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Math Help - Rate of change / Directional Derivatives

  1. #1
    Newbie piyourface166's Avatar
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    Rate of change / Directional Derivatives

    I can't quite figure out where to begin / find an equation for this problem.... I need a good push in the right direction, thanks ahead of time.
    The temperature at a point (x, y, z) is given by the following equation where T is measured in C and x, y, z in meters. T(x, y, z) = 200e^((-x^2)(-3y^2)(-9z^2))

    (a) Find the rate of change of temperature at the point P(2, -1, 2) in the direction towards the point (3, -3, 3).

    (b) In which direction does the temperature increase fastest at P?

    (c) Find the maximum rate of increase at P.
    What equation do I need to use, in terms of the given variables in my problem??
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  2. #2
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    Quote Originally Posted by piyourface166 View Post
    [INDENT]The temperature at a point (x, y, z) is given by the following equation where T is measured in C and x, y, z in meters. T(x, y, z) = 200e^((-x^2)(-3y^2)(-9z^2))

    (a) Find the rate of change of temperature at the point P(2, -1, 2) in the direction towards the point (3, -3, 3).
    Let u=<3,-3,3>-<2,-1,2>.

    For a) you want \dfrac{{\nabla T(2, - 1,2) \cdot u}}{{\left\| u \right\|}}.
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    Newbie piyourface166's Avatar
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    when you have ||u|| does that equal - abs of root(6) - because vector is <1,-2,1>? The ||vector|| always confused me. I know what |vector| asks, but the second '||' throws me off.
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    Some authors (older ones particularity) do use |u| for length of a vector.
    More modern notation is ||u|| to distinguish from absolute value as in ||\alpha u||=(|\alpha|)||u||.
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  5. #5
    Newbie piyourface166's Avatar
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    thank you so much that has been bothering me. Ok so I got [-518400e^(-432)] / root (6) - for part a, which I'm fairly confident is correct. So for part b - which direction does the temperature increase the fastest at P - we only have two points, which can only give us one vector and two directions, to P from the second point and away from P towards the second point correct? But that doesn't seem correct to me.
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  6. #6
    Newbie piyourface166's Avatar
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    I got the problem. Thank you for all your help!
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