finalfall05 - Math 192 Final Exam Dec 8 2005 7:00-9:30pm...

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Math 192 Final Exam. Dec 8 , 2005 , 7:00-9:30pm You are allowed one 8 . 5 × 11 sheet of paper with information on both sides. You are NOT allowed calculators, the text, or any other book or notes. SHOW ALL WORK! WRITE YOUR LECTURE NUMBER AND TA’S NAME ON YOUR EXAM BOOK. 1) Let f ( x, y ) = 1 - x 2 2 - y 2 16 . a) (5 Points). Find the equation of the tangent plane to the graph of f above the point (1 , 2). b) (5 Points). Using linear approximation, estimate the value of f at the point (0 . 8 , 2 . 4). 2) Let F = - + x be a vector field in the plane. Let C be a simple closed curve enclosing a domain D . a) (7 Points). If C is parametrized in a counterclockwise direction, show that 1 2 C F · T ds = Area of D. b) (5 Points). Use Part (a) to compute the area of the ellipse parameterized by r ( t ) = ( a cos t ) ı + ( b sin t ) , where a and b are constants, and 0 t 2 π . 3) (12 Points). Find the area of the region that lies outside of the circle r = 1 and inside the circle r = 2 sin θ .
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