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Contour map and Intersection points

  1. Jan 6, 2007 #1
    1. The problem statement, all variables and given/known data

    (i) Drawing a contour map for the function h(x.y) = -12-4x^2+16x-y^2-8y
    (ii) (Continuing from i) at the point (1,-1,7) which direction to move to have
    the maximum increase in height?

    (iii) Find the point closest to the origin on the curve of intersection of the
    plane 2y + 4z = 5 and the cone z^2= 4x^2 + 4y^2.

    2. Relevant equations

    (i) z = -12-4x^2+16x-y^2-8y (then Im stuck)

    (iii) I got f = z <--(not sure if this is right),
    g = 2y +4z = 5
    and h = z^2= 4x^2 + 4y^2.

    then gradient f = lambda * gradient g + mu * gradient h

    3. The attempt at a solution

    then from some caluculation i got two intersection point of (0,5/18,5/9) and (0, 1/10, 1/20) with (0, 1/10, 1/20) the point closest to the origin.
  2. jcsd
  3. Jan 6, 2007 #2


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    Science Advisor

    Do you understand what a "contour map" IS? No, don't set h(x,y)= z. A contour map of a function of two variables is a graph is the xy-coordinates system. Set h(x,y)= equal to a number of different constants and graph each of them. (Looks to me like you will have a number of different hyperbolas with the same asymptotes.)

    Okay, that's the "Lagrange multiplier" method.

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