Question

Consider the following. f(x, y, z) = x2yz − xyz8, P(6, −1, 1), u = 0, 4 5 , − 3 5 (a) Find the gradient of f. ∇f(x, y, z) = (b) Evaluate the gradient at the point P. ∇f(6, −1, 1) = (c) Find the rate of change of f at P in the direction of the vector u. Duf(6, −1, 1) =

Answer 1

a) ∇f(x, y, z) = (2xyz − yz⁸, x²z − xz⁸, x²y − 8xyz⁷)

b) ∇f(6, −1, 1) = (-11, 30, 12)

c) Duf(6, −1, 1) = 84/5

Explanation:

Given

f(x, y, z) = x²yz − xyz⁸

P(6, −1, 1)

u = (0, 4/5, − 3/5)

a) ∇f(x, y, z) = ?

We apply

fx(x, y, z) = 2xyz − yz⁸

fy(x, y, z) = x²z − xz⁸

fz(x, y, z) = x²y − 8xyz⁷

then

∇f(x, y, z) = (2xyz − yz⁸, x²z − xz⁸, x²y − 8xyz⁷)

b) ∇f(6, −1, 1) = (2*6*(-1)*1 − (-1)(1)⁸, (6)²(1) − (6)(1)⁸, (6)²(-1) − 8(6)(-1)(1)⁷)

⇒ ∇f(6, −1, 1) = (-11, 30, 12)

c) Duf(6, −1, 1) = ∇f(6, −1, 1)*(0, 4/5, − 3/5) = (-11, 30, 12)*(0, 4/5, − 3/5)

⇒ Duf(6, −1, 1) = 0 + 24 - 36/5 = 84/5