C\303\241lculo IILi\303\247\303\243o 30: Polin\303\264mios Ortogonais Propriedade da Ortogonalidade de fun\303\247\303\265es. Polin\303\264mios de Legendre. C\303\241lculo dos polin\303\264mios de Legendre. Expans\303\243o de uma fun\303\247\303\243o em termos de polin\303\264mios de Legendre. orthopoly, P Sejam f e g fun\303\247\303\265es definidas em [a, b]. Definimos o produto 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=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 . Seja 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 um polin\303\264mio de grau n e 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 um polin\303\264mio de grau m, definidos em [a, b], tal que: 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 , sendo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== uma constante e 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 para m = n e 0 caso contr\303\241rio. Um conjunto de tais polin\303\264mios constitui um conjunto de polin\303\264mios ortogonais. Estes polin\303\264mios surgem como solu\303\247\303\265es para algumas equa\303\247\303\265es diferenciais importantes da f\303\255sica. Tamb\303\251m podem ser deduzidos diretamente de sua defini\303\247\303\243o e de algumas rela\303\247\303\265es de recorr\303\252ncia, e podem ser calculados por fun\303\247\303\265es geradoras. Uma fun\303\247\303\243o geradora \303\251 uma fun\303\247\303\243o cuja s\303\251rie de pot\303\252ncias apresenta tais polin\303\264mios como coeficientes. Este conjunto de fun\303\247\303\265es ortogonais s\303\243o os polin\303\264mios de Legendre. Este conjunto de polin\303\264mios \303\251 completo. Ou seja, todo polin\303\264mio pode ser expresso como uma combina\303\247\303\243o linear de polin\303\264mios de Legendre. Devido ao teorema de Weierstrass e a completude do conjunto de polin\303\264mios de Legendre, qualquer fun\303\247\303\243o pode ser aproximada por alguma combina\303\247\303\243o linear de polin\303\264mios de Legendre, com a precis\303\243o desejada. Mostraremos a seguir como calcular os polin\303\264mios de Legendre e como expressar uma fun\303\247\303\243o, definida e cont\303\255nua em [-1,1] , como uma combina\303\247\303\243o linear de polin\303\264mios de Legendre. Os polin\303\264mios de Legendre podem ser calculados a partir de sua defini\303\247\303\243o, como mostramos a seguir. Para isso, deve-se ter um conjunto de polin\303\264mios, 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 , tal que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYjUSFGJy1GIzYlLUklbXN1YkdGJDYlLUYxNiVRIlBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Iy1GMTYlUSJuRidGPEY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GQFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlIvJSlzdHJldGNoeUdGUi8lKnN5bW1ldHJpY0dGUi8lKGxhcmdlb3BHRlIvJS5tb3ZhYmxlbGltaXRzR0ZSLyUnYWNjZW50R0ZSLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGW28tRiw2JC1GIzYjLUYxNiVRInhGJ0Y8Rj9GTi1GSzYtUSIsRidGTkZQL0ZURj5GVUZXRllGZW5GZ25GaW4vRl1vUSwwLjMzMzMzMzNlbUYnLUklbXN1cEdGJDYlRmJvLUYxNiVRIm1GJ0Y8Rj8vJTFzdXBlcnNjcmlwdHNoaWZ0R0ZJRk4vJSVvcGVuR1EzJkxlZnRBbmdsZUJyYWNrZXQ7RicvJSZjbG9zZUdRNCZSaWdodEFuZ2xlQnJhY2tldDtGJw===0 para m<n e 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== . LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== \303\251 normalmente definido como 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 . Quando valores diferentes de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== s\303\243o selecionados, os polin\303\264mios resultantes diferem dos polin\303\264mios padr\303\265es por um fator multiplicativo. Ex. 1 Encontrar o polin\303\264mio de Legendre de grau 5. restart: p[5]:=sum(a[k]*x^k,k=0..5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicEdGKDYjIiImLC4qJkkieEdGKEYwJkkiYUdGKEYvIiIiRjYqJkYzIiIlJkY1NiNGOEY2RjYqJkYzIiIkJkY1NiNGPEY2RjYqJkYzIiIjJkY1NiNGQEY2RjYqJkYzRjYmRjU2I0Y2RjZGNiZGNTYjIiIhRjY3I0Yx

for k to 5 do eq||k:=int(p[5]*x^(k-1),x=-1..1)=0 od;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTFHRigvLCgmSSJhR0YoNiMiIiUjIiIjIiImJkYxNiNGNSNGNSIiJCZGMTYjIiIhRjVGPTcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTJHRigvLCgmSSJhR0YoNiMiIiYjIiIjIiIoJkYxNiMiIiQjRjVGMyZGMTYjIiIiI0Y1RjkiIiE3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTNHRigvLCgmSSJhR0YoNiMiIiUjIiIjIiIoJkYxNiNGNSNGNSIiJiZGMTYjIiIhI0Y1IiIkRj03I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTRHRigvLCgmSSJhR0YoNiMiIiYjIiIjIiIqJkYxNiMiIiQjRjUiIigmRjE2IyIiIiNGNUYzIiIhNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTVHRigvLCgmSSJhR0YoNiMiIiUjIiIjIiIqJkYxNiNGNSNGNSIiKCZGMTYjIiIhI0Y1IiImRj03I0Yu

Como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiNUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn deve ser ortogonal a cada polin\303\264mio de grau menor que 5, em particular LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYlUSJuRidGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw== | LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNUYnRj5GKw== , basta considerar o termo de quinto grau em um fator de 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 em 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. eq||6:=int(p[5]*a[5]*x^5,x=-1..1)=2/11;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTZHRigvLCgqJCZJImFHRig2IyIiJiIiIyNGNSIjNiomJkYyNiMiIiQiIiJGMUY8I0Y1IiIqKiYmRjI2I0Y8RjxGMUY8I0Y1IiIoRjY3I0Yu

S:={seq(eq||k,k=1..6)}: v:={seq(a[k],k=0..5)}: c:=solve(S,v);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJjR0YoNiQ8KC8mSSJhR0YoNiMiIiFGNC8mRjI2IyIiIiMiIzoiIikvJkYyNiMiIiNGNC8mRjI2IyIiJCMhI04iIiUvJkYyNiNGRkY0LyZGMjYjIiImIyIjakY7PChGMC9GNiMhIzpGO0Y8L0ZBIyIjTkZGRkcvRksjISNqRjs3JEYvRlA=

assign(c[1]): p[5]:=p[5];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkicEdGKDYjIiImLCgqJEkieEdGKEYwIyIjaiIiKSokRjMiIiQjISNOIiIlRjMjIiM6RjY3I0Yx

Os polin\303\264mios de Legendre podem ser calculados por 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkEvJSlzdHJldGNoeUdGQS8lKnN5bW1ldHJpY0dGQS8lKGxhcmdlb3BHRkEvJS5tb3ZhYmxlbGltaXRzR0ZBLyUnYWNjZW50R0ZBLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGUC1JI21uR0YkNiRRIjFGJ0Y8RjwtSSNtaUdGJDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvRj1RJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GIzYlLUZYNiNRIUYnLUYjNiYtRjE2JS1GVDYkUSIyRidGPEZXRmpuLUY5Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y8Rj9GQkZERkZGSEZKRkwvRk9RJjAuMGVtRicvRlJGXXAtRiM2JEZXLUY5Ni1RIiFGJ0Y8Rj9GQkZERkZGSEZKRkwvRk9RLDAuMTExMTExMWVtRicvRlJGZXBGX29GX28vJS5saW5ldGhpY2tuZXNzR0ZWLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRltxLyUpYmV2ZWxsZWRHRkE= 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 . restart: p[5](x):=simplify((-1)^5/(2^5*5!)*diff((1-x^2)^5,x$5));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+LSZJInBHRig2IyIiJjYjSSJ4R0YoLCgqJEYzRjEjIiNqIiIpKiRGMyIiJCMhI04iIiVGMyMiIzpGODcjRjQ=

O Maple cont\303\251m a fun\303\247\303\243o P, para determina\303\247\303\243o de polin\303\264mios de Legendre. P with(orthopoly): P(5,x);

LCgqJEkieEc2IiIiJiMiI2oiIikqJEYkIiIkIyEjTiIiJUYkIyIjOkYp

Seja f(x) uma fun\303\247\303\243o definida em [-1,1], e 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 os polin\303\264mios de Legendre de grau n. Ent\303\243o, se f(x) puder ser expandido em s\303\251rie de pot\303\252ncias, tamb\303\251m poder\303\241 ser expandido em uma s\303\251rie de polin\303\264mios de Legendre.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. Multiplicar pelo m-\303\251simo polin\303\264mio de Legendre, 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. Para isso, deve-se utilizar a defini\303\247\303\243o de produto interno e a defini\303\247\303\243o de ortogonalidade, acima.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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KS1JKG1zdWJzdXBHRiQ2Jy1JI21vR0YkNi1RJiZpbnQ7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiM2JC1GNTYtUSomdW1pbnVzMDtGJ0Y4RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GVS1JI21uR0YkNiRRIjFGJ0Y4RlcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKy1GIzYnLUkrbXVuZGVyb3ZlckdGJDYnLUY1Ni9RJiZTdW07RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnRjgvSSttc2VtYW50aWNzR0YkUSZpbmVydEYnRjtGPi9GQVEldHJ1ZUYnRkIvRkVGam8vRkdGam9GSEZKL0ZOUSwwLjE2NjY2NjdlbUYnLUYjNiUtRiw2JVEibkYnLyUnaXRhbGljR0Zqby9GOVEnaXRhbGljRictRjU2LVEiPUYnRjhGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZccS1GWDYkRmpuRjgtRiw2JVEoJmluZmluO0YnL0ZlcEY9RjhGSC8lLGFjY2VudHVuZGVyR0Y9RistSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJDUuMEYnLyUmZGVwdGhHRltyLyUqbGluZWJyZWFrR1ElYXV0b0YnLUYjNigtSSVtc3ViR0YkNiUtRiw2JVEiYUYnRmRwRmZwLUYjNiNGYXBGaG4tRjU2LVExJkludmlzaWJsZVRpbWVzO0YnRjhGO0Y+RkBGQkZERkZGSEZKRk0tRiM2JS1GZ3I2JS1GLDYlUSJQRidGZHBGZnBGXHNGaG4tRjU2LVEwJkFwcGx5RnVuY3Rpb247RidGOEY7Rj5GQEZCRkRGRkZIRkpGTS1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInhGJ0ZkcEZmcEY4Rl5zLUYjNiUtRmdyNiVGZXMtRiM2Iy1GLDYlUSJtRidGZHBGZnBGaG5GaHNGW3RGK0YrRistRmdxNiZGaXEvRl1yUSYwLjNlbUYnRl9yRmFyLUY1Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRjhGO0Y+RkBGQkZERkZGSEZKRk1GYHRGKw== = 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 = 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 = 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Resultam da\303\255 os valores dos coeficientes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm1GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==.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 = 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 Ex. 2 Expandir 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 em polin\303\264mios de Legendre de grau maior que tr\303\252s. restart: with(orthopoly): f:=exp(sin(x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoLUkkZXhwR0YlNiMtSSRzaW5HRiU2I0kieEdGKDcjRi4=

a:=n->int(P(n,x)*f,x=-1..1)/Int(P(n,x)^2,x=-1..1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJhR0YoZio2I0kibkdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiYtSSRpbnRHRiU2JComLUkiUEdGKDYkOSRJInhHRigiIiJJImZHRihGPi9GPTshIiJGPkY+LUkkSW50R0YlNiQqJEY5IiIjRkBGQkYoRihGKDcjRi4=

Expans\303\243o em Polin\303\264mios de Legendre LPE:=Sum(a(n)*P(n,x),n=0..3);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRMUEVHRigtSSRTdW1HRiU2JCooLUkkaW50R0YlNiQqJi1JIlBHRig2JEkibkdGKEkieEdGKCIiIi1JJGV4cEdGJTYjLUkkc2luR0YlNiNGOkY7L0Y6OyEiIkY7RjstSSRJbnRHRiU2JCokRjYiIiNGQkZERjZGOy9GOTsiIiEiIiQ3I0Yu

A resposta do Maple \303\251 um pouco confusa. O operador integral no numerador n\303\243o opera no polin\303\264mio mais a direita, 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, uma vez que dx delimita as fun\303\247\303\265es sobre as quais a integral opera. LPE:=evalf(%);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRMUEVHRigsKiQiK3gjKW81NSEiKiIiIkkieEdGKCQiKypITUQsIkYxKiRGMyIiIyQiKysmcHMjUiEjNSokRjMiIiQkIStzTEdPaSEjNjcjRi4=

N\303\243o h\303\241 como mudar as regras de avalia\303\247\303\243o do Maple e exibir os polin\303\264mios de Legendre com seus coeficientes. No entanto, \303\251 poss\303\255vel visualiz\303\241-los separadamente. A:=[seq(evalf(a(k)),k=0..3)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJBR0YoNyYkIitnc2ZUNiEiKiQiKyopZjteKCohIzUkIitvJ3oiPUVGNCQhK1xMXiVcIyEjNjcjRi4=

LP:=[seq(P(n,x),n=0..3)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNMUEdGKDcmIiIiSSJ4R0YoLCYjISIiIiIjRi8qJEYwRjQjIiIkRjQsJiokRjBGNyMiIiZGNEYwIyEiJEY0NyNGLg==

\303\211 interessante comparar a expans\303\243o de Legendre de terceiro grau com a expans\303\243o em polin\303\264mio de Taylor de quarto grau. T:=taylor(f,x,5);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJUR0YoKy1JInhHRigiIiIiIiFGMEYwI0YwIiIjRjMjISIiIiIpIiIlLUkiT0dGJTYjRjAiIiY3I0Yu

T:=convert(T,polynom);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJUR0YoLCoiIiJGL0kieEdGKEYvKiRGMCIiIyNGL0YyKiRGMCIiJSMhIiIiIik3I0Yu

plot([f,LPE,T],x=-1..1,color=[black,red,green]);

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No intervalo [-1,1], a expans\303\243o de Legendre de terceiro grau encontra-se mais pr\303\263xima da fun\303\247\303\243o do que o polin\303\264mio de Taylor de grau quatro. evalf(int(abs(f-T),x=-1..1));

JCIrMi9gWD4hIzY=

evalf(int(abs(f-LPE),x=-1..1));

JCIrVnZLWTshIzY=

Obviamente, nem toda fun\303\247\303\243o pode ser convenientemente limitada ao intervalo 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. Se o intervalo de interesse \303\251 [a, b], este problema pode ser tratado por "mudan\303\247a de coordenadas". Define-se ent\303\243o uma nova coordenada, z, cujos valores variam linearmente sobre [-1,1] a medida que x varia sobre [a,b]. Isto \303\251 equivalente a encontrar a equa\303\247\303\243o da reta que passa pelos pontos (a,-1) e (b,1), sendo x a vari\303\241vel independente e z a vari\303\241vel dependente. restart: eq1:='(1-(-1))/(b-a)=(1-z)/(b-x)';

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTFHRigvLCQqJCwmSSJiR0YoIiIiSSJhR0YoISIiRjUiIiMqJiwmRjNGM0kiekdGKEY1RjMsJkYyRjNJInhHRihGNUY1NyNGLg==

eq2:=z=solve(eq1,z);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTJHRigvSSJ6R0YoKiYsKEkiYkdGKCIiIkkiYUdGKEYzSSJ4R0YoISIjRjMsJkYyISIiRjRGM0Y4NyNGLg==

eq3:=subs({a=1,b=100},eq2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTNHRigvSSJ6R0YoLCYjISQsIiIjKioiIiJJInhHRigjIiIjRjM3I0Yu

Z \303\251, claramente, uma fun\303\247\303\243o linear. Esta \303\251 igual a -1 em x = 1, e igual a 1 em x = 100. Agora, basta resolver a equa\303\247\303\243o para x e substituir na fun\303\247\303\243o desejada. X:=solve(eq3,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJYR0YoLCYjIiQsIiIiIyIiIkkiekdGKCMiIyoqRjE3I0Yu

O uso da letra mai\303\272scula X \303\251 necess\303\241rio, pois a letra min\303\272scula x \303\251 reservada para a vari\303\241vel. Aplica\303\247\303\265es 1. Alterando o valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== \303\251 poss\303\255vel gerar uma variedade infinita de fam\303\255lias de polin\303\264mios ortogonais. Suponhamos que, em vez de considerarmos 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 , exigimos que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdWJHRiQ2JS1GLDYlUSJLRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRiw2JVEibkYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiPUYnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGTS8lKXN0cmV0Y2h5R0ZNLyUqc3ltbWV0cmljR0ZNLyUobGFyZ2VvcEdGTS8lLm1vdmFibGVsaW1pdHNHRk0vJSdhY2NlbnRHRk0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zmbi1JI21uR0YkNiRRIjFGJ0ZJRis= para todo n. Calcular os primeiros cinco membros desta fam\303\255lia. 2. Expandir cada uma das fun\303\247\303\265es a seguir no intervalo [-1,1] como uma soma dos primeiros seis polin\303\264mios de Legendre. Tra\303\247ar o gr\303\241fico da fun\303\247\303\243o e sua expans\303\243o no intervalo dado. Encontrar o polin\303\264mio de Taylor com o grau adequado. Utilizando o m\303\251todo apresentado nesta li\303\247\303\243o, determinar qual expans\303\243o \303\251 mais precisa no intervalo dado. a) 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 b) 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 3. No intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictSSNtaUdGJDYlUSNQaUYnLyUnaXRhbGljR0Y9RjRGNC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJw==, expandir 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 como uma soma dos primeiros seis polin\303\264mios de Legendre. Tra\303\247ar o gr\303\241fico da fun\303\247\303\243o, juntamente com sua expans\303\243o. Encontrar o polin\303\264mio de Taylor de grau 6. Qual expans\303\243o \303\251 mais precisa no intervalo dado?