Problem 153 - Calculus Volume 3

Question

Find \(\lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y}\) for \(f(x, y)=-7 x-2 x y+7 y\).

Step-by-Step Solution

Answer

The limit is \(-2x + 7\).

Step 1: Understand the Problem

We need to find the limit \( \lim _{\Delta y \rightarrow 0} \frac{f(x, y+\Delta y)-f(x, y)}{\Delta y} \) for the given function \( f(x, y) = -7x - 2xy + 7y \). This expression is the partial derivative of \( f \) with respect to \( y \).

Step 2: Substitute \( y + \Delta y \) into the Function

Substitute \( y + \Delta y \) into the function instead of \( y \). This gives us \( f(x, y + \Delta y) = -7x - 2x(y + \Delta y) + 7(y + \Delta y) \).

Step 3: Simplify the Expression

Simplify the expression \( f(x, y + \Delta y) \). This gives:\[-7x - 2xy - 2x\Delta y + 7y + 7\Delta y\].

Step 4: Find \( f(x, y + \Delta y) - f(x, y) \)

Calculate the difference:\[(-7x - 2xy - 2x\Delta y + 7y + 7\Delta y) - (-7x - 2xy + 7y)\].Upon simplification, this becomes:\[-2x\Delta y + 7\Delta y\].

Step 5: Factor \( \Delta y \)

Factor out \( \Delta y \) from the expression:\[\Delta y(-2x + 7)\].

Step 6: Evaluate the Limit

Evaluate the limit as \( \Delta y \) approaches 0:\[\lim_{{\Delta y \to 0}} \frac{\Delta y(-2x + 7)}{\Delta y} = -2x + 7\].

Key Concepts

Multivariable Calculus

Multivariable calculus is a branch of mathematics that deals with functions of more than one variable. Unlike single-variable calculus, which focuses on functions with just one input, multivariable calculus explores how changes in more than one input affect the output. This allows us to study complex systems in which multiple factors might be interdependent.

• Applications: Used in diverse fields such as engineering, physics, and economics to solve real-world problems.
• Key Concepts: Includes topics such as partial derivatives, multiple integrals, and gradient vectors.

In this exercise, we're working with a function that depends on two variables, \(x\) and \(y\). Understanding how a slight change in one variable affects the function involves using partial derivatives.

Limit Definition

The limit definition is fundamental to calculus, describing how a function behaves as its inputs approach a particular value. In this problem, we're exploring the behavior of the function \(f(x, y) = -7x - 2xy + 7y\) as the variable \(\Delta y\) approaches zero.

• Limits analyze how functions change as inputs get infinitesimally small or large.
• The notation \(\lim_{\Delta y \to 0}\) expresses the idea of observing the function's behavior as \(\Delta y\) gets closer to zero.

For partial derivatives, limits help us to find the rate of change with respect to one variable while keeping others constant, providing a deeper understanding of the function's dynamics.

Function Differentiation

Differentiation is a crucial part of calculus, allowing us to find the rate at which one quantity changes with respect to another. For functions of multiple variables, we use partial differentiation, which focuses on one variable at a time while keeping others fixed.

• Purpose: Helps in understanding how a tiny change in one input affects the output of the function.
• Technique: Involves taking the derivative with respect to each variable individually, called partial derivatives.

In our exercise, we're particularly interested in differentiating the function \(f\) with respect to \(y\), while regarding \(x\) as a constant, allowing us to capture how variations in \(y\) influence \(f\).

Partial Derivatives with Respect to y

Partial derivatives are used to find the derivative of a multivariable function concerning one variable while holding the others constant. In this context, it's a measure of how the function \(f(x, y) = -7x - 2xy + 7y\) changes as \(y\) changes, with \(x\) fixed.

• Formula: Expressed as \(\frac{\partial f}{\partial y}\), reflecting differentiation with respect to \(y\).
• Application: Important for optimizing functions and understanding their behavior across dimensions.

In our exercise, applying the limit definition, we simplified the expression to derive \(-2x + 7\), indicating how \(f\) reacts to changes in \(y\) when \(x\) remains unchanged. This partial derivative is key to problems involving sensitivity analysis in engineering and other fields.