Problem 3 - Applied Calculus

Question

Find the partial derivatives in problems. The variables are restricted to a domain on which the function is defined. \(f_{x}\) and \(f_{y}\) if \(f(x, y)=x^{2}+2 x y+y^{3}\)

Step-by-Step Solution

Answer

The partial derivatives are \( f_x = 2x + 2y \) and \( f_y = 2x + 3y^2 \).

Step 1: Identify the Function

We have the function \( f(x, y) = x^2 + 2xy + y^3 \). Our goal is to find the partial derivatives \( f_x \) and \( f_y \).

Step 2: Compute the Partial Derivative with respect to x

To find \( f_x \), treat \( y \) as a constant and differentiate \( f(x, y) = x^2 + 2xy + y^3 \) with respect to \( x \). The derivative is:\[ f_x = \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial x}(2xy) + \frac{\partial}{\partial x}(y^3) = 2x + 2y + 0 \]Thus, \( f_x = 2x + 2y \).

Step 3: Compute the Partial Derivative with respect to y

To find \( f_y \), treat \( x \) as a constant and differentiate \( f(x, y) = x^2 + 2xy + y^3 \) with respect to \( y \). The derivative is:\[ f_y = \frac{\partial}{\partial y}(x^2) + \frac{\partial}{\partial y}(2xy) + \frac{\partial}{\partial y}(y^3) = 0 + 2x + 3y^2 \]Thus, \( f_y = 2x + 3y^2 \).

Key Concepts

differentiation

Differentiation is a fundamental concept in calculus that deals with finding the derivative of a function. The derivative of a function represents how the function's output changes as its input changes. In simpler terms, differentiation tells us the rate at which a function grows or shrinks. When we differentiate a function of a single variable, we find the function's derivative concerning that variable. This process helps us understand many real-world phenomena such as velocity, acceleration, and cost functions.

To differentiate a function efficiently, we use several rules or techniques such as:

• Power Rule: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
• Product Rule: If \( f(x) \) and \( g(x) \) are both differentiable, then \( (fg)' = f'g + fg' \).
• Quotient Rule: If \( f(x) \) and \( g(x) \) are both differentiable, where \( g(x) eq 0 \), then \[ \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} \].
• Chain Rule: If a function \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

Differentiation plays a crucial role in understanding the dynamics of any system that can be described mathematically.

multivariable calculus

Multivariable calculus extends the concepts of calculus, such as differentiation and integration, to functions of more than one variable. These functions can depend on two or more inputs, unlike single-variable calculus, where the function depends on only one input variable.

Functions in multivariable calculus can be denoted by \( f(x, y, z, ...) \), where each variable represents a dimension. This is particularly useful for modeling systems in fields like physics, engineering, and economics, where multiple factors influence outcomes.

• Gradient: The gradient of a multivariable function is a vector composed of its partial derivatives. It points in the direction of the greatest rate of increase of the function.
• Level Curves: These are curves along which the function has a constant value, giving insight into the function's behavior across a plane.
• Double and Triple Integrals: These allow us to calculate the volume under a surface in 3D space or mass, center of mass, etc., of objects.

By exploring multivariable calculus, we can analyze how systems change with multiple influences and better predict outcomes based on these interactions.

partial differentiation

Partial differentiation is a technique used in multivariable calculus when a function depends on several variables. It helps us understand how a function changes with respect to one specific variable, keeping others constant. This is crucial for optimizing functions where multiple variables play roles.

Given a function \( f(x, y) \), which depends on variables \( x \) and \( y \), the partial derivative with respect to \( x \), denoted \( f_x(x, y) \), is found by treating \( y \) as a constant and differentiating the function with respect to \( x \). Similarly, \( f_y(x, y) \) is found by treating \( x \) as constant and differentiating with respect to \( y \).

Here’s the recipe:

• Identify and write down the function.
• Treat the other variables as constant and differentiate the function with respect to the variable of interest.
• Simplify the expression to find the partial derivative.

This method is used in various applications, such as in economics to find marginal costs or in physics to determine force fields.