MATH 2431 – Summer 2012

Project 3 – l'Hôpital's Rule – Due: 5 July 2012

You must complete the project in a group of two students. Normally, all members of the group will receive the same grade; however, the instructor reserves the right to conduct individual interviews over the content of the project and to assign different grades to different members of the group.

Introduction

In this project, you will use explore how the graph of rational function, f(x)/g(x), compares with the graph of the f ′(x)/g ′(x) and f ′′(x)/g ′′(x), etc. You will need to describe the relationship of the graphs. You will be graded on the quality and clarity of your written presentation as well as the mathematical accuracy of your paper.

Labeling Your Work

To label your work, use comments. For example, use the following to start your project (after every line, use SHIFT-ENTER to execute):

(* Your Name *)
(* Partner's Name *)
(* Math2431-100 *)
(* Summer 2012 *)
(* Project 3 *)

The Rational Function

Consider the rational function f(x) = (2x³ + 5x²)/(xsin(x)). Define the function as follows:

numerator: h[x_]:=2x^3 + 5x^2
denominator: g[x_]:=x Sin[x]
rational function: f[x_]:=h[x]/g[x]
etc.

Questions (clearly label the questions and your work)

1. Answer the following questions about f(x) = (2x³ + 5x²)/(xsin(x)) (2 points):
   • Evaluate h[0] and g[0] and plot f[x] near x = 0. What do you notice (i.e., what is the limit of f(x) as x approaches 0)?
   • Evaluate h'[0] and g'[0] and plot j[x]:=h'[x]/g'[x] near x = 0. What do you notice?
   • Evaluate h''[0] and g''[0] and plot k[x]:=h''[x]/g''[x] near x = 0. What do you notice?
   • Evaluate h'''[0] and g'''[0] and plot l[x]:=h'''[x]/g'''[x] near x = 0. What do you notice? Why is this different from the above answers?
2. Repeat the above questions for f(x) = (1 – cos(x))/x² (2 points)
3. Repeat the above questions for f(x) = (x⁴)/(5x – sin(5x)) (1 point)

Please follow these guidelines when preparing your report:

• Find a partner with whom you will complete the project. One person in the project needs to send the instructor an email in iCollege and cc: your partner. Type your partner's name and the project number on the Subject line of the email. This email is due on or before 10:00 a.m. on 2 July (1 point deducted if not completed).
• Each function must be graphed in order and clearly labeled.
• The Mathematica code is to be submitted by email in iCollege on or before 10:00 a.m. on the due date.
• The answers to every question, e.g., "What do you notice ...?" must be typed (you may handwrite any unusual mathematical symbols, e.g., ∞) and submitted in class on or before 10:00 a.m. on the due date.
• Use mathematically correct notation.
• Do not use any cover page or report cover.
• Turn in one paper per group. The instructor will keep all papers. Make a copy for your files before turning in your paper.
• Clearly explain your reasoning. Use complete, grammatically-correct sentences or complete mathematical sentences.