M23final2002-sol

# M23final2002-sol - Mathematics 23 Final Exam Solutions This...

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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 [email protected] . 1. The cross product of any two vectors on the plane gives us a vector which is perpen- dicular to the plane. 1 , 2 , 1 × 0 , - 2 , 0 = 2 , 0 , - 2 . 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) = 1 , 4 . 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 = 2 π 0 π/ 4 0 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 . Solving the first three equations for λ we find yz = xz 3 = xy 2 . Since none of x, y and z are zero we have x = 3 y = 2 x . Now x + 3 y + 2 z = 6 implies that x = 2, y = 2 / 3 and z = 1. So the maximum volume is 4 / 3.

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