Problem 113 - Calculus 3

Question

For the following exercises, calculate the partial derivative using the limit definitions only. \(\frac{\partial z}{\partial y}\) for \(z=x^{2}-3 x y+y^{2}\)

Step-by-Step Solution

Answer

The partial derivative \(\frac{\partial z}{\partial y}\) is \(-3x + 2y\).

Step 1: Identify the Partial Derivative to be Found

We are asked to determine \(\frac{\partial z}{\partial y}\) for the function \(z = x^2 - 3xy + y^2\). This means we hold \(x\) constant and find the rate of change of \(z\) with respect to \(y\).

Step 2: Use the Definition of the Partial Derivative

The partial derivative \(\frac{\partial z}{\partial y}\) is defined as:\[\frac{\partial z}{\partial y} = \lim_{h \to 0} \frac{z(x, y+h) - z(x, y)}{h}\]Substitute \(z(x, y) = x^2 - 3xy + y^2\) into this formula.

Step 3: Substitute and Simplify the Expression

Calculate \(z(x, y+h)\):\[z(x, y+h) = x^2 - 3x(y+h) + (y+h)^2 = x^2 - 3xy - 3xh + y^2 + 2yh + h^2\]Thus, the expression becomes:\[\frac{z(x, y+h) - z(x,y)}{h} = \frac{(x^2 - 3xy - 3xh + y^2 + 2yh + h^2) - (x^2 - 3xy + y^2)}{h}\]

Step 4: Simplify the Resulting Expression

Simplify the numerator:\[= -3xh + 2yh + h^2\]Divide by \(h\):\[= \frac{-3xh + 2yh + h^2}{h} = -3x + 2y + h\]

Step 5: Apply the Limit

Take the limit as \(h\) approaches 0:\[\frac{\partial z}{\partial y} = \lim_{h \to 0} (-3x + 2y + h) = -3x + 2y\]The limit calculation confirms that the partial derivative is \(-3x + 2y\).

Key Concepts

Limit Definition in Partial Derivatives

The limit definition plays a crucial role in understanding partial derivatives. It's the mathematical way to describe how a function changes as the variables change by an infinitesimally small amount. In our exercise, we are finding \(\frac{\partial z}{\partial y}\) of the function \(z = x^2 - 3xy + y^2\). This requires us to use the limit definition of a partial derivative: \[\frac{\partial z}{\partial y} = \lim_{h \to 0} \frac{z(x,y+h) - z(x,y)}{h}\] This formula tells us how the function \(z\) changes with respect to \(y\) while keeping \(x\) constant. Here's a simple breakdown of why the limit is taken: - **Increment \(h\):** It's a tiny change or increment in the variable \(y\). - **Difference Quotient:** The expression \(\frac{z(x,y+h) - z(x,y)}{h}\) measures how \(z\) changes with the small change in \(y\). - **Taking the Limit:** As \(h\) approaches zero, this expression gives the exact rate at which \(z\) changes with respect to \(y\) at that point. Using limits ensures that our derivative is accurate even for very tiny changes. This method applies to all functions, ensuring we can calculate derivatives in a consistent manner.

Rate of Change in Multivariable Functions

When dealing with multivariable functions, we often want to know how one variable affects another. That's where the concept of 'rate of change' comes in. For single-variable functions, the rate of change is straightforward, but with multiple variables, it becomes more complex because each variable can influence the outcome differently.In our exercise, we calculate the rate of change of \(z\) with respect to \(y\) while considering \(x\) as a constant. This means: - **Focus on \(y\):** We investigate how small changes in \(y\) can significantly influence \(z\), affecting the overall shape and behavior of the function.- **Holding \(x\) Constant:** By not changing \(x\), we isolate the effects of \(y\) on \(z\) and simplify the problem to one dimension. This helps in understanding the unique influence of \(y\) on \(z\). - **Multivariable Interaction:** Each variable's interaction in a multivariable function can tell us different things about the function's behavior. For instance, changing \(y\) while holding \(x\) constant might have a different rate of change compared to changing both \(x\) and \(y\) simultaneously.The rate of change is essential for modeling real-world phenomena where variables often depend on each other. For instance, in physics or economics, understanding these rates can lead to better predictions and solutions.

Understanding Multivariable Functions

Multivariable functions like \(z = x^2 - 3xy + y^2\) involve more than one input variable, and dealing with them introduces fascinating complexities. These functions are seen everywhere in the real world, from physics equations describing motion to financial models predicting market trends.Here's what you need to know about multivariable functions: - **Input Variables:** Instead of a single variable specter like \(x\) in one-variable functions, multivariable functions have several, such as \(x\) and \(y\). - **Output:** These functions output a single value, determined by the combination of input variables.- **Partial Derivatives:** Unlike regular derivatives, partial derivatives tell us how the function changes as one specific variable changes, holding others constant. This is crucial for isolating the rate of change related to individual variables.Understanding multivariable functions is like seeing the world in more dimensions. In mathematics, it lets us visualize complex surfaces and shapes. It provides insight into interactions among variables, leading to a greater understanding of dynamic systems. This understanding is foundational for studying calculus in higher dimensions and applications across numerous scientific fields.