Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=4 t-2, y=2 t ; 0 \leq t \leq 3 $$

Step-by-Step Solution

Answer

The curve is a line segment from (-2, 0) to (10, 6); it is simple but not closed. The Cartesian equation is \( y = \frac{x}{2} + 1 \).

Step 1: Understanding the Parameterization

The given equation is parameterized with respect to the variable \( t \) where \( x = 4t - 2 \) and \( y = 2t \). The parameter \( t \) varies from 0 to 3.

Step 2: Graph the Curve

To graph the curve, calculate the coordinates of some key points by substituting different values of \( t \) into the parameter equations. For instance, if \( t = 0 \), \( x = -2 \) and \( y = 0 \). If \( t = 3 \), \( x = 10 \) and \( y = 6 \). Identify that this is a line segment.

Step 3: Determine if the Curve is Closed and Simple

The path is linear from \( t = 0 \) to \( t = 3 \) corresponding to \((x, y) = (-2, 0) \to (10, 6)\). Since the starting point does not connect to the endpoint and does not intersect itself, the curve is not closed but is simple.

Step 4: Eliminate Parameter

To find the Cartesian equation, first solve one of the equations for \( t \). From \( y = 2t \), we get \( t = \frac{y}{2} \). Substituting into \( x = 4t - 2 \) yields \( x = 4 \left( \frac{y}{2} \right) - 2 = 2y - 2 \). Rewrite as \( x + 2 = 2y \).

Step 5: Simplify to Cartesian Form

The Cartesian form of the equation is \( x + 2 = 2y \). Rearrange it to the standard form of a straight line: \( y = \frac{x}{2} + 1 \).

Key Concepts

Graphing Parametric Curves

When we discuss graphing parametric curves, we're dealing with a way to represent a curve by a set of equations that involve a third variable, often referred to as the parameter. For example, when we have equations such as \( x = 4t - 2 \) and \( y = 2t \), the variable \( t \) serves as our parameter. In this case, as \( t \) varies over a specified interval, it defines points on the curve. The problem specifies \( t \) between 0 and 3. Each value of \( t \) delivers a pair \((x, y)\) that can be plotted. Creating a graph involves picking several values of \( t \) within this interval, calculating corresponding \( x \) and \( y \) values, and plotting those pairs. For this specific parameterization, points calculated from \( t = 0, 1.5, \text{and} 3 \) could be \((-2, 0), (4, 3), \text{and} (10, 6)\) respectively. These points lie along a straight line segment in this case, which is the graph for this parametric curve.

Closed and Simple Curves

Determining if a curve is closed or simple is vital in understanding its behavior and properties. A closed curve means that the curve starts and ends at the same point, effectively forming a loop. A simple curve implies that the curve does not cross itself at any point. In our example, the curve follows a linear path from the point \((x, y) = (-2, 0)\) to \((x, y) = (10, 6)\) as \( t \) goes from 0 to 3. Since the starting and ending points are not the same, the curve is not closed. Also, because it follows a straight line without intersection, it qualifies as simple.

Cartesian Equation

Translating parametric equations into a Cartesian equation helps to simplify the analysis and visualization of a curve. In the realm of parametric equations, this involves eliminating the parameter to express relationships directly between \( x \) and \( y \). In our given problem, we already know \( x = 4t - 2 \) and \( y = 2t \). To eliminate the parameter \( t \), solve the equation \( y = 2t \) for \( t \), giving us \( t = \frac{y}{2} \). Substituting this back into the \( x \) equation gives \( x = 4 \left( \frac{y}{2} \right) - 2 \), which simplifies to \( x = 2y - 2 \). This can further be rearranged to the standard linear Cartesian form: \( y = \frac{x}{2} + 1 \). The obtained line equation offers a direct formula to plot the line graph without referring to a parameter.

Eliminating the Parameter

Eliminating the parameter in parametric equations involves reducing the equations to a single expression that links \( x \) and \( y \) without reference to the parameter. This process provides clarity and ease of function analysis. In our step-by-step solution, we first solved \( y = 2t \) to express \( t \) as \( t = \frac{y}{2} \). This conversion allows substituting \( t \) in the \( x \) equation \( x = 4t - 2 \), obtaining \( x = 4 \left( \frac{y}{2} \right) - 2 \), simplifying further to form \( x = 2y - 2 \). This transformation results in a Cartesian equation,\( y = \frac{x}{2} + 1 \), effectively eliminating the parameter while maintaining the relation between \( x \) and \( y \), thus simplifying graphing and further mathematical operations.