Problem 25 - Calculus and its applications

Question

Find \(f_{x}, f_{y},\) and \(f_{\lambda}\). $$f(x, y, \lambda)=x^{2}+y^{2}-\lambda(10 x+2 y-4)$$

Step-by-Step Solution

Answer

\(f_{x} = 2x - 10\lambda\), \(f_{y} = 2y - 2\lambda\), \(f_{\lambda} = -(10x + 2y - 4)\).

Step 1: Find the Partial Derivative with Respect to x

To find \(f_{x}\), take the partial derivative of the function \(f(x, y, \lambda) = x^2 + y^2 - \lambda(10x + 2y - 4)\) with respect to \(x\). First, differentiate \(x^2\) to get \(2x\). Next, identify terms containing \(x\) in \(-\lambda(10x + 2y - 4)\); this is \(10x\cdot(-\lambda)\). The derivative with respect to \(x\) is \(-10\lambda\). Therefore, the partial derivative is \(f_{x} = 2x - 10\lambda\).

Step 2: Find the Partial Derivative with Respect to y

To find \(f_{y}\), take the partial derivative of the function with respect to \(y\). Differentiate \(y^2\) with respect to \(y\) to get \(2y\). From \(-\lambda(10x + 2y - 4)\), identify the term \(2y\cdot(-\lambda)\). The derivative with respect to \(y\) is \(-2\lambda\). Hence, the partial derivative is \(f_{y} = 2y - 2\lambda\).

Step 3: Find the Partial Derivative with Respect to \(\lambda\)

To find \(f_{\lambda}\), take the partial derivative of the function with respect to \(\lambda\). This involves differentiating \(-\lambda(10x + 2y - 4)\) with respect to \(\lambda\). The result is the negative of \((10x + 2y - 4)\) because each term is multiplied directly by \(-\lambda\). Therefore, \(f_{\lambda} = -(10x + 2y - 4)\).

Key Concepts

Partial Derivatives

Partial derivatives form a cornerstone in multivariable calculus, especially when dealing with functions involving several variables. In our scenario, the function is given as \( f(x, y, \lambda) = x^2 + y^2 - \lambda(10x + 2y - 4) \). To find the partial derivatives of this function, we consider how \( f \) changes as each independent variable \( x \), \( y \), and \( \lambda \) changes individually, while the other variables are held constant.

• Partial Derivative with Respect to \( x \): To solve for \( f_x \), we differentiate \( x^2 \), resulting in \( 2x \). The term \(-\lambda(10x + 2y - 4)\) contains \( x \) in \( 10x(-\lambda) \), contributing \(-10\lambda\) to the partial derivative, hence, \( f_{x} = 2x - 10\lambda \).
• Partial Derivative with Respect to \( y \): For \( f_y \), we differentiate \( y^2 \), resulting in \( 2y \). The term \(-\lambda(10x + 2y - 4)\) has \( y \) in \( 2y(-\lambda) \), contributing \(-2\lambda\) to the partial derivative, so \( f_{y} = 2y - 2\lambda \).
• Partial Derivative with Respect to \( \lambda \): To solve for \( f_\lambda \), we focus on the linear term \(-\lambda(10x + 2y - 4)\) and find its derivative relative to \( \lambda \), producing \( -(10x + 2y - 4) \), hence \( f_{\lambda} = -(10x + 2y - 4) \).

Partial derivatives allow us to analyze the function's behavior in relation to each variable individually, providing valuable insights for further analysis.

Multivariable Calculus

The realm of multivariable calculus opens up as we delve into functions with more than one variable, such as our function \( f(x, y, \lambda) \). Understanding functions within multivariable calculus involves looking at several partial derivatives, assessing how changes in the variables affect the outcome of the function.

Considerations in multivariable calculus include using tools like the gradient vector, often formed from partial derivatives, to evaluate how the function behaves across a multidimensional space.

Points where all partial derivatives are zero could be extremal points—local minima, maxima, or saddle points—critical for solving or optimizing the function. Our step-by-step solution involves calculating derivatives \( f_x \), \( f_y \), and \( f_\lambda \), which systematically prepare us for applying these concepts in optimization or constraint problems. This structured approach highlights how changes in each independent variable influence the function's values.

Using multivariable calculus, one can delve deeper into more complex problems in physics, engineering, and economics, tackling challenges where multiple variables simultaneously influence the outcomes.

Optimization Problem

Optimization problems are prevalent in mathematics, engineering, and beyond, often solved using multivariable calculus. In context, we dealt with an optimization problem using the Lagrange multipliers method, focusing on maximizing or minimizing a function subject to constraints.

Here, the function \( f(x, y, \lambda) \) incorporates the Lagrange multiplier \( \lambda \), reflecting constraints set by \( 10x + 2y = 4 \). The method of Lagrange multipliers involves setting up supplementary conditions or equations. These conditions help in finding optimal values that are subject to the given constraints.

• The main idea is finding the values of \( x \), \( y \), and \( \lambda \) that satisfy both the original function and the constraint condition.
• The method involves solving the system of equations formed by setting the gradients of the target function and constraints in parallel.

Using this technique provides systematic solutions even when direct optimization seems complicated due to existing constraints. By understanding both partial derivatives and the broader scope of multivariable calculus, students can effectively address and solve complex optimization challenges.