Problem 27 - Calculus Volume 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

Plot the function \( 2 \cos t^2 \) vs \( 2 - \sqrt{t} \) for \( t \) in [0, 3].

Step 1: Understand the Vector-Valued Function

The given vector-valued function is \( \mathbf{r}(t) = 2 \cos t^2 \mathbf{i} + (2 - \sqrt{t}) \mathbf{j} \). This function consists of two components: the \( i \)-component \( 2 \cos t^2 \) and the \( j \)-component \( 2 - \sqrt{t} \). For visualizing this function, we will plot it in the plane, with \( t \) being the parameter that changes and traces out a path.

Step 2: Choose the Range for t

Select a suitable range for the parameter \( t \). Since the function involves \( \sqrt{t} \), \( t \) needs to be non-negative. A reasonable range to explore the nature of this path could be from \( t = 0 \) to \( t = 3 \). This should provide enough details to understand the shape of the graph.

Step 3: Plot the Components of the Function

Using a graphing utility (such as Desmos, GeoGebra, or a graphing calculator), plot the function \( 2 \cos t^2 \) versus \( 2 - \sqrt{t} \) for \( t \) ranging from 0 to 3. On the graph, the horizontal axis will represent the \( i \)-component \( 2 \cos t^2 \), and the vertical axis will represent the \( j \)-component \( 2 - \sqrt{t} \).

Step 4: Interpret the Graph

Once plotted, the graph will represent the trajectory traced by the vector-valued function \( \mathbf{r}(t) \) as \( t \) changes within the chosen range. It may show oscillatory behavior in the horizontal direction due to the cosine function and a decreasing trend in the vertical direction due to the \( 2 - \sqrt{t} \).

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that enable you to visualize mathematical functions easily. They come in various forms, such as online graphing calculators, like Desmos or GeoGebra, as well as handheld graphing calculators. Each type offers a user-friendly interface where you input the function you wish to plot. This simplicity is especially beneficial when working with complex functions, such as vector-valued functions, because it allows you to quickly see the shape and behavior of these functions in a graphical form.

When using a graphing utility to sketch a vector-valued function, you need to enter the component functions separately. The tool then automatically generates the path that the vector-valued function traces. This process not only makes it easier to interpret the resulting graph but also helps you gain insights into how each component influences the overall trajectory of the function.

Component Functions

In the context of vector-valued functions, component functions play a crucial role. A vector-valued function consists of multiple scalar functions, each forming a part of a vector. In our example, the vector-valued function is represented as \( \mathbf{r}(t) = 2 \cos t^2 \mathbf{i} + (2 - \sqrt{t}) \mathbf{j} \). Here, the component functions are \( 2 \cos t^2 \) and \( 2 - \sqrt{t} \).

These component functions determine how the vector function behaves as the parameter \( t \) changes over a specified range. The \( i \)-component, \( 2 \cos t^2 \), dictates horizontal movements, reflecting the oscillatory properties of the cosine function. Meanwhile, the \( j \)-component, \( 2 - \sqrt{t} \), affects vertical movements, introducing a gradual decrease as \( t \) increases. Understanding how each component function operates is vital for visualizing and interpreting the entire vector path.

Parameterization

Parameterization is the process of defining a set of functions to represent a geometric object or path. In vector-valued functions, it involves introducing a parameter, usually denoted by \( t \), which helps trace the curve or path on a plane as it varies.

In our exercise, the parameter \( t \) is non-negative due to the presence of \( \sqrt{t} \) in the \( j \)-component. This parameter allows us to explore the function's behavior by selecting a range, such as \( t = 0 \) to \( t = 3 \), allowing for a comprehensive view of the vector path.

Parameterization simplifies the graphing process as it breaks down a complex problem into manageable parts. By understanding how the parameter affects the vector path, we can gain deeper insights into the motion and trajectory represented by the function.

Function Plotting

Plotting functions, especially vector-valued ones, is essential for visually interpreting complex mathematical expressions. In this context, plotting means visually representing the components of a vector-valued function on coordinate axes.

When plotting \( \mathbf{r}(t) = 2 \cos t^2 \mathbf{i} + (2 - \sqrt{t}) \mathbf{j} \), we graph the \( i \)-component on the horizontal axis and the \( j \)-component on the vertical axis. This creates a visual path that the vector traces as the parameter \( t \) varies. The graph then displays characteristics like oscillations from the cosine function or changes due to the square root.

Accurate function plotting allows you to explore the path's unique features, providing clarity on how component functions and parameterization together influence the entire curve. By learning to effectively plot these graphs, you develop a stronger foundation in understanding and analyzing vector-valued functions.