Problem 1 - Calculus III

Question

Give the component functions \(x=f(t)\) and \(y=g(t)\) for the vector-valued function \(\mathbf{r}(t)=3 \sec t \mathbf{i}+2 \tan t \mathbf{j}\)

Step-by-Step Solution

Answer

The component functions are \( x = 3 \sec t \) and \( y = 2 \tan t \).

Step 1: Understanding the Vector-Valued Function

The given vector-valued function is \( \mathbf{r}(t) = 3 \sec t \mathbf{i} + 2 \tan t \mathbf{j} \), where \( \mathbf{r}(t) \) is expressed as a sum of its components in the \( \mathbf{i} \) and \( \mathbf{j} \) directions.

Step 2: Identifying the Component in the i-Direction

The portion of the vector function \( \mathbf{r}(t) \) in the \( \mathbf{i} \) direction is \( 3 \sec t \). This is the component function \( x = f(t) \).

Step 3: Identifying the Component in the j-Direction

The portion of the vector function \( \mathbf{r}(t) \) in the \( \mathbf{j} \) direction is \( 2 \tan t \). This is the component function \( y = g(t) \).

Step 4: Writing the Component Functions

Now, we can write the component functions as follows: \( x = f(t) = 3 \sec t \) and \( y = g(t) = 2 \tan t \). These functions represent the parametric form of the vector function \( \mathbf{r}(t) \).

Key Concepts

Component Functions

Vector-valued functions are expressions that have multiple components, usually aligning along coordinate axes. In this context, a function like \( \mathbf{r}(t) = 3 \sec t \mathbf{i} + 2 \tan t \mathbf{j} \) is broken down into its individual components in the i and j directions. Each section of this vector corresponds to a specific mathematical operation or function that describes movement or change in one dimension.

For a vector function of the form \( \mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} \), each component can be extracted separately:

• The component function for the i-direction is represented as \( x = f(t) \).
• The component function for the j-direction is represented as \( y = g(t) \).

In our example, these component functions are given as \( x = 3 \sec t \) and \( y = 2 \tan t \). Understanding component functions is essential since they allow us to study the behavior of each part of the vector separately, which can simplify complex analyses.

• The benefit of using component functions lies in their simplicity, which helps when calculating limits, derivatives, or integrals component-wise.
• Additionally, component functions aid visualization. They can tell us how each dimension behaves as \( t \) varies.

Parametric Equations

Parametric equations offer a way of defining a set of related quantities as explicit functions of a number of independent parameters. In this particular exercise, we have the parametric equations coming from the component functions of the vector. They tell us how the position on the coordinate plane changes over time or another parameter.

• For the i-direction, the parametric equation is \( x = 3 \sec t \).
• For the j-direction, the parametric equation is \( y = 2 \tan t \).

This setup of using parametric equations allows for greater flexibility, especially when dealing with curves or paths that wouldn’t be readily describable by functions in x or y alone.

By allowing both equations to change with the same parameter \( t \), parametric equations effectively trace out a path as \( t \) changes over a specified domain. This is especially useful in physics, engineering, and computer graphics for modeling motions such as projectile paths or rotations.

Trigonometric Functions

The component functions \( x = 3 \sec t \) and \( y = 2 \tan t \) involve trigonometric functions, specifically secant and tangent. These functions are essential in describing periodic phenomena due to their links with angles and rotations.

• The secant function, \( \sec t = \frac{1}{\cos t} \), reflects the reciprocal nature of the cosine function.
• The tangent function, \( \tan t = \frac{\sin t}{\cos t} \), highlights the ratio of the sine to cosine functions and is useful for describing slopes or directing angles in circular motion.

Using trigonometric functions in parametric equations allows for a wide spectrum of transformations, like circles and ellipses, to be easily expressed.

In this scenario, they beautify the parametric path defined by the vector-valued function, offering a way to incorporate angles directly into calculations. Understanding how these functions interact—whether through their periodicity or asymptotic behaviors—is vital for appreciating how complex paths unfold as parameters vary. Consequently, these trigonometric component functions provide both flexibility and depth to our analysis of vector functions.