Maximizing the Rate of Change of the Function f(x, y, z) ln(xy^2z^3) in the Direction of (1, -2, -3)

In this article, we will explore the process of finding the maximum rate of change of the function f(x, y, z) ln(xy^2z^3) in the direction of the vector (1, -2, -3). This involves several steps that we will detail below.

Step 1: Compute the Gradient of the Function

The gradient of the function, denoted by ?f, is a vector that points in the direction of the maximum rate of change of the function. For the given function f(x, y, z) ln(xy^2z^3), we need to calculate the partial derivatives with respect to x, y, and z.

First, let's re-write the function for clarity:

f(x, y, z) ln(xy^2z^3)

Now, we compute the partial derivatives:

Therefore, the gradient of the function is:

?f1x,2y,3z

Step 2: Normalize the Direction Vector

The direction vector mathbf{v} (1, -2, -3) needs to be normalized to find the unit vector in this direction.

The magnitude of the vector is:

|mathbf{v}|1 -22 -3214

The normalized vector mathbf{u} is:

mathbf{u}114,-214,-314

Step 3: Calculate the Maximum Rate of Change

The maximum rate of change of the function in the direction of mathbf{v} is given by the dot product of the gradient and the normalized direction vector.

Mathematically, this can be expressed as:

Maximum Rate of Change?f mathbf{u}

Substituting the values, we get:

Maximum Rate of Change1x,2y,3z 114,-214,-314

Calculating the dot product:

1x114-4xy-9xz114

The maximum rate of change of f(x, y, z) ln(xy^2z^3) in the direction of (1, -2, -3) is:

114

To get the exact numerical value, specific values for x, y, and z are required.

Conclusion

The process of finding the maximum rate of change of a multivariate function involves computing the gradient, normalizing the given direction vector, and then calculating the dot product. For the function f(x, y, z) ln(xy^2z^3) in the direction (1, -2, -3), the maximum rate of change is given by the formula derived above.

This process is crucial in various fields such as physics, engineering, and economics, where understanding the direction and rate of change of a function is essential for optimization and analysis.