Problem 35 - Calculus Early Transcendentals

Question

\(29-38\) Determine the set of points at which the function is continuous. $$f(x, y, z)=\frac{\sqrt{y}}{x^{2}-y^{2}+z^{2}}$$

Step-by-Step Solution

Answer

Continuous when \(x^2 - y^2 + z^2 \neq 0\) and \(y \geq 0\).

Step 1: Identify Denominator Restrictions

The function is \[f(x, y, z)=\frac{\sqrt{y}}{x^{2}-y^{2}+z^{2}}.\]The denominator term is \(x^2 - y^2 + z^2\). This term cannot be equal to zero, as division by zero is undefined. Hence, the function is discontinuous where the denominator equals zero. Identify the restriction:\[x^{2} - y^{2} + z^{2} eq 0.\]

Step 2: Determine Numerator Restrictions

The numerator term is \(\sqrt{y}\). The square root function is defined only for non-negative values, so \(y \geq 0\) is a restriction. Thus, the function is undefined for negative values of \(y\).

Step 3: Define the Set of Continuous Points

Combine the restrictions from Steps 1 and 2 to identify the set where the function is continuous:- \(x^{2} - y^{2} + z^{2} eq 0\)- \(y \geq 0\)The function \(f(x, y, z)\) is continuous for points \((x, y, z)\) that satisfy both conditions.

Key Concepts

Continuity

Understanding continuity in multivariable calculus is crucial because it tells us where a function behaves predictably — where it doesn't "jump" or become undefined. For a function of multiple variables like \(f(x, y, z) = \frac{\sqrt{y}}{x^2 - y^2 + z^2}\), continuity relies on both the numerator and the denominator behaving as expected. This means:

• The denominator should not be zero because division by zero is undefined. If it were zero, we would not be able to determine a value for \(f(x, y, z)\).
• The numerator \(\sqrt{y}\) involves a square root, which is only defined when \(y\) is non-negative. When \(y < 0\), the square root becomes imaginary, making the function discontinuous.

By identifying and considering these aspects, we can effectively analyze where such a function is continuous or not. The function is continuous precisely where these potentially troublesome elements of denominator zero and negative square roots are not present.

Functions of Multiple Variables

In multivariable calculus, functions can express relationships among more than one variable. A function like \(f(x, y, z) = \frac{\sqrt{y}}{x^2 - y^2 + z^2}\) involves three variables: \(x\), \(y\), and \(z\). Analyzing such functions requires considering relationships in a three-dimensional space. Here, the output depends upon all three inputs, contributing to complex relationships.The function’s behavior changes based on the value combination of \(x\), \(y\), and \(z\). While examining this function:

• Numerator Influence: As the numerator \(\sqrt{y}\) requires non-negative \(y\), it influences the valid domain.
• Denominator Influence: The expression \(x^2 - y^2 + z^2\) cannot equal zero, also affecting which \((x, y, z)\) values keep the function defined.

Understanding how different values impact the function’s continuity and defining valid input points is key to working with such multivariable functions.

Restrictions on Domains

Discovering restrictions on a domain involves finding values that make the function undefined or behave unpredictably. For our given function, \(f(x, y, z) = \frac{\sqrt{y}}{x^2 - y^2 + z^2}\), carefully examining such restrictions is an essential step in analysis.With this function, we observe several restrictions:

• The denominator \(x^2 - y^2 + z^2\) cannot be zero. If it becomes zero, the function doesn’t produce a real number output.
• The numerator involves the square root of \(y\), imposing \(y \geq 0\) because negative square roots are not real numbers in standard calculus contexts.

To express the domain properly, we consider all these restrictions, painting a clear picture of where the function is real and defined. Restrictions help us understand where the function is continuous by outlining where it is not. These conditions, when not met, cause discontinuities, thus providing clear boundaries for function analysis and realistic problem-solving in calculus.