Problem 15 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of the given equation.\(r=1 / \theta\) (reciprocal spiral)

Step-by-Step Solution

Answer

Draw a polar graph with key points, illustrating the spiral shape showing the inverse relationship between r and θ.

Step 1: Understand the equation

The given equation is a polar equation, where the radius r is inversely proportional to the angle θ. In polar coordinates, the radius (r) and the angle (θ) define the position of a point.

Step 2: Know the Range of θ

The angle θ can take any positive value. This is important because it will affect the radius r.

Step 3: Inverse Relationship

Since the relationship is given by the reciprocal function, as θ increases, the radius r decreases. Mathematically, it means that when θ approaches 0, r approaches infinity.

Step 4: Plot Key Points

Calculate a few key points to plot on the polar graph. For example: - When θ=1, r=1.- When θ=2, r=0.5- When θ=π, r=1/π.

Step 5: Draw the Graph

Plot the points found in Step 4 on polar grid paper. Connect these points smoothly, remembering that as θ gets smaller, r gets larger, and as θ grows larger, r gets smaller. This will create a spiral shape.

Step 6: Final Touches

Review the graph to ensure it smoothly transitions according to the function. The spiral should be tighter as it moves outward because r decreases faster as θ increases.

Key Concepts

Reciprocal Spiral

The concept of a reciprocal spiral comes from the polar equation given as \(r = \frac{1}{\theta}\). This equation describes how the radius \(r\) is inversely proportional to the angle \(\theta\).

Understanding this relationship is crucial for identifying how the points are plotted on a polar graph. This type of graph is often referred to as a reciprocal spiral due to its spiral-like appearance. As \(\theta\) increases, the radius \(r\) decreases smoothly, forming a spiraling curve that tightens as it moves outward.

• Reciprocal means one number is divided by another.
• Spiral means a curve that keeps winding around a central point.

Therefore, as \(\theta\) gets larger, \(r\) gets smaller, creating a spiral.

Inverse Relationship

An inverse relationship in mathematics refers to a situation where one quantity increases as another quantity decreases. In the context of a polar graph described by \(r = \frac{1}{\theta}\), this means:

• When \(\theta\) is small, \(r\) is large.
• When \(\theta\) is large, \(r\) is small.

This relationship implies that the radius \(r\) and the angle \(\theta\) change in opposite directions. Such a relationship is fundamental to understanding how to plot a reciprocal spiral.

As \(\theta\) approaches zero, \(r\) approaches infinity, and as \(\theta\) increases, the radius \(r\) continues to decrease towards zero. This behavior explains why the spiral coils more tightly as you move outward from the center.

Polar Graph Points

When plotting a polar equation, placing points accurately on the graph ensures the correct shape and behavior of the curve. Each point on a polar graph is determined by its polar coordinates: the radius \(r\) and the angle \(\theta\).

For the equation \(r = \frac{1}{\theta}\), some key points calculated for plotting include:

• When \(\theta = 1\), \(r = 1\).
• When \(\theta = 2\), \(r = 0.5\).
• When \(\theta = \pi\), \(r = \frac{1}{\pi}\).

Start by plotting these points on polar grid paper. Remember the inverse relationship – as you increase \(\theta\), the radius \(r\) will get smaller. This inverse proportionality helps guide the plotting of additional points to fully shape the reciprocal spiral.

Plotting Polar Equations

Plotting polar equations involves translating mathematical relationships into visual graphs. Here’s a simple step-by-step guide to plotting polar equations, using \(r = \frac{1}{\theta}\) as an example:

1. **Understand the Equation:** Recognize the type of relationship - here it’s an inverse relationship
2. **Range of \(\theta\):** Know that \(\theta\) can take any positive value.
3. **Calculate Key Points:** Substitute different values for \(\theta\) to get corresponding \(r\).

Examples include:

• \(\theta = 1\), \(r = 1\)
• When \(\theta = 2\), \(r = 0.5\)
• When \(\theta = \pi\), \(r = \frac{1}{\pi}\)

4. **Plot on Polar Grid Paper:** Place the points on the grid and draw the curve smoothly. Remember, the graph should spiral inward as \(\theta\) increases.

5. **Review and Adjust:** Check that the points create a smooth and accurate curve. Take special note as \(\theta\) approaches zero, where \(r\) should head toward infinity. This practice helps in visualizing complex functions and understanding the geometry of inverse relationships in polar coordinates.