Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

Investigate the family of curves defined by the polar equations \(r=|\cos n \theta|\), where \(n\) is some positive integer. How do the number of leaves depend on \(n\) ?

Step-by-Step Solution

Answer

If \( n \) is even, there are \( 2n \) leaves; if \( n \) is odd, there are \( n \) leaves.

Step 1: Understand the Polar Equation

The given equation is a polar equation in the form \( r = |\cos n\theta| \). The variable \( n \) is a positive integer that will affect the shape of the curve.

Step 2: Evaluate the Behavior of \( \cos n\theta \)

The function \( \cos n\theta \) oscillates between -1 and 1. The absolute value \( |\cos n\theta| \) modifies this to oscillate between 0 and 1. Thus, the graph will only consist of non-negative values of \( r \), reflecting symmetry in the radial direction.

Step 3: Determine the Symmetry and Shape

As \( n\theta \) varies from 0 to \( 2\pi \), the function \( \cos n\theta \) completes \( n \) full cycles. This is because \( n \) multiples determine how many times the cosine function completes a cycle in the range.

Step 4: Analyze the Petal Structure for Even and Odd \( n \)

- If \( n \) is even, each complete cycle of \( \cos n\theta \) will create two petals because the absolute value introduces symmetry about both the x-axis and in the radial direction.- If \( n \) is odd, the function will have \( n \) petals because each petal will be distinctly formed without any double occurrence within one cycle.

Step 5: Conclude the Relationship between \( n \) and the Number of Leaves

For an even \( n \), the number of leaves will be \( 2n \). For an odd \( n \), the number of leaves will equal \( n \). These results stem from the symmetry induced by the absolute value and the pattern of the cosine function.

Key Concepts

Understanding Curve Symmetry

In polar equations, such as the one defined by \( r = |\cos n\theta| \), symmetry plays a significant role. The symmetry of a curve helps us to predict and draw the curve efficiently.

For the equation \( r = |\cos n\theta| \), symmetry arises from the absolute value and the trigonometric nature of cosine. When considering the graph of this equation, we note that:

• The absolute value \( |\cos n\theta| \) implies that the curve reflects across the origin. This is known as radial symmetry.
• Symmetry across the x-axis comes from the even nature of the cosine function. Negative values of \( \cos n\theta \) are turned positive by the absolute value, allowing the curve to maintain x-axis symmetry.

Symmetry is important because it helps in reducing the effort needed to graph complex polar equations. Instead of plotting all points, understanding symmetry allows us to draw only part of the curve and mirror it across appropriate axes or around the origin.

Examining Petal Structure

The petal structure in polar graphs like \( r = |\cos n\theta| \) is a captivating aspect. The varying integer \( n \) in our equation dictates the appearance of 'petals,' or lobes, on the graph.

Here's how it works:

• When \( n \) is even, each cycle of \( \cos n\theta \) produces two mirrored petals. For instance, an equation with \( n = 2 \) results in four petals, due to each petal being mirrored across the x-axis.
• When \( n \) is odd, each petal appears only once in the cycle. Thus, \( n = 3 \) would result in three distinct petals occurring once around the full \( 2\pi \) rotation.

This petal structure is deeply tied to the periodic nature of trigonometric functions and the effects of radial symmetry introduced by the absolute value. Understanding these patterns makes it easier to predict and draw polar graphs with varying integers \( n \).

Role of Trigonometric Functions

Trigonometric functions, such as cosine, are at the heart of understanding polar equations. The function \( \cos n\theta \) exhibits predictable, periodic behavior essential for plotting polar curves.

Key characteristics of trigonometric functions in this context include:

• Periodicity: \( \cos n\theta \) completes \( n \) full cycles as \( \theta \) sweeps through \( 0 \) to \( 2\pi \). This behavior results in multiple duplicated patterns, like petals in our graph.
• Even spacing: Each cycle equals \( \frac{2\pi}{n} \), regularly dividing the circle into equal segments, contributing to the formation of evenly spaced petals on the graph.

By understanding these trigonometric properties, students can predict the behavior of polar equations over their input domain. This knowledge helps in both graphing and analyzing polar equations more effectively.

Engaging in Mathematical Analysis

Mathematical analysis involves breaking down and exploring the specifics of equations like \( r = |\cos n\theta| \) to grasp their underlying patterns and behaviors. For polar equations, analysis often focuses on elements like symmetry and periodicity.

When analyzing \( r = |\cos n\theta| \), consider the following:

• Structure: Classify the number of petals based on whether \( n \) is even or odd. An even value doubles the count due to symmetry, while an odd value maps directly to the number of petals.
• Symmetry Impact: Recognize how the absolute value function and the cosine function’s inherent symmetry reduce complexity, allowing predefined rules for sketching.

Through analysis, not only can we graph these complex equations effectively, but we gain insights into how modifications of parameters like \( n \) alter the overall shape and structure of polar curves. This analytical approach ensures a deeper comprehension of mathematical concepts presented in polar coordinate systems.