Lesson 2.1: Parametric Curves

In this lesson, you will learn about parametric curves by their basic definitions, parameterization curves, derivative for parametric curves, and analyzing parametric equations.

View all of the instructional videos, read the provided readings, and complete the practice exercises/activities. These will help you master the objectives for this lesson. Pay attention to additional comments provided.

Lesson Objectives

Upon completion of lesson 2.1, you will be able to:

• Analyze and plot parametric equations

• Parameterize curves

• Manipulate parametric equations

• Find derivative for parametric curves

• Model and solve applications

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this lesson.

1. Introduction to Parametric Equations [8:52]

   1. Source: bullcleo1 from YouTube

1. Parametric Curves [8:45]

   1. Source: patrickJMT from YouTube

• 1. Note: Example 1: The presenter should have picked a few more points to help to see the path. 3 points is not enough unless you know on which curve the path is supposed to lie.

1. Parametric Equations: Converting to Cartesian [part 1] [7:02]

   1. Source: ExamSolutions from YouTube

1. Parametric Equations: Converting to Cartesian [part 2] [6:23]

   1. Source: ExamSolutions from YouTube

1. The Derivative of Parametric Equations [10:03]

2. Source: bullcleo1 from YouTube

1. The Second Derivative of Parametric Equations – Part 1 of 2 [9:59]

   1. Source: bullcleo1 from YouTube

1. The Second Derivative of Parametric Equations – Part 2 of 2 [5:01]

   1. Source: bullcleo1 from YouTube

1. Derivatives of Parametric Functions - Clip 1 [5:50]

   1. Source: patrickJMT from YouTube

1. Derivatives of Parametric Functions - Clip 3 [3:11]

   1. Source: patrickJMT from YouTube

1. Parameterize Any Circle [9:03]

   1. Source: Linda Fahlberg-Stojanovska from YouTube

• 1. Note: It shows you how to parameterize any circle if the orientation is NOT specified and if the path is exactly once around the circle. You may need to modify the parameterization in order to fit specific orientation and adjust the values of the parameter to control the length of the path.

1. Cycloid - Part 1 [4:58]

   1. Source: DemystifyingMath from YouTube

• 1. Note: The derivation of the parametric equations of a Cycloid is incomplete. The presenter shows you the discussion of one case, where θ is obtuse. However, you should be able to verify it works for all θ based on the knowledge that you learned from your trigonometry class.

1. The Cycloid [3:43]

   1. Source: Jill Britton from YouTube

• 1. This is a very good silent animation. Do check out the link below:

  2. Cycloid

Lesson Readings

The following required readings cover the content for this lesson. As you go through each reading, pay close attention to the content that will help you learn the objectives for this lesson.

1. Parametric Equations and Curves

2. Paul's Online Math Notes

3. Parametric Equations

   1. HMC Mathematics Online Tutorial

4. An Introduction to Parametric Equations

5. by Jackie Ruff (The University of Georgia)

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

1. Drill- Tangent Lines and Parametric Curves (problems with solutions)

   1. Visual Calculus

2. Parametric Equations Quiz (4 problems with answers)

3. The University of Sydney, School of Mathematics and Statistics

4. The text may not be displayed properly on the browser, but you still can see the problems.

Additional Resources

Below are additional resources that help reinforce the content for this lesson.

1. Graphing Parametric Equations

   1. In order to see the orientation of the parametric curve, you should 1) set t_min and t_max to have the same value; 2) enter the parametric equations; 3) increase the t_max by clicking the + button. Use this applet to practice parameterizing simple curve (such as circles, ellipses, line segments) with specified initial position, orientation, and terminal position.

2. 9.2 Calculus with Parametric Curves

• 1. Parametric Equations (with 3 practice problems and solutions)

  2. 17Calculus

3. Parametric Equations

• 1. WyzAnt|Turoring

4. Parametric Equations

5. Parametric Equations: Differentiation (video) [9:46]

• 1. Source: ExamSolutions

1. Construction of Cycloid (video) [0:46]

   1. Source: santhoshdarling450

1. Parametric Equations (video)

   1. Source: Khan Academy

   2. Clip 1

• 1. Clip 2

1. Parametric Curves – Finding Second Derivatives (video) [4:37]

   1. Source: patrickJMT

Homework Assignment

Do Homework 2.1 on MyMathLab.

Show Your Work Assignment

Show Your Work assignment will be given via Assignment Tool on Laulima. Please check your email for Laulima announcement.

Lesson 2.2: Lines and Curves in Space -->