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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Describe the motion of a particle with position (x, y) as t varies in the given interval.

22. x = sin t, y = cos2 t, −2πt ≤ 2π

To determine

The motion of a particle with position (x,y) as t varies.

Explanation

Given data:

The parametric equation for x is as below.

x=sint (1)

The parametric equation for y is as below.

y=cos2t

y=1sin2t (2)

Calculation:

Substituting sin2t=x in Equation (2) ,

y=1x2

The value of t is increased from 2π to 2π with a step value of 1 and substituted in the parametric equations x=sint and y=cos2t to obtain the value of x and y respectively

Determine the starting point (x1,y1) of the particle.

Substitute 2π for t in equation (1),

x1=sint=sin(2π)x1=0

Substitute π for t in equation (2),

y1=cos2t=cos2(π)y1=1

When t proceeds from 2π to π . The particle begin at the point (0,1)

Determine the midpoint (x2,y2) of the particle.

Substitute π2 for t in the equation (1),

x2=sint=sin(π2)x2=1

Substitute π2 for t in the equation (2),

y2=cos2t=cos2(π2)y2=0

When t proceeds from 2π to π

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