Problem 81 - Contemporary Precalculus

Question

Sketch the graph of the equation. $$r=1 / \theta \quad(\theta > 0)$$

Step-by-Step Solution

Answer

Based on the analysis and solution provided for sketching the graph of the polar equation $$r=1 / \theta \quad(\theta > 0)$$, describe the main features of the graph as well as the behavior of the curve as $$\theta$$ increases.

Step 1: Identifying initial positions and direction

We begin the sketch by plotting a few points when $$\theta$$ is small and positive, such as $$\theta = \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$$. This would give us an idea of the initial positions and direction of the graph.

Step 2: Plotting more points

As we increase $$\theta$$, the value of $$r$$ continues to get smaller. For example, when $$\theta = \pi$$, $$r = \frac{1}{\pi}$$ and when $$\theta = 2\pi$$, $$r = \frac{1}{2\pi}$$. The graph shows that the curve spirals inwards towards the origin as $$\theta$$ increases.

Step 3: Behavior as $$\theta$$ approaches infinity

As $$\theta$$ approaches infinity, $$r$$ approaches 0, meaning the graph gets closer and closer to the origin. The curve converges to the origin but never reaches it since $$r$$ will always be greater than 0 as long as $$\theta > 0$$.

Step 4: Final Sketch

Combining all the information, we can now sketch the graph of the polar equation $$r=1 / \theta \quad(\theta > 0)$$. It starts at a certain distance from the origin when $$\theta$$ is small and spirals inwards, getting closer and closer to the origin as $$\theta$$ increases, without ever touching the origin.

Key Concepts

Spirals

In mathematics, spirals are fascinating curves that twist around a point with increasing or decreasing distance from that point. They come in various forms, such as Archimedean, logarithmic, and hyperbolic spirals, each defined by a unique mathematical equation.

In the case of the polar equation \( r = \frac{1}{\theta} \) where \( \theta > 0 \), we are looking at a hyperbolic spiral. As \( \theta \) increases towards infinity, the value of \( r \) gets smaller, causing the curve to spiral inwards towards the origin. This inwards spiral illustrates how each point on the curve is getting closer to the origin as \( \theta \) grows larger.

Spirals are not only beautiful but also have practical applications. They appear in nature, such as in the patterns of shells or galaxies, and in human-made objects like spirals springs or the spiral binding of a notebook.

• Spiral patterns can result from simple mathematical rules.
• They are often seen in both nature and technology.

Graph Sketching

Sketching a graph, especially in polar coordinates, involves visualizing how values of \( r \) change with \( \theta \). It’s about bringing equations to life and understanding their behavior.

For the equation \( r=\frac{1}{\theta} \), we start by plotting initial points with small \( \theta \), like \( \frac{\pi}{6}, \frac{\pi}{4}, \) and \( \frac{\pi}{3} \). These help set the foundation by showing an initial outward position on the polar graph.
As \( \theta \) increases, \( r \) decreases, pointing towards the origin. The graph starts from the first plotted point and draws a spiral, continuously contracting as \( \theta \) keeps growing.

Key steps in graph sketching:

• Start with a few strategic points to understand initial direction.
• Observe how changes in \( \theta \) affect \( r \) and adjust the curve accordingly.
• Consider the behavior of the curve at limiting values, here as \( \theta \) goes to infinity.

Limits as Theta approaches Infinity

Understanding limits, especially as a variable approaches infinity, is crucial in mathematics for determining the behavior of functions. When \( \theta \) climbs higher and higher in our polar equation \( r = \frac{1}{\theta} \), it directly affects the value of \( r \).

As \( \theta \) moves towards infinity, \( \frac{1}{\theta} \) approaches zero. This indicates that \( r \), the radius from the origin, shrinks towards zero while \( \theta \) grows. Despite this shrinkage, \( r \) never truly reaches zero. Because \( \theta \) is always positive, \( r \) remains a positive value, getting infinitesimally small, but not reaching the origin.

This concept is significant in understanding asymptotic behavior—how curves approach a line or point but never quite touch it:

• The limit tells us how the spiral gets infinitely close to the origin without touching it.
• It demonstrates the power of limits in determining behavior at extremities.