6. The gradient of a function g(x, y, z) at the point (-3,4,5) is (-2, 1, 2). (a

Question

6. The gradient of a function g(x, y, z) at the point (-3,4,5) is (-2, 1, 2). (a) Find the values of the partial derivatives gx, gy, and gz at the point (-3,4, 5). (b) Find the maximum rate of change of g at the point (-3, 4, 5) and the unit vector in the direction which the maximum rate of change occurs. (c) Find the rate of change of g at the point (-3, 4, 5) in the direction of the point (-1,8,1). (Note: in the direction of the point (-1,8,1) means in the direction of the displacement vector from (-3,4,5) to (-1,8, 1) .)

Explanation / Answer

a)just use the components of the gradient.
g_x (-3, 4, 5) = -2
g_y (-3, 4, 5) = 1
g_z (-3, 4, 5) = 2.

B) b. max rate of change: magnitude of gradient
direction: gradient/(magnitude of gradient)

Use the direction of the gradient for the maximal rate of change.

u = ?g(-3, 4, 5)/||?g(-3, 4, 5)|| = <-2, 1, 2>/3.

So, Du f(-3, 4, 5)
= ?g(-3, 4, 5)

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