Problem 5 - Calculus 3

Question

For the following exercises, find the domain of the function. $$ V(x, y)=4 x^{2}+y^{2} $$

Step-by-Step Solution

Answer

The domain is all real numbers: \((x, y) \in \mathbb{R}^2\).

Step 1: Understand the Function Type

The function given is \( V(x, y) = 4x^2 + y^2 \). This is a multivariable quadratic function that involves both \( x \) and \( y \), and it includes only polynomial terms of \( x \) and \( y \).

Step 2: Identify the Restrictions

Polynomials are defined for all real numbers. There are no fractions, square roots, logarithms, or any other operations that limit the input values of \( x \) and \( y \).

Step 3: Determine the Domain

Since there are no restrictions on \( x \) or \( y \), the domain includes every possible combination of \( x \) and \( y \) values. Thus, both \( x \) and \( y \) can be any real numbers.

Key Concepts

Quadratic Functions

Quadratic functions, like the one in this exercise, are a specific type of polynomial function that involve terms with variables raised to the second power. When dealing with multivariable functions, a quadratic function might look something like \( V(x, y) = ax^2 + by^2 + cxy + dx + ey + f \), where \( a, b, c, d, e, \) and \( f \) are constants. Here, each term is either solely dependent on one variable squared or is a linear combination of the variables.

• The primary characteristic of quadratic functions is the presence of terms like \( x^2 \) or \( y^2 \).
• These functions create parabolic shapes when graphed in two dimensions.
• In multivariable quadratic functions, these parabolas form a surface in three-dimensional space.

For the given function \( V(x, y) = 4x^2 + y^2 \), it is a multivariable quadratic function where each variable is only squared and positive, meaning the graph will show an upward-opening bowl shape. Understanding these basic features helps in determining domains, as quadratic terms can accept any real numbers without restriction.

Polynomial Functions

Polynomial functions consist of terms that include variables raised to whole number powers and coefficients. These functions can vary from simple linear polynomials to more complex multivariable polynomials like the one in this exercise. Each term in a polynomial is a product of a constant and variables raised to non-negative integer powers.

• For example, a term like \( 4x^2 \) is a polynomial term where the variable \( x \) is squared.
• Polynomial terms like \( y^2 \) in \( 4x^2 + y^2 \) do not introduce any restrictions on inputs, as they are well defined for all real numbers.

In the context of multivariable polynomial functions, each variable behaves independently, and there are no divisions by zero or taking square roots of negative numbers to worry about.Therefore, there are no restrictions caused by polynomial terms for finding the domain, making it straightforward, as seen in this problem.

Real Numbers

Real numbers are a fundamental part of mathematics, representing a continuous unbroken line of values that include all rational and irrational numbers. Real numbers encompass natural numbers, whole numbers, integers, fractions, and even numbers like \( \pi \) and square root of non-perfect squares.

• In the exercise provided, both \( x \) and \( y \) are real numbers.
• This means that they can take any value on the number line, from negative infinity to positive infinity.
• In the context of functions, particularly polynomials, being defined over real numbers means that there are no inherent restrictions on the values that \( x \) and \( y \) can assume.

Understanding that the domain includes all real numbers simplifies many problems in mathematics, like the given function \( V(x, y) = 4x^2 + y^2 \). It reaffirms that the domain of this function covers all possible real values of \( x \) and \( y \). This broadens the scope and possibilities when analyzing or applying this function.