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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.1.43
Capitolo 12, Problema 12.1.43

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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Identify the given curve equation: \(y = 2x^{2} - 4\) and the domain for \(x\) is \(-1 \leq x \leq 5\).
Choose a parameter to represent \(x\). A natural choice is to let the parameter \(t\) equal \(x\), so set \(x = t\).
Express \(y\) in terms of the parameter \(t\) by substituting \(x = t\) into the equation: \(y = 2t^{2} - 4\).
Write the parametric equations as \(x = t\) and \(y = 2t^{2} - 4\) with the parameter interval \(-1 \leq t \leq 5\).
Verify that the parametric equations describe the segment of the parabola by checking that as \(t\) varies over the interval, the points \((x,y)\) trace the curve segment.

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Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a third variable, called the parameter. Instead of y as a function of x, both x and y are defined in terms of a parameter t, allowing more flexible representations of curves, including segments and complex shapes.
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Parameterizing Equations

Interval of the Parameter

The interval of the parameter defines the range of values that the parameter t can take, which corresponds to the portion of the curve being described. Choosing the correct interval ensures the parametric equations represent only the desired segment of the curve, such as restricting x between -1 and 5.
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Eliminating the Parameter

Converting Cartesian to Parametric Form

To convert a Cartesian equation like y = 2x² - 4 into parametric form, assign the parameter t to x (e.g., x = t), then express y in terms of t using the original equation (y = 2t² - 4). This method directly translates the curve into parametric equations, with t varying over the specified x-interval.
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Eliminating the Parameter
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77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.


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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.


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Multiple descriptions Which of the following parametric equations describe the same curve?

a. x = 2t², y = 4 + t; -4 ≤ t ≤ 4

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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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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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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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