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Find the direction in which the maximum rate of change occurs for the function f(x,y)=2xsin(xy) at the point (2,3).

User Seldridge
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Final answer:

The direction of maximum rate of change for the function f(x,y)=2xsin(xy) at the point (2,3) can be found by calculating the gradient vector.

Step-by-step explanation:

The direction of maximum rate of change for a function can be found by calculating the gradient vector. In this case, we have the function f(x,y) = 2xsin(xy).

First, we find the partial derivatives of f with respect to x and y:

∂f/∂x = 2sin(xy) + 2xycos(xy)

∂f/∂y = 2x²cos(xy)

Next, we evaluate these partial derivatives at the point (2,3):

∂f/∂x(2,3) = 2sin(6) + 12cos(6) ≈ 10.88

∂f/∂y(2,3) = 8cos(6) ≈ 7.95

Thus, the maximum rate of change occurs in the direction given by the gradient vector, which is approximately (10.88, 7.95).

User Satyajeet
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