Problem 23 - Calculus III

Question

Eliminate the parameter \(t\), write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. $$ \mathbf{r}(t)=t^{3} \mathbf{i}+2 t \mathbf{j} $$

Step-by-Step Solution

Answer

The Cartesian equation is \( y = 2x^{1/3} \). The graph is a cubic root curve, stretched vertically by a factor of 2.

Step 1: Identify the Parametric Equations

The given vector-valued function is \( \mathbf{r}(t) = t^3 \mathbf{i} + 2t \mathbf{j} \). This can be separated into parametric equations: \( x(t) = t^3 \) and \( y(t) = 2t \). We will use these equations to eliminate the parameter \( t \).

Step 2: Solve for t in terms of x or y

From the parametric equation \( x = t^3 \), solve for \( t \) in terms of \( x \). We have \( t = x^{1/3} \).

Step 3: Substitute t into the y equation

Substitute \( t = x^{1/3} \) into the equation for \( y \): \( y = 2t \). This gives \( y = 2x^{1/3} \).

Step 4: Write the Cartesian Equation

The equation in Cartesian coordinates, obtained by eliminating the parameter \( t \), is \( y = 2x^{1/3} \).

Step 5: Graph the Cartesian Equation

Graph the equation \( y = 2x^{1/3} \). This represents the cube root function with a vertical stretch factor of 2. The graph passes through the origin and extends into both the positive and negative directions for \( x \) values. The curve increases gently as \( x \) becomes positive and goes to negative values for negative \( x \).

Key Concepts

Vector-Valued Functions

Vector-valued functions are mathematical expressions that use vectors instead of scalar variables to define a relationship. In essence, these functions map a parameter, commonly denoted as \( t \), to a vector. This vector can be composed of components along the \( x \), \( y \), and sometimes \( z \) axes. For instance, consider the vector-valued function \( \mathbf{r}(t) = t^3 \mathbf{i} + 2t \mathbf{j} \). This expression can be understood as a function providing us with

• an \( x \)-component: \( t^3 \)
• a \( y \)-component: \( 2t \)

Each value of \( t \) gives us a point in space defined by these components. This makes vector-valued functions an essential tool in analyzing paths and motions in physics and engineering. They allow us to easily break down complex motions and visualize them in terms of simpler, linear components.

Cartesian Coordinates

Cartesian coordinates are a system used to uniquely determine each point in a plane or space by a set of numerical coordinates. These coordinates represent points through a pair (or triplet in three dimensions) of perpendicular lines, called axes. Commonly, the horizontal axis is the \( x \)-axis, and the vertical is the \( y \)-axis. In this exercise, we will focus on converting a vector-valued function into a Cartesian equation.By manipulating vector components, such as \( x(t) = t^3 \) and \( y(t) = 2t \), we transform them to eliminate the parameter \( t \) and express the function purely in terms of \( x \) and \( y \). Such a transformation results in a Cartesian equation, "flattening" the path described by the vector into a more visually interpretable format on a two-dimensional plane, like \( y = 2x^{1/3} \). Transforming vector-valued functions into Cartesian coordinates simplifies graphing and visual analysis.

Graphing Parametric Equations

Graphing parametric equations involves plotting the relationship defined by parametric expressions. Once we eliminate the parameter \( t \) from the equations, we transform the function into a familiar \( y = f(x) \) format, which can be easier to graph on the Cartesian plane.For example, let's consider the contrived example of the equation \( y = 2x^{1/3} \), derived from the original parametric equations \( x(t) = t^3 \) and \( y(t) = 2t \).

• This equation represents a stretched cube root function.
• It passes through the origin, meaning that when \( x = 0 \), \( y \) is also zero.
• The curve stretches toward both positive and negative values of \( x \).

The equation increases gently along positive \( x \)-values and decreases symmetrically as \( x \) becomes negative. By graphing it, you can better understand the behavior and shape of vector-valued paths in different mathematical and physical applications.