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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 12.3.14
Capitolo 12, Problema 12.3.14

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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Recall that for a polar curve given by \(r = f(\theta)\), the slope of the tangent line \(\frac{dy}{dx}\) can be found using the formulas for \(x\) and \(y\) in terms of \(r\) and \(\theta\): \(x = r \cos \theta\) and \(y = r \sin \theta\).
Express \(x\) and \(y\) as functions of \(\theta\): \(x(\theta) = (4 + \sin \theta) \cos \theta\) and \(y(\theta) = (4 + \sin \theta) \sin \theta\).
Find the derivatives \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) by applying the product rule to each: \(\frac{dx}{d\theta} = \frac{d}{d\theta}[(4 + \sin \theta) \cos \theta]\) and \(\frac{dy}{d\theta} = \frac{d}{d\theta}[(4 + \sin \theta) \sin \theta]\).
Use the chain rule and product rule to compute these derivatives explicitly, remembering that \(\frac{d}{d\theta}(\sin \theta) = \cos \theta\) and \(\frac{d}{d\theta}(\cos \theta) = -\sin \theta\).
Calculate the slope of the tangent line at each given point by evaluating \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\) at the corresponding \(\theta\) values: \(\theta = 0\) for the point \((4,0)\) and \(\theta = \frac{3\pi}{2}\) for the point \((3, \frac{3\pi}{2})\).

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Polar Coordinates and Polar Curves

Polar coordinates represent points using a radius r and an angle θ from the positive x-axis. Polar curves are equations expressed as r = f(θ), describing how the radius changes with the angle. Understanding this system is essential to analyze points and curves given in polar form.
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Intro to Polar Coordinates

Slope of Tangent Lines in Polar Coordinates

The slope of a tangent line to a polar curve at a point is found by converting the polar equation to Cartesian coordinates or using the formula dy/dx = (dr/dθ sin θ + r cos θ) / (dr/dθ cos θ - r sin θ). This formula relates derivatives of r with respect to θ to the slope in the xy-plane.
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Percorso guidato
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Slopes of Tangent Lines

Differentiation with Respect to θ

To find the slope of the tangent line, you must differentiate r = 4 + sin θ with respect to θ. This involves applying basic differentiation rules to trigonometric functions, which is crucial for computing dr/dθ and subsequently the slope dy/dx.
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Finding Differentials
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Domanda del libro di testo

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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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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37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r cos θ = -4

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

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