Question

Use a computer algebra system to graph the vector-valued function \(\mathbf{r}(t) .\) For each \(\mathbf{u}(t)\) make a conjecture about the transformation (if any) of the graph of \(\mathbf{r}(t) .\) Use a computer algebra system to verify your conjecture. \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\frac{1}{2} t^{3} \mathbf{k}\) (a) \(\mathbf{u}(t)=t \mathbf{i}+\left(t^{2}-2\right) \mathbf{j}+\frac{1}{2} t^{3} \mathbf{k}\) (b) \(\mathbf{u}(t)=t^{2} \mathbf{i}+t \mathbf{j}+\frac{1}{2} t^{3} \mathbf{k}\) (c) \(\mathbf{u}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\left(\frac{1}{2} t^{3}+4\right) \mathbf{k}\) (d) \(\mathbf{u}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\frac{1}{8} t^{3} \mathbf{k}\) (e) \(\mathbf{u}(t)=(-t) \mathbf{i}+(-t)^{2} \mathbf{j}+\frac{1}{2}(-t)^{3} \mathbf{k}\)

Short answer

The transformations for the respective functions are as follows: (a) The graph is shifted 2 units down in the y-direction. (b) The graph is reflected around the y=z plane and scaled along x-axis. (c) The graph is shifted 4 units up in the z-direction. (d) The graph is scaled along the z-axis by a factor of 1/4. (e) The graph is reflected in the x=y plane.

Step-by-step solution

Preparation

We first start by preparing to graph the function. The original vector-valued function is given as \(\mathbf{r}(t) = t \mathbf{i} + t^2 \mathbf{j} + \frac{1}{2} t^3 \mathbf{k}\). We will be comparing this graph to the graphs of each transformation function \(\mathbf{u}(t)\), and making observations about how the graph has transformed.

Graphing and Observing (a)

Now, graph the first transformation function \(\mathbf{u}(t) = t \mathbf{i} + (t^2 - 2) \mathbf{j} + \frac{1}{2} t^3 \mathbf{k}\). Compare it to the original function graph. Observe that there is a vertical shift of 2 units down in the y-direction.

Graphing and Observing (b)

Next, graph the second transformation function \(\mathbf{u}(t) = t^2 \mathbf{i} + t \mathbf{j} + \frac{1}{2} t^3 \mathbf{k}\). Upon comparing to the original function, the graph appears to be a reflection around the y=z plane and a scaling along the x-axis.

Graphing and Observing (c)

Then, graph the third transformation function \(\mathbf{u}(t) = t \mathbf{i} + t^2 \mathbf{j} + \left( \frac{1}{2} t^3 + 4\right) \mathbf{k}\). Comparing to the original function, we can observe that there is a vertical shift of 4 units up in the z-direction.

Graphing and Observing (d)

Next, graph the fourth transformation function \(\mathbf{u}(t) = t \mathbf{i} + t^2 \mathbf{j} + \frac{1}{8} t^3 \mathbf{k}\). Upon comparing this with the original function, it can be observed there's been a scaling along the z-axis by a factor of 1/4.

Graphing and Observing (e)

Lastly, graph the final transformation function \(\mathbf{u}(t) = -t \mathbf{i} + (-t)^2 \mathbf{j} + \frac{1}{2} (-t)^3 \mathbf{k}\). Comparing this graph to the original, it can be observed that there is a reflection in the x=y plane.

Key concepts

Computer Algebra System

A Computer Algebra System (CAS) is a software program designed to simplify the process of solving complex mathematical equations. These programs are particularly useful for operations like graphing vector-valued functions, as they can handle the multi-dimensional nature of vectors that typical graphing calculators cannot.

A CAS enables users to input vector equations symbolically and then automatically processes the symbols to provide visual representations. For instance, in our exercise, the vector-valued function \(\mathbf{r}(t) = t \mathbf{i} + t^2 \mathbf{j} + \frac{1}{2} t^3 \mathbf{k}\) can be plotted in three dimensions to show the trajectory that a particle would follow over time in a 3D space. Likewise, transformations of \(\mathbf{r}(t)\) can be visualized, and their graph transformations compared.

The use of a CAS not only aids in visualization but also in verification. By inputting different vector-valued functions \(\mathbf{u}(t)\), students can test their conjectures about graph transformations and instantly see the outcomes, facilitating a deeper understanding of vector calculus and transformations.

Graph Transformation

Graph transformation involves altering the appearance of a graph in a systematic way. These transformations can include shifts, stretches, shrinks, and reflections. In the context of vector calculus, visualizing how vector-valued functions are transformed becomes crucial to understanding their characteristics.

For example, considering the function \( \mathbf{u}(t) = t \mathbf{i} + (t^2 - 2) \mathbf{j} + \frac{1}{2} t^3 \mathbf{k} \), it has undergone a vertical downwards shift in comparison to \(\mathbf{r}(t)\), specifically in the \(y\)-direction. Other transformations like reflection across a plane or scaling can be immediately identified by comparing the transformed graph to the original.

Understanding these transformations can greatly enhance a student's ability to analyze and interpret complex vector behaviors. It's essential to recognize how each component of the vector-valued function affects the graph's shape and orientation in the 3D coordinate system.

Vector Calculus

Vector calculus is an essential branch of mathematics that deals with vector fields and differential and integral calculus of vector functions. It allows us to describe physical phenomena where both magnitude and direction are necessary, such as fluid flow, electromagnetic fields, and motion in space.

The vector-valued function \(\mathbf{r}(t)\) in our exercise represents a path in a three-dimensional space as a function of time, \(t\). Each component of the vector maps to a coordinate axis, contributing to the curve's trajectory. By altering these components, you change the path, which can represent different physical scenarios.

For a firm grasp of vector calculus, students need to understand how changes to vector components affect the overall vector field and how calculating derivatives and integrals of vectors give us important information about the behavior of physical systems, such as velocity, acceleration, and flux across a surface.