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final

# final - Math 31BH Winter 2011 Final Exam Name Student ID...

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Math 31BH - Winter 2011 - Final Exam Name: Student ID: Instructions: During the test, you may not use books or notes. Read each question carefully, and show all your work. Answers with no explanation will receive no credit, even if they are correct. There are 11 questions which are worth 115 points. You have 180 minutes to complete the test. Question Score Maximum 1 10 2 10 3 8 4 10 5 8 6 10 7 10 8 10 9 8 10 18 11 13 Total 115

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Problem 1. [ 10 points. ] Find the critical points of the function f ( x, y ) = 1 2 x 2 + 3 2 y 2 - xy 3 and determine their nature.
Problem 2. [ 11 points. ] Consider the function f ( x, y ) = y 2 x 4 . (i) [4] Draw the level curves of f . (ii) [3] At the point P (1 , 1), find the unit direction of steepest increase for f . (iiii) [4] For the vector ~u = 4 ~ i +3 ~ j 5 , find the directional derivative of f at P in the direction ~u , and use it to estimate f ((1 , 1) + . 01 ~u ) .

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Problem 3. [ 8 points. ] Show that the function f : R 2 R 2 given by f ( x, y ) = ( e - x + e 2 y , e 3 y + e 2 x ) is locally invertible at every point.
Problem 4. [ 10 points. ] Consider the region R given by x 2 + 4 y 2 100 .

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final - Math 31BH Winter 2011 Final Exam Name Student ID...

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