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 dome-shaped surface starting at \( z = 1 \) and opening upwards, known as a paraboloid.

Step 1: Identify the Equation Type

The given equation is in the form \( z = \sqrt{x^2 + y^2 + 1} \). This equation resembles the form of a paraboloid, albeit modified due to the square root. Typically, paraboloids take the form \( z = \sqrt{x^2 + y^2} \), but here we have \(+1\) inside the square root.

Step 2: Analyze the Shape

Since \( z = \sqrt{x^2 + y^2 + 1} \) always yields real, positive values (as square roots do), the surface remains positive along the z-axis. The term \( +1 \) results in an upward shift along the z-axis such that at \( x = 0 \) and \( y = 0 \), \( z = 1 \). This indicates a paraboloid that does not pass through the origin but rather starts from \( z = 1 \).

Step 3: Sketch the Graph

Visualize the graph starting from the point \( (0, 0, 1) \). As \( x \) and \( y \) move away from zero, \( z \) increases due to the added distance squared, forming an upward-opening surface resembling a paraboloid dome. At every point on the surface, the value of \( z \) is determined by moving out radially in every direction from \( (0, 0, 1) \), creating a smooth surface without sharp edges.

Key Concepts

Paraboloid

A paraboloid is a three-dimensional surface that can be generated by rotating a parabola around its axis. In simpler terms, it's like a bowl or a dome shape. Paraboloids come in two primary forms: elliptic and hyperbolic. This exercise focuses on an elliptic paraboloid, which has a convex, upward-opening shape.Imagine taking a standard parabola (like the graph of a quadratic function) and revolving it around its vertical axis. The result is a surface that extends infinitely in three dimensions.

• In affine form, a standard elliptic paraboloid equation is either:
  • Vertical: \( z = x^2 + y^2 \)
  • Horizontal: \( y = x^2 + z^2 \)

In our exercise, the equation \( z = \sqrt{x^2 + y^2 + 1} \) looks like a paraboloid but is modified by the square root, indicating a soft, rounded shape rather than a typical paraboloid's sharper contour.

Square Root Function

The square root function is crucial when working with graphs involving non-linear shapes. For any positive number \( a \), \( \sqrt{a} \) gives a value that, when multiplied by itself, returns \( a \). In graphs, this function causes data points to rise sharply and then flatten as they move along the axis.In the equation \( z = \sqrt{x^2 + y^2 + 1} \), the square root provides smoothing by changing angular curves into softer arcs. This rounding is important because it adjusts how the graph can be understood—transforming a sharp parabola into a gentler, dome-like surface in the 3D space.

• Square root functions are inherently non-negative; thus, \( z \) is always positive.
  • This means no negative values will exist along the \( z \)-axis, reinforcing an upward-facing graph.
• The structure \( +1 \) inside the function ensures the minimum starting point is \( z = 1 \) when \( x \) and \( y \) are zero.

Graphical Analysis

Graphical analysis helps us interpret and visualize mathematical functions into tangible shapes and forms. By effectively analyzing the given equation, we derive its geometry in a three-dimensional space, allowing us to "see" what the function represents.For the function \( z=\sqrt{x^2+y^2+1} \), the graph represents a 3D surface, which can be interpreted and analyzed like any visual data:

• The graph's vertex is at \( (0, 0, 1) \), indicating where the shape starts.
• As \( x \) and \( y \) increase, the graph opens upwards symmetrically in all directions, since the values under the square root grow larger.
• Understanding the transformation from 2D to 3D involves seeing how each point translates in each dimension.

The key aspect of graphical analysis here is identifying that each point on the surface emanates radially from the vertex, creating a continuous, smooth shape. This forms the basis for understanding more complex shapes and surfaces.

3D Coordinate System

The 3D coordinate system is a method for locating points in three-dimensional space using three coordinates: \( x \), \( y \), and \( z \).Unlike a 2D plane (which only uses \( x \) and \( y \)), the 3D system allows for depiction of volume and depth—exceedingly important in accurately representing figures like the given paraboloid.

• Points are represented as \( (x, y, z) \).
• The positive \( z \)-axis indicates movement away from the origin "upwards," while the \( x \)-axis and \( y \)-axis are your horizontal and vertical planes.
• This system enables mathematicians and scientists to categorize objects with an accurate depiction of dimensions.

Viewing this system helps in graphing functions like the equation \( z = \sqrt{x^2 + y^2 + 1} \), allowing analysts to visualize complex surfaces effectively. Understanding these attributes simplifies plotting multi-variable functions, which form the core of graphing in 3D space.