Problem 10 - The Calculus with Analytic Geometry

Question

In Exercises 8 through 17, determine the region of continuity of \(f\) and draw a sketch showing as a shaded region in \(R^{2}\) the region of continuity of \(f\). f(x, y)=\frac{x^{2}+y^{2}}{\sqrt{9-x^{2}-y^{2}}}

Step-by-Step Solution

Answer

The region of continuity is inside the circle \(x^{2} + y^{2} < 9\).

Step 1 - Identify the Domain

To determine the region of continuity for the function \(f(x, y)=\frac{x^{2}+y^{2}}{\sqrt{9-x^{2}-y^{2}}}\), we first need to identify its domain. The denominator \(\sqrt{9-x^{2}-y^{2}}\) must be positive, so the expression inside the square root must be positive, \(9 - x^{2} - y^{2} > 0\).

Step 2 - Inequality Simplification

Re-write the inequality \(9 - x^{2} - y^{2} > 0\). This simplifies to \(x^{2} + y^{2} < 9\). This inequality represents a circular region in the \(xy\)-plane with radius 3 and centered at the origin (0,0).

Step 3 - Determine Continuity

The function \(f(x, y)\) is continuous where the denominator is non-zero and the square root is defined and positive. Because we determined that \(x^{2} + y^{2} < 9\), the function \(f(x, y)\) is continuous inside the circle of radius 3 centered at the origin.

Step 4 - Sketch the Region

Draw the region representing the circle with radius 3 centered at the origin on the \(xy\)-plane. The region of continuity is the interior of this circle. Shade this area to represent the region where \(f(x, y)\) is continuous.

Key Concepts

calculus

Calculus is a branch of mathematics that studies change and motion. It involves derivatives and integrals, which are used to solve complex problems in a range of fields. In this exercise, we focus on the concept of continuity.

A function is continuous if it is smooth and unbroken. To determine the continuity of a function like your provided multivariable function, we must ensure that there are no gaps, jumps, or undefined points. For instance, for the function given, we examined the denominator to ensure it is positive. This ensured that the values are not undefined and contributed to the continuity of the function across its domain.

analytic geometry

Analytic geometry, sometimes called coordinate geometry, is the study of geometry using a coordinate system. This method allows us to describe geometric shapes algebraically.

In this exercise, we translate the inequality for the function into a geometric shape. When we identified the domain for our function, we rewrote the inequality to find that it describes a circle with a radius of 3 and center at the origin. You can visualize geometric problems, which makes understanding the region of continuity much easier.

multivariable functions

Multivariable functions involve more than one variable, an extension from single-variable functions. Calculating and understanding these functions require evaluating how function values change with these variables.

For instance, the function provided, \( f(x, y)=\frac{x^{2}+y^{2}}{\sqrt{9-x^{2}-y^{2}}}\), is dependent on both x and y. To find the region of continuity for this function, we examined how it behaves as both x and y change. We found that the function is continuous within a circle of radius 3, centered at the origin, encompassing all points where \(x^{2} + y^{2} < 9\).

domain and range

The domain of a function is the set of all possible input values (x, y in this case) for which the function is defined. The range is the set of all possible output values.

In this exercise, we determined that the domain is a circle centered at the origin with a radius of 3. This results from ensuring the denominator of our function stays positive. The domain translated to geometric terms helps visualize where our function maintains continuity. By understanding the domain, we ensure the function behaves correctly within this region.