MATH 2431 – Summer 2012

Project 1 – Algebra – Due: 19 June 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.

There are several algebra concepts that are extremely important for successfully completing calculus I. In this project, you will learn some basic mathematica functions, you will use these mathematica functions to answer some algebra questions, and you will make some conclusions from your work on this project. 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 Name *)

(* Partner's Name *)

(* Math2431-100 *)

(* Summer 2012 *)

(* Project 1 *)

There are several methods to evaluate functions. The method here is probably the most useful (after every line, use SHIFT-ENTER to execute). Example:
Evaluate f(x) = 3x^{2} – 2x + 1
at x = 2:

`f[x_] := 3x^2 – 2x + 1`

f[2]

What is useful about this approach, we can quickly evaluate at several values for x. Evaluate f(x) = 3x^{2} – 2x + 1
at x = –2, 0, 1/2, 5 (after every line, use SHIFT-ENTER to execute):

`x = {-2,0,1/2,5}`

f[x]

There are several methods to graph functions. The method here is probably the most useful (after every line, use SHIFT-ENTER to execute). Example:
Graph g(x) = 2x^{3} + 4:

`g[x_] := 2x^3 + 4`

Plot[g[x],{x,-10,10}]

Look at the graph. It is difficult to see the y-intercept of the function on the graph. We can add an argument to the `Plot`

function to change the range of y-values on the graph:

`Plot[g[x],{x,-10,10},PlotRange->{-8,8}]`

What is useful about this approach, we can quickly graph several functions simultaneously. Example: Plot f(x) and
g(x) on the same set of axes (NOTE the use of { } around the two functions):

`Plot[{f[x],g[x]},{x,-10,10}, PlotRange -> {-8, 8}]`

Example: Factor 3x^{2} – 27:

`Factor[3x^2-27]`

Example: Solve x^{2} – x = 2:

`Solve[x^2-x==-2,x]`

However, `Solve`

gives real AND complex solutions. We only need real roots in calculus I, so use `Reduce`

(the output `false`

means there are no real solutions):

`Reduce[x^2-x==-2,x]`

Lastly, to find decimal approximations to roots, use NSolve. Example:

`NSolve[x^2-x==5,x]`

Once you are finished solving, use `Clear[x]`

to remove any stored values.

Complete the following using mathematica:

- (1 point) Solve x
^{2}– 3x = –2 using the following methods:- the
`Solve`

function and, if needed, the`NSolve`

function - store x
^{2}– 3x into f(x) and store –2 into g(x) and`Plot`

both functions on the same set of axes. - rearranging the equation to x
^{2}– 3x + 2 = 0 and using- the
`Plot`

function - the
`Factor`

function

- the

`Evaluate`

function - the
- (1 point) Repeat the above steps for x
^{3}= 3x – 1. - (1 point) Repeat the above steps for 2x
^{3}+ x^{2}= –2. - (1 point) Repeat the above steps for x
^{2}+ x + 1 = 0. - (1 point) Write a detailed summary of your comparison of the four methods to find a solution.

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 14 June (1 point deducted if not completed).
- Each question must be answered in order and clearly labeled.
- The Mathematica code [for Questions #1 – 4] is to be submitted by email in iCollege on or before 10:00 a.m. on the due date.
- The detailed summary [for Question #5] must be typed and submitted in class on or before 10:00 a.m. on the due date.
- 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.