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final - 18.02 FINAL EXAM BJORN POONEN 1 out of 25 2 out of...

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18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) Please turn cell phones off completely and put them away. No books, notes, or electronic devices are permitted during this exam. Generally, you must show your work to receive credit. Name: Student ID number: Recitation leader’s last name: Recitation time (e.g., 10am): 1 out of 25 2 out of 15 3 out of 15 4 out of 15 5 out of 15 6 out of 10 7 out of 25 8 out of 20 9 out of 20 10 out of 25 11 out of 25 12 out of 20 13 out of 20 Total out of 250

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1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please do not use the abbreviations T and F.) No explanations are required in this problem. (a) (5 pts.) If f ( x, y ) is a continuously differentiable function on R 2 , and 2 f ∂x 2 = 0 at every point of R 2 , then there exist constants a and b such that f ( x, y ) = ax + b for all x and y . (b) (5 pts.) The annulus in R 2 defined by 9 x 2 + y 2 16 is simply connected. (c) (5 pts.) If f ( x, y ) is a function whose second derivatives exist and are continuous everywhere on R 2 , and f (0 , 0) = f x (0 , 0) = f y (0 , 0) = f xx (0 , 0) = f yy (0 , 0) = 0 and f xy (0 , 0) = 0, then f has a saddle point at (0 , 0).
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final - 18.02 FINAL EXAM BJORN POONEN 1 out of 25 2 out of...

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