Problem 24 - Calculus and its applications

Question

Find \(f_{x}, f_{y},\) and \(f_{\lambda}\). $$f(x, y, \lambda)=9 x y-\lambda(3 x-y+7)$$

Step-by-Step Solution

Answer

The derivatives are: \( f_x = 9y - 3\lambda \), \( f_y = 9x + \lambda \), \( f_\lambda = -(3x - y + 7) \).

Step 1: Identify the Function Components

First, let's identify the function components: The function given is \( f(x, y, \lambda) = 9xy - \lambda(3x-y+7) \). This function includes three variables: \( x \), \( y \), and \( \lambda \). The goal is to find the partial derivatives of \( f \) with respect to each of these variables.

Step 2: Compute the Partial Derivative with Respect to x (f_x)

To find \( f_x \), differentiate \( f(x, y, \lambda) \) with respect to \( x \). Treat \( y \) and \( \lambda \) as constants:\[ f_x = \frac{\partial}{\partial x}(9xy - \lambda(3x - y + 7)) = 9y - 3\lambda. \]

Step 3: Compute the Partial Derivative with Respect to y (f_y)

To find \( f_y \), differentiate \( f(x, y, \lambda) \) with respect to \( y \). Treat \( x \) and \( \lambda \) as constants:\[ f_y = \frac{\partial}{\partial y}(9xy - \lambda(3x - y + 7)) = 9x + \lambda. \]

Step 4: Compute the Partial Derivative with Respect to \( \lambda \) (f_\lambda)

To find \( f_\lambda \), differentiate \( f(x, y, \lambda) \) with respect to \( \lambda \). Treat \( x \) and \( y \) as constants:\[ f_\lambda = \frac{\partial}{\partial \lambda}(9xy - \lambda(3x - y + 7)) = -(3x - y + 7). \]

Key Concepts

Multivariable Calculus

Multivariable calculus deals with functions that depend on more than one variable. In our exercise, the function has the form \( f(x, y, \lambda) = 9xy - \lambda(3x-y+7) \), which means it involves three variables: \( x \), \( y \), and \( \lambda \). In multivariable calculus, it's common to explore how changes in individual variables affect the outcome of a function.

Rather than taking the derivative in one dimension, we now take partial derivatives for each variable, holding the others constant. Here's why:

• Focus on Variable Effects: By taking partial derivatives, we can focus on how one specific variable influences the function, keeping all other variables fixed.
• Real-World Problems: Many practical problems involve several variables. Multivariable functions help describe real phenomena in physics, engineering, and economics.
• Gradient Vector: The collection of all partial derivatives forms the gradient, a vector that shows the direction of the steepest increase of the function.

Understanding these basics provides a foundation for more complex concepts in calculus, such as Lagrange multipliers and optimization problems.

Lagrange Multipliers

Lagrange multipliers are a strategy used in multivariable calculus for finding the local maxima and minima of a function subject to equality constraints. This problem is particularly useful in optimizing functions with constraints. In the function we consider, \( f(x, y, \lambda) = 9xy - \lambda(3x-y+7) \), \( \lambda \) acts as a Lagrange multiplier.

Here's how the method works:

• Understand Constraints: To optimize a function in the presence of constraints, Lagrange multipliers add a term to the function, which includes the constraints multiplied by the Lagrange multiplier.
• Create a New Function: The constraint \((3x-y+7)\) in our exercise is a condition that needs to be satisfied while optimizing \(9xy\).
• Find Critical Points: By taking partial derivatives of the adjusted function with respect to each variable (including \( \lambda \)), we find necessary conditions for optimization.

Using Lagrange multipliers simplifies solving constrained optimization problems, making it easier to find optimal solutions under given restrictions.

Calculus Functions

In calculus, functions are expressions that relate inputs to outputs. For a function with multiple variables, like our example \( f(x, y, \lambda) = 9xy - \lambda(3x-y+7) \), we aim to understand how these variables interact within the function. Functions in calculus can be categorized in several ways:

• Linear vs. Non-linear : Functions can be linear or non-linear based on the degree of the variables involved. In our instance, the function is non-linear due to the product term \( 9xy \).
• Polynomial Functions: These are functions expressed as a sum of powers in one or more variables, like \(9xy\) in our case.
• Differentiability: This determines if we can calculate the derivatives of functions at various points. Our function is differentiable, allowing us to take partial derivatives efficiently.

Functions are the building blocks of calculus, providing deep insights into mathematical relationships and simplifying complex real-world problems.