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midterm_w07

# midterm_w07 - MATH237 MIDTERM EXAM WINTER 2007[8 1 Consider...

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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 ) negationslash = (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] a). From the point (1 , 1) in which direction does the function value increase most rapidly? Specify the direction as a unit vector. [3] b). From the point (1 , 1) , indicate any and all directions in which the rate of change is equal to zero. Specify directions as unit vectors. [5] c). From the point (1 , 1) , indicate any and all directions in which the rate of
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