Question

Given f(x,y,z)=xyz², Find the directional derivative at (1,2,1) in the direction of v= ⟨−1,−2,2⟩.

Answer 1

The directional derivative at (1, 2, 1) in the direction of v = ⟨−1,−2,2⟩ is 2.

Step-by-step explanation:

The directional derivative of a function f(x, y, z) in the direction of a vector v is given by the dot product of the gradient of f with v. To find the directional derivative at (1, 2, 1) in the direction of v = ⟨−1,−2,2⟩, we need to calculate the gradient of f and then take the dot product with v.

The gradient of f is ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) = (yz², xz², 2xyz).

The directional derivative is given by D_vf = ∇f · v = (yz²)(-1) + (xz²)(-2) + (2xyz)(2) = -y - 2xz + 4xyz. Substituting the point (1, 2, 1) into the equation, we get D_vf = -2 - 2(1)(1) + 4(1)(2)(1) = 2.