211-2008&amp;2009-3-M20-August2009

# 211-2008&2009-3-M20-August2009 - kuwait university...

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k u w a i t u n i v e r s i t y Math 211 Second Midterm Exam 1 August 2009 Calculators and mobile phones are not allowed. 1. Let f ( x,y ) = xy ln( x 2 + y 2 - 4) . [4 pts] (a) Find and sketch the domain of f . (b) Find and sketch the level curve passing through the point P (3 , 0) . 2. Show that the following limit does not exist: [2 pts] lim ( x,y ) (0 , 0) 3 x 4 y 2 7 x 8 + y 4 3. Find the linearization of the function [2 pts] f ( x,y ) = x 2 2 + y 2 4 + xy + 3 cos( x - 2) - 3 y + 4 at the point P (2 , 1) . 4. Given the function [4 pts] f ( x,y ) = x 3 cos x - 4 y 3 x 2 + y 2 if ( x,y ) 6 = (0 , 0) 0 if ( x,y ) = (0 , 0) (a) Use the deﬁnition of partial derivative to ﬁnd f y (0 , 0) . (b) Determine whether f y ( x,y ) is continuous at (0 , 0) . 5. Let the directional derivatives of f ( x,y ) at the point P (1 , 0) in the direction of the vectors v = h 3 , 4 i and w = h 4 , - 3 i be equal to 1 and - 1, respectively. [4 pts] (a) Find the gradient vector of f at the point P . (b) Show that - 2 D u f (1 , 0) 2 for all unit vectors u = h a,b i . 6. Let

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## This note was uploaded on 02/23/2010 for the course CAL C 0410211 taught by Professor Deparpment during the Spring '10 term at Kuwait University.

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211-2008&2009-3-M20-August2009 - kuwait university...

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