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Vectors, Tangent Planes, Gradients, Derivatives and Rate of Change

1) F(x, y, z) = xyz, denote the directional derivative of f at the point (x0, y0, z0) along the vector v by Lvf(x0, y0, z0).
a. Find the gradient ∇f(1, 2, 3) ≡ grad f(1,2,3)
b. Find Lvf(1, 2, 3), where v = (-1, -2, 4)
c. Find Luf(1, 2, 3), where u is the unit vector u = (2/3, -2/3, 1/3)
d. Find the direction w, such that Lwf(1, 2, 3) is greater than or equal to Luf(1, 2, 3) for any unit
vector u. In other words, direction of w is that of the fastest increase of f at any point (1, 2, 3).

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Vectors, Tangent Planes, Gradients, Derivatives and Rate of Change are investigated. The solution is detailed and well presented.

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