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Math Help - Directional Derivatives

  1. #1
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    Directional Derivatives

    Let f(x,y) - x^2y + xy^2. Find all vectors u such that Duf(1, -2) is exactly one-half the maximal possible directional derivative.
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  2. #2
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    Quote Originally Posted by kiddopop View Post
    Let f(x,y) - x^2y + xy^2. Find all vectors u such that Duf(1, -2) is exactly one-half the maximal possible directional derivative.
    I assume you mean f(x,y)= x^2y+ xy^2

    First find \nabla f(x,y) and ||\nabla f(x,y)||. That norm [b]is[/tex] the "maximal possible directional derivative[/tex].

    The directional derivative in the direction of vector \vec{u} is \nabla f(x,y)\cdot \frac{\vec{u}}{||\vec{u}||}

    Of course, \frac{\vec{u}}{||\vec{u}||} is just the unit vector in the direction of \vec{u} and that can always be written as cos(\theta)\vec{i}+ sin(\theta)\vec{j} where \theta is the angle \vec{u} makes with the x-axis.

    That is, \vec{u} is any vector making angle \theta with the x-axis where \nabla f \cdot cos(\theta)\vec{i}+ sin(\theta)\vec{j}= \frac{1}{2}||\nabla f|| which is the same as f_x cos(\theta)+ f_y sin(\theta)= \frac{1}{2}\sqrt{(f_x)^2+ (f_y)^2}.
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