Ch. 11 - Parametric Equations and Polar Coordinates

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 11 - Parametric Equations and Polar Coordinates

Problem 11.PE.2

Chapter 11, Problem 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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Textbook Question

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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Textbook Question

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

Textbook Question

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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Textbook Question

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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Textbook Question