short_test_1_soln

short_test_1_soln - Math 237 Short Test 1 Wed Oct 1 2008...

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Math 237 Short Test # 1 Wed, Oct 1, 2008 Student Name (Print Legibly) Student ID Number Please indicate your section square Section 01 D. Wolczuk (2:30) square Section 02 Y. Hao (2:30) square Section 03 D. Wolczuk (10:30) square Section 04 N. Spronk (8:30) Your answers must be stated in a clear and logical form and you must justify all of your steps in order to receive full marks.
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2 1. Short Answer Problems [2] a) If f has continuous partial derivatives at a point ( a, b ) what are two things you can conclude about f at ( a, b )? Solution: Since f has continuous partial derivatives f is differentiable at ( a, b ). Since f is differentiable at ( a, b ), f is continuous at ( a, b ). [2] b) State the precise definition of a limit. Solution: If for every ǫ > 0 there exists a δ > 0 such that | f ( x, y ) L | < ǫ whenever 0 < bardbl ( x, y ) ( a, b ) bardbl < δ then lim ( x,y ) ( a,b ) f ( x, y ) = L . [2] c) State the squeeze theorem. Solution: If there exists a function B ( x, y ) such that | f ( x, y ) L | ≤ B ( x, y ) for all ( x, y ) in some neighborhood of ( a, b ) except possible at ( a, b ) and lim ( x,y ) ( a,b ) B ( x, y
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