Question

Sketch the vector-valued function on the given interval. $$ \vec{r}(t)=\left\langle t^{2}, t^{3}\right\rangle, \text { for }-2 \leq t \leq 2 $$

Short answer

Sketch the curve passing through (4, -8), (1, -1), (0, 0), (1, 1), and (4, 8) for \(-2 \leq t \leq 2\).

Step-by-step solution

Understand the Function Components

The vector-valued function is given as \( \vec{r}(t) = \langle t^2, t^3 \rangle \). This indicates that for each value of \( t \), the position vector is defined by the \( x \)-component \( t^2 \) and \( y \)-component \( t^3 \). We need to sketch the curve that results from varying \( t \) in the interval \([-2, 2]\).

Evaluate the Endpoints

First, let's calculate the vector at the endpoints of the interval. For \( t = -2 \), \( \vec{r}(-2) = \langle (-2)^2, (-2)^3 \rangle = \langle 4, -8 \rangle \). For \( t = 2 \), \( \vec{r}(2) = \langle 2^2, 2^3 \rangle = \langle 4, 8 \rangle \). These points are critical in understanding the direction and orientation of the curve.

Calculate Intermediate Values

Choose intermediate values in the interval, such as \( t = -1, 0, 1 \). For \( t = -1 \), \( \vec{r}(-1) = \langle (-1)^2, (-1)^3 \rangle = \langle 1, -1 \rangle \). For \( t = 0 \), \( \vec{r}(0) = \langle 0^2, 0^3 \rangle = \langle 0, 0 \rangle \). For \( t = 1 \), \( \vec{r}(1) = \langle 1^2, 1^3 \rangle = \langle 1, 1 \rangle \).

Plotting the Points

Plot the calculated points on a coordinate plane: \((-2, 4, -8)\), \((-1, 1, -1)\), \((0, 0, 0)\), \((1, 1, 1)\), and \((2, 4, 8)\). These points will help visualize the vector curve. The curve should pass through these points and represent the parametric relationship.

Sketch the Curve

The vector curve derived from \( \vec{r}(t) = \langle t^2, t^3 \rangle \) generally represents the shape of a cubic function viewed in a parametric form. It is symmetric about the y-axis. Starting from \( \langle 4, -8 \rangle \), the curve passes through the origin \( \langle 0, 0 \rangle \) and ends at \( \langle 4, 8 \rangle \), smoothly transitioning through the calculated points.

Key concepts

Parametric Equations

Parametric equations describe a way to represent curves in terms of parameters, rather than relying solely on the traditional relation between two variables. In the original exercise, the vector-valued function \( \vec{r}(t) = \langle t^2, t^3 \rangle \) is expressed using parametric equations where each component of the vector depends on the parameter \( t \).
This representation provides distinct advantages:

• It allows for more complex curve forms beyond the simple \( y = f(x) \) relationship.
• Both components—\( x = t^2 \) and \( y = t^3 \)—are independently defined, offering a broader perspective of how the curve evolves.
• This method facilitates the representation of motion or other multidimensional phenomena where time \( t \) or another variable serves as a parameter.

Understanding parametric equations aids in visualizing relationships between variables that might not be apparent in a single equation format. By examining the equations, you notice how changes in \( t \) directly affect both the \( x \) and \( y \) coordinates, providing insight into the shape of the curve.

Curve Sketching

Curve sketching involves plotting the behavior of a function based on calculations at specific points or using mathematical properties. With vector-valued functions like in this exercise, sketching requires:

• Plotting specific calculated points for selected parameter values (critical for understanding curve direction and shape).
• Observing symmetry, smoothness, and where the curve intersects with axes.
• Understanding the general shape influenced by both parametric components.

In this scenario, for each value of \( t \) between \(-2\) and \(2\), points such as \((-2, 4, -8)\) and \((0, 0, 0)\) are found by evaluating \( t^2 \) and \( t^3 \).
Plot these points, join them smoothly respecting the computed direction and curve properties. Sketch fans out from \((-2, 4, -8)\) through the central point \((0, 0, 0)\) curving upwards to \((2, 4, 8)\). This highlights the cubic nature of \( t^3 \) seen from variable transformations in parametric forms.

Vector Components

Vector components are the individual parts of a vector, each representing a direction in space. In the context of a vector-valued function, like \( \vec{r}(t) = \langle t^2, t^3 \rangle \), each component corresponds to a dimension on a graph.

• The horizontal component \( t^2 \) dictates how the function moves along the x-axis, rendering symmetrical properties about this axis.
• The vertical component \( t^3 \) affects movement along the y-axis, introducing more complexity through its cubic nature.

Each part of this vector must be evaluated separately and correctly interpreted to accurately draw the curve. Together, these components create a comprehensible picture of the curve's path through space and help predict behavior over intervals. Recognizing this decomposition aids in mastering the interpretation of vector-valued functions and how they visually manifest as curves, wiggling synonyms among dimensions.