Problem 15 - Thomas Calculus

Question

In Exercises \(1-16,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$y=x^{2}, \quad z=0$$

Step-by-Step Solution

Answer

A parabolic curve in the xy-plane (the plane where \(z = 0\)).

Step 1: Understand the First Equation

The equation \(y = x^2\) represents a parabola in the xy-plane. This parabola opens upwards and has its vertex at the origin (0,0). It includes all points \((x, y)\) where \(y\) is the square of \(x\).

Step 2: Understand the Second Equation

The equation \(z = 0\) implies that we are looking for points on the xy-plane, where the z-coordinate is zero. This essentially restricts the set of points to the plane where \(z = 0\).

Step 3: Combine the Equations

Combining the equations \(y = x^2\) and \(z = 0\), we limit the points to those on the parabola \(y = x^2\) which lie in the xy-plane (since \(z = 0\)).

Step 4: Describe the Geometric Shape

This set of points forms a 2D parabolic curve within the xy-plane. Since \(z = 0\), this curve does not extend into other planes and remains on the xy-plane, looking like a standard parabola.

Key Concepts

Parabola

Let's start by understanding what a parabola is. In mathematics, a parabola is a U-shaped curve that is symmetric. The simplest form of a parabola is represented by the equation \( y = x^2 \). Here, the curve opens upwards, and the lowest point, called the vertex, is located at the origin, (0,0).
The key characteristics of a parabola:

• It is symmetric, meaning if you fold it along the line that goes through the vertex, both sides will match perfectly.
• The vertex is the point where the curve turns.
• It contains all points \((x, y)\) that satisfy the equation \(y = x^2\); meaning for every \(x\) value, \(y\) is its square.

When visualizing a parabola, think of the way a ball thrown in the air travels. This path, under the influence of gravity, creates a parabolic shape. In a coordinate plane, it shows all possible points that lie on this particular curve.

XY-Plane

Next, we need to talk about what the xy-plane is. When you're working with 3D space, the xy-plane is essentially a flat surface where any point has a z-coordinate of 0. Imagine a piece of paper lying on a table, that is analogous to the xy-plane.
To understand it better, remember:

• The xy-plane is a 2D plane embedded in a 3D coordinate system.
• It includes all points \((x, y, 0)\), meaning while \(x\) and \(y\) can be any real numbers, \(z\) is always zero.
• This plane divides the 3D space into two halves, being part of the larger XYZ coordinate system where each axis (X, Y, Z) is perpendicular to each other.

The xy-plane acts like a foundation within a 3D space upon which you can map various shapes, like our parabola, projecting it into this flat surface.

Coordinate Geometry

Now, let's dive into how coordinate geometry helps us. This branch of geometry uses a coordinate system to describe the relationships between points, lines, and shapes. It's about understanding how mathematical equations translate into geometric figures.
In our context:

• Coordinate geometry allows us to depict shapes like a parabola using equations. For example, \( y = x^2 \) describes a parabola geometrically.
• By understanding the equations \(y = x^2\) and \(z = 0\), we are able to combine algebraic expressions with geometric visualization.
• It provides a visual framework to comprehend how these mathematical concepts play out in space.

In real-world applications, coordinate geometry can help with everything from designing graphics in computer programming to planning routes in GPS systems. Mastering it is essential for both understanding geometry in depth and applying mathematical principles to solve practical problems.