91Ó°ÊÓ

A piecewise function is a type of function that is defined by different expressions depending on the region of the domain. For example, in the function given, \( f(x, y)= \{ y^{3} \text{ if } y \geq 0 \text{ and } -y^{2} \text{ if } y < 0 \} \), the expression used for calculation depends on the value of \( y \). This makes the function piecewise because different equations are used for different parts of the domain.
Piecewise functions are often used in mathematics to handle scenarios where a rule changes at certain conditions. They resemble how many real-life processes work; for instance, tax rates or shipping costs might change depending on income or weight. Understanding the domain where each piece operates is crucial to working accurately with piecewise functions.

• For \( y \geq 0 \), the piece of the function is \( y^3 \).
• For \( y < 0 \), the piece of the function is \( -y^2 \).

Mixed partial derivatives involve taking the partial derivative of a function with respect to multiple variables. In the given exercise, the mixed partial derivatives are \( f_{xy} \) and \( f_{yx} \), which involve differentiating first with respect to one variable (\( x \) or \( y \)) and then the other.
Mixed partial derivatives can provide insights into how different variables of a function influence each other. For most functions with continuous second partial derivatives, as guaranteed by Clairaut's theorem, we expect \( f_{xy} \) to be equal to \( f_{yx} \).
In this exercise's context:

• \( f_{xy} \) and \( f_{yx} \) are both 0, as the function does not explicitly depend on \( x \).
• This matches the expectation from Clairaut's theorem where well-behaved functions (smooth enough) will have equal mixed partial derivatives, as is the case here with a value of 0 across all points in the domain \( \mathbb{R}^2 \).

Understanding the domain of a function is vital as it defines the set of input values for which the function is valid. The domain tells us where each part of a piecewise function is applicable. For the function given, the domain accommodates all real numbers, \( (x, y) \in \mathbb{R}^2 \), meaning both \( x \) and \( y \) are real numbers within this two-dimensional plane.
For each partial derivative calculated in the exercise:

• The domain of \( f_x \) is \( \mathbb{R}^2 \); it remains 0 across all \( y \).
• The domain of \( f_y \) is \( \mathbb{R}^2 \); the function changes with \( y \) following the piecewise rules.
• Similarly, the domains for \( f_{xy} \) and \( f_{yx} \) are also \( \mathbb{R}^2 \), reflecting that the results are universally 0 across the entire plane of real numbers.

Understanding the domain is like understanding the stage where the mathematics acts. It is crucial when solving or graphing functions as well as when determining where formulas apply.