Problem 21 - Applied Calculus

Question

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$f(x, y)=x^{2} y$$

Step-by-Step Solution

Answer

The second-order partial derivatives confirm that the mixed partials are equal: \( f_{xy} = f_{yx} = 2x \).

Step 1: Understand the Function

We are given the function \( f(x, y) = x^2 y \). This is a function of two variables, \( x \) and \( y \), and we need to find the second-order partial derivatives.

Step 2: Find First-Order Partial Derivatives

Find the first-order partial derivatives with respect to \( x \) and \( y \).\Partial derivative with respect to \( x \): \[ f_x = \frac{\partial}{\partial x}(x^2 y) = 2xy \]Partial derivative with respect to \( y \):\[ f_y = \frac{\partial}{\partial y}(x^2 y) = x^2 \]

Step 3: Compute Second-Order Partial Derivative \( f_{xx} \)

Differentiate \( f_x = 2xy \) with respect to \( x \) again to find \( f_{xx} \).\[ f_{xx} = \frac{\partial}{\partial x}(2xy) = 2y\]

Step 4: Compute Second-Order Partial Derivative \( f_{yy} \)

Differentiate \( f_y = x^2 \) with respect to \( y \) to find \( f_{yy} \).\[ f_{yy} = \frac{\partial}{\partial y}(x^2) = 0\]

Step 5: Compute Mixed Partial Derivative \( f_{xy} \)

Differentiate \( f_x = 2xy \) with respect to \( y \) to find \( f_{xy} \).\[ f_{xy} = \frac{\partial}{\partial y}(2xy) = 2x\]

Step 6: Compute Mixed Partial Derivative \( f_{yx} \)

Differentiate \( f_y = x^2 \) with respect to \( x \) to find \( f_{yx} \).\[ f_{yx} = \frac{\partial}{\partial x}(x^2) = 2x\]

Step 7: Confirm Equality of Mixed Partials

Compare \( f_{xy} \) and \( f_{yx} \).\[ f_{xy} = 2x \quad \text{and} \quad f_{yx} = 2x\]Since \( f_{xy} = f_{yx} \), the mixed partial derivatives are equal, confirming Clairaut's theorem.

Key Concepts

Partial Derivatives

In multivariable calculus, understanding partial derivatives is key. They are similar to regular derivatives but are used for functions with more than one variable. When we write the function \( f(x, y) = x^2 y \), we mean that \( f \) depends on both \( x \) and \( y \). Partial derivatives help us understand how \( f \) changes when we tweak one variable while holding others constant.
Partial derivatives are denoted by \( \frac{\partial}{\partial x} \) for changes in \( x \) and \( \frac{\partial}{\partial y} \) for changes in \( y \).

• For instance, the partial derivative of \( f(x, y) = x^2 y \) with respect to \( x \) is \( 2xy \), showing how \( f \) changes as \( x \) varies.
• Similarly, the partial with respect to \( y \), \( x^2 \), indicates the change in \( f \) when \( y \) changes.

Getting comfortable with these derivatives paves the way for deeper understanding of functions of multiple variables.

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus. It deals with functions that have more than one input, such as \( f(x, y) \).
Imagine a landscape where both \( x \) and \( y \) represent directions, like east-west and north-south. Here, \( f(x, y) \) represents the height at different points on this landscape.

• The beauty of multivariable calculus lies in its ability to analyze how combinations of variables affect our function.
• It extends familiar ideas such as tangents and slopes to higher dimensions, finding critical points where functions reach maxima, minima, or saddle points.

In our exercise, computing second-order partials involves taking derivatives of derivatives, further exploring this landscape's curvature and shape. Analyzing such changes is crucial for fields like physics and engineering, aiding in understanding complex systems.

Function of Two Variables

Understanding functions of two variables involves grasping how they behave over a plane. In \( f(x, y) = x^2 y \), there are two inputs, \( x \) and \( y \), which determine the output of the function.
In practical applications:

• Variables such as \( x \) and \( y \) might represent dimensions, time, temperature, or any other measurable factors.
• The output could represent a value influenced by these variables, such as pressure, height, or velocity.

To explore such functions, we use partial derivatives to see the effect of changes in one variable.Calculating second-order partial derivatives, as in our exercise, helps us understand the rate of change not just in one dimension, but across different planes. This gives a more comprehensive understanding of the function's behavior and the interactions between the variables.