(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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). 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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. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 82250, 2232]*) (*NotebookOutlinePosition[ 82914, 2255]*) (* CellTagsIndexPosition[ 82870, 2251]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Lab 1A: Introduction to Labs and ", FontSize->18, FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox["Mathematica", FontSize->18, 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[TextData[StyleBox["Introduction to the Labs", FontSize->16]], "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\nAs you move on, you'll find \ there are commands in the lab for you to run. It would also be useful to \ open another notebook while you read the lab so that you can do your own work \ there. (Go to the File menu and choose \"New\" to do this.)\n\nThere are \ also a number of exercises for you to work on in the labs. To help you \ distinguish these from rhetorical questions, or things that we just want you \ to do on your own, we've formatted the labs so that \"official\" exercises \ are always in a box with a reddish background. (On some computers the \ background is more pink than red.) Here's an example:" }], "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["\<\ Note that you won't always have to turn in every exercise, although \ it would be a good idea to work on all of them. Your TA will tell you at an \ appropriate time which solutions you need to hand in for each lab. Usually 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. 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["\<\ In fact, here's the first \"real\" gray box, with an important \ message that you should keep in mind throughout the semester:\ \>", "Text"], Cell[TextData[{ StyleBox["The computer labs are an important part of this course!", FontWeight->"Bold"], " \n\nWith only two lectures per week, your instructors have to pick \ lecture material very carefully. Sometimes they might leave out certain \ concepts with the knowledge that they will be covered in the labs. In other \ words, these labs are one of the ways you will learn the material in this \ course.\n\nYou should also note that the lab assignments make up a \ significant part of your grade, so you should not take them lightly. Many of \ you probably never had to read your calculus book. At most, you may have \ glanced through the examples to find out how to do a certain homework \ problem. (Lest you think I'm accusing you, let me admit right now that I and \ most of your instructors probably did exactly the same thing in ", StyleBox["our", FontSlant->"Italic"], " Calculus classes!) ", "This approach will ", StyleBox["not", FontSlant->"Italic"], " work well with these labs. If you look at the exercises first, you might \ find yourself completely lost. We ", StyleBox["highly", FontSlant->"Italic"], " recommend you read each lab thoroughly before trying the exercises. In \ some cases this might mean re-reading a paragraph a number of times before it \ makes sense.\n\nYour solutions to lab exercises will be written up much more \ carefully than normal homework assignments. This isn't a writing-intensive \ course, so you don't have to turn in ten pages per problem, but we do expect \ clear writing, reasonable mathematical justification for your work, pictures, \ and so on. A good rule of thumb is that your solution should be a like a \ detailed textbook example. Your TA will show you examples of what we expect \ before you hand in your first lab assignment." }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], 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[{ StyleBox["Introduction to ", FontSize->16], StyleBox["Mathematica", FontSize->16, FontSlant->"Italic"], StyleBox[" - Arithmetic, Functions, and Graphs", FontSize->16] }], "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[{ "From now on, whenever you run across a command as you read, you can assume \ it's meant as an example for you. You should evaluate it, even if you're not \ specifically told to do so.\n", "\n", StyleBox["Mathematica", FontSlant->"Italic"], " uses the normal operators +, -, /, and * for arithmetic operations, and ^ \ for exponents." }], "Text"], Cell[BoxData[{ \(3*3\), "\[IndentingNewLine]", \(3\ ^2\)}], "Input"], Cell[TextData[{ "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 be very careful to remember the * \ in between them.\ \>", "Text"], Cell[BoxData[ \(a*b\)], "Input"], Cell["\<\ Evaluate this next cell to see what happens if you forget the \ *.\ \>", "Text"], Cell[BoxData[ \(ab\)], "Input"], Cell[TextData[{ "Mathematica returns \"ab\" because there is nothing between the letters in \ the input cell, so it doesn't know you're trying to multiply to different \ variables together. Instead, 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 left out the underscore, you'll probably have to clear the variable \ name (as in ", StyleBox["Clear[x]", FontWeight->"Bold"], ") before redefining the function." }], "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[CellGroupData[{ Cell["Example", "Subsubsection"], Cell[TextData[{ "It'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"] }, Open ]] }, 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["Loading New Commands", "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has so many commands that most computers would slow to a crawl if \ everything were automatically loaded. To make things a little faster, many \ commands in ", StyleBox["Mathematica", FontSlant->"Italic"], " are contained in \"packages.\" Occasionally this semester we're going to \ use some of these commands. Rather than have you learn the complexities of \ loading packages, we've assembled everything into one notebook, called \ \"math2374.nb,\" which your TA should have directed you to download into your \ home directory. If you haven't downloaded math2374.nb from the course web \ page yet, you should do so now.\n\nThe math2374.nb file contains commands \ which automatically load every command we'll need this semester. You should \ follow these steps to load them:\n\n(1) Open math2374.nb in ", StyleBox["Mathematica", FontSlant->"Italic"], ". Do this now by choosing \"Open\" under the File menu.\n(2) Click on the \ button which says, \"Click here to Load the Math 2374 Commands.\"\n\nOnce \ everything works, a gray box will appear with a confirmation message. At \ this point you can close math2374.nb if you like, to avoid cluttering up your \ mailbox. Don't bother saving the changes; the only change is the appearance \ of the box, and you probably don't want to save multiple copies of that \ anyway!\n\nFrom now on, you should follow the steps above ", StyleBox["before", FontSlant->"Italic"], " starting any labs, or doing any work on your own. If you don't many of \ the commands you try to use will fail. One of the commands in the next \ section, ", StyleBox["ImplicitPlot", FontWeight->"Bold"], ", needs to be loaded, so make sure you've completed these steps before you \ move on." }], "Text"] }, 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[CellGroupData[{ Cell[TextData[StyleBox["Options", FontWeight->"Bold"]], "Subsubsection"], Cell[TextData[{ "Occasionally you will want to use optional arguments when drawing graphs. \ Options generally come at the end of a command 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\nYou'll \ learn more options in Lab 1B next week." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Plotting Implicit Functions", "Subsubsection"], Cell[TextData[{ "In 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"], ". (Remember from before, this command needs to be loaded using the \ math2374.nb file before you evaluate the next cell.)" }], "Text"], Cell[BoxData[ \(ImplicitPlot[ x^2\ + \ y^2\ \[Equal] \ 1, \ {x, \(-1\), 1}, \ {y, \(-1\), 1}]\)], "Input"], Cell[TextData[{ "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"] }, Open ]] }, 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 disk space 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[TextData[StyleBox["Single Variable Calculus with Mathematica", FontSize->16]], "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 focus on them very much this year, but in a few weeks \ you'll have to use them to compute 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[CellGroupData[{ Cell[BoxData[{ \(f[x_] = Sin[x]/x\), "\[IndentingNewLine]", \(Plot[f[x], {x, \(-Pi\), Pi}]\)}], "Input"], Cell[BoxData[ \(Sin[x]\/x\)], "Output"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.151576 0.0147151 0.588605 [ [.04527 .00222 -6 -9 ] [.04527 .00222 6 0 ] [.19685 .00222 -6 -9 ] [.19685 .00222 6 0 ] [.34842 .00222 -6 -9 ] [.34842 .00222 6 0 ] [.65158 .00222 -3 -9 ] [.65158 .00222 3 0 ] [.80315 .00222 -3 -9 ] [.80315 .00222 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Now check what ", StyleBox["Mathematica", FontSlant->"Italic"], " thinks the limit of f(x) is as x\[RightArrow]0:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[f[x], x \[Rule] 0]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], 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[TextData[StyleBox["Pseudo-Exercises", FontSize->16]], "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 tell your TA you are finished, and \ s/he might 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 = 1/2. 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"], " doesn't help here, because it doesn't work with trigonometric functions. \ 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.) I made minor modifications in January 2004 to reflect the \ changes in lab exercises and the use of math2374.nb\n\nThis lab is copyright \ 2002, 2004 by Jonathan Rogness (rogness@math.umn.edu) and is protected by the \ Creative Commons Attribution-NonCommercial-ShareAlike License. You can find \ more information on this license at \ http://creativecommons.org/licenses/by-nc-sa/1.0/\n\nAlthough it's not \ specifically required by the license, I'd appreciate it if you let me know if \ you use parts of our labs, just so I can keep track of it. Please send me \ any questions or comments!" }], "Text"] }, Closed]] }, Closed]] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, ScreenStyleEnvironment->"Working", WindowSize->{607, 692}, WindowMargins->{{Automatic, 21}, {Automatic, 8}} ] (******************************************************************* Cached data follows. 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