Question

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

Question: Explain how to approximate the change in a function \(f(x,y)\) when the independent variables change from \((a, b)\) to \((a+\Delta x, b+\Delta y)\) using the Total Differential method. Answer: To approximate the change in a function \(f(x,y)\) when the independent variables change from \((a, b)\) to \((a+\Delta x, b+\Delta y)\), we need to compute the total differential of the function using partial derivatives. The Total Differential method involves computing the partial derivatives \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\), and then finding the total differential as \(df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy\). By plugging in the changes \(dx = \Delta x\) and \(dy = \Delta y\), we can approximate the change in the function as \(df \approx \Delta f = f(a+\Delta x, b+\Delta y) - f(a,b)\).

Step-by-step solution

Understand the problem

Given a function \(f(x,y)\), we need to approximate the change in \(f\) when the independent variables \((x, y)\) change from \((a, b)\) to \((a+\Delta x, b+\Delta y)\). We will use the Total Differential method to achieve this.

Compute partial derivatives

Compute the partial derivatives of \(f\) with respect to \(x\) and \(y\), denoted as \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\), respectively.

Compute the total differential

To find the total differential of the function \(f\), denoted as \(df\), we will use the following formula: \(df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy\). Here, \(dx = \Delta x\) and \(dy = \Delta y\) are the changes in the independent variables \(x\) and \(y\).

Approximate the change in the function

Now, we will plug in the changes in the independent variables \(dx\) and \(dy\), and compute the total differential \(df\). The total differential \(df\) will give us an approximate value for the change in the function, when changing the independent variables from \((a, b)\) to \((a+\Delta x, b+\Delta y)\): \(df \approx \Delta f = f(a+\Delta x, b+\Delta y) - f(a,b)\).

Key concepts

Partial Derivatives

Partial derivatives are the backbone of understanding how functions change with respect to their variables. In a multi-variable function like \(f(x, y)\), a partial derivative measures how the function changes in one variable while keeping the other constant.
For example, the partial derivative of \(f\) with respect to \(x\), denoted \(\frac{\partial f}{\partial x}\), identifies the rate at which \(f\) changes as \(x\) increases, assuming that \(y\) does not change. Similarly, \(\frac{\partial f}{\partial y}\) describes the rate of change of \(f\) with changes in \(y\), keeping \(x\) constant.
Calculating partial derivatives is essential when dealing with variable changes in multivariable calculus. They help us precisely estimate the change in function value due to small changes in the variables involved.

Change in Function

When we talk about the change in a function, we refer to how its value alters due to variations in its inputs. For a function \(f(x, y)\), if the inputs change from \((a, b)\) to \((a+\Delta x, b+\Delta y)\), the change can be denoted as \(\Delta f\).
This change \(\Delta f\) can be viewed as the difference in function values: \(f(a+\Delta x, b+\Delta y) - f(a, b)\). However, calculating this directly might be complex for intricate functions.
Using methods like the total differential allows us to approximate \(\Delta f\) efficiently, especially when \(\Delta x\) and \(\Delta y\) are small. This approximation helps in simplifying the analysis and predictions of how a function behaves with changing inputs.

Approximating Function Changes

Approximating changes in a function is often necessary when dealing with small variations in the independent variables. This is where the concept of the total differential comes into play.
The total differential, \(df\), gives an approximate change of the function \(f\) based on its partial derivatives. The formula \(df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy\) provides a linear approximation of how the function changes when \(x\) changes by \(dx\) and \(y\) changes by \(dy\).
By substituting \(dx = \Delta x\) and \(dy = \Delta y\) into this formula, we can approximate \(\Delta f\), which represents the change in function value. This is notably useful in predicting small but significant adjustments in situations where exact computations might be too cumbersome. Thus, total differential allows for a streamlined means of estimating function changes effectively.