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.6

Chapter 11, Problem 11.PE.6

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 = 4 cos t, y = 9 sin t, 0 ≤ t ≤ 2π

Verified step by step guidance

Start with the given parametric equations: \(x = 4 \cos t\) and \(y = 9 \sin t\), where \(0 \leq t \leq 2\pi\).

Recall the Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\). We will use this to eliminate the parameter \(t\).

Express \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\): from \(x = 4 \cos t\), we get \(\cos t = \frac{x}{4}\); from \(y = 9 \sin t\), we get \(\sin t = \frac{y}{9}\).

Substitute these into the Pythagorean identity: \(\left(\frac{x}{4}\right)^2 + \left(\frac{y}{9}\right)^2 = 1\).

This equation represents the Cartesian form of the particle's path, which is an ellipse. The parameter interval \(0 \leq t \leq 2\pi\) means the particle traces the entire ellipse once, and the direction of motion corresponds to increasing \(t\).

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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.

Conversion to Cartesian Equation

To identify the particle’s path, parametric equations are combined to eliminate the parameter t, resulting in a Cartesian equation relating x and y directly. This often involves using trigonometric identities or algebraic manipulation to rewrite the curve in standard form.

Graphing and Direction of Motion

Graphing the Cartesian equation shows the shape of the path, while the parameter interval and parametric equations indicate the portion traced and the direction of motion. Understanding how t changes helps to mark the start, end, and direction along the curve.

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Find the eccentricities of the ellipses and hyperbolas in Exercises 59–62. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch.

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Find parametric equations for the given curve.

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