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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 12.1.15
Chapter 12, Problem 12.1.15

15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.


x = 3 + t, y = 1 − t; 0 ≤ t ≤ 1

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1
Identify the given parametric equations: \(x = 3 + t\) and \(y = 1 - t\), with the parameter \(t\) ranging from \(0\) to \(1\).
To eliminate the parameter \(t\), solve one of the equations for \(t\). For example, from \(x = 3 + t\), isolate \(t\) to get \(t = x - 3\).
Substitute the expression for \(t\) into the other equation: replace \(t\) in \(y = 1 - t\) with \(x - 3\), resulting in \(y = 1 - (x - 3)\).
Simplify the equation to express \(y\) solely in terms of \(x\): \(y = 1 - x + 3\), which simplifies further to \(y = 4 - x\).
Interpret the curve: the equation \(y = 4 - x\) represents a straight line. The parameter \(t\) increases from \(0\) to \(1\), so the curve starts at the point when \(t=0\) (which is \((3,1)\)) and ends at \(t=1\) (which is \((4,0)\)). This indicates the positive orientation of the curve is from \((3,1)\) to \((4,0)\).

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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 the description of more complex curves and motions.
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Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove t, resulting in a direct relationship between x and y. This often requires solving one equation for t and substituting into the other, yielding a Cartesian equation of the curve.
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Eliminating the Parameter

Curve Orientation and Interval of Parameter

The orientation of a parametric curve is determined by the direction in which the parameter t increases. The interval for t specifies the portion of the curve traced, and understanding this helps describe the curve's direction and endpoints.
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Finding Area Between Curves that Cross on the Interval
Related Practice
Textbook Question

Given three polar coordinate representations for the origin.

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Textbook Question

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


The region inside the limaçon r = 2 + cos θ

Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


x = 3 cos t, y = 3 sin t; π ≤ t ≤ 2π

Textbook Question

15–22. Sets in polar coordinates Sketch the following sets of points.


2 ≤ r ≤ 8

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Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


4x = -y²

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Textbook Question

63–74. Arc length of polar curves Find the length of the following polar curves.


The complete circle r = a sin θ, where a > 0