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Unformatted text preview: Fall 2007 ARE211 Problem Set #12 Answer key First CompStat Problem Set (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 + x 2 2 = 3 and use the Envelope theorem (starting from = 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 . Ans: From problem 3 on the preceding problem set we know that the Lagrangian was: L = x 2 1 + x 2 2 + (3 x 2 1 x 1 x 2 x 2 2 ) Lets parameterize the Lagrangian accordingly to estimate the change. L = x 2 1 + x 2 2 + (3 x 2 1 x 1 x 2 x 2 2 ) Initially, = 1 . From the envelope theorem we know that the first derivative of the squared distance M 2 ( ) w.r.t the parameter is dM 2 ( ) d = L = x 2 2 Using a first order Taylor expansion we know that: M 2 ( + d ) M 2 ( ) + M 2 d = M 2 ( ) x 2 2 d In the following = 1 ,d = . 1 (1) x (1) 1 = 1 , x (1) 2 = 1 , (1) = 2 3 , M ( = 1) = 2 1 . 4142 M 2 (0 . 9) 2 2 3 * 1 2 * ( . 1) = 31 15 M (0 . 9) = radicalbig M 2 (0 . 9) = radicalBig 31 15 1 . 4376 (2) x (2) 1 = 1 , x (2) 2 = 1 , (2) = 2 3 , M ( = 1) = 2 1 . 4142 M 2 (0 . 9) 2 2 3 * ( 1) 2 * ( . 1) = 31 15 M (0 . 9) = radicalbig M 2 (0 . 9) = radicalBig 31 15 1 . 4376 (3) x (3) 1 = 3 , x (3) 2 = 3 , (3) = 2 , M ( = 1) = 6 2 . 4495 M 2 (0 . 9) 6 2 * ( 3) 2 ( . 1) = 6 . 6 M (0 . 9) = radicalbig M 2 (0 . 9) = 6 . 6 2 . 5690 1 2 (4) x (4) 1 = 3 , x (4) 2 = 3 , (4) = 2 , M ( = 1) = 6 2 . 4495 M 2 (0 . 9) 6 2 * 3 2 ( . 1) = 6 . 6 M (0 . 9) = radicalbig M 2 (0 . 9) = 6 . 6 2 . 5690 3 (2) a) Prove that the expression 2 x 3 + x 5 = 17 defines x implicitly as a function of . Ans: The partial derivative of the expression w.r.t. x is f x = 3 x 2 + 5 x 4 Evaluated at the point ( , x ) = (5 , 2) : f x = 3 * 5 * 4+5 * 16 = 20 negationslash = 0 Since f ( , x ) x negationslash = 0 there exists a neighborhood around ( , x ) where x can be written as an implicit function of , i.e., x = g ( ) . in a neighborhood of ( , x ) = (5 , 2) b) Estimate the xvalue which corresponds to = 4 . 8 using a first order approximation. Ans: The partial derivative of f w.r.t. is f = 2  x 3 Evaluated at the point ( , x ) = (5 , 2) : f = 2 * 5 8 = 2 From the implicit function theorem we know g = f f x = 2 20 = . 1 Using a first order Taylor expansion: g ( + d ) g ( ) + g d Hence for = 5 ,d = . 2 , g (4 . 8) 2 . 1 * ( . 2) = 2 . 02 4 (3) Consider the equation 3 1 + 3 2 2 + 4 1 x 2 3 x 2 2 = 1. Does this equation define= 1....
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 Fall '07
 Simon
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