Q2mat237 - 4 4. [4 marks] Using the definition (formulate...

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1. Let be defined by . R R F 2 : ) )( 2 ( ) , ( 2 2 2 x y y x y x F + = (a) [5 marks] Determine whether the set S = is a smooth curve. } 0 ) , ( : ) , {( ) 0 ( = = y x F y x F Near which points of S is S the graph of a function 1 C ) ( y f x = ? (b) [5 marks] Suppose the surface S is parametrized by , i.e. S = im f . ) , , ( ) , ( 2 v u v u e v u v u + = + f Find the vector normal to the surface S at the point S ) 2 , 2 , 1 ( and the equation of the tangent plane at this point. 2
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2. Let S be the intersection of the paraboloid and the plane . 1 2 2 + + = y x z 1 2 + = x z (a) [5 marks] Find a parametrization of S. (b) [3 marks] Find a parametric equation for the tangent line to S at the point . ) 5 , 0 , 2 ( 3
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3. Let . ) , 2 ( ) , ( ) , ( xy y x y x v u = = f (a) [3 marks] Which regions in the xy - plane map onto the rectangle in the uv - plane given by ? Draw a picture of them. 2 1 , 1 0 v u (b) [5 marks] Does the inverse of f exist in a neighborhood of the point ) 1 , 1 ( ? Justify, formulating precisely the theorem you have used.
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Unformatted text preview: 4 4. [4 marks] Using the definition (formulate it), prove that the set } 2 , 2 : ) , {( 2 = + = x y x R y x S has zero content. 5 5. ( Test bonus question ) Let ) ( 2 x y xf xy u + = . (a) [3 marks] Determine the constant k such that kxy u u y u x y x = + . (b) [1 mark] Find the directional derivative of u at the point (1, 1) in the direction ( 5 4 , 5 3 ) if it is given that 1 ) 1 ( ) 1 ( = = f f . (c) [1 mark] Fill the blanks: Suppose f is . ..on an .. set S and M f ) ( x for every S x . Then the inequality holds for all S b a , (which is a corollary to the Mean Value Theorem). 6...
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Q2mat237 - 4 4. [4 marks] Using the definition (formulate...

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