Problem 80 - Calculus Early Transcendentals

Question

A family of curves is given by the equations \(r=1+c \sin n \theta\) where \(c\) is a real number and \(n\) is a positive integer. How does the graph change as \(n\) increases? How does it change as \(c\) changes? Illustrate by graphing enough members of the family to support your conclusions.

Step-by-Step Solution

Answer

Increasing \( n \) increases complexity with more petals; increasing \( c \) enlarges petal size.

Step 1: Understand the Equation

The given equation is in polar form: \( r = 1 + c \sin n \theta \). It represents a family of curves, where each curve is determined by the parameters \( c \) and \( n \). The parameter \( c \) affects the amplitude of the sine wave, while \( n \) affects the frequency or the number of cycles the curve makes around the origin.

Step 2: Analyze the Effect of Parameter \( n \)

As \( n \) increases, the term \( \sin n \theta \) causes the graph to complete more cycles or oscillations between 0 and \( 2\pi \). Specifically, the curve will have \( n \) symmetric 'petals' or lobes when \( n \) is odd, and \( 2n \) petals when \( n \) is even. This means the graph becomes more intricate, adding more lobes with higher \( n \).

Step 3: Analyze the Effect of Parameter \( c \)

The parameter \( c \) affects the size of the petals. When \( c = 0 \), we have a circle of radius 1. As \( c \) increases to positive values, the amplitude of the lobing increases, enlarging the size of the petals. If \( c \) is negative, the effect is similar but reflects the graph along the radial axis, still increasing petal size but in an inverted manner.

Step 4: Illustrate with Graphs

To visualize these changes, graph several curves like:\- For constant \( c = 1 \):\ - \( n = 2 \): 4-petaled curve.\ - \( n = 3 \): 3-petaled curve.\ - \( n = 4 \): 8-petaled curve.\- For constant \( n = 3 \):\ - \( c = 0.5 \): smaller, tight 3-petaled curve.\ - \( c = 2 \): large, expanded 3-petaled curve.\These examples demonstrate how increasing \( n \) increases the number of petals, creating more complexity, while changing \( c \) alters their size.

Key Concepts

Parametric Curves

Parametric curves are a way to represent geometric shapes using parameters. In the context of polar coordinates, equations like \( r = 1 + c \sin n \theta \) describe such curves by mapping a parameter \( \theta \) to coordinates \( (r, \theta) \). Here, \( \theta \) is considered as a moving parameter, tracing the curve as it varies. This approach helps in visualizing complex patterns by breaking down circular motions.

Important characteristics of parametric curves include how they change with different parameter values. In our polar equation, \( c \) and \( n \) control different aspects:

• \( c \) changes the size or amplitude, affecting the overall appearance of the petals or lobes of the curve.
• \( n \) affects the frequency, determining the number of symmetrical lobes around the origin.

As you plug different values into \( \theta \), you can observe the resulting set of coordinates trace out an intricate pattern. Understanding these parameters allows you to predict and manipulate the shape of the curve effectively.

Trigonometric Functions

Trigonometric functions, such as \( \sin \theta \) and \( \cos \theta \), are crucial in describing rotational and oscillating behaviors. In polar coordinates, these functions are the foundation for circles, spirals, and more complex shapes.

For the given problem, \( \sin n \theta \) plays a significant role in determining the shape's symmetry and repetition. The sine function oscillates between -1 and 1, meaning the term \( c \sin n \theta \) will vary the radial distance \( r \) from the origin, in a periodic fashion.

• When \( c \) is positive, greater amplitudes result in larger petals.
• The repetition or frequency factor \( n \) dictates how many times the sine wave completes its cycle as \( \theta \) varies from 0 to \( 2\pi \).

Understanding these trigonometric influences allows you to predict the number of petals (lobes) the curve will have and dynamically illustrate their size based on parameter changes.

Graphical Analysis

Graphical analysis involves interpreting the visual representation of mathematical equations and understanding their underlying relationships. When analyzing polar graphs, parameters such as \( c \) and \( n \) significantly impact the shape and size of the curves.

When you sketch graphs for different values of \( c \) and \( n \), you notice clear patterns:

• As \( n \) increases, the number of lobes increases, with \( n \) lobes for odd values and \( 2n \) lobes for even values of \( n \).
• When \( c \) transitions from negative to positive, the petals expand and reflect along the radial axis, altering the graph's orientation but retaining its key features.

Using software or graphing calculators to plot these curves can help visualize these properties vividly. By graphing, you observe consistent symmetrical patterns, informed by rotational symmetries inherent in the trigonometric components. This hands-on analysis reinforces theoretical insights into the behavior of polar curves.