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 11.1.10

Capitolo 11, Problema 11.1.10

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 = 1 + sin t, y = cos t − 2, 0 ≤ t ≤ π

Guida verificata passo dopo passo

Start with the given parametric equations: \(x = 1 + \sin t\) and \(y = \cos t - 2\), where \(0 \leq t \leq \pi\).

Isolate the trigonometric functions from the parametric equations: from \(x = 1 + \sin t\), we get \(\sin t = x - 1\); from \(y = \cos t - 2\), we get \(\cos t = y + 2\).

Recall the Pythagorean identity for sine and cosine: \(\sin^2 t + \cos^2 t = 1\).

Substitute the expressions for \(\sin t\) and \(\cos t\) into the identity: \((x - 1)^2 + (y + 2)^2 = 1\).

This equation represents the Cartesian form of the particle's path, which is a circle centered at \((1, -2)\) with radius 1. The parameter interval \(0 \leq t \leq \pi\) indicates the particle moves along the upper half of this circle, starting at \(t=0\) and ending at \(t=\pi\).

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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

Eliminating the Parameter to Find Cartesian Equations

To find a Cartesian equation from parametric equations, the parameter t is eliminated by using algebraic manipulation or trigonometric identities. This process converts the parametric form into a single equation relating x and y, describing the particle’s path in the xy-plane.

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Eliminate Parameter: Equations with Trig

Graphing and Interpreting Parametric Curves

Graphing parametric curves involves plotting points (x(t), y(t)) over the given parameter interval. Understanding the direction of motion and the portion of the curve traced requires analyzing how x and y change as t increases, which helps visualize the particle’s trajectory and orientation.

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Percorso guidato

06:49

Differentiation of Parametric Curves

Pratica correlata

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Implicitly Defined Parametrizations

Assuming that the equations in Exercises 15−20 define x and y implicitly as differentiable functions x=f(t), y=g(t), find the slope of the curve x=f(t), y=g(t) at the given value of t.

x sin t + 2x = t, t sin t − 2t = y, t = π

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Polar to Cartesian Equations

Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.

r² = 4r sin θ

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Domanda del libro di testo

Finding Parametric Equations

In Exercises 31–36, find a parametrization for the curve.

the ray (half line) with initial point (-1,2) that passes through the point (0,0)

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