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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.R.4
Capitolo 12, Problema 12.R.4

3–6. Eliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.
x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π

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Identify the parametric equations given: \(x = \sin t - 3\) and \(y = \cos t + 6\), with the parameter \(t\) ranging from \(0\) to \(\pi\).
Recall the Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\). This will help eliminate the parameter \(t\) by expressing \(x\) and \(y\) in terms of sine and cosine.
Express \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\): from \(x = \sin t - 3\), we get \(\sin t = x + 3\); from \(y = \cos t + 6\), we get \(\cos t = y - 6\).
Substitute these expressions into the Pythagorean identity: \((x + 3)^2 + (y - 6)^2 = 1\). This is the Cartesian equation of the curve.
Interpret the equation: it represents a circle centered at \((-3, 6)\) with radius \(1\). The parameter \(t\) from \(0\) to \(\pi\) corresponds to the upper half of the circle, and the positive orientation follows the direction of increasing \(t\) from \(0\) to \(\pi\).

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Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. Understanding how x and y depend on t allows us to describe the curve's path and behavior over the given interval.
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Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations to form a single equation in x and y. This process often uses algebraic manipulation and trigonometric identities to remove t, yielding a Cartesian equation that describes the curve.
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Eliminating the Parameter

Geometric Description and Orientation

After finding the Cartesian form, interpreting the curve geometrically involves identifying its shape (e.g., circle, ellipse) and position. Positive orientation refers to the direction in which the curve is traced as the parameter increases, important for understanding motion along the curve.
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Geometric Sequences - Recursive Formula
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65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (0, ±2) and directrices y = ±1

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53–57. Conic sections

c. Find the eccentricity of the curve.

x²/4 + y²/25 = 1

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7–8. Parametric curves and tangent lines

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

x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3

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Polar conversion Consider the equation r=4/(sinθ+cosθ). 


a. Convert the equation to Cartesian coordinates and identify the curve it describes.  

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Jake’s response Jake responds to Liz (Exercise 33) with a graph that shows his love for her is infinite. Sketch each of the following curves. Which one should Jake send to Liz to get an infinity symbol?

b. r=(½)+sinθ

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42–43. Intersection points Find the intersection points of the following curves.

r= √(cos3t) and r= √(sin3t)

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