Problem 32 - Precalculus with Calculus Previews

Question

The graph of the given equation is a spiral. Sketch its graph. $$ r \theta=\pi, \theta>0 \text { (hyperbolic) } $$

Step-by-Step Solution

Answer

The hyperbolic spiral starts far from the origin and winds inward as \( \theta \) increases, getting tighter around the origin.

Step 1: Convert Polar Equation to Cartesian Coordinates

The given equation is in polar form: \( r \theta = \pi \). First, solve for \( r \) in terms of \( \theta \):\[ r=\frac{\pi}{\theta}\]We know that in polar coordinates, \( x=r\cos(\theta) \) and \( y=r\sin(\theta) \). Substitute \( r \) from the equation:\[ x = \frac{\pi}{\theta} \cos(\theta),\quad y = \frac{\pi}{\theta} \sin(\theta)\]

Step 2: Study Behavior of the Spiral

Analyze the behavior of \( x = \frac{\pi}{\theta} \cos(\theta) \) and \( y = \frac{\pi}{\theta} \sin(\theta) \) as \( \theta \to \infty \) and \( \theta \to 0^+ \).As \( \theta \to 0^+ \), \( r \to \infty \) which leads to points moving away from the origin. As \( \theta \to \infty \), \( r \to 0 \), which suggests points move towards the origin. These observations confirm that the spiral begins far from the origin and winds inwardly as \( \theta \) increases.

Step 3: Draw the Spiral

Using the behavior analysis, begin sketching the spiral by marking key points and direction:- Start with a large radius when \( \theta \) is near zero, plotting points further from the origin.- Gradually decrease the radius of plotted points as \( \theta \) increases.- Ensure the spiral wraps around (counterclockwise) and tightens towards the origin.This forms a hyperbolic spiral depicted with decreasing radii as it approaches the origin.

Key Concepts

Polar Coordinates

Polar coordinates are a system used to describe a position on a plane using a distance and an angle. Think about it as giving directions: you would say how far to go and at what angle. In mathematics, this system is particularly useful for curves like spirals.

• The distance from the origin is represented as \( r \).
• The angle from the positive x-axis is \( \theta \).

To describe the position of a point, we use the pair \( (r, \theta) \). This can be handy when sketching circular or spiral shapes since it naturally fits into these patterns.

Cartesian Coordinates

Cartesian coordinates are probably what you first learned about graphing. They use an x-axis and y-axis to locate a point on a plane. For curves, this system maps those points using algebraic expressions. We can convert between polar and Cartesian coordinates because each system serves different analytical purposes. For instance:

• To find the x-coordinate: \( x = r \cos(\theta) \)
• To find the y-coordinate: \( y = r \sin(\theta) \)

These formulas allow for transforming spirals, or any other polar curves, into the familiar Cartesian plane expression.

Graph Sketching

Graph sketching is where creativity meets analysis in math. To sketch the hyperbolic spiral given by the equation \( r\theta = \pi \), understanding the transition from polar to Cartesian coordinates becomes vital.

• Start by marking points for small values of \( \theta \), where \( r \) becomes large, meaning the points are far from the origin.
• As \( \theta \) grows larger, the radius \( r \) decreases, moving the points closer to the origin.
• Ensuring smooth curves by avoiding abrupt changes helps form a clear image of the spiral pattern.

This practice makes complex equations more visual and understandable.

Behavior Analysis

Analyzing the behavior of the hyperbolic spiral involves looking at how the spiral moves as \( \theta \) changes.

• As \( \theta \to 0^+ \), \( r \to \infty \), which means the spiral starts far from the origin.
• As \( \theta \to \infty \), \( r \to 0 \), showing that the spiral approaches the origin and gets tighter.
• This movement indicates a counterclockwise winding inward pattern.

Understanding these behaviors is crucial for predicting how such a spiral will appear without having to compute many points.

Spiral Equation

The given spiral equation \( r \theta = \pi \) specifies a unique pattern known as a hyperbolic spiral.In this form:

• \( r \) and \( \theta \) are inversely proportional, meaning as one increases, the other decreases.
• This equation allows us to see how points fan out as distance increases and eventually come closer as the angle increases.

Hyperbolic spirals are fascinating in mathematics because of this inward winding while retaining a regular, ever-tightening path. Such equations inspire and challenge us to understand their geometric intricacies.