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 11.1.14

Chapter 11, Problem 11.1.14

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=√(t+1), y=√t, t ≥ 0

Verified step by step guidance

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

Express both \(x\) and \(y\) in terms of \(t\) squared to eliminate the square roots: \(x^2 = t + 1\) and \(y^2 = t\).

Use the expression for \(y^2\) to substitute for \(t\) in the equation for \(x^2\): \(x^2 = y^2 + 1\).

Rewrite the equation to isolate terms and get the Cartesian form: \(x^2 - y^2 = 1\).

Analyze the parameter interval \(t \geq 0\) to determine the portion of the graph traced by the particle and the direction of motion by considering how \(x\) and \(y\) change as \(t\) increases.

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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 denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex paths and motions in the plane.

Eliminating the Parameter to Find Cartesian Equations

To find a Cartesian equation from parametric equations, solve one parametric 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.

Graphing and Interpreting the Particle's Path and Direction

After finding the Cartesian equation, graph it to visualize the path. Use the parameter interval to determine which portion of the curve is traced and analyze how the parameter changes to indicate the direction of motion along the path.

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Related Practice

Textbook Question

Surface Area

Find the areas of the surfaces generated by revolving the curves in Exercises 31-34 about the indicated axes.

x = t + √2, y = (t²/2) + √2t, −√2 ≤ t ≤ √2; y−axis

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Textbook Question

Finding Polar Areas

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

Shared by the circles r = 1 and r = 2 sin θ

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Textbook Question

Cartesian to Polar Equations

Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.

(x + 2)² + (y − 5)² = 16"

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Textbook Question

Hyperbolas and Eccentricity

In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.

y² − x² = 4

Textbook Question

Lines

Sketch the lines in Exercises 45–48 and find Cartesian equations for them.

r cos (θ + π/3) = 2

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Textbook Question

Hyperbolas and Eccentricity

Exercises 25–28 give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation in Cartesian coordinates.

Eccentricity: 1.25

Foci: (0, ±5)