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1692 - Department of Mathematics Math 203 The City College...

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Department of Mathematics The City College of New York Math 203 Final Exam Fall 2005 No calculators permitted. Answers may be left in terms of radicals, π , e , etc. and do not need to be simplified unless stated otherwise. Each question is worth 10 points. Part I. Answer all 7 questions 1. a) For the curve given parametrically by the equations x = sin( πt ); y = e t ; z = e - t , compute the parametric equations of the tangent line at the point (0 , 1 , 1). b) Compute all unit vectors normal to the plane which contains the points (0 , 1 , 1) , (1 , - 1 , 0) , and (1 , 0 , 2) and compute an equation of the plane. 2. For parts a), b) and c) let f ( x, y ) = x 2 - 3 xy . a) At the point with x = 1 and y = - 1 compute the unit vector pointing in the direction of greatest increase of the function f ( x, y ) and compute the rate of increase in that direction. b) Compute an equation for the plane tangent to the surface given by the equation z = f ( x, y ) at the point in space with x = 1 and y = - 1. c) Find the rate at which f ( x, y ) is changing at (1 , - 1) in the direction toward the point (5 , 2).
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