Problem 101 - Calculus Early Transcendentals

Question

Graph the following spirals. Indicate the direction in which the spiral is generated as \(\theta\) increases, where \(\theta>0 .\) Let \(a=1\) and \(a=-1\). Logarithmic spiral: \(r=e^{a \theta}\)

Step-by-Step Solution

Answer

Short Answer: For the logarithmic spiral with equation \(r = e^{a\theta}\), when \(a=1\), the spiral moves in a counterclockwise direction as the angle θ increases, and when \(a=-1\), the spiral moves in a clockwise direction as the angle θ increases.

Step 1: Understand the given equation of the spiral

The equation of the logarithmic spiral is given as \(r = e^{a\theta}\), where 'r' is the distance from the origin, 'a' is a constant, and θ is the polar angle. The spiral's direction will depend on the sign of 'a'.

Step 2: Determine the equation with the given values of 'a'

For the first case, let \(a=1\). The equation becomes: \(r = e^{\theta}\) For the second case, let \(a=-1\). The equation becomes: \(r = e^{-\theta}\)

Step 3: Determine the direction of the spirals

The direction of the spiral is determined by the rate at which r changes with respect to θ. Let's analyze this by finding \(\frac{dr}{d\theta}\) for both cases: Case 1: \(a=1\) \(\frac{dr}{d\theta} = \frac{d}{d\theta}(e^{\theta}) = e^{\theta}\) Since \(e^{\theta}\) is always positive (as θ is greater than 0), the spiral increases in radius as θ increases. Case 2: \(a=-1\) \(\frac{dr}{d\theta} = \frac{d}{d\theta}(e^{-\theta}) = -e^{-\theta}\) As \(e^{-\theta}\) is always positive and we have the negative sign, the overall value is negative. This indicates that the radius of the spiral decreases as θ increases.

Step 4: Graph the spirals

To graph the spirals, we will plot them in polar coordinates using the equations derived in Step 2. For \(r = e^{\theta}\), the spiral starts at the origin \((0,0)\) and increases in radius as θ increases. The spiral moves in a counterclockwise direction. For \(r = e^{-\theta}\), the spiral also starts at the origin \((0,0)\), but its radius decreases as θ increases, which means it moves towards the origin. The spiral moves in a clockwise direction. In conclusion, for the logarithmic spiral \(r = e^{a\theta}\), the direction in which the spiral is generated depends on the value of 'a'. If \(a=1\), the spiral moves in a counterclockwise direction, and if \(a=-1\), the spiral moves in a clockwise direction.

Key Concepts

Understanding Polar Coordinates

In polar coordinates, we describe a point's location on a plane using the distance from the origin and the angle from a reference direction. This system is different from the more familiar Cartesian coordinates, which use x and y coordinates to define a point. In polar coordinates:

• \(r\) represents the radius, which is the distance from the origin to the point.
• \(\theta\) is the angle measured from the positive x-axis, usually in radians.

To plot a point in polar coordinates, you first move \(r\) units from the origin in the direction of \(\theta\). This coordinate system is particularly useful for graphing curves like spirals, where the relationship between \(r\) and \(\theta\) defines the shape.

Determining Spiral Direction

The direction in which a spiral is generated depends on how \(r\) changes as \(\theta\) increases. For the logarithmic spiral given by \(r = e^{a\theta}\), the direction is influenced by the value of \(a\):

• If \(a=1\), the expression \(e^{\theta}\) increases as \(\theta\) increases, meaning the spiral expands outward, moving in a counterclockwise direction.
• If \(a=-1\), \(e^{-\theta}\) decreases, so the spiral contracts inward, moving in a clockwise direction.

This behavior is explained by examining the derivative \(\frac{dr}{d\theta}\), which indicates how the radius changes with respect to the angle. A positive derivative means the radius is growing, and a negative derivative means it is shrinking.

Application of Differential Equations

Differential equations play a crucial role in understanding the behavior of spirals. By finding \(\frac{dr}{d\theta}\), we determine how the radius changes as \(\theta\) varies. For a logarithmic spiral, we have:

• For \(a=1\), \(\frac{dr}{d\theta} = e^{\theta}\), which is always positive. This implies continuous growth in radius.
• For \(a=-1\), \(\frac{dr}{d\theta} = -e^{-\theta}\), leading to a negative result, indicating decreasing radius.

By analyzing these equations, students can predict the spiral's behavior as \(\theta\) changes. This connection between the rate of change and geometry is central to many mathematical models.

Techniques for Graphing Spirals

Graphing spirals involves plotting points in polar coordinates based on the relationship \(r = e^{a\theta}\). Follow these steps:

• Start at the origin, \((0,0)\).
• For \(a=1\), as \(\theta\) increases, plot points further from the origin, tracing a counterclockwise path.
• For \(a=-1\), as \(\theta\) increases, plot points closer to the origin, resulting in a clockwise path.

Using software or graphing tools can aid in visualizing these patterns, but understanding the polar coordinate system and how it transforms the equation into a path is key to mastering graphing of spirals.