Problem 32 - Precalculus with Limits

Question

In Exercises 29-34, use a graphing utility to graph the polar equation. Identify the graph. \(r=\dfrac{4}{1-2\ \cos\ \theta}\)

Step-by-Step Solution

Answer

The graph of the given polar equation \(r=\dfrac{4}{1-2\ \cos\ \theta}\) represents the Lemniscate of Bernoulli.

Step 1: Understand Polar Coordinates

Polar coordinates are a two-dimensional coordinate system, where each point on the plane is determined by an angle and a distance from the origin. In polar coordinates, a point is represented by the radius (r) and the angle (θ). Here, the given polar equation is \(r=\dfrac{4}{1-2\ \cos\ \theta}\). This needs to be graphed on a polar coordinate plane.

Step 2: Graphing the Equation

Now we need to graph our equation. Once we have our graphing utility, we set our graph type to polar mode. Enter the equation \(r=\dfrac{4}{1-2\ \cos\ \theta}\) into the graphing utility. Next, let the graphing utility graph the entered equation.

Step 3: Interpret the Graph

Now, observe and analyze the graph. The plotted graph for the equation \(r=\dfrac{4}{1-2\ \cos\ \theta}\) is a Lemniscate of Bernoulli. This is a figure 8 shaped curve, named after Jacob Bernoulli who discovered it. The graphical representation is unique which can be identified as the Lemniscate of Bernoulli.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help us visualize and understand mathematical equations better. They can plot complex equations in different coordinate systems, like polar or Cartesian coordinates.

To graph the polar equation, \(r=\dfrac{4}{1-2\ \cos\ \theta}\), it's essential to use a graphing utility. These utilities allow you to input the equation and automatically generate a graph, making the visualization process quick and easy.

Here's how you can use a graphing utility for polar equations:

• Switch the utility to polar mode. This mode is specifically tailored to handle angles and radii.
• Input the equation exactly as it appears to ensure accuracy.
• Observe the output graph, which should provide a clear visual representation of the equation.

Using graphing utilities not only simplifies the graphing process but also provides accurate representations to analyze the properties of the graph effectively.

Lemniscate of Bernoulli

The Lemniscate of Bernoulli is a fascinating and intricate shape that's best understood visually. It's recognized by its distinct figure-8 shape and is a classic example in the study of polar equations.

This curve was named after Jacob Bernoulli, a notable mathematician in the late 17th century. He made essential contributions to understanding curves of this type.

To identify a Lemniscate of Bernoulli:

• Look for the characteristic figure-eight pattern. This is its most distinctive and recognizable feature.
• Ensure the equation matches the form commonly associated with lemniscates, often involving trigonometric functions like cosine or sine.
• Note the symmetry around the origin, which is another telling characteristic of the lemniscate.

The lemniscate is more than just a beautiful curve; it provides insight into complex systems described by polar equations and shows how these can result in unique geometric forms.

Polar Equations

Polar equations are a vital part of trigonometry and geometry, allowing for the representation of curves using radius and angle in a coordinate plane.

A polar equation expresses relation in terms of \(r\), the radius, and \(\theta\), the angle. Unlike Cartesian coordinates, which rely on \(x\) and \(y\) axes, polar coordinates allow for a more natural description of curves emanating from a central point.

Key characteristics of polar equations include:

• The radius \(r\) varies as a function of angle \(\theta\), letting curves with radial symmetry be expressed naturally.
• Choosing values of \(\theta\) to plot can provide significant information about the shape and behavior of the curve.
• Polar equations can describe not just circles, but a variety of curves like spirals and figure-eights (such as the Lemniscate of Bernoulli).

Understanding polar equations opens the door to exploring various natural phenomena and engineering concepts where radial symmetry is an inherent feature.