lista3 - As listas de exerccios podem ser encontradas nos...

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As listas de exercícios podem ser encontradas nos seguintes endereços : www.mat.ufmg.br/calculoI ou na pasta J18, no xerox (sala1036) TERCEIRA LISTA DE EXERCÍCIOS 1. Derive: a ) y = 3x 6 + 9 x – 3 b ) y = 9 5 x c ) x x y 9 10 7 6 = d ) x x x x y 4 7 2 5 + = 2. Calcule ( ) h h h 6 6 0 9 9 lim + . 3. Calcule o h h h cos 1 lim 0 . 4. Calcule 3 3 lim 2000 2000 3 x x x . Como esse limite se relaciona com uma derivada? 5. Determine a equação da reta tangente ao gráfico de x x y = 3 5 , no ponto de abscissa x = 64. 6. Determine a equação da reta r tangente ao gráfico de y = x 2 + 3 x + 1 e que é paralela à reta de equação y = 4 x + 7 . 7. Determine as tangentes horizontais ao gráfico de 5 6 2 5 3 2 3 + + = x x x y . 8. Mostre que a reta de equação y = x é tangente à curva de equação y = x 3 6 x 2 + 8 x . Encontre o ponto de tangência. Respostas: 1) a ) . 9 18 5 + = x dx dy b ) . 9 5 9 14 x dx dy = c ) . 2 9 7 60 3 7 x x dx dy + = d ) . 2 45 7 9 11 7 2 x x dx dy = 2) . 3) 0. 5 9 6 × 4) Esse limite é igual a 1999 3 2000 3 2000 × = = x dx dx . 5) 3 2060 48 1277 = x y . 6) . 4 3 4 + = x y 7) 3 29 = y em 2 = x e 2 19 = y em 3 = x . 8) . ( ) 3 , 3
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9. Considere a função dada por . a ) Encontre uma relação entre a , b e c para que f seja contínua em x = 1. b ) Determine os valores de a , b e c para que f seja derivável em x = 1. > + + = < = 1 1 2 1 3 ) ( 2 x se c bx x x se x se ax x f 10. Derive: a ) y = e 2 x +5 b ) y = x cos 1 . c ) . Qual é o domínio dessa função? Qual é o domínio da derivada y’? ) ) ( ln ( sen x y = d ) e ) 7 4 ) 9 3 5 ( + = x x y ) 7 2 1 ( e 3 2 3 4 + + = + x x x y x f ) g ) h ) y = ln( x ) i ) j ) k ) y = ln(cos x ) 9 5 4 2 ) 3 2 4 ( ) 1 3 ( + + + = x x x x y x xe y = ( ) ( x y sen ln tg e = ) x y ln e = 11. Mostre que h ( t ) = | t 3 | não é derivável em t = 3.
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