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204Final082905

# 204Final082905 - Economics 204Final E 6-9pm Each question...

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Economics 204–Final Exam–August 29, 2005, 6-9pm Each question is worth 20% of the total Please use separate bluebooks for each of the three Parts Part I 1. (a) Define the term “contraction.” (b) State the Contraction Mapping Theorem. (c) Give the following portion of the proof of the Contraction Mapping Theorem: Given a contraction, start with an arbitrary point, and show how to construct a sequence of points and prove that the sequence converges to a limit which is a candidate fixed point. You need not prove that the limit is in fact a fixed point. 2. Suppose T L ( R n , R n ) is a linear transformation, and let V be any basis of R n . Show that λ is an eigenvalue of T if and only if λ is an eigenvalue of Mtx V ( T ). Part II 3. Consider the function f ( x, y ) = x 3 + y 3 + 6 x 2 + 8 y 2 2 3 xy + (2 3 15) x + (2 3 19) y (a) Compute the first order conditions for a local maximum or minimum of f . Show that the first order conditions are satisfied at the point ( x 0 , y 0 ) = (1 , 1).

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204Final082905 - Economics 204Final E 6-9pm Each question...

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