Problem 20 - Applied Calculus

Question

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 mixed partial derivatives are equal: \( f_{xy} = f_{yx} = 2x \).

Step 1: First-Order Partial Derivatives

To start finding the second-order partial derivatives, we need to first calculate the first-order partial derivatives. The first-order partial derivative of \( f(x, y) = x^2 y \) with respect to \( x \) is found by treating \( y \) as a constant:\[ f_x = \frac{\partial}{\partial x}(x^2 y) = 2xy \]Similarly, the first-order partial derivative with respect to \( y \) is found by treating \( x \) as a constant:\[ f_y = \frac{\partial}{\partial y}(x^2 y) = x^2 \]

Step 2: Calculate f_xx

Now, compute the second partial derivative of \( f \) with respect to \( x \) by differentiating \( f_x \) with respect to \( x \):\[ f_{xx} = \frac{\partial}{\partial x}(2xy) = 2y \]

Step 3: Calculate f_yy

Next, compute the second partial derivative of \( f \) with respect to \( y \) by differentiating \( f_y \) with respect to \( y \):\[ f_{yy} = \frac{\partial}{\partial y}(x^2) = 0 \]

Step 4: Calculate f_xy

Now calculate the mixed partial derivative first taking \( x \), then \( y \):\[ f_{xy} = \frac{\partial}{\partial y}(2xy) = 2x \]

Step 5: Calculate f_yx

Finally, calculate the mixed partial derivative first taking \( y \), then \( x \):\[ f_{yx} = \frac{\partial}{\partial x}(x^2) = 2x \]

Step 6: Confirm Equality of Mixed Partials

The mixed partial derivatives \( f_{xy} \) and \( f_{yx} \) have both been found to be \( 2x \). Thus, they are equal, confirming Clairaut's theorem for this function.

Key Concepts

Partial Differentiation

Partial differentiation is a technique used in calculus to find the rate at which a function changes along one variable, while keeping the other variable(s) constant. This method is particularly useful in multivariable calculus, where functions depend on two or more variables. In the context of the given function \( f(x, y) = x^2 y \), we find the partial derivatives by treating all other variables as constants.
To find the partial derivative of \( f \) with respect to \( x \), denoted as \( f_x \), we consider \( y \) as a constant:

• \( f_x = \frac{\partial}{\partial x}(x^2 y) = 2xy \)

Next, we differentiate with respect to \( y \), holding \( x \) constant:

• \( f_y = \frac{\partial}{\partial y}(x^2 y) = x^2 \)

Understanding partial differentiation is vital as it forms the foundation for finding second-order partial derivatives.

Clairaut's Theorem

Clairaut's theorem, also known as the symmetry of mixed partial derivatives, states that if the mixed partial derivatives of a function are continuous, they are equal. In simple terms, for a function \( f \), the order of differentiation does not matter, so long as the mixed second derivatives exist and are continuous.
This theorem is particularly useful in confirming calculations in problems involving second-order partial derivatives.
In the exercise, we confirm Clairaut's theorem by calculating the mixed partial derivatives \( f_{xy} \) and \( f_{yx} \). Both were found to be \( 2x \) in this case, demonstrating their equality and consistency with Clairaut's symmetry principle. Utilizing Clairaut's theorem can simplify verification steps in complex calculus problems.

Mixed Partial Derivatives

Mixed partial derivatives involve taking the derivative of a function with respect to one variable and then another. They are crucial in understanding the behavior of functions with multiple variables.
In our context, after obtaining the first-order partial derivatives, we calculated:

• \( f_{xy} = \frac{\partial}{\partial y}(2xy) = 2x \)
• \( f_{yx} = \frac{\partial}{\partial x}(x^2) = 2x \)

Both calculations yield \( 2x \), illustrating that these mixed derivatives are identical for the function given, in agreement with Clairaut's theorem. Mixed partials help in determining the curvature and structure of multivariable functions. They are integral in fields like physics and engineering where optimization and changes in variables play a critical role.

Calculus Problem Solving

Calculus problem solving often involves applying a series of systematic steps to approach a solution. In dealing with functions of multiple variables, understanding the role of partial derivatives is crucial.
In solving for second-order partial derivatives, the following systematic steps are typically followed:

• Calculate first-order partial derivatives with respect to each variable.
• Proceed to second-order derivatives by further differentiating the first-order results.
• Compute mixed partial derivatives by altering the order of differentiation to verify consistency.
• Utilize Clairaut's theorem to confirm the equality of mixed partials if the function's conditions allow.

This approach helps in accurately solving and verifying multivariable calculus problems. Such skills are indispensable in advanced studies and professional applications involving dynamic systems and models.