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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 12.3.25
Capitolo 12, Problema 12.3.25

25–28. Horizontal and vertical tangents Find the points at which the following polar curves have horizontal or vertical tangent lines.
r = 4 cos θ

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Recall that for a polar curve given by \(r = f(\theta)\), the Cartesian coordinates are \(x = r \cos \theta\) and \(y = r \sin \theta\).
To find the slope of the tangent line, compute \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\). This requires finding \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) using the product and chain rules.
Calculate \(\frac{dx}{d\theta} = \frac{d}{d\theta} (r \cos \theta) = \frac{dr}{d\theta} \cos \theta - r \sin \theta\) and \(\frac{dy}{d\theta} = \frac{d}{d\theta} (r \sin \theta) = \frac{dr}{d\theta} \sin \theta + r \cos \theta\).
Find \(\frac{dr}{d\theta}\) by differentiating \(r = 4 \cos \theta\), so \(\frac{dr}{d\theta} = -4 \sin \theta\).
Set the numerator of \(\frac{dy}{dx}\) equal to zero to find horizontal tangents (where \(\frac{dy}{d\theta} = 0\) and \(\frac{dx}{d\theta} \neq 0\)), and set the denominator equal to zero to find vertical tangents (where \(\frac{dx}{d\theta} = 0\) and \(\frac{dy}{d\theta} \neq 0\)). Solve these equations for \(\theta\) and then find the corresponding points \((r, \theta)\).

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Polar Coordinates and Polar Curves

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian coordinates (x, y). Polar curves are equations expressed in terms of r and θ, describing shapes based on the distance from the origin and the angle. Understanding how to interpret and plot these curves is essential for analyzing their properties.
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Intro to Polar Coordinates

Derivatives of Polar Curves and Tangent Lines

To find tangent lines to polar curves, we convert the polar equation to parametric form (x = r cos θ, y = r sin θ) and compute dy/dx using derivatives with respect to θ. The slope of the tangent line is given by (dy/dθ) / (dx/dθ). This derivative helps identify where the tangent is horizontal or vertical.
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Percorso guidato
05:13
Slopes of Tangent Lines

Conditions for Horizontal and Vertical Tangents

A horizontal tangent occurs where the slope dy/dx = 0, meaning dy/dθ = 0 while dx/dθ ≠ 0. A vertical tangent occurs where the slope is undefined, meaning dx/dθ = 0 while dy/dθ ≠ 0. Applying these conditions to the derivatives of the polar curve reveals the points with horizontal or vertical tangents.
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Percorso guidato
5:28
Horizontal Parabolas
Pratica correlata
Domanda del libro di testo

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 lower half of the circle centered at (−2, 2) with radius 6, oriented in the counterclockwise direction

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Domanda del libro di testo

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.


x=t ²−1, y=t ³ +t; t=2

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Domanda del libro di testo

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.

y² - x²/64 = 1; (6, -5/4)

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Domanda del libro di testo

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.


(1, 2π/3)

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Domanda del libro di testo

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


1 < r < 2 and π/6 ≤ θ ≤ π/3

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Domanda del libro di testo

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=tan t, y=sec ² t−1 

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