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Directional Derivatives

  1. Mar 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Assume f(1,1,1)=3 and f(1.1,1.2,1.1)=3.1

    a) Which directional derivative Duf at (1,1,1) can be estimated from this information? Give vector u

    b) Estimate the directional derivative in part a

    2. Relevant equations

    Duf = del f (dot product) vector u
    del f = ([itex]\partial[/itex]f/[itex]\partial[/itex]x, [itex]\partial[/itex]f/[itex]\partial[/itex]y)




    3. The attempt at a solution

    So far I've been able to get unit vector
    u = <1.1-1, 1.2-1, 1.1-1>/[itex]\sqrt{.1^2+.2^2+.1^2}[/itex] = <0.41, 0.82, 0.41>

    I've been rolling it around in my head but I can't think of a way to obtain del f.
    How would I get any thing resembling the partial derivatives of the unknown function, if all I know are points? I understand del f is the vector pointing in the direction of greatest change.



    Edit: I have made another push at an answer.
    I figured del f= <partial f/ partial x, partial f/ partial y, partial f/partial x>
    using partial f = 3.1-1=.1
    partial x = .1
    partial y= .2
    partial z= .1

    del f= <.1/.1, .1/.2, .1/.1>=<1, 1/2, 1>

    so the directional derivative would be
    <1, 1/2, 1>dot<.41, .82, .41> =1.23
     
    Last edited: Mar 12, 2012
  2. jcsd
  3. Mar 12, 2012 #2

    Fredrik

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    What I would have used is that
    $$D_{\vec u}f(\vec x)=\lim_{t\to 0}\frac{f(\vec x+t\vec u)-f(\vec x)}{t} \approx\frac{f(\vec x+t\vec u)-f(\vec x)}{t}$$ when t is small. But your approach should work too. You should however avoid notations that suggest that ∂f, ∂x etc. are numbers.
     
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