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.6

Capitolo 11, Problema 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

Guida verificata passo dopo passo

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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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.

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Parameterizing Equations

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.

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08:02

Parameterizing Equations

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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Percorso guidato

5:37

Introduction to Cotangent Graph

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Finding Parametric Equations and Tangent Lines

Find parametric equations for the given curve.

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Graphing Conic Sections

Exercises 63-68 give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.

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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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Graphing Conic Sections

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.

5y² − 4x² = 20

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Polar to Cartesian Equations

Sketch the lines in Exercises 23-28. Also, find a Cartesian equation for each line.

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Finding Parametric Equations and Tangent Lines

Find parametric equations for the given curve.

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