Problem 31 - Calculus Early Transcendentals

Question

\(29-32\) Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve. $$ \mathbf{r}(t)=\langle t, t \sin t, t \cos t\rangle $$

Step-by-Step Solution

Answer

Graph the curve with a parameter range \(-4\pi\) to \(4\pi\) to reveal its spiral nature.

Step 1: Understand the Vector Equation

The vector equation \( \mathbf{r}(t) = \langle t, t \sin t, t \cos t \rangle \) describes a three-dimensional parametric curve. Here, \( t \) is the parameter, and it defines the components of the vector as functions of \( t \): the x-component as \( t \), the y-component as \( t \sin t \), and the z-component as \( t \cos t \).

Step 2: Determine a Suitable Parameter Domain

We need to choose a parameter domain for \( t \) that reveals the nature of the curve. Since \( \sin t \) and \( \cos t \) are periodic with a period of \( 2\pi \), choosing \( t \) from a few multiples of \( 2\pi \) will display the repeating pattern. A typical range could be \( -4\pi \) to \( 4\pi \).

Step 3: Set Up the Graphing Program

Use graphing software capable of plotting 3D parametric equations. Input the functions for each coordinate: x as \( t \), y as \( t \sin t \), and z as \( t \cos t \). Set the parameter range from \( -4\pi \) to \( 4\pi \).

Step 4: Choose Viewpoints for Better Visualization

Adjust the view angles and perspective in your graphing software to expose the nature of the curve. Typically, a 3D rotational view helps in visualizing how the components interact and forms the path of the curve.

Step 5: Analyze the Graph

Observe the graph: the curve typically spirals outward, as both the y and z components form circular paths due to \( \sin t \) and \( \cos t \), while the x-component increases or decreases linearly, creating a spiral due to the parametric definition.

Key Concepts

Vector Equations and their Importance

A vector equation is a fundamental concept in mathematics and physics. It provides a way to describe a parametric curve in three-dimensional space using a parameter, typically denoted as \( t \). The vector equation \( \mathbf{r}(t) = \langle t, t \sin t, t \cos t \rangle \) specifies the position of a point on a curve for each value of \( t \). Here, \( t \) acts as an independent variable that generates the x, y, and z coordinates of the curve.
This approach transforms complex path descriptions into simple, manageable expressions:

• x-component: \( t \), indicating linear progression.
• y-component: \( t \sin t \), combining linear and sinusoidal patterns.
• z-component: \( t \cos t \), adding another dimension of oscillation.

The ability to use such equations helps us understand intricate paths, like spirals or helices, by breaking them into basic, interpretable components.

Understanding the Parameter Domain

In parametric equations, the parameter domain determines the range of values that \( t \) takes, affecting how much of the curve you view. Choosing an appropriate domain is crucial to reveal the true shape and behavior of the curve.
For the vector equation \( \mathbf{r}(t) = \langle t, t \sin t, t \cos t \rangle \), the trigonometric functions \( \sin t \) and \( \cos t \) have a period of \( 2\pi \). Therefore, selecting \( t \) from multiples of \( 2\pi \) allows you to observe repeating patterns or relationships.

• A parameter range like \( -4\pi \) to \( 4\pi \) is sufficient for observing several cycles of the \( \sin \) and \( \cos \) functions.

By defining this range effectively, one can identify how the curve evolves and repeats over different sections of its path.

Utilizing Graphing Software

Graphing software is an invaluable tool for visualizing 3D parametric curves. It allows students to input their vector equations and adjust parameters to better understand the curves they are studying. Using graphing software with the vector equation \( \mathbf{r}(t) = \langle t, t \sin t, t \cos t \rangle \), follow these steps to plot your curve:

• Input the x, y, and z components of your vector equation.
• Set the parameter range, for instance, from \( -4\pi \) to \( 4\pi \).
• Choose a suitable viewing angle: A 3D rotational view is often the best choice.

This allows a comprehensive display of the curve's shape and path, making complex interactions easier to study.
Graphing takes theoretical math into a visible realm, clarifying results and adding depth to your learning experience.

Role of Trigonometric Functions

Trigonometric functions \( \sin t \) and \( \cos t \) play a significant role in shaping parametric curves like \( \mathbf{r}(t) = \langle t, t \sin t, t \cos t \rangle \). These functions are periodic with a period of \( 2\pi \), displaying oscillations that contribute to the recurring structure of the curve.

• \( \sin t \) influences the y-component with an oscillating effect.
• \( \cos t \) gives the z-component a similar oscillating trajectory.

By combining these trigonometric functions with linear terms, parametric equations can form spirals, circles, and other fascinating shapes.
Understanding these functions lets you predict and interpret how the y and z components interact with each other and the linear x-component, offering insights into the behavior of complex three-dimensional curves.