midterm_w07 - MATH237 - MIDTERM EXAM - WINTER 2007 [8] 1....

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MATH237 - MIDTERM EXAM - WINTER 2007 [8] 1. Consider the function f ( x, y ) = 2 + x 2 + y 2 . [4] a). Sketch the level curves of f ( x, y ) , f ( x, y ) = C for C = 2 , 3 , 4 and indi- cate how the level curves change as C is increased. [4] b). Sketch the cross-section where y = 0 , i.e. z = f ( x, 0) , in the xz -plane and sketch the graph z = f ( x, y ) in three dimensions. [9] 2. Consider the function f ( x, y ) = 2 x 2 + 9 y 2 . [5] a). Find ∂ f ∂x (0 , 0) if it exists or verify that it does not exist. Is f differentiable at (0 , 0) ? [4] b). Show that f is differentiable for all ( x, y ) n = (0 , 0) . Justify your answer. [8] 3. Consider the function f ( x, y ) = 1 + sin ( xy ) y . [4] a). Find the linear approximation to f at (0 , 1) . [4] b). Use the linear approximation to approximate f (0 . 1 , 0 . 8) . [15] 4. Consider the function g ( x, y ) = 5 x 2 y 2 . [4]
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This note was uploaded on 07/28/2009 for the course MATH MATH137 taught by Professor Oancea during the Spring '08 term at Waterloo.

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