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}{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 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 1 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 0 "" 0 "" {TEXT 256 19 "Aula Pr\341tica sobre " } }{PARA 0 "" 0 "" {TEXT 272 19 "Integra\347\343o Num\351rica" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 30 "C\341lculo \+ Num\351rico, Turma A, UnB" }}{PARA 257 "" 0 "" {TEXT -1 27 "Prof. Maur icio Ayala-Rinc\363n" }}{PARA 257 "" 0 "" {TEXT -1 19 "16 de junho de \+ 2005" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 25 "Nome: " }}{PARA 256 "" 0 "" {TEXT -1 10 "Matr\355c ula:" }}{PARA 256 "" 0 "" {TEXT -1 6 "Nome: " }}{PARA 256 "" 0 "" {TEXT -1 10 "Matr\355cula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Aspetos te\363ric os devem ser consultados na biblio" }{TEXT 18 0 "" }{TEXT -1 21 "grafi a da disciplina." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 22 "I. Regra dos Trap\351zios" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "A integral de uma fun\347\343o " } {TEXT 274 1 "f" }{TEXT -1 14 " no intervalo " }{TEXT 275 5 "[a,b]" } {TEXT -1 65 " \351 aproximada pela \341rea de trap\351zios. O interva lo \351 dividido em " }{TEXT 276 1 "m" }{TEXT -1 15 " sub-intervalos" }}{PARA 0 "" 0 "" {TEXT -1 21 "de igual espa\347amento " }{TEXT 277 1 "h" }{TEXT -1 3 " ( " }{TEXT 278 11 "(b-a)/h = m" }{TEXT -1 15 " ), co m pontos " }{TEXT 279 2 "a=" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT 280 2 ", " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT 281 9 ", ..., b=" }{XPPEDIT 18 0 "x[m];" "6#&%\"xG6#%\"mG" }{TEXT -1 29 ", e para cada sub-intervalo [" }{XPPEDIT 18 0 "x[i];" "6#&%\"xG6#% \"iG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[i+1];" "6#&%\"xG6#,&%\"iG\"\" \"F(F(" }{TEXT -1 18 "], soma-se a \341rea " }}{PARA 0 "" 0 "" {TEXT -1 32 "do trap\351zio de altura h e bases " }{TEXT 284 2 "f(" } {XPPEDIT 18 0 "x[i];" "6#&%\"xG6#%\"iG" }{TEXT 285 1 ")" }{TEXT -1 3 " e " }{TEXT 286 2 "f(" }{XPPEDIT 18 0 "x[i+1]" "6#&%\"xG6#,&%\"iG\"\" \"F(F(" }{TEXT 287 1 ")" }{TEXT -1 11 "; i. e., " }{TEXT 282 5 "h (f (" }{XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT 288 4 ")+f(" } {XPPEDIT 18 0 "x[i+1]" "6#&%\"xG6#,&%\"iG\"\"\"F(F(" }{TEXT 283 4 "))/ 2" }{TEXT -1 38 ". No total, obt\351m-se a seguinte soma:" }}{PARA 0 "" 0 "" {TEXT -1 55 " \+ " }{XPPEDIT 18 0 "Sum(1/2*(f(x[i])+f(x[i+1]))*h,i = 0 .. m-1);" "6#-%$SumG6$**\"\"\"F'\"\"#!\"\",&-%\"fG6#&%\"xG6#%\"iGF'-F,6#&F/6#,&F 1F'F'F'F'F'%\"hGF'/F1;\"\"!,&%\"mGF'F'F)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "que pode ser escrita como:" }}{PARA 0 "" 0 "" {TEXT -1 54 " \+ " }{XPPEDIT 18 0 "h*(1/2*f(x[0])+sum(f(x[i]),i \+ = 1 .. m-1)+1/2*f(x[m]));" "6#*&%\"hG\"\"\",(*(F%F%\"\"#!\"\"-%\"fG6#& %\"xG6#\"\"!F%F%-%$sumG6$-F+6#&F.6#%\"iG/F8;F%,&%\"mGF%F%F)F%*(F%F%F(F )-F+6#&F.6#F " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "trapezios := proc (f,a,b,m) \n local h, acum, i; \n h:= (b-a)/m; \n acum := (f(a)+f( b))/2; \n for i to m-1 do acum := acum + f(a+h*i) end do; \n acum : = evalf(acum * h); \nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*t rapeziosGf*6&%\"fG%\"aG%\"bG%\"mG6%%\"hG%%acumG%\"iG6\"F/C&>8$*&,&9&\" \"\"9%!\"\"F69'F8>8%,&-9$6#F7#F6\"\"#*&F@F6-F>6#F5F6F6?(8&F6F6,&F9F6F6 F8%%trueG>F;,&F;F6-F>6#,&F7F6*&F2F6FFF6F6F6>F;-%&evalfG6#*&F;F6F2F6F/F /F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 26 "1. Utilize o procedimento " }{TEXT 260 9 "trapezios" }{TEXT 261 61 " para aproximar com dez subintervalos a integral das fun\347 \365es:" }}{PARA 0 "" 0 "" {TEXT 297 16 " 1.1 " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT 292 14 " no intervalo " }{TEXT 298 6 "[0,10]" }{TEXT 299 2 "; " }}{PARA 0 "" 0 "" {TEXT 300 15 " \+ 1.2 " }{TEXT 293 3 "exp" }{TEXT 294 14 " no intervalo " }{TEXT 295 6 "[0,1];" }{TEXT 296 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := x -> x^2; \ntrapezios(f ,0,10,10); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$% )operatorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$N$\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "trapezios(exp,0,1,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#\\8 (> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 26 "2. Util ize o procedimento " }{TEXT 262 9 "trapezios" }{TEXT 263 39 " para apr oximar a integral das fun\347\365es:" }}{PARA 256 "" 0 "" {TEXT 301 18 " 2.1 " }{TEXT 305 3 "sin" }{TEXT 306 14 " no interval o " }{TEXT 303 3 "[0," }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 304 1 "] " }{TEXT 302 3 "; " }}{PARA 256 "" 0 "" {TEXT 307 18 " 2 .2 " }{TEXT 308 3 "cos" }{TEXT 309 14 " no intervalo " }{TEXT 310 2 "[ -" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT 316 3 "/2," }{XPPEDIT 18 0 "Pi ;" "6#%#PiG" }{TEXT 311 3 "/2]" }{TEXT 312 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "trapezios(s in,0,Pi,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+QN_$)>!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "trapezios(cos,-Pi/2,Pi/2,10) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+QN_$)>!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 27 "Erro da regra dos trap\351zios" }}{PARA 0 "" 0 "" {TEXT -1 50 "O erro deste m\351todo \351 dado por \+ " }{XPPEDIT 18 0 "E[tr] = -1/12*m*h^3*diff(f(xi),`$`(xi,2));" "6#/ &%\"EG6#%#trG,$*,\"\"\"F*\"#7!\"\"%\"mGF*%\"hG\"\"$-%%diffG6$-%\"fG6#% #xiG-%\"$G6$F6\"\"#F*F," }}{PARA 0 "" 0 "" {TEXT -1 6 "onde " } {XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 19 " est\341 no intervalo " } {TEXT 313 5 "(a,b)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "Ass im, o erro pode ser estimado por " }{XPPEDIT 18 0 "abs(E[t r]) <= 1/12*(b-a)*h^2*M[2];" "6#1-%$absG6#&%\"EG6#%#trG*,\"\"\"F,\"#7! \"\",&%\"bGF,%\"aGF.F,%\"hG\"\"#&%\"MG6#F3F," }}{PARA 0 "" 0 "" {TEXT -1 5 "onde " }{XPPEDIT 18 0 "M[2];" "6#&%\"MG6#\"\"#" }{TEXT -1 16 " \+ \351 o m\341ximo de " }{XPPEDIT 18 0 "abs(diff(f(x),`$`(x,2)));" "6#- %$absG6#-%%diffG6$-%\"fG6#%\"xG-%\"$G6$F,\"\"#" }{TEXT -1 9 " para um \+ " }{TEXT 314 1 "x" }{TEXT -1 14 " no intervalo " }{TEXT 315 5 "[a,b]" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 107 "3. Calcule o n\372mero m\355nimo de sub-intervalos ne cess\341rios para computar as integrais das quatro fun\347\365es das" }}{PARA 256 "" 0 "" {TEXT -1 110 "quest\365es anteriores utilizando a \+ regra do trap\351zio com precis\343o 0.00001. Tente calcular tamb \351m as integrais " }}{PARA 256 "" 0 "" {TEXT -1 19 "com dita preciss \343o." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "m >= evalf(sqrt(10^5*10^(3)*2/12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#1$\"+1H[#3%!\"'%\"mG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "trapezios(f,0,10,4083);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LMLLL!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "m >= evalf(sqrt(10^5*exp(1)/12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#1$\"+ >(p]]\"!\"(%\"mG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "trapezi os(exp,0,1,151);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2\")G= " 0 "" {MPLTEXT 1 0 33 "m >= evalf(sqrt(10^5*Pi^3* 1/12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#1$\"+aZ;$3&!\"(%\"mG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "trapezios(sin,0,Pi,509);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[O****>!\"*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 30 "trapezios(cos,-Pi/2,Pi/2,509);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+_O****>!\"*" }}}{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 264 24 "II. Regra 1/3 de Simpson" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 105 "Do mesmo modo que a regra dos trap \351zios este m\351todo utiliza f\363rmulas de Newton-Cotes, que s\343 o da forma " }}{PARA 0 "" 0 "" {TEXT -1 41 " \+ " }{XPPEDIT 18 0 "A[0]*f(x[0])+A[1]*f(x[1]);" "6#,&*& &%\"AG6#\"\"!\"\"\"-%\"fG6#&%\"xG6#F(F)F)*&&F&6#F)F)-F+6#&F.6#F)F)F)" }{TEXT -1 9 " + ... + " }{XPPEDIT 18 0 "A[m]*f(x[m]);" "6#*&&%\"AG6#% \"mG\"\"\"-%\"fG6#&%\"xG6#F'F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "onde os " }{XPPEDIT 18 0 "A[i];" "6#&%\"AG6#%\"iG" }{TEXT -1 60 "'s s\343o coeficientes adequados, para aproximar a integral de \+ " }{TEXT 317 1 "f" }{TEXT -1 15 " no intervalo " }{TEXT 320 9 "[a,b] \+ = [" }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT 318 1 "," } {XPPEDIT 18 0 "x[m];" "6#&%\"xG6#%\"mG" }{TEXT 319 1 "]" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "A r egra dos trap\351zios realiza uma interpola\347\343o linear em cada su b-intervalo. Por sua vez, a regra 1/3 de Simpson realiza uma" }} {PARA 0 "" 0 "" {TEXT -1 67 "interpola\347\343o quadr\341tica em inter valos cont\355guos; i.e., para pontos " }{XPPEDIT 18 0 "x[i-1];" "6#&% \"xG6#,&%\"iG\"\"\"F(!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[i];" "6 #&%\"xG6#%\"iG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[i+1];" "6#&%\"xG6#, &%\"iG\"\"\"F(F(" }{TEXT -1 6 ", com " }{TEXT 321 9 "i=1..m-1." } {TEXT -1 28 " Nestes dois subintervalos" }}{PARA 0 "" 0 "" {TEXT -1 99 "adjacentes a integral ser\341 aproximada pela integral do polin \364mio quadr\341tico que passa pelos pontos " }{TEXT 322 1 "(" } {XPPEDIT 18 0 "x[i-1];" "6#&%\"xG6#,&%\"iG\"\"\"F(!\"\"" }{TEXT -1 2 " , " }{TEXT 323 2 "f(" }{XPPEDIT 18 0 "x[i-1];" "6#&%\"xG6#,&%\"iG\"\" \"F(!\"\"" }{TEXT 325 5 ")), (" }{XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG " }{TEXT 328 4 ", f(" }{XPPEDIT 18 0 "x[i]" "6#&%\"xG6#%\"iG" }{TEXT 329 3 ")) " }{TEXT -1 1 "e" }}{PARA 0 "" 0 "" {TEXT 324 1 "(" } {XPPEDIT 18 0 "x[i+1]" "6#&%\"xG6#,&%\"iG\"\"\"F(F(" }{TEXT 326 4 ", f (" }{XPPEDIT 18 0 "x[i+1]" "6#&%\"xG6#,&%\"iG\"\"\"F(F(" }{TEXT 327 2 "))" }{TEXT -1 26 ", que resulta na express\343o" }}{PARA 0 "" 0 "" {TEXT -1 67 " \+ " }{XPPEDIT 18 0 "Int(f(x),x = x[i-1] .. x[i+1]) = 1/3*h*(f( x[i-1])+4*f(x[i])+f(x[i+1]));" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;&F*6#,&% \"iG\"\"\"F1!\"\"&F*6#,&F0F1F1F1**F1F1\"\"$F2%\"hGF1,(-F(6#&F*6#,&F0F1 F1F2F1*&\"\"%F1-F(6#&F*6#F0F1F1-F(6#&F*6#,&F0F1F1F1F1F1" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Assi m, supondo que o n\372mero de sub-intervalos \351 par, teriamos no tot al uma aproxima\347\343o da integral para o intervalo " }{TEXT 330 5 " [a,b]" }{TEXT -1 5 " dada" }}{PARA 0 "" 0 "" {TEXT -1 16 "pela express \343o: " }}{PARA 0 "" 0 "" {TEXT -1 45 " \+ " }{XPPEDIT 18 0 "Int(f(x),x = x[0] .. x[m]) = sum(1/3* h*(f(x[i-1])+4*f(x[i])+f(x[i+1])),i = 1 .. m-1);" "6#/-%$IntG6$-%\"fG6 #%\"xG/F*;&F*6#\"\"!&F*6#%\"mG-%$sumG6$**\"\"\"F7\"\"$!\"\"%\"hGF7,(-F (6#&F*6#,&%\"iGF7F7F9F7*&\"\"%F7-F(6#&F*6#FAF7F7-F(6#&F*6#,&FAF7F7F7F7 F7/FA;F7,&F2F7F7F9" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "que pode ser tamb\351m escrita como: \+ " }}{PARA 0 "" 0 "" {TEXT -1 57 " \+ " }{XPPEDIT 18 0 "1/3*h*(a^2+b^2+2*sum(x[2 *i]^2,i = 1 .. 1/2*m-1)+4*sum(x[2*i+1]^2,i = 0 .. 1/2*m-1));" "6#**\" \"\"F$\"\"$!\"\"%\"hGF$,**$%\"aG\"\"#F$*$%\"bGF+F$*&F+F$-%$sumG6$*$&% \"xG6#*&F+F$%\"iGF$F+/F7;F$,&*(F$F$F+F&%\"mGF$F$F$F&F$F$*&\"\"%F$-F06$ *$&F46#,&*&F+F$F7F$F$F$F$F+/F7;\"\"!,&*(F$F$F+F&F " 0 "" {MPLTEXT 1 0 445 "umter cosimpson := proc(f,a,b,m) \n local h, acum, i;\n if m mod 2 = 0 th en \n h:= (b-a)/m; \n acum := 0; \n for i from 0 to m/2-1 do \n acum := acum + f(a+h*(2*i+1)); \n end do; \n acum := 2 * acum; \n for i to m/2-1 do \n acum := acum + f(a+h*(2*i)); \+ \n end do; \n acum := 2 * acum; \n acum := evalf((acum + f(a)+f(b)) * h/3);\n else print(`Deve fornecer um n\372mero de subi ntervalos par`); \n end if;\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%/umtercosimpsonGf*6&%\"fG%\"aG%\"bG%\"mG6%%\"hG%%acumG%\"iG6\" F/@%/-%$modG6$9'\"\"#\"\"!C)>8$*&,&9&\"\"\"9%!\"\"F>F5F@>8%F7?(8&F7F>, &F5#F>F6F>F@%%trueG>FB,&FBF>-9$6#,&F?F>*&F:F>,&FDF6F>F>F>F>F>>FB,$FBF6 ?(FDF>F>FEFG>FB,&FBF>-FK6#,&F?F>*(F6F>F:F>FDF>F>F>>FBFQ>FB-%&evalfG6#, $*&,(FBF>-FK6#F?F>-FK6#F=F>F>F:F>#F>\"\"$-%&printG6#%MDeve~fornecer~um ~n|ezmero~de~subintervalos~parGF/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 26 "4. Utilize o procediment o " }{TEXT 266 14 "umtercosimpson" }{TEXT 267 39 " para aproximar a in tegral das fun\347\365es:" }}{PARA 0 "" 0 "" {TEXT 354 16 " \+ 4.1 " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT 349 14 " no inter valo " }{TEXT 355 6 "[0,10]" }{TEXT 356 2 "; " }}{PARA 0 "" 0 "" {TEXT 357 15 " 4.2 " }{TEXT 350 3 "exp" }{TEXT 351 14 " no i ntervalo " }{TEXT 352 6 "[0,1];" }{TEXT 353 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f := x -> x ^2; \numtercosimpson(f,0,10,10); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LLLLL!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "umtercosimpson(exp,0,1,10); " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+#y#G= " 0 "" {MPLTEXT 1 0 28 "umtercosimpson(sin,0,Pi,10); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=&4,+#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "umtercosimpson(cos,-Pi/2,Pi/2,10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+<&4,+#!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 28 "Erro da regra 1/3 de Simpson" }}{PARA 0 "" 0 "" {TEXT -1 32 "O erro deste m\351todo \351 dado por " }{XPPEDIT 18 0 "E [SR] = -1/180*m*h^5*diff(f(xi),`$`(xi,4));" "6#/&%\"EG6#%#SRG,$*,\"\" \"F*\"$!=!\"\"%\"mGF*%\"hG\"\"&-%%diffG6$-%\"fG6#%#xiG-%\"$G6$F6\"\"%F *F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "onde " }{XPPEDIT 18 0 "xi;" "6#%#xiG" }{TEXT -1 19 " est\341 no intervalo " }{TEXT 358 5 "(a,b)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "Assim, o erro pode ser estimado por " }{XPPEDIT 18 0 "abs(E[SR]) <= 1/180*m*h^5*M[ 4];" "6#1-%$absG6#&%\"EG6#%#SRG*,\"\"\"F,\"$!=!\"\"%\"mGF,%\"hG\"\"&&% \"MG6#\"\"%F," }}{PARA 0 "" 0 "" {TEXT -1 5 "onde " }{XPPEDIT 18 0 "M[ 4];" "6#&%\"MG6#\"\"%" }{TEXT -1 16 " \351 o m\341ximo de " } {XPPEDIT 18 0 "abs(diff(f(x),`$`(x,4)));" "6#-%$absG6#-%%diffG6$-%\"fG 6#%\"xG-%\"$G6$F,\"\"%" }{TEXT -1 9 " para um " }{TEXT 359 1 "x" } {TEXT -1 14 " no intervalo " }{TEXT 360 5 "[a,b]" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 107 "6. Calcule o n\372mero m\355nimo de sub-int ervalos necess\341rios para computar as integrais das quatro fun\347 \365es das" }}{PARA 256 "" 0 "" {TEXT -1 106 "quest\365es anteriores u tilizando a regra 1/3 de Simpson com precis\343o 0.00001. Calcule ta mb\351m as integrais " }}{PARA 256 "" 0 "" {TEXT -1 86 "com dita preci ss\343o. Verifique o ganho significativo em rela\347\343o \340 regra \+ dos trap\351zios." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Note que o erro para polin\364mios lineares, quadr\341tic os e c\372bicos \351 nulo! " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "m >= sqrt(sqrt(10^ 5*10^5*0/180));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#1\"\"!%\"mG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "umtercosimpson(f,0,1,2);" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LLLLL!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "m >= evalf(sqrt(sqrt(10^5*1^ 5*exp(1)/180)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#1$\"+Xx$QB'!\"*%\" mG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "umtercosimpson(exp,0, 1,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bTG= " 0 "" {MPLTEXT 1 0 40 "m >= evalf(sqrt(sqrt(10^5*Pi^5*1/18 0)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#1$\"+SedI?!\")%\"mG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "umtercosimpson(sin,0,Pi,22); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+KY++?!\"*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "umtercosimpson(cos,-Pi/2,Pi/2,22);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+KY++?!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 57 "Para as integrais das quatro fun\347\365es at\351 \+ agora tratadas (" }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 1 " ," }{TEXT 361 15 " sin, cos e exp" }{TEXT -1 35 ") conhecemos bem expr ess\364es que s\343o" }}{PARA 0 "" 0 "" {TEXT -1 112 "facilmente calcu ladas anal\355ticamente. Mas ent\343o qual o sentido de integra\347 \343o num\351rica? A integra\347\343o num\351rica \351" }}{PARA 0 "" 0 "" {TEXT -1 107 "essencial quando n\343o podemos expressar a integra l como uma combina\347\343o de fun\347\365es elementares. P. ex. tent e" }}{PARA 0 "" 0 "" {TEXT -1 10 "integrar " }{XPPEDIT 18 0 "Int(exp( -x^2),x);" "6#-%$IntG6$-%$expG6#,$*$%\"xG\"\"#!\"\"F+" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 13 "7. \+ Aproxime " }{XPPEDIT 18 0 "Int(exp(-x^2),x = 0 .. 1);" "6#-%$IntG6$-%$ expG6#,$*$%\"xG\"\"#!\"\"/F+;\"\"!\"\"\"" }{TEXT 271 56 " com erro m \341ximo 0.00001 utilizando ambas as regras do:" }}{PARA 0 "" 0 "" {TEXT 362 31 " 7.1 trap\351zio e " }}{PARA 0 "" 0 "" {TEXT 363 34 " 7.2 1/3 de Simpson." }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 99 "gg := x -> exp(-x^2);\ndif2gg := unapply(dif f(gg(x), x$2),x);\ndif4gg := unapply(diff(gg(x), x$4),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ggGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$exp G6#,$*$)9$\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 'dif2ggGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%$expG6#,$*$)9$\"\"#\"\" \"!\"\"!\"#*(\"\"%F5F2F5F-F5F5F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'dif4ggGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(-%$expG6#,$*$)9$\"\"# \"\"\"!\"\"\"#7*(\"#[F5F2F5F-F5F6*(\"#;F5)F3\"\"%F5F-F5F5F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot([gg(x),dif2gg(x),dif4gg (x)],x=-1..1,color=[blue,red,green]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$!\"\"\"\"!$\"3MBWr6WzyO!#= 7$$!3ommm;p0k&*F-$\"3<9mmF0K1SF-7$$!3wKL$3t*\\F-7$ $!3\"QLL3i.9!zF-$\"3'=o:MqXiN&F-7$$!3\"ommT!R=0vF-$\"3'3c5,*yR$p&F-7$$ !3u****\\P8#\\4(F-$\"3aCa&=)f%[/'F-7$$!3+nm;/siqmF-$\"3LB,'4)yT3kF-7$$ !3[++](y$pZiF-$\"3#e?5'f*)GonF-7$$!33LLL$yaE\"eF-$\"3kv%pzD$)G8(F-7$$! 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F\363rmulas de Newton-Cotes de" }}{PARA 0 "" 0 "" {TEXT -1 6 "ordem " }{TEXT 364 2 "n " }{TEXT -1 44 "s\343o obtidas integrand o as Formas de Lagrange" }{TEXT 365 1 " " }{TEXT -1 34 "do polin\364mi o interpolador de grau " }{TEXT 366 2 "n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "Iremos deduzir f\363rmulas de New ton-Cotes utilizando as facilidades de manuseio simb\363lico do Maple. Para isto," }}{PARA 0 "" 0 "" {TEXT -1 57 "inicialmente revisamos fu n\347\365es para gera\347\343o dos fatores (" }{TEXT 367 4 "Lagr" } {TEXT -1 24 ") e Formas de Lagrange (" }{TEXT 368 13 "FormaLagrange" } {TEXT -1 2 "):" }}{PARA 0 "" 0 "" {TEXT -1 25 " \+ " }{XPPEDIT 18 0 "f(x[0])*L[0]+f(x[1])*L[1];" "6#,&*&-%\"fG6#&%\"xG6 #\"\"!\"\"\"&%\"LG6#F+F,F,*&-F&6#&F)6#F,F,&F.6#F,F,F," }{TEXT -1 8 "+ \+ ... + " }{XPPEDIT 18 0 "f(x[n])*L[n];" "6#*&-%\"fG6#&%\"xG6#%\"nG\"\" \"&%\"LG6#F*F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 576 "Lagr := proc (k, a, b, n, x ) # k varia entre 0 e n.\n local num, den, i, h; # A vari \341vel num para numerador e den para denominador. \n h:= (b-a)/n; nu m:= 1; den:= 1; \n for i from 0 to k-1 do \n num := num*(x-(a+h*i) ); \n den := den*((a+k*h)-(a+h*i)); \n end do; \n for i from k+1 \+ to n do \n num := num*(x-(a+h*i)); \n den := den*((a+k*h)-(a+h*i )); \n end do; \n num/den; \nend proc:\nFormaLagrange := proc(f,a,b, n,x)\n local pol, i, h; \n pol := 0; h := (b-a)/n;\n for i from 0 t o n do \n pol := pol + f(a+h*i)*Lagr(i,a,b,n,x); \n end do; \n po l \nend proc:" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Observe, p.ex., a computa\347\343o da Forma de Lagra nge no intervalo " }{TEXT 371 6 "[a,b] " }{TEXT -1 10 "de ordem 1" } {TEXT 369 1 " " }{TEXT -1 16 "para uma fun\347\343o " }{TEXT 370 2 "f: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:='f'; FormaLagrang e(f,a,b,1,x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(-%\"f G6#%\"aG\"\"\",&%\"xGF)%\"bG!\"\"F),&F,F-F(F)F-F)*(-F&6#F,F),&F+F)F(F- F),&F,F)F(F-F-F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 114 "Utilizando este procedimento \351 poss\355vel ded uzir as f\363rmulas de Newton-Cotes da regra do trap\351zio e 1/3 de S impson:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Regra do Trap\351zio:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "factor(simplify(int(FormaLagrange(f ,a,b,1,x),x=a..b)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&-%\"fG6# %\"bG\"\"\"-F'6#%\"aGF*F*,&F)!\"\"F-F*F*#F/\"\"#" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Observe que, neste caso " }{TEXT 372 12 "h = (b-a). " }{TEXT -1 14 "Assim, temos " } {XPPEDIT 18 0 "-1/2*(f(b)+f(a))*(-b+a) = 1/2*h*(f(b)+f(a));" "6#/,$** \"\"\"F&\"\"#!\"\",&-%\"fG6#%\"bGF&-F+6#%\"aGF&F&,&F-F(F0F&F&F(**F&F&F 'F(%\"hGF&,&-F+6#F-F&-F+6#F0F&F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 21 "Regra 1/3 de Simpson:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "factor(simp lify(int(FormaLagrange(f,a,b,2,x),x=a..b))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(-%\"fG6#,&%\"aG#\"\"\"\"\"#*&F+F,%\"bGF,F,\"\"%-F '6#F/F,-F'6#F*F,F,,&F/!\"\"F*F,F,#F6\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Neste caso, " }{TEXT 373 14 "h = (b-a)/2. " }{TEXT -1 13 "Assim, temos " }{XPPEDIT 18 0 "-1/6* (4*f(1/2*a+1/2*b)+f(b)+f(a))*(-b+a) = 1/3*h*(4*f(1/2*a+1/2*b)+f(b)+f(a ));" "6#/,$**\"\"\"F&\"\"'!\"\",(*&\"\"%F&-%\"fG6#,&*(F&F&\"\"#F(%\"aG F&F&*(F&F&F1F(%\"bGF&F&F&F&-F-6#F4F&-F-6#F2F&F&,&F4F(F2F&F&F(**F&F&\" \"$F(%\"hGF&,(*&F+F&-F-6#,&*(F&F&F1F(F2F&F&*(F&F&F1F(F4F&F&F&F&-F-6#F4 F&-F-6#F2F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Observe que a substitu i\347\343o de " }{TEXT 382 7 "(b-a)/m" }{TEXT -1 5 " por " }{TEXT 383 1 "h" }{TEXT -1 69 " pode ser realizada utilizando os operadores subs \+ e algsubs do Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "factor(algsubs(( b-a)=h,factor(simplify(int(FormaLagrange(f,a,b,1,x),x=a..b)))));\nfact or(algsubs((b-a)/2=h,factor(simplify(int(FormaLagrange(f,a,b,2,x),x=a. .b)))));\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&-%\"fG6#%\"bG\"\"\"-F'6#,&F)F*%\"hG!\"\"F*F*F.F*# F*\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(-%\"fG6#,&%\"bG\"\"\" %\"hG!\"\"\"\"%-F'6#F*F+-F'6#,&F*F+*&\"\"#F+F,F+F-F+F+F,F+#F+\"\"$" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 374 95 "8. Deduzir, utilizando este mecanismo, as f\363rmulas d e Newton-Cotes de ordem 3 e 4 e superiores." }}{PARA 0 "" 0 "" {TEXT 375 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "factor(simplify (int(FormaLagrange(f,a,b,3,x),x=a..b)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,*-%\"fG6#%\"aG\"\"\"-F'6#%\"bGF**&\"\"$F*-F'6#,&F)#\"\"#F/* &#F*F/F*F-F*F*F*F**&F/F*-F'6#,&F)F6*&F3F*F-F*F*F*F*F*,&F-!\"\"F)F*F*#F =\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Neste caso, " }{TEXT 376 14 "h = (b-a)/3. " }{TEXT -1 48 "Assim, temos esta \372ltima express\343o coincide com\n" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "-1/8*(f(a)+f(b)+3*f(2/3*a+1/3*b)+3*f(1/3*a+2/ 3*b))*(-b+a) = 3/8*h*(f(a)+f(b)+3*f(2/3*a+1/3*b)+3*f(1/3*a+2/3*b));" " 6#/,$**\"\"\"F&\"\")!\"\",*-%\"fG6#%\"aGF&-F+6#%\"bGF&*&\"\"$F&-F+6#,& *(\"\"#F&F2F(F-F&F&*(F&F&F2F(F0F&F&F&F&*&F2F&-F+6#,&*(F&F&F2F(F-F&F&*( F7F&F2F(F0F&F&F&F&F&,&F0F(F-F&F&F(**F2F&F'F(%\"hGF&,*-F+6#F-F&-F+6#F0F &*&F2F&-F+6#,&*(F7F&F2F(F-F&F&*(F&F&F2F(F0F&F&F&F&*&F2F&-F+6#,&*(F&F&F 2F(F-F&F&*(F7F&F2F(F0F&F&F&F&F&" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "-1/8*(f(a)+f(b)+3*f(2/3*a+1/3*b)+3*f(1/3*a+2/3* b))*(-b+a) = 3*h/8*(f(a)+f(b)+3*f(2/3*a+1/3*b)+3*f(1/3*a+2/3*b));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&,*-%\"fG6#%\"aG\"\"\"-F(6#%\"bGF+ *&\"\"$F+-F(6#,&F*#\"\"#F0*&#F+F0F+F.F+F+F+F+*&F0F+-F(6#,&F*F7*&F4F+F. F+F+F+F+F+,&F.!\"\"F*F+F+#F>\"\"),$*&%\"hGF+F&F+#F0F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "factor(simplify(int(FormaLagrange(f ,a,b,4,x),x=a..b)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,,-%\"fG6# ,&%\"aG#\"\"$\"\"%*&#\"\"\"F-F0%\"bGF0F0\"#K*&F2F0-F'6#,&F*F/*&F+F0F1F 0F0F0F0*&\"\"(F0-F'6#F1F0F0*&\"#7F0-F'6#,&F*#F0\"\"#*&FAF0F1F0F0F0F0*& F9F0-F'6#F*F0F0F0,&F1!\"\"F*F0F0#FH\"#!*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Neste caso, " }{TEXT 377 14 "h = (b-a)/4. " }{TEXT -1 51 "Assim, temos que esta \372ltima expr ess\343o coincide com" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " " }{XPPEDIT 18 0 "2/45*h*(32*f( 3/4*a+1/4*b)+7*f(a)+7*f(b)+12*f(1/2*a+1/2*b)+32*f(1/4*a+3/4*b));" "6#* *\"\"#\"\"\"\"#X!\"\"%\"hGF%,,*&\"#KF%-%\"fG6#,&*(\"\"$F%\"\"%F'%\"aGF %F%*(F%F%F2F'%\"bGF%F%F%F%*&\"\"(F%-F-6#F3F%F%*&F7F%-F-6#F5F%F%*&\"#7F %-F-6#,&*(F%F%F$F'F3F%F%*(F%F%F$F'F5F%F%F%F%*&F+F%-F-6#,&*(F%F%F2F'F3F %F%*(F1F%F2F'F5F%F%F%F%F%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 41 "Novamente, observe que a substitui\347\343o de " } {TEXT 378 7 "(b-a)/3" }{TEXT -1 5 " por " }{TEXT 379 11 "h e (b-a)/4" }{TEXT -1 71 " podem ser realizadas utilizando os operadores subs e al gsubs do Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 165 "factor(algsubs((b-a)/3=h,factor(simplify(int( FormaLagrange(f,a,b,3,x),x=a..b)))));\nfactor(algsubs((b-a)/4=h,factor (simplify(int(FormaLagrange(f,a,b,4,x),x=a..b)))));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&,*-%\"fG6#,&%\"bG\"\"\"*&\"\"$F+%\"hGF+!\"\"F+-F' 6#F*F+*&F-F+-F'6#,&F*F+*&\"\"#F+F.F+F/F+F+*&F-F+-F'6#,&F*F+F.F/F+F+F+F .F+#F-\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,,-%\"fG6#,&%\"bG\" \"\"*&\"\"$F+%\"hGF+!\"\"\"#K*&F0F+-F'6#,&F*F+F.F/F+F+*&\"\"(F+-F'6#F* F+F+*&\"#7F+-F'6#,&F*F+*&\"\"#F+F.F+F/F+F+*&F6F+-F'6#,&F*F+*&\"\"%F+F. F+F/F+F+F+F.F+#F?\"#X" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 123 "Observe que juntando as id\351ias preced entes, podemos definir um procedimento, que ger\341 a f\363rmula de Ne wton-Cotes de ordem n:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 384 50 "9. Utilizando estas id\351ias defina um \+ procedimento " }{TEXT 386 19 "NewtonCotes(ordem) " }{TEXT 385 49 "que \+ compute f\363rmulas de Newton-Cotes de qualquer " }{TEXT 388 5 "ordem " }{TEXT 387 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "NewtonCotes := proc(ordem)\n local a,b,h,f,x;\n a: ='a';b:='b';h:='h';f:='f';x:='x'; # Reassinamos valores simbolicos as \+ variaveis\n factor(algsubs((b-a)/ordem=h,\n factor(si mplify(int(FormaLagrange(f,a,b,ordem,x),x=a..b)))));\nend proc;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%,NewtonCotesGf*6#%&ordemG6'%\"aG%\"b G%\"hG%\"fG%\"xG6\"F.C(>8$.F1>8%.F4>8&.F7>8'.F:>8(.F=-%'factorG6#-%(al gsubsG6$/*&,&F4\"\"\"F1!\"\"FH9$FIF7-F@6#-%)simplifyG6#-%$intG6$-%.For maLagrangeG6'F:F1F4FJF=/F=;F1F4F.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Observe, ent\343o, a dedu \347\343o das f\363rmulas de Newton-Cotes para diferentes ordens." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "NewtonCotes(1); NewtonCotes(2);\nNewtonCotes(3);NewtonCotes(4); \nNewtonCotes(5);NewtonCotes(6);\nNewtonCotes(7);NewtonCotes(8);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&-%\"fG6#,&%\"aG\"\"\"%\"hGF+F+-F '6#F*F+F+F,F+#F+\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(-%\"fG6 #,&%\"aG\"\"\"%\"hGF+\"\"%-F'6#,&F*F+*&\"\"#F+F,F+F+F+-F'6#F*F+F+F,F+# F+\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,*-%\"fG6#,&%\"aG\"\"\" *&\"\"#F+%\"hGF+F+\"\"$-F'6#F*F+-F'6#,&F*F+*&F/F+F.F+F+F+*&F/F+-F'6#,& F*F+F.F+F+F+F+F.F+#F/\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,,-% \"fG6#,&%\"bG\"\"\"*&\"\"%F+%\"hGF+!\"\"\"\"(*&\"#KF+-F'6#,&F*F+*&\"\" $F+F.F+F/F+F+*&F2F+-F'6#,&F*F+F.F/F+F+*&F0F+-F'6#F*F+F+*&\"#7F+-F'6#,& F*F+*&\"\"#F+F.F+F/F+F+F+F.F+#FE\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*&,.-%\"fG6#,&%\"aG\"\"\"*&\"\"#F+%\"hGF+F+\"#]*&\"#>F+-F'6#,&F*F+ *&\"\"&F+F.F+F+F+F+*&F1F+-F'6#F*F+F+*&\"#vF+-F'6#,&F*F+F.F+F+F+*&F;F+- F'6#,&F*F+*&\"\"%F+F.F+F+F+F+*&F/F+-F'6#,&F*F+*&\"\"$F+F.F+F+F+F+F+F.F +#F6\"$)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,0-%\"fG6#,&%\"aG\"\" \"*&\"\"$F+%\"hGF+F+\"$s#*&\"#FF+-F'6#,&F*F+*&\"\"%F+F.F+F+F+F+*&\"#TF +-F'6#,&F*F+*&\"\"'F+F.F+F+F+F+*&F1F+-F'6#,&F*F+*&\"\"#F+F.F+F+F+F+*&F 8F+-F'6#F*F+F+*&\"$;#F+-F'6#,&F*F+F.F+F+F+*&FHF+-F'6#,&F*F+*&\"\"&F+F. F+F+F+F+F+F.F+#F+\"$S\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,2-%\"f G6#,&%\"aG\"\"\"%\"hGF+\"%xN*&\"$^(F+-F'6#F*F+F+*&\"%B8F+-F'6#,&F*F+*& \"\"&F+F,F+F+F+F+*&\"%*)HF+-F'6#,&F*F+*&\"\"$F+F,F+F+F+F+*&F3F+-F'6#,& F*F+*&\"\"#F+F,F+F+F+F+*&F-F+-F'6#,&F*F+*&\"\"'F+F,F+F+F+F+*&F/F+-F'6# ,&F*F+*&\"\"(F+F,F+F+F+F+*&F:F+-F'6#,&F*F+*&\"\"%F+F,F+F+F+F+F+F,F+#FQ \"&!G<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,4-%\"fG6#,&%\"aG\"\"\"* &\"\"%F+%\"hGF+F+\"%SX*&\"$*)*F+-F'6#F*F+!\"\"*&\"$G*F+-F'6#,&F*F+*&\" \"'F+F.F+F+F+F+*&\"&'\\5F+-F'6#,&F*F+*&\"\"&F+F.F+F+F+F4*&\"%))eF+-F'6 #,&F*F+*&\"\"(F+F.F+F+F+F4*&F1F+-F'6#,&F*F+*&\"\")F+F.F+F+F+F4*&FDF+-F '6#,&F*F+F.F+F+F4*&F6F+-F'6#,&F*F+*&\"\"#F+F.F+F+F+F+*&F=F+-F'6#,&F*F+ *&\"\"$F+F.F+F+F+F4F+F.F+#!\"%\"&vT\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "D e maneira similar, podem ser utilizadas as facilidades de manuseio sim b\363lico do Maple para deduzir os erros destas f\363rmulas de" }} {PARA 0 "" 0 "" {TEXT -1 13 "Newton-Cotes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "70 2 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }