91Ó°ÊÓ

In calculus, the gradient of a function represents the direction and rate of the steepest ascent of that function. Think of hiking up a hill: the gradient points you to the steepest path upwards. It is a vector that comprises all the partial derivatives with respect to each variable.

To calculate the gradient, we find the partial derivatives of the function for each variable. In our exercise, the function is given as f(x, y) = x \(\cos 3y\). The partial derivative with respect to x is \(\cos 3y\), and with respect to y, it is -3x \sin 3y. Hence, the gradient vector is \([\cos 3y, -3x \sin 3y]\).

At the point (2,0), this becomes \([1, 0]\), indicating that the steepest ascent at this point is in the direction of the x-axis and the rate of ascent (the magnitude of the gradient) is 1.

Partial derivatives are used to measure how a function changes as only one of its input variables changes, holding the others constant. They are essentially the slopes of the function graph along the respective axes.

In our problem, we deal with the function f(x, y), which has two input variables: x and y. Thus, we calculate two partial derivatives: \(\frac{{\partial f}}{{\partial x}}\), which tells us how the function changes with x when y is held constant, and \(\frac{{\partial f}}{{\partial y}}\), indicating how it changes with y when x is held constant.

The partial derivatives we obtained are \(\cos 3y\) for x, and -3x \sin 3y for y. Importantly, these derivatives give us the components of the gradient vector, which is central to understanding the behavior of the function.

Determining the maximum and minimum rates of change of a function at a given point is akin to finding where you are on a hill. The maximum rate of change is akin to knowing the steepest way up, and the minimum rate is the steepest way down. These rates correspond to the magnitude of the gradient vector and its negative, respectively.

In the context of our example, the maximum rate of change at the point (2,0) occurs in the direction of the gradient [1, 0] which we evaluated earlier. The magnitude of this gradient, 1, gives us the maximum rate.

The minimum rate of change occurs in the opposite direction, [-1, 0], but interestingly, its magnitude is also 1, because we are dealing with a flat slope in the direction of y. This tells us that the function is not changing at all in the direction of the y-axis at the point (2,0).