Final answer:
The maximum rate of change of the function f(x, y) = 8 sin(xy) at the point (0, 3) is 24, and it occurs in the direction of the positive x-axis.
Step-by-step explanation:
Maximum Rate of Change and Direction
To find the maximum rate of change of the function f(x, y) = 8 sin(xy) at the point (0, 3), we calculate the gradient of f, which is a vector of the partial derivatives of f with respect to x and y. The maximum rate of change is given by the magnitude of the gradient vector at the point in question. First, calculate the partial derivatives:
- fx(x, y) = 8y cos(xy)
- fy(x, y) = 8x cos(xy)
Now evaluate the partial derivatives at the point (0, 3):
- fx(0, 3) = 8(3) cos(0) = 24
- fy(0, 3) = 8(0) cos(0) = 0
The gradient vector is ∇f = <24, 0> and its magnitude is |24| = 24, which is the maximum rate of change. The direction is in the direction of the positive x-axis since the gradient has a zero y-component at this point.
The direction of the maximum rate of change is given by the unit vector of the gradient, which, in this case, is simply <1, 0> or along the x-axis.