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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 12.3.29
Capitolo 12, Problema 12.3.29

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

Guida verificata passo dopo passo
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Start by setting the two polar equations equal to each other to find the intersection points algebraically: \[2 \cos \theta = 1 + \cos \theta\].
Rearrange the equation to isolate terms: \[2 \cos \theta - \cos \theta = 1 \implies \cos \theta = 1\].
Solve for \[\theta\] where \[\cos \theta = 1\]. Recall that \[\cos \theta = 1\] at \[\theta = 0\] (and at multiples of \[2\pi\], but consider the principal values in the interval \[[0, 2\pi)\]).
Substitute \[\theta = 0\] back into either original equation to find the corresponding \[r\] value for the intersection point.
To find any additional intersection points that may not be captured by the algebraic method, use a graphical approach by plotting both curves \[r = 2 \cos \theta\] and \[r = 1 + \cos \theta\] over the interval \[\theta \in [0, 2\pi)\] and visually identify where they intersect.

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Polar Coordinates and Equations

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how to interpret and manipulate polar equations like r = 2 cos θ is essential for finding points where two curves intersect.
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Intro to Polar Coordinates

Solving Systems of Equations

Finding intersection points involves solving the system formed by the two given equations. In this case, equate r = 2 cos θ and r = 1 + cos θ, then solve algebraically for θ and r to find common solutions representing intersection points.
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Solving Logarithmic Equations

Graphical Analysis of Polar Curves

Graphing polar curves helps visualize intersections that may be difficult to find algebraically, especially when multiple solutions or complex angles are involved. Plotting r = 2 cos θ and r = 1 + cos θ reveals additional intersection points and confirms algebraic results.
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Percorso guidato
09:04
Slope of Polar Curves
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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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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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49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


(x - 1)² + y² = 1

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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 = 2θ; (π/2, π/4)

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