(************** Content-type: application/mathematica **************
Mathematica-Compatible Notebook
This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.
To get the notebook into a Mathematica-compatible application, do
one of the following:
* Save the data starting with the line of stars above into a file
with a name ending in .nb, then open the file inside the
application;
* Copy the data starting with the line of stars above to the
clipboard, then use the Paste menu command inside the application.
Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode. Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).
NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
try to use invalid cache data.
For more information on notebooks and Mathematica-compatible
applications, contact Wolfram Research:
web: http://www.wolfram.com
email: info@wolfram.com
phone: +1-217-398-0700 (U.S.)
Notebook reader applications are available free of charge from
Wolfram Research.
*******************************************************************)
(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[ 43412, 1245]*)
(*NotebookOutlinePosition[ 44077, 1268]*)
(* CellTagsIndexPosition[ 44033, 1264]*)
(*WindowFrame->Normal*)
Notebook[{
Cell[TextData[{
StyleBox["Lab 1A: Introduction to Labs and ",
FontSize->24,
FontWeight->"Bold",
FontVariations->{"Underline"->True}],
StyleBox["Mathematica",
FontSize->24,
FontWeight->"Bold",
FontSlant->"Italic",
FontVariations->{"Underline"->True}],
"\nMath 2374 - University of Minnesota\nhttp://www.math.umn.edu/math2374\n\
Questions to: rogness@math.umn.edu"
}], "Text",
CellFrame->True,
TextAlignment->Center,
FontColor->GrayLevel[1],
Background->RGBColor[0, 0, 1]],
Cell[CellGroupData[{
Cell["Introduction to the Labs", "Section"],
Cell[TextData[{
"This semester you will spend a significant amount of time working on the \
computers. We've written a number of labs which should help illustrate many \
of the concepts we'll talk about. Sometimes we'll use the computer to draw \
pretty pictures, which the computer is extremely good at, so you can \
understand a certain idea. Other times we'll give you an interesting problem \
to work on which includes some long and technical computations, and would \
therefore be difficult to do by hand; with the computer doing the number \
crunching (and sometimes even the calculus) for you, you can concentrate on \
understanding the ideas and not worrying about evaluating an ugly integral \
which requires three integration by parts, trigonometry substitutions, and an \
extra u-substitution for good measure.\n\nMost of the time you'll be using ",
StyleBox["Mathematica",
FontSlant->"Italic"],
", the program you're using to view this notebook right now. Because this \
is an Institute of Technology course, and nearly all of our students are \
enrolled in the IT, we'll assume a basic level of computer knowledge. \
Although we use Linux, which is quite different from Windows or Macintosh \
computers, the interface in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" is very similar to most other applications you can run on any modern \
system. We won't assume you have a working knowledge of Linux, but once \
you're using ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" or a web browser, we expect that you will be comfortable working with \
pull-down menus, windows with scroll bars, etc. If you're worried about this \
you should talk to your TA and we'll try to help you improve your computer \
skills. For now all you have to do is read.\n\nUsually we'll work on a \
different lab each week, but in general you'll only have to turn something in \
every two weeks. If you look on the syllabus you'll notice that most of the \
labs are in two pieces, as in \"Lab 2A\" and \"Lab 2B.\" This means you \
should hand in the exercises from these two labs together in one report. \
These lab assignments will be due the week after you work on them. For \
example, the exercises in labs 2A and 2B will be due in lab the next week, \
when you'll start working on lab 3. Your TA will generally remind you when \
labs are due, but if you have any questions you should ask. \n\nYou'll be \
asked to do a lot of things in the labs. A lot of times there will be \
commands in the lab notebook for you to run, but you should probably have \
another notebook open while you read the lab so you can do your own work \
there. (Go to the File menu and choose \"New\" to do this.)\n\nWe will want \
you to hand in some of the problems, but others are just for you do to on \
your own. To help you distinguish, we've formatted the labs so that you only \
need to hand in problems which are inside a box with a reddish background:"
}], "Text"],
Cell[TextData[{
StyleBox["Fake Exercise 1",
FontSize->16,
FontWeight->"Bold"],
"\n\nIf this were a real exercise, this message would be followed by \
instructions about what to do..."
}], "Text",
CellFrame->True,
Background->RGBColor[1, 0.501961, 0.501961]],
Cell["There's another type of colored box that you'll see as well:", "Text"],
Cell["\<\
Boxes with a gray background generally contain important information, \
warnings about potential pitfalls, or hints on how to use certain commands.\
\>", "Text",
CellFrame->True,
Background->GrayLevel[0.849989]],
Cell[TextData[{
"If you scroll down, you'll see that there doesn't seem to be much of \
anything there. That's because the other sections in this lab are ",
StyleBox["collapsed",
FontSlant->"Italic"],
". If you look on the right side of this window, you'll see that there are \
little blue lines which bracket the text and the colored boxes. These blue \
brackets represent ",
StyleBox["cells",
FontSlant->"Italic"],
", which are the basic units of a ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" notebook. Cells can contain things such as text, commands, formulas, and \
pictures. Cells can also be grouped together in sections, which is done by \
having a big bracket which includes all of the cells. You should see a long \
blue line to the right of all these cells; this is the \"section bracket.\"\n\
\nIf you were to double click on it, this Introduction would collapse. \
(Don't do this quite yet!) All you would see is the cell with the title of \
the section, the little blue bracket for that cell, and then another blue \
bracket to the right. This second bracket would have a little arrow on the \
bottom. Any time you see this arrow on a cell it means there are cells below \
which have been collapsed and are hidden from view. To get them back, you \
just double click on the outer bracket (the one with the arrow on it). Try \
collapsing this Introduction section, and then open it back up again. If you \
can't get it back, ask your TA for help.\n\nUsually when you open a lab, all \
of the sections (including the Introduction) will be collapsed. This lets \
you see sort of a \"Table of Contents\" so you know what you'll be doing. We \
left the introduction to this lab open so that you wouldn't open the first \
lab and not know what to do.\n\nOne last note before you start working: a few \
semesters ago we spent a lot of time revising these labs, and we'd really \
appreciate feedback from you. If you think a lab really helped you \
understand a topic, let us know. If you think a lab is boring and dull, and \
needs to be changed, tell us. (And you don't have to wait until the end of \
the semester to give us these comments.) We have lots of ideas about what \
should be done in the labs, but the final measure of success is whether or \
not you learn from them, so your opinion really does matter!\n\nNow you can \
go on to the actual lab. Remember, double click on the outer bracket of a \
section or sub-section to expand it."
}], "Text",
TextJustification->0]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[{
"Introduction to ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" - Arithmetic, Functions, and Graphs"
}], "Section"],
Cell[TextData[{
"As we alluded to above, ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" is a very powerful program. If you own a graphing calculator, you may as \
well put it away. Even a TI-89 or TI-92 is out of its league here. ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" can do everything they can do, and then some. And some more. And then a \
lot more. The purpose of this lab is to get you comfortable with ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". We'll start with the easy stuff -- such as how to add two numbers -- \
and move on to more complicated things. In the next section we'll show you \
how to do single variable calculus with ",
StyleBox["Mathematica",
FontSlant->"Italic"],
", i.e. everything you learned how to do last year."
}], "Text"],
Cell[CellGroupData[{
Cell["Arithmetic and Variables", "Subsection"],
Cell[TextData[{
"As mentioned above, the basic unit of a ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" notebook is a ",
StyleBox["cell",
FontSlant->"Italic"],
". You're currently reading a text cell, which we can use to document what \
we're doing, but the real work is done in \"input\" cells. To run a command \
(or \"evaluate a cell\") you have to use the keyboard or the mouse to \
position the cursor anywhere in the input line and hit either (1) \
Shift+Enter, where enter is the normal Enter key, or (2) the Enter key on the \
numeric keypad. If you use option #2, you do ",
StyleBox["not",
FontSlant->"Italic"],
" have to press the shift key. \n\nPractice by evaluating these cells:"
}], "Text"],
Cell[BoxData[
\(2 + 2\)], "Input"],
Cell[BoxData[
\(35/7\)], "Input"],
Cell[TextData[{
StyleBox["Mathematica",
FontSlant->"Italic"],
" uses the normal operators +, -, /, and * for arithmetic operations, and ^ \
for exponents. You can also put a space between two numbers to multiply \
them:"
}], "Text"],
Cell[BoxData[{
\(3*9\), "\[IndentingNewLine]",
\(3\ 9\)}], "Input"],
Cell[TextData[{
"(Note that there's a space there, so the second input is ",
StyleBox["not",
FontSlant->"Italic"],
" \"39\", and so the output \"27\" is correct.) As you can see, you can \
put multiple commands in a single input cell by hitting Enter (without the \
shift key!) and putting a new command on the next line. ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" will return the output in the same order. If you want to suppress the \
output of a command, put a semicolon after it. (If you use a semicolon, you \
can put the next command on the same line, so the third line of input here is \
valid:)"
}], "Text"],
Cell[BoxData[{
\(\(9/3;\)\), "\[IndentingNewLine]",
\(6^2\)}], "Input"],
Cell[BoxData[
\(12*12; \ 5 + 1; \ 3/2\)], "Input"],
Cell[TextData[{
StyleBox["Mathematica",
FontSlant->"Italic"],
" does most of its work symbolically, which is why the last output was a \
fraction instead of the decimal 1.5. Special constants like \[Pi] and \
\[ExponentialE] (the symbol for ",
StyleBox["e",
FontSlant->"Italic"],
") are treated as such; ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" does not replace \[Pi] with a number such as 3.14159. You can enter \
these constants like this:"
}], "Text"],
Cell[BoxData[{
\(Pi\), "\[IndentingNewLine]",
\(E\)}], "Input"],
Cell["\<\
You can use variables and assign values to them. For reasons that will be \
clear later, you should only use lower case letters in your variable names.\
\>", "Text"],
Cell[BoxData[{
\(a = 2; \ b = 3;\ \), "\[IndentingNewLine]",
\(a\), "\[IndentingNewLine]",
\(b\), "\[IndentingNewLine]",
\(a + b\)}], "Input"],
Cell["\<\
If you want to multiply variables, you can either use * or a space in between \
them:\
\>", "Text"],
Cell[BoxData[{
\(a*b\), "\[IndentingNewLine]",
\(a\ b\)}], "Input"],
Cell["\<\
It's probably best to use *, because if you use the space you might \
accidentally forget it, and then you won't get the right answer:\
\>", "Text"],
Cell[BoxData[
\(ab\)], "Input"],
Cell[TextData[{
"Mathematica returns \"ab\" because there is no space between the letters \
in the input cell, so it assumes you're asking for the value of a new \
variable named \"ab.\" You haven't given ab a value yet, so Mathematica just \
returns the variable itself.\n\nIf you're done using variables you can erase \
them from memory using the ",
StyleBox["Clear[ ]",
FontWeight->"Bold"],
" command. This is sometimes useful before you use variables, as well; you \
can clear them just in case they were used for something else before"
}], "Text"],
Cell[BoxData[
\(Clear[a, b]\)], "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell["Functions", "Subsection"],
Cell[TextData[{
"In order to do anything really interesting, we need to use functions. \
Functions which are part of ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" are always capitalized, and always use square brackets, [ and ], around \
their arguments. For example, here's the square root function:"
}], "Text"],
Cell[BoxData[
\(Sqrt[5]\)], "Input"],
Cell[TextData[{
"Remember, ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" does things symbolically unless we tell it otherwise, so it returns ",
Cell[BoxData[
\(TraditionalForm\`\@5\)]],
" instead of 2.23607. If you want to see a decimal approximation of a \
number, use the function ",
StyleBox["N",
FontWeight->"Bold"],
":"
}], "Text"],
Cell[BoxData[
\(N[Sqrt[5]]\)], "Input"],
Cell["\<\
If you get an answer to a problem and want a numeric value for it, you don't \
have to type the answer again. You can use the symbol %, which refers back \
to the most recent output:\
\>", "Text"],
Cell[BoxData[
\(Sqrt[10]\)], "Input"],
Cell[BoxData[
\(N[%]\)], "Input"],
Cell[TextData[{
"The other way to force ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" to give you a decimal answer is to start out with a decimal number, i.e. \
\"5.0\" instead of \"5\" \[LongDash] in fact, you can simply type \"5.\" as \
shown here:"
}], "Text"],
Cell[BoxData[
\(Sqrt[5. ]\)], "Input"],
Cell[TextData[{
"You could probably guess the names of some other common functions, such as \
",
StyleBox["Sin",
FontWeight->"Bold"],
", ",
StyleBox["Cos",
FontWeight->"Bold"],
", ",
StyleBox["Tan",
FontWeight->"Bold"],
", ",
StyleBox["Log",
FontWeight->"Bold"],
", and ",
StyleBox["Exp",
FontWeight->"Bold"],
". (For people who haven't taken computer science classes, Exp[number] is \
a common notation for ",
Cell[BoxData[
\(TraditionalForm\`e\^number\)]],
".) To see if you understand how to use functions, you should try to \
evaluate sine and cosine at 0, \[Pi]/2, and \[Pi] in another notebook \
window."
}], "Text"],
Cell[TextData[{
StyleBox["Warning!",
FontSize->14,
FontWeight->"Bold"],
" You must remember that ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" functions are capitalized and use ",
StyleBox["square",
FontWeight->"Bold"],
" brackets. Also remember that you ",
StyleBox["must",
FontWeight->"Bold"],
" capitalize Pi if you want the number \[Pi]. For example, all of these \
commands are incorrect:\n\n",
StyleBox["Sin(0)\ncos[0]\nTan[pi]\nsin(pi)",
FontFamily->"Courier",
FontWeight->"Bold"],
StyleBox["\n",
FontFamily->"Courier"],
"\nThe last one is really bad; there are ",
StyleBox["three",
FontSlant->"Italic"],
" mistakes! (See if you can find them.)\n\nForgetting to capitalize \
functions like ",
StyleBox["Sin",
FontWeight->"Bold"],
" and ",
StyleBox["Cos,",
FontWeight->"Bold"],
" and using ( ) instead of [ ], are ",
StyleBox["by far",
FontSlant->"Italic"],
" the most common mistakes students make well into the semester. During \
the first few weeks of the course, it's very common for people to call us to \
their computer and say, \"This isn't working,\" and the problem is that they \
typed ",
StyleBox["sin",
FontWeight->"Bold"],
" instead of ",
StyleBox["Sin",
FontWeight->"Bold"],
", or ",
StyleBox["Sin(Pi)",
FontWeight->"Bold"],
" instead of ",
StyleBox["Sin[Pi],",
FontWeight->"Bold"],
" etc. \n\nIf you have a problem with the computer, you should always feel \
free to ask us for help. Especially during these first few weeks, however, \
you will usually save yourself (and us) some time by carefully \
double-checking your brackets and capitalization; that's very likely the \
problem. We realize it takes a while to get use to how syntax-sensitive ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" is, but never fear\[LongDash]in a few weeks you will get used to the \
syntax and everything will go much smoother."
}], "Text",
CellFrame->True,
Background->GrayLevel[0.849989]],
Cell[TextData[{
"Some ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" functions can actually grind out algebra problems for you. For example, \
suppose you're trying to find the intersection of the parabola y=",
Cell[BoxData[
\(TraditionalForm\`\((x - 1)\)\^2\)]],
"+2 with the line y = x + 5. You could set these two equations equal and \
solve for x, or you can have ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" do it for you: (Note that we have replaced = with ==. You must do this \
or ",
StyleBox["Solve",
FontWeight->"Bold"],
" won't work.)"
}], "Text"],
Cell[BoxData[
\(Solve[\((x - 1)\)^2\ + 2\ \[Equal] \ x + 5, \ x]\)], "Input"],
Cell[TextData[{
"Another useful function is ",
StyleBox["Simplify",
FontWeight->"Bold"],
", which can take ugly expressions and make them much nicer."
}], "Text"],
Cell[BoxData[
\(6 x \((x + 2)\)/Sqrt[2]\ + \((6 + \ Pi)\)/Sqrt[2] -
Sqrt[2]*\ Pi/2\)], "Input"],
Cell[BoxData[
\(Simplify[%]\)], "Input"],
Cell[BoxData[
\(Simplify[Cos[x]^2\ + \ Sin[x]^2]\)], "Input"],
Cell["\<\
Often we'll ask you to simplify your answers before you hand in an \
assignment. Even if we forget, you still should!\
\>", "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Help Browser", "Subsection"],
Cell[TextData[{
"There is one very important resource for you, called the Help Browser. \
You can find it under the Help menu above. If you want to know how to do \
something you should check there first. Sometimes the help files are a \
little hard to understand, especially if you don't have much experience with \
",
StyleBox["Mathematica",
FontSlant->"Italic"],
", so you can always ask your TA for help. However, if you haven't looked \
it up, you should be prepared for us to answer with, \"Check the help browser \
and let me know if it doesn't make sense.\"\n\nAs a test, open the help \
browser and see if you can figure out how to get ",
StyleBox["Mathematica ",
FontSlant->"Italic"],
"to find \[VerticalSeparator]x\[VerticalSeparator], the absolute value of \
x. (Suggestion: search for \"absolute value.\") Check your work by \
computing the absolute values of 3 and -3.\n\nHere's a tip: many pages in the \
help browser include examples, which can be very instructive. To see these \
example you have to click on the little triangle next to the words \"Further \
Examples.\""
}], "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Defining your Own Functions", "Subsection"],
Cell[TextData[{
"Very often we'll want to work with our own functions, such as ",
Cell[BoxData[
\(TraditionalForm\`f(x) = x\^2\)]],
". We can do this by using the following input:"
}], "Text"],
Cell[BoxData[
\(f[x_] = x^2\)], "Input"],
Cell[TextData[{
"Note the underscore after the x on the left hand side. You ",
StyleBox["must",
FontSlant->"Italic"],
" include the underscore after the x on the left hand side inside the \
bracket, but you should ",
StyleBox["never",
FontSlant->"Italic"],
" include it on the right hand side! You don't really need to know the \
reason for this, but roughly speaking, the underscore tells ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" that the thing inside the brackets is a variable that can take on any \
value."
}], "Text"],
Cell[TextData[{
"Forgetting the underscore is another very common problem during the first \
month of the class. If you're having a problem with a function that you \
defined on your own, double check that you've used the underscore correctly. \
If you've messed up you'll probably have to clear the variable name before \
redefining the function, e.g. do ",
StyleBox["Clear[x]",
FontWeight->"Bold"],
" in this case."
}], "Text",
CellFrame->True,
Background->GrayLevel[0.849989]],
Cell[TextData[{
"You can choose your own favorite name for a function when you define it, \
but you should only use lowercase letters. The reason for this, and for why \
we recommend you only use lowercase variables, is that all of the internal ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" functions are capitalized. If you only use lowercase functions, you \
don't have to worry about a conflict with something that is already defined.\n\
\nOnce we've defined a function, we can do all sorts of cool things with it. \
You can input numbers or symbols -- or even whole expressions -- into a \
function:"
}], "Text"],
Cell[BoxData[{
\(f[4]\), "\[IndentingNewLine]",
\(f[Pi]\), "\[IndentingNewLine]",
\(f[\((1 + t)\)]\), "\[IndentingNewLine]",
\(f[Sin[t*Pi]]\)}], "Input"],
Cell[TextData[{
"Functions can have more than one argument, and in fact most of our \
functions this semester will. (Hence the name of the class, \"Multivariable \
Calculus.\") Also note that when you define a function, ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" returns the definition as output unless you use a semicolon after it:"
}], "Text"],
Cell[BoxData[{
\(\(g[u_, v_] = u/v;\)\), "\[IndentingNewLine]",
\(g[2, 3]\), "\[IndentingNewLine]",
\(g[5 Pi, \((x - 9)\)^6]\)}], "Input"],
Cell[TextData[{
"\n",
StyleBox["Example",
FontSize->16,
FontWeight->"Bold"],
"\n\nIt's not imperative that you do this problem, but if you have the time \
it would probably be very helpful. Recall that if you want to solve the \
equation ",
Cell[BoxData[
\(TraditionalForm\`ax\^2 + bx\ + \ c\ = \ 0\)]],
", you can use the quadratic formula, which says \n\n",
StyleBox["x = ",
FontSize->14],
Cell[BoxData[
FormBox[
StyleBox[
FractionBox[
RowBox[{\(-b\), " ", "\[PlusMinus]", " ",
FormBox[
SqrtBox[
FormBox[\(b\^2 - 4 ac\),
"TraditionalForm"]],
"TraditionalForm"]}], \(2 a\)],
FontSize->18], TraditionalForm]]],
". \n\nDefine a function f[a_,b_,c_] which returns one root, and another \
function g[a_,b_,c_] which returns the other root. (There are two roots \
because of the \[PlusMinus] sign.) To see if you've done everything \
correctly, try to find the two roots of ",
Cell[BoxData[
\(TraditionalForm\`2 x\^2 + 8 x - 1 = 0\)]],
". (The numeric approximations of the roots, found using the ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" function ",
StyleBox["N[ ]",
FontWeight->"Bold"],
", are -4.12132 and 0.12132.)"
}], "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Vectors", "Subsection"],
Cell[TextData[{
"Depending on your textbooks, you have probably seen vectors written in \
various ways, such as (1,2), ",
Cell[BoxData[
\(TraditionalForm\`\((1, 2)\)\&\[RightVector]\)]],
", or \[LeftAngleBracket]1,2\[RightAngleBracket]. In ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" vectors are written with curly brackets. Here we define two vectors, ",
Cell[BoxData[
\(TraditionalForm\`v\&\[RightVector]\)]],
" and ",
Cell[BoxData[
\(TraditionalForm\`u\&\[RightVector]\)]],
". We add them, we multiply ",
Cell[BoxData[
\(TraditionalForm\`v\&\[RightVector]\)]],
" by a scalar number, and we compute the dot product ",
Cell[BoxData[
\(TraditionalForm\`v\&\[RightVector]\)]],
"\[CenterDot]",
Cell[BoxData[
\(TraditionalForm\`u\&\[RightVector]\)]],
". (The dot product is written as a period.) Make sure the output here \
makes sense to you. Note that we've used semicolons after the definition \
of",
Cell[BoxData[
\(TraditionalForm\`v\&\[RightVector]\)]],
" and ",
Cell[BoxData[
\(TraditionalForm\`u\&\[RightVector]\)]],
", so they are not displayed."
}], "Text"],
Cell[BoxData[{
\(v = {1, 2}; \ u = {4, 4};\), "\[IndentingNewLine]",
\(v + u\), "\[IndentingNewLine]",
\(2 v\), "\[IndentingNewLine]",
\(v . u\)}], "Input"],
Cell[TextData[{
"Three-dimensional vectors are possible, and in fact we can make a vector \
with as many dimensions as we like. Here is a three dimensional vector, and \
two nine dimensional vectors as well. To see that ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" actually treats these as vectors, you should insert a command into this \
cell to compute ",
Cell[BoxData[
\(TraditionalForm\`v\&\[RightVector]\[CenterDot]w\&\[RightVector]\)]],
"."
}], "Text"],
Cell[BoxData[{
\(u = {4, 2, \(-1\)}\), "\[IndentingNewLine]",
\(v = {1, 2, 3, 4, 5, 6, 7, 8, 9}\), "\[IndentingNewLine]",
\(w = {0, 0, 0, 0, 0, 0, 0, 0, 1}\)}], "Input"],
Cell[TextData[{
StyleBox["Review of Brackets in ",
FontWeight->"Bold"],
StyleBox["Mathematica",
FontWeight->"Bold",
FontSlant->"Italic"],
"\n\nRemember, using the wrong kind of brackets is the number one cause of \
problems for most students. To help you keep them straight, let's review:\n\n\
",
StyleBox["( and )",
FontWeight->"Bold"],
" : used to enter mathematical expressions, e.g. (x+1)^2, or 1/(x-2).\n\n",
StyleBox["[ and ]",
FontWeight->"Bold"],
" : used with functions, e.g. f[x_] = x^2, or Sin[x].\n\n",
StyleBox["{ and }",
FontWeight->"Bold"],
" : used to denote vectors, e.g. {2,-3}, or {x, Sin[x]}.\n\n(The last \
example is a ",
StyleBox["vector-valued function",
FontSlant->"Italic"],
", a function of x whose value is a vector.)"
}], "Text",
CellFrame->True,
Background->GrayLevel[0.849989]]
}, Closed]],
Cell[CellGroupData[{
Cell["Drawing Graphs", "Subsection"],
Cell[TextData[{
"There are ",
StyleBox["many",
FontSlant->"Italic"],
" commands you can use to produce pictures in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". Today we're going to learn two of them, and you will be introduced to \
others in the rest of lab 1 and in lab 2.\n\nIf we have a function y=f(x), \
the easiest way to graph it is with the ",
StyleBox["Plot",
FontWeight->"Bold"],
" command. The syntax is ",
StyleBox["Plot[",
FontWeight->"Bold"],
" function, {x, xmin, xmax}",
StyleBox["]",
FontWeight->"Bold"],
". Note that expressions such as {x, xmin, xmax} will be very common this \
semester. Basically it means you want to let x range from xmin to xmax."
}], "Text"],
Cell[BoxData[{
\(\(f[x_] = x^2;\)\), "\[IndentingNewLine]",
\(Plot[f[x], {x, \(-1\), 3}]\)}], "Input"],
Cell[TextData[{
"You don't have to name a function before you can graph it. You can simply \
enter the function into the ",
StyleBox["Plot",
FontWeight->"Bold"],
" command."
}], "Text"],
Cell[BoxData[
\(Plot[x\ \ Sin[1/x], {x, \(-0.3\), 0.3}]\)], "Input"],
Cell["\<\
You can name plots so you can refer to them later as well:
\
\>", "Text"],
Cell[BoxData[{
\(plot1 = Plot[Sin[x], {x, 0, 2 Pi}]\), "\[IndentingNewLine]",
\(plot2 = Plot[Cos[x], {x, 0, 2 Pi}]\)}], "Input"],
Cell[TextData[{
"If you've named a graph and you want to display it again later, you can \
use the ",
StyleBox["Show",
FontWeight->"Bold"],
" command. You can give the command the names of multiple plots, and it \
will show them together:"
}], "Text"],
Cell[BoxData[
\(Show[plot1, plot2]\)], "Input"],
Cell[TextData[{
StyleBox["Options",
FontWeight->"Bold"],
"\n\nOccasionally you will want to use optional arguments when drawing \
graphs. Options generally come at the end and have the form \"OptionName\
\[RightArrow]Setting.\" [You can type the \[RightArrow] as (hyphen)(greater \
than), \[Dash]\[Succeeds]]. For example, the option Axes\[RightArrow]False \
will prevent ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" from showing the x- and y- axes in a graph. This option works with ",
StyleBox["Plot",
FontWeight->"Bold"],
" and ",
StyleBox["Show",
FontWeight->"Bold"],
". Try adding it to the ",
StyleBox["Show",
FontWeight->"Bold"],
" command above and re-evaluating it. (You need to add a comma after \
\"plot2\" before you can add the option.) Did the axes disappear?\n\nIn \
order to use ",
StyleBox["Plot",
FontWeight->"Bold"],
", you need to be able to solve your equation for y. If you want to graph \
an equation such as ",
Cell[BoxData[
\(TraditionalForm\`\(\(x\^2 + y\^2 = 1\)\(,\)\)\)]],
" we need to use something different. You might recall that equations like \
these are called ",
StyleBox["implicit",
FontSlant->"Italic"],
" functions, because we can't ",
StyleBox["explicitly",
FontSlant->"Italic"],
" solve for y in terms of x. If you try to solve this equation for y, you \
get y = \[PlusMinus]",
Cell[BoxData[
\(TraditionalForm\`\@\(1 - x\^2\)\)]],
", and ",
StyleBox["Plot",
FontWeight->"Bold"],
" will complain if you give it a function with a \[PlusMinus] in it. (Try \
it and see! You can copy and paste the function into a ",
StyleBox["Plot",
FontWeight->"Bold"],
" command, so you don't have to figure out how to type the \[PlusMinus] \
symbol.)\n\nTo plot the graph of an implicit function we can use a command \
called ",
StyleBox["ImplicitPlot",
FontWeight->"Bold"],
". The problem is that this command is not usually loaded when ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" starts up. Before we can graph the circle, we need to tell ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" to load the package which contains the ",
StyleBox["ImplicitPlot",
FontWeight->"Bold"],
" command:"
}], "Text"],
Cell[BoxData[{
\(Needs["\"]\), "\[IndentingNewLine]",
\(ImplicitPlot[
x^2\ + \ y^2\ \[Equal] \ 1, \ {x, \(-1\), 1}, \ {y, \(-1\),
1}]\)}], "Input"],
Cell[TextData[{
"The first command loads the package. Note that the single quote used in \
this command is not the normal one; it is the one located on the upper left \
hand side of your keyboard, just below the Esc key. (On most computers, at \
least.) Once you have loaded the package you can use ",
StyleBox["ImplicitPlot",
FontWeight->"Bold"],
" without using this first command again."
}], "Text"],
Cell[TextData[{
"The second command plots the circle. The syntax is very similar to the ",
StyleBox["Plot",
FontWeight->"Bold"],
" command, except you have to give ranges for both x and y! Also notice \
that the equal sign in the original equation is replaced by == when you type \
the function into the ",
StyleBox["ImplicitPlot",
FontWeight->"Bold"],
" command.\n\nTo test yourself, try to plot the ellipse given by the \
following equation:"
}], "Text"],
Cell[BoxData[
\(x\^2\/3\^2 + \(\(\ \)\(y\^2\)\)\/4\^2 = \ 1\)], "DisplayFormula"],
Cell["\<\
Remember, you don't have to reload the ImplicitPlot package if you've already \
loaded it to draw the circle!\
\>", "Text"],
Cell[TextData[{
StyleBox["Warning",
FontSize->14,
FontWeight->"Bold"],
": If you try to use a command in a package ",
StyleBox["before",
FontSlant->"Italic"],
" you load the package you will really mess things up. Loading the package \
and then trying the command again will only make it worse. Repeat to \
yourself: \"I ",
StyleBox["must",
FontSlant->"Italic"],
" load a package before using any of its commands, or I will create a big \
headache for my TA.\" For this reason it's probably best for you to load any \
necessary packages right away at the top of your notebooks so that you can \
get this step out of the way."
}], "Text",
CellFrame->True,
Background->GrayLevel[0.849989]]
}, Closed]],
Cell[CellGroupData[{
Cell["Saving Notebooks", "Subsection"],
Cell[TextData[{
"Once you're finished working, you'll usually want to save your notebook so \
you don't lose your work. You can do this through the File menu with either \
Save or Save As. ",
StyleBox["Please note",
FontWeight->"Bold"],
": output, and particularly graphics output, takes up a tremendous amount \
of memory and, if you save notebooks with graphics, they will quickly get to \
be so large that you will use up your disk quota and be barred from using the \
computer. This is especially true in later labs, where we will create \
animations. If you save a notebook with an animation, it will take up \
several megabytes of disk space.\n\nSo, before you save a notebook, you \
should always go to the Kernel menu and choose \"Delete All output.\" This \
will leave all of your commands intact, but delete all of the answers and \
graphics from ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". If you load a notebook that was saved after deleting all output, you \
can run all of the commands automatically by going to the Kernel menu again \
and choosing Evaluation : Evaluate Notebook."
}], "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Single Variable Calculus with Mathematica", "Section"],
Cell[TextData[{
"Some of what we'll do this semester in Multivariable calculus is more or \
less the same as what you learned to do last year with functions of one \
variable. In the last part of this introduction we'll show you how to use ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" to do single variable calculus."
}], "Text"],
Cell[CellGroupData[{
Cell["Limits", "Subsection"],
Cell[TextData[{
"You should have learned something about limits during your first year of \
calculus. We won't use them too much this year, but there will be one lab \
where you have to compute limits of a function, and you'll have to use limits \
to calculate so-called \"partial derivatives\" by the definition. ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" can compute limits using a function named, not surprisingly, ",
StyleBox["Limit",
FontWeight->"Bold"],
".\n\nFirst let's define a function and plot its graph around x=0."
}], "Text"],
Cell[BoxData[{
\(f[x_] = Sin[x]/x\), "\[IndentingNewLine]",
\(Plot[f[x], {x, \(-Pi\), Pi}]\)}], "Input"],
Cell[TextData[{
"Although you can't tell by looking at the graph, you know that f[0] is \
undefined because you can't divide by zero. The limit of this function as x\
\[RightArrow]0 is a very important limit in calculus; your teacher may have \
spent a fair amount of time proving that it's equal to one. ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" can compute this very quickly:"
}], "Text"],
Cell[BoxData[
\(Limit[f[x], x -> 0]\)], "Input"],
Cell[TextData[{
"We need to be a little careful because ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" isn't really computing a two-sided limit. In this limit it didn't matter \
because, as you can see by examining the picture, the limit is equal to 1 \
whether you approach x=0 from the left side or from the right side. Let's \
look at a function where it ",
StyleBox["will",
FontSlant->"Italic"],
" matter!"
}], "Text"],
Cell[BoxData[{
\(f[x_] = Abs[x]/x\), "\[IndentingNewLine]",
\(Plot[f[x], {x, \(-1\), 1}]\)}], "Input"],
Cell[TextData[{
"First, you should look at the function definition and its graph to make \
sure you understand what's going on; f(x) = 1 whenever x is positive, f(x) = \
-1 whenever x is negative, and f(0) is undefined. Now check what ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" thinks the limit of f(x) is as x\[RightArrow]0:"
}], "Text"],
Cell[BoxData[
\(Limit[f[x], x \[Rule] 0]\)], "Input"],
Cell[TextData[{
"So you can see that, by default, ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" computes limits by approaching from the right hand side. You should use \
the Help Browser to find out how to compute this limit from the left hand \
side, so the answer is -1. (Look up \"limit.\")"
}], "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Derivatives", "Subsection"],
Cell[TextData[{
StyleBox["Mathematica",
FontSlant->"Italic"],
" can differentiate just about everything you can throw at it. There are a \
few different commands you can use, but the easiest is just named ",
StyleBox["D",
FontWeight->"Bold"],
"."
}], "Text"],
Cell[BoxData[
\(D[x^6, \ x]\)], "Input"],
Cell[TextData[{
"The first argument of ",
StyleBox["D",
FontWeight->"Bold"],
" is the function you wish to differentiate. The second argument is the \
variable. Obviously in this example it's clear that the variable has to be \
x, but you need to tell ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" anyway. Here's an example where ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" will automatically do the product and quotient rules for you:"
}], "Text"],
Cell[BoxData[{
\(f[x_] = Sin[x] Cos[\(-x^2\)]/ArcTan[1/x]\), "\[IndentingNewLine]",
\(D[f[x], x]\)}], "Input"],
Cell[TextData[{
"That's fairly ugly, and this is a good example of when you should try to \
use ",
StyleBox["Simplify",
FontWeight->"Bold"],
" to make your answers nicer. (In this case, it turns out, it doesn't help \
much.)\n\n",
StyleBox["D",
FontWeight->"Bold"],
" can also do multiple derivatives. If you want the ",
Cell[BoxData[
\(TraditionalForm\`n\^th\)]],
"derivative with respect to x, replace the argument \"x\" with {x,n}:"
}], "Text"],
Cell[BoxData[
\(D[x^6, {x, 3}]\)], "Input"],
Cell[BoxData[
\(D[Log[x], {x, 4}]\)], "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell["Integration", "Subsection"],
Cell[TextData[{
StyleBox["Mathematica",
FontSlant->"Italic"],
" can do both indefinite and definite integrals using the same command, ",
StyleBox["Integrate",
FontWeight->"Bold"],
". ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" will automatically try u-substitutions, perform integration by parts, and \
do all kinds of other tricks. For example, to compute\n\n\t\t",
Cell[BoxData[
\(TraditionalForm\`\[Integral]\(x\^2\) Sin[x] \[DifferentialD]x\)],
FontSize->16],
"\n\nyou would type:"
}], "Text"],
Cell[BoxData[
\(Integrate[x^2\ Sin[x], x]\)], "Input"],
Cell[TextData[{
"But if you wanted to compute the definite integral\n\n\t\t",
Cell[BoxData[
\(TraditionalForm\`\[Integral]\_0\%\[Pi]\( x\^2\)
Sin[x] \[DifferentialD]x\)],
FontSize->16],
"\n"
}], "Text"],
Cell["you would replace \"x\" with \"{x,0,Pi}\":", "Text"],
Cell[BoxData[
\(Integrate[x^2\ Sin[x], \ {x, 0, Pi}]\)], "Input"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Pseudo-Exercises", "Section"],
Cell[TextData[{
"These are some exercises from single variable calculus for you to work on \
so that you get used to the commands you've learned above. It's one thing to \
read about ",
StyleBox["Integrate",
FontWeight->"Bold"],
" or ",
StyleBox["Plot",
FontWeight->"Bold"],
"; it's another to use them. You do ",
StyleBox["not",
FontSlant->"Italic"],
" have to turn these problems in. We're not interested in seeing if you \
can do single variable calculus. (Or, more accurately, we're already \
assuming you can, and if you can't you should talk to us quickly. A little \
rust is ok, but if you don't know what a tangent line is, we may have a \
problem.)\n\nWhen you are done with these exercises, you are finished, and \
you can start Lab 1B if you wish. (But if you tell your TA you are finished, \
s/he will ask to see your work for these exercises and the tasks given in the \
text of the lab above.)"
}], "Text"],
Cell[TextData[{
StyleBox["Exercise 1",
FontSize->14,
FontWeight->"Bold"],
"\n\nThere are two points on the circle ",
Cell[BoxData[
\(TraditionalForm\`x\^2 + y\^2 = 4\)]],
" where y = 0.5. Find those points, and then find the lines which are \
tangent to the circle at those two points. Graph the circle along with the \
two lines to verify that they are indeed tangent. \n\nYou may use ",
StyleBox["Solve",
FontWeight->"Bold"],
" to find the points, or do it by hand if you wish. It's probably easiest \
to find the equations to the tangent lines by hand. You will have to use ",
StyleBox["ImplicitPlot",
FontWeight->"Bold"],
" to graph the circle, and you can use ",
StyleBox["Plot",
FontWeight->"Bold"],
" to plot the lines. Then you can use ",
StyleBox["Show",
FontWeight->"Bold"],
" to display all three graphs together.\n\n",
StyleBox["Exercise 2",
FontSize->14,
FontWeight->"Bold"],
StyleBox["\n",
FontSize->14],
"\nConsider the following functions:\n\nf(x) = 4 + ",
Cell[BoxData[
\(TraditionalForm\`Sin[\[Pi]\ x]\)]],
" / 2\n\ng(x) = ",
Cell[BoxData[
\(TraditionalForm\`\((x - 2)\)\^2\)]],
"\n\nFind the area of the region enclosed by the graphs of these two \
functions.\n\nHint: First you need to find the points where the graphs of f \
and g intersect. ",
StyleBox["Solve",
FontWeight->"Bold"],
" will probably not be of much use, but you can try it. You should plot \
both functions, see if you can estimate visually where the graphs intersect, \
and then verify this by plugging in the appropriate values of x into f and g. \
Then you need to use ",
StyleBox["Integrate",
FontWeight->"Bold"],
" to find the area."
}], "Text",
CellFrame->True,
Background->GrayLevel[0.849989]],
Cell[CellGroupData[{
Cell["Credits", "Subsection"],
Cell[TextData[{
"This lab was written entirely from scratch in January 2002. Our previous \
Lab 1 started with three dimensional graphs and went from there, without much \
of an introduction to ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" itself. (And students who had lab before lecture didn't know what a 3D \
plot was, anyway.) Please send me any questions or comments!\n\nCopyright \
2002,2003 by Jonathan Rogness\nLicensed under GNU GPL (and FDL as applicable) \
to the University of Minnesota School of Mathematics.\nrogness@math.umn.edu"
}], "Text"]
}, Closed]]
}, Closed]]
},
FrontEndVersion->"4.1 for X",
ScreenRectangle->{{0, 1600}, {0, 1200}},
ScreenStyleEnvironment->"Working",
WindowSize->{573, 620},
WindowMargins->{{88, Automatic}, {Automatic, 25}}
]
(*******************************************************************
Cached data follows. If you edit this Notebook file directly, not
using Mathematica, you must remove the line containing CacheID at
the top of the file. The cache data will then be recreated when
you save this file from within Mathematica.
*******************************************************************)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[1705, 50, 512, 16, 114, "Text"],
Cell[CellGroupData[{
Cell[2242, 70, 43, 0, 60, "Section"],
Cell[2288, 72, 2992, 45, 626, "Text"],
Cell[5283, 119, 273, 8, 92, "Text"],
Cell[5559, 129, 76, 0, 32, "Text"],
Cell[5638, 131, 225, 5, 66, "Text"],
Cell[5866, 138, 2542, 40, 536, "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell[8445, 183, 145, 5, 89, "Section"],
Cell[8593, 190, 819, 18, 140, "Text"],
Cell[CellGroupData[{
Cell[9437, 212, 46, 0, 38, "Subsection"],
Cell[9486, 214, 737, 16, 158, "Text"],
Cell[10226, 232, 38, 1, 27, "Input"],
Cell[10267, 235, 37, 1, 27, "Input"],
Cell[10307, 238, 241, 6, 50, "Text"],
Cell[10551, 246, 76, 2, 43, "Input"],
Cell[10630, 250, 645, 13, 104, "Text"],
Cell[11278, 265, 80, 2, 43, "Input"],
Cell[11361, 269, 54, 1, 27, "Input"],
Cell[11418, 272, 490, 13, 86, "Text"],
Cell[11911, 287, 72, 2, 43, "Input"],
Cell[11986, 291, 176, 3, 50, "Text"],
Cell[12165, 296, 161, 4, 75, "Input"],
Cell[12329, 302, 109, 3, 32, "Text"],
Cell[12441, 307, 76, 2, 43, "Input"],
Cell[12520, 311, 158, 3, 50, "Text"],
Cell[12681, 316, 35, 1, 27, "Input"],
Cell[12719, 319, 564, 10, 140, "Text"],
Cell[13286, 331, 44, 1, 27, "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell[13367, 337, 31, 0, 30, "Subsection"],
Cell[13401, 339, 329, 7, 68, "Text"],
Cell[13733, 348, 40, 1, 27, "Input"],
Cell[13776, 351, 370, 12, 51, "Text"],
Cell[14149, 365, 43, 1, 27, "Input"],
Cell[14195, 368, 208, 4, 50, "Text"],
Cell[14406, 374, 41, 1, 27, "Input"],
Cell[14450, 377, 37, 1, 27, "Input"],
Cell[14490, 380, 278, 7, 50, "Text"],
Cell[14771, 389, 42, 1, 27, "Input"],
Cell[14816, 392, 676, 24, 86, "Text"],
Cell[15495, 418, 2041, 59, 405, "Text"],
Cell[17539, 479, 608, 17, 86, "Text"],
Cell[18150, 498, 83, 1, 27, "Input"],
Cell[18236, 501, 172, 5, 32, "Text"],
Cell[18411, 508, 108, 2, 27, "Input"],
Cell[18522, 512, 44, 1, 27, "Input"],
Cell[18569, 515, 66, 1, 27, "Input"],
Cell[18638, 518, 142, 3, 50, "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell[18817, 526, 34, 0, 30, "Subsection"],
Cell[18854, 528, 1125, 20, 242, "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell[20016, 553, 49, 0, 30, "Subsection"],
Cell[20068, 555, 205, 5, 52, "Text"],
Cell[20276, 562, 44, 1, 30, "Input"],
Cell[20323, 565, 560, 14, 90, "Text"],
Cell[20886, 581, 495, 11, 106, "Text"],
Cell[21384, 594, 634, 11, 147, "Text"],
Cell[22021, 607, 172, 4, 90, "Input"],
Cell[22196, 613, 365, 7, 71, "Text"],
Cell[22564, 622, 153, 3, 70, "Input"],
Cell[22720, 627, 1338, 37, 276, "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell[24095, 669, 29, 0, 30, "Subsection"],
Cell[24127, 671, 1167, 32, 90, "Text"],
Cell[25297, 705, 176, 4, 90, "Input"],
Cell[25476, 711, 486, 11, 71, "Text"],
Cell[25965, 724, 183, 3, 70, "Input"],
Cell[26151, 729, 863, 24, 239, "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell[27051, 758, 36, 0, 30, "Subsection"],
Cell[27090, 760, 735, 20, 158, "Text"],
Cell[27828, 782, 111, 2, 43, "Input"],
Cell[27942, 786, 196, 6, 50, "Text"],
Cell[28141, 794, 72, 1, 27, "Input"],
Cell[28216, 797, 83, 3, 50, "Text"],
Cell[28302, 802, 139, 2, 43, "Input"],
Cell[28444, 806, 262, 7, 68, "Text"],
Cell[28709, 815, 51, 1, 27, "Input"],
Cell[28763, 818, 2285, 61, 393, "Text"],
Cell[31051, 881, 198, 4, 43, "Input"],
Cell[31252, 887, 416, 8, 86, "Text"],
Cell[31671, 897, 476, 11, 122, "Text"],
Cell[32150, 910, 85, 1, 40, "DisplayFormula"],
Cell[32238, 913, 133, 3, 50, "Text"],
Cell[32374, 918, 724, 18, 139, "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell[33135, 941, 38, 0, 30, "Subsection"],
Cell[33176, 943, 1140, 20, 248, "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[34365, 969, 60, 0, 40, "Section"],
Cell[34428, 971, 346, 7, 68, "Text"],
Cell[CellGroupData[{
Cell[34799, 982, 28, 0, 45, "Subsection"],
Cell[34830, 984, 567, 11, 109, "Text"],
Cell[35400, 997, 113, 2, 50, "Input"],
Cell[35516, 1001, 411, 8, 71, "Text"],
Cell[35930, 1011, 52, 1, 30, "Input"],
Cell[35985, 1014, 446, 11, 52, "Text"],
Cell[36434, 1027, 111, 2, 50, "Input"],
Cell[36548, 1031, 357, 7, 52, "Text"],
Cell[36908, 1040, 57, 1, 30, "Input"],
Cell[36968, 1043, 323, 7, 52, "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell[37328, 1055, 33, 0, 29, "Subsection"],
Cell[37364, 1057, 275, 8, 33, "Text"],
Cell[37642, 1067, 44, 1, 30, "Input"],
Cell[37689, 1070, 489, 13, 71, "Text"],
Cell[38181, 1085, 119, 2, 50, "Input"],
Cell[38303, 1089, 477, 13, 90, "Text"],
Cell[38783, 1104, 47, 1, 30, "Input"],
Cell[38833, 1107, 50, 1, 30, "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell[38920, 1113, 33, 0, 29, "Subsection"],
Cell[38956, 1115, 544, 15, 130, "Text"],
Cell[39503, 1132, 58, 1, 30, "Input"],
Cell[39564, 1135, 229, 7, 93, "Text"],
Cell[39796, 1144, 58, 0, 33, "Text"],
Cell[39857, 1146, 69, 1, 30, "Input"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[39975, 1153, 35, 0, 40, "Section"],
Cell[40013, 1155, 952, 20, 194, "Text"],
Cell[40968, 1177, 1808, 48, 482, "Text"],
Cell[CellGroupData[{
Cell[42801, 1229, 29, 0, 45, "Subsection"],
Cell[42833, 1231, 551, 10, 158, "Text"]
}, Closed]]
}, Closed]]
}
]
*)
(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)