M23final2002-sol

M23final2002-sol - Mathematics 23 Final Exam December 14,...

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Mathematics 23 Final Exam December 14, 2002 Solutions This document was created on December 11, 2007 at 1:36pm. These solutions are presented without warranty. If you find any typos or errors, please inform Ken at ken.marks@lehigh.edu . 1. The cross product of any two vectors on the plane gives us a vector which is perpen- dicular to the plane. h 1 , 2 , 1 i × h 0 , - 2 , 0 i = h 2 , 0 , - 2 i . 2. (a) Maximum rate of change at (1 , 0) is |∇ f (1 , 0) | = 1 + 16 = 17. (b) Direction of maximum rate of change is f (1 , 0) = h 1 , 4 i . 3. In spherical coordinates the cone is φ = π/ 4 and the sphere is ρ = cos φ . Thus the volume of the solid is V = Z 2 π 0 Z π/ 4 0 Z cos φ 0 ρ 2 sin φ dρ dφ dθ. 4. Use Lagrange multipliers to maximize f ( x, y, z ) = xyz subject to the constraint g ( x, y, z ) = x + 3 y + 2 z = 6. Note that x, y, z are all nonzero. We solve the sys- tem of equations yz = λ, xz = 3 λ, xy = 2 λ, x + 3 y + 2 z = 6 .
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M23final2002-sol - Mathematics 23 Final Exam December 14,...

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