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Determine the highest and lowest elevation on a path

  1. Oct 27, 2016 #1
    1. The problem statement, all variables and given/known data
    Determine the highest and lowest elevations given by the height
    z = f(x,y) = 1 - (1/16)x^2 - (1/9)y^2
    on the path r(t) = <2cos(t), 3sin(t)>. The xy position on the path at time t is given by r(t).

    2. Relevant equations
    Lagrange Multipliers
    Partial derivatives

    3. The attempt at a solution
    I was going to try using the Lagrange Multiplier method to find the extreme values of the function f(x,y), but I would need the xy position to be given as an equation and in x and y terms. I am confused as to how to do this because we are given a position vector instead, in terms of t.

    Could anyone help me out and guide me with how to do this?
     
  2. jcsd
  3. Oct 27, 2016 #2

    Mark44

    Staff: Mentor

    On the path, x(t) = 2cos(t), and y(t) = 3sin(t). So you could write f(x, y) as a function of t alone, and then use the derivative to find the maximum and minimum values. That's how I would approach this problem.
     
  4. Oct 27, 2016 #3

    Simon Bridge

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    Before choosing the method, understand the problem.
    The path is being given to you as a position vector that varies with t. What shape is the path?
    As you travel along that path, how does z vary?

    [edit] mark44 beat me to it.
     
  5. Oct 27, 2016 #4
    That sounds like a good idea, I should have thought of that. But then I am confused as to how to turn the position vector into a function that I can use to find extreme values (min/max).
     
  6. Oct 27, 2016 #5

    LCKurtz

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    Here's what Mark44 said above:
    Try that.
     
  7. Oct 27, 2016 #6

    Simon Bridge

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    If you are having trouble understanding mark44's suggestion, try the focussing questions in post #3.
    It is very difficult to tell you more detail without effectively telling you the answer too.
     
  8. Oct 28, 2016 #7
    Ya I get that, but to use Lagrange Multipliers, I need two sets of equations, and plugging in the 2cost and 3sint into the f(x,y) equation still only leaves me with one equation to use to find a max and min.
     
  9. Oct 28, 2016 #8

    Simon Bridge

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    So don't use Lagrange multipliers... have you heard of the 1st and second derivative tests?
     
  10. Oct 28, 2016 #9

    Ray Vickson

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    I don't see why you would want or need to use Lagrange multipliers in this problem. Lagrange multipliers are designed to deal with certain types of problems, and this is not one of them.
     
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