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psCompStat1Qu - Fall 2007 Problem Set#12 First CompStat...

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Fall 2007 ARE211 Problem Set #12 First CompStat Problem Set Due date: Dec 04 (1) In problem 3 on your last problem set, you found the maximum and minimum distance from the origin to the ellipse x 2 1 + x 1 x 2 + x 2 2 = 3. Generalize this problem to “minimize/maximize the distance from the origin to the ellipse x 2 1 + x 1 x 2 + αx 2 2 = 3” and use the Envelope theorem (starting from α = 1) to estimate the maximum and minimum distance from the origin to the following ellipse, x 2 1 + x 1 x 2 + 0 . 9 x 2 2 = 3 . (2) a) Prove that the expression α 2 - αx 3 + x 5 = 17 defines x implicitly as a function of α . in a neighborhood of (¯ α, ¯ x ) = (5 , 2) b) Estimate the x -value which corresponds to α = 4 . 8 using a first order approximation. (3) Consider the equation α 3 1 + 3 α 2 2 + 4 α 1 x 2 - 3 x 2 α 2 = 1. Does this equation define x as an implicit function of α 1 , α 2 a) in a neighborhood of ( ¯ α 1 , ¯ α 2 ) = (1 , 1) b) in a neighborhood of ( ¯ α 1 , ¯ α 2 ) = (1 , 0) c) in a neighborhood of ( ¯ α 1 , ¯ α 2 ) = (0 . 5 , 0) If so, compute δx δα 1 and δx δα 2 at this point.
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