Problem 21 - University Calculus: Early Transcendentals

Question

Sketch the surfaces PARABOLOIDS AND CONES $$z=x^{2}+4 y^{2}$$

Step-by-Step Solution

Answer

The surface is an upward-opening elliptical paraboloid.

Step 1: Identify Surface Type

The given equation is of the form \( z = x^2 + 4y^2 \). This is an equation of an elliptical paraboloid. This specific surface is a type of quadratic surface with one variable squared in terms of both \(x\) and \(y\), and \(z\) is a function of these squares.

Step 2: Analyze Cross Sections

Consider cross-sections parallel to the \(xy\)-plane by fixing \(z = c\). This gives us the equation \(c = x^2 + 4y^2\), indicating an ellipse in the \(xy\)-plane for each positive \(c\). The equation's form confirms that the cross sections are ellipses.

Step 3: Examine Axis Intersections

Find the intersections with the axes. When \(x = 0\), the equation becomes \(z = 4y^2\), a parabola in the \(yz\)-plane. When \(y = 0\), the equation becomes \(z = x^2\), another parabola in the \(xz\)-plane.

Step 4: Describe Surface Orientation

The surface opens upward because the coefficients of \(x^2\) and \(y^2\) are positive, meaning \(z\) increases as either \(x\) or \(y\) move away from the origin. The shape becomes wider along the \(y\)-axis compared to the \(x\)-axis due to the coefficient \(4\) in front of \(y^2\).

Key Concepts

Quadratic Surfaces

Quadratic surfaces are a fascinating category of three-dimensional shapes described by second-degree polynomial equations. These surfaces can have various forms, such as ellipsoids, hyperboloids, and paraboloids. Each form has its unique characteristics and equations that define them. One notable type of quadratic surface is the elliptical paraboloid, which is the shape we have when dealing with the equation \( z = x^2 + 4y^2 \).

• Ellipsoids have equations like \( x^2/a^2 + y^2/b^2 + z^2/c^2 = 1 \).
• Hyperboloids come in one or two sheet varieties, featuring more complex equations.
• Elliptical paraboloids, as with our example, have the general structure \( z = ax^2 + by^2 \).

Quadratic surfaces are incredibly useful in fields such as physics and architecture, where they describe phenomena like gravitational fields or the optimal design of structures.

Cross Sections

Understanding cross sections is crucial for visualizing complex three-dimensional objects like curved surfaces. A cross section is essentially a 'slice' of a 3D shape that helps us understand its structure.
When we analyze cross sections of an elliptical paraboloid (e.g., \( z = x^2 + 4y^2 \)), we usually fix one variable to see the two-dimensional shape that emerges:

• By setting \( z = c \), we are effectively slicing parallel to the \(xy\)-plane.
• This transforms the equation into \( c = x^2 + 4y^2 \), which is the formula for an ellipse, confirming that each cross section parallel to this plane is elliptical.

This method helps in breaking down and understanding how a surface behaves and its spatial symmetry, providing insights into the surface's behavior and properties.

Parabola in yz-plane

To understand the parabola's form in the \(yz\)-plane, we set \( x = 0 \) in the elliptical paraboloid equation \( z = x^2 + 4y^2 \). This simplifies the equation to \( z = 4y^2 \), highlighting a parabolic curve.
This parabolic symmetry shows that the surface has a consistent upward opening in the \(yz\)-plane. Some points to remember:

• The vertex of this parabola is at the origin (0,0).
• The parabola opens upwards along the \(z\)-axis because of the positive coefficient 4, which leads to steeper slopes away from the vertex.

Visualizing how surfaces intersect the plane helps one grasp the vertical curvature and how this surface progresses along different planes.

Parabola in xz-plane

Similarly, examining the parabola in the \(xz\)-plane involves setting \( y = 0 \) in the given equation \( z = x^2 + 4y^2 \). This reduces it to \( z = x^2 \), another recognizable parabolic shape.
Here are key details related to this plane:

• The resulting parabola, \( z = x^2 \), has its vertex at the origin (0,0).
• It opens upwards in line with the \(z\)-axis due to the positive coefficient 1 before \(x^2\), indicating a symmetrical spread across this plane.

Understanding these characteristics helps in picturing how this elliptical paraboloid tends to rise infinitely as it moves away from the origin in any direction along the \( x \) or \( y \) axes, always maintaining its open canopy-like structure.