(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 60239, 1788] NotebookOptionsPosition[ 54250, 1609] NotebookOutlinePosition[ 54622, 1625] CellTagsIndexPosition[ 54579, 1622] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) 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[ RowBox[{"2", "+", "2"}]], "Input"], Cell[BoxData[ RowBox[{"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[{ RowBox[{"3", "*", "3"}], "\[IndentingNewLine]", RowBox[{"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[{ RowBox[{ RowBox[{"9", "/", "3"}], ";"}], "\[IndentingNewLine]", RowBox[{"6", "^", "2"}]}], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"12", "*", "12"}], ";", " ", RowBox[{"5", "+", "1"}], ";", " ", RowBox[{"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[{ RowBox[{ RowBox[{ RowBox[{"a", "=", "2"}], ";", " ", RowBox[{"b", "=", "3"}], ";"}], " "}], "\[IndentingNewLine]", "a", "\[IndentingNewLine]", "b", "\ \[IndentingNewLine]", RowBox[{"a", "+", "b"}]}], "Input"], Cell["\<\ If you want to multiply variables be very careful to remember the * in \ between them.\ \>", "Text"], Cell[BoxData[ RowBox[{"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 two 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", CellChangeTimes->{{3.442061710034564*^9, 3.4420617103499947`*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"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[ RowBox[{"Sqrt", "[", "5", "]"}]], "Input"], Cell[TextData[{ "Remember, ", StyleBox["Mathematica", FontSlant->"Italic"], " does things symbolically unless we tell it otherwise, so it returns ", Cell[BoxData[ FormBox[ SqrtBox["5"], TraditionalForm]]], " 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[ RowBox[{"N", "[", RowBox[{"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[ RowBox[{"Sqrt", "[", "10", "]"}]], "Input"], Cell[BoxData[ RowBox[{"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[ RowBox[{"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[ FormBox[ SuperscriptBox["e", "number"], TraditionalForm]]], ".) 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[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], "2"], TraditionalForm]]], "+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[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], "^", "2"}], " ", "+", "2"}], " ", "\[Equal]", " ", RowBox[{"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[ RowBox[{ RowBox[{"6", "x", RowBox[{ RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], "/", RowBox[{"Sqrt", "[", "2", "]"}]}]}], " ", "+", RowBox[{ RowBox[{"(", RowBox[{"6", "+", " ", "Pi"}], ")"}], "/", RowBox[{"Sqrt", "[", "2", "]"}]}], "-", RowBox[{ RowBox[{"Sqrt", "[", "2", "]"}], "*", " ", RowBox[{"Pi", "/", "2"}]}]}]], "Input"], Cell[BoxData[ RowBox[{"Simplify", "[", "%", "]"}]], "Input"], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "^", "2"}], " ", "+", " ", RowBox[{ RowBox[{"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["Documentation Center", "Subsection", CellChangeTimes->{{3.4420617225899353`*^9, 3.442061727461833*^9}}], 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 \ Documentation Center and let me know if it doesn't make sense.\"\n\nAs a \ test, open the Documentation Center 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 \ Documentation Center include examples, which can be very instructive. To see \ these examples you have to click on the little triangles to expand those \ sections." }], "Text", CellChangeTimes->{{3.442061748495924*^9, 3.442061824378501*^9}}] }, 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[ FormBox[ RowBox[{ RowBox[{"f", "(", "x", ")"}], "=", SuperscriptBox["x", "2"]}], TraditionalForm]]], ". We can do this by using the following input:" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x_", "]"}], "=", RowBox[{"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\n\ Once 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[{ RowBox[{"f", "[", "4", "]"}], "\[IndentingNewLine]", RowBox[{"f", "[", "Pi", "]"}], "\[IndentingNewLine]", RowBox[{"f", "[", RowBox[{"(", RowBox[{"1", "+", "t"}], ")"}], "]"}], "\[IndentingNewLine]", RowBox[{"f", "[", RowBox[{"Sin", "[", RowBox[{"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[{ RowBox[{ RowBox[{ RowBox[{"g", "[", RowBox[{"u_", ",", "v_"}], "]"}], "=", RowBox[{"u", "/", "v"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"g", "[", RowBox[{"2", ",", "3"}], "]"}], "\[IndentingNewLine]", RowBox[{"g", "[", RowBox[{ RowBox[{"5", "Pi"}], ",", RowBox[{ RowBox[{"(", RowBox[{"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[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["ax", "2"], "+", "bx", " ", "+", " ", "c"}], " ", "=", " ", "0"}], TraditionalForm]]], ", you can use the quadratic formula, which says \n\n", StyleBox["x = ", FontSize->14], Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ RowBox[{"-", "b"}], " ", "\[PlusMinus]", " ", FormBox[ SqrtBox[ FormBox[ RowBox[{ SuperscriptBox["b", "2"], "-", RowBox[{"4", "ac"}]}], TraditionalForm]], TraditionalForm]}], RowBox[{"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[ FormBox[ RowBox[{ RowBox[{ RowBox[{"2", SuperscriptBox["x", "2"]}], "+", RowBox[{"8", "x"}], "-", "1"}], "=", "0"}], TraditionalForm]]], ". (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[ FormBox[ OverscriptBox[ RowBox[{"(", RowBox[{"1", ",", "2"}], ")"}], "\[RightVector]"], TraditionalForm]]], ", or \[LeftAngleBracket]1,2\[RightAngleBracket]. In ", StyleBox["Mathematica", FontSlant->"Italic"], " vectors are written with curly brackets. Here we define two vectors, ", Cell[BoxData[ FormBox[ OverscriptBox["v", "\[RightVector]"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ OverscriptBox["u", "\[RightVector]"], TraditionalForm]]], ". We add them, we multiply ", Cell[BoxData[ FormBox[ OverscriptBox["v", "\[RightVector]"], TraditionalForm]]], " by a scalar number, and we compute the dot product ", Cell[BoxData[ FormBox[ OverscriptBox["v", "\[RightVector]"], TraditionalForm]]], "\[CenterDot]", Cell[BoxData[ FormBox[ OverscriptBox["u", "\[RightVector]"], TraditionalForm]]], ". (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[ FormBox[ OverscriptBox["v", "\[RightVector]"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ OverscriptBox["u", "\[RightVector]"], TraditionalForm]]], ", so they are not displayed." }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"v", "=", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}]}], ";", " ", RowBox[{"u", "=", RowBox[{"{", RowBox[{"4", ",", "4"}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"v", "+", "u"}], "\[IndentingNewLine]", RowBox[{"2", "v"}], "\[IndentingNewLine]", RowBox[{"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[ FormBox[ RowBox[{ OverscriptBox["v", "\[RightVector]"], "\[CenterDot]", OverscriptBox["w", "\[RightVector]"]}], TraditionalForm]]], "." }], "Text"], Cell[BoxData[{ RowBox[{"u", "=", RowBox[{"{", RowBox[{"4", ",", "2", ",", RowBox[{"-", "1"}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{"v", "=", RowBox[{"{", RowBox[{ "1", ",", "2", ",", "3", ",", "4", ",", "5", ",", "6", ",", "7", ",", "8", ",", "9"}], "}"}]}], "\[IndentingNewLine]", RowBox[{"w", "=", RowBox[{"{", RowBox[{ "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 some 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 contains commands which automatically load every \ package we'll need this semester.\n\nThere aren't any extra packages required \ for this lab, so you don't need the math2374.nb file yet. Later this \ semester your TA will show you how to download and run the file. For the \ record, you can follow these directions:\n\n(1) Download math2374.nb from the \ course web page.\n\n(2) Open math2374.nb in ", StyleBox["Mathematica", FontSlant->"Italic"], ". Do this now by choosing \"Open\" under the File menu.\n\n(3) 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!" }], "Text", CellChangeTimes->{{3.409325295117556*^9, 3.40932529534059*^9}, { 3.40932550459217*^9, 3.409325644999607*^9}, {3.409326226786976*^9, 3.409326228842386*^9}}] }, 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[{ RowBox[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], "=", RowBox[{"x", "^", "2"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"f", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "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[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"x", " ", "*", RowBox[{"Sin", "[", RowBox[{"1", "/", "x"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "0.3"}], ",", "0.3"}], "}"}]}], "]"}]], "Input"], Cell["You can name plots so you can refer to them later as well:", "Text"], Cell[BoxData[{ RowBox[{"plot1", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"plot2", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}]}], "Input"], Cell[TextData[{ "If you end a ", StyleBox["Plot", FontWeight->"Bold"], " command with a semi-colon, ", StyleBox["Mathematica", FontSlant->"Italic"], " will create the graph, but the output will be surpressed. This is useful \ if you want to construct multiple graphs for later use. For example, here \ are the same ", StyleBox["Plot", FontWeight->"Bold"], " commands from above, with semi-colons instead." }], "Text", CellChangeTimes->{{3.409325676056955*^9, 3.409325778104255*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"plot1", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"plot2", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]}], ";"}]}], "Input", CellChangeTimes->{{3.409325765388139*^9, 3.409325765566768*^9}}], Cell[TextData[{ "If you later decide you actually wanted to see the first graph, just ask ", StyleBox["Mathematica", FontSlant->"Italic"], " to show you plot1:" }], "Text", CellChangeTimes->{{3.409325788941339*^9, 3.409325809621117*^9}}], Cell[BoxData["plot1"], "Input", CellChangeTimes->{{3.409325783696211*^9, 3.409325785322375*^9}}], Cell[TextData[{ "You can also use the ", StyleBox["Show", FontWeight->"Bold"], " command:" }], "Text", CellChangeTimes->{{3.409325820715033*^9, 3.409325826541169*^9}}], Cell[BoxData[ RowBox[{"Show", "[", "plot2", "]"}]], "Input", CellChangeTimes->{{3.409325828041954*^9, 3.409325829866596*^9}}], Cell[TextData[{ "The ", StyleBox["Show", FontWeight->"Bold"], " command is also useful for combined different graphs. You can give the \ command the names of multiple plots, and it will show them together:" }], "Text", CellChangeTimes->{{3.409325779572087*^9, 3.409325782045253*^9}, { 3.409325814487222*^9, 3.409325856085892*^9}}], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"plot1", ",", "plot2"}], "]"}]], "Input", CellChangeTimes->{{3.409325343449441*^9, 3.409325351652108*^9}}], 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"] }, Closed]], 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[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], "=", "1"}], ","}], TraditionalForm]]], " 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[ FormBox[ SqrtBox[ RowBox[{"1", "-", SuperscriptBox["x", "2"]}]], TraditionalForm]]], ", 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["ContourPlot", FontWeight->"Bold"], ". (In older editions of Mathematica implicit functions were graphed with \ ", StyleBox["ImplicitPlot", FontWeight->"Bold"], ", whose name made more sense, but nowadays ", StyleBox["ImplicitPlot", FontWeight->"Bold"], " is obsolete.)" }], "Text", CellChangeTimes->{{3.409325380082925*^9, 3.409325488583071*^9}}], Cell[BoxData[ RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], "\[Equal]", "1"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "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[ RowBox[{ RowBox[{ FractionBox[ SuperscriptBox["x", "2"], SuperscriptBox["3", "2"]], "+", FractionBox[ RowBox[{" ", SuperscriptBox["y", "2"]}], SuperscriptBox["4", "2"]]}], "=", " ", "1"}]], "DisplayFormula"] }, Closed]] }, Open ]], 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 can take up several \ megabytes of disk space.\n\nSo, before you save a notebook, you should always \ go to the Cell 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", CellChangeTimes->{{3.409326098816974*^9, 3.409326116213823*^9}}] }, 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[BoxData[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], "=", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "/", "x"}]}], "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"f", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "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[ RowBox[{"Limit", "[", RowBox[{ RowBox[{"f", "[", "x", "]"}], ",", RowBox[{"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[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], "=", RowBox[{ RowBox[{"Abs", "[", "x", "]"}], "/", "x"}]}], "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"f", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "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[ RowBox[{"Limit", "[", RowBox[{ RowBox[{"f", "[", "x", "]"}], ",", RowBox[{"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[ RowBox[{"D", "[", RowBox[{ RowBox[{"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[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], "=", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], RowBox[{ RowBox[{"Cos", "[", RowBox[{"-", RowBox[{"x", "^", "2"}]}], "]"}], "/", RowBox[{"ArcTan", "[", RowBox[{"1", "/", "x"}], "]"}]}]}]}], "\[IndentingNewLine]", RowBox[{"D", "[", RowBox[{ RowBox[{"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[ FormBox[ SuperscriptBox["n", "th"], TraditionalForm]]], "derivative with respect to x, replace the argument \"x\" with {x,n}:" }], "Text"], Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{"x", "^", "6"}], ",", RowBox[{"{", RowBox[{"x", ",", "3"}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"D", "[", RowBox[{ RowBox[{"Log", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "4"}], "}"}]}], "]"}]], "Input"], Cell["\<\ If you have a function of one variable, you can also use the \"prime\" \ notation as a shortcut to compute derivatives:\ \>", "Text", CellChangeTimes->{{3.40932591268033*^9, 3.409325941129724*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], "=", RowBox[{"x", "^", "4"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "'"}], "[", "x", "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "''"}], "[", "x", "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "'''"}], "[", "x", "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "''''"}], "[", "x", "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "'''''"}], "[", "x", "]"}]}], "Input", CellChangeTimes->{{3.409325942537409*^9, 3.409325963244009*^9}}], Cell["\<\ Most of the functions in this class will have more than one variable, so this \ shortcut isn't always useful.\ \>", "Text", CellChangeTimes->{{3.409325988700607*^9, 3.40932600310342*^9}}] }, 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[ FormBox[ RowBox[{"\[Integral]", RowBox[{ SuperscriptBox["x", "2"], RowBox[{"Sin", "[", "x", "]"}], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]], FontSize->16], "\n\nyou would type:" }], "Text"], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], " ", RowBox[{"Sin", "[", "x", "]"}]}], ",", "x"}], "]"}]], "Input"], Cell[TextData[{ "But if you wanted to compute the definite integral\n\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[{ SuperscriptBox["x", "2"], RowBox[{"Sin", "[", "x", "]"}], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]], FontSize->16], "\n" }], "Text"], Cell["you would replace \"x\" with \"{x,0,Pi}\":", "Text"], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], " ", RowBox[{"Sin", "[", "x", "]"}]}], ",", " ", RowBox[{"{", RowBox[{"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\n\ When 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[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], "=", "4"}], TraditionalForm]]], " 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["ContourPlot", 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[ FormBox[ RowBox[{"Sin", "[", RowBox[{"\[Pi]", " ", "x"}], "]"}], TraditionalForm]]], " / 2\n\ng(x) = ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", "2"}], ")"}], "2"], TraditionalForm]]], "\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, CellChangeTimes->{{3.442061837744852*^9, 3.442061839378462*^9}}, 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. More modifications were \ made in January 2007 to make the lab compatible with ", StyleBox["Mathematica", FontSlant->"Italic"], " 6.0.\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\n\ Although 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", CellChangeTimes->{{3.409326036662891*^9, 3.40932605253547*^9}}] }, Closed]] }, Closed]] }, ScreenStyleEnvironment->"Working", WindowSize->{607, 692}, WindowMargins->{{36, Automatic}, {Automatic, 10}}, FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (April 20, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[568, 21, 491, 16, 93, "Text"], Cell[CellGroupData[{ Cell[1084, 41, 78, 1, 63, "Section"], Cell[1165, 44, 2423, 38, 416, "Text"], Cell[3591, 84, 265, 8, 76, "Text"], Cell[3859, 94, 929, 16, 191, "Text"], 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