Problem 30 - Applied Calculus

Question

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$Q=5 p_{1}^{2} p_{2}^{-1}, \quad p_{2} \neq 0$$

Step-by-Step Solution

Answer

The mixed partial derivatives are equal: \( \frac{\partial^2 Q}{\partial p_1 \partial p_2} = \frac{\partial^2 Q}{\partial p_2 \partial p_1} \).

Step 1: Identifying the Problem

The function given is \( Q = 5p_1^2 p_2^{-1} \), which is a function of two variables \( p_1 \) and \( p_2 \). We are asked to compute all second-order partial derivatives and verify that the mixed partials are equal.

Step 2: First Partial Derivatives

First, compute the partial derivatives with respect to each variable. For \( p_1 \), we get: \( \frac{\partial Q}{\partial p_1} = 10p_1 p_2^{-1} \). For \( p_2 \), apply the power rule: \( \frac{\partial Q}{\partial p_2} = -5p_1^2 p_2^{-2} \).

Step 3: Second Partial Derivative with respect to \( p_1 \)

Next, compute the second partial derivative with respect to \( p_1 \). Begin by differentiating \( \frac{\partial Q}{\partial p_1} = 10p_1 p_2^{-1} \) with respect to \( p_1 \): \( \frac{\partial^2 Q}{\partial p_1^2} = 10p_2^{-1} \).

Step 4: Second Partial Derivative with respect to \( p_2 \)

Now compute the second partial derivative with respect to \( p_2 \). Differentiate \( \frac{\partial Q}{\partial p_2} = -5p_1^2 p_2^{-2} \) with respect to \( p_2 \): \( \frac{\partial^2 Q}{\partial p_2^2} = 10p_1^2 p_2^{-3} \).

Step 5: Mixed Partial Derivative \( \frac{\partial^2 Q}{\partial p_1 \partial p_2} \)

Compute the mixed partial derivative by differentiating \( \frac{\partial Q}{\partial p_1} = 10p_1 p_2^{-1} \) with respect to \( p_2 \): \( \frac{\partial^2 Q}{\partial p_1 \partial p_2} = -10p_1 p_2^{-2} \).

Step 6: Mixed Partial Derivative \( \frac{\partial^2 Q}{\partial p_2 \partial p_1} \)

Compute the mixed partial derivative by differentiating \( \frac{\partial Q}{\partial p_2} = -5p_1^2 p_2^{-2} \) with respect to \( p_1 \): \( \frac{\partial^2 Q}{\partial p_2 \partial p_1} = -10p_1 p_2^{-2} \).

Step 7: Verification of Equality of Mixed Partials

Observe that the mixed partials \( \frac{\partial^2 Q}{\partial p_1 \partial p_2} = -10p_1 p_2^{-2} \) and \( \frac{\partial^2 Q}{\partial p_2 \partial p_1} = -10p_1 p_2^{-2} \) are equal, confirming the equality of mixed partial derivatives.

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept when dealing with multivariable functions. A multivariable function is a function that relies on more than one variable. In this exercise, the function given is \( Q = 5 p_1^2 p_2^{-1} \), where the function depends on the variables \( p_1 \) and \( p_2 \).

To find a partial derivative, you choose one variable to differentiate with respect to, while treating all other variables as constants. For example:

• When you compute the partial derivative of \( Q \) with respect to \( p_1 \), you treat \( p_2 \) as a constant.
• The resulting expression describes how \( Q \) changes as \( p_1 \) changes, while \( p_2 \) remains constant.

This process isolates the effect of one specific variable on the overall behavior of the function. Understanding partial derivatives is essential in fields like economics, engineering, and physics where variables can depend on multiple factors.

Mixed Partials

Mixed partial derivatives are derivatives that involve differentiating a multivariable function with respect to two different variables. For mixed partials, you take a partial derivative with respect to one variable and then take another partial derivative with respect to a different variable.

For the function given \( Q = 5 p_1^2 p_2^{-1} \):

• The operation \( \frac{\partial^2 Q}{\partial p_1 \partial p_2} \) involves taking the partial derivative of \( Q \) with respect to \( p_1 \) first, followed by a partial derivative with respect to \( p_2 \).
• Conversely, \( \frac{\partial^2 Q}{\partial p_2 \partial p_1} \) starts with \( p_2 \) and then differentiates with respect to \( p_1 \).

A noteworthy aspect of mixed partials is that they are equal if the function is continuous and the mixed derivatives are continuous around the point of differentiation. This is known as Clairaut's theorem and it's a vital check for consistent results in calculus applications.

Second-Order Derivatives

Second-order derivatives are simply the derivatives of derivatives. For a function that depends on more than one variable, second-order partial derivatives can give you much more insight into how the function behaves.

For our function \( Q = 5 p_1^2 p_2^{-1} \), we have:

• \( \frac{\partial^2 Q}{\partial p_1^2} \) tells us about the concavity or convexity of \( Q \) with respect to \( p_1 \).
• \( \frac{\partial^2 Q}{\partial p_2^2} \) does the same but with respect to \( p_2 \).
• The mixed second-order derivatives further help understand interactions between the variables \( p_1 \) and \( p_2 \).

Second-order derivatives play a key role in optimization problems and ensures stability in systems across different fields of study.

Multivariable Functions

Multivariable functions are functions with more than one input variable. These functions model a vast array of real-world scenarios where outcomes depend on several changing factors.

In the given exercise, \( Q = 5 p_1^2 p_2^{-1} \) serves as a classic example of a multivariable function, where \( Q \) depends on both \( p_1 \) and \( p_2 \).

• Understanding how each variable influences the function individually, through partial derivatives, is key.
• Exploring how variables influence each other via mixed partials adds another layer of comprehension.

Grasping these concepts allows a better understanding of models in economics, physical sciences, and complex systems, where numerous driving factors are at play.