Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

Graph the curve \(r=\cos (8 \theta / 5)\) using the parametric graphing facility of a graphing calculator or computer. Notice that it is necessary to determine the proper domain for \(\theta\). Assuming that you start at \(\theta=0\), you have to determine the value of \(\theta\) that makes the curve start to repeat itself. Explain why the correct domain is \(0 \leq \theta \leq 10 \pi\).

Step-by-Step Solution

Answer

The domain \(0 \leq \theta \leq 10\pi\) ensures one full repetition for \(r=\cos(\frac{8\theta}{5})\).

Step 1: Understanding the function

The equation given is in polar form: \(r = \cos(\frac{8\theta}{5})\). We are tasked to graph this function using a parametric graphing calculator or tool. We need to find out the range of \(\theta\) that will fully display the curve before it starts repeating.

Step 2: Understanding polar repetitions

The periodic nature of the cosine function tells us that \(\cos(x)\) repeats every \(2\pi\). In our function, \(\cos(\frac{8\theta}{5})\) will complete a cycle and repeat every time \(\frac{8\theta}{5} = 2\pi\). Thus, solving for \(\theta\) gives \(\theta = \frac{5}{4} \times 2\pi = \frac{5\pi}{2}\).

Step 3: Finding the number of repeats

To determine when the curve repeats itself entirely, we need \(8\theta/5\) to reach an integer multiple of \(2\pi\) that covers all lobes of the graph. For \(\cos(k\theta)\) where \(k\) is a rational number, we multiply the basic repeat angle by the denominator of \(k\) to ensure all lobes are covered. Since the effective number of lobes is the LCM of the numerator and denominator (8, 5 = 40), \(\theta = 10\pi\) fully completes and repeats the plot.

Step 4: Verify the domain

Check that \(0 \leq \theta \leq 10\pi\) indeed brings \(\cos\left(\frac{8\theta}{5}\right)\) through all lobes. Each cycle of the cosine function opens certain curves, and stretching the \(\theta\)-scope up to \(10\pi\) ensures a complete coverage based on the rational \'8/5\', completing multiple, fully overlapping cycles.

Step 5: Conclusion of the chosen domain

The domain \(0 \leq \theta \leq 10\pi\) covers the graph repeating exactly once for the non-terminating rational \(k=8/5\) in terms of polar cycles. Every specific lobe like those in a rose curve completes when this equation sweeps \(10 \pi\) radians, showing both symmetry and full extent.

Key Concepts

Parametric Graphing

Parametric graphing is a way to represent mathematical functions in terms of parameters, usually with a pair of equations that define a set of coordinates. Instead of dealing with just x and y coordinates, parametric equations use a third variable, often called t or, in polar coordinates, θ. This method allows for plotting complex curves that may not be possible to describe with a single equation in standard y=f(x) form.

• In polar coordinates, points are determined based on their distance from the origin, denoted as r, and the angle from a reference direction, denoted as θ.
• Parametric graphing is especially useful in engineering, physics, and computer graphics, where complex movements and paths need precise descriptions.
• Using a graphing calculator, students can observe changes in the graph dynamically by adjusting the parameter values. This aids in better understanding of how parametric equations work.

By utilizing parametric graphing tools, students can grasp complex relationships in functions and explore mathematics beyond traditional algebra.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, commonly denoted as cos(x). It describes how the x-coordinate of a point on a unit circle changes as it moves around the circle. In our given problem, the function is expressed as r = cos(8θ/5).

• The cosine function is known for its wave-like shape and periodic nature, which means it repeats itself at regular intervals.
• In the context of the provided exercise, the cosine function is incorporated in a polar equation. Here, the angle θ plays a crucial role in how r values change.
• Understanding modifications to the cosine function's period involves recognizing changes in its argument, as in the fraction 8θ/5 in our exercise.

The cosine function in parametric form often controls the smooth, continuous shapes we see in polar graph representations.

Periodic Function

A periodic function is any function that repeats its values at regular intervals or periods. Understanding periodic behavior is essential in many fields, such as

• signal processing,
• quantum physics,
• and even in understanding natural cycles.

For the cosine function, this periodicity is seen every 2π radians. In our equation r = cos(8θ/5), understanding the periodicity helps us to find the correct domain for plotting. We need θ to make sure all the peculiar cycles of the function are visible without overlap.

• In our problem, the unique period of the modified cosine function is determined by setting the expression 8θ/5 equal to integer multiples of 2π.
• Knowing the period allows us to determine when the curve will repeat, helping us choose the domain 0 ≤ θ ≤ 10π correctly.
• This ensures the graph drawn truly represents all the cycles inherent in the function without redundancies.

Recognizing and calculating periodic functions are key skills in mathematics that help predict and map behaviors in both theoretical and real-world scenarios.

Graphing Calculator

Graphing calculators are powerful tools for students to visualize mathematical concepts. They are especially useful for plotting parametric and polar coordinates, where visualizing complex curves is necessary.

• Graphing calculators take parameters and quickly plot functions, displaying results that are often difficult to imagine or draw by hand.
• These calculators allow for dynamic interaction with the math problem, as students can adjust values and see immediate impacts on the graph.
• In our exercise, a graphing calculator was necessary to precisely visualize how the curve r = cos(8θ/5) develops as θ varies from 0 to 10π.

Graphing calculators make learning more interactive and help students verify their mathematical intuition with tangible, visible results.