204 Midterm 2003 - University of California, Berkeley...

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University of California, Berkeley Economics 204–First Midterm Test Tuesday August 25, 2003; 6-9pm Each question is worth 20% of the total Please use separate bluebooks for Parts I and II Part I 1. Prove that n X k =1 k = n ( n +1) 2 2. Consider the function f ( x, y )= e x 2 6 xy + y 2 Recall that d dz e z = e z . (a) Compute the Frst order conditions for a local maximum or minimum of f . ±ind the unique ( x ,y )a twh ichtheF r s t order conditions are satisFed. (b) ±ind the second order Taylor series expansion of f at the point ( x ,y ) determined in part (a); your answer should in- volve a symmetric matrix A representing the quadratic terms in the expansion. (c) Diagonalize the matrix A you found in part (b). ±ind an or- thonormal basis { v 1 ,v 2 } of R 2 such that the quadratic terms of the Taylor expansion can be written as g (( x ,y )+ γ 1 v 1 + γ 2 v 2 )= λ 1 ( γ 1 ) 2 + λ 2 ( γ
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This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at University of California, Berkeley.

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204 Midterm 2003 - University of California, Berkeley...

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