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251sol7

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Unformatted text preview: Page 1 Section 14.7 Page 2 Section 14.7 Page 3 Section 14.7 Page 4 Section 14.7 Page 1 Section 14.8 Page 2 Section 14.8 Page 1 Section 15.1 Page 2 Section 15.1 Assignment 7 Multivariate Calculus Math 251 (Fall 2010) A1. Let C be the curve which consists of all the points on the hyperbolic paraboloid x 2 − y 2 = z which are at distance √ 2 from the origin. Find all the points on C which are closest to the point P (0 , , 1), and find all the points on C which are farthest from P (0 , , 1). Hints: Set up Lagrange equations for as on Page 939 of Stewart. To solve them, first try to manipulate two of the equations involving λ and μ . Solution We are asked to optimize the distance from a point ( x, y, z ) on C to the point P (0 , , 1). It is easier to optimize the squared distance f ( x, y, z ) = x 2 + y 2 + ( z − 1) 2 , subject to being on the sphere of radius 2, which is the level surface g ( x, y, z ) = x 2 + y 2 + z 2 = 2 , (1) and subject to lying on the hyperbolic paraboloid h (...
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