Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

Describe the largest set \(S\) on which it is correct to say that \(f\) is continuous. \(f(x, y)=\left(4-x^{2}-y^{2}\right)^{-1 / 2}\)

Step-by-Step Solution

Answer

The largest set \( S \) is the open disk \( \{ (x, y) : x^2 + y^2 < 4 \} \).

Step 1: Identify Conditions for Continuity

For a function to be continuous, it must be defined at that point or region. The function \( f(x, y) = (4 - x^2 - y^2)^{-1/2} \) is defined only when the expression inside the square root is positive. Thus, \( 4 - x^2 - y^2 > 0 \).

Step 2: Rewrite the Inequality

The inequality \( 4 - x^2 - y^2 > 0 \) can be rewritten as \( x^2 + y^2 < 4 \). This inequality describes a region in the \(xy\)-plane, specifically the interior of a circle centered at the origin with a radius of 2.

Step 3: Define the Set S

The set \( S \) on which \( f(x, y) \) is continuous is the set of all points \( (x,y) \) such that \( x^2 + y^2 < 4 \). This is because for these points, the function \( f(x, y) \) is defined and therefore continuous.

Step 4: Describe the Largest Set S

The largest set \( S \) is the open disk centered at the origin with a radius less than 2. Formally, this can be described as \( S = \{ (x, y) \,|\, x^2 + y^2 < 4 \} \).

Key Concepts

Inequalities in Calculus

In calculus, inequalities help us understand the regions in which certain functions behave or are defined. An inequality often indicates where a function is valid or continuous. For a function like \( f(x, y) = (4 - x^2 - y^2)^{-1/2} \), it is critical that the expression under the square root remains positive to ensure the function is well-defined and continuous. Therefore, we analyze the inequality \( 4 - x^2 - y^2 > 0 \).

This inequality tells us the set of points \( (x, y) \) where this function can operate correctly. Solving the inequality gives \( x^2 + y^2 < 4 \), which provides a clear criterion for continuity. While solving, always remember:

• To find where the expression is positive, place the expression on one side and zero on the other.
• Ensure each step maintains the proper mathematical logic to avoid errors.
• These inequalities lead to geometric insights, showing the region of validity such as circles or disks.

Open Disk in Cartesian Plane

The concept of an open disk in the Cartesian plane is pivotal in understanding where functions like \( f(x, y) = (4 - x^2 - y^2)^{-1/2} \) are continuous. An open disk is defined by all points \( (x, y) \) that satisfy a certain inequality like \( x^2 + y^2 < r^2 \).

In our context, the inequality \( x^2 + y^2 < 4 \) describes an open disk centered at the origin with a radius of 2. This means that all points within, but not on, the boundary defined by the circle \( x^2 + y^2 = 4 \) are part of this open disk.

Here are some characteristics of the open disk:

• All points inside the boundary are included, but the boundary itself is excluded.
• It allows us to describe the largest area where the function can be continuous.
• Visualizing this in the \(xy\)-plane helps in understanding function limits and regions of continuity.

Conditions for Function Definition

For a function to be defined across a certain region, specific conditions must be met. With \( f(x, y) = (4 - x^2 - y^2)^{-1/2} \), the function is defined only when the expression inside the square root is positive. This yields the condition \( 4 - x^2 - y^2 > 0 \), which implies that the domain of definition does not include points on the boundary \((x^2 + y^2 = 4)\), making it essential that the set of points considered must satisfy \( x^2 + y^2 < 4 \).

When determining where a function is defined, bear in mind:

• Any terms under square roots need to be strictly positive.
• Denominators must not be zero to ensure the function remains meaningful.
• Checking these conditions is critical in ascertaining both the continuity and validity of the function in a particular domain.

Understanding these conditions ensures the correct interpretation of functions and their continuity across different regions.