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7 April, 03:45

Find a unit vector in the direction in which f increases most rapidly at P and give the rate of chance of f in that direction; find a unit vector in the direction in which f decreases most rapidly at P and give the rate of change of f in that direction.

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  1. 7 April, 06:16
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    There's a part of the question missing and it is:

    f (x, y) = 4{x (^3) }{y^ (2) }; P (-1,1)

    Answer:

    A) Unit vector = 4 (3i - 2j) / (√13)

    B) The rate of change;

    |Δf (1, - 1) | = 4 / (√13)

    Explanation:

    First of all, f increases rapidly in the positive direction of Δf (x, y)

    Now;

    [differentiation of the x item alone] to get;

    fx (x, y) = 12{x (^2) }{y^ (2) }

    So at (1,-1), fx (x, y) = 12

    Similarly, [differentiation of the y item alone] to get; fy (x, y) =

    8{x (^3) }{y}

    At (1,-1), fy (x, y) = - 8

    Therefore, Δf (1, - 1) = 12i - 8j

    Simplifying this, vector along gradient = 4 (3i - 2j)

    Unit vector = 4 (3i - 2j) / (√ (3^2) + (-2^2) = 4 (3i - 2j) / (√13)

    Therefore, the rate of change;

    |Δf (1, - 1) | = 4 / (√13)
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