final_fall07_2401

# final_fall07_2401 - Math 2401 M1 M2 M3 Final Total 40...

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Math 2401 M1 M2 M3 Final Total: 40 points Section: Name: Student Number: (1) (a) Parametrize the curve C : x 2 + 4 y 2 = 36 with a vector valued function r . (2 points). (b) Find the tangent and normal vectors of curve C . (2 points). (c) Find the curvature and arc length of C . (There is no need to evaluate the definite integral). (2 points). (2)(a) Find the tangent plane equation at a point ( x 0 , y 0 , z 0 ) of the surface S : z = x 2 + y 2 + xy - x - y , and find a vector in the xy -plane along which z decreases the fastest on the surface. (2 points). (b) The surface S intersects the cylinder y = x 2 in a space curve C . Find a parametrization for C and find the tangent vector of C . (2 points). (c) The surface S intersects the cylinder y = x 2 + sin ( xy ) in a space curve C 1 . Find the tangent vector of C 1 . (1 point). (d) Minimize 2 x - 3 y + 5 z on ( x - 1) 2 + y 2 + z 2 = 19. (2 points). (3) (a) Evaluate R R Ω xdxdy where Ω is the parallelgram bounded by x + y = - 1, x + y = 1, x = 2 y and x = 2 y + 1. (2 points). (b) Evaluate R R R T ( x 2 + y 2 + z 2 ) dxdydz where T is the solid x 2 + y 2 + z 2 1, z 2 x 2 + y 2 . (2 points).

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