Certamen 1 - Matemáticas III (2009).pdf - UTFSM Departamento de Matem´ atica Valparaiso 15 de abril de 2009 MAT-023 2009-1 Resoluci´on Certamen 1 1

Certamen 1 - Matemáticas III (2009).pdf - UTFSM...

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UTFSM Departamento de Matem´ atica. Valparaiso, 15 de abril de 2009. MAT-023. 2009-1. Resoluci´on Certamen 1 1. Sea g : R 2 −→ R una funci´ on diferenciable tal que g (3 , 5) = π/ 2, y g (3 , 5) = ( a, b ). Considere la ecuaci´ on x 2 + y 2 + xz 2 + cos( g ( zx, y + z )) = 14 . a ) Determine los valores de los n´umeros reales a y b para los cuales existe una funci´ on diferenciable z = z ( x, y ) soluci´on de la ecuaci´ on dada cerca del punto (1 , 2 , 3). Soluci´ on : Sea F ( x, y, z ) = x 2 + y 2 + xz 2 + cos( g ( zx, y + z )) 14. Es claro que F es diferenciable y que F (1 , 2 , 3) = 0. Entonces F z ( x, y, z ) = 2 xz sen( g ( zx, y + z ))( xg 1 + g 2 ) y por tanto F z (1 , 2 , 3) = 6 sen( π/ 2)( a + b ) = 6 ( a + b ) . Por el teorema de la funci´ on impl´ ıcita es posible despejar z = z ( x, y ) si a + b 6 negationslash = 0. OTRA FORMA: Derivando con respecto a x la ecuaci´ on F ( x, y, z ) = 0: 2 x + z 2 + 2 xzz x sen( g ( xz, y + z ))(( z + xz x ) g 1 + z x g 2 ) = 0 Despejando z x = zg 1 sen( g ( xz, y + z )) 2 x z 2 ( xg 1 + g 2 ) sen( g ( xz, y + z )) 2 xz Entonces z x (1 , 2) = 3 a sen( π/ 2) 2 9 ( a + b ) sen( π/ 2) 6 = 3 a 11 6 ( a + b ) por lo que es posible despejar z = z ( x, y ) si a + b negationslash = 6. b ) Si a y b toman valores dados en el inciso anterior, obtenga la ecuaci´ on del plano tangente de z = z ( x, y ) en (1 , 2 , 3). Se calcula z x (1 , 2) como arriba, y analogamente z y (1 , 2). O bien, z y = F y F z = 2 y sen( g ( xz, y + z )) g 2 F z . Entonces z y (1 , 2) = b 4 6 ( a + b ) . Por lo tanto la ecuaci´ on del plano tangente en (1 , 2 , 3) es z 3 = 3 a 11 6 ( a + b ) ( x 1) + b 4 6 ( a + b ) ( y 2) .
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2. Una empresa estima que su utilidad mensual est´ a dada por U ( x, y, z ) = 18 x x + 4 + 48 y y + 6 + 3 x + 3 y + 5 z 45 donde x , y , z representan el gasto por radio, televisi´ on y prensa escrita, respectivamente. Determine la cantidad a asignar a cada medio de manera que se maximice la utilidad, si el presupuesto mensual no puede superar los 10 millones de pesos.
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