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# 1693 - 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 2006 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) Compute an equation for the plane which contains the point (1,0,1) and the line given parametrically by the equations x = 2 t ; y = 2 + t ; z = 2 - t . b) Compute the directional derivative of the function F ( x, y, z ) = xz 2 - (2 y - 1) 2 at the point (1 , 2 , 3) in the direction of the vector < 1 , 0 , - 1 > . 2. For parts (a) and (b) let f ( x, y ) = e x y - 3 tan( x ). a) Compute the unit vector pointing in the direction of greatest increase of the function f at the point (0 , - 1) 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 = 0 and y = - 1. 3. Evaluate Z Z T x 2 y dA , where T is the first quadrant region bounded by the curve with equations y = x 3 and lines y = 8 and y = 8 x . Include a labeled sketch of the region

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