91Ó°ÊÓ

When dealing with multivariable functions like \( f(x, y, z) = xy + 2yz - 3xz \), we often need to understand how the function behaves in response to changes in each variable. This is where partial derivatives come into play. A partial derivative measures how a function changes as just one of the variables changes, while all other variables are held constant.

For example:

• The partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} = y - 3z \). This tells us how \( f \) changes as \( x \) changes, keeping \( y \) and \( z \) constant.
• The partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = x + 2z \), showing how \( f \) reacts to changes in \( y \).
• The partial derivative with respect to \( z \) is \( \frac{\partial f}{\partial z} = 2y - 3x \), indicating changes in \( f \) due to adjustments in \( z \).

Partial derivatives are essential in forming the linear approximation of a multivariable function, like when we create a linearization around a specific point.

In linear approximations, error estimation is crucial to gauge how well our linearization approximates the original function. The goal is to find the maximum potential error when substituting the linear function for the actual one within a defined region. This is often calculated using the second derivatives of the function.

The error can be expressed as a function of the maximum value of the second derivatives within the small region. For example, if

• we have second derivatives \( \frac{\partial^2 f}{\partial x^2} = 0 \), \( \frac{\partial^2 f}{\partial y^2} = 0 \), and \( \frac{\partial^2 f}{\partial z^2} = 0 \), which indicate uniform behavior in changes with respect to each variable individually,
• then coupling terms like \( \frac{\partial^2 f}{\partial x \partial y} = 1 \) show interconnectedness between \( x \) and \( y \).

The principle formula used to describe the error bound is \( E \leq \frac{1}{2} \max |\text{Second derivatives}| (\Delta x^2 + \Delta y^2 + \Delta z^2) \). This ensures that given tiny shifts in each variable, the error remains small.

Second derivatives help us understand the curvature of a function and are central to estimating the error in linearization. They denote how the rate of change of a function's slope varies with respect to its variables.
For the function \( f(x, y, z) \), these second derivatives include cross partial derivatives like \( \frac{\partial^2 f}{\partial x \partial y} \), indicating how two variables simultaneously influence the rate of change.

• A second derivative such as \( \frac{\partial^2 f}{\partial x^2} \), if it were not zero, would indicate the rate of change of the \( \frac{\partial f}{\partial x} \) itself changing as \( x \) changes.
• In our example, all pure second derivatives are zero, meaning that each variable alone has no additional curvature effects beyond the first derivative information.

By examining these second derivatives, especially those with the highest absolute values, we can estimate how steep or flat the function's graph is around a point and thus set bounds on the error for linear approximations in a small region near that point.