Problem 38 - Calculus an Applied Approach

Question

Describe the traces of the surface in the given planes. $$ \text { Surface } \quad \text { Planes } $$ $$ y=x^{2}+z^{2} \quad x y \text { -plane }, y=1, y z \text { -plane } $$

Step-by-Step Solution

Answer

The traces of the surface \(y = x² + z²\) are: a parabola \(y = x²\) in the xy-plane, a circle with radius 1 in the plane \(y = 1\), and a parabola \(y = z²\) in the yz-plane.

Step 1: Find the trace in the xy-plane

To find the trace of the surface in the xy-plane, substitute \( z = 0 \) in the equation of the surface. This gives the equation: \( y = x² + 0 \) = \( x² \) which is a parabola.

Step 2: Find the trace in the plane y = 1

The plane \( y = 1 \) is parallel to the xz-plane. To find the trace of the surface in this plane, substitute \( y = 1 \) in the equation of the surface. This gives the equation: \( 1 = x² + z² \) which is a circle of radius 1 centered on the origin in the xz-plane.

Step 3: Find the trace in the yz-plane

Finally, to find the trace of the surface in the yz-plane, substitute \( x = 0 \) in the equation of the surface. This gives the equation: \( y = 0 + z² \) = \( z² \). This is a parabola in the yz-plane, opening up.

Key Concepts

Understanding Parabolas

A parabola is a symmetrical, U-shaped curve that appears in many mathematical contexts. It's most commonly recognized as the graph of a quadratic function. In the simplest form, a parabola can be described with the equation \( y = x^2 \). This notation means that for every value of \( x \), \( y \) is equal to \( x \) squared. The vertex of this simple parabola is at the origin, which is the lowest or highest point of the curve, depending on its orientation.

• The graph opens upwards if the coefficient of \( x^2 \) is positive, like in \( y = x^2 \).
• If the equation is \( y = -x^2 \), the parabola opens downwards.
• Parabolas can also open sideways, but the basic form must change to \( x = y^2 \) or similar.

In the context of coordinate planes, the parabola \( y = x^2 \) seen in the xy-plane reflects a perfectly symmetric arc centered around the y-axis. This feature makes it a key concept for understanding quadratic equations overall.

Exploring Circles

Circles are another fundamental shape in geometry, defined as the set of all points in a plane that are the same distance from a given point called the center. The equation of a circle in standard form is \( x^2 + z^2 = r^2 \), where \( r \) is the radius of the circle. Here’s a quick breakdown:

• The center of this particular circle is at the origin \((0, 0)\) in the xz-plane, according to the equation \( 1 = x^2 + z^2 \).
• The radius \( r = 1 \), meaning each point on the circle is 1 unit away from the center.
• This specific circle exists when the plane is parallel to the xz-plane and intersected by the line where \( y = 1 \).

Circles like these are crucial for understanding rotational symmetry and are often encountered in real-world applications involving waves, lenses, and more complex geometrical shapes that incorporate rotational elements.

Understanding Coordinate Planes

Coordinate planes are essential for locating and describing geometrical shapes within a space. In three-dimensional geometry, we typically discuss the xy-plane, xz-plane, and yz-plane. Let’s break these down:

• The **xy-plane** is a flat surface where each point is defined by an \( (x, y) \) pair, and \( z = 0 \). It’s often used for two-dimensional graphs like lines and parabolas.
• The **yz-plane** includes the coordinates \( (y, z) \), with \( x = 0 \), typically used to explore how shapes like parabolas behave in three dimensions.
• The **xz-plane** involves \( (x, z) \) pairs with \( y = 0 \). This plane helps visualize elements like circles and hyperbolas.

Coordinate planes are pivotal for visualizing and understanding spatial relationships among various shapes. By analyzing an object’s traces on these planes, you can predict its structure and behavior in a three-dimensional space, leading to insights about its geometry and applications.