Problem 402 - Calculus Volume 3

Question

For the following exercises, find the largest interval of continuity for the function. \(f(x, y)=x^{3} \sin ^{-1}(y)\)

Step-by-Step Solution

Answer

The largest interval of continuity is when \(-1 \leq y \leq 1\) for all real \(x\).

Step 1: Identify Continuity Conditions for Each Component

The function involves two main components: \(x^3\) and \(\sin^{-1}(y)\). \(x^3\) is a polynomial function, which is continuous for all real number \(x\). \(\sin^{-1}(y)\), also known as \(\arcsin(y)\), is continuous for \(y\) values in \([-1, 1]\). We must consider these conditions for continuity.

Step 2: Combine Conditions for Continuity

For the function \(f(x, y) = x^3 \sin^{-1}(y)\) to be continuous, \(x\) must be any real number and \(y\) must satisfy the condition \(-1 \leq y \leq 1\). Therefore, the largest interval of continuity for the function in terms of \(y\) is within the closed interval from \(-1\) to \(1\). There is no restriction on \(x\), which means it can be any real number.

Key Concepts

Largest Interval of Continuity in Multivariable Functions

Continuity is a fundamental concept in mathematics that determines where a function behaves without "breaks" or interruptions. In multivariable functions, this involves examining each variable separately to identify where the function remains continuous.

When we talk about the largest interval of continuity, we focus on the range of values that keeps the function continuous. For the function \(f(x, y) = x^3 \sin^{-1}(y)\), we must analyze both \(x^3\) and \(\sin^{-1}(y)\).

• \(x^3\) - This is a polynomial function, continuous for all real numbers. It does not impose any restrictions.
• \(\sin^{-1}(y)\) - Known as the arcsine function, it is only continuous over the closed interval \([-1, 1]\).

Thus, the largest interval of continuity is determined by the most restrictive component. Here, the arcsine function limits \(y\) to \([-1, 1]\), securing the continuous nature of the given function for all real \(x\).

Understanding the Arcsin Function

The arcsin function, or inverse sine function, is essential in trigonometry. It gives the angle with a given sine value. Mathematically, it's denoted as \(\sin^{-1}(y)\) or \(\arcsin(y)\). This function maps its input \(y\) back to an angle between \(-\pi/2\) and \(\pi/2\).

In terms of continuity:

• Arcsin is not defined for any \(y\) less than \(-1\) or greater than \(1\).
• Within the interval \([-1, 1]\), every input yields a unique and continuous output.

When combined with other functions, the domains of those functions must accommodate this restriction. For \(f(x, y) = x^3 \sin^{-1}(y)\), this explains why the function's continuity condition requires \(-1 \leq y \leq 1\).

The Continuous Nature of Polynomial Functions

Polynomial functions, such as \(x^3\), are a bedrock in mathematics due to their predictability and comprehensive continuity across all real numbers. Defined as equations involving terms of varying powers of \(x\), polynomial functions are continuous and differentiable everywhere.

Key properties include:

• Smooth and unbroken curves, meaning no gaps or undefined points.
• They allow both positive and negative infinity as inputs without compromising continuity.

When analyzing multivariable functions, polynomial components often do not restrict the domain. Thus, for a function like \(f(x, y) = x^3 \sin^{-1}(y)\), \(x^3\)'s continuity plays supportive background role. The real storytelling happens with constraints put by other non-polynomial parts, such as the arcsine function discussed before.