{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 58 "Um exemplo de uma Equa \347\343o de Diferen\347a Nascida de uma EDO." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 197 "J\341 vimos que h\341 pa ralelos entre a an\341lise de equa\347\365es de diferen\347as (ED's) e diferenciais (EDO's). Vamos ver agora um exemplo concreto de como uma EDO pode inspirar uma ED que aproxima sua solu\347\343o." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Suponha que queira mos resolver a EDO" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {XPPEDIT 18 0 "diff(y(t),t)=sin(y(t))+t^2" "6#/-%%diffG6$-%\"yG6#%\" tGF*,&-%$sinG6#-F(6#F*\"\"\"*$F*\"\"#F1" }{TEXT -1 3 ", " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "y(0) = -Pi/2;" "6#/-%\"yG6#\"\"!,$*&%#PiG\" \"\"\"\"#!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "Note que a EDO acima nem \351 da forma " }{XPPEDIT 18 0 "diff(y(t),t)=a(t)*y(t)+b(t)" "6#/-%%diffG6$-%\"yG6#%\" tGF*,&*&-%\"aG6#F*\"\"\"-F(6#F*F0F0-%\"bG6#F*F0" }{TEXT -1 45 ", para \+ qual temos uma f\363rmula, nem separ\341vel." }}{PARA 0 "" 0 "" {TEXT -1 72 "N\343o somos s\363 n\363s que n\343o sabemos a solu\347\343o. O Maple tamb\351m n\343o consegue:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "restart;with(plots):with(DETools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "# define edo\nedo:=diff(y(t),t)=sin(y(t)) +t^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$edoG/-%%diffG6$-%\"yG6#%\" tGF,,&-%$sinG6#F)\"\"\"*$)F,\"\"#F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "# tenta resolver diretamente\ndsolve([edo,y(0)=-Pi/2] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Vamos supor que estamos int eressados num valor espec\355fico da fun\347\343o y, digamos, y(2)." } }{PARA 0 "" 0 "" {TEXT -1 92 "Ao menos sabemos quanto vale y em t=0. N \343o s\363 isso. Usando a pr\363pria EDO, sabemos tamb\351m que" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "eval( diff(y(t),t),t = 0) = sen(y(0));" "6#/-%%evalG6$-%%diffG6$-%\"yG6#%\"t GF-/F-\"\"!-%$senG6#-F+6#F/" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sen(-Pi /2)" "6#-%$senG6#,$*&%#PiG\"\"\"\"\"#!\"\"F+" }{TEXT -1 11 ", ou seja, " }{XPPEDIT 18 0 "eval(diff(y(t),t),t=0)=-1" "6#/-%%evalG6$-%%diffG6$ -%\"yG6#%\"tGF-/F-\"\"!,$\"\"\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "A id\351ia agora \351 \+ dividir o intervalo [0,2] em N subintervalos e supor que a fun\347\343 o y em cada subintervalo \351 uma reta." }}{PARA 0 "" 0 "" {TEXT -1 77 "J\341 sabemos como come\347ar o desenho, j\341 que y(0) e sua deri vada s\343o conhecidos." }}{PARA 0 "" 0 "" {TEXT -1 242 "Dando uma rou bada, at\351 d\341 para mandar o Maple fazer o desenho em que eu estou pensando (olhe s\363 para a figura, ignorando o comando do Maple). Se dividirmos o intervalo em N=4 subintervalos de tamanho h=2/N, estamos falando da seguinte figura:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "#n\343o olhe para c\341, s \363 para o desenho!\n\nN:=4:g1:=DEplot(edo,y(t),t=0..2,y=-3..0,[y(0)= -Pi/2],method=classical[foreuler],stepsize=2/N,arrows=none,linecolor=b lue,thickness=1,axes=boxed):\ndisplay(g1);" }}{PARA 13 "" 1 "" {GLPLOT2D 307 223 223 {PLOTDATA 2 "6&-%'CURVESG6&7S7$$\"\"!F)$!/\\zEjz q:!#87$$\"+nmmmT!#6$!3>+\\Y$*HY7;!#<7$$\"+MLLL$)F0$!3/+\\8g'HTl\"F37$$ \"++++]7!#5$!35+\\zEjz&p\"F37$$\"+nmmm;F<$!3>+\\\\$*HYPg'H\"zF37$$\"+-++]PF<$!37+\\*pK'zX>F37$$ \"+pmmmTF<$!3)***[p$*HY()>F37$$\"+OLL$e%F<$!3G+\\Rg'H\"H?F37$$\"+.+++] F<$!39+\\4Fjzq?F37$$\"+qmm;aF<$!3/$>;/gXp4#F37$$\"+PLLLeF<$!3D6#\\Q([4 B@F37$$\"+/++]iF<$!3YHAGZTC\\@F37$$\"+rmmmmF<$!3mZ_r?MRv@F37$$\"+QLL$3 (F<$!3(eE[TpU:?#F37$$\"+0+++vF<$!32%G\"en>pFAF37$$\"+smm;zF<$!3G-V,T7% QD#F37$$\"+RLLL$)F<$!3[?tW90**zAF37$$\"+1++]()F<$!3pQ.)yyRhI#F37$$\"+t mmm\"*F<$!3*oN881*GKBF37$$\"+SLL$e*F<$!35vjuM$Q%eBF37$$\"+,+++5!\"*$!3 -owO3we%Q#F37$$\"+ommT5F`s$!3;%Gc,UN:P#F37$$\"+NLL$3\"F`s$!3qNd)GB$[eB F37$$\"+-++D6F`s$!3y'=:c/JaM#F37$$\"+pmmm6F`s$!3(yjW$e)yBL#F37$$\"+OLL 37F`s$!3T*3u5nE$>BF37$$\"+.++]7F`s$!3]SN!Q[uiI#F37$$\"+qmm\"H\"F`s$!3- #*H`'HAKH#F37$$\"+PLLL8F`s$!37VCE4,#z6nAF 37$$\"+rmm;9F`s$!3wX8sMd1aAF37$$\"+QLLe9F`s$!3%oz]ua85C#F37$$\"+0+++:F `s$!3$zC!=g8'zA#F37$$\"+smmT:F`s$!3Oq;6))**>n@F37$$\"+RLL$e\"F`s$!3_%> oF37 $$\"+SLL3F37$$\"+2++]F`s$!3c&Q?5pZ.i\"F37$$\"+ULLe>F`s$!3]4pnCjef:F3 7$$\"\"#F)$!3G))yXr\\#))\\\"F3-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%&S TYLEG6#%%LINEG-%*THICKNESSG6#\"\"\"-%+AXESLABELSG6$Q\"t6\"Q%y(t)F^\\l- %*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F(Fgz;$!\"$F)F(" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 76 "Vamos ver como obter\355amos os cinco pontos que determ inam a curva azul acima." }}{PARA 0 "" 0 "" {TEXT -1 43 "O primeiro \+ \351 f\341cil, j\341 que \351 simplesmente (" }{XPPEDIT 18 0 "t[0]" "6 #&%\"tG6#\"\"!" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y[0]" "6#&%\"yG6#\" \"!" }{TEXT -1 9 ") = (0 , " }{XPPEDIT 18 0 "-Pi/2" "6#,$*&%#PiG\"\"\" \"\"#!\"\"F(" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 58 "Como es tamos assumindo que a fun\347\343o y(t) \351 uma reta entre " } {XPPEDIT 18 0 "t[0]" "6#&%\"tG6#\"\"!" }{TEXT -1 3 " e " }{XPPEDIT 18 0 "t[1]=t[0]+h" "6#/&%\"tG6#\"\"\",&&F%6#\"\"!F'%\"hGF'" }{TEXT -1 40 ", cujo coeficiente linear \351 dado por y'(" }{XPPEDIT 18 0 "t[0]" "6 #&%\"tG6#\"\"!" }{TEXT -1 30 ")=-1, tamb\351m sabemos o ponto (" } {XPPEDIT 18 0 "t[1]" "6#&%\"tG6#\"\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 12 "). De fato, " }{XPPEDIT 18 0 "y[1]=y[0]+(-1)*h" "6#/&%\"yG6#\"\"\",&&F%6#\"\"!F'*&,$F'!\"\"F'% \"hGF'F'" }{TEXT -1 20 ". Fazendo as contas:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "t[0]:=0;y[0]:=-Pi/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"yG6#\"\"!,$*&\"\"#!\"\"%#PiG\"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "h:=2/N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG#\"\" \"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "y[1]:=y[0]-h;" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "t[1]:=t[0]+h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"\",&*&\"\"#!\"\"%#PiGF'F+#F'F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#\"\"\"#F'\"\"#" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 163 "# gr\341fico desse segmento em vermelho\ntk :=3:\ng2:=pointplot([[t[0],y[0]],[t[1],y[1]]],connect=true,color=red,t hickness=tk):\ndisplay(g2,view=[0..2,-3..0],axes=boxed);" }}{PARA 13 " " 1 "" {GLPLOT2D 278 192 192 {PLOTDATA 2 "6%-%'CURVESG6%7$7$$\"\"!F)$! +Fjzq:!\"*7$$\"+++++]!#5$!+Fjzq?F,-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F( -%*THICKNESSG6#\"\"$-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F)\"\"#;!\"$F)" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Vamos sobrepor as duas figuras" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(g2,g1);" }}{PARA 13 "" 1 "" {GLPLOT2D 302 196 196 {PLOTDATA 2 "6'-%'CURVESG6%7$7$$\"\"!F)$!+Fjzq:!\"*7$$\"+++++]!#5$!+Fj zq?F,-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"$-F$6&7S7$ F($!/\\zEjzq:!#87$$\"+nmmmT!#6$!3>+\\Y$*HY7;!#<7$$\"+MLLL$)FH$!3/+\\8g 'HTl\"FK7$$\"++++]7F0$!35+\\zEjz&p\"FK7$$\"+nmmm;F0$!3>+\\\\$*HYPg'H\"zFK7$$\"+-++]PF0$!37+\\*pK 'zX>FK7$$\"+pmmmTF0$!3)***[p$*HY()>FK7$$\"+OLL$e%F0$!3G+\\Rg'H\"H?FK7$ $\"+.+++]F0$!39+\\4Fjzq?FK7$$\"+qmm;aF0$!3/$>;/gXp4#FK7$$\"+PLLLeF0$!3 D6#\\Q([4B@FK7$$\"+/++]iF0$!3YHAGZTC\\@FK7$$\"+rmmmmF0$!3mZ_r?MRv@FK7$ $\"+QLL$3(F0$!3(eE[TpU:?#FK7$$\"+0+++vF0$!32%G\"en>pFAFK7$$\"+smm;zF0$ !3G-V,T7%QD#FK7$$\"+RLLL$)F0$!3[?tW90**zAFK7$$\"+1++]()F0$!3pQ.)yyRhI# FK7$$\"+tmmm\"*F0$!3*oN881*GKBFK7$$\"+SLL$e*F0$!35vjuM$Q%eBFK7$$\"+,++ +5F,$!3-owO3we%Q#FK7$$\"+ommT5F,$!3;%Gc,UN:P#FK7$$\"+NLL$3\"F,$!3qNd)G B$[eBFK7$$\"+-++D6F,$!3y'=:c/JaM#FK7$$\"+pmmm6F,$!3(yjW$e)yBL#FK7$$\"+ OLL37F,$!3T*3u5nE$>BFK7$$\"+.++]7F,$!3]SN!Q[uiI#FK7$$\"+qmm\"H\"F,$!3- #*H`'HAKH#FK7$$\"+PLLL8F,$!37VCE4,#z6nAFK7 $$\"+rmm;9F,$!3wX8sMd1aAFK7$$\"+QLLe9F,$!3%oz]ua85C#FK7$$\"+0+++:F,$!3 $zC!=g8'zA#FK7$$\"+smmT:F,$!3Oq;6))**>n@FK7$$\"+RLL$e\"F,$!3_%>oFK7$$\"+SLL3 FK7$$\"+2++]F,$!3c&Q?5pZ.i\"FK7$$\"+ULLe>F,$!3]4pnCjef:FK7$$\"\"#F)$!3G)) yXr\\#))\\\"FK-F46&F6F(F(F7-%&STYLEG6#%%LINEG-F;6#\"\"\"-%+AXESLABELSG 6$Q\"t6\"Q%y(t)F^]l-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F(F]\\l;$!\"$F)F(" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Ok, at\351 agora parece que estamos indo bem. Mas como continuar? Afinal na largada tudo foi \+ t\343o f\341cil?" }}{PARA 0 "" 0 "" {TEXT -1 84 "Ora, vamos assumir ag ora que de novo estamos resolvendo nossa EDO, mas a partir de (" } {XPPEDIT 18 0 "t[1]" "6#&%\"tG6#\"\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT -1 79 ") e que a solu\347\343o a inda \351 um segmento de reta. Basta ent\343o descobrir quem \351 y'( " }{XPPEDIT 18 0 "t[1]" "6#&%\"tG6#\"\"\"" }{TEXT -1 16 ") para calcul ar " }{XPPEDIT 18 0 "y[2]" "6#&%\"yG6#\"\"#" }{TEXT -1 2 " (" } {XPPEDIT 18 0 "t[2]=t[1]+h" "6#/&%\"tG6#\"\"#,&&F%6#\"\"\"F+%\"hGF+" } {TEXT -1 56 " j\341 sabemos). Mas de novo, como a EDO continua valendo , " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "eval(diff(y(t),t),t = t[1]) = sen (y(t[1]))+t[1]^2;" "6#/-%%evalG6$-%%diffG6$-%\"yG6#%\"tGF-/F-&F-6#\"\" \",&-%$senG6#-F+6#&F-6#F1F1*$&F-6#F1\"\"#F1" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "sen(y[1])+t[1]^2;" "6#,&-%$senG6#&%\"yG6#\"\"\"F**$)&% \"tG6#F*\"\"#F*F*" }{TEXT -1 24 " , que podemos calcular:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m[1]:=sin(y[1])+t[1]^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"mG6#\"\"\",&-%$cosG6##F'\"\"#!\"\"#F'\" \"%F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "E da\355, " }{XPPEDIT 18 0 "y[2]=y[1]+m[1]*h" "6#/&%\"yG6#\"\"#,&&F%6#\"\"\"F+*&&%\"mG6#F+F+%\" hGF+F+" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "t [2]:=t[1]+h;\ny[2]:= y[1]+m[1]*h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%\"tG6#\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"#, (*&F'!\"\"%#PiG\"\"\"F*#\"\"$\"\")F**&#F,F'F,-%$cosG6##F,F'F,F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "# sobrepondo o segundo segm ento\ng3:=pointplot([[t[1],y[1]],[t[2],y[2]]],connect=true,color=green ,thickness=tk):\ndisplay(g1,g2,g3,view=[0..2,-3..0],axes=boxed);" }} {PARA 13 "" 1 "" {GLPLOT2D 264 174 174 {PLOTDATA 2 "6(-%'CURVESG6&7S7$ $\"\"!F)$!/\\zEjzq:!#87$$\"+nmmmT!#6$!3>+\\Y$*HY7;!#<7$$\"+MLLL$)F0$!3 /+\\8g'HTl\"F37$$\"++++]7!#5$!35+\\zEjz&p\"F37$$\"+nmmm;F<$!3>+\\\\$*H YPg'H\"zF37$$\"+-++]PF<$!3 7+\\*pK'zX>F37$$\"+pmmmTF<$!3)***[p$*HY()>F37$$\"+OLL$e%F<$!3G+\\Rg'H \"H?F37$$\"+.+++]F<$!39+\\4Fjzq?F37$$\"+qmm;aF<$!3/$>;/gXp4#F37$$\"+PL LLeF<$!3D6#\\Q([4B@F37$$\"+/++]iF<$!3YHAGZTC\\@F37$$\"+rmmmmF<$!3mZ_r? MRv@F37$$\"+QLL$3(F<$!3(eE[TpU:?#F37$$\"+0+++vF<$!32%G\"en>pFAF37$$\"+ smm;zF<$!3G-V,T7%QD#F37$$\"+RLLL$)F<$!3[?tW90**zAF37$$\"+1++]()F<$!3pQ .)yyRhI#F37$$\"+tmmm\"*F<$!3*oN881*GKBF37$$\"+SLL$e*F<$!35vjuM$Q%eBF37 $$\"+,+++5!\"*$!3-owO3we%Q#F37$$\"+ommT5F`s$!3;%Gc,UN:P#F37$$\"+NLL$3 \"F`s$!3qNd)GB$[eBF37$$\"+-++D6F`s$!3y'=:c/JaM#F37$$\"+pmmm6F`s$!3(yjW $e)yBL#F37$$\"+OLL37F`s$!3T*3u5nE$>BF37$$\"+.++]7F`s$!3]SN!Q[uiI#F37$$ \"+qmm\"H\"F`s$!3-#*H`'HAKH#F37$$\"+PLLL8F`s$!37VCE4,#z6nAF37$$\"+rmm;9F`s$!3wX8sMd1aAF37$$\"+QLLe9F`s$!3%oz]ua 85C#F37$$\"+0+++:F`s$!3$zC!=g8'zA#F37$$\"+smmT:F`s$!3Oq;6))**>n@F37$$ \"+RLL$e\"F`s$!3_%>oF37$$\"+SLL3F37$$\"+2++]F`s$!3c&Q?5pZ.i\"F37$$\"+ULL e>F`s$!3]4pnCjef:F37$$\"\"#F)$!3G))yXr\\#))\\\"F3-%'COLOURG6&%$RGBGF(F ($\"*++++\"!\")-%&STYLEG6#%%LINEG-%*THICKNESSG6#\"\"\"-F$6%7$7$F($!+Fj zq:F`s7$$\"+++++]F<$!+Fjzq?F`s-F\\[l6&F^[lF_[lF(F(-Fg[l6#\"\"$-F$6%7$F `\\l7$$Fi[lF)$!+3we%Q#F`s-F\\[l6&F^[lF(F_[lF(Fg\\l-%+AXESLABELSG6$Q\"t 6\"Q%y(t)Fg]l-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F)Fhz;!\"$F)" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "Ora, pelo visto aquele p rimeiro desenho que o Maple fez \351 exatamente o que estamos obtendo. De fato, esse processo de aproximar a resposta de uma EDO tem o nome \+ de " }{TEXT 257 15 "M\351todo de Euler" }{TEXT -1 2 " (" }{TEXT 258 8 "foreuler" }{TEXT -1 8 " vem de " }{TEXT 259 13 "forward Euler" } {TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "E o que tem a ver esse processo com ED's? Tem tudo a ver. " }}{PARA 0 "" 0 "" {TEXT -1 44 "Note que estamos resolvendo uma EDO d a forma" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(y(t),t)=f(t,y(t))" "6 #/-%%diffG6$-%\"yG6#%\"tGF*-%\"fG6$F*-F(6#F*" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 20 "onde f \351 conhecida. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Para evitar confus\343o entre a fun\347\343o cont\355nua y e os n\372meros que v\355nhamos ch amando de " }{XPPEDIT 18 0 "y[n]" "6#&%\"yG6#%\"nG" }{TEXT -1 28 ", va mos definir a sequ\352ncia " }{XPPEDIT 18 0 "upsilon[n];" "6#&%(upsilo nG6#%\"nG" }{TEXT -1 56 " (se l\352 \372psilon) que pretende aproximar y de forma a ser" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "upsilon[n]=y(t[0 ]+n*h)" "6#/&%(upsilonG6#%\"nG-%\"yG6#,&&%\"tG6#\"\"!\"\"\"*&F'F0%\"hG F0F0" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 62 "vemos que o que e st\341vamos fazendo corresponde a resolver a ED:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "upsilon[n+1]=upsilon [n]+m[n]*h" "6#/&%(upsilonG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)*&&%\"mG6#F(F )%\"hGF)F)" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "upsilon[0]=y[0]" "6#/&% (upsilonG6#\"\"!&%\"yG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Note que " }{XPPEDIT 18 0 "m[n] " "6#&%\"mG6#%\"nG" }{TEXT -1 28 " faz o papel da derivada y'(" } {XPPEDIT 18 0 "t[n]" "6#&%\"tG6#%\"nG" }{TEXT -1 16 "), que deve ser \+ " }{XPPEDIT 18 0 "f(t[n],upsilon[n])" "6#-%\"fG6$&%\"tG6#%\"nG&%(upsil onG6#F)" }{TEXT -1 52 ". Ou seja, substituindo todos esses dados, a ED fica" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "upsilon[n+1] = upsilon[n]+f(t[n],upsilon[n])*h;" "6#/&%(upsilonG 6#,&%\"nG\"\"\"F)F),&&F%6#F(F)*&-%\"fG6$&%\"tG6#F(&F%6#F(F)%\"hGF)F)" }{TEXT -1 19 " , " }{XPPEDIT 18 0 "upsilon[0]=y[0]" "6 #/&%(upsilonG6#\"\"!&%\"yG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Vamos resolver isso no Ma ple, para isso usando que " }{XPPEDIT 18 0 "t[n]=n*h" "6#/&%\"tG6#%\"n G*&F'\"\"\"%\"hGF)" }{TEXT -1 97 ", onde h j\341 foi calculado. Lembre mos tamb\351m que quando n=4 chegamos ao ponto onde desejamos, t=2." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ed:=upsilon(n+1)=upsilon(n)+(sin(upsilon(n))+(n*h)^2)*h;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#edG/-%(upsilonG6#,&%\"nG\"\"\"F+F+, (-F'6#F*F+*&#F+\"\"#F+-%$sinG6#F-F+F+*&\"\")!\"\"F*F1F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "rsolve(\{ed,upsilon(0)=y[0]\},upsil on(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'rsolveG6$<$/-%(upsilonG6 #\"\"!,$*&\"\"#!\"\"%#PiG\"\"\"F//-F)6#,&%\"nGF1F1F1,(-F)6#F6F1*&#F1F. F1-%$sinG6#F8F1F1*&\"\")F/F6F.F1F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "O Maple n\343o tem uma f\363rmula fechada para essa ED (nem ni ngu\351m). Vamos mandar ent\343o ele gerar os pontos que nos interessa m. Para isso, podemos usar um loop:" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "upsilon[0]:=y[0]:\nfor i from 0 to 3 do\n upsilon[i+1]:=upsilon[i]+(sin(upsilon[i])+(i*h)^2)* h;\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "# gr\341fic o dos pontos\npontos:=[seq([j*h,upsilon[j]],j=0..4)];\nG:=pointplot(po ntos,connect=true,thickness=tk,color=orange):\ndisplay(G,view=[0..2,-3 ..0],axes=boxed);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'pontosG7'7$\" \"!,$*&\"\"#!\"\"%#PiG\"\"\"F+7$#F-F*,&*&F*F+F,F-F+#F-F*F+7$F-,(*&F*F+ F,F-F+#\"\"$\"\")F+*&#F-F*F--%$cosG6#F/F-F+7$#F7F*,**&F*F+F,F-F+#F-F8F -*&#F-F*F-F;F-F+*&#F-F*F--F<6#,&#F7F8F-*&F/F-F;F-F-F-F+7$F*,,*&F*F+F,F -F+#\"\"&\"\"%F-*&#F-F*F-F;F-F+*&#F-F*F-FGF-F+*&#F-F*F--F<6#,(#F-F8F+* &F/F-F;F-F-*&F/F-FGF-F-F-F+" }}{PARA 13 "" 1 "" {GLPLOT2D 292 182 182 {PLOTDATA 2 "6%-%'CURVESG6%7'7$$\"\"!F)$!+Fjzq:!\"*7$$\"+++++]!#5$!+Fj zq?F,7$$\"\"\"F)$!+3we%Q#F,7$$\"+++++:F,$!+i8'zA#F,7$$\"\"#F)$!+s\\#)) \\\"F,-%'COLOURG6&%$RGBG$\")+++!)!\")$\")Vyg>FHFI-%*THICKNESSG6#\"\"$- %*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F)F?;!\"$F)" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Novamente, vamos dar a cara a tapa, sobrepondo os gr\341f icos." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display(g1,G,view= [0..2,-3..0],axes=boxed);" }}{PARA 13 "" 1 "" {GLPLOT2D 287 210 210 {PLOTDATA 2 "6'-%'CURVESG6&7S7$$\"\"!F)$!/\\zEjzq:!#87$$\"+nmmmT!#6$!3 >+\\Y$*HY7;!#<7$$\"+MLLL$)F0$!3/+\\8g'HTl\"F37$$\"++++]7!#5$!35+\\zEjz &p\"F37$$\"+nmmm;F<$!3>+\\\\$*HYPg'H\"zF37$$\"+-++]PF<$!37+\\*pK'zX>F37$$\"+pmmmTF<$!3)***[p$*HY ()>F37$$\"+OLL$e%F<$!3G+\\Rg'H\"H?F37$$\"+.+++]F<$!39+\\4Fjzq?F37$$\"+ qmm;aF<$!3/$>;/gXp4#F37$$\"+PLLLeF<$!3D6#\\Q([4B@F37$$\"+/++]iF<$!3YHA GZTC\\@F37$$\"+rmmmmF<$!3mZ_r?MRv@F37$$\"+QLL$3(F<$!3(eE[TpU:?#F37$$\" +0+++vF<$!32%G\"en>pFAF37$$\"+smm;zF<$!3G-V,T7%QD#F37$$\"+RLLL$)F<$!3[ ?tW90**zAF37$$\"+1++]()F<$!3pQ.)yyRhI#F37$$\"+tmmm\"*F<$!3*oN881*GKBF3 7$$\"+SLL$e*F<$!35vjuM$Q%eBF37$$\"+,+++5!\"*$!3-owO3we%Q#F37$$\"+ommT5 F`s$!3;%Gc,UN:P#F37$$\"+NLL$3\"F`s$!3qNd)GB$[eBF37$$\"+-++D6F`s$!3y'=: c/JaM#F37$$\"+pmmm6F`s$!3(yjW$e)yBL#F37$$\"+OLL37F`s$!3T*3u5nE$>BF37$$ \"+.++]7F`s$!3]SN!Q[uiI#F37$$\"+qmm\"H\"F`s$!3-#*H`'HAKH#F37$$\"+PLLL8 F`s$!37VCE4,#z6nAF37$$\"+rmm;9F`s$!3wX8sM d1aAF37$$\"+QLLe9F`s$!3%oz]ua85C#F37$$\"+0+++:F`s$!3$zC!=g8'zA#F37$$\" +smmT:F`s$!3Oq;6))**>n@F37$$\"+RLL$e\"F`s$!3_%>oF37$$\"+SLL3F37$$\"+2++] F`s$!3c&Q?5pZ.i\"F37$$\"+ULLe>F`s$!3]4pnCjef:F37$$\"\"#F)$!3G))yXr\\#) )\\\"F3-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%&STYLEG6#%%LINEG-%*THICKN ESSG6#\"\"\"-F$6%7'7$F($!+Fjzq:F`s7$$\"+++++]F<$!+Fjzq?F`s7$$Fi[lF)$!+ 3we%Q#F`s7$$\"+++++:F`s$!+i8'zA#F`s7$Fgz$!+s\\#))\\\"F`s-F\\[l6&F^[l$ \")+++!)Fa[l$\")Vyg>Fa[lFe]l-Fg[l6#\"\"$-%+AXESLABELSG6$Q\"t6\"Q%y(t)F ^^l-%*AXESSTYLEG6#%$BOXG-%%VIEWG6$;F)Fhz;!\"$F)" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 43.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Xeque mate!" }}{PARA 0 "" 0 "" {TEXT -1 149 "Ou quase, j\341 que a solu\347\343o que o Maple retorna simpelsme nte dando um DEplot sem as op\347\365es para for\347ar o m\351todo ao \+ ser esse m\351todo tosco \351 bem melhor:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 149 "unassign('y','t');\ng:=DEplot(edo,y(t),t=0..2,y=-3 ..0,[y(0)=-Pi/2],arrows=none,linecolor=blue,thickness=1,axes=boxed):\n display(g,G,view=[0..2,-3..0]);" }}{PARA 13 "" 1 "" {GLPLOT2D 314 202 202 {PLOTDATA 2 "6'-%'CURVESG6&7S7$$\"\"!F)$!/\\zEjzq:!#87$$\"+nmmmT!# 6$!30lO6YoU7;!#<7$$\"+MLLL$)F0$!3)fndBuSQl\"F37$$\"++++]7!#5$!3a\"[[CZ C[p\"F37$$\"+nmmm;F<$!3x%46w5l^t\"F37$$\"+MLL$3#F<$!3C^cy[OluF37$$\"+pmmmTF<$!3]szD=Ac_>F37$$ \"+OLL$e%F<$!3?WU_$p-I)>F37$$\"+.+++]F<$!3&yn#p!)=O6?F37$$\"+qmm;aF<$! 3FbV*=c,v.#F37$$\"+PLLLeF<$!3/`y+'p#Hh?F37$$\"+/++]iF<$!31M#f'=ah#3#F3 7$$\"+rmmmmF<$!3$Q0AA/f85#F37$$\"+QLL$3(F<$!3y&)\\.A4U<@F37$$\"+0+++vF <$!3Py_)[Z/28#F37$$\"+smm;zF<$!3\\PK76H7T@F37$$\"+RLLL$)F<$!3+\"Hr'=Jf [@F37$$\"+1++]()F<$!3))RYxb^.`@F37$$\"+tmmm\"*F<$!3%GE,5DsV:#F37$$\"+S LL$e*F<$!3#)=>C.3`_@F37$$\"+,+++5!\"*$!3Y19:\"QSu9#F37$$\"+ommT5F`s$!3 @YDtK'H!R@F37$$\"+NLL$3\"F`s$!3%p$32x!Gs7#F37$$\"+-++D6F`s$!3\"**Q7j=k >6#F37$$\"+pmmm6F`s$!3SAUYA/;$4#F37$$\"+OLL37F`s$!3SgURPKtq?F37$$\"+.+ +]7F`s$!3HL8$G2$fW?F37$$\"+qmm\"H\"F`s$!3OhZW$4WY,#F37$$\"+PLLL8F`s$!3 C7c.l*y2)>F37$$\"+/++v8F`s$!3*3aakqzG%>F37$$\"+rmm;9F`s$!3lkO7A4\"3!>F 37$$\"+QLLe9F`s$!3GK!>Ye+$zu\"F37$$\"+RLL$e\"F`s$!3u:,R)H3uo\"F37$$\"+1++D;F`s$!3l0 l()oEp@;F37$$\"+tmmm;F`s$!3QRM?.VZ]:F37$$\"+SLL3F`s$! 3Ho&=uo!4O)*!#=7$$\"+ULLe>F`s$!3V>j-ww)Qg)Faz7$$\"\"#F)$!3U)*esJ#*yosF az-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%&STYLEG6#%%LINEG-%*THICKNESSG6 #\"\"\"-F$6%7'7$F($!+Fjzq:F`s7$$\"+++++]F<$!+Fjzq?F`s7$$Fj[lF)$!+3we%Q #F`s7$$\"+++++:F`s$!+i8'zA#F`s7$Fhz$!+s\\#))\\\"F`s-F][l6&F_[l$\")+++! )Fb[l$\")Vyg>Fb[lFf]l-Fh[l6#\"\"$-%+AXESLABELSG6$Q\"t6\"Q%y(t)F_^l-%*A XESSTYLEG6#%$BOXG-%%VIEWG6$;F)Fiz;!\"$F)" 1 2 0 1 10 0 2 9 1 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Para contornar isso, podemos tentar resolver no ssa ED com mais subdivis\365es. Tomando, digamos N=100:" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "N:=1 00;h:=2/N;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG#\"\"\"\"#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "upsilon[0]:=evalf(-Pi/2):\nfor i from 0 to N do\n upsilon[i+1]:=upsilon[i]+(sin(upsilon[i])+(i*h)^2)*h;\nend do:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "# gr\341fico dos pontos\npo ntos2:=[seq([j*h,upsilon[j]],j=0..N)]:\nG2:=pointplot(pontos2,connect= true,thickness=tk,color=green):\ndisplay(G2,g,view=[0..2,-3..0],axes=b oxed);" }}{PARA 13 "" 1 "" {GLPLOT2D 290 254 254 {PLOTDATA 2 "6'-%'CUR VESG6%7aq7$$\"\"!F)$!+Fjzq:!\"*7$$\"+++++?!#6$!+Fjz!f\"F,7$$\"+++++SF0 $!+FVy5;F,7$$\"+++++gF0$!+RjtI;F,7$$\"+++++!)F0$!+A%G1l\"F,7$$\"+++++5 !#5$!+CnVq;F,7$$\"+++++7FE$!+Cv8!p\"F,7$$\"+++++9FE$!+qsq4F,7$$\"+++++QFE$!+Ud8E>F,7$$F5FE$!+6 $)*>%>F,7$$\"+++++UFE$!+^hVd>F,7$$\"+++++WFE$!+yF,7$$\"+++++YFE$!+ O#op)>F,7$$\"+++++[FE$!++\"H5+#F,7$$\"+++++]FE$!+u%)f9?F,7$$\"+++++_FE $!+()4mF?F,7$$\"+++++aFE$!+(y,-/#F,7$$\"+++++cFE$!+Nl?_?F,7$$\"+++++eF E$!+#RhO1#F,7$$F:FE$!+4Ibu?F,7$$\"+++++iFE$!+:&o[3#F,7$$\"+++++kFE$!+- bf%4#F,7$$\"+++++mFE$!+1?s.@F,7$$\"+++++oFE$!+$\\OA6#F,7$$\"+++++qFE$! +Sy7?@F,7$$\"+++++sFE$!+:`QF@F,7$$\"+++++uFE$!+c&)*R8#F,7$$\"+++++wFE$ !+_v&*R@F,7$$\"+++++yFE$!+=EDX@F,7$$F?FE$!+vV()\\@F,7$$\"+++++#)FE$!+E P\"Q:#F,7$$\"+++++%)FE$!+J=1d@F,7$$\"+++++')FE$!+$35'f@F,7$$\"+++++))F E$!+%3]9;#F,7$$\"+++++!*FE$!+?Odi@F,7$$\"+++++#*FE$!+JE(H;#F,7$$\"++++ +%*FE$!+*=RE;#F,7$$\"+++++'*FE$!+oach@F,7$$\"+++++)*FE$!+=Puf@F,7$$\" \"\"F)$!+Ni;d@F,7$$\"++++?5F,$!+K`#Q:#F,7$$\"++++S5F,$!+4Lr\\@F,7$$\"+ +++g5F,$!+BC#[9#F,7$$\"++++!3\"F,$!+a[9R@F,7$$\"+++++6F,$!+uEnK@F,7$$ \"++++?6F,$!+8yRD@F,7$$\"++++S6F,$!+??J<@F,7$$\"++++g6F,$!+JoS3@F,7$$ \"++++!=\"F,$!+ENn)4#F,7$$FJF,$!+\"4.\")3#F,7$$\"++++?7F,$!+whow?F,7$$ \"++++S7F,$!+[ITk?F,7$$\"++++g7F,$!+ZNF^?F,7$$\"++++!G\"F,$!+OqDP?F,7$ $\"+++++8F,$!+[BNA?F,7$$\"++++?8F,$!+Kxa1?F,7$$\"++++S8F,$!+&zI)*)>F,7 $$\"++++g8F,$!+R%)=s>F,7$$\"++++!Q\"F,$!+*z1O&>F,7$$FOF,$!+p62M>F,7$$ \"++++?9F,$!+Kfc8>F,7$$\"++++S9F,$!+![u?*=F,7$$\"++++g9F,$!+K\"z&p=F,7 $$\"++++![\"F,$!+U51Y=F,7$$\"+++++:F,$!+3,]@=F,7$$\"++++?:F,$!+p[(ez\" F,7$$\"++++S:F,$!+,C;pQ8u\"F,7$$\"++++!e\"F,$!+eg P7d^Z\"F,7$$\"++++SF,$!+\"yIR1\"F,7$$\"++++?>F,$!+*=;#45F,7$$ \"++++S>F,$!+J[;C&*FE7$$\"++++g>F,$!+#e3W$*)FE7$$\"++++!)>F,$!+a^$>K)F E7$$\"\"#F)$!+Vrt&o(FE-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-%*THICKNESS G6#\"\"$-F$6&7S7$F($!/\\zEjzq:!#87$$\"+nmmmTF0$!30lO6YoU7;!#<7$$\"+MLL L$)F0$!3)fndBuSQl\"F\\\\m7$$\"++++]7FE$!3a\"[[CZC[p\"F\\\\m7$$\"+nmmm; FE$!3x%46w5l^t\"F\\\\m7$$\"+MLL$3#FE$!3C^cy[OluF\\\\m7$$\"+pmmmTFE$!3]szD=Ac_> F\\\\m7$$\"+OLL$e%FE$!3?WU_$p-I)>F\\\\m7$$\"+.+++]FE$!3&yn#p!)=O6?F\\ \\m7$$\"+qmm;aFE$!3FbV*=c,v.#F\\\\m7$$\"+PLLLeFE$!3/`y+'p#Hh?F\\\\m7$$ \"+/++]iFE$!31M#f'=ah#3#F\\\\m7$$\"+rmmmmFE$!3$Q0AA/f85#F\\\\m7$$\"+QL L$3(FE$!3y&)\\.A4U<@F\\\\m7$$\"+0+++vFE$!3Py_)[Z/28#F\\\\m7$$\"+smm;zF E$!3\\PK76H7T@F\\\\m7$$\"+RLLL$)FE$!3+\"Hr'=Jf[@F\\\\m7$$\"+1++]()FE$! 3))RYxb^.`@F\\\\m7$$\"+tmmm\"*FE$!3%GE,5DsV:#F\\\\m7$$\"+SLL$e*FE$!3#) =>C.3`_@F\\\\m7$$\"+,+++5F,$!3Y19:\"QSu9#F\\\\m7$$\"+ommT5F,$!3@YDtK'H !R@F\\\\m7$$\"+NLL$3\"F,$!3%p$32x!Gs7#F\\\\m7$$\"+-++D6F,$!3\"**Q7j=k> 6#F\\\\m7$$\"+pmmm6F,$!3SAUYA/;$4#F\\\\m7$$\"+OLL37F,$!3SgURPKtq?F\\\\ m7$$\"+.++]7F,$!3HL8$G2$fW?F\\\\m7$$\"+qmm\"H\"F,$!3OhZW$4WY,#F\\\\m7$ $\"+PLLL8F,$!3C7c.l*y2)>F\\\\m7$$\"+/++v8F,$!3*3aakqzG%>F\\\\m7$$\"+rm m;9F,$!3lkO7A4\"3!>F\\\\m7$$\"+QLLe9F,$!3GK!>Ye+$zu\"F\\\\m7$$\"+RLL$e\"F,$!3 u:,R)H3uo\"F\\\\m7$$\"+1++D;F,$!3l0l()oEp@;F\\\\m7$$\"+tmmm;F,$!3QRM?. VZ]:F\\\\m7$$\"+SLL3F,$!3Ho&=uo!4O)*! #=7$$\"+ULLe>F,$!3V>j-ww)Qg)F^jm7$Fajl$!3U)*esJ#*yosF^jm-Ffjl6&FhjlF(F (Fijl-%&STYLEG6#%%LINEG-F][m6#F\\[l-%+AXESLABELSG6$Q\"t6\"Q%y(t)Fc[n-% *AXESSTYLEG6#%$BOXG-%%VIEWG6$;F)Fbjl;!\"$F)" 1 2 0 1 10 0 2 9 1 2 2 1.000000 46.000000 44.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "a situa\347\343o fica bem melhor." }}}}{MARK "2 6 0 0" 11 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }