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204Midterm083004

# 204Midterm083004 - the set of all subsets of X has 2 n...

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Economics 204–First Midterm Test–August 30, 2004, 6-9pm Each question is worth 20% of the total Please use separate bluebooks for Parts I and II Part I 1. State and prove a theorem on the uniqueness of limits of sequences in metric spaces. 2. Consider the function f ( x, y ) = x 3 + y 3 + 2 x 2 2 xy y 2 3 x 6 y (a) Compute the first order conditions for a local maximum or minimum of f . Verify these are satisfied at the point ( x 0 , y 0 ) = (1 , 2). (b) Compute D 2 f ( x 0 , y 0 ) and give the quadratic Taylor series for f at the point ( x 0 , y 0 ). (c) Find the eigenvalues of D 2 f ( x 0 , y 0 ) and determine whether f has a local max, a local min, or a saddle at ( x 0 , y 0 ). (d) Find an orthonormal basis for R 2 consisting of eigenvectors D 2 f ( x 0 , y 0 ). Rewrite the quadratic Taylor series for f at the point ( x 0 , y 0 ) in terms of this basis. (e) Use the Taylor series found in part (d) to describe the approximate shape of the level sets of f near the point ( x 0 , y 0 ). 3. Prove that if a set X has n elements, then 2 X
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Unformatted text preview: , the set of all subsets of X , has 2 n elements. Hint: use induction. Part II 4. Suppose X,Y,Z are Fnite-dimensional vector spaces over R with bases U,V,W re-spectively, S ∈ L ( X,Y ) and T ∈ L ( Y,Z ). Summarize the relationships among S , T , T ◦ S , and their matrix representations using a commutative diagram. 1 Explain the interpretation of the diagram. 5. Consider the metric space ( X,d ), where X = Q ∩ [ , 1 ], Q is the set of all rational numbers, and d is the usual Euclidean metric d ( x,y ) = | x − y | . Show that ( X,d ) is not compact by exhibiting an open cover of X that has no Fnite subcover. 1 If you don’t remember the commutative diagram given in class and the handout, don’t panic. Think through the relationships and explain them, if possible with a diagram. 1...
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