Problem 71 - Calculus Early Transcendentals

Question

\(71-76\) Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. $$r=1+2 \sin (\theta / 2) \quad$$ (nephroid of Freeth)

Step-by-Step Solution

Answer

Graph the curve in the interval \( \theta = 0 \) to \( \theta = 4\pi \) to see the entire nephroid.

Step 1: Identify the Polar Equation

The given polar equation is \( r = 1 + 2 \sin(\theta/2) \). This is a variation of the standard form of a nephroid, a type of limaçon, modified by the division by two in the angle.

Step 2: Determine the Parameter Interval

To ensure we graph the entire curve, we need to determine the interval for \( \theta \). Since the sine function has a period of \( 2\pi \), dividing \( \theta \) by 2 increases this period to \( 4\pi \). Thus, the parameter interval \( [0, 4\pi] \) will cover the entire nephroid.

Step 3: Prepare for Graphing

Set your graphing device to polar mode. Input the equation \( r = 1 + 2 \sin(\theta/2) \) and ensure that the device is ready to plot curves in the range \( \theta = 0 \) to \( \theta = 4\pi \).

Step 4: Graph the Polar Curve

Use the graphing device to plot the curve. Ensure that your graphing interval covers \( \theta = 0 \) to \( \theta = 4\pi \). The graph should resemble a nephroid shape, completing two loops that reflect the effect of \( \sin(\theta/2) \).

Step 5: Analyze the Graph

Observe the completed graph. It should show a rounded figure with two arches reflecting the sine function's influence over the interval. The nephroid should be symmetric about the polar axis, as expected with \( \theta/2 \) modulation.

Key Concepts

Graphing Polar Curves

Polar curves are fascinating mathematical representations where each point on a plane is determined by an angle and a distance from the origin. The formula for polar coordinates is given as \( r = f(\theta) \), where \( r \) is the radius, and \( \theta \) is the angle measured from the polar axis. To graph such curves, we typically follow a structured approach:

• First, identify the given polar equation.
• Determine an appropriate parameter interval for \( \theta \).
• Use a graphing tool set to polar mode.
• Finally, plot the points to visualize the curve.

Using these steps, we ensure that we accurately portray the characteristics of the curve over its complete range.

Nephroid of Freeth

The Nephroid of Freeth is a captivating curve within the family of limaçons, notable for its distinct circular features. A nephroid typically presents as a two-cusped curve, exemplified by the equation \( r = a + b \sin(\theta) \) or similar forms. In our exercise, the equation provided is \( r = 1 + 2 \sin(\theta/2) \), a variant called Nephroid of Freeth.
This unique form gives rise to a double-looped graph, due to the modulation within the sine function. Hence, it is crucial to fully develop the sine wave's effects, ensuring a complete representation in the polar graph. This curve remains symmetrical, showcasing its aesthetic appeal and mathematical intrigue.

Parameter Interval

Choosing the correct parameter interval is vital to graph an entire polar curve. The goal is to cover a range of \( \theta \) values that encompasses all aspects of the curve. In the case of \( r = 1 + 2 \sin(\theta/2) \), the sine function inside the equation affects the interval period.
Since the basic period of \( \sin(\theta) \) is \( 2\pi \), dividing \( \theta \) by 2 effectively doubles this period to \( 4\pi \). Therefore, the appropriate parameter interval is \([0, 4\pi]\). Setting \( \theta \) to range from 0 to \( 4\pi \) ensures that the nephroid curve is entirely captured, reflecting its symmetry and looping behavior.

Sine Function Modulation

Sine function modulation plays a key role in shaping polar curves. When a sine function is part of the equation, like \( \sin(\theta/2) \) in our example, it directly affects the length and shape of the curve. The modulation can alter the basic sine wave, influencing its frequency and period.

• Reducing the angle within the sine function, as with \( \theta/2 \), alters the frequency, effectively stretching the curve.
• This stretching results in a wider period of \( 4\pi \), as seen in the nephroid example.
• Moreover, the amplitude of 2 in \( 2 \sin(\theta/2) \) affects the curve’s maximum and minimum radii.

Together, these variations create rich, intricate shapes like the Nephroid of Freeth, illustrating the harmony between mathematical theory and visual representation.