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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 12.2.13
Capitolo 12, Problema 12.2.13

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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Recall that a point in polar coordinates is given as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
Plot the point \((-4, \frac{3\pi}{2})\) by first considering the angle \(\frac{3\pi}{2}\), which corresponds to the downward direction along the negative y-axis.
Since \(r\) is negative, move in the opposite direction of the angle \(\frac{3\pi}{2}\). This means you move 4 units upward (opposite to downward) along the positive y-axis.
To find two alternative representations, use the fact that adding \(2\pi\) to the angle does not change the point, and changing the sign of \(r\) while adding \(\pi\) to the angle gives the same point. So, the alternatives are:
1) \((r, \theta + 2\pi) = (-4, \frac{3\pi}{2} + 2\pi)\) and 2) \((-r, \theta + \pi) = (4, \frac{3\pi}{2} + \pi)\). Write these explicitly as your alternative polar coordinates.

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Polar Coordinates System

Polar coordinates represent points in a plane using a radius and an angle, denoted as (r, θ). The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. This system is useful for describing locations in circular or rotational contexts.
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Intro to Polar Coordinates

Graphing Points in Polar Coordinates

To graph a point (r, θ), start at the origin, rotate counterclockwise by angle θ, then move outward (or inward if r is negative) by distance r. Negative radius values mean moving in the opposite direction of the angle, which affects the point's position on the plane.
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Intro to Polar Coordinates

Alternative Representations of Polar Coordinates

A single point can have multiple polar coordinate representations by adding or subtracting full rotations (2π) to the angle or by changing the sign of the radius and adjusting the angle by π. For example, (r, θ) is equivalent to (-r, θ + π) and (r, θ + 2πk) for any integer k.
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Intro to Polar Coordinates
Pratica correlata
Domanda del libro di testo

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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90–94. Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.

The length of the latus rectum of a hyperbola centered at the origin is (2b²)/a = 2b√(1 - e²)

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Domanda del libro di testo

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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39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 


A hyperbola with vertices (±4, 0) and foci (±6, 0)

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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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