Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

a parametric representation of a curve is given. $$ x=3 t, y=2 t ;-\infty

Step-by-Step Solution

Answer

The Cartesian equation of the curve is \(y = \frac{2}{3}x\).

Step 1: Understanding Parametric Equations

The given parametric equations are \(x = 3t\) and \(y = 2t\). These equations express \(x\) and \(y\) as functions of the parameter \(t\). Our job is to eliminate \(t\) to find a relationship between \(x\) and \(y\).

Step 2: Solving for t in one of the equations

Choose one of the equations to solve for \(t\). Let’s use \(x = 3t\). To isolate \(t\), divide both sides by 3 to get \(t = \frac{x}{3}\).

Step 3: Substitute t into the other equation

Now, substitute the expression we found for \(t\) in terms of \(x\) into the other equation. So, substitute \(t = \frac{x}{3}\) into \(y = 2t\):\[ y = 2 \left(\frac{x}{3}\right) = \frac{2x}{3}.\]

Step 4: Presenting the Cartesian Equation

We now have \(y\) in terms of \(x\): \(y = \frac{2}{3}x\). This equation represents the curve in the Cartesian (\(x, y\)) coordinate system, showing the relationship without the parameter \(t\).

Key Concepts

Parametric Representation

In mathematics, parametric representation is a powerful method to define curves using a set of equations. Instead of describing curves in the traditional X-Y coordinate system, parametric equations use one or more independent variables, known as parameters, to express the coordinates. For example, in the given problem, we have parametric equations:

• \( x = 3t \)
• \( y = 2t \)

These equations use the parameter \( t \) to define both \( x \) and \( y \). This allows for a more flexible depiction of curves. A common range of such parameters is \( -\infty < t < \infty \). The parameter can be thought of as a storyteller narrating the movement along the curve. It's important to understand that each value of \( t \) corresponds to a unique point on this curve. Breaking up complex curves into simpler, manageable segments through parametrics can often simplify analysis and graphing.

Cartesian Equation

A Cartesian equation is what most of us are familiar with when describing curves. It is a direct relationship between \( x \) and \( y \) that does not depend on any other parameter. For the exercise, after finding the parametric equations, next was to derive the Cartesian equation. Using the steps provided, the process started by expressing one of the variables in terms of \( t \), choosing \( x = 3t \). From this, \( t \) was isolated, giving \( t = \frac{x}{3} \).Substituting this into the equation for \( y \), we have:\[y = 2 \left(\frac{x}{3}\right) = \frac{2x}{3}\]This cross-elimination of \( t \) provides a clear functional relationship \( y = \frac{2}{3}x \) , representing the curve directly on the Cartesian plane. The Cartesian form is favored when you want to depict a curve without involving additional parameters.

Eliminating Parameter

One of the main goals in working with parametric equations is often to eliminate the parameter, yielding a Cartesian equation. This process transforms parametric forms into more widely recognized Cartesian forms. To eliminate the parameter \( t \), we start by solving one of the parametric equations for \( t \). In this exercise, \( x = 3t \) is chosen, leading to:\[t = \frac{x}{3}\]Next, substituting this expression into the second equation, \( y = 2t \), provides:\[y = 2\left(\frac{x}{3}\right) = \frac{2x}{3}\] This results in \( y = \frac{2}{3}x \), eliminating \( t \) completely. Such a method is essential for simplifying expressions. It turns complex parametric curves into manageable mathematical expressions that can be easily graphed or further analyzed. Eliminating parameters can also reveal hidden characteristics of the curve, such as slopes or intercepts, assisting in better understanding and interpretation of the mathematical function.