Problem 46 - 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 ).$$f(x, y, z)=2 x y z-3 x z+4 y z.$$

Step-by-Step Solution

Answer

Answer: The domain of the function $$f(x, y, z)$$ is all points in the three-dimensional space, represented as $$\{(x, y, z) | x, y, z \in \mathbb{R}\}$$.

Step 1: Identify the type of function

We have a function $$f(x, y, z) = 2xyz - 3xz + 4yz$$ which is a polynomial function in three variables.

Step 2: Determine if there are any restrictions on the variables

Since the function is a polynomial, there are no restrictions on the variables, i.e., we don't have any division by zero or negative numbers inside square roots or logarithms in this case.

Step 3: Define the domain

Since there are no restrictions on the variables and the function is defined for all real numbers, the domain of the function $$f(x, y, z)$$ is all points in the three-dimensional space. We can represent this mathematically as: $$\{(x, y, z) | x, y, z \in \mathbb{R}\}$$

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving sums and products of variables raised to whole number powers. The basic structure of a polynomial involves coefficients, variables, and exponents, all summed up together. Whether in one variable, like \(x^2 + 3x + 2\), or in multiple variables, such as \(2xyz - 3xz + 4yz\), the principle remains the same.Here are some key characteristics of polynomial functions:

• Simple structure: They consist only of terms where the variables are raised to non-negative integer powers, making them easy to work with.
• Smooth curves: Their graphs are continuous and smooth, with no breaks or corners.
• Derivatives: They are easy to differentiate, which is crucial for calculus applications.
• Solving: Polynomial equations can be solved using several methods such as factoring, using the quadratic formula, or employing numerical techniques like synthetic division.

Polynomial functions are fundamental in mathematics because they set the groundwork for understanding more complex functions.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions with more than one variable. While single-variable calculus primarily deals with changes along a line, multivariable calculus explores rates of change in higher dimensions. This is crucial for understanding systems and phenomena in physics, engineering, and other sciences that involve more than one variable. Key aspects of multivariable calculus include:

• Dimensions: Functions can have multiple inputs and outputs, often mapped in multi-dimensional space.
• Partial Derivatives: These represent the rate of change of a function concerning one variable while keeping others constant.
• Gradient Vectors: These vectors point in the direction of the steepest increase of the function and are composed of partial derivatives.
• Applications: Used widely in optimization, motion studies, and other fields involving complex systems.

Multivariable calculus provides the tools to understand, describe, and model behaviors in a multi-dimensional world.

Domain of a Function

The domain of a function is the complete set of possible input values (or "independent variables") for which the function is defined. It essentially tells us where the function can "live" without running into any issues such as division by zero or taking square roots of negative numbers.To identify the domain for different types of functions:

• Polynomial functions: These generally have a domain of all real numbers because they are defined for any value. For example, the polynomial function \(f(x, y, z) = 2xyz - 3xz + 4yz\) has no restrictions and hence, its domain is all of \((x, y, z) \in \mathbb{R}^3\).
• Rational functions: Exclude values that make the denominator zero.
• Radical functions: Ensure the radicand is non-negative, if applicable.
• Logarithmic functions: Ensure the argument of the logarithm is positive.

Understanding the domain is crucial for grasping the full scope of a function and where it applies.