# exam2f02 - Name: Recitation: Multivariable Calculus Math...

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Name: Recitation: Multivariable Calculus Math 215 Fall 2002 Second Exam November 14, 2002 Please do all your work in this booklet and show all the steps . Write your ﬁnal answer in the corresponding box . Calculators are not allowed. One 3x5 note-card (both sides) is allowed. Problem Possible points Score 1 15 2 15 3 10 4 10 5 15 6 15 7 10 8 10 Total 100

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2 Problem 1. (15 pts.) Find and classify all critical points of the function f ( x, y ) = 6 x 2 - 2 x 3 + 3 y 2 + 6 xy + 31 Answer: ˛ ˛ ˛ ˛ ˛ 15
3 Problem 2. (15 pts.) Find the shortest and the longest distances between the points on the ellipse 17 x 2 + 12 xy + 8 y 2 = 100 and the origin. At what points are these distances attained? x y Answer: (Shortest) Answer: (Longest) ˛ ˛ ˛ ˛ ˛ 15

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4 Problem 3. (10 pts.) Use the level curves of the function f ( x, y ) on the graph on the right to approximate ZZ R f ( x, y ) dA over the rectangle R = [ - 1 , 2] × [ - 1 , 1] by Riemann Sums. Explain your setup.
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## This note was uploaded on 11/03/2008 for the course MATH 215 taught by Professor Fish during the Fall '08 term at University of Michigan.

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exam2f02 - Name: Recitation: Multivariable Calculus Math...

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