Results 1 to 9 of 9

Math Help - Directional Derivatives 2

  1. #1
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1

    Directional Derivatives 2

    Compute the directional derivative of the following function along unit vectors at the indicated points in directions parallel to the given vector:

    f(x,y) = x^y, \ (x_0,y_0) = (e,e), \ \bold{d} = 5\bold{i} + 12\bold{j}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,803
    Thanks
    1692
    Awards
    1
    What are the partials and their values at (e,e)?
    What is the unit vector in the direction of d?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1
    The partials are:

    \frac{\partial{f}}{\partial{x}} = yx^{y-1}

    and

    \frac{\partial{f}}{\partial{y}} = x^y \log{x}

    So, at (e, e) you get:

    For the 1st one: ee^{e-1} = e^e

    For the 2nd one: e^e \log{e}

    And the unit vector of \bold{d} is:

    \bold{v} = \frac{5}{13}\bold{i} + \frac{12}{13}\bold{j}

    The parallel part is what confuses me...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,803
    Thanks
    1692
    Awards
    1
    First \ln (e) = 1.
    You are correct about v it is a unit vector in the same direction as d.
    Parallel means a scalar multiple.

    Now what is the directional derivative?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1
    So, I need to use \nabla f(\bold{x}) \cdot \alpha\bold{v} = \alpha[\nabla f(\bold{x})] \cdot \bold{v}

    Right?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,803
    Thanks
    1692
    Awards
    1
    I give you a very cautious yes is may be correct.
    But because v is a unit vector then \alpha = 1.
    You did this in your other posting. So why is this different?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1
    If parallel means a scalar multiple, then \alpha\bold{v}, where \alpha is some number. Would that not cover the parallel vectors? I think this is the difference between this problem and the one I posted before.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,803
    Thanks
    1692
    Awards
    1
    Quote Originally Posted by Aryth View Post
    If parallel means a scalar multiple, then \alpha\bold{v}, where \alpha is some number. Would that not cover the parallel vectors? I think this is the difference between this problem and the one I posted before.
    You are overthinking this problem!
    If we want a directional derivative parallel to i + 2j - 2k we change that vector to the following: \frac{1}{3}\left( {i + 2j - 2k} \right).
    You make the vector into a unit vector.
    Again I say that you are over thinking this problem.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1
    I got it. Thanks. I figured it out, Haha.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Directional Derivatives
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 13th 2010, 01:23 PM
  2. Directional derivatives
    Posted in the Calculus Forum
    Replies: 4
    Last Post: November 18th 2009, 03:18 PM
  3. Directional Derivatives
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 7th 2009, 11:45 PM
  4. Directional derivatives
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 2nd 2008, 01:53 AM
  5. Directional derivatives
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 5th 2008, 02:51 PM

Search Tags


/mathhelpforum @mathhelpforum