MATH 2431 – Summer 2012

Project 4 – Newton's Method – Due: 9 July 2011

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

Identifying patterns is important in mathematics. In this project, you will use explore how the tangent line is used to estimate the x-intercept [i.e., when y = 0] of f(x) by graphing the function and various tangent lines. You will need to describe the pattern formed by the tangent lines. 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 Names *)
(* Math2431-100 *)
(* Summer 2012 *)
(* Project 4 *)

The Tangent Line to a Function

There are three required values needed to write the equation of the tangent line to a function: 1. x = a, 2. f(a), and 3. f ′(a). The equation of the tangent line is y = f(a) + f ′(a)(xa). Lastly, you will need to graph f(x) with the various tangent lines using Plot[{f[x],tangent1,tangent2,etc.},{x,x0,x1},PlotRange->{y0,y1}]

Estimating the x-intercept of f(x)

1a. Find the equation of the tangent line to f(x) = x5 – 2x – 2 at a = 2 and plot f(x) and the tangent line using the window [0, 3] × [–10, 50] as follows:

Store f(x) using f[x_] := x^5 - 2 x - 2
Find tangent line to f(x) at x = a and store into tangent1 using tangent1 = f[2] + f'[2] (x - 2)
Graph f(x) and the tangent line using Plot[{f[x], tangent1}, {x, 0, 3}, PlotRange -> {-10, 50}]

1b. Find another equation of the tangent line to f(x) using the x-intercept of tangent1 and plot f(x) and both tangent lines using the window [0, 3] × [–10, 50] as follows:

Set y = 0 in tangent1 and solve for x using x1 = Solve[tangent1 == 0, x] // N. Round this number to 5 decimal places and record the value.
Find next tangent line to f(x) at x = x1 and store into tangent2 using tangent2 = f[x1] + f'[x1] (x - x1) [note: manually type in the value for x1]
Graph f(x) and both tangent lines using Plot[{f[x], tangent1, tangent2}, {x, 0, 3}, PlotRange -> {-10, 50}]

1c. Repeat step 1b at least two more times to find x2, tangent3, x3, tangent4, etc.

1d. What you notice about the graph of f(x) and the tangent lines. Explain. Set the last tangent line to 0 and solve for x. What is this x-value? Explain.

2. Repeat all the above steps with f(x) = x2 – 2 starting with a = 2.* (2 points)

3. Repeat all the above steps with f(x) = xex starting with a = 2.* (1 point)

4. Repeat all the above steps with f(x) = x3 – 2x + 2 starting with a = 0.* (1 point)
* Note: Choose an appropriate window for the graphs.

Questions (clearly label the questions and your work)

Please follow these guidelines when preparing your report: