Problem 290 - Calculus 3

Question

For the following exercises, find the gradient vector at the indicated point. $$ f(x, y)=x y^{2}-y x^{2}, P(-1,1) $$

Step-by-Step Solution

Answer

The gradient vector at \( P(-1, 1) \) is \( (3, -3) \).

Step 1: Understand the Function

The function given is a multivariable function, \( f(x, y) = xy^2 - yx^2 \). Our goal is to find the gradient vector of this function at a specific point, \( P(-1, 1) \). The gradient vector \( abla f(x, y) \) will capture the direction of the steepest ascent.

Step 2: Compute Partial Derivatives

Find the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \).\( f_x(x, y) = \frac{\partial}{\partial x}(xy^2 - yx^2) = y^2 - 2yx \).\( f_y(x, y) = \frac{\partial}{\partial y}(xy^2 - yx^2) = 2xy - x^2 \).

Step 3: Evaluate at the Point P(-1,1)

Substitute \( x = -1 \) and \( y = 1 \) into the partial derivatives found in Step 2:\( f_x(-1, 1) = (1)^2 - 2(-1)(1) = 1 + 2 = 3 \).\( f_y(-1, 1) = 2(-1)(1) - (-1)^2 = -2 - 1 = -3 \).

Step 4: Form the Gradient Vector

The gradient vector \( abla f(x, y) \) is given by \( (f_x, f_y) \). Thus, at \( P(-1, 1) \), the gradient vector is \( (3, -3) \).

Key Concepts

The Nature of Multivariable Functions

In mathematics, a multivariable function is one that has more than one input variable. Unlike single-variable functions, which deal with just one changing quantity, multivariable functions involve two or more variables. This allows such functions to model complex systems with several influencing factors, making them most invaluable in fields like physics, economics, and engineering.
For example, in the function \( f(x, y) = xy^2 - yx^2 \), both \( x \) and \( y \) are variables that jointly determine the value of \( f \).
This added complexity allows us to study how changes in each variable individually, and together, influence the overall outcome of the function.

Working with multivariable functions involves:

• Understanding how each variable affects the function's behavior.
• Calculating derivatives, like partial derivatives, to analyze rates of change.

These functions extend our ability to model the real world, providing a powerful framework for analysis.

Understanding Partial Derivatives

Partial derivatives are a fascinating tool in calculus, particularly when dealing with multivariable functions. Simply put, a partial derivative measures how a function changes as one of its input variables changes, while all other variables are held constant.
When we calculate the partial derivative of \( f(x, y) = xy^2 - yx^2 \) with respect to \( x \), we get \( f_x(x, y) = y^2 - 2yx \).
Similarly, with respect to \( y \), we derive \( f_y(x, y) = 2xy - x^2 \).

This process involves taking the derivative of the function as though it is a single-variable function, treating all other variables as constants. Partial derivatives help us understand the sensitivity of the function's output due to small changes in one variable, which is crucial in optimization and approximations.

To work efficiently with partial derivatives:

• Identify which variable to differentiate with respect to and treat others as constants.
• Apply the standard rules of differentiation, focusing on the variable of interest.
• Use the results to construct the gradient vector to analyze the function's behavior further.

The Gradient Vector and Direction of Steepest Ascent

The gradient vector is a powerful concept in calculus, especially when examining the behavior of multivariable functions. It combines all the partial derivatives of a function, providing a vector that points in the direction of the steepest ascent from a given point.
For the function \( f(x, y) = xy^2 - yx^2 \), the gradient vector is denoted by \( abla f(x, y) \) and is composed of its partial derivatives: \( (f_x, f_y) \).
At any given point \( (x, y) \), this vector \( abla f(x, y) \) reveals the direction in which the function increases most rapidly. For instance, at \( P(-1, 1) \), the gradient vector is \( (3, -3) \), indicating that moving in this direction will increase the value of the function most effectively.

Understanding the gradient vector involves:

• Computing the partial derivatives for the function's variables.
• Forming the vector \( (f_x, f_y) \) to pinpoint the direction of greatest change.
• Analyzing its impact on the function's growth and evaluating it at specific points to understand practical implications.

The gradient vector is invaluable for optimizing functions and understanding their behavior across different values.