Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Describe the largest set \(S\) on which it is correct to say that \(f\) is continuous. \(f(x, y)=\ln \left(1-x^{2}-y^{2}\right)\)

Step-by-Step Solution

Answer

The largest set \(S\) is the open unit disk: \(\{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 < 1 \} \).

Step 1: Identify the Domain of the Natural Logarithm Function

The natural logarithm function, \(\ln(z)\), is defined only when \(z > 0\). For the function \(f(x, y) = \ln(1 - x^2 - y^2)\) to be continuous, the expression inside the logarithm must be positive: \(1 - x^2 - y^2 > 0\).

Step 2: Determine the Inequality Constraint

From the condition \(1 - x^2 - y^2 > 0\), we can rearrange this to \(x^2 + y^2 < 1\). This describes the set of points \((x, y)\) that lie inside the unit circle but not on its boundary.

Step 3: Define the Set S

The largest set \(S\) refers to the maximum domain where the function \(f(x, y)\) remains continuous. From the inequality \(x^2 + y^2 < 1\), the largest set \(S\) where \(f\) is continuous is the open unit disk, expressed as \(\{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 < 1 \} \).

Key Concepts

Domain of Functions

Understanding the domain of a function is crucial for determining where a function is defined and behaves correctly. In the context of multivariable functions like \( f(x, y) = \ln(1 - x^2 - y^2) \), identifying the domain involves recognizing the values of \( x \) and \( y \) that make the expression within the logarithm valid. Here, we rely on the property of the natural logarithm function, which is only defined for positive numbers. Therefore, the expression \( 1 - x^2 - y^2 \) must be greater than zero. This domain constraint ensures that the function outputs real numbers and is continuous without any breaks or undefined points.

To find the domain of \( f \), solve the inequality \( 1 - x^2 - y^2 > 0 \). Rearranging, we get \( x^2 + y^2 < 1 \), which describes an open set. In mathematical terms, this represents the points inside a circle with a radius of 1, centered at the origin in the coordinate plane. This is known as the open unit disk. The boundary of this circle, where \( x^2 + y^2 = 1 \), is excluded from the domain because the logarithm becomes zero, which is undefined.

Inequality Constraints

Inequality constraints determine limits within which a function operates correctly. They define permissible conditions for variables that lead to valid values for the function. For our function \( f(x, y) = \ln(1 - x^2 - y^2) \), the inequality constraint is derived from the logarithmic condition \( 1 - x^2 - y^2 > 0 \).

Let's break down the steps:

• The inequality \( x^2 + y^2 < 1 \) arises from rearranging the condition \( 1 - x^2 - y^2 > 0 \). This means that for any valid pair \((x, y)\), their squared sum must be less than 1.
• This condition translates geographically into a circle on the coordinate plane. It encompasses all the points within the circle, but not the boundary itself.

Understanding this inequality key is to recognizing which inputs \((x, y)\) are valid. Visualizing the constraint as an open unit disk helps assure the function's continuity by ensuring no divisors or boundaries destabilize its definition.

Natural Logarithm Functions

The natural logarithm, often written as \( \ln(z) \), is a special logarithmic function based on the constant \( e \), approximately equal to 2.718. This function is defined only for positive real numbers, which directly influences the conditions placed on any expression within its argument.

For the function in question, \( f(x, y) = \ln(1 - x^2 - y^2) \), understanding the natural logarithm's nature is essential. It dictates that its argument \( 1 - x^2 - y^2 \) must remain positive, as negative or zero values inside the logarithm are undefined.

Here's why positivity is crucial:

• Inside the logarithm, negative values would imply taking the log of a non-positive number, which is undefined in the realm of real numbers.
• Logarithms approaching zero as an argument lead to undefined results and can cause breaks in continuity.

In summary, paying attention to the natural logarithm properties ensures that \( f(x, y) \) is defined and continuous within its domain. Always remember, the natural logarithm demands positivity, setting a mathematical foundation to explore and solve problems involving log-based functions.