Math13S09final_sol

# Math13S09final_sol - Tufts University Department of...

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Tufts University Math 13 Department of Mathematics Spring 2009 Solutions to Final 1. True or False? (a) F: cylinder (b) T (c) F (d) F: maybe saddle (e) T 2. (a) The vector < 3 , 0 , 2 > , from the origin to the point (3 , 0 , 2), as well as the direction vector < 1 , 2 , 1 > of the line are in the plane, so a normal is given by vn = < 3 , 0 , 2 > × < 1 , 2 , 1 > = < 4 , 5 , 6 > . Therefore, we have line given parametri- cally by x ( t ) = 3 4 t , y ( t ) = 5 t , z ( t ) = 2+6 t , (b) Let f ( x,y,z ) = e x ln y + z 3 , then the gradient is normal to level surfaces so we have v f = < e x ln y, e x y , 3 z 2 > | (3 , 1 , 2) = < 0 ,e 3 , 12 > is normal to the surface and tangent plane. Thus an equation for the tangent plane is e 3 y + 12 z = e 3 + 24. 3. x 2 y 2 z 2 = 1 is a hyperboloid of one sheet with the x -axis as the axis of symmetry. x 2 y 2 z 2 = 0 is a cone with the x -axis as the axis of symmetry. x 2 y 2 z 2 = 1 is a hyperboloid of two sheets with the x -axis as the axis of symmetry. x y z x 2 y 2 z 2 = 1 x y z x 2 y 2 z 2 = 0 x y z x 2 y 2 z 2 = 1 1

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4. For double integrals we think of the integrating the height of boxes over a region in the plane.
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