prelim2_spring2006solutions - Math 192 Prelim 2 Solutions...

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Math 192 Prelim 2 Solutions Spring 2006 1. i) f ( x, y, z ) = cos xy + e yz + ln xz → ∇ f = ( - y sin xy + 1 x ) ~ i + ( - x sin xy + ze yz ) ~ j + ( ye yz + 1 z ) ~ k f (1 , 0 , 1 / 2) = ~ i + 1 2 ~ j +2 ~ k, ~u = ~ PQ/ | ~ PQ | = ( ~ i +2 ~ j +2 ~ k ) / 3. Therefore, D ~u f | P = f · ~u = 1 / 3+1 / 3+4 / 3 = 2. ii) f increases most rapidly in the direction of f and the value of the derivative in that direction is |∇ f | = p 1 + 1 / 4 + 4 = 21 / 2. 2. Solve ( a 2) 2 = 4 a 2 cos 2 θ 1 / 2 = cos 2 θ θ = π/ 6. Area = 4 Z π/ 6 0 Z 2 a cos 2 θ a 2 r drdθ = 2 Z π/ 6 0 (4 a 2 cos 2 θ - 2 a 2 ) = 2 a 2 [2 sin 2 θ - 2 θ ] π/ 6 0 = 2 a 2 ( 3 - π/ 3). 3. Let f = xyz and g = x 2 + 2 y 2 + 3 z 2 . Then f = yz ~ i + xz ~ j + xy ~ k and g = 2 x ~ i + 4 y ~ j + 6 z ~ k . So, f (1 , 1 , 1) = ~ i + ~ j + ~ k and g (1 , 1 , 1) = 2 ~ i + 4 ~ j + 6 ~ k , which are orthogonal to the level surfaces f = 1 and g = 6 respectively. The tangent line is parallel to
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