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Chapter 11: Problem 27

Find a parametric description for the given oriented curve. the directed line segment from (-2,-1) to (3,-4)the curve \(y=4-x^{2}\) from (-2,0) to (2,0) .

Short Answer

Expert verified
Line segment: \(x(t) = -2 + 5t, y(t) = -1 - 3t\) for \(t \in [0,1]\); Parabola: \(x(t) = t, y(t) = 4-t^2\) for \(t \in [-2,2]\).

Step by step solution

01

Curve Definition

First, identify the two components of the exercise. You need to express two curves parametrically: the line segment from \((-2, -1)\) to \((3, -4)\), and the parabolic section of the curve \(y=4-x^2\) from \(x = -2\) to \(x = 2\).
02

Parametric Equations for the Line Segment

The general form of a parametric equation for a line from point \((x_0, y_0)\) to point \((x_1, y_1)\) is \((x, y) = (1-t)(x_0, y_0) + t(x_1, y_1)\) for \(t\) in \([0,1]\). Substituting our points, \((-2, -1)\) and \((3, -4)\), results in:\[ x(t) = -2 + 5t \]\[ y(t) = -1 - 3t \]These equations describe the line segment from \((-2, -1)\) to \((3, -4)\).
03

Parametric Equations for the Parabola

For the parabola section, already given by \(y = 4 - x^2\), use \(x = t\) as a parameter directly, since \(x\) varies from \(-2\) to \(2\), and the equation for \(y\) remains:\[ y(t) = 4 - t^2 \]This results in parametric equations for the portion of the parabola from \((-2, 0)\) to \((2, 0)\):\[ x(t) = t, \; y(t) = 4-t^2 \]for \(t\) in \([-2, 2]\).
04

Combine both Descriptions

Summarize the parametric descriptions. For the line segment, the parametric equations are: \[ x_1(t) = -2 + 5t \]\[ y_1(t) = -1 - 3t \] for \(t\) in \([0,1]\). For the parabola, the parametric equations are: \[ x_2(t) = t, \; y_2(t) = 4-t^2 \]for \(t\) in \([-2, 2]\).

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Key Concepts

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

Line Segment
A line segment is the shortest path between two points. In this exercise, the line segment connects the points (-2,-1) and (3,-4). It is crucial to understand that line segments are straight, and have a finite length, defined by the distance between their endpoints. When working with parametric equations for a line segment, we use a parameter, often denoted as \( t \), to smoothly transition between these endpoints.
  • To describe a line segment parametrically, we identify the starting and ending points and interpolate between them.
  • The general form is \((x,y) = (1-t)(x_0,y_0) + t(x_1,y_1)\)
In our case, we have:- \( x(t) = -2 + 5t \) - \( y(t) = -1 - 3t \)for \( t \) in the interval \([0,1]\), indicating the transition from start to end.
Parabola
The concept of a parabola is fundamentally linked to its equation, which is usually of the form \( y = ax^2 + bx + c \). Here, the parabola in question is given by \( y = 4 - x^2 \). Parabolas have a characteristic symmetrical U-shape, curving upwards or downwards depending on the coefficient of the \( x^2 \) term.
  • In this exercise, the parabola extends from \( x = -2 \) to \( x = 2 \).
  • The vertex, or highest/lowest point, is at the origin for this parabola, with \( y = 4 \) at \( x = 0 \).
For the portion between the points (-2,0) and (2,0), the parametric description is straightforward as: - \( x(t) = t \) - \( y(t) = 4 - t^2 \)where \( t \) directly represents the \( x \) value, spanning from \(-2\) to \(2\).
Parametric Description
Parametric description is a powerful way to represent curves, allowing the expression of both \( x \) and \( y \) coordinates as functions of a third variable, typically denoted as \( t \). This is especially useful when describing complex shapes that cannot be easily captured with a single equation in \( x \) and \( y \).
  • In the context of line segments and curves, parametric equations provide flexibility by specifying movement along the path through a parameter.
  • It enables us to describe orientation and direction as \( t \) typically progresses from 0 to 1 or other specified ranges.
  • Changing the range of \( t \) effectively chooses different parts of the curve we want to focus on.
In the exercise, the parametric approach is used for both the line segment and parabola. Each segment has unique limits for \( t \), representing distinct sections of the curve.
Oriented Curve
An oriented curve is not just a path in space but has a direction associated with it. Thus, when we speak about an oriented curve, we refer to both the line segment and the parabola, acknowledging how one moves from start to end.
  • In this exercise, the orientation for the line segment describes a path from (-2,-1) to (3,-4).
  • For the parabolic section, the orientation aligns with increasing \( t \), starting at \( x = -2 \) and moving through to \( x = 2 \).
Understanding the orientation is essential as it dictates how we navigate the curve, influencing how we perceive and interpret the path. This is critical in fields like physics and engineering, where such curves often model moving objects.

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Most popular questions from this chapter

Use the pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities.\(\bullet \vec{v} \cdot \vec{w}\) \( \bullet \operatorname{proj}_{\vec{w}}(\vec{v})\) \( \bullet\)The angle \(\theta\) (in degrees) between \(\vec{v}\) and \(\vec{w}\) \( \bullet\) \(\vec{q}=\vec{v}-\operatorname{proj}_{\vec{w}}(\vec{v})\) (Show that \(\left.\vec{q} \cdot \vec{w}=0 .\right)\) $$ \vec{v}=\frac{3}{2} \hat{\imath}+\frac{3}{2} \hat{\jmath} \text { and } \vec{w}=\hat{\imath}-\hat{\jmath} $$

Find a parametric description for the given oriented curve. the ellipse \(9 x^{2}+4 y^{2}+24 y=0\), oriented clockwise (Shift the parameter so \(t=0\) corresponds to (0,0) .)

Convert the equation from polar coordinates into rectangular coordinates. \(r=-\sqrt{5} \csc (\theta)\)

Use the pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities.\(\bullet \vec{v} \cdot \vec{w}\) \( \bullet \operatorname{proj}_{\vec{w}}(\vec{v})\) \( \bullet\)The angle \(\theta\) (in degrees) between \(\vec{v}\) and \(\vec{w}\) \( \bullet\) \(\vec{q}=\vec{v}-\operatorname{proj}_{\vec{w}}(\vec{v})\) (Show that \(\left.\vec{q} \cdot \vec{w}=0 .\right)\) $$ \vec{v}=\langle-8,3\rangle \text { and } \vec{w}=\langle 2,6\rangle $$

Plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=t^{2}+2 t+1 \\ y=t+1 \end{array}\right. \text { for } t \leq 1 $$
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