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Minimizing a directional derivative

  1. Nov 26, 2005 #1

    Pengwuino

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    I know that you determine maximum value of a directional derivative at a point by finding...

    [tex]
    |\nabla f(a_0 ,b_0 )|
    [/tex]

    But how do you find the minimum value?

    I'm also kinda wondering exactly what a gradient is. It seems like if you have the gradient equation and a point... all you are getting is a single vector and a single rate of change... doesn't seem all that useful.
     
    Last edited: Nov 26, 2005
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  3. Nov 26, 2005 #2

    HallsofIvy

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    If going in the direction [itex]\theta[/itex] makes the derivative a maximum, what happens if you go in the exact opposite direction?
     
  4. Nov 26, 2005 #3

    Pengwuino

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    ... its the opposite and parallel vector isn't it (as in <1,1> is to <-1,-1>) hahaha... oh man.... i wonder why my professor made a big fuss about it on our homework assignment then...
     
  5. Nov 26, 2005 #4

    HallsofIvy

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    By the way- it also follows that the directional derivative is 0 in the direction perpendicular to the gradient. That's useful property: the gradient is always perpendicular to level curves.
     
  6. Nov 26, 2005 #5

    Pengwuino

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    Wow thats helpful since I now have to figure out 2 directions where the rate of change is 0 at a certain point. Problem is, I don't remember how to figure out what vector that would be.... I have....

    [tex]f(p,q) = qe^{ - p} + pe^{ - q} [/tex]

    at (0,0)

    I got….

    [tex]
    \nabla f(p,q) = \frac{{\partial f}}{{\partial p}}i + \frac{{\partial f}}{{\partial q}}j \\ [/tex]
    [tex] \frac{{\partial f}}{{\partial p}} = e^{ - q} - e^{ - p} q \\ [/tex]
    [tex] \frac{{\partial f}}{{\partial q}} = e^{ - p} - e^{ - q} p \\ [/tex]
    [tex]\nabla f(p,q) = (e^{ - q} - e^{ - p} q)i + (e^{ - p} - e^{ - q} p)j \\ [/tex]
    [tex]\nabla f(0,0) = < 1,1 > \\ [/tex]
    [tex] |\nabla f(0,0)| = \sqrt 2 \\
    [/tex]

    I assume this all means that at (0,0), the maximum slope is a vector of <1,1> with a rate of increase of [tex]\sqrt 2 [/tex]

    YES, STUPID LATEX, TAKE THAT!!! How do you get it to automatically go to a new line without having to tex and /tex after every single line?

    So.... is my assumption correct or am i missing the point of gradients all together?
     
    Last edited: Nov 26, 2005
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