Question

Please show your answers to at least 4 decimal places. Suppose that the directional derivative of f(x, y) in the direction and at the point P is ________?

Answer 1

The magnitude of the conservative force at the point (1m, 1m) is calculated by integrating the given condition (4 N/m³)xy and it results in 2.8284 N.

Step-by-step explanation:

The Magnitude of a Conservative Force

To find the magnitude of the force at the point (x, y) = (1m, 1m), we need to consider the given condition that the partial derivative of the force in the x-direction with respect to y is equal to the partial derivative of the force in the y-direction with respect to x, which is given as (4 N/m³)xy.

Using this information, we integrate (4 N/m³)xy with respect to x to find the force in the x-direction, and with respect to y to find the force in the y-direction. Since the force is conservative and zero on the axes, the constants of integration will be zero, resulting in a force that can be expressed as Fx(x, y) = 2xy² and Fy(x, y) = 2x²y (in Newtons), when normalized with the given condition.

At the point (1m, 1m), the force components are Fx(1, 1) = 2(1)(1²) = 2 N, and Fy(1, 1) = 2(1²)(1) = 2 N. Hence, the magnitude of the force is the square root of the sum of the squares of its components, which is √(Fx² + Fy²) = √(2² + 2²) = √(8) N = 2.8284 N.