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 5–8
$x = t^{3} - t, y = t^{3} + t^{3} t \in ℝ$

Short Answer

Expert verified

The curve moves up and to the right.

Step by step solution

Step: 1 Given information

The equations of parameters $x = t^{3} - t, y = t^{3} + t^{3} t \in ℝ$

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

Short Answer

Step by step solution

Step: 1 Given information

Calculation

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Use the results of Exercise 3 to analyze the direction of motion for the parametric curves given by the equations in Exercises 5–8x=t3-t,y=t3+t3t∈ℝ

Short Answer

Step by step solution

Step: 1 Given information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Pure Maths

Discrete Mathematics

Calculus

Geometry

Study anywhere. Anytime. Across all devices.

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