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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.1.29a
Capitolo 11, Problema 11.1.29a

Finding Parametric Equations


Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².


a. once clockwise.


(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

Guida verificata passo dopo passo
1
Recall that the equation of the circle is given by \(x^{2} + y^{2} = a^{2}\). This describes a circle centered at the origin with radius \(a\).
To find parametric equations for the motion of a particle on this circle, use the standard parameterization involving trigonometric functions: \(x = a \cos(t)\) and \(y = a \sin(t)\), where \(t\) is the parameter.
Since the particle starts at the point \((a, 0)\), determine the value of \(t\) at the start. Substitute \(x = a\) and \(y = 0\) into the parametric equations to find the initial parameter value.
To trace the circle once clockwise, consider the direction of motion. The standard parameterization with \(t\) increasing from \(0\) to \(2\pi\) traces the circle counterclockwise. To reverse the direction, modify the parameterization to \(x = a \cos(t)\) and \(y = -a \sin(t)\).
Set the parameter interval for \(t\) to cover one full revolution, typically \(t\) going from \(0\) to \(2\pi\), to ensure the particle traces the entire circle once in the clockwise direction.

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Parametric Equations of a Circle

Parametric equations express the coordinates of points on a curve as functions of a parameter, often time. For a circle of radius a centered at the origin, a common parametrization is x = a cos(t), y = a sin(t), where t varies over an interval to trace the circle.
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06:03
Parameterizing Equations of Circles & Ellipses

Direction of Motion (Clockwise vs Counterclockwise)

The direction in which a particle moves along a curve depends on how the parameter affects the coordinates. For a circle, using x = a cos(t), y = a sin(t) traces counterclockwise motion, while x = a cos(t), y = -a sin(t) reverses the direction to clockwise.
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Percorso guidato
06:29
Derivatives Applied To Velocity

Parameter Interval and Initial Point

Choosing the correct parameter interval ensures the particle starts at the given point and completes the desired path. For starting at (a, 0), setting t = 0 at this point and letting t vary from 0 to 2π traces the full circle once.
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Percorso guidato
05:03
Initial Value Problems
Pratica correlata
Domanda del libro di testo

Polar Coordinates


Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.


a. (2, π/2)

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Graphing Conic Sections


Sketch the parabolas in Exercises 55–58. Include the focus and directrix in each sketch.


y² = −(8/3)x

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Cartesian to Polar Equations


Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.


x² + y² + 5y = 0

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Domanda del libro di testo

Cartesian to Polar Coordinates


Find the polar coordinates, 0 ≤ θ < 2π and r ≥ 0, of the following points given in Cartesian coordinates.


b. (-3,0)

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Domanda del libro di testo

Shifting Conic Sections


You may wish to review Section 1.2 before solving Exercises 39-56.


The hyperbola (y²/4) − (x²/5) = 1 is shifted 2 units down to generate the hyperbola (y + 2)²/4 − x²/5 = 1.

a. Find the center, foci, vertices, and asymptotes of the new hyperbola.

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Cycloid


a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).

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