Results 1 to 2 of 2

Math Help - Directional Derivative in the direction of steepest descent.

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    21

    Directional Derivative in the direction of steepest descent.

    Hi

    I just want to check that I am solving this correctly.

    You are given a function f(x, y, z) = x^2 + 3xy + 2z and asked to find the directional derivative at the point (2, 0, -1) in the direction of the steepest descent.

    Partial Derivatives at (2, 0, -1):
     f_x = 2x + 3y = 4
     f_y = 3x = 6
     f_z = 2

    Now this gives you the direction of the steepest descent : -\nabla f=(-4, -6, -2).

    So you then find the unit vector:  {u} = (\frac{-2}{\sqrt{14}}, \frac{-3}{\sqrt{14}}, \frac{-1}{\sqrt{14}} )

    And then you dot product the above with with the partial derivatives giving:

    D_u f = u \bullet (2x + 3y, 3x, 2) = \frac{-13x}{\sqrt{14}} - \frac{6y}{\sqrt{14}} - \frac{2}{\sqrt{14}}

    Is that the directional derivative? Or do I then have to substitute in the point (2, 0, -1) ?

    I would appreciate if someone could verify the above for me, or correct it (more likely!).

    Also, is the magnitude of the rate of steepest descent  \mid u \mid = \sqrt{14} ?

    Thanks a lot.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    That is correct up to the final step, and yes, you do then have to substitute in the point (2,0,1).

    Also, the magnitude of the rate of steepest descent is 2\sqrt{14}, because that is what you divided \nabla f by in order to get a unit vector.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: March 26th 2011, 10:38 PM
  2. Replies: 0
    Last Post: November 20th 2009, 05:28 AM
  3. Path of steepest descent
    Posted in the Calculus Forum
    Replies: 4
    Last Post: October 13th 2009, 03:15 PM
  4. Gradient - Steepest Descent
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 12th 2009, 05:52 PM
  5. steepest descent
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 8th 2006, 11:34 AM

Search Tags


/mathhelpforum @mathhelpforum