midterm_sol - Midterm 1 of Math 254.01 (Au10) October 18,...

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Midterm 1 of Math 254.01 (Au10) October 18, 2010 The Ohio State University 12:30pm-1:18pm Including the cover sheet, this exam consists of seven (7) pages and six (6) problems and is worth a total of 100 points. The point value of each problem is indicated. To obtain full credit, you must have the correct answers along with relevant supporting work to jus- tify them. Partial credit will be given based on the work that is shown. However, answers without sufficient supporting work will receive no credit. Also, you must complete the exam in 48 minutes. Good luck! Name: Signature: OSU Internet Username: Problem # Score Problem # Score 1 4 (15 pts) (18 pts) 2 5 (12 pts) (20 pts) 3 6 (15 pts) (20 pts)
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1. (a) (5) Calculate lim ( x,y ) (0 , 0) 5 x 2 y x 4 + 4 y 2 , if it exists, or show that the limit does not exist. Solution: Along the curve C 1 : y = 0 , 5 x 2 y x 4 + 4 y 2 = 0 for all ( x,y ) 6 = (0 , 0) . Hence 5 x 2 y x 4 + 4 y 2 0 = L 1 as ( x,y ) (0 , 0) along C 1 . On the other hand, along the curve C 2 : y = x 2 , 5 x 2 y x 4 + 4 y 2 = 5 x 4 x 4 + 4 x 4 = 1 for all ( x,y ) 6 = (0 , 0) . Hence 5 x 2 y x 4 + 4 y 2 1 = L 2 as ( x,y ) (0 , 0) along C 2 . Since L 1 6 = L 2 , lim ( x,y ) (0 , 0) 5 x 2 y x 4 + 4 y 2 does not exist. (b) (2) Is the function f ( x,y ) = 5 x 2 y x 4 + 4 y 2 if ( x,y ) 6 = (0 , 0) 0 if ( x,y ) = (0 , 0) continuous? Justify your answer. Solution:
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midterm_sol - Midterm 1 of Math 254.01 (Au10) October 18,...

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