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Chapter 9: Q 6 (page 721)

Use the results of Exercise $3$ 3 to analyze the direction of
motion for the parametric curves given by the equations in
Exercises $5 - 8$
$x = \sin t, y = \cos t, t 鈭� 鈩�$

Short Answer

Expert verified

The required answer is the direction of motion is clockwise.

Step by step solution

01

Given information

The parametric equations :

$x = \sin t, y = \cos t, t 鈭� 鈩�$

02

Find the parametric equations that move in different directions.

The function $x = \sin t$

Differentiate in terms of $t$

$x^{1} (t) = \frac{d}{d t} (\sin t) x^{1} (t) = \cos t x^{1} (t) > 0$

The function $y = \cos t$

Differentiate in terms of $t$

$y^{'} (t) = \frac{d}{d t} (\cos t) y^{1} (t) = - \sin t < 0$

That $x^{1} (t) > 0 a n d y (t) < 0$

03

Find the direction of motion for the parametric curves given by the equations

When $t = 0$

$(x, y) = (\sin t, \cos t) (x, y) = (\sin 0, \cos 0) (x, y) = (0, 0)$

The value of $x$ increases as $t$ increases while the value of $y$ decreases

Therefore the direction of motion is clockwise.

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Most popular questions from this chapter

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.

$胃 = 蟺$

Use polar coordinates to graph the conics in Exercises 44鈥�51.

$r = \frac{1}{1 - \cos 胃}$

Use polar coordinates to graph the conics in Exercises 44鈥�51.

$r = \frac{2}{2 + \sin 胃}$

Use Cartesian coordinates to express the equations for the ellipses determined by the conditions specified in Exercises 32鈥�37.

$foci (0, 卤伪), minor axis 2 伪$

In exercise 26-30 Find a definite integral that represents the length of the specified polar curve and then find the exact value of integral

$T h e s p i r a l r = e^{伪胃} f o r 0 鈮� 胃鈮� 2 蟺$

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