Math 23 final2002 - Mathematics 23 Final Exam Answer all questions and show all work Simplify your answers 1 Find a vector perpendicular to the plane

Math 23 final2002 - Mathematics 23 Final Exam Answer all...

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Mathematics 23 Final Exam, December 14, 2002 Answer all questions and show all work . Simplify your answers. 1. Find a vector perpendicular to the plane containing the points (1 , 2 , 1) , (2 , 4 , 2), and (1 , 0 , 1). 2. For the function f ( x, y ) = xe - 2 y + 6 y , find a. the maximum rate of change of f at (1 , 0). b. the direction in which the maximum rate of change at (1 , 0) occurs. 3. Let E be the solid above z = x 2 + y 2 and below x 2 + y 2 + z 2 = z . Set up an integral (in spherical coordinates) which gives the volume of E . 4. Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 3 y + 2 z = 6. 5. a. Find the tangent plane to the ellipsoid x 2 + 2 y 2 + 3 z 2 = 1 at the point (1 , 0 , 0). b. Find the points on the ellipsoid x 2 + 2 y 2 + 3 z 2 = 1 where the tangent plane is parallel to the plane 3 x - y + 3 z = 1. 6. a. Find the length of the curve C given by r ( t ) = h cos t, sin t, t i , 0 t 1. b. What is the curvature of C at r ( t o ), where t o is an arbitrary point 0 t o 1?

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