Problem 24 - 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. \(z=1-y, \quad\) no restriction on \(x\) b. \(z=y^{3}, \quad x=2\)

Step-by-Step Solution

Answer

(a) Plane \( z = 1 - y \), parallel to \( x \) axis; (b) Curve \( z = y^3 \) at \( x = 2 \).

Step 1: Understand the equations and inequalities

Read the given equations carefully. For part (a), you have the equation \( z = 1-y \) with no restriction on \( x \). For part (b), both \( z = y^3 \) and \( x = 2 \).

Step 2: Interpret equation for part (a)

The equation \( z = 1-y \) represents a plane in three-dimensional space. Since there is no restriction on \( x \), this plane is parallel to the \( x \)-axis and spans infinitely along the \( x \)-axis. Any point \((x, y, z)\) that satisfies this equation lies on the plane.

Step 3: Define the set of points for part (a)

The set of points that satisfy \( z = 1 - y \) with no restriction on \( x \) is an infinite plane parallel to the \( x \)-axis. You can choose any value for \( x \), and the corresponding \( y \) and \( z \) follow the equation \( z = 1 - y \).

Step 4: Interpret equation for part (b)

The equations \( z = y^3 \) and \( x = 2 \) describe a curve in three-dimensional space. The equation \( x = 2 \) fixes \( x \) along the plane where \( x = 2 \), and \( z = y^3 \) describes the relationship between \( z \) and \( y \).

Step 5: Define the set of points for part (b)

The set of points for part (b) forms a curve where \( z = y^3 \) on the plane at \( x = 2 \). Thus, any point can be represented as \((2, y, y^3)\) in 3D space, tracing out a curve as \( y \) varies.

Key Concepts

Understanding Planes in 3D Geometry

Planes in three-dimensional geometry are flat surfaces that extend infinitely in two dimensions. A plane is determined by equations like the one given in the problem, such as \( z = 1 - y \). This specific equation represents a plane parallel to the x-axis, indicating that there is no restriction on how far the plane extends in that direction.

Here are some important points about planes in 3D geometry:

• A plane is determined by a single equation involving three coordinates \( x \), \( y \), and \( z \).
• The equation \( z = 1 - y \) shows that for every \( y \) value, the corresponding \( z \) is uniquely defined, while \( x \) can be any value.
• This characteristic creates a plane that covers all points \((x, y, z)\) fulfilling the given relationship.

Identifying a plane in space helps visualize and understand the spatial relationships and the infinite nature of some geometrical surfaces in three dimensions.

Exploring Curves in a 3D Space

Curves in three-dimensional geometry offer a fascinating perspective as they differ significantly from lines or planes. In this exercise, the curve is defined by \( z = y^3 \) and \( x = 2 \), effectively pinning the curve along a specific plane where \( x \) remains constant. This creates a unique type of line that has curvature across the y-z dimensions but is restricted in the x-dimension.

Let's break down the concept of curves in 3D:

• The fixed value \( x = 2 \) implies the curve doesn't extend along the x-axis, and it remains on the plane where \( x = 2 \).
• On this plane, \( z = y^3 \) governs the behavior of the curve, allowing it to take on various forms as \( y \) changes.
• The expression \((2, y, y^3)\) describes every point along this curve, showcasing how \( z \) evolves with \( y \).

Understanding curves is crucial as they represent paths, trajectories or shapes in 3D space, playing a significant role in mathematical modeling and physical world simulations.

Navigating Coordinate Systems

Coordinate systems are fundamental in describing positions and movements in space, whether in two or three dimensions. In 3D geometry, coordinate systems help locate points such as those given in the problem: \( (x, y, z) \). Each point is defined by its position along the x, y, and z axes, akin to a unique address in space.

Here are essential concepts about coordinate systems:

• In a 3D Cartesian coordinate system, each point is represented by three values \((x, y, z)\) denoting their respective positions on the x, y, and z axes.
• The axes themselves are perpendicular to each other, creating an orthogonal grid that describes every point in space unambiguously.
• Understanding equations in terms of coordinates allows us to interpret spatial relationships, as seen with the planes and curves discussed earlier.

Mastery of coordinate systems is vital for studying geometry and applying mathematical principles to real-world problems, including physics and engineering applications.