Question

f(x, y, z) = x^2yz - xyz^3, P(2, -1, 1), u = (0, 4/5, -3/5) (a) Find the gradient of f. ∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z) (b) Evaluate the gradient at the point P. ∇f(2, -1, 1) = (∂f/∂x, ∂f/∂y, ∂f/∂z) at (2, -1, 1) (c) Find the rate of change of f at P in the direction of the vector u. Duf(2, -1, 1) = ∇f(2, -1, 1) · u

Name: fx y z x2yz xyz3 p2 1 1 u 0 4 5 3 5 a find the gradient of f fx y z b evaluate the gradient at the point p f2 1 1 c find the rate of change of f at p in the direction of the vector u duf2 1 80375
Uploaded: 2023-09-09T17:38:01-08:00
Duration: 2 min 8 s
Channel: Adi S
Description: fx y z x2yz xyz3 p2 1 1 u 0 4 5 3 5 a find the gradient of f fx y z b evaluate the gradient at the point p f2 1 1 c find the rate of change of f at p in the direction of the vector u duf2 1 80375

          f(x, y, z) = x^2yz - xyz^3, P(2, -1, 1), u = (0, 4/5, -3/5)

(a) Find the gradient of f.
∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)

(b) Evaluate the gradient at the point P.
∇f(2, -1, 1) = (∂f/∂x, ∂f/∂y, ∂f/∂z) at (2, -1, 1)

(c) Find the rate of change of f at P in the direction of the vector u.
Duf(2, -1, 1) = ∇f(2, -1, 1) · u

Added by Kathleen P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/09/2023

Step 1

The gradient of f, denoted as ∇f(x, y, z), is given by (∂f/∂x, ∂f/∂y, ∂f/∂z). Calculating the partial derivatives: ∂f/∂x = 2xyz - yz^3 ∂f/∂y = x^2z - xz^3 ∂f/∂z = x^2y - 3xyz^2 Therefore, the gradient of f is ∇f(x, y, z) = (2xyz - yz^3, x^2z - xz^3, x^2y - Show more…

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f(x, y, z) = x^2yz - xyz^3, P(2, -1, 1), u = (0, 4/5, -3/5) (a) Find the gradient of f. ∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z) (b) Evaluate the gradient at the point P. ∇f(2, -1, 1) = (∂f/∂x, ∂f/∂y, ∂f/∂z) at (2, -1, 1) (c) Find the rate of change of f at P in the direction of the vector u. Duf(2, -1, 1) = ∇f(2, -1, 1) · u