Problem 27 - Calculus 3

Question

Use a graphing utility to sketch each of the following vector-valued functions: $$ \mathbf{r}(t)=2 \cos t^{2} \mathbf{i}+(2-\sqrt{t}) \mathbf{j} $$

Step-by-Step Solution

Answer

Use a graphing utility to plot the function with \( t \) in \( [0, 5] \) and observe its trajectory.

Step 1: Understand the Components of the Vector-Valued Function

The vector-valued function given is \( \mathbf{r}(t) = 2 \cos(t^2) \mathbf{i} + (2 - \sqrt{t}) \mathbf{j} \). It has an \( x \)-component \( x(t) = 2 \cos(t^2) \) and a \( y \)-component \( y(t) = 2 - \sqrt{t} \). We'll plot these components on a graph to visualize the trajectory of the function for values of \( t \).

Step 2: Set the Range for Values of t

Decide a practical range for \( t \) based on the function components. Since \( \sqrt{t} \) is used, \( t \) must be non-negative. For \( t \) in the range \( [0, 5] \), both components are well-defined and capture the behavior of the function.

Step 3: Plot the Vector Function's Trajectory

Using a graphing utility, plot the vector-valued function by calculating \( x(t) \) and \( y(t) \) for several points of \( t \) within the range \( [0, 5] \). Plot each resultant pair \( (x(t), y(t)) \) on a plane.

Step 4: Analyze the Graph

Examine the graph plotted by the graphing utility. The curve will not resemble a typical function graph but rather the path traced by the vector-valued function. Note how the trigonometric and square root components affect the shape of the trajectory.

Key Concepts

Vector Components

In the realm of vector-valued functions, understanding the individual components is crucial. A vector-valued function consists of separate components that define its behavior with respect to each dimension. For the function \( \mathbf{r}(t) = 2 \cos(t^2) \mathbf{i} + (2 - \sqrt{t}) \mathbf{j} \), we identify two distinct parts:
\( x(t) = 2 \cos(t^2) \) and \( y(t) = 2 - \sqrt{t} \).

• \( x(t) \) is the horizontal component and influences the left-right movement.
• \( y(t) \) is the vertical component and affects the up-down motion.

Grasping these separate vector components enables us to predict and understand the overall direction and shape of the vector's path. By examining each piece independently, we can better visualize how the vector will behave.

Graphing Utilities

Graphing utilities are powerful tools that aid us in visualizing mathematical functions, especially vector-valued functions which might not have simple analytical solutions. To use a graphing utility:
Identify your desired range for the parameter \( t \). Since \( t \) appears under a square root in \( y(t) \), it's limited to non-negative values. A typical choice might be \( 0 \leq t \leq 5 \).

• Select your points strategically within this range.
• Calculate the corresponding \( x(t) \) and \( y(t) \) values.
• Input these pairs into your utility to plot.

This plotting process helps to reveal the vector's trajectory and offers insights into the function's behavior across the chosen range.

Trigonometric Functions

Trigonometric functions play a significant role in shaping the behavior of vector components. In the function \( \mathbf{r}(t) = 2 \cos(t^2) \mathbf{i} + (2 - \sqrt{t}) \mathbf{j} \), the trigonometric component is \( 2 \cos(t^2) \). This contributes a periodic aspect to the function, influencing how \( x(t) \) oscillates as \( t \) changes.

• The cosine function typically oscillates between -1 and 1, but it's scaled by 2 here.
• As \( t^2 \) increases, the function displays rapid oscillations.
• This periodic behavior can result in multiple directional changes in the trajectory.

Being familiar with such functions enables us to anticipate how the graph might look and interpret changes in direction as \( t \) evolves.

Function Trajectory

The trajectory of a vector-valued function is the path that its components trace on a coordinate plane as the parameter \( t \) varies. For \( \mathbf{r}(t) = 2 \cos(t^2) \mathbf{i} + (2 - \sqrt{t}) \mathbf{j} \), the trajectory reveals how the vector moves through space. This path provides a visual representation of the function's dynamism.

• The spiral or oscillating nature of \( x(t) = 2 \cos(t^2) \) may create wave-like paths horizontally.
• The descent aspect of \( 2 - \sqrt{t} \) usually causes a downward bending of the trajectory.
• Observing the trajectory helps in understanding how different components combine to form complex paths.

This visualization is invaluable for comprehensive insights into the geometric behavior of vector-valued functions.