Chapter 9: Q.7 (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.
$x = e^{t}, y = \ln t, t > 0$

Short Answer

Expert verified

The curve rises and goes to the right.

Step by step solution

Step: 1 Given information

Consider the parametric equations $x = e^{t}, y = \ln t, t > 0$

Calculation

Consider the parametric equations $x = e^{'}, y = \ln t, t > 0$

The goal is to examine the parametric equations' direction of motion.

Take the function $x = e^{'} .$

Differentiate with respect to $t .$

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

$x^{'} (t) = e^{'} > 0$

Further calculation

Take the function now. $y = \ln t$ .

different in terms of $t$ .

$y^{3} (t) = \frac{d}{d t} (\ln t) y^{3} (t) = \frac{1}{t} > 0$

Thus, $x^{'} (t) > 0$ and $y^{'} (t) > 0 .$

The value of $t$ , the derivatives of the functions $x (t)$ , $y (t)$ are also growing. As a response, the curve goes up and to the right throughout its motion.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use the results of Exercise 3 to analyze the direction of motion for the parametric curves given by the equations in Exercises 5–8.
$x = e^{t}, y = \ln t, t > 0$

Short Answer

Step by step solution

Step: 1 Given information

Calculation

Further calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Logic and Functions

Calculus

Probability and Statistics

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=et,y=lnt,t>0

Short Answer

Step by step solution

Step: 1 Given information

Calculation

Further calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Logic and Functions

Calculus

Probability and Statistics

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.
$x = e^{t}, y = \ln t, t > 0$