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                    Mathematica-Compatible Notebook

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Cell[CellGroupData[{
Cell["\<\
Although Mathematica is a powerful programming language that can model system \
dynamics once you know the syntax and have a good knowledge of Mathematica \
functions, on a more basic level the software is an excellent graphing \
calculator.  In other words, Mathematica is only as complicated as the \
calculations you want to evaluate or the programs you want to write.  Let's \
get started!!
\
\>", "Subsubtitle"],

Cell[CellGroupData[{

Cell["Numbers and General Stuff To Know", "Section"],

Cell[CellGroupData[{

Cell["\<\
To evaluate a line of code or a command, simply click within that cell and \
press SHIFT+ENTER..This should create a line of output in most cases.  Try \
evaluating this line:\
\>", "Subsection"],

Cell[BoxData[
    \(3 + 5 \((8)\)\)], "Input"]
}, Open  ]],

Cell["\<\
If for some reason you would like to cut/copy and then paste the previous \
input,3+5(8), to another part of your document simply cut and paste like you \
would in WORD.  However, you must wait for the cursor to be horizontal...then \
click and paste (this will create a new cell in which you can paste and/or \
type input).  Try to copy and paste the line of code in the previous cell \
into a new cell below.\
\>", "Subsection"],

Cell[CellGroupData[{

Cell["\<\
This document is called a notebook because there is text and inputto be \
calculated.  To vary the Format of the text, select the blue bracket around \
the cell of text you would like to change by clicking on the blue bracket.  \
Then select Format from the Menu, and then under Style there are various \
options.  Try changing the format of the following text:\
\>", "Subsection"],

Cell[BoxData[
    \(Change\ the\ format\ of\ the\ text\ in\ this\ \(\(cell\)\(.\)\)\)], \
"Input"]
}, Open  ]],

Cell["\<\
There are probably going to points in working with Mathematica that you will \
want to save your work.  Just like Word, select Save (or Save As)from the \
File menu and give the file a descriptive name.\
\>", "Subsection"],

Cell["\<\
To look up a function and its properties, type a ? before the name of the \
function.  Use * as a wildcard.  Try \"?Table\", \"?*Table\", and \"?Table*\" \
and compare output.\
\>", "Subsection"],

Cell["\<\
Another note before we move on...when you open a saved Notebook (or file), if \
you would like to use previous calculations in that file to continue your \
work, you must re-evaluate each cell.  The file sort of \"forgets\" what it \
previously computed even though the output lines are still there.  Also, each \
time you change an aspect of the input, you must re-evaluate in order to get \
the new output.  The output doesn't immediately change as it would in \
Excel.\
\>", "Subsection"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Calculations", "Section"],

Cell["\<\
Now let's work with some calculations.  Choose the Basic Input pallette from \
Pallettes under the File menu.  These tools will help us type in some of the \
input.  Notice that there are other pallettes which are more specialized and \
may be of more use to you as you gain proficiency with Mathematica and thus \
choose to evaluate more complicated calculations.\
\>", "Subsection"],

Cell["\<\
Find the sum of 1/4 and 1/6.  You can write fractions as I did, or you can \
use the pallete.  Mathematica gives the most accurate answer it can (in this \
case, not a decimal).  If you wanted a decimal answer, add \"//N\" to the end \
of your expression. \
\>", "Subsection"],

Cell[CellGroupData[{

Cell["\<\
\"E\" is little e (2.71...) and Log[3] takes the NATURAL log of 3.  Notice \
that most Mathematica functions begin in a capital letter (later we will want \
to name our variables in lowercase so that the names don't conflict with \
pre-existing function names).  Evaluate:\
\>", "Subsection"],

Cell[BoxData[
    \(E^1\)], "Input"],

Cell[BoxData[
    \(Log[4]\)], "Input"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Graphing Functions", "Section"],

Cell[CellGroupData[{

Cell["\<\
Graphing functions is a good application for Mathematica.  In the following \
steps we will graph a function, x^2+1.\
\>", "Subsection"],

Cell["\<\
STEP 1:  Clear the function (let's define the function using f[x])\
\>", "Text"],

Cell["\<\
STEP 2:  Define the function.  Don't forget the underscore or the colon!\
\>", "Text"],

Cell["\<\
STEP 3: Evaluate the function at 7.  Be sure to use square brackets.  Symbols \
can also be evaluated...try evaluating \"3z\" as well.\
\>", "Text"],

Cell["\<\
STEP 4: For practice, repeat these steps for the function p(t)=100e^0.1t  \
(start by clearing p, not f).\
\>", "Text"],

Cell["\<\
STEP 5: Use the plot function.  The following is how you would plot f[x]. \
\>", "Text"],

Cell[BoxData[
    \(Plot[f[x], {x, \(-2\), 2}, PlotRange \[Rule] {0, 4}, 
      AspectRatio \[Rule] 1, AxesLabel \[Rule] {"\<x\>", "\<y\>"}]\)], "Input"],

Cell["\<\
Play around with the numbers and other variables and you will discover the \
function of each of the commands within Plot.For example,PlotRange sets the \
range (or y coordinates),the numbers in the first line set the domain over \
which the function is graphed.The AspectRatio changes the proportions of the \
axes,and AxesLabel labels the axes.Now try to plot p[t].You can retype the \
function after Plot or use the symbols p[t].\
\>", "Text"],

Cell["\<\
You can plot both functions on the same graph, and give each function its own \
color.  Here's how you would do that:\
\>", "Text"],

Cell[BoxData[
    \(Plot[{f[x], p[x]}, {x, \(-3\), 
        3}, \[IndentingNewLine]PlotStyle \[Rule] {RGBColor[1, 0, 0], 
          RGBColor[0, 0, 1]}]\)], "Input"],

Cell["\<\
The commands are pretty straight-forward, the syntax can be tedious however.  \
RGBColor indicates the amount of red, green, and blue.  The values for these \
colors are floating point numbers between 0 and 1.  f[x] should be red and \
p[t] should be blue.  Now define h[x] and add another function to the graph \
making it green.\
\>", "Text"],

Cell["\<\
Colors are not the only way you can differentiate among graphed functions.  \
Copy the previous code and change the function RGBColor to Thickness.  Inside \
the square brackets put a decimal number (this number is a fraction of the \
graph's size)..0.05 is very thick.  Evaluate again and observe the output.\
\>", "Text"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Fitting Data", "Section"],

Cell[CellGroupData[{

Cell["\<\
Now let's pretend that we have a set of data to which you would like to fit a \
line (or other type of function).  We will want to define a list which \
contains the data, plot that data,and then fit the line.\
\>", "Subsection"],

Cell[CellGroupData[{

Cell["STEP 1: Define a list.", "Subsubsection"],

Cell[BoxData[
    \(\(t = {{\(-3\), 2}, {2, 2}, {\(-1\), \(-1\)}, {4, 3}, {0, 
            0}};\)\)], "Input"],

Cell["\<\
The semicolon suspends the output from displaying.  Remove the semicolon and \
evaluate, then compare the results.\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
STEP 2:  Plot your data. mydataplotted is the assigned name to the scatter \
plot.  The following lines of code will give you a scatter plot.  Play with \
the numbers and observe the changes.\
\>", "Subsubsection"],

Cell[BoxData[
    \(mydataplotted = 
      ListPlot[t, \[IndentingNewLine]AxesLabel \[Rule] {"\<x\>", "\<y\>"}, \
\[IndentingNewLine]AspectRatio \[Rule] 
          1, \[IndentingNewLine]PlotRange \[Rule] {\(-5\), 5}]\)], "Input"],

Cell["\<\
Now add this line of code to make the points magenta and bigger:
      PlotStyle\[Rule]{PointSize[0.03],RGBColor[1,0,1]},\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
STEP 3:  Fit the data
Using Fit, we can find the least-squares fit to a list of data,t, as a linear \
combination of x and 1.  Thus the fit is a linear expression of the form \
ax+b.  The following code accomplishes this fit:\
\>", "Subsubsection"],

Cell[BoxData[
    \(fitline = Fit[t, {x, 1}, x]\)], "Input"],

Cell["To plot the equation of this line of best fit:", "Text"],

Cell[BoxData[
    \(plottedfitline = Plot[fitline, {x, \(-3\), 4}]\)], "Input"],

Cell["\<\
Using the function Show, we can display the best fit line and the scatter \
plot of the data in the list on the same coordinate system.\
\>", "Text"],

Cell[BoxData[
    \(Show[mydataplotted, plottedfitline]\)], "Input"],

Cell["\<\
To take this a step further:  Fit t to a cubic ax^3+bx^2+cx+d by using the \
Fit command and four terms--x^3,x^2,x,and 1 in braces,{}.\
\>", "Text"],

Cell[BoxData[
    \(fitline2 = Fit[t, {x^3, x^2, x, 1}, x]\)], "Input"],

Cell["This yields the equation of the line of best fit.", "Text"],

Cell[BoxData[
    \(plottedfitline2 = Plot[{fitline2}, {x, \(-10\), 10}]\)], "Input"],

Cell["This plots the best fit equation and names that plot.", "Text"],

Cell[BoxData[
    \(Show[mydataplotted, plottedfitline2]\)], "Input"],

Cell["\<\
This shows the scatter plot of data as well as the line of best fit.  \
\>", "Text"]
}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Check this out:", "Section"],

Cell[BoxData[
    \(3*5*8\)], "Input"],

Cell[BoxData[
    \(3\ 5\ 8\)], "Input"],

Cell["\<\
Turns out you don't need to put the multiplaction signs in, just put spaces \
between the numbers and Mathematica will assume that you want to multiply.\
\>", "Text"],

Cell[BoxData[
    \(Round[4.5]\)], "Input"],

Cell[BoxData[
    \(Round[4.501]\)], "Input"],

Cell["\<\
This function rounds to the closest integer to the number in brackets.\
\>", "Text"],

Cell[BoxData[
    \(FactorInteger[567]\)], "Input"],

Cell["\<\
This function lists the prime factors of the number in brackets.\
\>", "Text"],

Cell[BoxData[
    \(Mod[11, 7]\)], "Input"],

Cell["\<\
This function takes the first number modulo the second number (remainder on \
division of the first number by the second)\
\>", "Text"],

Cell[BoxData[
    \(Abs[\(-6\)]\)], "Input"],

Cell["\<\
This function takes the absolute value of the number in brackets.\
\>", "Text"],

Cell[BoxData[
    \(\(25!\)\)], "Input"],

Cell["\<\
Wow..Mathematica can do all sorts of stuff...even much more...\
\>", "Text"],

Cell[BoxData[
    \(Log[10, 7]\)], "Input"],

Cell["\<\
This line takes the Log BASE10 of 7.  The base is always the first number.\
\>", "Text"],

Cell[BoxData[
    \(Sqrt[4]\)], "Input"],

Cell["\<\
This function takes the square root of the number in brackets.\
\>", "Text"],

Cell[BoxData[
    \(Sin[3  \[Pi] + 5] // N\)], "Input"],

Cell["Wow..even geometric calculations are easily done!", "Text"],

Cell[BoxData[
    \(ArcTan[\[Pi]] // N\)], "Input"],

Cell["For all of you \"arc\" enemies of Bush...", "Text"],

Cell[BoxData[
    \(%\)], "Input"],

Cell["\<\
Typing the percent sign as the input generates the previous output, or the \
last result generated.\
\>", "Text"],

Cell[BoxData[
    \(%%%\)], "Input"],

Cell["That gives me the third previous result.", "Text"],

Cell["\<\
In case you startrunning a long calculation and you want to stop the program, \
use alt-comma or control-c (sometimes command-comma depending on your \
machine)\
\>", "Text"],

Cell[BoxData[
    \(xy + 2  x^2  y + y^2  x^2 - 2  y^2  x^2\)], "Input"],

Cell["\<\
Mathematica can will simplify the latter expression (adding like terms), and \
it can expand the following expression by multiplying out products and \
powers, writing the result as a sum of terms.\
\>", "Text"],

Cell[BoxData[
    \(Expand[\((x + 2  y + 1)\) \((x - 2)\)^2]\)], "Input"],

Cell["\<\
Mathematica can also factor algebraic expressions by writing the expression \
as a product of minimal factors:\
\>", "Text"],

Cell[BoxData[
    \(Factor[x^2 - 9]\)], "Input"],

Cell["Now let's use the \"%\" in a clever way:", "Text"],

Cell[BoxData[
    \(Expand[%]\)], "Input"],

Cell["\<\
Simplify is a function which finds the simplest form of the expression by \
applying various standard algebraic transformations...different than \
Factor.\
\>", "Text"],

Cell[BoxData[
    \(Simplify[x^2 + 2  x + 1]\)], "Input"],

Cell["\<\
You can also take derivatives and integrals (definite or indefinite).\
\>", "Text"],

Cell[BoxData[
    \(D[x^2 - x + 1, x]\)], "Input"],

Cell[BoxData[
    \(Integrate[x^3, x]\)], "Input"],

Cell[BoxData[
    \(Integrate[x^3, {x, 0, 5}]\)], "Input"],

Cell["\<\
Sum x^2 with respect to x with the min being 1 and the max being 5.\
\>", "Text"],

Cell[BoxData[
    \(Sum[x^2, {x, 1, 5}]\)], "Input"],

Cell["We can take it to the limit...one more time.", "Text"],

Cell[BoxData[
    \(Limit[x^3 - x + 11, x \[Rule] 5]\)], "Input"],

Cell[BoxData[
    \(Limit[x^3 - x + 11, x \[Rule] 0]\)], "Input"],

Cell[BoxData[
    \(Limit[x^3 - x + 11, x \[Rule] \(-Infinity\)]\)], "Input"],

Cell["\<\
You can find an approximate numerical solution using FindRoot, and giving a \
starting value for x.\
\>", "Text"],

Cell[BoxData[
    \(FindRoot[Cos[x] \[Equal] x, {x, 1}]\)], "Input"],

Cell["If you're looking for a solution to simultaneous equations:", "Text"],

Cell[BoxData[
    \(Solve[{x^2 + y^2 \[Equal] 1, x + 3  y \[Equal] 0}, {x, y}]\)], "Input"],

Cell["Here's a look at how you could do a 3D plot:", "Text"],

Cell[BoxData[
    \(Plot3D[Sin[x\ y], {x, 0, 3}, {y, 0, 3}]\)], "Input"],

Cell[BoxData[
    \(ParametricPlot3D[{Sin[t], Cos[t], t/3}, {t, 0, 15}]\)], "Input"],

Cell[BoxData[
    \(ParametricPlot3D[{Cos[t] \((3 + Cos[u])\), Sin[t] \((3 + Cos[u])\), 
        Sin[u]}, {t, 0, 2  Pi}, {u, 0, 2  Pi}]\)], "Input"],

Cell["", "Text"]
}, Open  ]]
}, Open  ]]
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(*******************************************************************
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