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mid1-b - Math 31BH Winter 2011 Midterm I Name Student ID...

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Math 31BH - Winter 2011 - Midterm I Name: Student ID: Instructions: Please print your name and student ID. 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 5 questions which are worth 50 points, and a bonus question. You have 50 minutes to complete the test. Question Score Maximum 1 8 2 16 3 11 4 9 5 6 EXTRA CREDIT 5 Total 50+5
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Problem 1. [ 8 points ] For each of the following limits, either find its value or explain why it does not exist: (i) [4] lim x,y 0 ( x 2 + y 2 ) 2 2 x 2 +3 y 2 (ii) [4] lim x,y 0 x 3 y x 4 + y 4
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Problem 2. [ 16 points ] Consider the function f ( x, y ) = x 2 sin(2 y - 2 x ) and the point P (1 , π 2 + 1) . (i) [4] Find the gradient of f at the point P . (ii) [4] Calculate the directional derivative of f at P in the direction ~u = ~ 3 i + 4 ~ j 5 .
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(iii) [4] What is the unit direction of steepest decrease for the function f ? What is the rate of decrease? (iv) [4] Find the equation to the tangent plane to the graph of f at the point ( P, f ( P ))
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Problem 3. [ 11 points ] Consider the function f ( x, y ) = ( | y | , if | x | ≤ | y | | x | , if | y | < | x | . (i) [4] Draw the cross section of the graph of
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