Problem 48 - Precalculus with Limits

Question

In Exercises 41-48, find a set of parametric equations for the rectangular equation using (a) \(t=x\) and (b) \(t=2-x\). \(y=\dfrac{1}{2x}\)

Step-by-Step Solution

Answer

The set of parametric equations using (a) \(t=x\) are \[x = t\] and \(y=\dfrac{1}{2t}\]. On the other hand, using (b) \(t=2-x\) yields \[x=2-t\] and \(y=\dfrac{1}{2(2-t)}\].

Step 1: Conversion to Parametric Equation using \(t=x\)

In (a), we are given that \(t=x\). Thus, we can directly replace \(x\) in the given equation with \(t\), which means \(y=\dfrac{1}{2t}\). So, the set of parametric equations using \(t=x\) is \[x = t\] and \(y=\dfrac{1}{2t}\].

Step 2: Conversion to Parametric Equation using \(t=2-x\)

In (b), we are given \(t=2-x\). In this case, we solve for \(x\), which gives us \(x=2-t\). Substituting \(x=2-t\) into the given equation yields \(y=\dfrac{1}{2(2-t)}\). Thus, the set of parametric equations using \(t=2-x\) is \[x=2-t\] and \(y=\dfrac{1}{2(2-t)}\].

Key Concepts

Rectangular Equation

A rectangular equation is a standard form of expressing relationships between two variables, typically using Cartesian coordinates. In such equations, we solve for one variable in terms of another, usually represented as an equation like \( y = \frac{1}{2x} \). The objective here is to express the relationship in terms of both \( x \) and \( y \) on the Cartesian plane.
Understanding rectangular equations is crucial as they form the basis for representing lines, circles, and many other shapes in standard math contexts. They can be identified as equations where variables are not dependent on a third variable like a parameter.

Parametric Form

When an equation is in parametric form, it expresses the coordinates \( x \) and \( y \) as functions of a third variable, typically \( t \). This creates a pair of equations that describe the same line, curve, or figure as the original rectangular equation. For example, with the rectangular equation \( y = \frac{1}{2x} \), the parametric equations could be represented as \( x = t \) and \( y = \frac{1}{2t} \).
Parametric forms are quite useful in scenarios where there's a dynamic aspect to the movement between points, like simulating gradual motion along a curve. You often encounter parametric forms in physics and engineering, where they provide a more intuitive understanding of complex paths, motions, or transformations.

Conversion to Parametric Equations

Converting a rectangular equation into a parametric form involves introducing a third variable, typically \( t \), which allows us to express \( x \) and \( y \) independently of each other. To convert \( y = \frac{1}{2x} \) to parametric equations, we follow the given cases:

• Using \( t = x \): Substitute \( t \) for \( x \), then the equations become \( x = t \) and \( y = \frac{1}{2t} \).
• Using \( t = 2 - x \): Rearrange to find \( x = 2 - t \) and substitute this into the equation to get \( y = \frac{1}{2(2 - t)} \).

These conversions help in representing curves more flexibly, enabling more advanced geometrical transformations and analysis.

Precalculus Problem Solving

In the context of precalculus, problem solving with different forms of equations is a core skill. You learn to manipulate equations to suit the problem's needs, particularly with parametric forms. This involves:

• Identifying the most suitable form of an equation for the problem at hand.
• Using parametric equations to analyze motion, as these equations naturally unfold variables over a parameter, making it easy to trace paths and curves.
• Practicing conversions between rectangular and parametric forms to understand their applications in various real-world contexts.

Precalculus often lays the groundwork for calculus by reinforcing these skills, allowing for seamless transition to more advanced concepts including derivatives and integrals, where parametrization becomes essential.