Question

Calculate the directional derivative in the direction of v at the given point. Remember to normalize the direction vector. f(x, y, z) = z² − xy², v = −1, 2, 2 , P = (6, 3, 6)_____

Answer 1

To calculate the directional derivative in the direction of v at the given point, find the gradient vector of the function f(x,y,z), normalize the direction vector v, and then take the dot product of the gradient vector and the normalized direction vector.

Step-by-step explanation:

To calculate the directional derivative in the direction of v at the given point, we first need to find the gradient vector of the function f(x,y,z).

The gradient vector is given by ∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z).

In this case, f(x, y, z) = z² - xy², so the gradient vector is ∇f(x, y, z) = (0 - y², 0 - 2xy, 2z).

Next, we need to normalize the direction vector v. The normalized direction vector is given by V = (v₁/||v||, v₂/||v||, v₃/||v||), where ||v|| is the magnitude of v.

In this case, v = (-1, 2, 2), so ||v|| = sqrt((-1)² + 2² + 2²) = sqrt(9) = 3. Therefore, V = (-1/3, 2/3, 2/3).

Finally, the directional derivative in the direction of v at the given point P = (6, 3, 6) is given by the dot product of the gradient vector and the normalized direction vector: Dᵥf(P) = ∇f(P) ⋅ V = (0 - (3)², 0 - 2(6)(3), 2(6)) ⋅ (-1/3, 2/3, 2/3).

Calculating the dot product, we get Dᵥf(P) = (-9, -36, 12) ⋅ (-1/3, 2/3, 2/3) = -26.