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Math Help - Directional derivatives

  1. #1
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    Directional derivatives

    The question:
    A skier is on a mountain described by the equation h(x, y) = 2000 - x^4/10^8 - y^2/10^2 at the point (100, 1). He skis down the mountain, always moving in the direction of the steepest descent.
    a) In what direction does he start moving?
    b) Describe the curve along which he skis. [You will need to solve a separable first order ODE.] [HINT: Let the skier's position at time t be (x(t), y(t), z(t)) where z(t) = h(x(t), y(t)). What does the condition say about the skier's velocity in the xy-plane, ( \dot{x}, \dot{y})?]

    My attempt:
    I can almost solve a), I ended up getting \frac{10^{-2}}{2}(-2i - j) which is in the opposite direction. I'm not sure why it's the wrong direction, my working is:

    P = (100, 1)
    \frac{dh}{dx} = \frac{-4x^3}{10^8}
    \frac{dh}{dy} = \frac{-2y}{10^2}

    \nabla f(x, y) = \frac{-4x^3}{10^8}i + \frac{-2y}{10^2}j

    When I subbed in the point, I got my answer. Where am I getting the signs wrong?

    For b), I have no idea how to attempt it. I think I'm missing some intuition here. Any ideas?

    Thanks.
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  2. #2
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    No ideas?
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  3. #3
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    My answer for part (a) is slightly different.

    h_x = -\dfrac{4x^3}{10^8} \implies h_{x}(100,1) = -\dfrac{4}{10^2}

    h_y = -\dfrac{2y}{10^2} \implies h_{y}(100,1) = -\dfrac{2}{10^2}

    \nabla h(100,1) = -\dfrac{4}{10^2} \mathbf{i} - \dfrac{2}{10^2} \mathbf{j} = -\dfrac{2}{10^2}\left(2\mathbf{i} + \mathbf{j}\right) = -\dfrac{1}{50}\left(2\mathbf{i} + \mathbf{j}\right)

    Why is there division by 2 (instead of multiplication by 2) in your answer?
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  4. #4
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    Quote Originally Posted by Glitch View Post
    No ideas?
    Do you realize that any form of 'bumping" is out-lawed at this site?
    You can be banded for doing so.
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  5. #5
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    Quote Originally Posted by Plato View Post
    Do you realize that any form of 'bumping" is out-lawed at this site?
    You can be banded for doing so.
    Really? That's a bit harsh. All I wanted to know is whether people had even considered helping. If the question is too difficult, I can ask elsewhere.
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  6. #6
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    Quote Originally Posted by Glitch View Post
    When I subbed in the point, I got my answer. Where am I getting the signs wrong?
    Your calculations show that the skier starts moving UP the mountain in the direction of the vector (-2,-1). ie. (2,1) DOWN the mountain.

    For b), I have no idea how to attempt it. I think I'm missing some intuition here. Any ideas?
    For (b): dx/dt = k 4x^3/10^8, dy/dt= k 2y/10^2.

    So dy/dx=10^6 y / 2x^3

    Integrate etc and you should end up with y=e^((-2.5 *10^5)/x^2 + 25)...
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