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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 12.1.50
Capitolo 12, Problema 12.1.50

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 left half of the parabola y=x ² +1, originating at (0, 1)

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Identify the given curve: the parabola is defined by the equation \(y = x^2 + 1\).
Since we want parametric equations, choose a parameter, typically \(t\), to represent \(x\). Let \(x = t\).
Express \(y\) in terms of \(t\) using the original equation: \(y = t^2 + 1\).
Determine the interval for \(t\) to represent the left half of the parabola. The left half corresponds to \(x \leq 0\), so \(t \leq 0\).
Write the parametric equations and specify the interval: \(x = t\), \(y = t^2 + 1\), with \(t \in (-\infty, 0]\).

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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 are defined in terms of t, allowing more flexibility in describing curves, especially those that are not functions in the traditional sense.
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Parameterizing Equations

Parabola and Its Properties

A parabola is a curve defined by a quadratic equation, such as y = x² + 1. Understanding its shape and symmetry is essential; here, the parabola opens upward with vertex at (0,1). The 'left half' refers to the portion where x ≤ 0, which guides the choice of parameter intervals.
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Properties of Parabolas

Parameter Interval and Curve Orientation

Choosing an appropriate interval for the parameter ensures the parametric equations trace the desired portion of the curve. For the left half of the parabola, the parameter should cover values corresponding to x ≤ 0, and the interval defines the start and end points, such as originating at (0,1).
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Finding Area Between Curves that Cross on the Interval
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57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.


r = 1/(2 - 2 sin θ)

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Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)

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11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 4 + sin θ; (4, 0) and (3, 3π/2)

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31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

A parabola with focus at (3, 0)

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9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.


(-4, 3π/2)

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Spiral arc length Consider the spiral r=4θ, for θ≥0.


a. Use a trigonometric substitution to find the length of the spiral, for 0≤θ≤√8.

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