Problem 37 - Calculus III

Question

Consider the curve described by the vector-valued function \(\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}\). [T] Let \(r(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+0.3 \sin (2 t) \mathbf{k}\). Use technology to graph the curve (called the roller-coaster curve) over the interval \([0,2 \pi)\). Choose at least two views to determine the peaks and valleys.

Step-by-Step Solution

Answer

Use a graphing tool to visualize the curve from different angles and spot peaks and valleys based on the \( z \)-value extrema.

Step 1: Understanding the Curve Function

The given vector-valued function for the curve is \( \mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k} \). This expression describes a three-dimensional curve where \( t \) varies over an interval, creating a path in space.

Step 2: Setting Up the Graphing Tool

To visualize the curve, use a graphing technology, such as GeoGebra, Desmos, or a similar software, that allows for 3D plotting. Enter the expressions for each component of the vector \( x(t) = 50 e^{-t} \cos t \), \( y(t) = 50 e^{-t} \sin t \), and \( z(t) = 5 - 5 e^{-t} \) to graph it over the interval \([0, 2\pi)\).

Step 3: Analyzing the Views to Find Peaks and Valleys

Graph the vector \( \mathbf{r}(t) = \cos t \mathbf{i}+\sin t \mathbf{j}+0.3 \sin (2 t) \mathbf{k} \) using the same software over the interval \([0,2\pi)\). Use the software's 'rotate' feature to explore different angles, helping you identify the highest points (peaks) and lowest points (valleys) of the curve.

Step 4: Confirming Peaks and Valleys Positions

Identify specific values of \( t \) where the peaks and valleys occur by observing the graph's maximum and minimum \( z \)-values. Peaks are where the \( z \)-component (height) reaches local maximums, and valleys are where it attains local minimums. Compare these with the pattern given by the term \( 0.3 \sin(2t) \).

Key Concepts

3D Curve Plotting

When dealing with vector-valued functions, plotting a curve in three dimensions starts by understanding the expression that describes the path. Consider the function \( \mathbf{r}(t) = (50 e^{-t} \cos t) \mathbf{i} + (50 e^{-t} \sin t) \mathbf{j} + (5 - 5 e^{-t}) \mathbf{k} \). This represents a curve in space, where the position of a point on the curve depends on the parameter \( t \). As \( t \) varies, the values of \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) components change consequently, tracing a specific path in a 3D space.
One efficient way to visualize such curves is through the use of graphing tools that specialize in 3D plots. These tools help translate the mathematical expression into a graphical form that's easier to interpret. Typically, students or practitioners look for software that can handle these types of plots well, such as GeoGebra or Desmos, which offer a user-friendly interface to deal with complex equations.

• Start by decomposing the vector function into its constituent parts: \( x(t) = 50 e^{-t} \cos t \), \( y(t) = 50 e^{-t} \sin t \), and \( z(t) = 5 - 5 e^{-t} \).
• Ensure you set the parameter \( t \) to span the required interval, here \([0, 2\pi)\).

With these settings, a precise plot of the 3D curve can be achieved, allowing further analysis and exploration.

Graphing Technology

Graphing technology revolutionizes the way we examine and interact with complex mathematical functions. These technologies are designed to convert variables and equations into interactive visual experiences. They not only help in plotting curves but also provide dynamic features that enhance learning and understanding.
For example, when tackling the plot of \( \mathbf{r}(t) \),

• Input the components directly into the software, respecting the structure of \( x(t), y(t), \) and \( z(t) \).
• Most graphing technologies offer a variety of tools to customize the view, including zoom, rotation, and shifting capabilities.

These functions allow users to manipulate the 3D view of the curve, enabling them to observe its unique characteristics from multiple angles. Moreover, features like overlays and contrasting colors enhance clarity and distinction, making complex structures easier to dissect. Students can leverage these technologies to predict behaviors of curves and verify their understanding by comparing visual outcomes with theoretical predictions.

Peaks and Valleys Identification

Identifying peaks and valleys in a 3D curve is an essential skill in understanding the nature of roller-coaster-like functions, often studied under vector-valued functions. To find these position points, you should focus on the \( z \)-component of the curve, which relates to height or elevation in the 3D space.
Using graphing tools, one can rotate and examine the plot for maximum and minimum heights:

• Peaks are identified when the \( z \)-value reaches its local maximum, which corresponds to the highest points of the curve.
• Valleys occur at the local minimum points, where \( z \) is at its lowest.

For deeper understanding, compare how the changes in the curve, particularly influenced by terms like \( 0.3 \sin(2t) \), contribute to these highs and lows.
By carefully manipulating the view with the software, you ensure that all critical points are visible, effectively highlighting positions where these peaks and valleys align. This analysis is crucial when it comes to drafting real-world paths and systems, like actual roller coasters, where the thrill comes from these varied elevations.