Problem 153 - Calculus 3

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Function

The given function is \( f(x, y) = -7x - 2xy + 7y \). We are to find the limit of a difference quotient with respect to \( y \).

Step 2: Determine \( f(x, y + \Delta y) \)

Calculate \( f(x, y + \Delta y) \) by substituting \( y + \Delta y \) into the function: \[f(x, y + \Delta y) = -7x - 2x(y + \Delta y) + 7(y + \Delta y). \]

Step 3: Simplify \( f(x, y + \Delta y) \)

Expand \( f(x, y + \Delta y) \): \[ f(x, y + \Delta y) = -7x - 2xy - 2x\Delta y + 7y + 7\Delta y. \]

Step 4: Compute the Difference Quotient

Find the numerator of the difference quotient: \[ f(x, y + \Delta y) - f(x, y) = (-7x - 2xy - 2x\Delta y + 7y + 7\Delta y) - (-7x - 2xy + 7y).\] Simplifying gives \[ -2x\Delta y + 7\Delta y.\] So, the difference quotient is \[ \frac{-2x\Delta y + 7\Delta y}{\Delta y}.\]

Step 5: Simplify the Difference Quotient

Factor \( \Delta y \) out of the numerator: \[ \frac{\Delta y(-2x + 7)}{\Delta y}.\]Cancel \( \Delta y \) from numerator and denominator: \[ -2x + 7. \]

Step 6: Evaluate the Limit

As \( \Delta y \to 0 \), the expression \(-2x + 7\) remains unchanged because it does not depend on \(\Delta y\). Thus, \[\lim _{\Delta y \to 0} \frac{f(x, y + \Delta y) - f(x, y)}{\Delta y} = -2x + 7.\]

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept used to understand rates of change in calculus. In the context of multivariable calculus, it helps in finding partial derivatives, which measure how a function changes as one of its variables changes, while keeping other variables constant.

When you have a function of several variables, say two variables like in this exercise, the difference quotient with respect to one variable is expressed as:

• \[ \frac{f(x, y + \Delta y) - f(x, y)}{\Delta y} \]

This expression calculates the average rate of change with respect to \( y \), while \( x \) is held constant. By evaluating this limit as \( \Delta y \to 0 \), we compute the exact rate of change, which is our partial derivative with respect to \( y \). As shown in the solution, simplifying the difference quotient often involves factoring and canceling terms, which helps simplify the expression before taking the limit.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions with more than one variable. Unlike single-variable calculus, which focuses on derivatives and integrals of functions with one variable, multivariable calculus deals with functions like \( f(x, y) \) that depend on two or more variables.

This area of calculus introduces several new elements:

• **Partial Derivatives:** Measures how a function changes as one variable changes, keeping all other variables constant.
• **Gradient:** A multidimensional extension of derivatives, pointing in the direction of greatest increase.
• **Differentials and Partial Derivatives:** Used to understand changes in multivariable functions with accuracy.

In our given exercise, we specifically deal with finding the partial derivative with respect to \( y \). This involves taking a slice of our function \( f(x, y) \) and investigating how it changes as \( y \) changes while \( x \) stays put. The focus on specific variables one by one is key in understanding the behavior of rough surfaces or multidimensional spaces.

Limit of a Function

The limit of a function is a crucial concept in calculus, providing insight into the behavior of a function as its variables approach some value. When dealing with multivariable functions, the concept is essential for defining derivatives and continuity.

In our exercise, we want to find:

• \[ \lim_{\Delta y \to 0} \frac{f(x, y + \Delta y) - f(x, y)}{\Delta y} \]

This expression explores how closely the difference quotient approximates a particular value as \( \Delta y \) approaches 0.
By evaluating this limit, we determine the exact partial derivative with respect to \( y \), which is a fundamental aspect of multivariable calculus.
The essence of taking this limit in the context of our problem allows us to understand the instantaneous rate of change in the function concerning one variable, capturing the precise slope or direction of change. Limits help us transition from average rates to exact, instantaneous changes, giving rise to the derivative.