Faculdade de Economia Universidade Nova de Lisboa Semestre de Inverno 2010/2011. Cálculo I. Caderno de exercícios 2

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1 Faculdade de Economia Universidade Nova de Lisboa Semestre de Inverno 2010/2011 Cálculo I Caderno de exercícios 2 Paulo Corte-Real Ernesto Freitas Claudia Alves David Antunes Silvia Guerra 1

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37 2.7 Sucessões (continuação) Exercícios Propostos 1. Considere a sucessão cujo termo geral é: U n = 3 0 x2n dx. (a) Determine os primeiros dois termos da sucessão U n. (b) Mostre que o termo geral da sucessão é dado por U n = 32n+1 2n+1. (c) Averigue se U n émonótona. (d) Resolva a inequação em n: U n < 1 2n Considere a sucessão v n dada por Resolução (a) Calcule v 1 e v 2. v n = ˆ n (b) Estude-a quanto à monotonia. (c) Calcule lim v n. 0 1 (x + 1) (x + 2) dx (d) Indique um minorante e um majorante de u n (a) U 1 = 3 0 x2 dx = (b) U n = 3 0 x2n dx = 3 x x 2n+1 2n+1 =9; U 2 = 3 0 x4 dx = 3 0 = 32n+1 0 2n+1 2n+1 = 32n+1 2n+1 3 x 5 5 = (c) U n+1 U n = 32n+3 2n+3 32n+1 2n+1 = (2n+1)32n+3 (2n+3)3 2n+1 (2n+3)(2n+1) 48n3 2n n (2n+3)(2n+1) = 32n (48n+18) (2n+3)(2n+1) > 0, n. U n estritamente crescente. == 54n32n n 6n.3 2n 9.3 2n (2n+3)(2n+1) = (d) U n < 1 2n+1 32n+1 2n+1 < 1 2n+1 32n+1 < 1 3 2n+1 < 3 0 2n +1< 0 n< 1 2. Não existe nenhuma ordem que resolva a inequação, pois lembramos n N. 1 (a) P (x+1)(x+2) = P 1 1 x+1 +P x+2 =ln x +1 ln x +2 +C =ln x+1 x+2+ C. v n = n 1 0 (x+1)(x+2) dx. v 1 = (x+1)(x+2) ln dx = x+1 1 x+2 = 0 ln 2 3 ln 1 2 =ln4 3. v 2 = (x+1)(x+2) ln dx = x+1 2 x+2 =ln 3 4 ln 1 2 = (b) ln

38 ˆ n+1 v n+1 v n = 0 ˆ 1 n (x + 1) (x + 2) dx 0.Logov n éestritamentecrescente. (c) n lim v n = lim 0 ln 1 2 =ln2 1 (x+1)(x+2) ln dx = lim ˆ 1 n+1 (x + 1) (x + 2) dx = n n x+1 x+2 0 =lim ln n+1 n+2 ln 1 2 (d) Sendo v n crescente, basta saber o primeiro termo e o seu limite. Minorante =ln 4 3 Majorante =ln2 Exercícios Resolvidos 1. Considere a sucessão cujo termo geral é: U n = 2 0 xn dx. (a) Determine os primeiros dois termos da sucessão U n. (b) Mostre que o termo geral da sucessão é dado por U n = 2n+1 n+1. (c) Averigue se U n émonótona. (d) Resolva a inequação em n: U n < 8 n Considere a sucessão v n dada por Resolução (a) Calcule v 1 e v 2. v n = ˆ 2n (b) Estude-a quanto à monotonia. (c) Calcule lim v n. 1 4 x 2 +3x dx (d) Indique um minorante e um majorante de v n. 1. (a) 2, 8/3 (c) Estritamente crescente (d) n =1 2. (a) 4 3 ln 8 5, 4 3 ln 16 7 (b) Estritamente crescente (c)ln (d) Min: 4 3 ln 8 5, Maj: ln dx > 0, n (x + 1) (x + 2) = 2

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