Question

Describe the change in accuracy of \(d z\) as an approximation of \(\Delta z\) as \(\Delta x\) and \(\Delta y\) increase.

Short answer

When \(\Delta x\) and \(\Delta y\) increase, the accuracy of \(dz\) as an approximation of \(\Delta z\) generally decreases, because larger changes make the linear approximation less accurate.

Step-by-step solution

Define the variables

In this context, \(\Delta z\) represents the actual change in \(z\) in response to changes in \(x\) and \(y\), while \(dz\) represents an approximation of that change. \(\Delta x\) and \(\Delta y\) correspondingly represent changes in \(x\) and \(y\).

General trend

As \(\Delta x\) and \(\Delta y\) increase, the difference between the actual change (\(\Delta z\)) and the approximate change (\(dz\)) generally grows. This is because the approximation \(dz\) is based on a linear approximation (namely, the derivative), which becomes less accurate as we move away from the point at which it was calculated.

Detailed explanation

When we have small changes in \(x\) and \(y\), the linear approximation given by \(dz\) is more accurate because the reality is closer to the local linearity assumed by the derivative. However, for larger changes in \(x\) and \(y\), the actual change \(\Delta z\) may be more influenced by the curvature of the function, making the linear approximation \(dz\) less accurate.