Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

a parametric representation of a curve is given. $$ x=4 t-2, y=2 t ; 0 \leq t \leq 3 $$

Step-by-Step Solution

Answer

The Cartesian equation is \(x = 2y - 2\), with \(0 \leq y \leq 6\).

Step 1: Identify the Parametric Equations

The parametric equations for the curve are given as \(x = 4t - 2\) and \(y = 2t\), where \(t\) is the parameter with the interval \(0 \leq t \leq 3\). These equations describe the coordinates \((x, y)\) of the points on the curve as the parameter \(t\) changes within the specified range.

Step 2: Eliminate the Parameter

To express the curve in terms of \(x\) and \(y\) only, we need to eliminate the parameter \(t\). Start with the equation for \(y\): \(y = 2t\). Solve for \(t\): \(t = \frac{y}{2}\). Substitute this expression into the equation for \(x\): \(x = 4\left(\frac{y}{2}\right) - 2\). Simplify this to obtain \(x = 2y - 2\).

Step 3: Determine the Range for y

Given the interval for \(t\), \(0 \leq t \leq 3\), we can determine the range of \(y\). Since \(y = 2t\), substituting \(t = 0\) gives \(y = 0\) and \(t = 3\) gives \(y = 6\). Therefore, \(y\) ranges from 0 to 6.

Step 4: Express the Final Cartesian Form

We have \(x = 2y - 2\) from the parameter elimination, valid for \(0 \leq y \leq 6\). This is the equation of the curve in Cartesian form.

Key Concepts

Curve Sketching

Parametric equations describe a curve using a third variable, usually denoted as a parameter like \(t\). In our exercise, each value of \(t\) from 0 to 3 corresponds to a specific point \((x, y)\) on the curve, defined by the equations \(x = 4t - 2\) and \(y = 2t\). Curve sketching involves plotting these points to visualize the path of the curve.

To sketch the curve:

• Choose several values for \(t\) within the given range, such as 0, 1, 2, and 3.
• Calculate the corresponding \(x\) and \(y\) values for each \(t\).
• Plot these points on a coordinate plane.

By connecting these points in order of increasing \(t\), you get a visual representation of the curve's path. The line or curve you draw shows how \((x, y)\) coordinates change as \(t\) varies.

Parameter Elimination

To express a parametric curve in only \(x\) and \(y\), you eliminate the parameter \(t\). In this exercise, the goal is to find a relationship between \(x\) and \(y\) without involving \(t\).

Here's how:

• Start with one of the parametric equations, like \(y = 2t\).
• Solve this equation for \(t\), yielding \(t = \frac{y}{2}\).
• Substitute this expression for \(t\) into the other equation, \(x = 4t - 2\), to remove \(t\).

This substitution gives \(x = 2y - 2\). This new equation describes the same curve but expressed in terms of only \(x\) and \(y\). This is useful for conceptualizing the curve in a traditional Cartesian plane.

Cartesian Form

The Cartesian form is simply a way of expressing a curve using standard polynomial equations in \(x\) and \(y\). After eliminating the parameter \(t\) in parametric equations, you are left with an equation like \(x = 2y - 2\). This is the Cartesian form of the original parametric curve.

The advantages of using the Cartesian form include:

• Recognizing common types of curves, such as lines, parabolas, ellipses, etc.
• Simplifying the process of finding intersections with other curves.
• Making it easier to compute derivatives for calculus problems.

The Cartesian equation \(x = 2y - 2\) represents a straight line, showing that the original parametric equations also describe a linear path.

Coordinate Geometry

Coordinate geometry is the study of geometric figures using the coordinate plane, which allows for algebraic manipulation of shapes and curves. In the context of our exercise, it provides a powerful way to analyze the shape and position of the curve determined by the parametric equations in the Cartesian plane.

Considerations in coordinate geometry:

• Understanding the curve's path as expressed by the parametric equations \((x = 4t - 2, y = 2t)\).
• Analyzing properties like slope and intercepts using the Cartesian form \(x = 2y - 2\).
• Visualizing the geometric meaning behind algebraic manipulations.

Understanding coordinate geometry helps in recognizing how parametric equations transition into familiar algebraic ones, breaking down potentially complex curves into understandable parts.