03spr-f - FINAL EXAM Math 51 Spring 2003 You have 3 hours...

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FINAL EXAM Math 51, Spring 2003. You have 3 hours. No notes, no books, no calculators. YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT Good luck! Name ID number 1. (/40 points) 2. (/40 points) 3. (/40 points) 4. (/40 points) 5. (/40 points) Bonus (/20 points) Total (/200 points) “On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination.” Signature: Circle your TA’s name: Byoung-du Kim (2 and 6) Ted Hwa (3 and 7) Jacob Shapiro (4 and 8) Ryan Vinroot (A02) Michel Grueneberg (A03) Circle your section meeting time: 11:00am 1:15pm 7pm 1
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1. Consider the function f : R 2 R 2 given by f pb x y BP = xy ( x 2 + y 2 ) 2 if b x y B n = b 0 0 B 0 if b x y B = b 0 0 B (a) Do ∂f ∂x and ∂f ∂y exist at the origin? If yes, compute them; if not, explain why. 2
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(b) Is the function f continuous at the origin? Explain your reasoning. (c) Is the function f diFerentiable at the origin? Explain your reasoning. 3
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2. (a) Suppose that a function f is diFerentiable at a given point -→ a . Use the following
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03spr-f - FINAL EXAM Math 51 Spring 2003 You have 3 hours...

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