Problem 31 - Calculus Volume 3

Question

Use a graphing utility to sketch each of the following vector-valued functions: \(\mathbf{r}(t)=\left\langle t, t^{2}\right\rangle ;\) from left to right

Step-by-Step Solution

Answer

Plot \( (x, y) = (t, t^2) \) for \( t \) from \( -3 \) to \( 3 \) to see a parabola.

Step 1: Understand the Vector-Valued Function

The given vector-valued function is \( \mathbf{r}(t)=\left\langle t, t^{2}\right\rangle \). This means that for each value of \( t \), the vector \( \mathbf{r} \) takes coordinates \( (x, y) = (t, t^2) \). This is effectively a parametrization of a parabola opening upwards.

Step 2: Identify the Graphing Range

To graph this function using a utility, you will need to decide on a range for \( t \). Commonly, for a parabola, you might want to see how it behaves from \( t = -c \) to \( t = c \), where \( c \) is a positive constant. Let's choose \( t \) from \( -3 \) to \( 3 \) as a typical example.

Step 3: Input the Function into a Graphing Utility

Using a graphing utility or software like Desmos or a graphing calculator, input the function by setting \( x = t \) and \( y = t^2 \). Ensure you input the range of \( t \) as discussed in Step 2. Adjust the window settings if necessary to fully view the curve.

Step 4: Sketch and Interpret the Graph

Once the function is graphed, you should see a parabolic curve starting from the bottom left (where \( t = -3 \)) to the top right (where \( t = 3 \)). Since the vector-valued function is parameterized by \( t \), the plot should traverse the parabola from left to right as \( t \) increases.

Key Concepts

Graphing Utilities

Graphing utilities are amazing tools that help us visualize algebraic and vector functions. They provide a clear representation of what mathematical equations look like in the form of graphs. These utilities can be software or calculators designed specifically for graphing purposes. When we deal with vector-valued functions, like in the given exercise, a graphing utility can plot the coordinates for different values of the parameter.

Trying a graphing utility, such as Desmos or GeoGebra, allows users to input parametrized equations easily. For our current function \( \mathbf{r}(t)=\left\langle t, t^{2}\right\rangle \), you'll input the parameter \( t \) and watch how the graph unfolds. Select the range of \( t \) you want to visualize—in this case, typically something like \(-3 \leq t \leq 3\). Adjusting the range and window settings helps capture the entire curve and any crucial details.

• Enhances understanding by providing a visual aid.
• Helps in checking the behavior of functions over different intervals.
• Enables students to experiment and modify parameters to see immediate changes.

Parametric Equations

Parametric equations are a powerful way to describe geometrical and physical phenomena in mathematics. Instead of expressively linking \( x \) to \( y \) through one equation, they use parameters, often denoted by \( t \), to express each variable.

For our vector-valued function, \( \mathbf{r}(t)=\left\langle t, t^{2}\right\rangle \), we see how \( x \) and \( y \) are parameterized separately: \( x = t \) and \( y = t^2 \). This suggests a relationship between \( x \) and \( y \) that maps out a shape—in this scenario, a parabola.

• Provides flexibility in detailing complex curves and paths.
• Useful in simulations and animations to describe a path or motion.
• By varying the parameter, you can navigate through the curves differently.

Think of parametric equations as a means of programming a journey over a curve rather than just presenting a static relationship between the coordinates.

Parabolas

Parabolas are one of the central figures in mathematics, often represented as the graph of quadratic functions. They have a distinct U or inverted-U shape depending on their orientation. In this exercise, the parabola is oriented upwards, forming from the equation \( y = x^2 \), readily displayed when \( x = t \) and \( y = t^2 \).

The fundamental properties of a parabola include:

• A vertex, the turning point, which is the origin \( (0, 0) \) in this case.
• A symmetry axis down its middle, providing mirror-image qualities.
• The direction of opening, which is upwards when the coefficient of \( t^2 \) is positive.

By understanding these properties, graphing becomes more intuitive. Even without plotting, recognizing the curve's features helps interpret the trajectory of the vector-valued function. Thus, understanding parabolas is not just significant for graphing utilities but also underpins many mathematical and theoretical applications.