Problem 28 - Calculus an Applied Approach

Question

Describe the region \(R\) in the \(x y\) -plane that corresponds to the domain of the function. $$ f(x, y)=y e^{1 / x} $$

Step-by-Step Solution

Answer

The region \(R\) in the \(x y\) -plane that corresponds to the domain of the function \(f(x, y)=y e^{1 / x}\) is the entire plane except for the y-axis where x = 0.

Step 1: Identify the function components

First, look at the given function \(f(x, y)=y e^{1 / x}\). The function has two components: \(y\) and \(e^{1 / x}\). Both \(y\) and \(e^{1 / x}\) are defined for all real numbers. However, remember that the exponential function \(e^{1 / x}\) is not defined for \(x=0\).

Step 2: Define the domain of the function

The domain of this function is the set of all values of \(x\) and \(y\) that make the function defined. Based on the observations from Step 1, the variable \(y\) can take any real value, while \(x\) can take any real value except 0. Hence, \(\text{Domain}(f) = (x, y) : x \in \mathbb{R} - \{0\}, y \in \mathbb{R}\)

Step 3: Define the region \(R\)

The region \(R\) in the \(x y\) -plane corresponding to the domain of the function is called the Cartesian product of the domains of the \(x\) and \(y\) components. Based on Step 2, this is all pairs \( (x, y) \) such that \(x\) is any real number except 0 and \(y\) is any real number. This is the entire plane, excluding the y-axis, where \(x=0\).

Key Concepts

Domain of a Function

The domain of a function defines the set of all possible input values for which the function is defined. Consider a function \(f(x, y)\). To find the domain, we look at each component of the function and check its restrictions on values.

• The function in question is \(f(x, y) = y e^{1/x}\).
• The exponential component \(e^{1/x}\) is undefined if \(x = 0\), because division by zero is not permissible.
• Meanwhile, \(y\) can take any real number.

Thus, combining these observations, the domain is all \(x\) and \(y\) such that \(x eq 0\) and \(y\) is any real number. In set notation, this domain is expressed as \((x, y) : x \in \mathbb{R} - \{0\}, y \in \mathbb{R}\). This means we permit all values for \(y\) and all non-zero values for \(x\).

The domain is important because it defines where the function can actually operate. For each point in the domain, the function returns a valid output, ensuring that we only consider inputs that do not lead to undefined behavior.

Exponential Function

An exponential function is a mathematical function of the form \(f(x) = a^x\), where \(a\) is a constant. Exponential functions have unique features, and they often appear in scientific applications like population growth and radioactive decay. In our function \(f(x, y) = y e^{1/x}\), the expression \(e^{1/x}\) is key.

• Here, \(e\) is known as Euler's number, approximately \(2.718\), which is the base of the natural logarithm.
• The function \(e^{1/x}\) grows rapidly as \(1/x\) becomes positive and decreases quickly as \(1/x\) becomes negative.
• Because the exponential part is undefined for \(x = 0\), this dictates the exclusion of zero in the domain of \(f(x, y)\).

Exponential functions are powerful tools in calculus and other areas because of their continuous growth and decay properties. Understanding their behavior is crucial when analyzing complex functions like \(f(x, y)\).

Overall, they often indicate situations where change occurs at a constantly increasing or decreasing rate, such as our function's behavior when approaching \(x=0\).

Cartesian Plane

The Cartesian plane is a two-dimensional plane defined by the intersection of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). It allows us to visually represent algebraic equations and functions. Each point on this plane is identified by a pair of coordinates \((x, y)\).

• In our function \(f(x, y) = y e^{1/x}\), the Cartesian plane helps map the domain \((x, y) : x \in \mathbb{R} - \{0\}, y \in \mathbb{R}\).
• All real number combinations except those where \(x = 0\) make up this domain region.
• The plane makes it easier to visualize which points are valid inputs by excluding the line where \(x = 0\).

With the Cartesian plane, complex relationships between variables can be easily visualized and conceptualized. This is particularly useful for multivariable calculus, where understanding the interaction between different variables is key.

By using the Cartesian coordinates, one can better understand how different inputs relate to their respective outputs in the realm of mathematical functions.