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.

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.

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 *)

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)(x–a).
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}]`

1a. Find the equation of the tangent line to
f(x) = x^{5} – 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) = x^{2} – 2 starting with a = 2.^{*} (2 points)

3. Repeat all the above steps with f(x) = xe^{–x} starting with a = 2.^{*} (1 point)

4. Repeat all the above steps with f(x) = x^{3} – 2x + 2 starting with a = 0.^{*} (1 point)
^{*} Note: Choose an appropriate window for the graphs.

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 3 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 part d for each step 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.