Problem 19 - University Calculus: Early Transcendentals

Question

Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. a. \(x^{2}+y^{2}+z^{2} \leq 1\) b. \(x^{2}+y^{2}+z^{2}>1\)

Step-by-Step Solution

Answer

a) Points inside or on the unit sphere. b) Points outside the unit sphere.

Step 1: Understand the equation of a sphere

The expression \(x^2 + y^2 + z^2 = 1\) represents a sphere centered at the origin (0, 0, 0) with a radius of 1. Any set of coordinates (x, y, z) that satisfy this equation lie on the surface of this sphere.

Step 2: Interpret Set a

The inequality \(x^2 + y^2 + z^2 \leq 1\) describes all the points (x, y, z) inside or on the surface of the sphere with radius 1 centered at the origin. This is because the inequality allows for values less than or equal to 1, filling the interior of the sphere.

Step 3: Interpret Set b

The inequality \(x^2 + y^2 + z^2 > 1\) describes all the points (x, y, z) outside the sphere with radius 1 centered at the origin. Any point that satisfies this inequality is not on or inside the sphere, as they exceed the radius distance from the origin.

Key Concepts

Sphere

In analytic geometry, a sphere is a fascinating three-dimensional object. It's like a fully round ball in space. The equation of a sphere involves three coordinates:

• The x-coordinate, representing the horizontal distance from the origin.
• The y-coordinate, representing the vertical distance.
• The z-coordinate, which accounts for the depth or height into the three-dimensional space.

The standard form for the equation of a sphere is given by \(x^2 + y^2 + z^2 = r^2\), where \(r\) is the radius of the sphere.
The equation tells us that every point on the surface of this sphere is equidistant from a central point called the center. In our case, the sphere is centered at the origin (0,0,0) with a radius of 1, implying the surface of the sphere possesses all points whose combined distance squared from the origin is exactly 1.
Understanding spheres is essential in various fields of study, including physics, engineering, and graphics, as it helps visualize and solve problems in three-dimensional space.

Inequalities

Inequalities in coordinate geometry allow us to identify regions of space rather than just surfaces or lines. They are essential for defining areas that extend beyond a simple boundary. Let's consider the inequalities provided in the problem:

• The inequality \(x^2 + y^2 + z^2 \leq 1\) defines all the points that are within or on the surface of the sphere.
• It includes points where their sum of squared distances from the origin is less than or equal to 1, hence covering the entire volume within the sphere.
• On the other hand, the inequality \(x^2 + y^2 + z^2 > 1\) defines points outside the sphere. These points have a combined squared distance from the origin greater than 1.

By using these inequalities, we can easily categorize points as being inside, on, or outside a sphere based on their coordinates.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a branch of mathematics that connects algebra and geometry. It involves using ordered pairs (or in three dimensions, ordered triples) to describe geometric figures. In the three-dimensional space, the coordinate system includes:

• The x-axis, or the horizontal line running left and right.
• The y-axis, which is vertically oriented.
• The z-axis, representing depth, adding the third dimension.

Together, these axes form a three-dimensional grid where every point in space can be represented by a set of three numbers \((x, y, z)\).
Coordinate geometry helps in expressing relationships between distances, angles, and shapes in space through algebraic equations. It is particularly useful in solving problems involving curves and surfaces, such as spheres, cones, and cylinders.
Understanding coordinate geometry is fundamental in various applications like navigation, computer graphics, and even in designing engineering projects where spatial reasoning is critical.