Problem 137 - Calculus Volume 3

Question

Let \(z=x^{2}+3 x y+2 y^{2} .\) Find \(\frac{\partial^{2} z}{\partial x^{2}}\) and \(\frac{\partial^{2} z}{\partial y^{2}}\).

Step-by-Step Solution

Answer

\(\frac{\partial^{2} z}{\partial x^{2}} = 2\) and \(\frac{\partial^{2} z}{\partial y^{2}} = 4\).

Step 1: Find the First Partial Derivative with Respect to x

First, we find the partial derivative of the function \(z\) with respect to \(x\). The function is \(z = x^{2} + 3xy + 2y^{2}\).The terms involving \(x\) are as follows:- For \(x^{2}\), the derivative with respect to \(x\) is \(2x\).- For \(3xy\), treating \(y\) as a constant, the derivative with respect to \(x\) is \(3y\).- \(2y^{2}\) is treated as constant and differentiates to 0.Therefore, the first partial derivative with respect to \(x\) is:\[ \frac{\partial z}{\partial x} = 2x + 3y \]

Step 2: Differentiate Again to Find \(\frac{\partial^{2} z}{\partial x^{2}}\)

Now we take the second derivative with respect to \(x\) of \(\frac{\partial z}{\partial x} = 2x + 3y\).- The derivative of \(2x\) with respect to \(x\) is 2.- The derivative of \(3y\) with respect to \(x\) is 0.Thus, we have:\[ \frac{\partial^{2} z}{\partial x^{2}} = 2 \]

Step 3: Find the First Partial Derivative with Respect to y

Next, we find the partial derivative of the function \(z\) with respect to \(y\). Start again with the original function \(z = x^{2} + 3xy + 2y^{2}\).- The derivative of \(x^{2}\) with respect to \(y\) is 0.- For \(3xy\), the derivative with respect to \(y\) is \(3x\).- For \(2y^{2}\), the derivative with respect to \(y\) is \(4y\).Thus, the first partial derivative with respect to \(y\) is:\[ \frac{\partial z}{\partial y} = 3x + 4y \]

Step 4: Differentiate Again to Find \(\frac{\partial^{2} z}{\partial y^{2}}\)

Finally, differentiate \(\frac{\partial z}{\partial y} = 3x + 4y\) with respect to \(y\).- The derivative of \(3x\) with respect to \(y\) is 0.- The derivative of \(4y\) with respect to \(y\) is 4.Thus, the second partial derivative with respect to \(y\) is:\[ \frac{\partial^{2} z}{\partial y^{2}} = 4 \]

Key Concepts

Partial Derivatives

Partial derivatives are an essential part of multivariable calculus, dealing with functions that have more than one variable. Imagine we have a function like \[ f(x, y, z) \]This function depends on multiple variables. We use partial derivatives to study the function's rate of change concerning one variable at a time while keeping the others constant.For example:

• If you take the partial derivative of \( f \) with respect to \( x \), you hold \( y \) and \( z \) constant, and find out how \( f \) changes as \( x \) changes.
• Similarly, a partial derivative with respect to \( y \) keeps \( x \) and \( z \) constant.

Partial derivatives appear in a lot of practical applications. They're widely used in physics, engineering, and economics for modeling and solving real-world problems.

Multivariable Calculus

Multivariable calculus extends concepts from basic calculus to functions involving several variables. Where single-variable calculus deals with functions in one-dimensional space,multivariable calculus handles functions mapped in multidimensional spaces.Some important topics include:

• Function of Several Variables: Unlike functions in simple calculus, these depend on two or more variables, e.g., \( f(x, y) \) or \( g(x, y, z) \).
• Partial Derivatives: These help us analyze the individual influence of each variable on the overall function.
• Gradient: A vector representing the direction and rate of the steepest ascent of a function.

Multivariable calculus allows us to understand quantities where regular calculus falls short. It is crucial for studying phenomena involving complex and interconnected relationships.

Differentiation

Differentiation is a significant process in calculus used to find the rate of change of one variable with respect to another. It's like finding the slope or steepness of a function at any point.In the context of single-variable calculus, you're typically used to dealing with functions of a single variable, finding their derivatives. But in multivariable calculus, we expand this concept by using partial derivatives.

• Normal Differentiation: Involves functions like \( f(x) \) where you find the derivative with respect to that one variable \( x \).
• Partial Differentiation: Deals with functions like \( f(x, y) \) where you calculate partial derivatives for each variable \( x \) and \( y \). Each variable is differentiated separately in relation to the function, while keeping others constant.

Understanding differentiation in many variables allows us to analyze functions more thoroughly, especially when dealing with complex problems.