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11Breview1sol

11Breview1sol - econ 11b ucsc ams 11b Review Questions 1...

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econ 11 b ucsc ams 11 b Review Questions 1 Solutions 1. Compute the indicated partial derivatives of the functions below. a. z = 3 x 2 + 4 xy - 5 y 2 - 4 x + 7 y - 2 , z x = 6 x + 4 y - 4 z y = 4 x - 10 y + 7 b. F ( u, v, w ) = 60 u 2 / 3 v 1 / 6 w 1 / 2 ∂F ∂u = 40 u - 1 / 3 v 1 / 6 w 1 / 2 2 F ∂w∂u = 20 u - 1 / 3 v 1 / 6 w - 1 / 2 c. w = x 2 z ln( y 2 + z 3 ) w x = 2 xz ln( y 2 + z 3 ) w y = 2 x 2 yz y 2 + z 3 w xx = 2 z ln( y 2 + z 3 ) w yz = 2 x 2 y ( y 2 + z 3 ) - 6 x 2 yz 3 ( y 2 + z 3 ) 2 = 2 x 2 ( y 3 - 2 yz 3 ) ( y 2 + z 3 ) 2 . w xyz = 4 x ( y 3 - 2 yz 3 ) ( y 2 + z 3 ) 2 , since w xyz = w yzx . d. f ( x, y, z ) = 2 x 3 yz 2 - 3 xy 3 z + 5 x 2 y 2 - 7 yz 5 + 11 x - 1; f x = 6 x 2 yz 2 - 3 y 3 z + 10 xy 2 + 11 . f y = 2 x 3 z 2 - 9 xy 2 z + 10 x 2 y - 7 z 5 . f zx = f xz = 12 x 2 yz - 3 y 3 . e. q ( u, v ) = u 2 v - 3 uv 3 2 u + 3 v ; ∂q ∂u = (2 uv - 3 v 3 )(2 u + 3 v ) - 2( u 2 v - 3 uv 3 ) (2 u + 3 v ) 2 = 2 u 2 v + 6 uv 2 - 9 v 4 (2 u + 3 v ) 2 . ∂q ∂v = ( u 2 - 9 uv 2 )(2 u + 3 v ) - 3( u 2 v - 3 uv 3 ) (2 u + 3 v ) 2 = 2 u 3 - 18 u 2 v 2 - 18 uv 3 (2 u + 3 v ) 2 . 2. dz dt t =0 = ∂z ∂x t =0 dx dt t =0 + ∂z ∂y t =0 dy dt t =0 = ( (6 xy 3 + 5 y 2 - 3) t =0 ) · 3 + ( (9 x 2 y 2 + 10 xy + 1) t =0 ) · 2 = ( - 12 + 5 - 3) · 3 + (36 - 20 + 1) · 2 = - 30 + 34 = 4.

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11Breview1sol - econ 11b ucsc ams 11b Review Questions 1...

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