Question

There is a curve known as the "Black Hole. " Use technology to plot $r=e^{-0.01\theta }$ for $-100\le \theta \le 100$.

Short answer

Plot the curve for $r=e^{-0.01\theta }$ using graphing software with $\theta$ from -100 to 100.

Step-by-step solution

Identify the Equation

The equation to plot is given as $r=e^{-0.01\theta }$, where $\theta$ is the independent variable that ranges from $-100\le \theta \le 100$. This equation represents a polar curve where $r$ is expressed in terms of $\theta$.

Understand the Plotting Range

In this problem, $\theta$ ranges from $-100$ to $100$. This defines the domain of the polar plot and dictates the scope of the graph. It's important to consider this range to ensure the curve is fully displayed.

Utilize Technology to Plot the Curve

To plot this curve, use graphing software or a graphing calculator capable of plotting polar equations. Enter the equation $r=e^{-0.01\theta }$ and set the $\theta$ range from $-100$ to $100$. This will generate the visualization of the curve known as the 'Black Hole.'

Analyze the Plot

Once plotted, the curve will spiral inward as $\theta$ increases, due to the exponential decay in the equation. This behavior results from the $e^{-0.01\theta }$ term, which decreases $r$ as $\theta$ increases. Observe the overall shape and confirm it matches the expected pattern based on the equation's nature.

Key concepts

Exponential Decay

Exponential decay is a mathematical concept where a quantity decreases at a rate proportional to its current value. In the context of the equation $r=e^{-0.01\theta }$, exponential decay is illustrated through how the value of $r$ changes as $\theta$ increases or decreases.
Some key points about exponential decay include:

• The base of the exponential function is Euler's number $e$, a constant approximately equal to 2.71828, which naturally arises in contexts involving growth and decay.
• The rate of decay is determined by the exponent. Here, $-0.01$ indicates a relatively slow rate of decay, implying that the curve will not lose its size too quickly.
• As $\theta$ increases, $e^{-0.01\theta }$ becomes smaller, meaning that $r$ decreases, making the curve spiral inward when plotted.
• Exponential decay is common in natural processes like radioactive decay and cooling processes, highlighting its practical applicability across various fields.

Understanding exponential decay helps visualize how values change over time and comprehend the behavior of the 'Black Hole' curve as $\theta$ shifts.

Graphing Technology

Graphing technology refers to the tools and devices that assist in plotting equations and visualizing mathematical concepts. In dealing with the assignment to plot $r=e^{-0.01\theta }$, graphing technology becomes essential for rendering the polar curve efficiently.
Here’s how graphing technology can aid in plotting:

• Graphing calculators and software, such as Desmos or GeoGebra, can quickly compute complex equations and plot them over specified domains.
• These technologies facilitate adjustments in parameters and viewing angles, permitting a comprehensive exploration of the graph's nature.
• By entering the function and setting the domain for $\theta$ from $-100$ to $100$, students can immediately visualize the polar curve and better understand its characteristics.
• Graphing technology saves time and reduces the potential for errors associated with manual plotting, especially important for intricate plots like spirals or complex 3D functions.

Incorporating graphing technology into learning helps bridge the gap between theoretical understanding and practical application, making it a valuable resource for students.

Polar Curve Plotting

Polar curve plotting involves graphing equations where the position of a point is determined by: the distance from the origin $r$, and the angle $\theta$ with respect to the positive x-axis. This differs from Cartesian plotting, where points are defined by x and y coordinates.
Some characteristics of polar curve plotting are:

• Polar coordinates are expressed as $(r,\theta )$, where $r$ is the radial distance from the origin, and $\theta$ is the angular displacement.
• The equation $r=e^{-0.01\theta }$ represents a polar curve where $\theta$ dictates how $r$ changes, resulting in the spiral 'Black Hole' shape.
• Unlike Cartesian plots, polar coordinates can extend infinitely around the origin as $\theta$ varies, facilitating the depiction of intricate patterns.
• Polar plots are often circular or spiral in nature, showcasing intriguing visual properties not seen in Cartesian graphs.

Polar curve plotting is a unique method that enriches comprehension of mathematical behaviors in circular paths and rotational symmetries, and it helps visual learners grasp complex functional relationships.