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Multivariable calculus yummy in my tummy

  1. Sep 18, 2007 #1
    Hi, my teacher gave us this problem, and he couldn't figure out why
    method was incorrect and why I got the answer I did.

    Given we know the gradient slope = <-56,1.886> at the point (2,0) on a
    surface f(x,y), in what direction, expressed as a unit vector, is f
    increasing most rapidly?

    I solved the problem like this:

    Max slope = magnitude of gradient slope
    (gradient slope) dot (unit vector) = 56.03
    -56.03Ux + 1.866Uy = 56.03
    sqt(Ux^2 + Uy^2) = 1

    Solving the system Ux= -.9977 Uy=.0672

    The only problem is, by definition, I should be able to get the unit
    vector by taking the gradient slope vector and dividing by the
    magnitude of the gradient vector. or,

    <-56/56.03, 1.886/56.03> = <Ux,Uy> = <-.9994,.0336>

    The weird thing is, the correct Uy value is almost exactly half of
    mine...what's going on???


    I tried a similiar technique for finding where the ∇f = 0

    <-56, 1.866> dot (unit vector) = 0

    -56.03Ux + 1.886Uy = 0
    sqt(Ux^2 + Uy^2) = 1

    Solving the system this time I got the correct answer, why here and
    not there?
  2. jcsd
  3. Sep 19, 2007 #2


    User Avatar
    Science Advisor

    I understand that the 56.03 is the magnitude of the given vector but the coefficient of Ux should be -56, not -56.03.

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