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Multivar calculus final (1)

  1. Jan 17, 2016 #1
    1. The problem statement, all variables and given/known data
    I had a rather tough multivariable calculus final, and the class was creamed by it. Apparently the professor thinks a 40% average is okay.

    But here is a portion of the problems. Please help me solve 'em. I'm not exactly sure if I'm taking the right path in these, but I did work on them to try and solve it.

    F. jjfzk73.png
    show that he5tvs9.png

    G. xyz=a3 encloses a tetrahedron with any of its tangent planes. Show that the area of the tetrahedron is constant.
    In my work, I sometimes just considered a3 to be any constant "c" for simplicity's sake.

    2. Relevant equations
    (Look..... there's a ton of relevant equations..)


    3. The attempt at a solution
    Here is my attempt at a solution.

    F.
    http://postimg.org/image/fif8ixl53/
    I'm actually pretty happy with this solution, seeing as I may have gotten it somewhere. But as for really figuring it out entirely I couldn't. I was just stuck at y/x and I can't really see how the equation works out.
    http://postimg.org/image/xudxf024b/
    There's my attempt at the end. I'm really not sure if this was mathematically allowed.

    G.
    http://postimg.org/image/o2myzhgw9/
    Here's my attempt at this......
    I really dunno what to do. It feels really simple, but at the time I tried I just couldn't do it.
    I know you can represent the area of a tet by multiplying xy(.5) and then z(.5) which gives the area but I ended up not being able to do that. I think. I couldn't figure out how to write the equations. You can see those equations in the bottom left- they're boxed. I don't know if those are right. I'm not sure if the denominator is interchangable with the other remaining variable. I tried to draw level curves with one variable fixed at 1 in those 3 graphs, but I'm not sure if those were "right". Etc etc...............



    Thanks.
     
  2. jcsd
  3. Jan 17, 2016 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    I would be willing to look at the solutions if you typed them out.
     
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