Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ z=\sqrt{x^{2}+y^{2}+1} $$

Step-by-Step Solution

Answer

The graph is a circular paraboloid opening upwards with the vertex at \((0, 0, 1)\).

Step 1: Identify the form of the equation

The given equation is \( z = \sqrt{x^2 + y^2 + 1} \). The format suggests that this is an equation representing a surface in three-dimensional space. It resembles the form \( z = \sqrt{x^2 + y^2} + 1 \), which is related to a type of paraboloid.

Step 2: Rewrite the equation

Modify the given equation by squaring both sides to eliminate the square root. We obtain \( z^2 = x^2 + y^2 + 1 \), which represents the surface of a paraboloid with its axis along the z-axis.

Step 3: Analyze the equation's structure

The equation \( z^2 = x^2 + y^2 + 1 \) indicates that this is a circular paraboloid opening upwards. The term \(+1\) suggests that the vertex of the paraboloid is at \(z=1\) when \(x=0\) and \(y=0\).

Step 4: Sketch the graph

Visualize the paraboloid by noting that it is rotationally symmetric about the z-axis because \(x^2 + y^2\) forms circles centered at the origin in the xy-plane for constant values of \(z\). Plot the vertex at points \((0, 0, 1)\) and draw upwards circular layers expanding as \(z\) increases.

Key Concepts

Paraboloid

A paraboloid is a three-dimensional surface that can be thought of as a 'bowl' shape. This shape is defined by its symmetry around an axis, and it can open upwards or downwards. The form similar to the standard parabola, but it involves two spatial directions (usually x and y), and an additional direction (z) which determines the height.

In the equation \(z^2 = x^2 + y^2 + 1\), we identify this as a circular paraboloid. The key feature of a circular paraboloid is that it is symmetric:

• Symmetric around the z-axis.
• Circular cross-sections parallel to the xy-plane.
• Vertex located at the point where all variables equal zero.

In this context, the vertex is shifted up to \(z = 1\) due to the +1, unlike the standard \(z^2 = x^2 + y^2\), where the vertex is at the origin. Understanding this translation in the z-direction is crucial to sketching these surfaces accurately.

Graphing 3D Equations

Graphing 3D equations can initially seem daunting, but breaking it down into simpler steps makes it easier. Let's go through this:

• Identify the Equation Type: Determine if the equation resembles any known surface, like a paraboloid, ellipsoid, etc. This enables you to predict the overall shape.
• Transform the Equation (if necessary): Sometimes rewriting the equation, like squaring both sides, makes its type more apparent.
• Note the Symmetries: Understand the symmetries involved - if the surface is rotationally symmetric about an axis, this simplifies the sketching process.
• Consider Cross-Sections: Planes parallel to one of the coordinates can create simplifiable slices, often circular or elliptical, helping visualize the full 3D shape.

To visualize and sketch, begin with recognizing features such as symmetry, intercepts, and any translations, all of which can transform a relatively abstract equation into a comprehensible shape.

3D Coordinate System

The 3D coordinate system is foundational for understanding and working with 3D surfaces. It's formed by three axes:

• x-axis: This horizontal axis often represents horizontal positions or distances.
• y-axis: Another horizontal axis, usually perpendicular to the x-axis, representing another horizontal dimension.
• z-axis: This vertical axis provides the third dimension, adding the concept of depth or height.

In three-dimensional graphing, every point reflects a location defined by three coordinates \((x, y, z)\). Surfaces, like paraboloids, occupy space within this framework.
In practice, understanding shifts and rotations within this system is vital for sketching and visualizing graphs. For example, a shift by +1 in the z-direction places the surface’s vertex above the xy-plane, as seen in the vertex of \(z^2 = x^2 + y^2 + 1\). Mastery of this system allows for accurate interpretation and representation of complex equations and their corresponding surfaces.