1]-f[0])/(x[1]-x[0]))/(x[2]-x[
0]);" "6#>-%%fsecG6#%%xastG*(\"\"#\"\"\",&*&,&&%\"fG6#\"\"#F*&F/6#\"\"
\"!\"\"F*,&&%\"xG6#\"\"#F*&F86#\"\"\"F5F5F**&,&&F/6#\"\"\"F*&F/6#\"\"!
F5F*,&&F86#\"\"\"F*&F86#FEF5F5F5F*,&&F86#\"\"#F*&F86#FEF5F5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "al caso particular siguiente." }}}
{EXCHG {PARA 257 "" 0 "" {TEXT -1 17 "Caso particular: " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[0] := xast-h;" "6#>&%
\"xG6#\"\"!,&%%xastG\"\"\"%\"hG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }{XPPEDIT 18 0 "x[1] := xast;" "6#>&%\"xG6#\"\"\"%%xastG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[2] := xast+h;" 
"6#>&%\"xG6#\"\"#,&%%xastG\"\"\"%\"hGF*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 59 "En este caso la f\363rmula de derivaci\363n num\351rica s
e escribe: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "fsec[x[1]]:=
(y[x[1]+h]-2*y[x[1]]+y[x[1]-h])/h^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>&%%fsecG6#\"\"$\"#=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Aplicam
os ahora la f\363rmula del error de derivaci\363n: " }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "rf := -1/12*h^2*`@@`(D,4)(f);
" "6#>%#rfG,$**\"\"\"\"\"\"\"#7!\"\"%\"hG\"\"#--%#@@G6$%\"DG\"\"%6#%\"
fGF(F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "La funci\363n es x^3:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^3;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"
\"$\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "dercua:=
diff(f(x),x,x,x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dercuaG\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rf:=(-h^2/12)*dercua;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rfG\"\"!" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 51 "Una cota del error en el intervalo considerado es: " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "abs(maximize(rf,x,\{x=x[0]
..x[2]\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 
21 "EJERCICIOS PROPUESTOS" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Ejer
cicio 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Consideramos el censo d
e poblaci\363n siguiente: " }}{PARA 0 "" 0 "" {TEXT -1 46 "A\361o:    \+
               10     20     30     40" }}{PARA 0 "" 0 "" {TEXT -1 44 
"Millones de hab.: 5       8       9       12" }}{PARA 0 "" 0 "" 
{TEXT -1 114 "\277Cu\341l ser\355a aproximadamente la poblaci\363n en \+
el a\361o 33? Para ello, emplear la interpolaci\363n polin\363mica de \+
Lagrange." }}}{EXCHG {PARA 269 "" 0 "" {TEXT -1 10 "Respuesta." }}
{PARA 0 "" 0 "" {TEXT -1 93 "Primero, almacenamos los datos conocidos \+
(soporte y valor que toma la funci\363n en el soporte)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "Entrada de los datos conocidos." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 
"" {TEXT 266 10 "Utilizamos" }{TEXT -1 36 " los polinomios de base de \+
Lagrange." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 46 "C\341lculo de los p
olinomios de base de Lagrange." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }{XPPEDIT 18 0 "L[i] := Product((x-x[j])/(x[i]-x[j]),j = 0 .. n)" "6
#>&%\"LG6#%\"iG-%(ProductG6$*&,&%\"xG\"\"\"&F-6#%\"jG!\"\"F.,&&F-6#F'F
.&F-6#F1F2F2/F1;\"\"!%\"nG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 
18 0 "j <> i" "6#0%\"jG%\"iG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 256 47 "C\341lculo del polinomio interpolador de Lagrange." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "p[n] := Sum(f[i]*
L[i],i = 0 .. n);" "6#>&%\"pG6#%\"nG-%$SumG6$*&&%\"fG6#%\"iG\"\"\"&%\"
LG6#F/F0/F/;\"\"!F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 142 "Ahora evaluamos el valor que toma el polinomio
 interpolador de Lagrange en x=33 para obtener el valor aproximado de \+
la poblaci\363n en el a\361o 33." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "
Ejercicio 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Consideramos el sop
orte: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[0] := 0;" "6
#>&%\"xG6#\"\"!F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[1
] := 1/10*Pi;" "6#>&%\"xG6#\"\"\"*(\"\"\"\"\"\"\"#5!\"\"%#PiGF*" }}
{PARA 0 "" 0 "" {TEXT -1 57 "En este soporte una funci\363n toma los v
alores siguientes: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "s
en(2);" "6#-%$senG6#\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 
18 0 "sen(2+1/2*Pi);" "6#-%$senG6#,&\"\"#\"\"\"*(\"\"\"F(\"\"#!\"\"%#P
iGF(F(" }}{PARA 0 "" 0 "" {TEXT -1 92 "a) Calcular los polinomios de b
ase de Lagrange en este soporte. Representarlos gr\341ficamente." }}
{PARA 0 "" 0 "" {TEXT -1 66 "b) Calcular el polinomio interpolador de \+
Lagrange en este soporte." }}{PARA 0 "" 0 "" {TEXT -1 31 "c) Sabiendo \+
que la funci\363n es: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 
0 "f(x) := sen(5*x+2);" "6#>-%\"fG6#%\"xG-%$senG6#,&*&\"\"&\"\"\"F'F.F
.\"\"#F." }}{PARA 0 "" 0 "" {TEXT -1 34 "determinar la expresi\363n de
l error." }}{PARA 0 "" 0 "" {TEXT -1 46 "d) Hallar una cota del error \+
de interpolaci\363n." }}{PARA 0 "" 0 "" {TEXT -1 88 "e) Representar gr
\341ficamente el polinomio interpolador y la funci\363n en una misma g
r\341fica." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 267 10 "Res
puesta." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Introducimos los datos conocidos.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 49 "a) C\341lculo de los polinom
ios de base de Lagrange." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "L[i] := Product((x-x[j])/(x[i]-x[j]),j = 0 .. n)" "6#>&
%\"LG6#%\"iG-%(ProductG6$*&,&%\"xG\"\"\"&F-6#%\"jG!\"\"F.,&&F-6#F'F.&F
-6#F1F2F2/F1;\"\"!%\"nG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 
0 "j <> i" "6#0%\"jG%\"iG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 50 "b) C\341lcul
o del polinomio interpolador de Lagrange." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 18 0 "p[n] := Sum(f[i]*L[i],i = 0 .. n);" "6#>
&%\"pG6#%\"nG-%$SumG6$*&&%\"fG6#%\"iG\"\"\"&%\"LG6#F/F0/F/;\"\"!F'" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 270 "" 0 "" {TEXT -1 23 "c) Expresi\363n del er
ror." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "La funci\363n es: " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "f(x) := sen(5*x+2)" "6#>
-%\"fG6#%\"xG-%$senG6#,&*&\"\"&\"\"\"F'F.F.\"\"#F." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 28 "E
rror real de interpolaci\363n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT -1 37 "Expresi\363n del error de interpolaci
\363n." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "E := P
roduct(x-x[i],i = 0 .. n)*`@@`(D,n+1)(f)/(n+1)!;" "6#>%\"EG*(-%(Produc
tG6$,&%\"xG\"\"\"&F*6#%\"iG!\"\"/F.;\"\"!%\"nGF+--%#@@G6$%\"DG,&F3F+\"
\"\"F+6#%\"fGF+-%*factorialG6#,&F3F+\"\"\"F+F/" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 17 "d) Cota de \+
error." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Evaluamos ahora su cota
 en el intervalo considerado." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
272 "" 0 "" {TEXT -1 26 "e) Representaci\363n gr\341fica." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 
"Ejercicio 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Al soporte del eje
rcicio anterior se le a\361ade el punto: " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{XPPEDIT 18 0 "x := 1/20*Pi;" "6#>%\"xG*(\"\"\"\"\"\"\"#?!\"\"%#
PiGF'" }}{PARA 0 "" 0 "" {TEXT -1 96 "a) Calcular los polinomios de ba
se de Lagrange en el nuevo soporte. Representarlos gr\341ficamente." }
}{PARA 0 "" 0 "" {TEXT -1 70 "b) Calcular el polinomio interpolador de
 Lagrange en el nuevo soporte." }}{PARA 0 "" 0 "" {TEXT -1 37 "c) Dete
rminar la expresi\363n del error." }}{PARA 0 "" 0 "" {TEXT -1 46 "d) H
allar una cota del error de interpolaci\363n." }}{PARA 0 "" 0 "" 
{TEXT -1 88 "e) Representar gr\341ficamente el polinomio interpolador \+
y la funci\363n en una misma gr\341fica." }}{PARA 0 "" 0 "" {TEXT -1 
70 "Concluir en cuanto a la mejora aportada al a\361adir un punto al s
oporte." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 10 "Respue
sta." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 33 "Introducimos los datos conocidos." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 256 49 "a) C\341lculo de los polinomios de base de Lagrange." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "L[i] := Product(
(x-x[j])/(x[i]-x[j]),j = 0 .. n)" "6#>&%\"LG6#%\"iG-%(ProductG6$*&,&%
\"xG\"\"\"&F-6#%\"jG!\"\"F.,&&F-6#F'F.&F-6#F1F2F2/F1;\"\"!%\"nG" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "j <> i" "6#0%\"jG%\"iG" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 50 "b) C\341lcul
o del polinomio interpolador de Lagrange." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 18 0 "p[n] := Sum(f[i]*L[i],i = 0 .. n);" "6#>
&%\"pG6#%\"nG-%$SumG6$*&&%\"fG6#%\"iG\"\"\"&%\"LG6#F/F0/F/;\"\"!F'" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 23 
"c) Expresi\363n del error." }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 28 "
Error real de interpolaci\363n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 257 "" 0 "" {TEXT -1 37 "Expresi\363n del error de interpolaci
\363n." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "E := P
roduct(x-x[i],i = 0 .. n)*`@@`(D,n+1)(f)/(n+1)!;" "6#>%\"EG*(-%(Produc
tG6$,&%\"xG\"\"\"&F*6#%\"iG!\"\"/F.;\"\"!%\"nGF+--%#@@G6$%\"DG,&F3F+\"
\"\"F+6#%\"fGF+-%*factorialG6#,&F3F+\"\"\"F+F/" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 17 "d) Cota de \+
error." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Evaluamos ahora su cota
 en el intervalo considerado." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 26 
"e) Representaci\363n gr\341fica." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 11 "Ejercicio 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Cons
ideramos el soporte: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 
"x[0] := 0;" "6#>&%\"xG6#\"\"!F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "x[1] := 1/4*Pi;" "6#>&%\"xG6#\"\"\"*(\"\"\"\"\"\"\"\"%!
\"\"%#PiGF*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[2] := 1
/2*Pi;" "6#>&%\"xG6#\"\"#*(\"\"\"\"\"\"\"\"#!\"\"%#PiGF*" }}{PARA 0 "
" 0 "" {TEXT -1 57 "En este soporte una funci\363n toma los valores si
guientes: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "0;" "6#\"
\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "1/2*sqrt(2);" "6#
*(\"\"\"\"\"\"\"\"#!\"\"-%%sqrtG6#\"\"#F%" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{XPPEDIT 18 0 "1;" "6#\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 96 "a) \+
Calcular el polinomio interpolador de Lagrange en este soporte mediant
e la f\363rmula de Newton." }}{PARA 0 "" 0 "" {TEXT -1 61 "Ahora se le
 a\361ade al soporte anterior los siguientes puntos: " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }{XPPEDIT 18 0 "x := 1/6*Pi;" "6#>%\"xG*(\"\"\"\"\"\"
\"\"'!\"\"%#PiGF'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x :
= 1/3*Pi;" "6#>%\"xG*(\"\"\"\"\"\"\"\"$!\"\"%#PiGF'" }}{PARA 0 "" 0 "
" {TEXT -1 54 "en los cuales la funci\363n toma los siguientes valores
: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"
\"\"\"\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "1/2*
sqrt(3);" "6#*(\"\"\"\"\"\"\"\"#!\"\"-%%sqrtG6#\"\"$F%" }}{PARA 0 "" 
0 "" {TEXT -1 102 "b) Calcular el polinomio interpolador de Lagrange e
n este nuevo soporte mediante la f\363rmula de Newton." }}{PARA 0 "" 
0 "" {TEXT -1 70 "c) Comparar (gr\341ficamente) los 2 polinomios anter
iores con la funci\363n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 
18 0 "sen(x);" "6#-%$senG6#%\"xG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 10 "Respuesta." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 33 "Introducimos los datos conocidos." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 256 50 "a) C\341lculo del polinomio inter
polador de Lagrange." }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 18 "F\363rm
ula de Newton." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 
0 "p[n] := f[0]+Sum(f[[x[0] .. x[i]]],i = 1 .. n)*Product(x-x[j],j = 0
 .. i-1);" "6#>&%\"pG6#%\"nG,&&%\"fG6#\"\"!\"\"\"*&-%$SumG6$&F*6#7#;&%
\"xG6#F,&F76#%\"iG/F;;\"\"\"F'F--%(ProductG6$,&F7F-&F76#%\"jG!\"\"/FE;
F,,&F;F-\"\"\"FFF-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "f[[x[0] .. x[n]]] := (f[[x[0] .. x[n-1]]]-f[[x[1] .. x[
n]]])/(x[0]-x[n])" "6#>&%\"fG6#7#;&%\"xG6#\"\"!&F*6#%\"nG*&,&&F%6#7#;&
F*6#F,&F*6#,&F/\"\"\"\"\"\"!\"\"F;&F%6#7#;&F*6#\"\"\"&F*6#F/F=F;,&&F*6
#F,F;&F*6#F/F=F=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "Construcci
\363n de la tabla de diferencias divididas y c\341lculo del polinomio \+
interpolador de Lagrange mediante la f\363rmula de Newton." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 256 50 "b) C\341lculo del polinomio inter
polador de Lagrange." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT -1 18 "F\363rmula de Newton." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 273 "" 0 "" {TEXT -1 30 "c) Comparaci\363n de pnw2 y pnw4
." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Ejercicio 5
" }}{EXCHG {PARA 0 "" 0 "" {TEXT 272 2 "a)" }{TEXT -1 38 " Consideramo
s el soporte de 2 puntos: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 
18 0 "x[0] := -h;" "6#>&%\"xG6#\"\"!,$%\"hG!\"\"" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 18 0 "x[1] := h;" "6#>&%\"xG6#\"\"\"%\"hG" }}
{PARA 0 "" 0 "" {TEXT -1 14 "y la funci\363n: " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 18 0 "f(x) := exp(x);" "6#>-%\"fG6#%\"xG-%$exp
G6#F'" }}{PARA 0 "" 0 "" {TEXT -1 232 "Hallar una aproximaci\363n de f
'(0) mediante la f\363rmula de derivaci\363n num\351rica de tipo inter
polatorio en el soporte dado para los siguientes valores de h: 0'1, 0'
01, 0'001, 0'0001, 0'00001, 0'000001, 0'0000001. Se utilizar\341n 8 d
\355gitos." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 10 "Res
puesta." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re
start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 71 "Aplicamos la f\363rmula de derivaci\363n num
\351rica en un soporte de 2 puntos  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }{XPPEDIT 18 0 "fprim(xast) := (f[1]-f[0])/(x[1]-x[0]);" "6#>-
%&fprimG6#%%xastG*&,&&%\"fG6#\"\"\"\"\"\"&F+6#\"\"!!\"\"F.,&&%\"xG6#\"
\"\"F.&F56#F1F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Para ello, e
mpezamos introduciendo los datos conocidos." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Ahora aplic
amos la f\363rmula para los distintos valores de h: " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 8 "Resumen:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
271 2 "b)" }{TEXT -1 44 " Consideramos ahora el soporte de 2 puntos: \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[0] := -h;" "6#>&%
\"xG6#\"\"!,$%\"hG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 
0 "x[1] := 0;" "6#>&%\"xG6#\"\"\"\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{XPPEDIT 18 0 "x[2] := h;" "6#>&%\"xG6#\"\"#%\"hG" }}{PARA 0 "" 0 "
" {TEXT -1 14 "y la funci\363n: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "f(x) := exp(x);" "6#>-%\"fG6#%\"xG-%$expG6#F'" }}{PARA 
0 "" 0 "" {TEXT -1 233 "Hallar una aproximaci\363n de f''(0) mediante \+
la f\363rmula de derivaci\363n num\351rica de tipo interpolatorio en e
l soporte dado para los siguientes valores de h: 0'1, 0'01, 0'001, 0'0
001, 0'00001, 0'000001, 0'0000001. Se utilizar\341n 8 d\355gitos." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 10 "Respuesta." }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 71 "Aplicamos la f\363rmula de derivaci\363n num\351rica en u
n soporte de 3 puntos  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "fsec(xast) := 2*(
(f[2]-f[1])/(x[2]-x[1])-(f[1]-f[0])/(x[1]-x[0]))/(x[2]-x[0]);" "6#>-%%
fsecG6#%%xastG*(\"\"#\"\"\",&*&,&&%\"fG6#\"\"#F*&F/6#\"\"\"!\"\"F*,&&%
\"xG6#\"\"#F*&F86#\"\"\"F5F5F**&,&&F/6#\"\"\"F*&F/6#\"\"!F5F*,&&F86#\"
\"\"F*&F86#FEF5F5F5F*,&&F86#\"\"#F*&F86#FEF5F5" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 55 "Para ello, empezamos introduciendo los datos conocid
os." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 60 "Ahora aplicamos la f\363rmula para los distintos valores \+
de h: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 8 "R
esumen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 11 "Ejercicio 6" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Consideramos
 el soporte de 5 puntos: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 
18 0 "x[0] := -2*h;" "6#>&%\"xG6#\"\"!,$*&\"\"#\"\"\"%\"hGF+!\"\"" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x[1] := -h;" "6#>&%\"xG6
#\"\"\",$%\"hG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x
[2] := 0;" "6#>&%\"xG6#\"\"#\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{XPPEDIT 18 0 "x[3] := h;" "6#>&%\"xG6#\"\"$%\"hG" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 18 0 "x[4] := 2*h;" "6#>&%\"xG6#\"\"%*&\"\"#\"
\"\"%\"hGF*" }}{PARA 0 "" 0 "" {TEXT -1 14 "y la funci\363n: " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "f(x) := cos(x);" "6#>-%
\"fG6#%\"xG-%$cosG6#F'" }}{PARA 0 "" 0 "" {TEXT -1 120 "Hallar una apr
oximaci\363n de f''(0) mediante la f\363rmula de derivaci\363n num\351
rica de tipo interpolatorio en el soporte dado: " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 18 0 "fsec(xast) := 1/12*(-f(xast-2*h)+16*f(xa
st-h)-30*f(xast)+16*f(xast+h)-f(xast+2*h))/(h^2);" "6#>-%%fsecG6#%%xas
tG**\"\"\"\"\"\"\"#7!\"\",,-%\"fG6#,&F'F**&\"\"#F*%\"hGF*F,F,*&\"#;F*-
F/6#,&F'F*F4F,F*F**&\"#IF*-F/6#F'F*F,*&\"#;F*-F/6#,&F'F*F4F*F*F*-F/6#,
&F'F**&\"\"#F*F4F*F*F,F**$F4\"\"#F," }}{PARA 0 "" 0 "" {TEXT -1 114 "p
ara los siguientes valores de h: 0'1, 0'01, 0'001, 0'0001, 0'00001, 0'
000001, 0'0000001. Se utilizar\341n 8 d\355gitos." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 258 10 "Respuesta." }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Apli
camos la f\363rmula de derivaci\363n num\351rica dada.  " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 55 "Para ello, empezamos introduciendo los da
tos conocidos." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Ahora aplicamos la f\363rmula para
 los distintos valores de h: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 256 8 "Resumen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}}{MARK "0 3 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }