Question

Find the directional derivative of f at the given point in the direction indicated by the angle θ. f(x, y)=2ye^(−x), (0,6), θ=2π/3.

Answer 1

The directional derivative of the function f(x, y) = 2ye^(-x) at the point (0,6) in the direction of θ = 2π/3 is √3.

Step-by-step explanation:

To find the directional derivative of the function f(x, y) = 2ye−x at the point (0,6) in the direction indicated by the angle θ = 2π/3, we follow these steps:

1. Compute the gradient of f, denoted as ∇f. This involves taking the partial derivatives of f with respect to x and y.
2. Normalize the direction vector associated with θ. Since the angle is given, we can obtain the direction vector u by u = (cos(θ), sin(θ)).
3. Compute the directional derivative at the given point by taking the dot product of the gradient of f at the point and the normalized direction vector u.

Here, the gradient of f is (−2ye−x, 2e−x), and at the point (0,6), this becomes (0, 2). The direction vector is u = (-1/2, √3/2), which is the direction indicated by θ = 2π/3. The directional derivative is thus the dot product of (0, 2) and (-1/2, √3/2), which equals (0 × -1/2) + (2 × √3/2) = √3.