# 1699 - 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 Spring 2008 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) A curve is given parametrically by the equations x = sin(4 t ); y = cos(4 t ); z = 2 t 3 / 2 . Compute the parametric equations of the tangent line at the point when t = 1. (b) Write an iterated integral, using polar coordinates, whcich can be used to ﬁnd the surface area of the portion of the graph of z = xy 3 which is above the region in the ﬁrst quadrant of the xy-plane inside the circle x 2 + y 2 = 1. YOU DO NOT HAVE TO EVALUATE THE INTEGRAL. 2. For parts (a),(b) and (c) let f ( x,y ) = x 2 - e xy . (a) At the point (3 , 0) compute the unit vector in the direction of maximum increase of the function f and compute the rate of increase in that direction. (b) Compute the directional derivative of the function

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1699 - Department of Mathematics Math 203 The City College...

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