Problem 22 - Essential Calculus Early Transcendentals

Question

Reduce the equation to one of the standard forms, classify the surface, and sketch it. \(4 x^{2}-y+2 z^{2}=0\)

Step-by-Step Solution

Answer

The equation represents an elliptic paraboloid.

Step 1: Arrange the Equation

First, we will start by trying to rearrange the given equation into a recognizable form. The given equation is:\[ 4x^2 - y + 2z^2 = 0 \]Let's rearrange it to clarify the conic form:\[ 4x^2 + 2z^2 = y \]

Step 2: Identify the Standard Form

The equation \( 4x^2 + 2z^2 = y \) resembles the standard form for a quadric surface, specifically an elliptic paraboloid, which is normally seen as:\[ y = Ax^2 + Bz^2 \]Here, \( A = 4 \) and \( B = 2 \), which matches the quadratic form of an elliptic paraboloid.

Step 3: Understand the Properties

Recognize that the equation represents an elliptic paraboloid opening along the positive y-axis, because \( y \) is isolated on one side of the equation and is equal to a positive combination of squares \( 4x^2 + 2z^2 \).

Step 4: Sketch the Surface

When sketching an elliptic paraboloid given by the standard form \( y = 4x^2 + 2z^2 \), it looks like a three-dimensional upward facing bowl centered around the origin. Here, the cross-sections perpendicular to the y-axis are ellipses, as defined by \( 4x^2 + 2z^2 = \text{constant} \), which come closer as \( y \) increases.

Key Concepts

Quadric Surfaces

Quadric surfaces are a fascinating family of 3D shapes described by second-degree polynomial equations in three variables, typically denoted as \( x, y, \) and \( z \).
These surfaces come in several types, such as ellipsoids, hyperboloids, and paraboloids, depending on the specific equation form.
Our task here focuses on a special type known as an elliptic paraboloid.
The equation we started with, \( 4x^2 - y + 2z^2 = 0 \), belongs to this group. By arranging it to \( y = 4x^2 + 2z^2 \), it's clear we have an elliptic paraboloid.

Some essential features of quadric surfaces include:

• Symmetry: Many are symmetric around one or more axes, simplifying understanding and sketching.
• Standard forms: Recognizing standard forms helps classify the surface easily. For elliptic paraboloids, the form is typically \( y = Ax^2 + Bz^2 \).
• Dimensionality: These 3D shapes expand our understanding of geometry beyond flat surfaces.

Learning about quadric surfaces enriches our ability to navigate complex geometrical concepts.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. They include circles, ellipses, parabolas, and hyperbolas. To connect these with quadric surfaces, one can imagine how cutting through these surfaces might yield familiar conic sections.
For example:

• An elliptic paraboloid can show elliptical cross-sections if cut parallel to its axis of symmetry.
• If the cut is parallel to one of the planes \( x = 0 \) or \( z = 0 \), then the cross-section may appear as a parabola.

Understanding conic sections in relation to quadric surfaces provides a richer grasp of both 2D and 3D geometry.
In our exercise, since the surface described is an elliptic paraboloid, the cross-sections parallel to the \( xz \)-plane are parabolas \( y = ext{constant} \), giving insight into its structure and how it relates to simpler geometric figures.

3D Geometry

In 3D geometry, visualizing shapes in space is key. With quadric surfaces like elliptic paraboloids, this skill helps grasp their dimensions and orientations.
The equation \( y = 4x^2 + 2z^2 \) describes a 3D shape symmetrical around the y-axis, known as an elliptic paraboloid. Visualizing this shape involves imagining a bowl that opens upward and becomes wider as \( y \) increases.
In simpler terms, a slice taken through different planes, such as:\- A horizontal cut (at constant \( y \)): forms an ellipse.
- Vertical cuts parallel to the xy or yz planes: reveal parabolic shapes.

This 3D visualization not only provides a fuller picture of the shape's appearance, but it also strengthens one's capability to solve problems involving 3D spaces.
By mastering how to draw and interpret these complex forms, one builds a stronger foundation in geometry and enhances spatial intelligence.