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# 237t2 - 1 Let the surface S be parametrized by f(u v =(2 u...

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1. Let the surface S be parametrized by ) , , 2 ( ) , ( 2 v u e u v u v u v + + + = - f and let ) 1 , 2 , 2 ( 0 P be a point in 3 R . (a) [2 marks] Show that the point 0 P lies on the surface S . (b) [5 marks] Write the equation for the tangent plane to S at the point 0 P in the form d cz by ax = + + , where d c b a , , , are constants. 2

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2. Let V be the set in the first octant in 3 R bounded by the surface 8 4 2 3 3 3 = + + z y x and the coordinate planes and let the map 3 3 : R R G be defined by ) , , ( ) , , ( 3 3 3 w v u w v u = G . (a) [7 marks] Apply the change of variables ) , , ( ) , , ( w v u z y x G = to the integral ∫ ∫ ∫ + + = V dV z y x I ) 1 ln( . Write the iterated integral for I in the new variables in such an order of integration that there are NO FRACTIONS in the limits of integration. DO NOT EVALUATE THE INTEGRAL (b) [2 marks] Explain briefly , why the given change of variables could be applied for I although 0 ) , , ( det = w v u D G on some subset of the inverse image of V under G . 3
3. (a) [3 marks] Formulate the Mean Value Theorem for Integrals.

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237t2 - 1 Let the surface S be parametrized by f(u v =(2 u...

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