LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Cap\303\255tulo 6-Linearidade Local Esta planilha analisa o plano tangente e a linearidade local. Zoom e a ideia principal: A id\303\251ia principal da linearidade local come\303\247a com uma observa\303\247\303\243o que a maioria dos alunos faz alguns dias depois de trabalhar com uma calculadora gr\303\241fica. "Se voc\303\252 aumentar o zoom em qualquer fun\303\247\303\243o, ela se tornar\303\241 uma linha reta". Essa observa\303\247\303\243o n\303\243o \303\251 muito correta, e as se\303\247\303\265es sobre continuidade e diferenciabilidade s\303\243o esfor\303\247os para examinar as exce\303\247\303\265es. O resultado que queremos \303\251 uma ligeira modifica\303\247\303\243o: Teorema: Para qualquer fun\303\247\303\243o de uma vari\303\241vel suficientemente agrad\303\241vel para podermos us\303\241-la em uma classe de c\303\241lculo, se ampliarmos o suficiente em qualquer lugar que n\303\243o seja um conjunto isolado de pontos problem\303\241ticos, o gr\303\241fico acabar\303\241 parecendo uma linha reta. Observe as palavras de manobra. Existem fun\303\247\303\265es que se comportam muito mal, mas n\303\243o as vemos nesta classe. Tais fun\303\247\303\265es mal comportadas s\303\243o abordadas no curso chamado an\303\241lise. Para nossas fun\303\247\303\265es, os locais onde a fun\303\247\303\243o n\303\243o se comporta bem s\303\243o isolados. Eles recebem aten\303\247\303\243o especial, j\303\241 que a maioria do nosso teorema falha nesses pontos. Exemplo 1: Mostre que a fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIj1GJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVi1JJW1zdXBHRiQ2JS1GLDYlUSJlRidGL0YyLUYjNiVGOkYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUZDNi1RIitGJ0ZARkZGSEZKRkxGTkZQRlIvRlVRLDAuMjIyMjIyMmVtRicvRlhGYm8tRiw2JVEkc2luRicvRjBGP0ZARjVGXm8tRlo2JUY6LUYjNiUtSSNtbkdGJDYkUSIzRidGQEYvRjJGW28tRiw2JVEkdGFuRidGZ29GQC1GNjYkLUYjNiVGaG9GPUZARkBGPUZA \303\251 localmente linear em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5.

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L0kieEc2IiIiJA==

O gr\303\241fico \303\251 muito n\303\243o linear quando del = 1. Quando alteramos o valor de del e reexecutamos, vemos que a fun\303\247\303\243o \303\251 quase linear quando del = .1 e muito linear quando del = 0.01. Como esse \303\251 um c\303\241lculo multivari\303\241vel, queremos observar a \303\263bvia generaliza\303\247\303\243o para fun\303\247\303\265es de duas vari\303\241veis: Teorema: Para qualquer fun\303\247\303\243o de duas vari\303\241veis \342\200\213\342\200\213suficientemente boas para podermos us\303\241-la em uma classe de c\303\241lculo, se ampliarmos o suficiente em qualquer lugar que n\303\243o seja um conjunto isolado de pontos problem\303\241ticos, o gr\303\241fico acabar\303\241 parecendo um plano. Observe que mantemos todas as palavras de manobra. A principal mudan\303\247a \303\251 que o gr\303\241fico de uma fun\303\247\303\243o linear em duas vari\303\241veis \342\200\213\342\200\213\303\251 um plano. Exemplo 2: Mostre que a fun\303\247\303\243o 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 \303\251 localmente linear em x = 3, y = 2.

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Parece uma boa aproxima\303\247\303\243o para uma regi\303\243o de \302\261 0,04. Exemplo 4: Fa\303\247a um gr\303\241fico da fun\303\247\303\243o 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 e seu plano tangente em uma pequena regi\303\243o pr\303\263xima de x = 3, y = 2 a mostre que \303\251 localmente linear.

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= O plano tangente parece ser uma boa aproxima\303\247\303\243o em uma regi\303\243o de \302\261 0,05. > Exerc\303\255cio 2: Seja c, d e h (x, y) como acima. Gr\303\241fico h (x, y) e seu plano tangente em uma regi\303\243o pequena o suficiente em torno de (c, d) que \303\251 \303\263bvio que o plano tangente fornece uma boa aproxima\303\247\303\243o para h (x, y). > Formul\303\241rios: Hora da pergunta regular: "E por que eu me importo?" ou sua vers\303\243o menos abrasiva "A que isso pode ser aplicado?" Os problemas que olhamos para este material t\303\252m v\303\241rios usos: 1) Quero aproximar rapidamente uma fun\303\247\303\243o perto de um ponto interessante. Isso obviamente funciona com polin\303\264mios. Tamb\303\251m funciona com fun\303\247\303\265es trigonom\303\251tricas perto de pontos que podemos avaliar. 2) Estou trabalhando com valores de entrada imprecisos. (Isso aconteceria sempre que eu estivesse fora da aula de matem\303\241tica e meus valores fossem medidos em um laborat\303\263rio.) Frequentemente, estou preocupado com o quanto uma pequena altera\303\247\303\243o nos valores de entrada alterar\303\241 as sa\303\255das. (Estou tentando criar barras de erro para a reda\303\247\303\243o do meu laborat\303\263rio.) Exerc\303\255cio 3: Seja c, d e h (x, y) como acima. Use a equa\303\247\303\243o do plano tangente para aproximar h (c + .01, d-.03). > Exerc\303\255cio 4: Seja c, d e h (x, y) como acima. Estime o poss\303\255vel erro se c e d forem medidos em 0,01. (Qual \303\251 a dist\303\242ncia m\303\241xima entre h (x, y) e o plano tangente nessa regi\303\243o?)