Problem 24 - Thomas Calculus

Question

Graph the limacons. Limacon ("lee-ma-sahn") is Old French for "snail." You will understand the name when you graph the limacons in Exercise 21. Equations for limacons have the form \(r=a \pm b \cos \theta\) or \(r=a \pm b \sin \theta .\) There are four basic shapes. Oval limacons a. \(r=2+\cos \theta \quad\) b. \(r=-2+\sin \theta\)

Step-by-Step Solution

Answer

Graph (a) is a loopless oval limacon; Graph (b) is a limacon with a loop.

Step 1: Identify the Form

The given equations are in the form of the polar equations for limacons: \( r = a \pm b \cos \theta \) or \( r = a \pm b \sin \theta \). For problem (a), we have \( r = 2 + \cos \theta \) which follows the form \( r = a + b \cos \theta \). For problem (b), we have \( r = -2 + \sin \theta \). Here it follows the form \( r = a + b \sin \theta \), with \( a = -2 \) and \( b = 1 \).

Step 2: Determine Parameter Values

For both equations, identify the values of \( a \) and \( b \). In (a), \( a = 2 \) and \( b = 1 \). In (b), \( a = -2 \) and \( b = 1 \). These parameters help in determining the shape and orientation of the limacon.

Step 3: Identify the Basic Shape

The shape of a limacon is categorized based on \( |a| \) and \( |b| \). For the equation \( r = 2 + \cos \theta \) where \( |a| > |b| \), the limacon has no loop and is generally an oval shape. For \( r = -2 + \sin \theta \) where \( |a| = |b| \), the limacon has an inner loop.

Step 4: Graph the Limacon for (a)

To graph \( r = 2 + \cos \theta \), note that it is symmetric about the polar axis (horizontal axis). Start by plotting points for selected values of \( \theta \) (e.g., \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \)) to form the oval shape. The maximum radius occurs at \( r = a + b = 3 \), and the minimum radius is \( r = a - b = 1 \).

Step 5: Graph the Limacon for (b)

For \( r = -2 + \sin \theta \), note that it is symmetric about the line \( \theta = \frac{\pi}{2} \) (vertical axis). As \( |a| = |b| \), plot points for values of \( \theta \) such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \) to identify the loop. The inner loop results in a maximum radius of 0 and a minimum negative radius of -3.

Key Concepts

Limacons

Limacons are a fascinating family of curves in polar coordinates. The name "limacon" means "snail" in Old French, and this is quite fitting when you observe their shapes. A limacon can have one of four distinct shapes based on parameters in its equation: dimpled, cardioid, with inner loop, or convex.

• The general form of a limacon is either \( r = a \pm b \cos \theta \) or \( r = a \pm b \sin \theta \).
• The characteristics of the limacon are determined by the relationship between \( a \) and \( b \).
• If \( |a| > |b| \), the curve is dimpled or convex, resembling an oval.
• If \( |a| = |b| \), it forms a cardioid shape, which looks like a heart.
• If \( |a| < |b| \), the limacon has an inner loop.

In the exercises, we specifically look at two forms: an oval limacon \( r = 2 + \cos \theta \) and one with an inner loop \( r = -2 + \sin \theta \). Observing these forms helps understand how changes in parameters affect the curve's shape.

Graphing Polar Equations

Graphing polar equations can feel challenging at first but breaks down into simpler steps with practice. To graph a polar equation like a limacon, start by identifying its form and parameters. Polar coordinates work a bit differently from Cartesian coordinates.

• A point in the plane is defined by a distance \( r \) from the origin and an angle \( \theta \) from the positive x-axis.
• The parameter \( r \) is calculated using the polar equation at various values of \( \theta \).
• Select several \( \theta \) values, such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \), to plot key points on the graph.

When graphing, start from zero and move counterclockwise for positive angles. Plot each point \( (r, \theta) \) and draw a smooth curve through them to form the limacon. Remember, \( r \) can be negative, which means it plots the point in the opposite direction. Repeated practice with different equations leads to more intuitive graphing.

Mathematical Symmetry

Symmetry is an important property in graphing polar equations, simplifying the process. A polar graph’s symmetry helps predict unseen parts of the curve once part of it is plotted. Limacons often exhibit symmetry due to their polar form.

Common symmetries in polar graphs include:

• Polar Axis (horizontal axis): If \( r(\theta) = r(-\theta) \), the graph is symmetric about the polar axis.
• Line \( \theta = \frac{\pi}{2} \) (vertical axis): If \( r(\theta) = -r(\pi-\theta) \), the graph is symmetric about this line.
• Origin: If \( r(\theta) = -r(\theta + \pi) \), the graph is symmetric about the origin.

In our examples, \( r = 2 + \cos \theta \) is symmetric about the polar axis, creating an easy reference when graphing. On the other hand, \( r = -2 + \sin \theta \) shows symmetry about the vertical line \( \theta = \frac{\pi}{2} \). Recognizing these symmetries makes plotting quicker and less error-prone.