﻿ GPC - John Weber - Math2431 Computer Project 3 - 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.

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

(* 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)