211-2008&2009-3-F10-August2009

211-2008&2009-3-F10-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 Final Exam 15 August 2009 Calculators and mobile phones are not allowed. 1. Answer as True or False and justify your answers: [6 pts] (a) If 0 < a n < 1 , then the sequence { a n } converges. (b) If a sequence { a n } converges, then the series a n converges. (c) n =1 n - cosh α converges for any real number α 6 = 0 . (d) R C (e - xy i - e xy j + e z k ) · d r = 0 along any closed curve C in R 3 . 2. Let F ( x,y,z ) = e ( x + y ) z + x - y + z + 1 . (a) Find ∂z ∂x and ∂z ∂y at the point ( x,y ) = ( - 1 , 1) , where z is defined implicitly as a function of ( x,y ) by the equation F ( x,y,z ) = 0 . [3 pts] (b) Write an equation for the tangent plane to the surface F ( x,y,z ) = 0 at the point P ( - 1 , 1 , 0) . [3 pts] 3. Find a R for which f ( x,y ) = ( a - 1) x 2 + ( a + 1) y 2 + 5 xy + x + 2 y has (i) local maximum values; (ii) local minimum values; (iii) saddle points. [3 pts] 4. Given ZZ D f ( x,y ) dA = Z 1 0 Z 2 y 0 f ( x,y ) dxdy + Z 3 1 Z 3 - y 0 f ( x,y ) dxdy (a) Sketch the region D and express the double integral as an iterated integral with reverse order of integration. [3 pts] (b) Evaluate the double integral for f ( x,y ) = cos(4 x - x 2 ) . [3 pts] 5. Find the volume of the solid tetrahedron bounded by the planes x + z - 1 = 0 , y = 2 x, y = 4 x, and z = 0 . [3 pts] 6. Evaluate the triple integral ZZZ E y dV , where E is the solid bounded by the paraboloid y = 4 x 2 + 4 z 2 and the plane y = 4 . [3 pts] 7. Evaluate R C xy ds, where C is the circle x 2 + y 2 + 2 x - 4 y - 4 = 0 . [3 pts] 8. Use Green’s Theorem to evaluate the line integral R C ( y +cosh x 2 ) dx +(2 x +sinh y 2 ) dy , where C is the boundary of the region enclosed by the parabolas y = x 2 and x = y 2 .
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211-2008&amp;amp;2009-3-F10-August2009 - kuwait university...

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