Question

Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y) = 5 sin(xy), (0, 7).

Answer 1

The maximum rate of change of the function f(x, y) = 5 sin(xy) at (0, 7) is 35, occurring in the direction of the x-axis.

Step-by-step explanation:

The maximum rate of change of a function f at a point is found by calculating the gradient of f and then determining its magnitude at that point. The direction of this maximum rate of change is given by the direction of the gradient vector. For the function f(x, y) = 5 sin(xy) at the point (0, 7), we first find the partial derivatives of f with respect to x and y to form the gradient vector. Then, we evaluate the gradient at the given point to find its magnitude, which represents the maximum rate of change.

To find the gradient of f, we calculate:

• fx = y · 5 cos(xy)
• fy = x · 5 cos(xy)

At the point (0, 7), these partial derivatives become:

• fx(0, 7) = 7 · 5 cos(0·7) = 35
• fy(0, 7) = 0 · 5 cos(0·7) = 0

The gradient vector at (0, 7) is therefore <35, 0>, and its magnitude is 35. This means the maximum rate of change of f at (0, 7) is 35, and it occurs in the direction of the gradient vector, which is along the x-axis.