Problem 52 - Calculus Early Transcendentals

Question

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y-, x z^{-}\), and \(y z\) -traces, when they exist. c. Sketch a graph of the surface. $$z=\frac{x^{2}}{4}+\frac{y^{2}}{9}$$

Step-by-Step Solution

Answer

Question: Sketch the surface represented by the equation $$z = \frac{x^2}{4} + \frac{y^2}{9}$$ and describe the intercepts and traces. Answer: The surface represented by the given equation is an elliptical paraboloid opening upward with vertex at the origin (0, 0, 0). The intercepts with the coordinate axes are all at the origin (0, 0, 0). The xy-trace is an ellipse centered at the origin, the xz-trace is a parabola opening upward, and the yz-trace is a parabola opening upward as well.

Step 1: Find the intercepts with the coordinate axes

To find the intercepts with the coordinate axes, we need to set each of the other variables equal to zero and solve for the remaining variable. The given equation is: $$z=\frac{x^{2}}{4}+\frac{y^{2}}{9}$$ For the x-axis intercepts, set y = 0 and z = 0: $$0=\frac{x^{2}}{4}$$ Solving for x gives: $$x=0$$ For the y-axis intercepts, set x = 0 and z = 0: $$0=\frac{y^{2}}{9}$$ Solving for y gives: $$y=0$$ For the z-axis intercepts, set x = 0 and y = 0: $$z=0$$ The intercepts with the coordinate axes are (0,0,0).

Step 2: Find the equations of the xy-, xz-, and yz-traces

To find the equations of the xy-, xz-, and yz-traces, we need to set the missing variable equal to zero and simplify the equation. For the xy-trace, set z = 0: $$0 = \frac{x^2}{4} + \frac{y^2}{9}$$ This gives the equation of an ellipse centered at the origin. For the xz-trace, set y = 0: $$z = \frac{x^2}{4}$$ This is the equation of a parabola with vertex at the origin and opening upward. For the yz-trace, set x = 0: $$z = \frac{y^2}{9}$$ This is the equation of another parabola with vertex at the origin and opening upward.

Step 3: Sketch a graph of the surface

To sketch the graph of the surface, we will plot the intercepts and traces we found earlier. The surface is symmetric across the xz- and yz-planes. The xy-trace is an ellipse centered at the origin. The xz- and yz-trace are both parabolas, both with a vertex at point (0,0,0) and opening upwards. The graph of the surface is an elliptical paraboloid opening upward, with a vertex at (0, 0, 0).

Key Concepts

Intercepts

Intercepts are where a surface or a curve crosses the coordinate axes. For the equation \[ z = \frac{x^2}{4} + \frac{y^2}{9} \],we determine intercepts by setting the other variables to zero and solving for the remaining one.

• **x-axis intercept**: Set both \( y = 0 \) and \( z = 0 \). The equation becomes \[0 = \frac{x^2}{4}\], simplifying to \( x = 0 \).
• **y-axis intercept**: Set \( x = 0 \) and \( z = 0 \). The equation becomes \[0 = \frac{y^2}{9}\], simplifying to \( y = 0 \).
• **z-axis intercept**: Set \( x = 0 \) and \( y = 0 \). The equation simply gives \( z = 0 \).

Thus, the only intercept of the surface with the axes is at the origin (0, 0, 0). This point is significant because it serves as the center of the elliptical paraboloid.

Traces

Traces are the curves formed when a surface intersects a plane parallel to the coordinate planes. With the equation \[ z = \frac{x^2}{4} + \frac{y^2}{9} \],we can find the xy-, xz-, and yz-traces by setting one variable to zero.

• **xy-trace**: Set \( z = 0 \). The equation becomes \[ 0 = \frac{x^2}{4} + \frac{y^2}{9} \], which represents an ellipse centered at the origin.
• **xz-trace**: Set \( y = 0 \). The equation becomes \[ z = \frac{x^2}{4} \], which is a parabola with its vertex at the origin, opening upwards.
• **yz-trace**: Set \( x = 0 \). This gives the equation \[ z = \frac{y^2}{9} \], another parabola with its vertex at the origin, also opening upwards.

Traces are useful in graphing the surface because they reveal the shape and orientation of the surface in three-dimensional space. The xy-trace shows symmetry around the origin, while the xz- and yz-traces indicate the parabolic nature of the surface.

Elliptical Paraboloid

An elliptical paraboloid is a quadric surface that resembles a stretched paraboloid. In this case, the equation \[ z = \frac{x^2}{4} + \frac{y^2}{9} \] describes an elliptical paraboloid opening upwards.

• **Vertex**: The vertex of the paraboloid is located at the origin \( (0,0,0) \), which is also the only intercept point.
• **Shape**: The surface is formed by stretching parabolas (which appear in the xz- and yz-traces) and an ellipse (the xy-trace) around the vertex.
• **Functionality**: Useful in physics and engineering, elliptical paraboloids can model satellite dishes and similar structures due to their reflective properties.

The symmetry and curves of an elliptical paraboloid make it easy to identify if one understands how the traces exhibit parabolas and ellipses, all centered around a common focal point at the vertex.