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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.3.20
Capitolo 12, Problema 12.3.20

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


r = 2θ; (π/2, π/4)

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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 \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), and then applying the chain rule to find \(\frac{dy}{d\theta}\) and \(\frac{dx}{d\theta}\).
Write down the expressions for \(x\) and \(y\) in terms of \(\theta\): \(x = r \cos(\theta) = 2\theta \cos(\theta)\) \(y = r \sin(\theta) = 2\theta \sin(\theta)\).
Differentiate both \(x\) and \(y\) with respect to \(\theta\) using the product rule: \(\frac{dx}{d\theta} = \frac{d}{d\theta} (2\theta \cos(\theta))\) \(\frac{dy}{d\theta} = \frac{d}{d\theta} (2\theta \sin(\theta))\).
Calculate the slope of the tangent line using the formula \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\).
Evaluate \(\frac{dy}{dx}\) at the given point \(\theta = \frac{\pi}{4}\) to find the slope of the tangent line at that point.

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Polar Coordinates and Curves

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how curves are defined in polar form, such as r = 2θ, is essential for analyzing their geometric properties and converting between coordinate systems.
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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 sinθ + r cosθ) / (dr/dθ cosθ - r sinθ). This allows determination of the tangent's slope without explicit Cartesian conversion.
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Percorso guidato
05:13
Slopes of Tangent Lines

Differentiation with Respect to θ

Calculating the slope requires differentiating r with respect to θ (dr/dθ). This derivative captures how the radius changes as the angle varies, which is crucial for applying the tangent slope formula in polar coordinates.
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Percorso guidato
05:53
Finding Differentials
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Domanda del libro di testo

85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)


A circular corral of unit radius is enclosed by a fence. A goat is outside the corral and tied to the fence with a rope of length 0≤a ≤ π (see figure). What is the area of the region (outside the corral) that the goat can reach?


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How does the eccentricity determine the type of conic section?

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29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.


r = 2 cos θ and r = 1 + cos θ

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31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.


(-4, 4√3)

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84. Arc length for polar curves: Prove that the length of the curve r = f(θ) for α ≤ θ ≤ β is

L = ∫(α to β) √(f(θ)² + f'(θ)²) dθ.

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45–60. Areas of regions Find the area of the following regions.


The region inside the rose r = 4 sin 2θ and inside the circle r = 2

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