Problem 107 - Calculus Early Transcendentals

Question

The butterfly curve of Example 8 is enhanced by adding a term: $$r=e^{\sin \theta}-2 \cos 4 \theta+\sin ^{5}(\theta / 12), \quad \text { for } 0 \leq \theta \leq 24 \pi$$ a. Graph the curve. b. Explain why the new term produces the observed effect. (Source: S. Wagon and E. Packel, Animating Calculus, Freeman, 1994)

Step-by-Step Solution

Answer

Answer: The additional term, \(\sin^{5}(\frac{\theta}{12})\), adds a slower frequency sinusoidal pattern to the original butterfly curve, causing it to have "bumps" or protrude outwards, depending on the values of the sinusoidal term. This distorts the curve's overall shape, creating an enhanced and more complex butterfly curve.

Step 1: Understanding the given formula for the butterfly curve

Given the formula for the butterfly curve is: $$r = e^{\sin \theta} - 2 \cos 4\theta + \sin^{5} (\frac{\theta}{12})$$ The polar coordinates are represented by the angle \(\theta\) and the distance from the origin, \(r\). The ranges are given by \(0 \leq \theta \leq 24\pi\).

Step 2: Graphing the Curve

To graph the curve, we can use mathematical software (such as Desmos, Geogebra or Matlab) that allows us to create graphs in polar coordinates. To create the graph, input the given equation for \(r\) and set the range of \(\theta\) to the given values, \(0 \leq \theta \leq 24\pi\). Now for the explanation part:

Step 3: Analyzing the Effect of the Added Term

We need to focus on the added term \(\sin^{5} (\frac{\theta}{12})\). The function of \(\sin x\) oscillates between -1 and 1. So it's safe to say that \(\sin^{5} x\) oscillates also between -1 and 1, but with odd powers of x, the sign would remain as of x (for -1 its -1, for 1 its still 1). Now the oscillation of \(\sin^{5}(\frac{\theta}{12})\) would be 12 times slower than the original sine function due to the division of the angle by 12. This would cause the entire curve to oscillate with the same frequency but the amplitude would now be affected by this sinusoidal term. Adding this term to the original butterfly curve which has the polar coordinate \(r = e^{\sin \theta} - 2 \cos 4\theta\) will result in the combined effects from both terms, causing the curve to have "bumps" or protrude outwards depending on the values of the sinusoidal term.

Step 4: Conclusion

The graph of the enhanced butterfly curve displays the effects of the additional term, which creates an oscillation of the curve's overall shape. The term \(\sin^{5}(\frac{\theta}{12})\) adds a sinusoidal pattern at a slower frequency compared to the original trigonometric functions in the equation, distorting the curve in the shape of bumps or protrusions.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to describe a point's location on a plane using two components: the angle \(\theta\) and the radial distance \(r\) from a central origin point. This system is especially useful for graphing curves that have circular or spiral characteristics. To get started with polar coordinates, you first choose an angle \(\theta\) that represents the direction from the origin. Then, move outwards from the origin by the distance \(r\).

• The angle \(\theta\) is typically measured in radians, but can also be in degrees.
• Distance \(r\) can vary, sometimes being negative, which means the point is in the opposite direction.

The beauty of using polar coordinates for complex curves like the butterfly curve lies in their ability to transform these points seamlessly into graphable shapes. Polar coordinates help illustrate beautiful symmetry and patterns not easily visible in Cartesian systems.

Trigonometric Functions

Trigonometric functions are the backbone of many mathematical models, especially when dealing with oscillating systems or those rooted in circular geometry. In the butterfly curve equation, we encounter both sine and cosine functions. Let's break them down:

• The \(\sin \theta\) part introduces an oscillation effect as \(\theta\) changes, which means periodic movement up and down.
• The term \(-2 \cos 4\theta\) contributes significant fluctuations due to its cosine nature. Multiplied by 4, it rapidly adjusts the curve's radial distance \(r\), adding detail and complexity.
• The additional term \(\sin^5(\frac{\theta}{12})\) further modulates the curve, which is a more nuanced adjustment due to its slower oscillation and higher power.

Each of these trigonometric functions combines to contribute remarkable and intricate details to the graph's structure. These functions help form the distinctive shapes that define many mathematical phenomena.

Graphing Software

Graphing software empowers users to visualize complex mathematical equations like the butterfly curve. It translates abstract mathematical concepts into understandable visuals. Popular tools include Desmos, Geogebra, and Matlab. Each tool provides intuitive interfaces for inputting equations in various formats, including polar coordinates.

• Students can input equations directly and adjust parameters such as \(\theta\)'s range for dynamic exploration.
• Real-time changes help learners see how varying elements like frequency affect the graph's shape.
• Additional features such as zoom, trace, and animation enhance understanding.

By using graphing software, students convert abstract equations into tangible graphs, improving comprehension and retention of mathematical concepts. These tools are indispensable in modern education, enabling interactive learning experiences.

Oscillation Effects

The oscillation effects in the butterfly curve are primarily driven by the trigonometric terms \(\sin \theta\), \(-2\cos 4\theta\), and \(\sin^5(\frac{\theta}{12})\). These oscillations cause repeated rising and falling patterns within the curve, lending it its unique and beautiful appearance.

Oscillation occurs when a point in the graph continuously moves up and down or forward and backward around a central axis of balance. Here’s how each function influences the oscillation:

• \(\sin \theta\) softens the oscillations, gradually changing as \(\theta\) travels from 0 to 24\(\pi\).
• \(-2\cos 4\theta\) adds sharp peaks and troughs, rapidly changing the radius due to its fourfold frequency.
• \(\sin^5(\frac{\theta}{12})\) provides additional subtle ripples with a slower oscillation frequency, giving more delicate adjustments.

The interaction of these oscillating elements produces the butterfly curve's telltale "bumps" or protrusions. Understanding these effects helps students grasp how mathematical terms combine to shape intricate patterns and dynamics present in nature and various fields of science and engineering.