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gma00108-lista-08 - DEPARTAMENTO DE MATEMTICA APLICADA GMA...

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GMA DEPARTAMENTO DE MATEMÁTICA APLICADA LISTA DE EXERC ´ ICIOS alculo I –A– Humberto Jos´ e Bortolossi http://www.professores.uff.br/hjbortol/ 08 Continuidade e o Teorema do Valor Intermedi´ ario [01] Sejam f : R R e g : R R fun¸c˜oes cont´ ınuas com f (3) = 5 e lim x 3 [2 f ( x ) g ( x )] = 4. Calcule g (3). [02] Explique porque cada uma das fun¸ c˜oes abaixo ´ e descont´ ınua no ponto indicado. (a) f ( x ) = 1 x 1 , se x = 1 , 2 , se x = 1 , no ponto p = 1. (b) f ( x ) = x 2 1 x + 1 , se x = 1 , 2 , se x = 1 , no ponto p = 1. (c) f ( x ) = x 2 2 x 8 x 4 , se x = 4 , 3 , se x = 4 , no ponto p = 4. (d) f ( x ) = 1 x, se x 2 , x 2 2 x, se x = 2 , no ponto p = 2. [03] Use a continuidade da fun¸ c˜ao seno para calcular lim x π sen( x + sen( x )). [04] Mostre que a fun¸ c˜ao f ( x ) = x 1 , se x < 3 , 5 x, se x 3 , ´ e cont´ ınua em R . [05] Verifique se existe a R tal que a fun¸c˜ao f ( x ) = 1 + a x, se x 0 , x 4 + 2 a, se x > 0 , seja cont´ ınua em R . [06] Determine um valor diferente de zero para a constante c de modo que a fun¸ c˜ao y = f ( x ) = tg( c · x ) x , se x < 0 , 3 · x + c 2 , se x 0 , seja cont´ ınua ponto p = 0. Justifique a sua resposta! [07] Seja f : R −→ R , tal que x 2 · cos 2 ( x ) f ( x ) x · sen( x ) , x ( π/ 2 , + π/ 2). Mostre que f ´ e cont´ ınua em x = 0. [08] (a) Mostre que a fun¸ c˜ao f ( x ) = x 3 + x 1 tem pelo menos uma raiz no intervalo [0 , 1].
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