Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

x=1-t^{2}, y=2 t, 0 \leq t \leq 1

Step-by-Step Solution

Answer

The Cartesian equation is \(x = 1 - \frac{y^2}{4}\) for \(0 \leq y \leq 2\).

Step 1: Parametric Equations

The given problem involves parametric equations where the variables \(x\) and \(y\) are expressed as functions of a parameter \(t\). Given equations are \(x = 1 - t^2\) and \(y = 2t\).

Step 2: Determine Range of \(t\)

The parameter \(t\) has a defined range of \(0 \leq t \leq 1\). This range will help in determining how \(x\) and \(y\) vary as \(t\) changes.

Step 3: Eliminate Parameter \(t\) to Find Cartesian Equation

To eliminate the parameter, solve one of the parametric equations for \(t\). From \(y = 2t\), we have \(t = \frac{y}{2}\). Substitute \(t = \frac{y}{2}\) into \(x = 1 - t^2\), giving the equation \(x = 1 - \left(\frac{y}{2}\right)^2\).

Step 4: Simplify the Cartesian Equation

Simplify the equation \(x = 1 - \frac{y^2}{4}\) to make it a neat Cartesian form. This equation describes the curve without the parameter \(t\).

Step 5: Determine the Values for \(y\) and \(x\) by Range of \(t\)

From given \(0 \leq t \leq 1\): when \(t=0\), \(y=0\) and \(x=1\), and when \(t=1\), \(y=2\) and \(x=0\). So as \(t\) varies from 0 to 1, \(y\) varies from 0 to 2, and \(x\) decreases from 1 to 0.

Key Concepts

Cartesian Equation

Parametric equations can be transformed into a single equation in terms of the variables involved, known as a Cartesian equation. In our given exercise, we start with the parametric equations: \(x = 1 - t^2\) and \(y = 2t\). The goal is to eliminate the parameter \(t\) to combine these into a Cartesian equation solely in terms of \(x\) and \(y\). This results in a more familiar form that can easily be graphed, understood, and analyzed without involving the parameter.

Parameter Elimination

Parameter elimination is the process of removing the parameter to express the relationship between \(x\) and \(y\) directly. In our example, we have \(y = 2t\) which can be rearranged to express \(t\) as \(t = \frac{y}{2}\).
Substituting \(t = \frac{y}{2}\) in the equation \(x = 1 - t^2\), we get \(x = 1 - \left(\frac{y}{2}\right)^2\).
This simplifies to the Cartesian equation \(x = 1 - \frac{y^2}{4}\). This equation now links \(x\) and \(y\) directly, without the parameter \(t\), giving a single equation to describe the curve.

Range of Parameter

The range of the parameter is crucial in determining the values that the variables \(x\) and \(y\) can take. For our exercise, \(t\) is confined to the range \(0 \leq t \leq 1\). This range affects how \(x\) and \(y\) change:

• When \(t=0\), substituting in, \(y=0\) and \(x=1\).
• When \(t=1\), substituting in, \(y=2\) and \(x=0\).

This range helps define the segment of the curve that the parametric equations trace out, thus revealing how the curve progresses as \(t\) varies.

Graphing Curves

Graphing the curves derived from parametric equations involves plotting the values of \(x\) and \(y\) as \(t\) changes within its specified range. Once we have the Cartesian equation \(x = 1 - \frac{y^2}{4}\), this curve can be plotted without needing to vary \(t\) explicitly.
The given range for \(t\) from 0 to 1 shows that \(y\) takes values from 0 to 2, and consequently, \(x\) decreases from 1 to 0. This is a typical methodogram to localize the curve segments of the equation within a cartesian plane.
With this method, any point on the curve corresponds to a specific \(t\) value, which provides a complete picture of the curve's path as represented by the equations.