(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' 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[ 39393, 1313]*) (*NotebookOutlinePosition[ 40113, 1338]*) (* CellTagsIndexPosition[ 40069, 1334]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "An Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " for Calculus " }], "Title", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True, Background->RGBColor[0, 1, 1]], Cell[CellGroupData[{ Cell["Eliminating some unnecessary warning messages.", "Subsection"], Cell[BoxData[{ \(\(Off[General::"\"];\)\), "\n", \(\(Off[General::"\"];\)\)}], "Input", InitializationCell->True] }, Closed]] }, Closed]], Cell["\<\ This notebook is by Steven Amgott. Please send any questions or comments to \ samgott1@swarthmore.edu. Feel free to use and distribute this notebook, but \ keep this author information in any copy you use or distribute.\ \>", "SmallText"], Cell[TextData[{ "What you are reading is called a ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook. It consists of text, ", StyleBox["Mathematica", FontSlant->"Italic"], " commands, graphics, and other types of cells. A cell is a ", StyleBox["Mathematica", FontSlant->"Italic"], " \"unit,\" bordered by a bracket on the right side of the screen. The \ words \"An Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " for Calculus\" above are in one cell, while the rest of the text you are \ currently reading is in another cell. What you are currently reading is a \ text cell." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Perhaps the most important type of cell is an input cell, such as the one \ below.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(2 + 2\)], "Input", AspectRatioFixed->True], Cell["\<\ You might notice that the brackets at the right bordering the two cells above \ are slightly different. The type of bracket is one way to see what type the \ cell is.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Input cells are your way to tell ", StyleBox["Mathematica", FontSlant->"Italic"], " to do something. If you click anywhere in the input cell containing 2+2 \ above, and press the ", StyleBox["enter", FontColor->RGBColor[0, 0, 1]], " key on the number pad at the lower right side of the keyboard (NOT the ", StyleBox["return", FontColor->RGBColor[0, 0, 1]], " key on a Macintosh or the ", StyleBox["Enter", FontColor->RGBColor[0, 0, 1]], " key next to the single and double quote marks on a Windows machine!), \ something should happen. ", StyleBox["Do so now, answering Yes to the initialization prompt. ", FontColor->RGBColor[1, 0, 1]], "(Note that pressing just the ", StyleBox["return", FontColor->RGBColor[0, 0, 1]], " key on a Macintosh or the ", StyleBox["Enter", FontColor->RGBColor[0, 0, 1]], " key next to the single and double quote marks on a Windows machine only \ produces a line break. You can, however, use the ", StyleBox["Shift", FontColor->RGBColor[0, 0, 1]], " and ", StyleBox["Enter", FontColor->RGBColor[0, 0, 1]], " keys on a Windows machine, or the ", StyleBox["shift", FontColor->RGBColor[0, 0, 1]], " and ", StyleBox["return", FontColor->RGBColor[0, 0, 1]], " keys on a Mac, simultaneously to evaluate a cell. In particular, that is \ how you evaluate a cell on a notebook computer without a number pad.)" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "What (hopefully) happened is that ", StyleBox["Mathematica", FontSlant->"Italic"], " started its Kernel, which is the portion of the program used to perform \ calculations. After a short(?) while, the arithmetic you requested should \ have been done, and the sum of 2 and 2 displayed. ", StyleBox["Mathematica", FontSlant->"Italic"], " often does not start the Kernel until you request a calculation to be \ done. (This is a memory and time saving feature, in case you are only typing \ in text and commands which you are not interested in evaluating immediately, \ but when you evaluate your first cell in a ", StyleBox["Mathematica", FontSlant->"Italic"], " session it then takes extra time to get the Kernel started.)" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "To create a new cell, simply move your cursor to a place just between two \ cells, before the first cell in the notebook, or after the last cell in the \ notebook. The cursor will change from vertical to horizontal. If you click \ the mouse, you will get a horizontal line across the window, marking a new \ cell into which you can type or paste. Once it is created, you can choose \ (or change) the type of a cell using the ", StyleBox["Format..Style", FontColor->RGBColor[0, 0, 1]], " menu item. (By default, ", StyleBox["Mathematica", FontSlant->"Italic"], " creates an input cell, unless you tell it otherwise. If the toolbar is \ open at the top of the notebook, you can also choose the cell type by using \ the drop-down box at the left of the toolbar.) ", StyleBox["Create a text cell just following this cell, and type something \ profound in it.", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[TextData[{ "By the way, if you forgot to choose the cell type for a cell when you \ created it, you can change the cell type by clicking on the cell bracket at \ the right of the cell, and either using the ", StyleBox["Format..Style", FontColor->RGBColor[0, 0, 1]], " menu item or the drop-down box on the toolbar (if it is present at the \ top of the notebook)." }], "Text"], Cell[TextData[{ "Note that there is a color scheme in this notebook. Things in ", StyleBox["magenta", FontColor->RGBColor[1, 0, 1]], " are steps you need to perform or values you may wish to change. Words in \ ", StyleBox["blue", FontColor->RGBColor[0, 0, 1]], " are files, menu items, or keys on the keyboard. Words in ", StyleBox["red", FontColor->RGBColor[1, 0, 0]], " are ", StyleBox["Mathematica", FontSlant->"Italic"], " commands, syntax, or names." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " allows you to \"group\" cells together. This gives a \"table of contents\ \" look to the screen. Groups of cells have their cell brackets enclosed by \ additional brackets. To open up a group of cells, you can double-click the \ bracket which includes a \"filled-in\" triangle at the bottom of the bracket. \ You can re-close the group by double-clicking that same bracket. For \ example, the cell you are currently reading is the first cell of a group of \ two cells. To open the group, ", StyleBox["double-click the bracket at the right of this cell that has the \ \"filled-in\" triangle (the rightmost bracket). ", FontColor->RGBColor[1, 0, 1]], "Then close the group by ", StyleBox["double-clicking the bracket enclosing both cells.", FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["Boo!!!", Evaluatable->False, AspectRatioFixed->True, FontSize->72, FontColor->RGBColor[0, 1, 0]]], "SmallText", Evaluatable->False, AspectRatioFixed->True, FontSize->12] }, Closed]], Cell[TextData[{ "The remainder of this notebook consists of groups of cells you will need \ to open in order to read. Notice that there is also a hierarchy of text \ cell types (Title, Subtitle, Section, Subsection, etc.) that are used in \ organizing notebooks. The cell types are used by ", StyleBox["Mathematica", FontSlant->"Italic"], " if the ", StyleBox["Automatic Grouping", FontColor->RGBColor[0, 0, 1]], " option is chosen in the ", StyleBox["Cell..Cell Grouping", FontColor->RGBColor[0, 0, 1]], " menu item. If the toolbar is open at the top of the notebook, you can \ easily see the type of cell by clicking in the cell, and looking in the \ drop-down box at the left of the toolbar." }], "Text"], Cell[TextData[{ "A continuation of the above introduction is available in the", Cell[BoxData[ FormBox[ ButtonBox[\(\(\ \)\(Introduction_to _Mathematica . nb\)\), ButtonData:>{"Introduction_to_Mathematica.nb", None}, ButtonStyle->"Hyperlink"], TraditionalForm]]], " tutorial in your course folder. If you have the time and interest, take \ a look. If you copied the entire contents of the course directory to the \ computer you are using, you can open this tutorial by clicking on the link \ above in this paragraph. (This will also apply to any links below.)" }], "Text"], Cell[CellGroupData[{ Cell["Imitating a graphing calculator (Arithmetic and plotting)", "Section"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has many of the capabilities of a graphing calculator (except, sadly, the \ tracing function). You can easily do arithmetic (", StyleBox["evaluate each of the following input cells", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(3 + 5\)], "Input"], Cell[BoxData[ \(4 - 9\)], "Input"], Cell[BoxData[ \(6*4\)], "Input"], Cell[BoxData[ \(6\ 4\)], "Input"], Cell[BoxData[ \(6\/4\)], "Input"], Cell[BoxData[ \(6\^4\)], "Input"], Cell[BoxData[ \(2\^1000\)], "Input"], Cell[TextData[{ "Notice that the third and fourth input cells above do the same thing. If \ you leave a space between two numbers, ", StyleBox["Mathematica", FontSlant->"Italic"], " treats it as multiplication. Also, although your graphing calculator \ could do most of these calculations (possibly giving decimal answers) it \ probably could not do the last one!" }], "Text"], Cell[TextData[{ "The fifth of these cells does not give a decimal answer, but instead gives \ an answer in rational number form. There are many different ways we could \ get ", StyleBox["Mathematica", FontSlant->"Italic"], " to give us a decimal answer instead of the rational number. Here are a \ few (", StyleBox["evaluate each of the following input cells", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(6. \/4\)], "Input"], Cell[BoxData[ \(N[6\/4]\)], "Input"], Cell[BoxData[ \(6\/4 // N\)], "Input"], Cell[TextData[{ "Unless you specify otherwise, ", StyleBox["Mathematica", FontSlant->"Italic"], " does exact arithmetic. The same applies to its built-in functions (", StyleBox["evaluate each of the following input cells", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(Sin[\[Pi]\/4]\)], "Input"], Cell[BoxData[ \(N[Sin[\[Pi]\/4]]\)], "Input"], Cell[BoxData[ \(Sin[1]\)], "Input"], Cell[BoxData[ \(N[Sin[1]]\)], "Input"], Cell[BoxData[ \(N[Sin[1], 30]\)], "Input"], Cell[TextData[{ "Notice that since there is no other simplified exact answer to ", StyleBox["Sin[1]", FontColor->RGBColor[1, 0, 0]], ", ", StyleBox["Mathematica", FontSlant->"Italic"], " simply returns what you have entered, unless you ask for an \ approximation. Also the last of the above input cells shows you how to get \ more decimal places in your approximation." }], "Text"], Cell[TextData[{ "Built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " functions and constants start with CAPITAL letters, and use square \ brackets ", StyleBox["[ ]", FontColor->RGBColor[1, 0, 0]], " to enclose the arguments. (This is the only place where you use square \ brackets.) As you can see above, the function sin(x) is denoted in ", StyleBox["Mathematica", FontSlant->"Italic"], " by ", StyleBox["Sin[x]", FontColor->RGBColor[1, 0, 0]], ". The number 3.1415.... is denoted by ", StyleBox["Pi", FontColor->RGBColor[1, 0, 0]], " or by the palette symbol ", StyleBox["\[Pi]", FontColor->RGBColor[1, 0, 0]], " , and 2.71828... by ", StyleBox["E", FontColor->RGBColor[1, 0, 0]], " or the palette symbol ", StyleBox["\[ExponentialE]", FontColor->RGBColor[1, 0, 0]], ". Not writing functions correctly is one of the biggest problems people \ have with using ", StyleBox["Mathematica", FontSlant->"Italic"], ". For a list some of the most common problems, see the file ", ButtonBox["top_ten.nb", ButtonData:>{"top_ten.nb", None}, ButtonStyle->"Hyperlink"], "." }], "Text"], Cell[TextData[{ "Many of the standard mathematical functions you would find on a graphing \ calculator are on the ", StyleBox["Calculus", FontColor->RGBColor[0, 0, 1]], " palette. Check the ", StyleBox["File...Palettes", FontColor->RGBColor[0, 0, 1]], " menu item to see if this palette is on the computer you are using. ", StyleBox["If so, open it and take a look at the buttons.", FontColor->RGBColor[1, 0, 1]], " If not, ask your instructor how you can open this palette on your \ computer." }], "Text"], Cell[TextData[{ "Note that ", StyleBox["Log[", FontColor->RGBColor[1, 0, 0]], StyleBox["\[Placeholder]", FontColor->RGBColor[1, 0, 0]], StyleBox["]", FontColor->RGBColor[1, 0, 0]], " means base ", StyleBox["\[ExponentialE]", FontColor->RGBColor[1, 0, 0]], " logarithm, not base 10. To get base 10 logarithms you use ", StyleBox["Log[10, \[Placeholder]]", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[TextData[{ "Exercise: Find a 20 significant figure decimal approximation to the \ natural log of 2. Also find one to the base 10 logarithm of 2. Create input \ cells below this one to do your work. (You can paste the syntax you need by \ clicking on buttons on the ", StyleBox["Calculus", FontColor->RGBColor[0, 0, 1]], " palette.)" }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[TextData[{ "Of course a graphing calculator also graphs. Here's how you can get ", StyleBox["Mathematica", FontSlant->"Italic"], " to produce a graph of a single-variable function or of several functions \ simultaneously (", StyleBox["evaluate each of the following input cells", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(Plot[Sin[x], {x, \(-10\), 10}]\)], "Input"], Cell[BoxData[ \(Plot[{Sin[x], Cos[x]}, {x, \(-10\), 10}, PlotStyle \[Rule] {RGBColor[1, 0, 0], RGBColor[0, 0, 1]}]\)], "Input"], Cell[TextData[{ "The ", StyleBox["PlotStyle", FontColor->RGBColor[1, 0, 0]], " option allows you to change the way the plot is drawn. In the above \ example it makes the sine graph ", StyleBox["red", FontColor->RGBColor[1, 0, 0]], " and the cosine graph ", StyleBox["blue", FontColor->RGBColor[0, 0, 1]], ". It is one of the many, many different options available for the ", StyleBox["Plot", FontColor->RGBColor[1, 0, 0]], " function. While this makes ", StyleBox["Mathematica", FontSlant->"Italic"], " more complicated to use than a graphing calculator, it provides much more \ flexibility in the form of the graph you create. To see some of the other \ options, evaluate the cell below. Don't panic when you see the list - most \ of the time you do not need to specify any options (", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(?? Plot\)], "Input"], Cell[TextData[{ "This might be a good time to mention the help features available in ", StyleBox["Mathematica", FontSlant->"Italic"], ". In addition to the use of ", StyleBox["??", FontColor->RGBColor[1, 0, 0]], ", there is also the ", StyleBox["Help Browser", FontColor->RGBColor[0, 0, 1]], ", available from the ", StyleBox["Help", FontColor->RGBColor[0, 0, 1]], " menu. You can either open it from the menu and then type in the item for \ which you want help, or you can highlight any function in your ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook and then choose the ", StyleBox["Find Selected Function", FontColor->RGBColor[0, 0, 1]], " item in the ", StyleBox["Help", FontColor->RGBColor[0, 0, 1]], " menu. ", StyleBox["Highlight the word ", FontColor->RGBColor[1, 0, 1]], StyleBox["Plot", FontColor->RGBColor[1, 0, 0]], StyleBox[" in anywhere in this notebook, and use ", FontColor->RGBColor[1, 0, 1]], StyleBox["Find Selected Function", FontColor->RGBColor[0, 0, 1]], StyleBox[" to get more information on the function.", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[TextData[{ "Unfortunately it is not possible to actually trace a plot in ", StyleBox["Mathematica", FontSlant->"Italic"], ". A close thing we can do is to read off the location where you have the \ cursor in a plot. To do this, ", StyleBox["evaluate the cell below and click somewhere in the plot", FontColor->RGBColor[1, 0, 1]], " (to get a box around it). ", StyleBox["Hold down the", FontColor->RGBColor[1, 0, 1]], " ", StyleBox["Ctrl", FontColor->RGBColor[0, 0, 1]], " ", StyleBox["key (Windows) or the", FontColor->RGBColor[1, 0, 1]], " ", StyleBox["Apple", FontColor->RGBColor[0, 0, 1]], StyleBox[" key (Macintosh) to change the cursor from two crossed double \ arrows to a crosshair, and move the cursor around the graph", FontColor->RGBColor[1, 0, 1]], ". The coordinates of the current cursor location are displayed in the \ lower left corner of the notebook window." }], "Text"], Cell[BoxData[ \(Plot[x\^2, {x, \(-2\), 2}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Some things most graphing calculators do not do (Algebra)", "Section"], Cell[TextData[{ "Unlike most graphing calculators, ", StyleBox["Mathematica", FontSlant->"Italic"], " has the ability to work symbolically. This means, for instance, that it \ can do many routines from Algebra and Calculus, as well as other areas of \ mathematics." }], "Text"], Cell[TextData[{ "Mathematica can do arithmetic on polynomials (", StyleBox["evaluate each of the following input cells", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(\((x\^2 - 3\ x + 7)\) + \((4\ x\^3 - 3\ x\^2 + x\ - \ 4)\)\)], "Input"], Cell[BoxData[ \(\((x\^2 - 3\ x + 7)\) \((4\ x\^3 - 3\ x\^2 + x\ - \ 4)\)\)], "Input"], Cell[TextData[{ "Hmm... The last one did not seem to give the product of the two \ polynomials. Actually, it did, but not in a \"simplified\" form. You will \ need to ", StyleBox["Expand", FontColor->RGBColor[1, 0, 0]], " the answer. Assuming the cell directly above this one is the last one \ you evaluated, you should ", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[BoxData[ \(Expand[%]\)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " uses ", StyleBox["%", FontColor->RGBColor[1, 0, 0]], " to refer to the output of the last calculation it performed. More \ generally, ", StyleBox["%n", FontColor->RGBColor[1, 0, 0]], " (or equivalently, ", StyleBox["Out[n]", FontColor->RGBColor[1, 0, 0]], ") can be used to refer to the output that is preceded by ", StyleBox["Out[n]:=", FontColor->RGBColor[1, 0, 0]], ". (Notice that each evaluated input and output line is preceded by \ something of the form ", StyleBox["In[#]:=", FontColor->RGBColor[1, 0, 0]], " and ", StyleBox["Out[#]:=", FontColor->RGBColor[1, 0, 0]], " respectively, where the # indicates the order of evaluation. For \ instance, a cell preceded by ", StyleBox["In[5]:=", FontColor->RGBColor[1, 0, 0]], " was the fifth input cell evaluated by ", StyleBox["Mathematica", FontSlant->"Italic"], " during the current session. If an input cell does not have something of \ the form ", StyleBox["In[#]:=", FontColor->RGBColor[1, 0, 0]], " at the beginning, then it has not been evaluated during the current \ session.)" }], "Text"], Cell[TextData[{ "Of course, dividing one polynomial by another does not usually result in a \ polynomial. Here are two cases, one which does and one which doesn't (", StyleBox["evaluate each of the following input cells", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(Simplify[\(x\^3 + 4\ x\^2 + x - 6\)\/\(x\^2 + x - 2\)]\)], "Input"], Cell[BoxData[ \(Simplify[\(x\^3 + 4\ x\^2 + x - 6\)\/\(x\^2 - 2 x - 3\)]\)], "Input"], Cell[TextData[{ "You should be able to say why by looking at the results of using ", StyleBox["Factor", FontColor->RGBColor[1, 0, 0]], " on the numerators and denominators (", StyleBox["evaluate each of the following input cells", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(Factor[x\^3 + 4\ x\^2 + x - 6]\)], "Input"], Cell[BoxData[ \(Factor[x\^2 + x - 2]\)], "Input"], Cell[BoxData[ \(Factor[x\^2 - 2 x - 3]\)], "Input"], Cell[TextData[{ "Exercise 1: Why did the first ", StyleBox["Simplify", FontColor->RGBColor[1, 0, 0]], " give a polynomial while the second did not?" }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[TextData[{ "By the way, many Algebraic functions are available in the ", StyleBox["AlgebraicManipulation", FontColor->RGBColor[0, 0, 1]], " palette. ", StyleBox["Open it using the ", FontColor->RGBColor[1, 0, 1]], StyleBox["File...Palette", FontColor->RGBColor[0, 0, 1]], StyleBox[" menu item.", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[TextData[{ "The buttons on this palette can be used to perform the operation \"in \ place.\" To see what this means, ", StyleBox["highlight the polynomial in the next cell and then click the ", FontColor->RGBColor[1, 0, 1]], StyleBox["Factor[", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(TraditionalForm\`\[Placeholder]\)], FontColor->RGBColor[1, 0, 0]], StyleBox["]", FontColor->RGBColor[1, 0, 0]], StyleBox[" button on the palette", FontColor->RGBColor[1, 0, 1]], "." }], "Text"], Cell[BoxData[ \(x\^2 + 7\ x\ - 8\)], "Input"], Cell[TextData[{ "This even works on portions of input cells. ", StyleBox["Try to factor the numerator of the next cell, leaving the \ denominator alone.", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[BoxData[ \(\(x\^4 + 2\ x\^3 - 5\ x\^2 - 4\ x + 6\)\/\(x\^2 + x - 2\)\)], "Input"], Cell[TextData[{ "Notice that ", StyleBox["Mathematica", FontSlant->"Italic"], " only ", StyleBox["Factor", FontColor->RGBColor[1, 0, 0]], "s when the numbers that appear are integers (although there are options to \ get around this somewhat). Another way to \"factor\" further is to use the \ ", StyleBox["Solve", FontColor->RGBColor[1, 0, 0]], " command." }], "Text"], Cell[BoxData[ \(Solve[x\^4 + 2\ x\^3 - 5\ x\^2 - 4\ x + 6 \[Equal] 0, x]\)], "Input"], Cell["\<\ Notice the use of the double equal sign == in the equation. This is because \ the equal sign = is used for assigning values to variables and functions (see \ the next section).\ \>", "Text"], Cell["\<\ Exercise 2: Given the output from the above cell, give the complete \ factorization of the polynomial.\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]] }, Closed]], Cell[CellGroupData[{ Cell["Assigning your own functions and variable values", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "We can assign values to variables using ", StyleBox["=", FontColor->RGBColor[1, 0, 0]], " or ", StyleBox[":=", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[{ \(Clear[a]\), "\[IndentingNewLine]", \(a = 4\)}], "Input"], Cell[BoxData[ \(a\)], "Input"], Cell[BoxData[{ \(Clear[b]\), "\[IndentingNewLine]", \(b := 3\)}], "Input"], Cell[BoxData[ \(b\)], "Input"], Cell[TextData[{ "You can use ", StyleBox["Clear", FontColor->RGBColor[1, 0, 0]], " to make sure that there any previous values assigned to a and b are \ erased from memory. It is not always necessary, but it is a good habit to \ form in order to avoid some types of problems that can otherwise happen." }], "Text"], Cell[TextData[{ StyleBox[":=", FontColor->RGBColor[1, 0, 0]], " is called \"delayed execution.\" The right hand side is not immediately \ evaluated by ", StyleBox["Mathematica", FontSlant->"Italic"], ", but is instead evaluated only when you later evaluate a cell containing \ the left hand side. Notice that ", StyleBox["Mathematica", FontSlant->"Italic"], " does not give any output when you evaluate the cell with ", StyleBox[":=", FontColor->RGBColor[1, 0, 0]], " in it. The difference from the first method is a little subtle, but you \ can see the effect using the following examples. ", StyleBox["Random[ ]", FontColor->RGBColor[1, 0, 0]], " is a ", StyleBox["Mathematica", FontSlant->"Italic"], " function choosing a pseudorandom number (", StyleBox["evaluate each of the following input cells", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[{ \(Clear[r1]\), "\[IndentingNewLine]", \(r1 = Random[\ ]\)}], "Input"], Cell[BoxData[ \(r1\)], "Input"], Cell[BoxData[ \(r1\)], "Input"], Cell[BoxData[{ \(Clear[r2]\), "\[IndentingNewLine]", \(r2 := Random[\ ]\)}], "Input"], Cell[BoxData[ \(r2\)], "Input"], Cell[BoxData[ \(r2\)], "Input"], Cell[TextData[{ "As you can see, if we use ", StyleBox["=", FontColor->RGBColor[1, 0, 0]], " there is a final assignment to the variable r1, and every time you input \ r1 you get the same answer. Using ", StyleBox[":=", FontColor->RGBColor[1, 0, 0]], ", ", StyleBox["Mathematica", FontSlant->"Italic"], " waits until you evaluate a cell with r2 in it, and then finds a value. \ You can get different answers each time you evaluate r2." }], "Text"], Cell[TextData[{ "Functions are often defined by mathematical formulas like f(x) = ", Cell[BoxData[ \(TraditionalForm\`x\^2\)]], " sin(", Cell[BoxData[ \(TraditionalForm\`1\/x\)]], "). To input this function into ", StyleBox["Mathematica", FontSlant->"Italic"], " and name it f[x] you can use the syntax in the following input cell (", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], ")." }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell[BoxData[{ \(Clear[f, x]\), "\n", \(f[x_] = x\^2\ Sin[1\/x]\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "Note the underscore after the x on the left. It has to be there. It tells \ ", StyleBox["Mathematica", FontSlant->"Italic"], " that x is only serving as a placeholder when you define the function \ rule, that there is one input to the function, and that you are defining a \ function rather than just a variable. Again, ", StyleBox["Clear[ ]", FontColor->RGBColor[1, 0, 0]], " is used just in case f and x had been given some previous meaning (they \ haven't)." }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell["\<\ Having defined the function f[x], you can now use it just like built-in \ functions.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell[BoxData[ \(f[ .5]\)], "Input", AspectRatioFixed->True], Cell[BoxData[{ \(Clear[t]\), "\n", \(f[t]\)}], "Input", AspectRatioFixed->True], Cell[BoxData[ \(f[x\^2]\)], "Input", AspectRatioFixed->True], Cell["\<\ Exercise 3: What happens if you try to find the value of f[x] at x=0? (Try \ it in an input cell you create just beneath this one.) What do you think the \ output means?\ \>", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[TextData[{ "Actually, you can also use the ", StyleBox[":=", FontColor->RGBColor[1, 0, 0]], " method for defining functions." }], "Text"], Cell[BoxData[{ \(Clear[new, z]\), "\n", \(new[z_] := z\/\(z\^2 + 1\)\)}], "Input"], Cell[BoxData[ \(new[2]\)], "Input"], Cell[BoxData[ \(new[x\^2]\)], "Input"], Cell[TextData[{ "Most user defined functions in ", StyleBox["Mathematica", FontSlant->"Italic"], " should start with a lower case letter to avoid any possible conflict with \ a built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " function. Although there are times when it is important to use ", StyleBox["=", FontColor->RGBColor[1, 0, 0]], " instead of ", StyleBox[":=", FontColor->RGBColor[1, 0, 0]], " and vice-versa, as a rule of thumb most people use ", StyleBox[":=", FontColor->RGBColor[1, 0, 0]], " in defining new functions." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Lists", "Section"], Cell[TextData[{ "In order to tell Mathematica that what you are typing in is a list, you \ enclose it in braces ", StyleBox["{ }", FontColor->RGBColor[1, 0, 0]], ". For example, here is a list of numbers assigned to \"ourlist\" (", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[{ \(Clear[ourlist]\), "\n", \(ourlist = {1, 2, 3, 4, 5}\)}], "Input"], Cell[TextData[{ "You can apply functions to lists the same way you apply functions to \ numbers (", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(Log[{1, 2, 3, 4, 5}]\)], "Input"], Cell[TextData[{ "Notice you can use the name you gave the list as well, provided you have \ evaluated the cell above defining \"ourlist\" (", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(Log[ourlist]\)], "Input"], Cell[TextData[{ "As you can see, Mathematica is doing everything symbolically. If you want \ numerical approximations to the result, we can use ", StyleBox["N[ ]", FontColor->RGBColor[1, 0, 0]], " or ", StyleBox["//N", FontColor->RGBColor[1, 0, 0]], " (", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(N[Log[{1, 2, 3, 4, 5}]]\)], "Input"], Cell[TextData[{ "As before, you can also get numerical approximations by including a \ decimal point as part of the number (", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(Log[{1. , 2. , 3. , 4. , 5. }]\)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will also do arithmetic operations on lists when the lists have the same \ size and shape, or when a constant is being used. For instance, to add 3 to \ everything in our list, provided you have evaluated the cell defining \ \"ourlist\", you can use (", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(3 + ourlist\)], "Input"], Cell[TextData[{ "Here are some other arithmetic operations (", StyleBox["evaluate each of the following input cells", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(5 + {0, 2, 4}\)], "Input"], Cell[BoxData[ \({1, 2, 3} + {4, 5, 6}\)], "Input"], Cell[BoxData[ \(2\ {1, 1, 2, 3, 5}\)], "Input"], Cell[BoxData[ \({1, 2, 3, 4, 5}\ {2, 4, 6, 8, 10}\)], "Input"], Cell[TextData[{ "One way to create lists is to use the ", StyleBox["Table", FontColor->RGBColor[1, 0, 0]], " command. For instance, you might want to study the pattern of \ coefficients in the powers of (x+y). You can do for the first five powers by \ evaluating" }], "Text"], Cell[BoxData[ \(Table[\((x + y)\)\^n, {n, 1, 5}]\)], "Input"], Cell[TextData[{ "There are two problems with this. One is that you need things multiplied \ out. ", StyleBox["Create a new input cell below this text cell, and fix this \ problem. Be sure to evaluate your cell.", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[TextData[{ "The other problem is that it is a bit messy to read the results. This can \ be fixed using the command ", StyleBox["TableForm", FontColor->RGBColor[1, 0, 0]], ". Assuming the most recent cell evaluated is the one in which you fixed \ the first problem, the next cell will fix the second (", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], ")." }], "Text"], Cell[BoxData[ \(TableForm[%]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Additional exercises", "Section"], Cell[CellGroupData[{ Cell["Exercise 4", "Subsection"], Cell[TextData[{ "For a quadratic polynomial with two distinct real roots, ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontSlant->"Plain"], "2"], TraditionalForm]]], " -3 x + 2 for example, it can be shown that the graph of the function \ changes direction exactly midway between the two roots of the polynomial. \ (This trick is sometime used in algebra to find the maximum or minimum of a \ quadratic polynomial.) 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