Problem 28 - Calculus Volume 3

Question

Use a graphing utility to sketch each of the following vector-valued functions: \([\mathrm{T}] \mathbf{r}(t)=\left\langle e^{\cos (3 t)}, e^{-\sin (t)}\right\rangle\)

Step-by-Step Solution

Answer

Plot the function using the graphing utility for \( t \) from \( -2\pi \) to \( 2\pi \).

Step 1: Understanding the Function

The given vector-valued function is \( \mathbf{r}(t) = \langle e^{\cos(3t)}, e^{-\sin(t)} \rangle \). This is a function that specifies a two-dimensional curve in terms of parameter \( t \). It provides the \( x \) and \( y \) coordinates in terms of exponential functions involving cosine and sine.

Step 2: Set Up the Graphing Utility

Open your preferred graphing utility or software that can handle parametric functions. Ensure that it supports plotting vector-valued functions where each component is defined separately.

Step 3: Input the Parametric Equations

Enter the components of the vector function into the graphing utility as separate equations: \( x(t) = e^{\cos(3t)} \) and \( y(t) = e^{-\sin(t)} \). Make sure the tool is set to interpret these as parametric equations.

Step 4: Define the Range for t

Determine a suitable range for \( t \). A common initial choice is \( t \) from \( -2\pi \) to \( 2\pi \) to ensure that typical features of trigonometric functions are captured. Set this range in the utility.

Step 5: Sketch and Analyze the Graph

Use the utility to sketch the graph based on the input equations and range. Analyze the graph to identify key features such as symmetry, length, and shape of the curve generated by \( \mathbf{r}(t) \). Explore different values of \( t \) beyond the initial range if necessary to fully understand the function's behavior.

Key Concepts

Parametric Equations

Parametric equations are a way to define a set of related functions that describe a geometric object, like a curve, using a parameter. Typically, we use a parameter, often denoted as \( t \), to express \( x \) and \( y \) coordinates as distinct functions of \( t \). This allows us to trace the path of a curve in the \( xy \)-plane without relying solely on \( y=f(x) \) formats. For example, a circle can be defined as \( x(t) = r\cos(t) \) and \( y(t) = r\sin(t) \), where \( t \) varies over some range to complete the circle.

Parametric forms are particularly useful for:

• Describing motions and paths.
• Plotting curves that loop or intersect themselves.
• Simplifying complex curve expressions.

This is exactly what is used in the exercise question to describe a two-dimensional curve using the exponential functions with respect to trigonometric components.

Graphing Utility

A graphing utility is any tool or software that helps visualize mathematical functions and data. For vector-valued or parametric functions, it becomes particularly useful because plotting by hand can be complex and requires precise calculation of multiple points. Examples include graphing calculators like the TI series, mathematical software like Desmos, GeoGebra, or MATLAB.

When dealing with parametric equations in graphing utilities, ensure:

• The utility can distinguish between standard and parametric forms.
• You input \( x(t) \) and \( y(t) \) as separate and interpret them with respect to the parameter \( t \).
• The parameter range \( t \) is set. This range controls how much of the curve you see.

These tools can graph complex shapes by computing data points for every slice of the parameter \( t \), thereby painting a clear picture of the function's behavior.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to the power of a variable. They are significant in modeling growth and decay processes. For example, the function \( f(x) = a^{x} \) is considered exponential if \( a > 0 \) and \( a eq 1 \). In the context of parametric equations, exponential terms can alter the shape and behavior of the resulting curve significantly.

Characteristics of exponential functions:

• They grow rapidly if the base is greater than 1.
• They decay towards zero if the base is between 0 and 1.
• They exhibit constant multiplicative rates of change.

In the exercise, \( e^{\cos(3t)} \) and \( e^{-\sin(t)} \) are the components of the parametric equations, using the base \( e \) which is approximately equal to 2.718, commonly used in natural exponential contexts.

Trigonometric Functions

Trigonometric functions are periodic and relate to the angles of triangles and the unit circle. They include sine, cosine, and tangent, among others. They are pivotal in describing wave patterns, cyclic phenomena, and oscillations.

Properties and applications of trigonometric functions include:

• Periodic nature, repeating values over regular intervals (e.g., \( 2\pi \) for sine and cosine).
• Describe the angle's ratio of sides in right triangles.
• Useful for modeling seasonal or repeating cycles, such as sound waves.

In our parametric curve, the cosine and sine functions alter the inputs to the exponential functions, essentially adding oscillatory behavior to the growth or decay process. For instance, \( \cos(3t) \) speeds up the standard cosine cycle, and \( -\sin(t) \) changes the growth in the exponential term.