Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

a parametric representation of a curve is given. $$ x=3 t-1, y=t ; 0 \leq t \leq 4 $$

Step-by-Step Solution

Answer

The Cartesian equation is \( x = 3y - 1 \) with \( 0 \leq y \leq 4 \).

Step 1: Identify the Parametric Equations

The given parametric equations are:\[ x = 3t - 1 \] and \[ y = t \] with the parameter \( t \) in the range \( 0 \leq t \leq 4 \). These equations describe the \( x \) and \( y \) components of the curve in relation to the parameter \( t \).

Step 2: Eliminate the Parameter

To find a Cartesian equation, we eliminate the parameter \( t \). From the equation \( y = t \), we clearly have \( t = y \). Substitute \( t = y \) into the equation for \( x \):\[ x = 3t - 1 \Rightarrow x = 3y - 1 \].

Step 3: Determine the Range of \( y \)

Since \( t \) ranges from \( 0 \) to \( 4 \), and \( y = t \), it follows that \( y \) also ranges from \( 0 \) to \( 4 \).

Step 4: Write the Cartesian Equation and Its Domain

The Cartesian equation for the curve is \( x = 3y - 1 \). The range of \( y \) provides the domain for the graph: \( 0 \leq y \leq 4 \).

Step 5: Verify the Parametric to Cartesian Conversion

Substitute various values of \( y \) within the range (e.g., \( y = 0, 2, 4 \)) into the Cartesian equation \( x = 3y - 1 \) to ensure they correspond to values obtained by substituting \( t \) in parametric equations. The points (\( x, y \)) such as \( (-1, 0), (5, 2), (11, 4) \) should lie on the line.

Key Concepts

Cartesian equation

A Cartesian equation is an equation that describes a curve or a surface in a plane using Cartesian coordinates, which are typically represented by the variables \( x \) and \( y \). These equations are crucial for understanding relationships between variables without needing a parameter. In parametric equations like the one given \( x = 3t - 1 \) and \( y = t \), each variable depends on a third variable, \( t \), known as the parameter.

By converting parametric equations into a Cartesian form (\( x = 3y - 1 \), in this case), we translate the curve's dependence from the parameter \( t \) directly into relationships between \( x \) and \( y \). This removal of the parameter offers a simplified and often linear representation of the curve. It's a more familiar representation that can be easily plotted and analyzed in two-dimensional space.

• The Cartesian form provides a direct relation between \( x \) and \( y \).
• It omits the parameter, aiding in easier graphing and comprehensive problem solving.

eliminate the parameter

Eliminating the parameter in a set of parametric equations is a technique used to find the Cartesian equation. This involves removing the parameter \( t \), the independent variable in the pair of parametric equations.

For instance, we start with the given parametric equations \( x = 3t - 1 \) and \( y = t \). To eliminate \( t \), first express \( t \) in terms of one of the variables. Since \( y = t \), we have \( t = y \).

Substituting this expression for \( t \) into the equation for \( x \) gives us:

• Substitute \( t = y \) into \( x = 3t - 1 \)
\[ x = 3(y) - 1 \]

This step results in an equation without the parameter, giving us \( x = 3y - 1 \), which is the desired Cartesian equation. By eliminating the parameter, we shift the problem-solving focus from handling equations determined by \( t \), to those independently described by \( x \) and \( y \).

This elimination helps:

• Transform parametric equations into a single, cohesive Cartesian equation.
• Simplify analysis by reducing reliance on the parameter.

curve representation

Curve representation discusses how a curve can be described mathematically using equations. In parametric form, a curve is represented using equations that express the coordinates \( x \) and \( y \) as functions of a parameter \( t \). This approach provides a dynamic way of describing points along the curve based on varying values of \( t \).

However, for easier visualization and understanding, we often prefer the Cartesian representation, which relates \( x \) directly to \( y \). The exercise showcases this transformation. By converting the parametric curve \( x = 3t - 1 \) and \( y = t \) into the Cartesian equation \( x = 3y - 1 \), the essence of the curve is retained in a more straightforward form.

Understanding curve representation helps with:

• Providing a clear mathematical description of a geometric shape.
• Facilitating easier graphing on a 2D plane.
• Allowing for greater mathematical manipulations and calculations.

Through such transformation from parametric to Cartesian, curve representation becomes not just about the formula but also about gaining insight into the nature and properties of the curve.