Question

Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y) = 4 sin(xy), (0, 3). What is the maximum rate of change and the direction vector?

Answer 1

The maximum rate of change of the function f(x, y) = 4 sin(xy) at the point (0, 3) is 12, and it occurs in the direction of the y-axis (vertical direction).

Step-by-step explanation:

The maximum rate of change of the function f(x, y) = 4 sin(xy) at the point (0, 3) can be found by taking the partial derivatives of the function with respect to x and y. The partial derivative with respect to x is 4y cos(xy) and the partial derivative with respect to y is 4x cos(xy). Plugging in the coordinates of the point (0, 3), we get a rate of change of 0 with respect to x and 12 cos(0) = 12 with respect to y.

Therefore, the maximum rate of change of f at the point (0, 3) is 12, and it occurs in the direction of the y-axis (vertical direction).

Answer 2

At the point (0, 3) for the function f(x, y) = 4 sin(xy), the maximum rate of change is 12, and the direction vector at which it occurs is along the positive x-axis.

Step-by-step explanation:

To find the maximum rate of change at a given point (0, 3) for the function f(x, y) = 4 sin(xy), we need to calculate the gradient vector ∇f at that point. The gradient vector ∇f represents the direction of the maximum rate of change, and its magnitude gives the maximum rate of change.

First, compute the partial derivatives of the function f(x, y) with respect to x and y:

∂f/∂x = 4y cos(xy)

∂f/∂y = 4x cos(xy)

Evaluate these partial derivatives at the point (0, 3):

∂f/∂x = 4 * 3 * cos(0*3) = 12

∂f/∂y = 4 * 0 * cos(0*3) = 0

Therefore, at the point (0, 3), the maximum rate of change is 12, and it occurs in the direction of the positive x-axis. This means that the function f(x, y) increases most rapidly in the direction of the x-axis at the given point, with a rate of change of 12 units per unit distance in that direction. This result is obtained from the gradient vector, indicating both the magnitude and direction of the steepest ascent of the function at the specified point.