{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 24 0 0 0 0 0 0 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 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 Output" -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 "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" -1 13 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" -1 256 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 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 54 "Equa\347\365es Diferenc iais e de Diferen\347as, uma Introdu\347\343o." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "# derivando uma express\343o\nexpr:=a*t^2-b;\ndi ff(expr,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,&*&%\"aG\"\"\" )%\"tG\"\"#F(F(%\"bG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"# \"\"\"%\"aGF&%\"tGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "# derivando um operador\nopd:=unapply(expr,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$opdGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&%\"aG\"\" \")9$\"\"#F/F/%\"bG!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "opd(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"bG!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "opd(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"aG\"\"\")%\"tG\"\"#F&F&%\"bG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(opd);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"tG6\"6$%)op eratorG%&arrowGF&,$*(\"\"#\"\"\"%\"aGF-9$F-F-F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "A maior parte do tempo, vou usar express\365es \+ e " }{TEXT 258 4 "diff" }{TEXT -1 25 " (em vez de operadores e " } {TEXT 257 1 "D" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "# escrevendo uma EDO\nedo:=diff(x,t)=a*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$edoG/\"\"!*&%\"aG\"\"\"%\"xGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "edo;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"! *&%\"aG\"\"\"%\"xGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "# n ote que temos que avisar EXPLICITAMENTE a depend\352ncia em t\nedo:=di ff(x(t),t)=a*x(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$edoG/-%%diffG 6$-%\"xG6#%\"tGF,*&%\"aG\"\"\"F)F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "edo;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-% \"xG6#%\"tGF**&%\"aG\"\"\"F'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# vamos verificar que exp(at) \351 solu\347\343o\nsubs(x(t)=ex p(a*t),edo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%$expG6#*& %\"aG\"\"\"%\"tGF,F-*&F+F,F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"aG\"\"\"-%$ex pG6#*&F%F&%\"tGF&F&F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "si mplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"aG\"\"\"-%$expG6#* &F%F&%\"tGF&F&F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Ok, j\341 se i substituir a fun\347\343o em uma edo e verificar se \351 solu\347 \343o. Vamos agora ver como resolver uma edo:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "dsolve(edo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"xG6#%\"tG*&%$_C1G\"\"\"-%$expG6#*&%\"aGF*F'F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Vamos fazer o gr\341fico de um exemplo mais con creto." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "a:=-2.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$!#@!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "edo;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$- %\"xG6#%\"tGF*,$*&$\"#@!\"\"\"\"\"F'F0F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# resolve com condi\347\343o inicial dada\ndsolve(\{e do,x(0)=10\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,$*&\" #5\"\"\"-%$expG6#,$*(\"#@F+F*!\"\"F'F+F2F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot( 10*exp(-21/10*t),t=0..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 290 326 326 {PLOTDATA 2 "6%-%'CURVESG6$7Z7$$\"\"!F)$\"#5F)7$ 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,Nh%oU)*p$G.\"F][l7$$\"31,++]Qk\\* )F2$\"3Wuk2D)=F)o!#D7$$\"3![LL3dg6<*F2$\"3F@H<*3xCK%F\\]l7$$\"3%ymmmw( Gp$*F2$\"3WRR?(\\h7&GF\\]l7$$\"3C++D\"oK0e*F2$\"3XJWG%>\"pH=F\\]l7$$\" 35,+v=5s#y*F2$\"3M;y&zw!o'>\"F\\]l7$F*$\"3U2>\"zUgDe(!#E-%'COLOURG6&%$ RGBG$\"*++++\"!\")F(F(-%+AXESLABELSG6$%\"tGQ!6\"-%%VIEWG6$;F(F*%(DEFAU LTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1 " }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "No modelo acima, se a>0, a p opula\347\343o cresce com limite infinito" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "edo:=diff(x(t),t)=2.1*x(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$edoG/-%%diffG6$-%\"xG6#%\"tGF,,$*&$\"#@!\"\"\"\"\"F) F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sol:=rhs(dsolve(\{e do,x(0)=10\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG,$*&\"#5\"\" \"-%$expG6#,$*(\"#@F(F'!\"\"%\"tGF(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "plot(sol,t=0..5);" }}{PARA 13 "" 1 "" {GLPLOT2D 307 329 329 {PLOTDATA 2 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FFhn$\"3SU!\\w%)eJ#HFhr7$$\"3uLLe*Gst!GFhn$\"3%y`n))y=Rj$Fhr7$$\"30+++ DRW9HFhn$\"3=`q!*\\%e,b%Fhr7$$\"3:++DJE>>IFhn$\"3OhSm;%*opcFhr7$$\"3F+ ]i!RU07$Fhn$\"3Y:erScU9qFhr7$$\"3+++v=S2LKFhn$\"3yo\"RGE'H%)))Fhr7$$\" 3Jmmm\"p)=MLFhn$\"3O0npeZg)4\"!#87$$\"3B++](=]@W$Fhn$\"3+3BUP\\^SF[v7$$\"3w**\\ilAFjSFhn$\"3U?7 %**f5!z]F[v7$$\"3yLLL$)*pp;%Fhn$\"3'4x6g24ZJ'F[v7$$\"3)RL$3xe,tUFhn$\" 3y1_[8u&)*)yF[v7$$\"3Cn;HdO=yVFhn$\"3Xy5Pb4vR)*F[v7$$\"3a+++D>#[Z%Fhn$ \"3_(*\\F6*o`?\"!#77$$\"3SnmT&G!e&e%Fhn$\"3?\"Red`:5_\"Fcy7$$\"3m+]P%3 7^j%Fhn$\"3cd0*o%yu(o\"Fcy7$$\"3#RLLL)Qk%o%Fhn$\"3%*>\")>*>dF(=Fcy7$$ \"3-n\"z>6but%Fhn$\"3u/)=of2C4#Fcy7$$\"37+]iSjE!z%Fhn$\"3Kvgio,#yL#Fcy 7$$\"3L+++DM\"3%[Fhn$\"3+0d5.mi*f#Fcy7$$\"3a+]P40O\"*[Fhn$\"3IpQ3j?v!* GFcy7$$\"3>]7.#Q?&=\\Fhn$\"3u())oF#3UgIFcy7$$\"3s+voa-oX\\Fhn$\"35D;[f ![+C$Fcy7$$\"3O]PMF,%G(\\Fhn$\"3#3W$[s#=-V$Fcy7$$\"\"&F)$\"3NkYUn-bJOF cy-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%+AXESLABELSG6$%\"tGQ!6\"-%%VIE WG6$;F(Fg\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 46.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Equa \347\343o log\355stica tenta resolver esse problema:" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "diff(y(t),t) = a*(1-y(t)/K)*y(t);" "6#/-%%diffG6$- %\"yG6#%\"tGF**(%\"aG\"\"\",&F-F-*&-%\"yG6#%\"tGF-%\"KG!\"\"F5F--F(6#F *F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 23 "com a,K>0 (consta ntes)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a:=2.1;\ny0:=10; \nK:=50;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"#@!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"KG\"#]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "edolog:=dif f(y(t),t)=a*(1-y(t)/K)*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'edo logG/-%%diffG6$-%\"yG6#%\"tGF,,$*($\"#@!\"\"\"\"\",&F2F2*&#F2\"#]F2F)F 2F1F2F)F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dsolve(\{edo log,y(0)=y0\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG,$*& \"#]\"\"\",&F+F+*&\"\"%F+-%$expG6#,$*(\"#@F+\"#5!\"\"F'F+F6F+F+F6F+" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "sollog:=rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sollogG,$*&\"#]\"\"\",&F(F(*&\"\"%F(-%$ex pG6#,$*(\"#@F(\"#5!\"\"%\"tGF(F3F(F(F3F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot(sollog,t=0..50);" }}{PARA 13 "" 1 "" {GLPLOT2D 364 272 272 {PLOTDATA 2 "6%-%'CURVESG6$7]o7$$\"\"!F)$\"#5F)7$$\"3ummTN @Ki8!#=$\"31=4RBe][7!#;7$$\"3\\LL$3FWYs#F/$\"3#*3vo;s1N:F27$$\"3'**** \\iSmp3%F/$\"3?OwyH2)[&=F27$$\"3)pmm;a)G\\aF/$\"3];g\"y(p.*>#F27$$\"3! HL$3x1h6oF/$\"3yxRT#*3sb$F27$ $\"35]i!*GER37FT$\"3#30w2'*o()z$F27$$\"3em\"z%\\v#pK\"FT$\"37i/xa076SF 27$$\"3G$3_+ZiaW\"FT$\"3/u=LD))*Q>%F27$$\"3)***\\i!R(*Rc\"FT$\"3Q8d2]< Q[VF27$$\"3RL3xJs1,=FT$\"3/c_JI&)e#e%F27$$\"3zmm\"H2P\"Q?FT$\"3[\"*\\[ kQpPZF27$$\"3YLek.pu/BFT$\"39)yDFh-n%[F27$$\"37+]PMnNrDFT$\"3i'>wABe7 \"\\F27$$\"3ymT5ll'z$GFT$\"31?-UO^\"*[\\F27$$\"3XLL$eRwX5$FT$\"3eK)pf) zoq\\F27$$\"3fLLL$eI8k$FT$\"3c>.!zvm/*\\F27$$\"3=ML$3x%3yTFT$\"3cRq*\\ x2p*\\F27$$\"3gmm\"z%4\\Y_FT$\"3Ey@ED=n**\\F27$$\"34LLeR-/PiFT$\"3%p7* *\\+f***\\F27$$\"3;++DcmpisFT$\"3yiA=V_****\\F27$$\"3vLLe*)>VB$)FT$\"3 o_([s[*****\\F27$$\"3o++DJbw!Q*FT$\"3?MGLW******\\F27$$\"3%ommTIOo/\"F 2$\"3O,'GV*******\\F27$$\"3^LL3_>jU6F2$\"3u%RT#********\\F27$$\"3E++]i ^Z]7F2$\"3w57#*********\\F27$$\"3/++](=h(e8F2$\"3U$*=**********\\F27$$ \"3A++]P[6j9F2$\"31%4***********\\F27$$\"3[L$e*[z(yb\"F2$\"3mv)******* ****\\F27$$\"3+nm;a/cq;F2$\"3k))************\\F27$$\"3mmmm;t,mF2Fav7$$\"3M +]i!f#=$3#F2Fav7$$\"37+](=xpe=#F2Fav7$$\"3-nm\"H28IH#F2Fav7$$\"3%om\"z pSS\"R#F2Fav7$$\"3cLL3_?`(\\#F2Fav7$$\"3fL$e*)>pxg#F2Fav7$$\"3D+]Pf4t. FF2Fav7$$\"3ZLLe*Gst!GF2Fav7$$\"39+++DRW9HF2Fav7$$\"3:++DJE>>IF2Fav7$$ \"35+]i!RU07$F2Fav7$$\"3$)***\\(=S2LKF2Fav7$$\"3nmmm\"p)=MLF2Fav7$$\"3 U++](=]@W$F2Fav7$$\"36L$e*[$z*RNF2Fav7$$\"3e++]iC$pk$F2Fav7$$\"3Sm;H2q cZPF2Fav7$$\"3Y+]7.\"fF&QF2Fav7$$\"3amm;/OgbRF2Fav7$$\"3I+]ilAFjSF2Fav 7$$\"3)RLLL)*pp;%F2Fav7$$\"3WLL3xe,tUF2Fav7$$\"3Wn;HdO=yVF2Fav7$$\"3a+ ++D>#[Z%F2Fav7$$\"3)om;aG!e&e%F2Fav7$$\"3wLLL$)Qk%o%F2Fav7$$\"3m+]iSjE !z%F2Fav7$$\"3u+]P40O\"*[F2Fav7$FavFav-%'COLOURG6&%$RGBG$\"*++++\"!\") F(F(-%+AXESLABELSG6$%\"tGQ!6\"-%%VIEWG6$;F(Fav%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "limit(sollog,t=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Ok, ao menos para os par\342metros acima a equa\347\343o log\355stica tem um limite finito." }}{PARA 0 "" 0 "" {TEXT -1 88 "Vimos que o Map le resolveu em forma fechada. Vamos ver se consegue para a e K quaisqu er." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "unassign('a','K','y0 ');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "edolog:=diff(y(t),t) =a*(1-y(t)/K)*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'edologG/-%%d iffG6$-%\"yG6#%\"tGF,*(%\"aG\"\"\",&F/F/*&F)F/%\"KG!\"\"F3F/F)F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dsolve(\{edolog,y(0)=y0\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG*(%#y0G\"\"\"%\"KGF* ,(F)F**&-%$expG6#,$*&%\"aGF*F'F*!\"\"F*F+F*F**&F.F*F)F*F4F4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "sol:=rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG*(%#y0G\"\"\"%\"KGF',(F&F'*&-%$expG6#,$*&% \"aGF'%\"tGF'!\"\"F'F(F'F'*&F+F'F&F'F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "# tentar limite sem dizer sinal de a\nlimit(sol,t=inf inity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limitG6$*(%#y0G\"\"\"%\" KGF(,(F'F(*&-%$expG6#,$*&%\"aGF(%\"tGF(!\"\"F(F)F(F(*&F,F(F'F(F3F3/F2% )infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(a>0); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(sol,t=infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"KG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Equa\347\365es de Diferen\347a" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Consideremos a recurs\343o abai xo" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "z (n+1)=alpha*z(n)+1" "6#/-%\"zG6#,&%\"nG\"\"\"F)F),&*&%&alphaGF)-F%6#F( F)F)F)F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "com z(0)=z0 (assumido conhecido)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "# se quis\351ssemos, digamos, z3\nz1:=alpha*z0+1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z1G,&*&%&alphaG\"\"\"%#z0GF(F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "z2:=alpha*z1+1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z2G,&*&%&alphaG\"\"\",&*&F'F(%#z0GF (F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "z3:=alph a*z2+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z3G,&*&%&alphaG\"\"\",&* &F'F(,&*&F'F(%#z0GF(F(F(F(F(F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "simplify(z3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,* *&)%&alphaG\"\"$\"\"\"%#z0GF(F(*$)F&\"\"#F(F(F&F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "# hmm. se fosse z(127) daria trabal ho. Loop\nfor k from 1 to 127 do\n z(k):=alpha*z(k-1)+1;\nend do:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(z(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&)%&alphaG\"\"$\"\"\"-%\"zG6#\"\"!F(F(*$) F&\"\"#F(F(F&F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simp lify(z(127));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,\\[l\"\"\"F$*$)%&alp haG\"#zF$F$*$)F'\"#5F$F$*$)F'\"#*)F$F$*$)F'\"#!*F$F$*$)F'\"#**F$F$*$)F '\"##*F$F$*$)F'\"#$)F$F$*$)F'\"#%)F$F$*$)F'\"#HF$F$*$)F'\"#gF$F$*$)F' \"#hF$F$*$)F'\"#nF$F$*$)F'\"#9F$F$*$)F'\"#IF$F$*$)F'\"#WF$F$*$)F'\"#XF $F$*$)F'\"#\")F$F$*$)F'\"$?\"F$F$*$)F'\"#:F$F$*$)F'\"#;F$F$*$)F'\"#DF$ F$*$)F'\"#SF$F$*$)F'\"#JF$F$*$)F'\"\"$F$F$*$)F'\"#TF$F$*$)F'\"#EF$F$*$ )F'\"#6F$F$*$)F'\"#7F$F$*$)F'\"#wF$F$*$)F'\"#PF$F$*$)F'\"#%*F$F$*$)F' \"#eF$F$*$)F'\"#iF$F$*$)F'\"#xF$F$*$)F'\"\"#F$F$*$)F'\"#!)F$F$F'F$*$)F '\"#qF$F$*$)F'\"#cF$F$*$)F'\"#dF$F$*$)F'\"#\\F$F$*$)F'\"#]F$F$*$)F'\"$ E\"F$F$*$)F'\"$5\"F$F$*$)F'\"#sF$F$*$)F'\"#FF$F$*$)F'\"#LF$F$*$)F'\"#K F$F$*$)F'\"$9\"F$F$*$)F'\"$@\"F$F$*$)F'\"#$*F$F$*$)F'\"$B\"F$F$*$)F'\" #lF$F$*$)F'\"#oF$F$*$)F'\"#uF$F$*$)F'\"#\"*F$F$*$)F'\"$,\"F$F$*$)F'\"$ 0\"F$F$*$)F'\"#)*F$F$*$)F'\"$8\"F$F$*$)F'\"$.\"F$F$*$)F'\"#8F$F$*$)F' \"#vF$F$*$)F'\"$:\"F$F$*$)F'\"#BF$F$*$)F'\"#CF$F$*$)F'\"#^F$F$*$)F'\"$ 4\"F$F$*$)F'\"#F$F$*$)F'\"$6\"F$F$*$)F'\"#?F$F$*$)F'\"#yF$F$*$)F'\"#_F $F$*$)F'\"#UF$F$*$)F'\"#GF$F$*$)F'\"$/\"F$F$*$)F'\"#'*F$F$*$)F'\"#jF$F $*$)F'\"#rF$F$*$)F'\"$;\"F$F$*$)F'\"#&)F$F$*$)F'\"$>\"F$F$*$)F'\"$-\"F $F$*$)F'\"#MF$F$*$)F'\"#[F$F$*$)F'\"#mF$F$*$)F'\"$3\"F$F$*$)F'\"#bF$F$ *$)F'\"#`F$F$*$)F'\"$A\"F$F$*$)F'\"$C\"F$F$*$)F'\"#')F$F$*&)F'\"$F\"F$ -%\"zG6#\"\"!F$F$*$)F'\"#AF$F$*$)F'\"#()F$F$*$)F'\"#tF$F$*$)F'\"#YF$F$ *$)F'\"#(*F$F$*$)F'\"#&*F$F$*$)F'\"#@F$F$*$)F'\"#fF$F$*$)F'\"\"%F$F$*$ )F'\"\"&F$F$*$)F'\"\"'F$F$*$)F'\"#kF$F$*$)F'\"#pF$F$*$)F'\"##)F$F$*$)F '\"#ZF$F$*$)F'\"#OF$F$*$)F'\"#VF$F$*$)F'\"#NF$F$*$)F'\"#aF$F$*$)F'\"$< \"F$F$*$)F'\"#RF$F$*$)F'\"$D\"F$F$*$)F'\"\"(F$F$*$)F'\"\")F$F$*$)F'\" \"*F$F$*$)F'\"$7\"F$F$*$)F'\"#))F$F$*$)F'\"$=\"F$F$*$)F'\"$1\"F$F$" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Fazendo esses casos acima, ficamo s tentados a achar que " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "z(n) = alpha ^n*z0+(alpha^n-1)/(alpha-1);" "6#/-%\"zG6#%\"nG,&*&)%&alphaGF'\"\"\"%# z0GF,F,*&,&)%&alphaG%\"nGF,F,!\"\"F,,&F+F,F,F3F3F," }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 84 "Para verificar isso, vamos substituir na \+ recurs\343o e ver se a fun\347\343o acima a resolve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rec:=x(n+1)=alpha*x(n)+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$recG/-%\"xG6#,&%\"nG\"\"\"F+F+,&*&%&alphaGF+-F' 6#F*F+F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "subs(x(n)=a lpha^n*z0+(alpha^(n)-1)/(alpha-1),\nx(n+1)=alpha^(n+1)*z0+(alpha^(n+1) -1)/(alpha-1),rec);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&)%&alphaG, &%\"nG\"\"\"F*F*F*%#z0GF*F**&,&F&F*F*!\"\"F*,&F'F*F*F.F.F*,&*&F'F*,&*& )F'F)F*F+F*F**&,&F4F*F*F.F*F/F.F*F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*& ,**&)%&alphaG,&%\"nG\"\"\"\"\"#F+F+%#z0GF+F+*&)F(,&F*F+F+F+F+F-F+!\"\" F/F+F+F1F+,&F(F+F+F1F1F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lhs(%)-rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "Ok, sabemos iterar na m\343o poucos valo res de z; fazer um loop para calcular valores mais altos (e.g. z(127)) . Mas como resolver de forma mais geral?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "rec;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#,&%\" nG\"\"\"F)F),&*&%&alphaGF)-F%6#F(F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "rsolve(rec,x(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,(*&-%\"xG6#\"\"!\"\"\")%&alphaG%\"nGF)F)*&,&F+F)F)!\"\"F/F*F)F)*&F) F)F.F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rsol:=%;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rsolG,(*&-%\"xG6#\"\"!\"\"\")%&alph aG%\"nGF+F+*&,&F-F+F+!\"\"F1F,F+F+*&F+F+F0F1F1" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "# achando x(127)\nsubs(n=127,rsol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%\"xG6#\"\"!\"\"\")%&alphaG\"$F\"F)F)*& ,&F+F)F)!\"\"F/F+F,F)*&F)F)F.F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "63 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }