﻿ GPC - John Weber - Math2431 Computer Project 1 - 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.

## Introduction

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):

(* Partner's Name *)
(* Math2431-100 *)
(* Summer 2012 *)
(* Project 1 *)

### Evaluating Functions

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) = 3x2 – 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) = 3x2 – 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]```

### Graphing Functions

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) = 2x3 + 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}]`

### Factoring Expressions

Example: Factor 3x2 – 27:

`Factor[3x^2-27]`

### Solving Equations

Example: Solve x2 – 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.

### Questions (clearly label the questions and your work)

Complete the following using mathematica:

1. (1 point) Solve x2 – 3x = –2 using the following methods:
• the `Solve` function and, if needed, the `NSolve` function
• store x2 – 3x into f(x) and store –2 into g(x) and `Plot` both functions on the same set of axes.
• rearranging the equation to x2 – 3x + 2 = 0 and using
• the `Plot` function
• the `Factor` function
Compare the four methods. Check your answers using the `Evaluate` function
2. (1 point) Repeat the above steps for x3 = 3x – 1.
3. (1 point) Repeat the above steps for 2x3 + x2 = –2.
4. (1 point) Repeat the above steps for x2 + x + 1 = 0.
5. (1 point) Write a detailed summary of your comparison of the four methods to find a solution.