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Chapter 13: Problem 6

Explain how to approximate the change in a function \(f\) when the independent variables change from \((a, b)\) to \((a+\Delta x, b+\Delta y)\)

Short Answer

Expert verified
Answer: To find the approximate change in the function, follow these steps: 1. Calculate the partial derivatives of \(f\) with respect to \(x\) and \(y\), denoted as \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\). 2. Evaluate the partial derivatives at point \((a, b)\): \(\frac{\partial f}{\partial x}(a, b)\) and \(\frac{\partial f}{\partial y}(a, b)\). 3. Use the linear approximation formula: \(\Delta f \approx \frac{\partial f}{\partial x}(a, b) \cdot \Delta x + \frac{\partial f}{\partial y}(a, b) \cdot \Delta y\). 4. Substitute the values of \(\Delta x\), \(\Delta y\), \(\frac{\partial f}{\partial x}(a, b)\), and \(\frac{\partial f}{\partial y}(a, b)\) into the formula to find the approximate change in \(f\).

Step by step solution

01

Find the partial derivative of \(f\) with respect to \(x\) and \(y\).

To find the approximate change in \(f\), we will first need to compute the partial derivatives of \(f\) with respect to \(x\) and \(y\). We'll denote these as \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\). These derivatives represent the rate of change of \(f\) along each axis.
02

Evaluate the partial derivatives at point \((a, b)\).

Next, we need to find the values of the partial derivatives at the point \((a, b)\). To do this, substitute the value of \(a\) into \(\frac{\partial f}{\partial x}\) and the value of \(b\) into \(\frac{\partial f}{\partial y}\): \(\frac{\partial f}{\partial x}(a, b)\) and \(\frac{\partial f}{\partial y}(a, b)\)
03

Apply the linear approximation formula.

Now, we will use the linear approximation formula to find an approximation for the change in the function \(f\) when moving from \((a, b)\) to \((a + \Delta x, b + \Delta y)\). The formula is: \(\Delta f \approx \frac{\partial f}{\partial x}(a, b) \cdot \Delta x + \frac{\partial f}{\partial y}(a, b) \cdot \Delta y\)
04

Substitute the values.

Finally, substitute the values of \(\Delta x\), \(\Delta y\), \(\frac{\partial f}{\partial x}(a, b)\), and \(\frac{\partial f}{\partial y}(a, b)\) into the linear approximation formula to get the approximation for the change in the function: \(\Delta f \approx \frac{\partial f}{\partial x}(a, b) \cdot \Delta x + \frac{\partial f}{\partial y}(a, b) \cdot \Delta y\) Now, you have found the approximate change in the function \(f\) when the independent variables change from \((a, b)\) to \((a+\Delta x, b+\Delta y)\).

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