Problem 29 - Calculus III

Question

Use a graphing utility to sketch each of the following vector-valued functions: $$ \text { [T] } \mathbf{r}(t)=\langle 2-\sin (2 t), 3+2 \cos t\rangle $$

Step-by-Step Solution

Answer

Graph the function \( \mathbf{r}(t) = \langle 2 - \sin(2t), 3 + 2\cos(t) \rangle \) using a parametric graphing utility over \( t = 0 \) to \( 2\pi \) to visualize its closed curve.

Step 1: Understand the Vector Function

The given function is a vector-valued function: \( \mathbf{r}(t) = \langle 2 - \sin(2t), 3 + 2\cos(t) \rangle \). It describes a curve in the 2-dimensional plane, with the x-component being \( x(t) = 2 - \sin(2t) \) and the y-component being \( y(t) = 3 + 2\cos(t) \).

Step 2: Analyze the Interval for t

Before graphing, determine the domain of \( t \). Although trigonometric functions are periodic, a common interval to consider is from \( t = 0 \) to \( t = 2\pi \), as this captures one complete cycle of both \( \sin(2t) \) and \( \cos(t) \).

Step 3: Compute Key Points

Compute some key points for various values of \( t \) within the interval, such as \( t = 0, \pi/2, \pi, 3\pi/2, \) and \( 2\pi \). - At \( t = 0 \), \( \mathbf{r}(0) = \langle 2 - \sin(0), 3 + 2\cos(0) \rangle = \langle 2, 5 \rangle \). - At \( t = \pi/2 \), \( \mathbf{r}(\pi/2) = \langle 2 - \sin(\pi), 3 + 2\cos(\pi/2) \rangle = \langle 2, 3 \rangle \).- Continue this for all key points.

Step 4: Graph Using a Graphing Utility

Enter the vector-valued function into a graphing utility capable of plotting parametric equations. Set the parameter \( t \) to range from 0 to \( 2\pi \). The utility will plot the curve defined by the vector-valued function. Note how the curve traces on the plane, especially where the significant points calculated lie on this curve.

Step 5: Interpret the Graph

Observe the graph's shape—it's periodic due to the sine and cosine functions. This particular function depicts a closed curve, as it returns to the starting point at the end of the interval. Note any symmetry or special features like peaks or troughs, based on the \( x(t) \) and \( y(t) \) equations.

Key Concepts

Parametric Equations

Parametric equations use a parameter to define a set of points. In a 2-dimensional space, these equations define the x and y coordinates as continuous functions of a third variable, known as the parameter, often denoted as \( t \). For a vector-valued function like \( \mathbf{r}(t) = \langle x(t), y(t) \rangle \), each component \( x(t) \) and \( y(t) \) is expressed independently in terms of the parameter \( t \).

• In the given exercise, \( x(t) = 2 - \sin(2t) \) and \( y(t) = 3 + 2\cos(t) \).
• The parameter \( t \) often represents time or another independent variable progressing along the curve.
• This method allows for the description of complex curves that might be challenging to express in standard Cartesian equations.

Exploring parametric equations is vital for understanding how curves behave as \( t \) changes. This approach is valuable in physics and engineering, where the trajectory or position of an object can be described over time.

Trigonometric Functions

Trigonometric functions like sine and cosine are foundational in describing periodic phenomena. They are frequently utilized in parametric equations involving circular or elliptical paths due to their inherent periodic properties.

• The sine function \( \sin(t) \) varies between -1 and 1, while cosine \( \cos(t) \) also ranges from -1 to 1. This periodicity results in curves that repeat at regular intervals.
• In the exercise, notice how \( \sin(2t) \) affects the x-component: the factor "2" inside the sine function changes the frequency, causing it to complete two cycles over the interval \( 0 \leq t \leq 2\pi \).
• The y-component is influenced by \( \cos(t) \), shifted and scaled by "3 + 2" to transform it for our specific function.

Understanding these trigonometric functions helps decipher and predict the curve's shape, including identifying aspects like symmetry, peaks, and troughs.

Graphing Utilities

Graphing utilities are powerful tools for visualizing complex mathematical functions. They allow you to plot equations and see the graphical representation instantly, which is crucial for understanding and interpreting vector-valued functions.

• Input the parametric equations directly: In this case, enter \( x(t) = 2 - \sin(2t) \) and \( y(t) = 3 + 2\cos(t) \) into the utility.
• Set the parameter \( t \) to span a specific interval. Often \( 0 \leq t \leq 2\pi \) is used for periodic functions to capture a full cycle.
• The utility plots the curve, revealing its shape, distinctive features, and behavior over the given range.

Graphing utilities enhance learning by providing a visual aid, which is essential for grasping the periodic and geometric nature of vector-valued functions. They enable a more intuitive understanding of how changes to the parameters affect the overall shape and path of the curve.