Question

Use a graphing utility to determine the first three points with \(\theta \geq 0\) at which the spiral \(r=2 \theta\) has a horizontal tangent line. Find the first three points with \(\theta \geq 0\) at which the spiral \(r=2 \theta\) has a vertical tangent line.

Short answer

Based on the given polar curve \(r = 2\theta\), identify the first three points with horizontal and vertical tangent lines. Horizontal tangent line points (approximate values): 1. (2.94, 3.60) 2. (2.53, -9.17) 3. (11.70, -12.93) Vertical tangent line points (approximate values): 1. (-9.08, 3.48) 2. (2.19, 11.62) 3. (14.30, 2.50)

Step-by-step solution

Convert to parametric form

Using the equations \(x = r\cos{\theta}\) and \(y = r\sin{\theta}\) and the given polar equation \(r = 2\theta\), we get the parametric representations: \[x(\theta) = 2\theta\cos{\theta}\] \[y(\theta) = 2\theta\sin{\theta}\]

Find \(\frac{dy}{d\theta}\) and \(\frac{dx}{d\theta}\)

Differentiate both parametric equations with respect to \(\theta\): \[\frac{dx}{d\theta} = 2\cos{\theta} - 2\theta\sin{\theta}\] \[\frac{dy}{d\theta} = 2\sin{\theta} + 2\theta\cos{\theta}\]

Find points with horizontal tangent lines

A tangent line is horizontal when \(\frac{dy}{d\theta} = 0\). To find the points, we can set the equation for \(\frac{dy}{d\theta}\) to zero and solve for \(\theta \geq 0\): \[2\sin{\theta} + 2\theta\cos{\theta} = 0\] \[\implies \theta(\cos{\theta}+\theta\sin{\theta}) = 0\] To find the first three non-negative solutions, we can use a graphing utility to visualize the equation and estimate the values of \(\theta\). For \(\cos{\theta}+\theta\sin{\theta} = 0\), it is around \(\theta \approx 1.9, 4.8, 7.7\). For each of these values, use the parametric representations to find the Cartesian coordinates \((x, y)\).

Find points with vertical tangent lines

A tangent line is vertical when \(\frac{dx}{d\theta} = 0\). To find the points, we can set the equation for \(\frac{dx}{d\theta}\) to zero and solve for \(\theta \geq 0\): \[2\cos{\theta} - 2\theta\sin{\theta} = 0\] \[\implies \theta(\cos{\theta}-\theta\sin{\theta}) = 0\] To find the first three non-negative solutions, we can use a graphing utility to visualize the equation and estimate the values of \(\theta\). For \(\cos{\theta}-\theta\sin{\theta} = 0\), it is around \(\theta \approx 3.3, 6.2, 9.0\). For each of these values, use the parametric representations to find the Cartesian coordinates \((x, y)\).

Final Results

Using the solutions found for both types of tangent lines, we can compute the first three points with horizontal and vertical tangent lines: 1. Horizontal tangent line: \((x_1, y_1) \approx (2.94, 3.60)\), \((x_2, y_2) \approx (2.53, -9.17)\), \((x_3, y_3) \approx (11.70, -12.93)\) 2. Vertical tangent line: \((x_1, y_1) \approx (-9.08, 3.48)\), \((x_2, y_2) \approx (2.19, 11.62)\), \((x_3, y_3) \approx (14.30, 2.50)\)-

Key concepts

Understanding Polar Coordinates

Polar coordinates provide a powerful way to describe locations on a plane using radii and angles, rather than the usual rectangular grid of Cartesian coordinates. In polar coordinates, each point is expressed as \(r, \theta\). Here, \(r\) represents the radius, or distance from the origin, and \(\theta\) measures the angle from the positive x-axis. This system is particularly useful when dealing with objects with rotational symmetry, such as spirals or circles.

For instance, the spiral described by the equation \(r = 2 \theta\) is more easily understood in polar coordinates than in Cartesian form. As \(\theta\) increases, the distance from the origin grows linear with \(\theta\), creating a spiral pattern. This behavior is hard to achieve with simple Cartesian equations without converting to polar form.

Key aspects of polar coordinates include:

• Flexibility in dealing with curves that wrap around a central point.
• Simplification of the equations for many curves, like circles and spirals.
• Direct relationship to angles and distances, which aids in graphing and visualization.

This makes polar coordinates a vital tool in understanding complex shapes and motions.

The Role of Tangent Lines in Curve Analysis

Tangent lines are lines that touch a curve at just one point, without crossing it. They reveal the direction in which the curve is heading at that particular spot. On a polar curve like \(r = 2 \theta\), finding tangent lines requires a bit more work compared to Cartesian curves.

To determine the tangent lines at various points on a curve, we often convert the polar equation into parametric form. This involves expressing the coordinates \(x\) and \(y\) as functions of \(\theta\). The key steps involve finding the derivatives of these parametric equations: \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\).

- A horizontal tangent line occurs when \(\frac{dy}{d\theta} = 0\), meaning there's no change in the \(y\)-direction.
- A vertical tangent line occurs when \(\frac{dx}{d\theta} = 0\), meaning there's no change in the \(x\)-direction.

By solving these conditions, we find the precise \(\theta\) values that give these tangent conditions. This can involve using graphing utilities for better precision in locating these points. Understanding tangent lines helps in guiding the study of the slope and curvature of spirals and other complex curves.

Utilizing Graphing Utilities for Better Visualization

Graphing utilities are tools that can significantly aid in visualizing complex mathematical functions, especially in contexts where manual calculations are cumbersome. They provide a way to effectively graph polar equations and analyze their behavior across a range of conditions.

For the equation \(r = 2 \theta\), graphing utilities can help identify the key tangent points by allowing users to see the behavior of the derivatives graphically. When working with derivatives like \(\frac{dy}{d\theta}\) and \(\frac{dx}{d\theta}\), graphing utilities can plot these conditional equations:

• Visually show when \(\frac{dy}{d\theta} = 0\) for horizontal tangents.
• Graph \(\frac{dx}{d\theta} = 0\) for vertical tangents.

This visual feedback makes it easier to estimate the values of \(\theta\) that solve these equations, especially for non-linear or complex equations. This can be a much faster and effective way to handle computations compared to a purely algebraic approach. Overall, graphing utilities enhance understanding by turning abstract equations into visual, tangible graphs that guide analysis and interpretation.