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.PE.2

Capitolo 11, Problema 11.PE.2

Identifying Parametric Equations in the Plane

Exercises 1–6 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 and indicate the direction of motion and the portion traced by the particle.

x = √t, y = 1 − √t, t ≥ 0

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Start with the given parametric equations: \(x = \sqrt{t}\) and \(y = 1 - \sqrt{t}\), where \(t \geq 0\).

Express \(\sqrt{t}\) in terms of \(x\) from the first equation: \(\sqrt{t} = x\).

Substitute \(\sqrt{t} = x\) into the second equation to eliminate the parameter \(t\): \(y = 1 - x\).

Recognize that the Cartesian equation of the path is \(y = 1 - x\), which is a straight line.

Determine the portion of the line traced by the particle by considering the domain of \(t\): since \(t \geq 0\), and \(x = \sqrt{t}\), then \(x \geq 0\). So the particle moves along the line \(y = 1 - x\) for \(x \geq 0\), starting at the point \((0,1)\) and moving in the direction of increasing \(x\).

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Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often time (t). Instead of y as a function of x, both x and y depend on t, allowing the description of more complex motions and paths in the plane.

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Parameterizing Equations

Eliminating the Parameter

To find the Cartesian equation from parametric equations, solve one equation for the parameter and substitute into the other. This process removes the parameter, yielding a direct relationship between x and y that describes the particle’s path.

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Eliminating the Parameter

Direction and Domain of the Parameter

The parameter interval defines the portion of the curve traced and the direction of motion. Understanding the range of t and how x and y change with t helps determine which part of the Cartesian curve corresponds to the particle’s path and its movement direction.

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Eliminating the Parameter

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Length in Polar Coordinates

Find the lengths of the curves given by the polar coordinate equations in Exercises 51–54.

r = √(1 + cos 2θ), −π/2 ≤ θ ≤ π/2

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Identifying Conic Sections

Complete the squares to identify the conic sections in Exercises 69-76. Find their foci, vertices, centers, and asymptotes (as appropriate). If the curve is a parabola, find its directrix as well.

x² + y² + 4x + 2y = 1

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Lengths of Curves

Find the lengths of the curves in Exercises 13–19.

x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/2

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Area in Polar Coordinates

Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.

Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ

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