Math13S09final_sol

Math13S09final_sol - Tufts University Department of...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
4. For double integrals we think of the integrating the height of boxes over a region in the plane.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern