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Chapter 9: Q.8 (page 721)

Use the results of Exercise 3 to analyze the direction of motion for the parametric curves given by the equations in Exercises 5鈥�8
$x = t^{3} - t, y = t^{3} + t^{3} t 鈭� 鈩�$

Short Answer

Expert verified

The curve moves up and to the right.

Step by step solution

01

Step: 1 Given information

The equations of parameters $x = t^{3} - t, y = t^{3} + t^{3} t 鈭� 鈩�$

02

Calculation

Consider the parametric equations $x = t^{3} - t, y = t^{3} + t, t > 0$ .

The objective is to analyze the direction of motion for the parametric equations.

Take the function $x = t^{3} - t$ .

Differentiate with respect to t.

Then, $x^{1} (t) = \frac{d}{d t} (t^{3} - t)$

$x^{1} (t) = \frac{d}{d t} t^{3} - \frac{d}{d t} t x^{1} (t) = 3 t^{2} - 1$

Now take the function $y = t^{3} + t$ .

Differentiate with respect to t.

$y^{'} (t) = \frac{d}{d t} (t^{3} + t)$

$y^{'} (t) = 3 t^{2} + 1$

Thus, $x^{1} (t) > 0$ and $y^{1} (t) > 0$ .

Further the derivatives of the functions $x (t), y (t)$ are increasing for the value of t.

As a result, the curve moves up and to the right in its motion.

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Most popular questions from this chapter

Use Cartesian coordinates to express the equations for the parabolas determined by the conditions specified in Exercises 22鈥�31.

$directrix x = x_{0}, focus (x_{1}, y_{1}), where x_{0} 鈮� x_{1}$ .

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

$r = 胃$

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

$r = 5 \sin 胃$

in exercise 31-36 find a definite integral that represents the length of the specified polar curve, and then use graphing calculator or computer algebra system to approximate the value of integral

The limacon $r = 2 + \cos 胃$

Use Cartesian coordinates to express the equations for the hyperbolas determined by the conditions specified in Exercises 38鈥�43.

$foci (卤 2, 0), directrices x = 卤 1$

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