Problem 30 - Essential Calculus Early Transcendentals

Question

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. \(x^{2}+y^{2}+z^{2}>2 z\)

Step-by-Step Solution

Answer

The region is the exterior of a sphere, centered at (0, 0, 1) with radius 1.

Step 1: Simplify the Inequality

Start by simplifying the inequality: \(x^2 + y^2 + z^2 > 2z\).Move \(2z\) to the left side to obtain \(x^2 + y^2 + z^2 - 2z > 0\).

Step 2: Complete the Square

For the term \(z^2 - 2z\), complete the square to make it easier to interpret geometrically. Write \(z^2 - 2z\) as \((z-1)^2 - 1\). This transforms the inequality into: \(x^2 + y^2 + (z-1)^2 - 1 > 0\).

Step 3: Rearrange the Terms

Rearrange the terms to make the inequality more understandable:\(x^2 + y^2 + (z-1)^2 > 1\).

Step 4: Interpret the Geometric Meaning

Recognize that \(x^2 + y^2 + (z-1)^2 = 1\) describes a sphere of radius 1 centered at the point \((0, 0, 1)\) in \(\mathbb{R}^3\).Therefore, the inequality \(x^2 + y^2 + (z-1)^2 > 1\) represents the set of all points outside this sphere.

Key Concepts

Completing the Square

Completing the square is a method used in algebra to transform a quadratic expression into a perfect square trinomial, making the equation or inequality easier to analyze and solve. Let's take a closer look at how this technique works in the context of geometry. In the given exercise, we start with the inequality x^2 + y^2 + z^2 > 2z . By rearranging terms as x^2 + y^2 + z^2 - 2z > 0 , we focus on the z^2 - 2z portion. To complete the square, we add and subtract the same constant, turning z^2 - 2z into (z - 1)^2 - 1 .

• Find a constant to add and subtract that makes a perfect square: Here, 1 completes the square for z^2 - 2z .
• Rewrite z^2 - 2z as (z-1)^2 - 1 to make it manageable in geometric analysis.

Completing the square allows us to represent quadratic expressions in a form that is easily interpretable as geometric figures, like circles and spheres. In this case, it helps us better understand the inequality in terms of a geometric region.

Geometric Interpretation

The concept of geometric interpretation involves visualizing algebraic equations and inequalities as shapes or regions in space. Once we completed the square in the inequality, we arrived at a more familiar form: x^2 + y^2 + (z-1)^2 > 1 . This new form is the key to understanding the spatial relationship in three-dimensional space. Visualizing the inequality, x^2 + y^2 + (z-1)^2 = 1 represents a sphere centered at the point (0, 0, 1) with a radius of 1. The inequality x^2 + y^2 + (z-1)^2 > 1 describes the region outside this sphere, because it includes all points beyond the surface of the sphere. By rearranging mathematical expressions through completing the square, we can map these into intuitive geometric concepts.

• The center of the sphere derives from shifting the z-term as z' = z - 1 .
• The radius, 1, comes from the equation's equality part, denoting points exactly on the sphere.

Understanding equations geometrically makes it easier to relate algebraic ideas to spatial intuition, giving us a more comprehensive view of the problem.

Sphere in Mathematics

A sphere in mathematics is a perfectly round geometric object in three-dimensional space, analogous to a circle in two dimensions. The equation of a sphere centered at a point (h, k, l) with radius r is given by (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 . In the exercise example, this sphere is centered at (0, 0, 1) and has a radius of 1. The inequality x^2 + y^2 + (z-1)^2 > 1 denotes all points lying outside this sphere.

• The center (0, 0, 1) indicates a vertical shift along the z-axis.
• The inequality implies an open set outside of the sphere surface.

Mathematically, spheres are important due to their symmetry and simple representation, making them useful in various areas such as physics, computer graphics, and geometry. Spheres serve as fundamental objects in analyzing three-dimensional geometric relationships and help in visually representing spatial inequalities like the one in this exercise. They offer a tangible way to understand abstract mathematical concepts.