test4 - MATH2004A Test 4 4:35 pm - 5:25 pm, Nov 19 Name:...

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MATH2004A – Test 4 – 4:35 pm - 5:25 pm, Nov 19 Name: Student Number: Total: 20 marks Closed book, no calculator! 1. (3 points) Evaluate the double integral RR D (sin x sin y ) dA , where D = { 0 x π 2 , 0 y π 2 . Solution: ZZ D (sin x sin y ) dA = Z π/ 2 0 Z π/ 2 0 (sin x sin y ) dxdy = Z π/ 2 0 (sin x ) dx Z π/ 2 0 (sin y ) dy = ( - cos x ) | π/ 2 0 ( - cos y ) | π/ 2 0 = 1 . 2. (9 points) Let f ( x,y ) = x 3 + 2 y 3 - 3 x 2 - 3 y 2 + 1. (i) [3 points] Calculate f x ( x,y ), f y ( x,y ), f xx ( x,y ), f xy ( x,y ), f yy ( x,y ). (ii) [2 points] Find the directional derivative of f ( x,y ) at the point (1 , - 1) in the direction of v = < - 6 , - 8 > . (iii) [2 points] Given that (2 , 1) is a critical point of f ( x,y ), determine if it is a local maximum, a local minimum or a saddle-point. (iv) [2 points] Let x = r cos t , y = r sin t . Use chain rule to find the first order partial derivative of f ( x,y ) with respect to t , i.e., ∂f ∂t (you don’t need to simplify the result). Solution: (i)
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This note was uploaded on 11/29/2010 for the course MATH 2004 taught by Professor Bulig during the Fall '09 term at Carleton CA.

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test4 - MATH2004A Test 4 4:35 pm - 5:25 pm, Nov 19 Name:...

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