Question

Use technology to plot \(r=e^{-0.1 \theta}\) for \(-10 \leq \theta \leq 10\).

Short answer

Use a polar graphing tool to plot the curve for \(r = e^{-0.1\theta}\) from \(\theta = -10\) to \(10\).

Step-by-step solution

Understanding the Polar Equation

The given equation is in polar form, where \(r\) is defined in terms of \(\theta\). Here, \(r = e^{-0.1 \theta}\), which describes a curve in polar coordinates. The negative exponent means \(r\) decreases as \(\theta\) increases.

Choosing a Graphing Tool

To plot this polar equation, choose any graphing software or calculator that supports polar plots. Common choices include Desmos, GeoGebra, or graphing calculators with polar capabilities.

Entering the Polar Equation

Input the equation \(r = e^{-0.1 \theta}\) into the graphing tool. Ensure you set the plotting mode to 'polar' to handle the \(r(\theta)\) format correctly.

Setting the Domain for \(\theta\)

Restrict the domain of \(\theta\) from \(-10\) to \(10\) as per the given range. This ensures that the plot only shows the section described by this range.

Viewing and Analyzing the Graph

Once the graph is plotted, observe the resulting polar curve. The graph should show how the value of \(r\) changes as \(\theta\) varies from \(-10\) to \(10\). The curve will start with larger values of \(r\) when \(\theta\) is negative and decrease as \(\theta\) approaches positive values.

Key concepts

Polar Coordinates

Polar coordinates offer a unique way of describing positions in a plane using a radius and angle rather than the traditional Cartesian coordinates of x and y. This system is incredibly useful for scenarios where circular or rotational motions are involved. In polar coordinates, a point is defined by two parameters:

• \(r\): The radius or distance from the origin (center of the plane).
• \(\theta\): The angle measured from the positive x-axis.

To understand a polar equation such as \(r = e^{-0.1 \theta}\), it’s important to visualize how \(r\) changes as \(\theta\) varies. Here, you can think of it as a spiral that tightens as \(\theta\) increases. Consequently, when graphing, you’ll observe that the radius or repetition pattern of the spiral becomes smaller. This is an essential concept in applications of polar coordinates, especially when dealing with oscillating or looping motions.

Graphing Technology

With the advancement in technology, graphing equations has become much easier. Using software tools like Desmos, GeoGebra, or specialized graphing calculators allows you to visualize complex equations without manual plotting. To graph an equation in polar coordinates, you need to ensure your tool supports polar mode. Enter the equation precisely, such as \(r = e^{-0.1 \theta}\), and adjust your settings to accommodate the required range for \(\theta\). This often involves:

• Selecting the correct graphing type, like polar or parametric.
• Setting the angle range, ensuring \(\theta\) spans from \(-10\) to \(10\), for this exercise.
• Adjusting the scale to better view the nuances of the graph, like zooming in to see more detail.

These steps aid in a comprehensive understanding of the behavior of equations, especially those that might not be intuitive at first glance. Technology thereby enhances learning by providing immediate visual feedback.

Exponential Functions

Exponential functions, denoted generally as \(f(x) = a^{bx}\), where \(a\) is a constant base and \(bx\) is the exponent, are crucial in many areas of mathematics and science. They depict growth or decay at a constant relative rate and appear frequently in natural processes.In the specific equation \(r = e^{-0.1 \theta}\), we deal with an exponential decay. Here's what to note:

• The base \(e\) is the natural exponential constant, approximately equal to 2.71828.
• The exponent \(-0.1 \theta\) indicates a decay factor, meaning \(r\) diminishes as \(\theta\) becomes larger.

Understanding exponential decay helps in grasping various natural phenomena, such as radioactive decay, cooling processes, and population reduction. It’s essential to comprehend how the rate of decrease or increase influences the shape of the graph. In this context, the exponential aspect of the polar equation results in a curved spiral as \(\theta\) is varied, visually demonstrating how quickly or slowly changes occur as \(\theta\) shifts.