Mt 217 Accelerated Calculus - Fall 2003

Project 1: Running Circles around Circles

September 3, 2003

DUE: September 18, 2003

The Purpose

Calculus is a study of functions. The purpose of this project is to deepen and strengthen your understanding of this important concept by investigating two families of functions defined parametrically. As you work on this project, you will begin to develop the same kind of familiarity with functions defined parametrically as you already have with polynomial and trigonometric functions.

The Resources

There are a number of resources you can draw on as you work on this project:

• Several group work assignments will be given out in class on September 3 and 4 which will help to get you started thinking parametrically.
• Reading assignment:

  Section 1.7: Parametric Curves, pages 75 - 81 of your textbook

• Supplementary reading assignments: (not required, but potentially very helpful)
  • Getting Started with Maple, Chapter 9: "Plotting Parametric Curves and Line Segments" (beginning on page 48)
  • Module 1.7 A: Parametric Curves (on the TEC disk which comes with your textbook)
  • Module 1.7 B: Families of Cycloids (which will be quite helpful as you do this investigation)

The Exploration

Do the investigations suggested by the Laboratory Project on page 83, Running Circles around Circles.

• We will discuss #1 (page 83) in class on Monday, September 8. You might find it helpful to have begun the reading assignments before then.
• Use Maple or your graphing calculator to do the investigations suggested in #2 - #6. Similar investigations are also suggested in Module 1.7 B, on the TEC disk.

The Paper

Write a 2- to 5-page paper on hypocycloids and epicycloids, in which you present the results of your investigations on these two families of curves and the importance of using parametric representations to investigate them. Your paper should address the following issues:

• Characteristics of each family, how to recognize them when presented as algebraic expressions, and how to sketch their graphs

For example: You can describe general characteristics of the graph of y = ax^3 + bx = x (a x^2 + b). You can tell me the general shape of the graph, the location of x- and y-intercepts, possibly find their maximum and minimum points. You should be able to do the same for hypocycloids and epicycloids.

• Role of the constants a and b in the equations for these curves

How do the values of a and b affect the shapes and/or placement of graphs in these two families? For example: You can tell me that the graph of y = a x^2 + b is a parabola opening up or down depending on the value of a, and with y-intercept of b.

• These investigations should raise some questions in your own mind about what a function is, and the whole idea behind the vertical line test. We are calling these curves functions defined by parametric equations. What is a function? Where is (are) the function(s) in this investigation? What happened to the vertical line test?

The audience for your paper: Write your paper so that someone who is taking a course in precalculus or algebra and trigonometry is able to read and understand it.

The Grading Rubric

I grade papers and projects on format, writing style, mathematical content and correctness, and evidence of synthesis and integration of the concepts.

• Format: 10%

Your paper should be professionally presented. It should be word-processed (or typed). This paper should include fairly nicely rendered illustrations. (These can be done using Maple.)

It is not necessary to type the mathematical equations; you may write these into your paper neatly by hand. However, Maple does have a feature which allows you to import equations into a word-processed document. Or you can type up your paper using the text editor in Maple. (See Appendix A in Getting Started with Maple.)

Think of this paper as something that you would be proud to take with you to a job interview. This kind of paper should be kept for your professional portfolio as evidence of your ability to do a mathematical investigation, and present your work for others to read.

• Writing Style: 15%

Pay attention to grammar, spelling, and sentence structure.

• Mathematical Content: 60%

Of course, the ideas you present should be mathematically correct.

• Synthesis / Integration: 15%

The depth of your responses to questions raised by these investigations will give an indication of how well you have integrated ideas about functions, curves represented by parametric equations, and these two families of cycloids.