Final answer:
To find the maximum rate of change of the function f(x,y)=ln(x² + y²) at the point (1,4) and the direction in which it occurs, we calculate the gradient of the function at that point. The gradient vector gives us the maximum rate of change and the direction.
Step-by-step explanation:
To find the maximum rate of change of the function f(x,y) = ln(x² + y²) at the point (1,4) and the direction in which it occurs, we need to calculate the gradient of the function at that point. The gradient vector will give us both the maximum rate of change and the direction. The gradient vector is given by:
∇f(x, y) = (∂f/∂x, ∂f/∂y)
Taking the partial derivatives with respect to x and y, we get:
∂f/∂x = 2x / (x² + y²)
∂f/∂y = 2y / (x² + y²)
Substituting the coordinates of the point (1, 4), we have:
∂f/∂x = 2 / (1² + 4²) = 2/17
∂f/∂y = 8 / (1² + 4²) = 8/17
Therefore, the gradient vector at the point (1, 4) is:
∇f(1, 4) = (2/17, 8/17)
The magnitude of the gradient vector gives us the maximum rate of change, which is:
|∇f(1, 4)| = sqrt((2/17)² + (8/17)²) ≈ 0.486
The direction in which the maximum rate of change occurs is in the direction of the gradient vector, which is (2/17, 8/17).