Question

Sketch the vector-valued function on the given interval. $$ \vec{r}(t)=\langle 2 \sec t, \tan t\rangle, \text { on }[-\pi, \pi] $$

Short answer

The graph has segments with asymptotes at \( t = \pm\frac{\pi}{2} \), creating distinct curves.

Step-by-step solution

Define the Vector-Valued Function

The given vector-valued function is \( \vec{r}(t) = \langle 2 \sec t, \tan t \rangle \). This function can be rewritten using the component functions: \( x(t) = 2 \sec t \) and \( y(t) = \tan t \). Our task is to sketch this function over the interval \([-\pi, \pi]\).

Determine the Domain of Each Component

The \( \sec t \) and \( \tan t \) functions have values that are undefined at certain points: specifically, where \( \cos t = 0 \) (because \( \sec t = \frac{1}{\cos t} \)). These points occur at \( t = -\frac{\pi}{2}, \frac{\pi}{2} \). Therefore, the vector-valued function is defined on \([-\pi, -\frac{\pi}{2}) \cup (-\frac{\pi}{2}, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi]\).

Analyze the Behavior of the Components

For \( x(t) = 2 \sec t \), as \( t \) approaches \( -\frac{\pi}{2} \) from the left or \( \frac{\pi}{2} \) from the right, \( \sec t \to \pm \infty \). Similarly, \( y(t) = \tan t \) approaches \( \pm \infty \) as \( t \) approaches these points. This indicates potential vertical asymptotes where these discontinuities occur.

Sketch the Graph

Plot \( x = 2\sec t \) and \( y = \tan t \) for the defined range of \( t \). The graph will have segments in each interval where the function is defined, and vertical asymptotes at \( t = \pm \frac{\pi}{2} \). The function will appear to approach infinity in both directions near these asymptotes, producing distinct curves.

Key concepts

Domain of a Function

In mathematics, the domain of a function refers to the set of all possible input values (such as time, angle, etc.) for which the function produces a valid output. For the given vector-valued function \( \vec{r}(t) = \langle 2 \sec t, \tan t \rangle \), we are tasked with understanding its domain.

• The function has two components: \( 2 \sec t \) and \( \tan t \).
• Both \( \sec t \) and \( \tan t \) derive from trigonometric functions and have specific points where they are undefined due to their denominators becoming zero.
• \( \sec t = \frac{1}{\cos t} \) is undefined wherever \( \cos t = 0 \), which occurs at \( t = -\frac{\pi}{2} \) and \( t = \frac{\pi}{2} \).

This means the vector-valued function cannot have these points in its domain. The domain thus forms segments that avoid these undefined points:

• \([-\pi, -\frac{\pi}{2}) \cup (-\frac{\pi}{2}, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi]\)

Understanding the domain is crucial since it tells us where the function is valid and can help in sketching the graph without any interruptions.

Component Functions

Vector-valued functions consist of multiple component functions, each corresponding to different dimensions of the vector. In the function \( \vec{r}(t) = \langle 2 \sec t, \tan t \rangle \), there are two component functions:

• \( x(t) = 2 \sec t \)
• \( y(t) = \tan t \)

These functions describe how the vector changes with the parameter \( t \).

• \( x(t) = 2 \sec t \) stretches the basic secant function by a factor of two.
• \( y(t) = \tan t \) represents the tangent function, which is well-known for its periodic and unbounded behavior.

Understanding each component separately is essential to predict how the full vector behaves in space. When plotting, each component is evaluated across the interval to determine the vector path.

Vertical Asymptotes

Vertical asymptotes are crucial in understanding the behavior of functions, particularly in graphs. They occur where the function tends towards infinity or negative infinity, leading to a distinct line that the graph approaches but never touches.For the functions \( x(t) = 2 \sec t \) and \( y(t) = \tan t \), vertical asymptotes occur where their respective trigonometric functions are undefined:

• When \( t = -\frac{\pi}{2} \) and \( t = \frac{\pi}{2} \), both component functions become undefined.
• At these points, both \( \sec t \) and \( \tan t \) tend towards infinity or negative infinity. Consequently, the vector \( \vec{r}(t) \) has vertical asymptotes at these points.

These vertical asymptotes are not just theoretical; they significantly influence how the vector behaves visually on the graph, indicating a dramatic increase in value as the point is approached, leading to the graph shooting upwards or downwards.

Behavior Analysis

Analyzing function behavior helps us understand how functions will behave across their domain, particularly near critical points such as asymptotes.For the vector-valued function \( \vec{r}(t) = \langle 2 \sec t, \tan t \rangle \), behavior analysis involves

• Understand that as \( t \to -\frac{\pi}{2} \) from the left, \( \sec t \) and \( \tan t \) both approach negative infinity.
• As \( t \to \frac{\pi}{2} \) from the right, both \( \sec t \) and \( \tan t \) approach positive infinity.
• This behavior results in curves that sharply dip or rise as they approach these critical \( t \) values.

Hence, the vector-valued function graph will consist of seemingly disconnected segments. This analysis is fundamental in creating an accurate sketch, illustrating how the vector behaves and transforms over the specified intervals.