Question

Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y) = 9 sin(xy), (0, 7)

Answer 1

The maximum rate of change of the function f(x, y) = 9 sin(xy) at point (0, 7) is 63, occurring in the direction of the positive x-axis.

Step-by-step explanation:

Finding the Maximum Rate of Change

The maximum rate of change of a function f at a given point and the direction in which it occurs is given by the gradient of f at that point, which is represented by the vector of partial derivatives. Specifically, for f(x, y) = 9 sin(xy) at the point (0, 7), we need to find the gradients ∇f and then evaluate it at (0, 7).

1. First, find the partial derivatives of f with respect to x and y:
   
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   • ∂f/∂x = 9y cos(xy)
   • 
     
   • ∂f/∂y = 9x cos(xy)
2. Evaluate the partial derivatives at the given point (0, 7):
   
   • 
     
   • ∂f/∂x(0,7) = 9*7*cos(0) = 63
   • 
     
   • ∂f/∂y(0,7) = 9*0*cos(0) = 0
3. The gradient vector at the point (0,7) is ∇f(0,7) = <63, 0>.
4. The maximum rate of change of f at the point (0, 7) is the magnitude of the gradient vector, which is 63.
5. The direction of the maximum rate of change is in the direction of the gradient vector, which is along the positive x-axis.

Hence, the maximum rate of change of f at the point (0,7) is 63, and it occurs in the direction of the positive x-axis.