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211-2007&amp;2008-2-M20-April2008

# 211-2007&amp;2008-2-M20-April2008 - Kuwait University...

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Kuwait University Math. 211 April 28, 2008 Dept. of Math. & Comp. Sci. Second Exam. Duration: 90 min. Answer all questions. Calculators and Mobile Phones are not allowed. 1. [3 pts.] Describe and sketch the domain of f ( x, y ) = q x - y 2 + ln( x - 2 y ) . 2. [2 pts.] Find the limit if it exists lim ( x,y ) (0 , 0) x 2 - tan( xy ) xy 3. [4 pts.] Let z = f ( x, y ), where x = r cos θ and y = r sin θ . Show that sin θ ∂z ∂r + 1 r cos θ ∂z ∂θ = ∂f ∂y . 4. [4 pts.] Let f ( x, y, z ) = x 2 y + 9 + z 2 . (a) Find the linear approximation of the function f at the point (1 , 4 , 4). (b) Using part(a), find an approximate value of (0 . 9) 2 4 . 1 + q 9 + (3 . 9) 2 . 5. [4 pts.] Let f ( x, y ) = x ln y + sin( πxy ). (a) Find the rate of change of f at the point P ( e, 1 e ) in the direction from P to Q (2 e, 0). (b) In what direction does f have maximum rate of change at P ? What is that maximum rate of change? 6. [4 pts.] Find the absolute extreme values of the function f ( x, y ) = x 2 + y 2 + x + y on the triangular region with vertices (0 , 0) , ( - 2 , 0) and (0 , - 2). 7. [4 pts.] Find the local extrema and saddle points of the function f ( x, y ) = x 2 + 2 xy + y 3 - y 4 4

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Kuwait University Math. 211 May 7, 2008 Dept. of Math.
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211-2007&amp;2008-2-M20-April2008 - Kuwait University...

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