Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=e^{\theta}, \theta \geq 0\) (logarithmic spiral)

Step-by-Step Solution

Answer

The logarithmic spiral \(r=e^\theta\) has no symmetry.

Step 1: Understand the Polar Equation

The given polar equation is \(r = e^\theta\). In polar coordinates, \(r\) represents the radius or the distance from the origin, and \(\theta\) is the angle from the positive x-axis. The equation describes a logarithmic spiral because \(r\) increases exponentially with \(\theta\).

Step 2: Plot Key Points

To sketch the graph, plot key points by calculating \(r\) for several values of \(\theta\). For example:- For \(\theta = 0\), \(r = e^0 = 1\).- For \(\theta = \frac{\pi}{4}\), \(r = e^{\frac{\pi}{4}}\).- For \(\theta = \frac{\pi}{2}\), \(r = e^{\frac{\pi}{2}}\).- For \(\theta = \pi\), \(r = e^{\pi}\).Continue this process to capture the curve's nature.

Step 3: Sketch the Curve

Using the points identified in Step 2, draw the curve. The curve should show a spiral starting at \(r=1\) when \(\theta=0\) and spiraling outward while \(r\) increases as \(\theta\) increases. This is typical of a logarithmic spiral.

Step 4: Verify Symmetry

To verify the symmetry of the graph, we consider typical symmetries:- **Polar Axis Symmetry:** Not applicable since \(r(e^{\theta}) eq -r(e^{\theta})\).- **Line \(\theta = \frac{\pi}{2}\):** Not applicable due to the nature of exponential growth, \(e^{\theta}\) isn’t equal to \(e^{\pi - \theta}\).- **Origin Symmetry:** Checked by replacing \(\theta\) with \(\theta+\pi\), which gives \(r = e^{\theta + \pi} = re^{\pi}\), not symmetrical with \(r\). Thus, the spiral has no symmetry with respect to these criteria.

Key Concepts

Logarithmic Spiral

The logarithmic spiral is a fascinating and beautiful curve that appears in many natural phenomena, such as the shapes of galaxies, hurricanes, and even certain seashells. It's defined by the polar equation \(r = e^{\theta}\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis.

One interesting property of the logarithmic spiral is that the angle between the tangent and the radial line at any point is constant. This gives the spiral its regular, self-similar shape, meaning each part of the spiral is a scaled version of any other part.

• The starting point is at \(r=1\) when \(\theta=0\).
• As \(\theta\) increases, \(r\) grows exponentially, leading to an ever-widening curve.
• It spirals outwards infinitely but never actually reaches the center.

Polar Coordinates

Polar coordinates are a system used to define positions on a plane using a radius and an angle. Unlike Cartesian coordinates which use \((x, y)\), polar coordinates specify a point's location by \((r, \theta)\).

- \(r\) is the radial distance from the origin (the pole).- \(\theta\) is the angle from the positive x-axis.This system is particularly useful for problems involving circles and spirals, where radial symmetry simplifies computations.

In our exercise, plotting the equation \(r = e^{\theta}\) involves converting these coordinates back to Cartesian form for visual representation. You calculate \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\) to plot on the Cartesian plane.

Graph Symmetry

Graph symmetry is a property that describes how a graph is reflected or rotated around axes or points. Detecting symmetry can simplify graph analysis.

• Polar Axis Symmetry: Occurs if the graph looks unchanged when rotated around the polar axis.
• Line Symmetry: Involves reflection across line \(\theta = \frac{\pi}{2}\).
• Origin Symmetry: The graph can be rotated \(180^\circ\) around the origin.

For the logarithmic spiral, none of these symmetries apply due to the nature of exponential growth. The equation \(r = e^{\theta}\) leads to an asymmetric spiral, growing in only one direction round.

Exponential Growth

Exponential growth refers to the process where the value of a function increases at a rate proportional to its current value. In the context of the logarithmic spiral, the equation \(r = e^{\theta}\) shows us that as \(\theta\) increases, \(r\) grows very rapidly.

Key characteristics of exponential growth in polar equations include:

• a steep curve that increases more sharply as \(\theta\) increases,
• the radius increasing infinitely, but the spiral never closes in on itself,
• no point of symmetry, making the curve unique and continuously expanding.

Understanding this exponential growth helps clarify why the spiral has its characteristic outward expanding shape.