Ch. 11 - Parametric Equations and Polar Coordinates

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 11 - Parametric Equations and Polar Coordinates

Problema 11.4.6

Capitolo 11, Problema 11.4.6

Symmetries and Polar Graphs

Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.

r = 1 + 2 sin θ

Guida verificata passo dopo passo

Recall that to identify symmetries of a polar curve given by \(r = f(\theta)\), we check for symmetry about the polar axis (the x-axis), the line \(\theta = \frac{\pi}{2}\) (the y-axis), and the pole (origin).

For symmetry about the polar axis, replace \(\theta\) by \(-\theta\) and see if the equation remains unchanged or can be manipulated to the original form. That is, check if \(r = 1 + 2 \sin(-\theta)\) simplifies to the original \(r\).

For symmetry about the line \(\theta = \frac{\pi}{2}\), replace \(\theta\) by \(\pi - \theta\) and check if \(r = 1 + 2 \sin(\pi - \theta)\) simplifies to the original \(r\).

For symmetry about the pole (origin), replace \(r\) by \(-r\) and \(\theta\) by \(\theta + \pi\), and check if \(-r = 1 + 2 \sin(\theta + \pi)\) can be rearranged to the original equation.

After determining the symmetries, sketch the curve by plotting points for various values of \(\theta\) between \(0\) and \(2\pi\), using the equation \(r = 1 + 2 \sin \theta\), and then reflect the curve according to the symmetries found.

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Polar Coordinates and Graphing

Polar coordinates represent points using a radius r and an angle θ, differing from Cartesian coordinates. Understanding how to plot points given r(θ) is essential for sketching curves like r = 1 + 2 sin θ, where r changes with θ from 0 to 2π.

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05:32

Intro to Polar Coordinates

Symmetry in Polar Graphs

Symmetry in polar graphs can occur about the polar axis, the line θ = π/2, or the pole (origin). Testing the function for transformations like replacing θ with -θ, π - θ, or θ + π helps identify these symmetries, which simplifies sketching and understanding the curve.

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05:32

Intro to Polar Coordinates

Trigonometric Functions in Polar Equations

Trigonometric functions like sine and cosine influence the shape and symmetry of polar graphs. For r = 1 + 2 sin θ, the sine term affects the radius depending on θ, creating characteristic shapes such as limacons, cardioids, or loops, which are important to recognize.

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Percorso guidato

6:04

Introduction to Trigonometric Functions

Pratica correlata

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Implicitly Defined Parametrizations

Assuming that the equations in Exercises 15−20 define x and y implicitly as differentiable functions x=f(t), y=g(t), find the slope of the curve x=f(t), y=g(t) at the given value of t.

x sin t + 2x = t, t sin t − 2t = y, t = π

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Finding Parametric Equations

In Exercises 31–36, find a parametrization for the curve.

the ray (half line) with initial point (-1,2) that passes through the point (0,0)

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Finding Parametric Equations

In Exercises 31–36, find a parametrization for the curve.

the line segment with endpoints (-1,3) and (3,-2)

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Finding Cartesian from Parametric Equations

Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

x = 1 + sin t, y = cos t − 2, 0 ≤ t ≤ π

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Finding Polar Areas

Find the areas of the regions in Exercises 9–18.

Inside the circle r = 4 sin θ and below the horizontal line r = 3 csc θ

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Theory and Examples

Tangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.

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