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Chapter 1: Problem 4

Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=3-t, y=2 t-3,1.5 \leq t \leq 3 $$

Short Answer

Expert verified
The curve is \( y = 3 - 2x \) and is oriented from (1.5, 0) to (0, 3).

Step by step solution

01

Identify Parametric Equations

The given parametric equations are \( x = 3 - t \) and \( y = 2t - 3 \). These express the variables \( x \) and \( y \) in terms of the parameter \( t \).
02

Eliminate the Parameter

Solve the equation \( x = 3 - t \) for \( t \). This gives us \( t = 3 - x \). Substitute this expression for \( t \) into the equation for \( y \): \[ y = 2(3 - x) - 3 \].
03

Simplify the Equation

Simplify the equation for \( y \): \( y = 2(3 - x) - 3 \) becomes \( y = 6 - 2x - 3 \). Simplify further to obtain \( y = 3 - 2x \). This is the Cartesian equation of the curve.
04

Determine Curve Orientation

To determine the orientation, substitute values of \( t \) within the given range \( 1.5 \leq t \leq 3 \). Determine the coordinates \((x, y)\) for the endpoints: \( t = 1.5 \) gives \( (1.5, 0) \) and \( t = 3 \) gives \( (0, 3) \). The curve moves from point \((1.5, 0)\) to \((0, 3)\) as \( t \) increases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Curve Sketching
Curve sketching is an essential skill that involves drawing a curve based on its equations, often transitioning from parametric to Cartesian form. Understanding how a curve behaves not only helps in visualizing the path it follows but also in analyzing various properties such as intersections, maxima, and minima. To sketch a curve defined by parametric equations:
  • First, plot a few key points by substituting values of the parameter into the equations to find corresponding points on the curve.
  • Identify critical points like where the curve begins and ends, along with any points of inflection or symmetry.
  • Once a few points are calculated, connect these dots smoothly, maintaining awareness of any restrictions imposed by the domain of the parameter.
Curve sketching allows you to see the whole picture of the path without needing a detailed graphing calculator, grounding the visual spatially and mathematically.
Parameter Elimination
Parameter elimination is a process used to convert parametric equations into a single Cartesian equation. This simplifies analyzing and interpreting the curve's properties using Cartesian coordinates rather than parametric forms. The process typically involves these steps:
  • Express one parameter (such as \( t \)) in terms of another variable (\( x \) or \( y \)).
  • Substitute this expression into the other parametric equation.
  • Simplify the resulting equation to isolate \( y \) as a function of \( x \) or vice versa. This yields the Cartesian equation of the curve, free of the parameter \( t \).
This process not only makes a curve easier to plot and study but also reveals the underlying geometric structure of the graph. In our exercise, parameter elimination changed our perspective from a time-dependent path to a clear geometric line \( y = 3 - 2x \).
Cartesian Equation
A Cartesian equation expresses relationships between \( x \) and \( y \) in the familiar coordinate plane. Unlike parametric equations that depend on a third variable \( t \), Cartesian equations provide a direct relation between two variables. Here's why they are useful:
  • They allow for straightforward graphing and identification of graph features, such as slope and intercepts.
  • Cartesian forms are often simpler to manipulate algebraically, supporting functions transformation and analysis.
  • Their standard form can easily help identify shapes: lines, circles, parabolas, etc.
In our example, converting the parametric equations to \( y = 3 - 2x \) shows that the curve is a straight line. This simple form confirms the line鈥檚 direction and slope, enhancing comprehension of the graph's behavior and relationship between \( x \) and \( y \).
Curve Orientation
Curve orientation describes the direction a curve moves along as the parameter increases. Understanding this is key to comprehending how shapes form and split in space. Determining curve orientation involves:
  • Substituting the parameter's values into the parametric equations to find corresponding \( (x, y) \) points.
  • Assessing the sequence these points connect as the parameter progresses, identifying the start and end.
  • Checking continuity and directionality between calculated points to ensure no abrupt shifts.
In solving our exercise, by initiating with \( t = 1.5 \) and ending with \( t = 3 \), we learned that the curve moves from \( (1.5, 0) \) to \( (0, 3) \). Recognizing orientation helps solidify comprehension of a curve鈥檚 real-world movement, turning formulas into more tangible figures.

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Most popular questions from this chapter

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x=2 t+1 \\ y=t^{2}-3 \end{array} $$

Find the area bounded by the curve \(x=\cos t, y=e^{t}, 0 \leq t \leq \frac{\pi}{2}\) and the lines \(y=1\) and \(x=0\).

Find the length of the curve \(x=e^{t}-t, y=4 e^{t / 2},-8 \leq t \leq 3 .\)

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=2 \cosh t, \quad y=4 \sinh t $$

Determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. $$ x=2 t, \quad y=t^{3}, \quad t=-1 $$
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