Problem 39 - Calculus 3

Question

For the following exercises, plot a graph of the function. $$ z=f(x, y)=\sqrt{x^{2}+y^{2}} $$

Step-by-Step Solution

Answer

Plot the function as a 3D cone with its vertex at the origin in the coordinate space.

Step 1: Understand the Function

The function given is a three-dimensional function defined as \( z = f(x, y) = \sqrt{x^2 + y^2} \). This expression represents a surface in a 3D space, where \( z \) is the dependent variable, and \( x \) and \( y \) are independent variables.

Step 2: Analyze Function Characteristics

The expression \( z = \sqrt{x^2 + y^2} \) is known as the equation of a cone with its vertex at the origin. This is because \( z \) is determined by the distance from the origin in the \( xy \)-plane. The further away a point \((x,y)\) is from the origin, the higher the value of \( z \). Since the square root function is used, \( z \) is always non-negative.

Step 3: Set Up the Axes for the Plot

To plot this function, set up a 3D coordinate system with axes labeled \( x \), \( y \), and \( z \). The \( x \)-axis and \( y \)-axis will be your horizontal plane, and the \( z \)-axis will be vertical.

Step 4: Plot the Surface

The surface described by \( z = \sqrt{x^2 + y^2} \) forms a circular cone with its tip at the origin \((0,0,0)\). For each point \((x,y)\), calculate the corresponding \( z \) value using the equation. Plot these points in the 3D coordinate system, ensuring that for every circle on the \( xy \)-plane with radius \( r \), \( z = r \).

Step 5: Sketch the Visual Representation

To visualize, imagine drawing circles in the \( xy \)-plane and lifting them upwards to form a cone shape above the plane. Start with small circles and gradually enlarge them as you move away from the origin. Connect these circles smoothly to form the surface of the cone.

Key Concepts

3D Plotting

Creating three-dimensional plots allows us to visually explore surfaces that depend on two variables. The function given, \( z = \sqrt{x^2 + y^2} \), represents a surface in space: a cone centered at the origin. Here's how we construct such a plot:

• First, define the domain. Choose a range for both \( x \) and \( y \), like \([-5, 5]\). This gives you a square area over which the surface will be plotted.
• Next, calculate the corresponding \( z \) value for each pair of \( x, y \). This is done by substituting \( x \, \text{and} \, y \) values into the function.
• Finally, assemble these points in a three-dimensional space. Each point \((x, y, z)\) is plotted, showing the height \( z \) above the \( xy \)-plane.

By plotting a grid of points and connecting them, you can visualize how the surface extends and understand its shape—like watching a cone rise from a plane. Use software tools like Python's Matplotlib to create a detailed and visually appealing 3D plot. This gives a clear picture of how the surface behaves across different regions.

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane. When we consider the equation \( z = \sqrt{x^2 + y^2} \), we are looking at the vertical slice of a cone in 3D space, creating circular cross-sections in the \( xy \)-plane.

• The appearance of these circles in conic sections gives insight into how the z-value increases as one moves away from the center.
• For this particular function, imagine an infinite number of such circles stacked upon one another, starting from the origin.
• This stacking forms a right circular cone, perfectly symmetrical around the z-axis.

Understanding conic sections like this helps you see the connection between algebraic equations and geometrical shapes. In this case, the equation defines a surface that mirrors symmetry and uniform distance from a central point—the essence of conic geometry.

Coordinate Systems

A coordinate system is crucial for defining spaces in mathematics and plotting functions like \( z = \sqrt{x^2 + y^2} \). The standard system used here is the Cartesian coordinate system, which uses three axes: \( x \), \( y \), and \( z \). Here's the breakdown:

• The \( x \)-axis and \( y \)-axis together form a plane (the \( xy \)-plane), representing the bottom space across which functions are defined.
• The \( z \)-axis is perpendicular to this plane, representing height or depth—essential for 3D plotting.
• Every point in this system is identified with coordinates \((x, y, z)\), expressing how far along each axis the point resides.

Using this three-dimensional space, we can depict complex surfaces. The system allows us to visualize how changes in \( x \) and \( y \) affect \( z \), providing a framework to explore intricate mathematical problems geometrically.