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Unformatted text preview: 1. (a) [6 marks] Suppose l is the tangent line to the curve at (1, 1, 1), ) , , ( ) ( : 3 2 t t t t C = g and is the plane tangent to the surface at . Find the 1 2 3 2 2 = + + z z y x y x ) 1 , , 1 ( equation of the line through (1, 1, 1) that is parallel to the plane and orthogonal to the line l . (b) [4 marks] Find a real valued function whose 0-level set is the image ) , , ( z y x F S of the map defined by 3 2 : R R f ) 2 1 , , ( ) , ( u v u v u v u + = f . 2 2. (a) [5 marks] Define the function by R R f 2 : + = ) ln( ) , ( 2 2 y x y y x f ) , ( ) , ( ) , ( ) , ( = y x for y x for . Let u = be a unit vector. Write, using the definition (as a limit!), the directional ) , ( b a derivative of f in the direction of u at the point (0, 0). In what direction(s), if any, does exists? Is f of class on f u 1 C 2 R ? Justify your answers with virtually no calculations....
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- Fall '09