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Chapter 12: Problem 92

Consider the family of lima莽ons \(r=1+b \cos \theta .\) Describe how the curves change as \(b \rightarrow \infty\)

Short Answer

Expert verified
Answer: As \(b \rightarrow \infty\), the lima莽on curve approaches a vertical line at \(r=0\) on the polar plane, with angles \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\).

Step by step solution

01

Understanding lima莽ons in polar coordinates

尝颈尘补莽辞苍蝉 are curves represented by the equation \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\) in polar coordinates, where \(a\) and \(b\) are constants, and \(\theta\) is the angle. In this exercise, we are considering the family of lima莽ons given by \(r = 1 + b \cos \theta\), where \(b\) is the variable parameter.
02

Cases for the value of b

These curves have different shapes depending on the value of \(b\): 1. If \(b < 1\), the lima莽on is a "diminishing" curve with a loop. 2. If \(b = 1\), the lima莽on is a cardioid. 3. If \(b > 1\), the lima莽on does not have a loop but has a "dent" or "cusp" on the inner part. 4. If \(b \rightarrow \infty\), we need to analyze the behavior of the lima莽on.
03

Analyzing the behavior of the lima莽on as b approaches infinity

To study the behavior of the lima莽on as \(b \rightarrow \infty\), we can observe the equation \(r = 1 + b \cos \theta\). As \(b\) becomes larger, the value of \(b \cos \theta\) will dominate the equation, and the constant term \(1\) becomes negligible. The equation looks like: \(r \approx b\cos\theta\). By dividing both sides of the equation by \(b\), we get \(\frac{r}{b} \approx \cos \theta\). As \(b \rightarrow \infty\), the fraction \(\frac{r}{b}\) approaches \(0\), so we have \(0 \approx \cos\theta\). The solution of this equation is \(\theta = \frac{\pi}{2}\) or \(\theta = \frac{3\pi}{2}\)
04

Concluding the behavior of the lima莽on as b approaches infinity

As \(b\) becomes larger, the lima莽on curve becomes closer and closer to a vertical line passing through \(r=0\), and with angles \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\). So, as \(b \rightarrow \infty\), the lima莽on curve approaches a vertical line at \(r=0\) on the polar plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

尝颈尘补莽辞苍蝉
尝颈尘补莽辞苍蝉 are an intriguing family of curves in polar coordinates identified by equations of the form \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\). Here, \(a\) and \(b\) are constants, and \(\theta\) is the angle in radians. These curves are known for their diverse shapes, potentially exhibiting loops, dimples, or resembling a circle depending on the respective values of \(a\) and \(b\).
The exercise in question focuses on a specific type of lima莽on where the equation is \(r = 1 + b \cos \theta\). This reveals how the parameter \(b\) drastically alters the curve's appearance:
  • If \(b < 1\), the curve manifests as a dimpling or looping structure, hinting at a smaller enclosed area.
  • If \(b = 1\), it transitions into a unique heart-shaped curve termed a "cardioid".
  • When \(b > 1\), the lima莽on loses its loop but retains a cusp-like appearance, forming a defined dimple.
The beauty of lima莽ons lies in their geometric versatility, showcasing a spectrum of forms with minimal adjustments to the constants involved.
Cardioid
Among the fascinating shapes lima莽ons take is the cardioid, which occurs specifically when \(b = 1\) in the equation \(r = 1 + b \cos \theta\). A cardioid, deriving its name from the Greek word for "heart," resembles the shape of a heart and is a particularly symmetrical and aesthetically pleasing curve.
This charm arises from its balanced properties:
  • All points on a cardioid are equidistant from the radial line, which contributes to its symmetry.
  • The curve lacks the internal loop seen in some other lima莽ons, maintaining a continuous and smooth outline.
The cardioid is often observed in optical and acoustical applications due to its focused properties, serving as a key concept in understanding polar plots and their symmetry.
Infinity Behavior
Understanding the behavior of lima莽ons as \(b\) approaches infinity provides insight into the fundamental dynamics of polar coordinates. As we analyze the equation \(r = 1 + b \cos \theta\), it becomes evident that as \(b\) increases significantly, the influence of the term \(b \cos \theta\) overshadows the constant term 1.
The behavior can be described as follows:
  • In the limit where \(b\) is extremely large, the equation simplifies to \(r \approx b \cos \theta\).
  • Analyzing this form, by dividing through by \(b\), we find \(\frac{r}{b} \approx \cos \theta\), which converges to 0 as \(b\) goes to infinity.
  • Thus, \(\cos \theta\) centers around 0, meaning \(\theta\) approaches \(\frac{\pi}{2}\) or \(\frac{3\pi}{2}\), indicating the curve approximates the vertical line at \(r = 0\).
Therefore, the lima莽on effectively transforms into a straight vertical line, emphasizing the dramatic changes in geometric structure as parameters evolve towards extreme values.

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Most popular questions from this chapter

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