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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.1.46
Chapter 12, Problem 12.1.46

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The horizontal line segment starting at P(8, 2) and ending at Q(−2, 2)

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Identify the key characteristics of the curve: it is a horizontal line segment starting at point P(8, 2) and ending at point Q(−2, 2). Since the y-coordinate is constant (y = 2) for all points on this segment, the parametric equations will reflect this.
Choose a parameter, say \(t\), to represent the position along the line segment. A common choice is to let \(t\) vary from 0 to 1, where \(t=0\) corresponds to the starting point P and \(t=1\) corresponds to the ending point Q.
Express the \(x\)-coordinate as a linear function of \(t\) that moves from 8 to −2. This can be written as \(x(t) = 8 + t(-2 - 8) = 8 - 10t\).
Since the \(y\)-coordinate is constant at 2, write \(y(t) = 2\) for all \(t\) in the interval.
State the parametric equations and the interval for \(t\): \(x(t) = 8 - 10t\), \(y(t) = 2\), with \(t\) in the interval \([0, 1]\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing more flexible descriptions of curves, including lines and segments.
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Representation of Line Segments

A line segment between two points can be represented parametrically by defining x and y as linear functions of a parameter t, typically ranging from 0 to 1. This parameterization moves from the start point to the end point as t increases.
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Parameter Interval

The parameter interval specifies the range of t values for which the parametric equations describe the desired portion of the curve. For line segments, t usually varies over a closed interval like [0,1] to cover the entire segment between two points.
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Related Practice
Textbook Question

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.


x = 2 cos t, y = 8 sin t; slope = -1

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Textbook Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r = sin 3θ

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33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.


The region inside one leaf of r = cos 3θ

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Find the slope of the parametric curve x=−2t ³ +1, y=3t ², for −∞<t<∞, at the point corresponding to t=2. 

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Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The segment of the parabola y=2x ²−4, where −1≤x≤5

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Textbook Question

93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.


An ellipse centered at (-2, -3) with major and minor axes of lengths 30 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

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