Problem 50 - Calculus Early Transcendentals

Question

Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin ).$$Q(x, y, z)=\frac{10}{1+x^{2}+y^{2}+4 z^{2}}.$$

Step-by-Step Solution

Answer

Short Answer: The domain of the function Q(x, y, z) is $$\mathbb{R}^3$$ since the denominator, $$1+x^2+y^2+4z^2$$, is always positive and never equal to zero. Therefore, there are no restrictions on the values of x, y, and z.

Step 1: Part 1: Check the denominator

To find the domain of the function Q(x, y, z), we'll first examine the denominator: $$1+x^{2}+y^{2}+4z^{2}.$$ This is a sum of squares, which means it will always be non-negative. We want to make sure it's never equal to zero. If $$1+x^2+y^2+4z^2=0$$, the equation can be rearranged as $$x^2+y^2+4z^2=-1$$. Notice that this equation is not possible because the left side of the equation represents a sum of squares, which can never be negative. This means the denominator can never be equal to zero.

Step 2: Part 2: Determine the domain

Since we have established that the denominator $$1+x^2+y^2+4z^2$$ is always positive, there are no restrictions on the values of x, y, and z. Therefore, the domain of Q(x, y, z) includes all possible values of x, y, and z. In other words, the domain of Q(x, y, z) is $$\mathbb{R}^3$$, which is the set of all ordered triples of real numbers (x, y, z).

Key Concepts

Multivariable Functions

Multivariable functions are a fascinating area of mathematics that deal with functions of more than one variable. For example, the function \( Q(x, y, z) = \frac{10}{1 + x^2 + y^2 + 4z^2} \) is a function of three variables: \( x \), \( y \), and \( z \). Multivariable functions extend the concept of functions from simple, single-variable cases to more complex, real-world scenarios where the outcome depends on several factors.

Understanding these functions is essential because:

• They enable us to model systems with multiple dimensions, such as the trajectory of a projectile, the distribution of temperature in a room, or even the stress on a structural beam.
• The domain in multivariable functions refers to the set of all possible input values for which the function is defined.

In this example, the domain is all points in \( \mathbb{R}^3 \), which means \( x \), \( y \), and \( z \) can be any real numbers. This function is defined every time the denominator is not zero, but since there's no value of \( x \), \( y \), and \( z \) that makes it zero, the domain is unrestricted for all real numbers.

Rational Functions

Rational functions are a type of function that can be expressed as the ratio of two polynomials. A simple form of rational functions looks like \( r(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials. For our function \( Q(x, y, z) = \frac{10}{1+x^2+y^2+4z^2} \), the numerator is 10, which is a constant polynomial, and the denominator is the polynomial \( 1+x^2+y^2+4z^2 \).

Rational functions are important because:

• They are widely used in mathematical modeling and analysis in various fields - from engineering to economics.
• Their domains often exclude points where the denominator equals zero, as division by zero is undefined.

In this function, \(1 + x^2 + y^2 + 4z^2\) equals zero is not possible, due to the sums being non-negative and starting with 1. Hence, there are no restrictions on \( x \), \( y \), and \( z \), maintaining an unrestricted domain.

Real Numbers

Real numbers are fundamental in mathematics, encompassing all numbers that can be found on the continuous number line. They include rational numbers (like fractions and integers) and irrational numbers (numbers that can't be expressed as a fraction, such as \( \pi \) or \( \sqrt{2} \)).

In the context of functions:

• The domain often consists of real numbers since they provide a complete and continuous set of inputs.
• They are crucial in defining spaces like \( \mathbb{R}^n \), which in our example is \( \mathbb{R}^3 \) for the inputs \( (x, y, z) \).

Whenever you are dealing with multivariables, the notion of real numbers helps define the extent to which each variable can vary. In our function \( Q(x, y, z) \), all three variables are real numbers, indicating that there is no restriction within the real number system on their values.