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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[ 576123, 18523]*) (*NotebookOutlinePosition[ 605321, 19475]*) (* CellTagsIndexPosition[ 602246, 19394]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " for Calculus Students.\n", StyleBox["(With Hyperlinks)", FontSize->16, FontWeight->"Plain"] }], "Title"], Cell["\<\ By Bruce W. Atkinson and David L. Foreman, Department of Mathematics and \ Computer Science, Samford University\ \>", "Author"], Cell[TextData[{ StyleBox["ABSTRACT", FontWeight->"Bold"], ". ", StyleBox["Mathematica", FontSlant->"Italic"], " is a very powerful computer program which allows the student to explore \ Calculus with tools far beyond those available on hand-held graphing \ calculators. However, as with most useful programs, there is a bit of a \ \"learning curve\" involved with getting comfortable to the point of being \ able to use the program without a considerable amount of outside help; this \ help could be in the form of a manual (the 1400+ page ", StyleBox["Mathematica", FontSlant->"Italic"], " Book!), a computerized help file (the Help Browser), a computer lab \ assistant, or an instructor. The purpose of this introduction is to, as \ quickly as possible, point out the features of ", StyleBox["Mathematica", FontSlant->"Italic"], " most useful to a calculus student and, simultaneously, to point out the \ most common \"pitfalls\" that, in the experience of the authors, have \ frustrated students as they have used ", StyleBox["Mathematica", FontSlant->"Italic"], " for various assignments and/or projects in their courses. The authors \ assume that the student reading this has a basic precalculus background; the \ illustrative examples in this introduction make use of this assumption.\n\n\ Hyperlinks. The words underlined in blue are hyperlinks to other locations in \ the tutorial. Simply click on a link to go there. Also, you will find \ scattered throughout the text a link to return you back to the table of \ contents. Also, there are occasional links to the ", ButtonBox["(troubleshooting)", ButtonData:>"Troubleshooting", ButtonStyle->"Hyperlink"], " section and to the ", Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]], "." }], "Abstract"], Cell[CellGroupData[{ Cell[TextData[StyleBox["CONTENTS", FontWeight->"Bold"]], "SectionFirst", FontWeight->"Plain", CellTags->"Contents"], Cell[TextData[{ "1. ", ButtonBox["Getting Started", ButtonData:>"Getting Started", ButtonStyle->"Hyperlink"], "\t\t\t\t\t\t\t", StyleBox["\n\t", FontWeight->"Plain"], ButtonBox["Starting the Program", ButtonData:>"Starting the Program", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\n", FontWeight->"Plain"], "2. ", ButtonBox["Introduction to Notebooks and Cells", ButtonData:>"Introduction to Notebooks and Cells", ButtonStyle->"Hyperlink"], "\t\t\t\t", StyleBox["\n\t", FontWeight->"Plain"], ButtonBox["Notebooks", ButtonData:>"Notebooks", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Saving Notebooks", ButtonData:>"Saving Notebooks", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["A First Calculation", ButtonData:>"A First Calculation", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Front End and Kernel\t", ButtonData:>"Front End and Kernel\t", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Cells", ButtonData:>"Cells\t", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Input and Output Numbers", ButtonData:>"Input and Output Numbers", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Starting a New Cell", ButtonData:>"Starting a New Cell", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Referring to Previous Calculations", ButtonData:>"Referring to Previous Calculations", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Text Cells", ButtonData:>"Text Cells", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Mathematical Formulas and Palettes", ButtonData:>"Mathematical Formulas and Palettes\t", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Graphics Cells\t", ButtonData:>"Graphics Cells\t", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Editing", ButtonData:>"Editing\t", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Printing", ButtonData:>"Printing", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\n", FontWeight->"Plain"], "3.", ButtonBox[" Basic Operations", ButtonData:>"Basic Operations", ButtonStyle->"Hyperlink"], "\t\t\t\t\t\t\t", StyleBox["\n\t", FontWeight->"Plain"], ButtonBox["Addition, Subtraction, Multiplication, and Division\t", ButtonData:>"Addition", ButtonStyle->"Hyperlink"], StyleBox["\t\n\t", FontWeight->"Plain"], ButtonBox["Exponentiation", ButtonData:>"Exponentiation", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Square Roots and Radicals", ButtonData:>"Square Roots and Radicals", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\t\n", FontWeight->"Plain"], "4. ", ButtonBox["Special Numbers\t", ButtonData:>"Special Numbers", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["The Number \[Pi]\t", ButtonData:>"The Number p", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["The Number \[ExponentialE]\t", ButtonData:>"The Number ?", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["The Complex Number \[ImaginaryI]", ButtonData:>"The Complex Number ?", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t\n", FontWeight->"Plain"], "5. ", ButtonBox["Using Variables", ButtonData:>"Using Variables", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Variable Names", ButtonData:>"Variable Names", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Assignment Statements", ButtonData:>"Assignment Statements", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Changing the Value of a Variable", ButtonData:>"Changing the Value of a Variable", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Calculations with Variables", ButtonData:>"Calculations with Variables", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\t\n", FontWeight->"Plain"], "6. ", ButtonBox["Built-In Functions", ButtonData:>"Built-In Functions", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Correct Form\t", ButtonData:>"Correct Form", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Some Examples", ButtonData:>"Some Examples", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t\n", FontWeight->"Plain"], "7.", ButtonBox[" Lists and Tables", ButtonData:>"Lists and Tables", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Lists", ButtonData:>"Lists\t", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Calculations with a List", ButtonData:>"Calculations with a List", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Extracting Elements from a List", ButtonData:>"Extracting Elements from a List", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Tables\t", ButtonData:>"Tables\t", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Table Form", ButtonData:>"Table Form", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["List of Lists", ButtonData:>"List of Lists", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Prepend and Append\t", ButtonData:>"Prepend and Append\t", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\t\n", FontWeight->"Plain"], "8. ", ButtonBox["Algebraic Expressions", ButtonData:>"Algebraic Expressions", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Grouping Symbols", ButtonData:>"Grouping Symbols", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["More Built-in Functions", ButtonData:>"More Built-in Functions", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Replacement Rules", ButtonData:>"Replacement Rules", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t\n", FontWeight->"Plain"], "9. ", ButtonBox["Review of Grouping Symbols", ButtonData:>"Review of Grouping Symbols", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\n", FontWeight->"Plain"], "10. ", ButtonBox["Solving Equations", ButtonData:>"Solving Equations", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Solve and FindRoot", ButtonData:>"Solve and FindRoot", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["An Application", ButtonData:>"An Application", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t\n", FontWeight->"Plain"], "11. ", ButtonBox["Graphing\t", ButtonData:>"Graphing\t", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["The Plot Function", ButtonData:>"The Plot Function", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Graphics Options", ButtonData:>"Graphics Options", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["PlotStyle and Graphics Directives", ButtonData:>"PlotStyle and Graphics Directives", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Multiple Graphs", ButtonData:>"Multiple Graphs", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Graphing a Set of Points", ButtonData:>"Graphing a Set of Points", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\t\n", FontWeight->"Plain"], "12. ", ButtonBox["Using Your Own Functions", ButtonData:>"Using Your Own Functions", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Defining Your Own Functions", ButtonData:>"Defining Your Own Functions", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Graphing Your Own Functions", ButtonData:>"Graphing Your Own Functions", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\t", FontWeight->"Plain"], ButtonBox["Piecewise Defined Functions", ButtonData:>"Piecewise Defined Functions", ButtonStyle->"Hyperlink"], StyleBox["\t\t\t\t\t\n\t\n", FontWeight->"Plain"], "13.", StyleBox[" ", FontWeight->"Plain"], ButtonBox["Troubleshooting", ButtonData:>"Troubleshooting", ButtonStyle->"Hyperlink"], "\t\t\t\t\t\t\t", StyleBox["\n\n", FontWeight->"Plain"], "14.", StyleBox[" ", FontWeight->"Plain"], ButtonBox["Index", ButtonData:>"Index", ButtonStyle->"Hyperlink"], StyleBox["\n\n", FontWeight->"Plain"], ButtonBox["Quick Reference\n", ButtonData:>"Quick reference", ButtonStyle->"Hyperlink"], StyleBox["\n\t\t\t\t\t\t\t\t", FontWeight->"Plain"] }], "Text", FontWeight->"Bold"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "1. Getting Started ", ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "SectionFirst", CellTags->"Getting Started"], Cell[TextData[{ "We will have to assume that you have some basic familiarity with using \ computers and, in particular, Windows. However, we will try to be very \ explicit with instructions even to the point, sometimes, of exactly which \ keystrokes and/or mouse clicks to use. Since this is meant to be a tutorial, \ you should be sitting in front of a computer that has ", StyleBox["Mathematica", FontSlant->"Italic"], " loaded. When you get to a line in this manual which starts with the word \ \"", StyleBox["PRACTICE", FontWeight->"Bold"], "\" then perform the instructions on your computer. Also, there will be an \ occasional line which starts with the word \"", StyleBox["WARNING", FontWeight->"Bold"], "\". Pay close attention to these since they indicate common problems.\n\n\ To see exactly what is described in this tutorial, we recommend that you only \ type things into the program when so instructed in the PRACTICE sections. \ Even if you find specific steps described prior to a PRACTICE section, do not \ do anything until you get to that PRACTICE section. For clarity, each \ PRACTICE section begins with the word, ", StyleBox["PRACTICE #", FontWeight->"Bold"], ", and ends with the words, ", StyleBox["END PRACTICE #", FontWeight->"Bold"], ", where # stands for the number of the particular PRACTICE. Some PRACTICE \ sections will refer to the immediately preceding paragraphs so that you can \ perform steps similar to those described in these paragraphs. ", StyleBox["Thus the material between PRACTICE sections is meant for reading \ only. ", FontWeight->"Bold"], "\n\nTo avoid confusion about exactly what should be typed, we will put in \ a slightly different font what should be typed. For example you might see the \ instruction: type ", StyleBox["Plot[x,{x,0,2}] ", FontFamily->"System"], " Note that the font for the what should be typed is more plain looking; we \ will usually not type a period at the end of a typing instruction unless that \ period is meant to be typed." }], "Text"], Cell[TextData[{ StyleBox["Starting the Program. ", FontWeight->"Bold"], "Of course, the first thing to do is to actually start the ", StyleBox["Mathematica", FontSlant->"Italic"], " program. Follow these simple steps: (just read these steps now, and you \ will use them in ", StyleBox["PRACTICE 1 ", FontWeight->"Bold"], "below)\n1. Click the Start button in the lower left-hand corner of the \ desktop.\n2. Move the pointer up to Programs; you should now see a menu of \ available programs appear automatically.\n3. Move the pointer over to the \ program group entitled ", StyleBox["Mathematica", FontSlant->"Italic"], " 4.0; you should now see a small menu appear.\n4. Move the pointer to the \ menu item, ", StyleBox["Mathematica", FontSlant->"Italic"], " 4.0; this item has the ", StyleBox["Mathematica ", FontSlant->"Italic"], "logo", StyleBox[" ", FontSlant->"Italic"], "(a fancy looking polyhedron) as its icon. Now simply click the icon. ", Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]], "\n\nWhen the program finally opens you should see a window at the top of \ your screen which consists of the Title Bar and the Menu Bar (typical of \ most Windows applications). You should also see a blank ", StyleBox["Mathematica", FontSlant->"Italic"], " window initially entitled Untitled-1, and (depending on the settings on \ your particular computer) a window consisting of a number of buttons; this \ window is called a ", StyleBox["palette", FontSlant->"Italic"], " (more about this later). It will be useful to display a ", StyleBox["Tool Bar", FontWeight->"Bold"], " at the top of the blank window by opening the Format menu and then going \ down to the bottom and clicking Show Tool Bar.\n\n", StyleBox["PRACTICE 1", FontWeight->"Bold"], ". Follow the above steps to start ", StyleBox["Mathematica,", FontSlant->"Italic"], " and to display the Tool Bar at the top of the blank window. ", StyleBox["END PRACTICE 1.", FontWeight->"Bold"], "\n\nYou are ready to use the program. Continue to the next section. ", ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text", CellTags->"Starting the Program"] }, Open ]], Cell[CellGroupData[{ Cell["2. Introduction to Notebooks and Cells", "Section", CellTags->"Introduction to Notebooks and Cells"], Cell[TextData[{ StyleBox["Notebooks. ", FontWeight->"Bold"], "The files created by ", StyleBox["Mathematica", FontSlant->"Italic"], " are called ", StyleBox["notebooks, ", FontWeight->"Bold"], " and always have the extension, .nb. For example, on our computer we have \ stored this particular introduction with the filename, Tutorial.nb. To open a \ ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook from a desired location (e.g., your personal floppy disk in the \ A: drive, the harddrive C:, or the network drive J:) first open the program \ as described above, click on the File menu in the menu bar, click on Open, \ and then, in the usual way as for other Windows programs, find your file \ followed by clicking on Open. (Note: Assuming that this is your first time \ with ", StyleBox["Mathematica", FontSlant->"Italic"], " there are no files for you to open. However, in the future you will have \ created a number of your own notebooks that you will need to open as \ described above.) ", Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Notebooks"], Cell[TextData[{ StyleBox["Saving Notebooks. ", FontWeight->"Bold"], "As pointed out above, when you start ", StyleBox["Mathematica", FontSlant->"Italic"], " without loading a previously saved notebook you get a blank window with \ the default title, Untitled-1. Presuming that you want to use that notebook \ and save it for future reference (e.g., to print and turn in as a homework \ assignment), it would be wise to immediately give your notebook a name and \ save it in a desired location (usually to your personal floppy disk). You can \ do this by clicking on the File menu, clicking on Save As, and, in the usual \ way, going to the desired location followed by typing a name for your \ notebook and clicking Save; you do not have to worry about typing the \ extension .nb since ", StyleBox["Mathematica", FontSlant->"Italic"], " will do so automatically.\n\n", StyleBox["PRACTICE 2. ", FontWeight->"Bold"], "Put your own floppy disk in the A: drive of your computer. Use the above \ procedure to save your notebook on your floppy disk with the your own last \ name. You should now see *.nb in the title bar of the window, where the \ asterisk stands for your last name, whatever it is; when you are eventually \ ready to quit all you have to do is click on the File menu and click Save in \ order to save any changes you have made to your notebook. \n", StyleBox["END PRACTICE 2. ", FontWeight->"Bold"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text", CellTags->"Saving Notebooks"], Cell[TextData[{ StyleBox["A First Calculation. ", FontWeight->"Bold"], "To get started we will now try a very basic calculation using ", StyleBox["Mathematica", FontSlant->"Italic"], ": find ", Cell[BoxData[ \(TraditionalForm\`3\ + \ 4\)]], ". \n\n", StyleBox["PRACTICE 3. ", FontWeight->"Bold"], "To do this move the pointer anywhere inside the window and click. The \ pointer looks like a horizontal line; this means we are ready for input. Now \ simply type ", StyleBox["3+4 ", FontFamily->"System"], " There is a vertical bracket on the right; this is called a ", StyleBox["cell bracket", FontWeight->"Bold"], ", and will be described below. To perform this simple calculation press \ Shift and (while holding Shift down) press Enter; this type of key \ combination will be denoted by Shift+Enter. Here is the result: ", Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"A First Calculation"], Cell[CellGroupData[{ Cell[BoxData[ \(3 + 4\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[ \(7\)], "Output", CellLabel->"Out[1]="] }, Open ]], Cell[TextData[{ StyleBox["END PRACTICE ", FontWeight->"Bold"], " 3.\n\nNOTE.", StyleBox[" Because of the style used in creating this tutorial, what you \ see on your screen will be in a slightly different format than what is \ printed in this tutorial. For example, this tutorial has the input and the \ output enclosed in a box for display purposes, but your screen will not have \ such a box. Hopefully, this variation in style will not cause any confusion. \ Also, for clarity in this tutorial, you will see the inputs in bold face and \ the outputs in regular face.\n\nNow you are ready to try something on your \ own.\n", FontWeight->"Plain"], "\n", StyleBox["PRACTICE 4.", FontWeight->"Bold"], StyleBox[" As described above, do the calculation: 4 + 5. ", FontWeight->"Plain"], StyleBox["END PRACTICE 4.", FontWeight->"Bold"], StyleBox["\n\nSeveral things have happened that now require some \ explanation. ", FontWeight->"Plain"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text"], Cell[TextData[{ StyleBox["(i) Front End and Kernel. ", FontWeight->"Bold"], StyleBox["You will notice that there are two ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" items on the taskbar at the bottom of your screen. One has the ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" icon and the word ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" followed by your notebook name, *.nb (partially cut off). The \ other has the icon encircled followed by the words ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" Kernel. This indicates that there are actually two programs \ running. The first item mentioned above is the ", FontWeight->"Plain"], "Front End", StyleBox[". The Front End is, among other things, a word processor; \ whenever you are typing in a ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" window you are using the Front End program. The second item \ mentioned above is the ", FontWeight->"Plain"], "Kernel", StyleBox[". The Kernel is the actual ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" program; this is where all the computation takes place. The \ Front End is automatically linked to the Kernel and it is the means by which \ the user communicates with the Kernel. For example, in the above simple \ calculation you typed 4+5 using the Front End. When you pressed Shift+Enter \ you communicated that calculation to the Kernel, and the Kernel returned the \ answer, 9, to the Front End. ", FontWeight->"Plain"] }], "Text", CellTags->"Front End and Kernel\t"], Cell[TextData[{ StyleBox["(ii) Cells.", FontWeight->"Bold"], " ", StyleBox["Before you pressed Shift+Enter for the first time you noticed the \ vertical bracket at the right of the window. The bracket is indicating a ", FontWeight->"Plain"], "cell", StyleBox[". The Front End produces a series of cells of different types, \ all of which have special cell brackets. In your example, the cell containing \ 4+5 is an ", FontWeight->"Plain"], "input cell", StyleBox[". All input cells have brackets with a little open triangle at \ the top. \n\n", FontWeight->"Plain"], StyleBox["PRACTICE 5. ", FontWeight->"Bold"], StyleBox["To see that the cell containing 4+5 really is an input cell, \ place you pointer anywhere in the cell and click. You will notice that the \ word Input is in the small window at the left edge of the Tool Bar; also note \ that the pointer has turned into a vertical bar which means it is possible to \ begin typing there. Now click on the cell containing the result of 9. You \ should now see the word Output in the small window at the left edge of the \ Tool Bar indicating that this is an ", FontWeight->"Plain"], "output cell", StyleBox[". Also, notice the special bracket for output cells. Both of \ these cells have only one line but it is typical to have cells with many \ lines; the cell brackets simply \"grow\" with the size of the cells, as you \ shall see. There are many other types of cells. To see a list of them, click \ on the drop-down menu button with the little down-arrow at the left of the \ Tool Bar. It is possible to scroll through the list, and, by clicking on a \ particular entry, change the current cell to another type. ", FontWeight->"Plain"], StyleBox["END PRACTICE 5. ", FontWeight->"Bold"], StyleBox["\n\nIn addition to input and output cells we shall also be \ interested in ", FontWeight->"Plain"], "text cells", StyleBox[" and ", FontWeight->"Plain"], "graphics cells", StyleBox["; these will be demonstrated below. Unless you change the cell \ type, ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" always assumes that you want an input cell; i.e., the input cell \ is the default cell type when starting a new cell. ", FontWeight->"Plain"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Cells\t"], Cell[TextData[{ StyleBox["(iii) Input and Output Numbers.", FontWeight->"Bold"], " ", StyleBox["When you pressed Shift+Enter the first time you ", FontWeight->"Plain"], "evaluated", StyleBox[" an input cell. Immediately at the left of the cell you saw an ", FontWeight->"Plain"], "input number", StyleBox[" that looked like this: In[1]:= . During a given session with ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" the program will keep track of all your inputs by labeling them \ chronologically with input numbers. These numbers will be valid for the \ duration of your session. However, when you quit the program, and then reload \ your notebook later all of your input cells will have no input numbers until \ you evaluate them again. ", FontWeight->"Plain"], "(Note: When opening a previously saved notebook, it is a good idea to \ evaluate each input cell in the original order to reproduce what you had done \ previously and to be able to continue by referring, if necessary, to previous \ inputs or outputs. You can evaluate an input cell by simply placing the \ pointer anywhere in the cell, clicking once, and then pressing Shift+Enter. ) \ ", StyleBox["You also saw at the left of the first output cell an ", FontWeight->"Plain"], "output number", StyleBox[", that looked like this: Out[1]=. These output numbers are very \ useful when you want to refer to the result of some previous calculation. \ Note that an input cell and corresponding output cell are grouped together \ with a ", FontWeight->"Plain"], "grouping bracket", StyleBox[". You can open and close groups of cells by double clicking on \ the outside grouping bracket. \n\n", FontWeight->"Plain"], StyleBox["PRACTICE 6.", FontWeight->"Bold"], StyleBox[" Place your pointer on the grouping bracket which groups your \ first input and output cells; notice how the pointer changes shape. Click \ once. You should notice that the bracket is \"highlighted\" in black. This is \ called ", FontWeight->"Plain"], "selecting", StyleBox[" the cell group; you can also select either the input or the \ output cell individually in a similar fashion. Now double click the grouping \ bracket. You should notice that the bracket now has changed to a ", FontWeight->"Plain"], "closed group bracket", StyleBox[", and has a solid triangle at the bottom. Also, only the first \ line of the first cell in the group shows. To open the group, double click on \ the closed group bracket. ", FontWeight->"Plain"], StyleBox["END PRACTICE 6. ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Input and Output Numbers"], Cell[TextData[{ StyleBox["Starting a New Cell", FontWeight->"Bold"], ". ", StyleBox["There are two ways to begin a new cell : 1. Press the down or up \ arrow key repeatedly until a horizontal line appears in the window indicating \ that when you type there will be a new cell appearing with its corresponding \ bracket ( when you have a number of cells, you can \"scroll\" by pressing \ repeatedly either the up or down arrow key); 2. Place the pointer where you \ want to start a new cell and click. The second method is more versatile since \ you can quickly insert a cell in between two existing cells; ", FontWeight->"Plain"], "you know you are ready to start a new cell when the pointer looks like a \ horizontal line. ", ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"], StyleBox["\n\n", FontWeight->"Plain"], StyleBox["PRACTICE 7.", FontWeight->"Bold"], " ", StyleBox["Click anywhere in the bottom-most output cell and then press the \ down arrow key. You will now see a horizontal line. This will automatically \ be an input cell because the default cell type is the input cell. Now type \ ", FontWeight->"Plain"], StyleBox["12 * 75 ", FontFamily->"System", FontWeight->"Plain"], StyleBox[" You will notice a new input cell (with bracket) has been \ started. Press Shift+Enter, and you will see a new output cell, new input and \ output numbers, and a new grouping bracket. ", FontWeight->"Plain"], StyleBox["END PRACTICE 7. ", FontWeight->"Bold"], StyleBox["\n\n", FontWeight->"Plain"], StyleBox["PRACTICE 8. ", FontWeight->"Bold"], StyleBox["You will now use the second method for starting a new cell. Find \ an output cell that is directly above an input cell. Place your pointer \ between these cells, making sure the pointer is horizontal. Now click to \ create a horizontal line indicating readiness for a new input cell. Now type \ ", FontWeight->"Plain"], StyleBox["22.5/7", FontFamily->"System", FontWeight->"Plain"], StyleBox[" followed by Shift+Enter. You now see a new input cell, a new \ output cell, new input and output numbers, and a new grouping bracket. ", FontWeight->"Plain"], StyleBox["END PRACTICE 8. ", FontWeight->"Bold"] }], "Text", CellTags->"Starting a New Cell"], Cell[TextData[{ StyleBox["WARNING. ", FontWeight->"Bold"], StyleBox["Notice that the numbers of the input and output cells are ", FontWeight->"Plain"], "not", StyleBox[" in order from top to bottom. This is because the input and \ output numbers indicate the chronological order in which you evaluated input \ cells. Usually you will have the top-to-bottom order the same as the \ chronological order, but, for editing purposes, it is occasionally useful to \ insert new input cells between existing cells. ", FontWeight->"Plain"], ButtonBox["(back to troubleshooting)", ButtonData:>"Troubleshooting", ButtonStyle->"Hyperlink"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Cell Order"], Cell[TextData[{ StyleBox["Referring to Previous Calculations.", FontWeight->"Bold"], " ", StyleBox["You can refer to the result of a previous calculation in two \ different ways: 1. % stands for the result of the preceding (in the \ chronological sense) calculation (similar to ANS on many calculators); 2. You \ can make reference to a specific output number.\n\n", FontWeight->"Plain"], StyleBox["PRACTICE 9.", FontWeight->"Bold"], " ", StyleBox[" Move the pointer just below the bottom-most output cell and \ click to make way for a new cell. Type ", FontWeight->"Plain"], StyleBox["% + 26", FontFamily->"System", FontWeight->"Plain"], StyleBox[" and evaluate. You will see the result of your previous output \ added to 26, which is not the same as the output directly above added to 26. \ (Note that when a calculation is completed, ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" automatically creates a horizontal line immediately below, \ assuming that that is where you want to start your next input cell; you can \ accept this situation or, as described above, start a new cell elsewhere.) To \ subtract the result of Out[1] from 100 type ", FontWeight->"Plain"], StyleBox["100 - Out[1] ", FontFamily->"System", FontWeight->"Plain"], StyleBox[" and then evaluate. ", FontWeight->"Plain"], StyleBox["END PRACTICE 9.\n\nNOTE. ", FontWeight->"Bold"], StyleBox[" For some reason you may have deleted the first output cell. Even \ in this case the program still remembers what it was and can use it in \ further evaluations.", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Bold"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"], StyleBox["\n", FontWeight->"Plain"] }], "Text", CellTags->"Referring to Previous Calculations"], Cell[TextData[{ StyleBox["Text Cells.", FontWeight->"Bold"], StyleBox[" As mentioned above the Front End is, among other things, a word \ processor. ", FontWeight->"Plain"], "Text cells ", StyleBox["are particularly suited for this purpose. Within a text cell you \ can do most things available in a word processor with the added advantage of \ being able to type mathematical formulas from the key board in the same \ format that you read them in your mathematics textbooks. There are several \ ways to start a text cell. Here is one that is fairly straightforward. All \ you have to do is to start a new cell in the manner described above, go to \ the Tool Bar and click on the drop-down menu button at the left (the one with \ the little downward pointing triangle), and then click on Text. When you \ start typing a new text cell will be created. ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]], StyleBox["\n\n", FontWeight->"Plain"], StyleBox["PRACTICE 10. ", FontWeight->"Bold"], StyleBox[" Start a new cell at the bottom of all the cells you have so far. \ Follow the above procedure to make it a text cell. Then type ", FontWeight->"Plain"], StyleBox["This is an example of a text cell. ", FontFamily->"System", FontWeight->"Plain"], StyleBox[" Note the special cell bracket for a text cell. You have many \ of the standard word processing options within a text cell. To see some of \ them click on the Format menu in the Menu Bar. Experiment with a few of the \ options. ", FontWeight->"Plain"], StyleBox["END PRACTICE 10. ", FontWeight->"Bold"] }], "Text", CellTags->"Text Cells"], Cell[TextData[{ "Mathematical Formulas and Palettes.", StyleBox[" Here is an example of a mathematical formula entered with \ \"two-dimensional\" input:\n", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(f(x, y) = \(x\^2 + y\^2\)\/\(sin\ \((x\ y)\)\)\), FontSize->12], TraditionalForm]], FontSize->16], ". \n", StyleBox["There are several ways to enter mathematical formulas. We will \ explain here the method using the Basic Input Palette. The default settings \ for the program should have made it so that the Basic Input Palette opens \ when you load the program; if you have it, it looks like a small vertical \ window with an array of buttons. (If you do not have it, simply click on the \ File menu, move your pointer to Palettes, then click on BasicInput.) Here are \ the steps used for the sample formula above. ", FontWeight->"Plain"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text", FontWeight->"Bold", CellTags->"Mathematical Formulas and Palettes\t"], Cell[TextData[{ StyleBox["PRACTICE 11. \n", FontWeight->"Bold"], StyleBox["1. Start a new text cell, as described above. In order to type a \ formula in a text cell you have to already have typed in something. To \ accomplish this for our purpose here, type the letter ", FontWeight->"Plain"], StyleBox["a", FontFamily->"System", FontWeight->"Plain"], StyleBox[" and then press Backspace. You should see a blinking vertical \ bar.\n2. Press Control+9 . This creates a highlighted rectangle with a small \ box in the middle. This box is a ", FontWeight->"Plain"], "field", StyleBox[", i.e. an area ready for input. Until you are done with the \ formula, it will be highlighted.\n3. Type ", FontWeight->"Plain"], StyleBox["f(x,y) =", FontFamily->"System", FontWeight->"Plain"], StyleBox[" Note that in the formula the letters are automatically \ italicized, which is typical of mathematical notation in textbooks. \n4. At \ this point we need to start a fraction. Go to the palette and click the top \ right-hand button which looks like a fraction with boxes in the numerator and \ the denominator. You will now notice a fraction with two fields; the darker \ (highlighted) field is the one ready for input. You can toggle between the \ fields by pressing the Tab key. In our example we want the sum of two powers \ in the numerator. To begin the first power press the upper left palette \ button. You will now see a power expression with a box for the base and a box \ for the exponent. Type ", FontWeight->"Plain"], StyleBox["x", FontFamily->"System", FontWeight->"Plain"], StyleBox[", press Tab, type ", FontWeight->"Plain"], StyleBox["2 ", FontFamily->"System", FontWeight->"Plain"], StyleBox[" To start the next power we must move back down to the line with \ the first base. To do this press Control+Space. Type + Then enter ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`y\^2\)]], StyleBox[" in the same fashion. We now have the desired numerator.\n", FontWeight->"Plain"], ButtonBox["(back to troubleshooting)", ButtonData:>"Troubleshooting", ButtonStyle->"Hyperlink"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Formulas"], Cell[TextData[{ StyleBox["WARNING", FontWeight->"Bold"], ".", StyleBox[" At some computers the default settings for the program have \ somehow been changed so as not to allow the above \"two-dimensional\" input \ from palettes. If this is the case you will get something pretty \ strange-looking when you press one of the palette buttons; definitely not \ what is expected. The fix for this is as follows: Quit ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" altogether. Start it again, but hold down Control+Shift \ (simultaneously) while it is loading. This should restore all the default \ settings. To test, try the following: press the upper left palette button, \ type: \"2\", press TAB, type \"3\", then press Shift+Enter. If all is well \ you should see 8 as the result. (Note: From now on, when you start the \ program you might want to try this simple test at the beginning to make sure \ the palette buttons are enabled.) ", FontWeight->"Plain"], ButtonBox["(back to troubleshooting)", ButtonData:>"Troubleshooting", ButtonStyle->"Hyperlink"], StyleBox["\n\n5. At this point in typing the formula we see just one field \ (a box) in the denominator. To move to that box press Tab; you can also move \ to fields by clicking the pointer in them. Type ", FontWeight->"Plain"], StyleBox["sin(x y)", FontFamily->"System", FontWeight->"Plain"], StyleBox[" Notice how the numerator and denominator are sized to each \ other and centered for a neat looking display. Of course when you are using a \ text cell, you may type anything you like just like any word processor, and \ thus in text cells you need not be careful about proper syntax (i.e. \ capitals, square brackets, etc., which are described later). In text cells it \ makes more sense to use standard notation you are used to reading in \ textbooks. Also, note that when using grouping symbols, such as parentheses, \ the left symbol is in a different color until it is closed by a right symbol; \ this should cut down on grouping symbol errors, one of the more common types \ of syntax errors.\n6. To finish the formula and make it possible to type \ regular text, press Control+0. The formula is now no longer highlighted.", FontWeight->"Plain"], "\n", StyleBox["\nThe top two rows of the palette allow you to use powers, \ fractions, and roots as you build formulas. The bottom row allows you to use \ subscripts, underwriting, overwriting, overbars, and overcarets. As well, you \ can type some standard mathematical symbols and a few greek letters. You \ should open some of the other palettes to see what else is available. ", FontWeight->"Plain"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"], StyleBox["\n\n", FontWeight->"Plain"], StyleBox["END PRACTICE 11", FontWeight->"Bold"], ".", StyleBox["\n\nAt this point ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" may seem a bit overwhelming. You may be saying to yourself, \ \"There is no way I can remember all these things!\" Fortunately, the entire \ manual for the program is online. To get to it you could click on Help on \ the Menu Bar, and then click on Help. For example, you can find out keyboard \ shortcuts for the palette buttons by searching the topic \"two-dimensional \ input\" in the master index. This will direct your attention to a particular \ section of the ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" Book. You should not be afraid to use the Help Browser to find \ information relating to anything in this tutorial. ", FontWeight->"Plain"], "It is recommended that you have this tutorial handy each time you use ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox[" ", FontWeight->"Plain"], "since most of the homework problems you do will make use of what is \ described here.", StyleBox["\n\n", FontWeight->"Plain"], StyleBox["PRACTICE 12", FontWeight->"Bold"], ".", StyleBox[" Start a new text cell and, using procedures similar to the above \ example, type the following mathematical sentence. (Note: As with most word \ processors, you can use Control+B to toggle on or off boldface and Control+I \ to toggle on or off italics. Remember the following useful control sequences: \ Control+9 begins a formula in a text cell with previous content, \ Control+Space moves the cursor to the preceding text line, and Control+0 \ ends a formula in a text cell. Also, you can put a formula in a text cell \ only after something has been typed in previously.)\n\nThe distance, ", FontWeight->"Plain"], StyleBox["d", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", between points ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\((x\_1, y\_1)\), FontWeight->"Plain"], TraditionalForm]]], " ", StyleBox["and ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\((x\_2, y\_2)\), FontWeight->"Plain"], TraditionalForm]]], " ", StyleBox["is given by ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(d = \@\(\((x\_1 - x\_2)\)\^2 + \((y\_1 - y\_2)\)\^2\)\), FontWeight->"Plain"], TraditionalForm]]], ".\n\n", StyleBox["Try a few other formulas of your own just to get comfortable with \ it.", FontWeight->"Plain"], StyleBox[" END PRACTICE 12.", FontWeight->"Bold"], "\n\nNOTE.", StyleBox[" When typing formulas in input cells it is not necessary to begin \ and end with Control+9 and Control+0. Use this idea in the next PRACTICE. 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", FontWeight->"Plain"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]], StyleBox["\n\n", FontWeight->"Plain"], StyleBox["PRACTICE 13.", FontWeight->"Bold"], " ", StyleBox["We will show the graph of ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`y\ = \ \(\(x\^2\)\(.\)\)\)]], " ", StyleBox["Start a new input cell and type ", FontWeight->"Plain"], StyleBox["Plot[", FontFamily->"System", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["x", FontFamily->"System"], "2"], TraditionalForm]], FontFamily->"System"], StyleBox[",", FontFamily->"System"], StyleBox["{x,-3,3}", FontFamily->"System", FontWeight->"Plain"], StyleBox["] ", FontWeight->"Plain"], " ", StyleBox["followed by Shift+Enter", FontWeight->"Plain"], ". ", StyleBox["(There will be more about the Plot function later.) 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13.", FontWeight->"Bold"], " \n\nYou will notice that three different cells are grouped together. The \ first and the third are the, by now, familiar input and output cells. The \ middle cell, which contains the actual graph, is a graphics cell. In the \ output cell you see -Graphics-. This indicates all the data that ", StyleBox["Mathematica", FontSlant->"Italic"], " actually uses to produce the graph, and there is a way to actually use \ this output cell to see all this data. However, we will rarely have occasion \ to look at this data. Thus, from now on we will do graphics by typing a \ semicolon at the very end of the corresponding input; a semicolon at the end \ of a command to ", StyleBox["Mathematica", FontSlant->"Italic"], " suppresses the output which means, in the case of graphics, that the \ output cell is suppressed but ", StyleBox["not", FontWeight->"Bold"], " the graphic itself.\n\n", StyleBox["IMPORTANT NOTE ABOUT PRACTICE SECTIONS.", FontWeight->"Bold"], " From now on you will see some practice sections which simply consist of \ an input and corresponding output or graphics. For example, the next PRACTICE \ section is like this. For these it is expected that on your computer you will \ copy exactly the given input and then evaluate. When you do you should see \ the exact same output or graphics as in this tutorial. Also, the PRACTICE \ sections with no specific input are designed so that you must decide on your \ own input for new examples. 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Also, the cell bracket for a graphics cell \ looks the same as that for a text cell.\n", StyleBox["END PRACTICE 14.", FontWeight->"Bold"] }], "Text", PageBreakAbove->False], Cell[TextData[{ StyleBox["PRACTICE 15. ", FontWeight->"Bold"], "Use the above procedure to graph ", Cell[BoxData[ \(TraditionalForm\`y\ = \ 2\^x\)]], " by typing ", Cell[BoxData[ FormBox[ StyleBox[\(\(Plot\)\([\)\(2\^x\)\), FontFamily->"System"], TraditionalForm]]], ",", StyleBox["{x,-3,3}] ", FontFamily->"System"], " ", Cell[BoxData[ \(TraditionalForm\`\ \)]], StyleBox[" ", FontWeight->"Bold"], " in the input cell.\n", StyleBox["END PRACTICE 15. ", FontWeight->"Bold"], "\n\n" }], "Text"], Cell[TextData[{ StyleBox["WARNING.", FontWeight->"Bold"], " Now that we have discussed input, output, text, and graphics cells, we \ should point out that a common mistake is to use the wrong type of cell for a \ given purpose. There are two common problems.\n1. Don't use an input cell as \ a text cell. If you do this it will look pretty strange.\n2. Don't try to \ evaluate a text cell. If you do the computer will beep at you because it does \ not understand what you are trying to do.\nThe remedy for using the wrong \ type of cell is to simply change its type by selecting the cell and using the \ drop-down menu available at the left of the Tool Bar. ", ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"], "\n", ButtonBox["(back to troubleshooting)", ButtonData:>"Troubleshooting", ButtonStyle->"Hyperlink"] }], "Text", CellTags->"Cell"], Cell[TextData[{ StyleBox["Editing.", FontWeight->"Bold"], " Just as with a word processor it will sometimes be necessary to edit your \ work in ", StyleBox["Mathematica", FontSlant->"Italic"], ". Within a single text cell you can edit in the same way as you do with a \ word processor; i.e., select text and use the Edit menu from the Menu Bar. It \ is also possible to cut, copy, or paste an entire cell. To do this select the \ cell's bracket by clicking once on the bracket, and then use the Edit menu.\n\ \n", StyleBox["PRACTICE 16.", FontWeight->"Bold"], " Use the above procedure to make a copy of one of your graphics cells and \ paste it between two cells of your choosing higher up in your notebook. ", StyleBox["END PRACTICE 16. ", FontWeight->"Bold"], "\n" }], "Text", CellTags->"Editing\t"], Cell[TextData[{ StyleBox["WARNING.", FontWeight->"Bold"], " There are the usual keyboard shortcuts for editing; e.g. use Control+C \ for copying. Note that Control+P is for printing and ", StyleBox["not", FontWeight->"Bold"], " pasting. ", ButtonBox["(back to troubleshooting)", ButtonData:>"Troubleshooting", ButtonStyle->"Hyperlink"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Pasting"], Cell[TextData[{ StyleBox["Printing", FontWeight->"Bold"], ". It will be necessary to print your work sometimes in order to hand in an \ assignment to your professor. To print the entire notebook simply use \ Control+P. If, however, you want to print one or more cells without printing \ the entire notebook you must first select the cells, and then choose Print \ Selection from the File menu. (Note: To select a series of cells for \ printing, go to the first cell you want and click its bracket; then hold down \ the shift key and click the last cell you want. This will cause all cells \ from your start to your finish to be simultaneously selected.)\n\n", StyleBox["PRACTICE 17. ", FontWeight->"Bold"], " Print one of your graphics cells. (Note: As with most applications it is \ possible to use different print options such as portrait or landscape \ printing. To use these options go to Printing Settings in the File menu.) \ Also, select any series of two or more cells and print them. ", StyleBox["END PRACTICE 17. ", FontWeight->"Bold"], " ", ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Printing"] }, Open ]], Cell[CellGroupData[{ Cell["3. Basic Operations", "Section", CellTags->"Basic Operations"], Cell[TextData[{ StyleBox["Addition, Subtraction, Multiplication, and Division. ", FontWeight->"Bold"], "In communicating with the Kernel, the symbols ", StyleBox["Mathematica", FontSlant->"Italic"], " uses for the operations of addition, subtraction, multiplication , and \ division are the usual symbols that you, no doubt, are accustomed to:\n\t+ \ for addition\n\t- for subtraction\n\t* for multiplication\n\t/ for \ division\n\t\nHowever, there are two alternate ways of denoting \ multiplication with ", StyleBox["Mathematica", FontSlant->"Italic"], ":\n1. You may use parentheses to denote multiplication. ", StyleBox["Mathematica", FontSlant->"Italic"], " will interpret x(y) as x times y. \n2. You may use the space bar to \ denote multiplication. ", StyleBox["Mathematica", FontSlant->"Italic"], " will interpret x y (note the space between x and y) as x times y.\n\n ", ButtonBox["(back to contents) ", ButtonData:>"Contents", ButtonStyle->"Hyperlink"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->{"Addition", "Subtraction", "Multiplication", "and Division"}], Cell[TextData[{ StyleBox["WARNING. ", FontWeight->"Bold"], "We will learn later in this manual that in ", StyleBox["Mathematica ", FontSlant->"Italic"], "a variable can be denoted by more than one letter. Thus ", StyleBox["Mathematica", FontSlant->"Italic"], " will interpret xy (with no space between x and y) as ", StyleBox["ONE", FontVariations->{"Underline"->True}], " variable and not as x times y. To denote x times y, you must type x y \ (with the space between x and y). ", StyleBox["Thus it is highly recommended that any time multiplication \ occurs, a space be used between the two factors. ", FontWeight->"Bold"], ButtonBox["(back to troubleshooting) ", ButtonData:>"Troubleshooting", ButtonStyle->"Hyperlink"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]], "\n\nDivision of x by y can also be denoted by the fraction ", Cell[BoxData[ \(TraditionalForm\`x\/y\)]], ". \n\n", StyleBox["PRACTICE 18. ", FontWeight->"Bold"], " Start a new input cell. ", StyleBox[" ", FontWeight->"Bold"], "Go to the Basic Input palette and click the top right-hand button which \ looks like a fraction with boxes in the numerator and denominator. Type ", StyleBox["x ", FontFamily->"System"], " in the numerator first; toggle to the denominator by pressing the Tab \ key; type ", StyleBox["y", FontFamily->"System"], " in the denominator. When finished, hold down Control+Space.\n", StyleBox["END PRACTICE 18. ", FontWeight->"Bold"], "\n\n", StyleBox["PRACTICE 19.", FontWeight->"Bold"], " Put the following in a new input cell: ", Cell[BoxData[ \(TraditionalForm\`\(3\ x - 7\)\/\(a\ b + 5\)\)]], ". ", StyleBox["END PRACTICE 19. ", FontWeight->"Bold"] }], "Text", CellTags->"Variables"], Cell[TextData[{ StyleBox["Exponentiation. ", FontWeight->"Bold"], "In communicating with the Kernel, there are several methods to denote \ exponentiation, or powers, in ", StyleBox["Mathematica", FontSlant->"Italic"], ". First of all, you may use the ^ symbol. When ", StyleBox["Mathematica", FontSlant->"Italic"], " reads x^y, it interprets it as x raised to the power y. ", Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]], "\n\nOf course, you are not accustomed to reading in printed texts (or even \ writing yourself) \nx^y to denote x raised to the power y. Instead you \ usually see the expression ", Cell[BoxData[ \(TraditionalForm\`x\^y\)]], ". \n\n", StyleBox["PRACTICE 20. ", FontWeight->"Bold"], " Start a new input cell. ", StyleBox[" ", FontWeight->"Bold"], "To type ", Cell[BoxData[ \(TraditionalForm\`x\^y\)]], " go to the Basic Input palette and click the top left-hand button which \ looks like an exponential expression with boxes in the base and in the \ exponent. Remember that each of these boxes is called a field. Type in the \ base ", StyleBox["x", FontFamily->"System"], " of the expression first. Then toggle to the exponent field by pressing \ the Tab key. Now type in the exponent ", StyleBox["y", FontFamily->"System"], " of the expression. When you have finished typing the expression, hold \ down Control+Space. Remember that in typing the expression ", Cell[BoxData[ \(TraditionalForm\`x\^y\)]], " within a text cell that already has some content, you must press \ Control+9 before using the palette to type ", Cell[BoxData[ \(TraditionalForm\`x\^y\)]], " and finally press Control+0. The Control+9 at the first and the \ Control+0 at the end are not necessary within an Input cell. \n", StyleBox["END PRACTICE 20. ", FontWeight->"Bold"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text", CellTags->"Exponentiation"], Cell[TextData[{ StyleBox["Square Roots and Radicals.", FontWeight->"Bold"], " \n\n", StyleBox["PRACTICE 21. ", FontWeight->"Bold"], "To denote a square root you will use the Basic Input palette. Start a \ new input cell. Click on the square root button. Then type in the expression \ under the square root (called the radicand). When finished, hold down \ Control+Space. Try this for the expression ", Cell[BoxData[ \(TraditionalForm\`\@\(sin\ x\)\)]], ". For radicals of other indices, such as cube roots, click on the \ right-hand button on the second row. First type in the radicand; toggle to \ the index of the radical by using the Tab key; type in the index. When \ finished, hold down Control+Space. Try this for the expression ", Cell[BoxData[ \(TraditionalForm\`\@x\%3\)]], " in a new input cell. ", StyleBox["END PRACTICE 21.", FontWeight->"Bold"], "\n\n", StyleBox["PRACTICE 22.", FontWeight->"Bold"], " Put the following in a new input cell: ", Cell[BoxData[ \(TraditionalForm\`\@\(3 - \@\(x\ y\)\%3\)\)]], " . ", StyleBox["END PRACTICE 22. ", FontWeight->"Bold"] }], "Text", CellTags->"Square Roots and Radicals"] }, Open ]], Cell[CellGroupData[{ Cell["4. Special Numbers", "Section", CellTags->"Special Numbers"], Cell[TextData[{ StyleBox["The number ", FontWeight->"Bold"], StyleBox["\[Pi]", FontSize->14, FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " The number \[Pi] can be denoted in two ways. First of all, ", StyleBox["Mathematica", FontSlant->"Italic"], " will interpret \"Pi\" as this number. Notice that the \"P\" must be \ capitalized! Also \[Pi] can be retrieved off the Basic Input palette be \ clicking on the \[Pi] button." }], "Text", CellTags->"The Number p"], Cell[TextData[{ StyleBox["The number ", FontWeight->"Bold"], StyleBox["\[ExponentialE]", FontSize->14, FontWeight->"Bold"], StyleBox[". ", FontWeight->"Bold"], "The number \[ExponentialE] can also be retrieved from the Basic Input \ palette by clicking on the \[ExponentialE] button. Or ", StyleBox["Mathematica", FontSlant->"Italic"], " will recognize the capital letter \"E\" as this number. ", Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"The Number ?"], Cell[TextData[{ StyleBox["WARNING. ", FontWeight->"Bold"], "The exponential expression that you are accustomed to writing as ", Cell[BoxData[ \(TraditionalForm\`\(\(e\^x\)\(,\)\)\)]], " in ", StyleBox["Mathematica", FontSlant->"Italic"], ", must be written as either ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)]], " (retrieveing \[ExponentialE] from the palette) or as ", Cell[BoxData[ \(TraditionalForm\`E\^x\)]], ". Notice that \[ExponentialE] retrieved from the palette appears a bit \ differently than the keyboard e. ", ButtonBox["(back to troubleshooting)", ButtonData:>"Troubleshooting", ButtonStyle->"Hyperlink"] }], "Text", CellTags->"Exponential Function"], Cell[TextData[{ StyleBox["The complex number i.", FontWeight->"Bold"], " The complex number ", StyleBox["i", FontSlant->"Italic"], " can also be retrieved from the Basic Input palette by clicking on the \ \[ImaginaryI] button. Or ", StyleBox["Mathematica", FontSlant->"Italic"], " will recognize the capital letter \"I\" as this number." }], "Text", CellTags->"The Complex Number ?"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "5. Using Variables ", ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Section", CellTags->"Using Variables"], Cell[TextData[{ "Variable Names.", StyleBox[" With ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", you can use almost any name for a variable, and there is no \ limit on the length of a variable name. But there are some restrictions on \ the choice of a variable name. Use only lower case for the first letter in \ the variable name, and the variable can never start with a number. ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", FontWeight->"Bold", CellTags->"Variable Names"], Cell[TextData[StyleBox["Assignment Statements. ", FontWeight->"Bold"]], "Text", CellTags->"Assignment Statements"], Cell[TextData[{ StyleBox["PRACTICE 23. ", FontWeight->"Bold"], StyleBox["Start a new input cell. ", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Bold"], StyleBox["To assign the value of 7.14 to the variable ", FontWeight->"Plain"], StyleBox["q", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", type ", FontWeight->"Plain"], StyleBox["q = 7.14", FontFamily->"System", FontWeight->"Plain"], StyleBox[" and then evaluate. This is what you will see:", FontWeight->"Plain"] }], "Text", PageBreakBelow->False], Cell[CellGroupData[{ Cell[BoxData[ \(q = 7.14\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[ StyleBox["7.13999999999999968`", StyleBoxAutoDelete->True, PrintPrecision->3]], "Output", CellLabel->"Out[1]="] }, Open ]], Cell[TextData[{ StyleBox[ "Note that the value is simply repeated in the output cell. Internally the \ computer has stored this value in the variable ", FontWeight->"Plain"], StyleBox["q", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". You can see your value again:", FontWeight->"Plain"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(q\)], "Input", CellLabel->"In[2]:="], Cell[BoxData[ StyleBox["7.13999999999999968`", StyleBoxAutoDelete->True, PrintPrecision->3]], "Output", CellLabel->"Out[2]="] }, Open ]], Cell[TextData[{ StyleBox[ "You may also assign an algebraic expression to a variable. For example, \ the Input statement s = 1 + ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`a\^2\)]], " ", StyleBox["will assign the algebraic expression 1 + ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`a\^2\)]], " ", StyleBox["to the variable ", FontWeight->"Plain"], StyleBox["s", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[".", FontWeight->"Plain"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(s = 1 + a\^2\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \(1 + a\^2\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(s\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \(1 + a\^2\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell[TextData[{ StyleBox["END PRACTICE 23. ", FontWeight->"Bold"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text", PageBreakAbove->False], Cell[TextData[{ StyleBox["Changing the Value of a Variable", FontWeight->"Bold"], ".", StyleBox[" Once you have assigned a value or algebraic expression to a \ variable, it will remain until you explicitly remove it or until you start a \ new ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" session. To change the value of a variable, simply type a new \ Input statement assigning a new value to that same variable. \n\n", FontWeight->"Plain"], StyleBox["PRACTICE 24. ", FontWeight->"Bold"], StyleBox["To remove the value assigned to the variable ", FontWeight->"Plain"], StyleBox["q", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", you could type ", FontWeight->"Plain"], StyleBox["Clear[q]", FontFamily->"System", FontWeight->"Plain"], StyleBox[" Typing ", FontWeight->"Plain"], StyleBox["Clear[", FontFamily->"System", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`q\_1\)], FontFamily->"System"], StyleBox[", ", FontFamily->"System"], Cell[BoxData[ \(TraditionalForm\`q\_2\)], FontFamily->"System"], StyleBox[", ..., ", FontFamily->"System"], Cell[BoxData[ \(TraditionalForm\`q\_k\)], FontFamily->"System"], StyleBox["]", FontFamily->"System", FontWeight->"Plain"], StyleBox[" will clear all of these variables. Also, typing ", FontWeight->"Plain"], StyleBox["q=.", FontFamily->"System", FontWeight->"Plain"], StyleBox[" (notice the period) will clear the variable ", FontWeight->"Plain"], StyleBox["q", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". For example: ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Changing the Value of a Variable"], Cell[BoxData[ \(q =. \)], "Input", CellLabel->"In[5]:=", CellFrame->1], Cell[TextData[StyleBox[ "Notice that this is an example of an input evaluation that has no \ corresponding output cell. This happens when the input command requires no \ output."]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(q\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \(q\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell[TextData[{ StyleBox["This last output cell verifies that ", FontWeight->"Plain"], StyleBox["q", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" no longer is assigned to a specific value. ", FontWeight->"Plain"], StyleBox["END PRACTICE 24.", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["WARNING", FontWeight->"Bold"], ". ", StyleBox["Forgetting about variable assignment statements made previously \ is a common mistake in ", FontWeight->"Plain"], StyleBox["Mathematica.", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" To avoid this mistake, you should remove variable values as \ soon as you have finished using them. It is also a good habit to always \ clear the value of a variable before defining that variable.\n", FontWeight->"Plain"] }], "Text"], Cell[TextData[{ StyleBox["Calculations with Variables", FontWeight->"Bold"], ".", StyleBox[" Once variables have been assigned values, calculations can be \ performed with these variables. Examine the following Input and Output \ statements: ", FontWeight->"Plain"] }], "Text", CellTags->"Calculations with Variables"], Cell[TextData[{ StyleBox["PRACTICE 25. ", FontWeight->"Bold"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", PageBreakBelow->False], Cell[CellGroupData[{ Cell[BoxData[ \(a = 23\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \(23\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b = 3.5\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ StyleBox["3.5`", StyleBoxAutoDelete->True, PrintPrecision->2]], "Output", CellLabel->"Out[5]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a\^3 - 2\ \@\(5\ b\)\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \(12158.6333997346618`\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x = 3 - c\^2\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \(3 - c\^2\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(2 - 3\ x + x\^3\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \(2 - 3\ \((3 - c\^2)\) + \((3 - c\^2)\)\^3\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[TextData[StyleBox["END PRACTICE 25.", FontWeight->"Bold"]], "Text", PageBreakAbove->False], Cell[TextData[{ StyleBox["PRACTICE 26.", FontWeight->"Bold"], " First, be sure to clear the variables a, b, and c. Then assign the \ values of 2.5, 7.2, and -1.9 to a, b, and c, respectively. Then calculate \ the following expression:\n", Cell[BoxData[ FormBox[ StyleBox[\(\(\(-b\) + \@\(b\^2 - \(\(4\)\(\ \)\(a\)\(\ \)\(c\)\(\ \ \)\)\)\)\/\(2\ a\)\), FontSize->16], TraditionalForm]]], ". ", StyleBox["END PRACTICE 26. ", FontWeight->"Bold"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["6. Built-In Functions", "Section", CellTags->"Built-In Functions"], Cell[TextData[{ StyleBox["Correct Form.", FontWeight->"Bold"], " ", StyleBox["Mathematica", FontSlant->"Italic"], " has hundreds of built-in functions for your use, and we will examine a \ few of them below. But first of all, you must remember that the name of each \ built-in function begins with a capital letter. Also, when a built-in \ function is used, certain inputs (also called arguments) must be supplied, \ and these inputs must be placed in square brackets [ ] after the function \ name. ", Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]], "\n" }], "Text", CellTags->"Correct Form"], Cell[TextData[{ StyleBox["Some Examples.", FontWeight->"Bold"], " \n1. Sin[z] gives the sine of z where z is in radians. The other \ built-in trigonometric functions are Cos[z], Tan[z], Cot[z], Sec[z], and \ Csc[z]. \n\n2. ArcSin[z] gives the arc sine of z (also called the \ inverse sine of z). z must be a real number between -1 and 1, inclusive, and \ the range will be from ", Cell[BoxData[ \(TraditionalForm\`\(-\[Pi]\)\/2\)]], " to ", Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)]], " in radians. The other built-in arc trig functions are ArcCos[z], \ ArcTan[z], AcrCot[z], ArcSec[z], and ArcCsc[x]. \n\n3. Log[z] gives the \ natural logarithm (logarithm to base \[ExponentialE]) of z.\n Log[b,z] \ gives the logarithm to base b of z.\n \n4. Abs[z] gives the absolute \ value of z.\n \n5. N[", StyleBox["expr", FontSlant->"Italic"], "] gives the numerical value of ", StyleBox["expr", FontSlant->"Italic"], ".\n \n \n6. Floor[x] gives the greatest integer less than or \ equal to x.\n \n7. Divisors[n] gives a list of the positive integers that \ divide n.\n \n8. Prime[n] gives the ", Cell[BoxData[ \(TraditionalForm\`n\^th\)]], " prime number.\n PrimePi[n] gives the number of prime numbers less \ than or equal to n. ", ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"], "\n \n", StyleBox["PRACTICE 27.", FontWeight->"Bold"], " For these exercises ", StyleBox["pay very close attention to using the correct ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" input", FontWeight->"Bold"], " as described above.\n1. Find the numerical value of ", StyleBox["sin ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\[Pi]\/9\)], FontSize->14], " and ln 10.\n2. Find the numerical value of ", Cell[BoxData[ \(TraditionalForm\`log\_5\)]], "73.5.\n3. Find all the positive integers that are divisors of 3000. ", StyleBox["END PRACTICE 27.", FontWeight->"Bold"] }], "Text", CellTags->"Some Examples"] }, Open ]], Cell[CellGroupData[{ Cell["7. Lists and Tables", "Section", CellTags->"Lists and Tables"], Cell[TextData[{ "Lists.", StyleBox[" In ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", a list is an ordered collection of objects. ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" uses { } to designate a list. Since order is important in a \ list, the list, {3, 9, 1, 17}, is different from the list, {9, 17, 3, 1}. \ Thus you could think of the list, {a, b}, with two elements as the ordered \ pair, (a, b), the list, {a, b, c}, with three elements as the ordered triple, \ (a, b, c), etc. ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", FontWeight->"Bold", CellTags->"Lists\t"], Cell[TextData[{ StyleBox["WARNING", FontWeight->"Bold"], ".", StyleBox[" You are accustomed to using { } to designate a set in which the \ order of the elements does not matter. However, in ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", { } will denote a list in which order is important.\n\nA list \ can be assigned to a variable using an assignment statement similar to those \ we studied previously. The Input statement ", FontWeight->"Plain"], StyleBox["v = {3, 7, 2, 11, 4}", FontFamily->"System", FontWeight->"Plain"], StyleBox[" will assign this list to the variable v.\n", FontWeight->"Plain"] }], "Text"], Cell[TextData[{ StyleBox["Calculations with a List.", FontWeight->"Bold"], StyleBox[" With ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", you can perform the same operations to each value in a list. \ Consider the following Input and Output statements: ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Calculations with a List"], Cell[TextData[StyleBox["PRACTICE 28.", FontWeight->"Bold"]], "Text", PageBreakBelow->False], Cell[BoxData[ \(Clear[t]\)], "Input", CellLabel->"In[9]:=", CellFrame->1], Cell[CellGroupData[{ Cell[BoxData[ \(t = {5, 10, 15, 20}\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \({5, 10, 15, 20}\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(t\^2 - 5\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \({20, 95, 220, 395}\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell[TextData[{ StyleBox["END PRACTICE 28. ", FontWeight->"Bold"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text", PageBreakAbove->False], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has taken each of the values in the list t, squared the value, and then \ subtracted 5. The resulting values are given in the list in the Output \ statement above.\n\nNow examine the following Input and Output statements:" }], "Text"], Cell[TextData[StyleBox["PRACTICE 29.", FontWeight->"Bold"]], "Text", PageBreakBelow->False], Cell[BoxData[ \(Clear[x]\)], "Input", CellLabel->"In[12]:=", CellFrame->1], Cell[CellGroupData[{ Cell[BoxData[ \(x\^t + t\ x\ - 1\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ \({\(-1\) + 5\ x + x\^5, \(-1\) + 10\ x + x\^10, \(-1\) + 15\ x + x\^15, \(-1\) + 20\ x + x\^20}\)], "Output", CellLabel->"Out[13]="] }, Open ]], Cell[TextData[StyleBox["END PRACTICE 29.", FontWeight->"Bold"]], "Text", PageBreakAbove->False], Cell[TextData[{ "Do you see that ", StyleBox["Mathematica", FontSlant->"Italic"], " has replaced the ", StyleBox["t ", FontSlant->"Italic"], "'s in the expression ", Cell[BoxData[ \(TraditionalForm\`x\^t\)]], " + t x - 1 with each of the values in the list ", StyleBox["t", FontSlant->"Italic"], ", thereby giving the new list in the Output statement above?" }], "Text"], Cell[TextData[{ StyleBox["Extracting Elements from a List.", FontWeight->"Bold"], " An element or several elements can be extracted (i.e., removed) from a \ list. The Input statement v[[ i ]] will extract the ", Cell[BoxData[ \(TraditionalForm\`i\^th\)]], " element from list v. Be sure that you use the double square brackets [[ \ ]] after the list name. The Input statement v[[ {", Cell[BoxData[ \(TraditionalForm\`i\_1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`i\_2\)]], ", ..., ", Cell[BoxData[ \(TraditionalForm\`i\_k\)]], "} ]] will extract the ", Cell[BoxData[ \(TraditionalForm\`i\_1\^th\)]], ", ", Cell[BoxData[ \(TraditionalForm\`i\_2\^th\)]], ", ... , and ", Cell[BoxData[ \(TraditionalForm\`i\_k\^th\)]], " elements from list v and write them as a list in that order." }], "Text", CellTags->"Extracting Elements from a List"], Cell[TextData[{ StyleBox["Tables.", FontWeight->"Bold"], " The built-in function Table can be used to create lists. For example, \ suppose you want to create a list of the values of ", Cell[BoxData[ \(TraditionalForm\`n\^3\)]], " as n varies from 1 to 10 in increments of 1. Then use the Input \ statement ", StyleBox["Table[", FontFamily->"System"], Cell[BoxData[ \(TraditionalForm\`n\^3\)], FontFamily->"System"], StyleBox[",{n, 1, 10}] ", FontFamily->"System"], " Actually, you do not need to type the 1, since by default ", StyleBox["Mathematica", FontSlant->"Italic"], " always uses 1 as the first value for the variable unless indicated \ otherwise. So the Input statement ", StyleBox["Table[", FontFamily->"System"], Cell[BoxData[ \(TraditionalForm\`n\^3\)], FontFamily->"System"], StyleBox[",{n, 10}]", FontFamily->"System"], " will suffice. ", Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Tables\t"], Cell[TextData[StyleBox["PRACTICE 30.", FontWeight->"Bold"]], "Text", PageBreakBelow->False], Cell[CellGroupData[{ Cell[BoxData[ \(Table[n\^3, {n, 10}]\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ \({1, 8, 27, 64, 125, 216, 343, 512, 729, 1000}\)], "Output", CellLabel->"Out[14]="] }, Open ]], Cell["\<\ For use with another example below we will assign the variable, v, to this \ list.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(v\ = \ %\)], "Input", CellLabel->"In[15]:="], Cell[BoxData[ \({1, 8, 27, 64, 125, 216, 343, 512, 729, 1000}\)], "Output", CellLabel->"Out[15]="] }, Open ]], Cell[TextData[{ StyleBox["END PRACTICE 30. ", FontWeight->"Bold"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text", PageBreakAbove->False], Cell[TextData[{ StyleBox["PRACTICE 31. ", FontWeight->"Bold"], StyleBox[ "You can also use increments different from 1. For example, suppose you \ want a list of the values of "], StyleBox[Cell[BoxData[ \(TraditionalForm\`\@y\)]]], StyleBox[" as "], StyleBox["y", FontSlant->"Italic"], StyleBox[ " varies from 0 to 5 but in increments of .25. Then use the Input \ statement "], StyleBox["Table[ ", FontFamily->"System"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\@y\)], FontFamily->"System"]], StyleBox[", {y, 0, 5, .25}]", FontFamily->"System"], StyleBox["."] }], "Text", PageBreakBelow->False], Cell[CellGroupData[{ Cell[BoxData[ \(Table[\@y, {y, 0, 5, .25}]\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \({0, 0.5`, 0.70710678118654755`, 0.866025403784438552`, 1.`, 1.1180339887498949`, 1.22474487139158894`, 1.32287565553229535`, 1.4142135623730951`, 1.5`, 1.58113883008418953`, 1.6583123951776999`, 1.73205080756887692`, 1.8027756377319946`, 1.87082869338697062`, 1.93649167310370877`, 2.`, 2.06155281280883029`, 2.12132034355964238`, 2.17944947177033698`, 2.2360679774997898`}\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell[TextData[StyleBox["END PRACTICE 31.", FontWeight->"Bold"]], "Text", PageBreakAbove->Automatic], Cell[TextData[{ "In summary,\n\t", StyleBox["Table[ f, {i, imax}]", FontFamily->"System"], " gives a list of the values of f as i varies from 1 to imax in increments \ of 1.\n\t", StyleBox["Table[ f, {i, imin, imax}]", FontFamily->"System"], " gives a list of the values of f as i varies from imin to imax in \ increments of 1.\n\t", StyleBox["Table[ f, {i, imin, imax, d}]", FontFamily->"System"], " gives a list of the values of f as i varies from imin to imax in \ increments of d.\n\n", StyleBox["PRACTICE 32.", FontWeight->"Bold"], " 1. Create a list of the numerical values of ln x as x varies from 1 to \ 10 in increments of 1. \t \n2. Create a list of the numerical \ values of ", Cell[BoxData[ \(TraditionalForm\`\@z\%3\)]], " as z varies from 10 to 15 in increments of .5 \n", StyleBox["END PRACTICE 32.", FontWeight->"Bold"], "\n" }], "Text"], Cell[TextData[{ StyleBox["TableForm.", FontWeight->"Bold"], " Often you would prefer that the values in a list appear in a vertical \ column. The Input statement TableForm[ v ] will take the values in list ", StyleBox["v", FontSlant->"Italic"], " and write them in that fashion. Recall that we defined list ", StyleBox["v", FontSlant->"Italic"], " before." }], "Text", CellTags->"Table Form"], Cell[TextData[{ StyleBox["WARNING.", FontWeight->"Bold"], " Remember that ", StyleBox["Mathematica", FontSlant->"Italic"], " is case-sensitive. Notice that both the \"T\" and \"F\" are in upper \ case in the built-in function TableForm. ", Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text"], Cell[TextData[StyleBox["PRACTICE 33.", FontWeight->"Bold"]], "Text", PageBreakBelow->False], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[v]\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ InterpretationBox[GridBox[{ {"1"}, {"8"}, {"27"}, {"64"}, {"125"}, {"216"}, {"343"}, {"512"}, {"729"}, {"1000"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {1, 8, 27, 64, 125, 216, 343, 512, 729, 1000}]]], "Output", CellLabel->"Out[16]//TableForm="] }, Open ]], Cell[TextData[{ StyleBox["END PRACTICE 33. ", FontWeight->"Bold"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text", PageBreakAbove->False], Cell[TextData[{ StyleBox["PRACTICE 34.", FontWeight->"Bold"], " Put one of the lists from PRACTICE 32 into table form. ", StyleBox["END PRACTICE 34.", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["List of Lists.", FontWeight->"Bold"], " Suppose, for example, that you wanted to create a list of the ordered \ pairs ( ", Cell[BoxData[ \(TraditionalForm\`b\^2\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\@b\)]], ") as b varies from 2 to 10 in increments of 1. Then each element in the \ list will be a list. So use the Input statement \n", StyleBox["Table[ {", FontFamily->"System"], Cell[BoxData[ \(TraditionalForm\`b\^2\)], FontFamily->"System"], StyleBox[", ", FontFamily->"System"], Cell[BoxData[ \(TraditionalForm\`\@b\)], FontFamily->"System"], StyleBox["}, {b, 2, 10}]", FontFamily->"System"] }], "Text", CellTags->"List of Lists"], Cell[TextData[StyleBox["PRACTICE 35.", FontWeight->"Bold"]], "Text", PageBreakBelow->False], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", " ", RowBox[{ RowBox[{"{", StyleBox[ RowBox[{ FormBox[\(b\^2\), "TraditionalForm"], ",", " ", FormBox[\(\@b\), "TraditionalForm"]}], FontSlant->"Italic"], "}"}], ",", " ", \({b, \ 2, \ 10}\)}], "]"}]], "Input", CellLabel->"In[17]:="], Cell[BoxData[ \({{4, \@2}, {9, \@3}, {16, 2}, {25, \@5}, {36, \@6}, {49, \@7}, {64, 2\ \@2}, {81, 3}, {100, \@10}}\)], "Output", CellLabel->"Out[17]="] }, Open ]], Cell[TextData[StyleBox["END PRACTICE 35.", FontWeight->"Bold"]], "Text", PageBreakAbove->False], Cell[TextData[{ StyleBox["PRACTICE 36.", FontWeight->"Bold"], " Create a list of the values of ", Cell[BoxData[ \(TraditionalForm\`\(\(3\ x\^3\)\(\ \)\(+\)\(\ \)\(1\)\(\ \)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`9\ x\^2\)]], " as x varies from 1 to 3 in increments of .25. Then put this list in \ table form. ", StyleBox["END PRACTICE 36. ", FontWeight->"Bold"], "\n\n", StyleBox["PRACTICE 37.", FontWeight->"Bold"], " Create, in table form, the values of sin ", StyleBox["x", FontSlant->"Italic"], ", cos ", StyleBox["x", FontSlant->"Italic"], ", and tan ", StyleBox["x", FontSlant->"Italic"], " as ", StyleBox["x", FontSlant->"Italic"], " varies from 0 to \[Pi] in increments of ", Cell[BoxData[ FormBox[ StyleBox[\(\[Pi]\/12\), FontSize->16], TraditionalForm]]], ". ", StyleBox["END PRACTICE 37.", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["Prepend and Append. ", FontWeight->"Bold"], "In the table you just created in the previous ", StyleBox["PRACTICE", FontWeight->"Bold"], ", it might look better to have the headings Sine, Cosine, and Tangent at \ the beginning of each column of the table. The built-in function Prepend can \ be used to add elements to the beginning of a list. The Input statement \ Prepend[ t, \"Heading\"] will add the word Heading to the first of list t. \ Suppose that list v consists of a list of lists, each of which contains three \ elements. Then the Input statement Prepend[ v, { \"Column #1\", \"Column \ #2\", \"Column #3\"}] will add these three headings to beginnings of these \ three lists. ", Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]], "\n\n", StyleBox["PRACTICE 38.", FontWeight->"Bold"], " Go back to the table you created in the previous PRACTICE and add the \ words Sine, Cosine, and Tangent as headings at the top of the three columns \ in the table. (Note: Be sure to place quotation marks around the words, \ \"Sine\", \"Cosine\", \"Tangent\".) ", StyleBox["END PRACTICE 38.", FontWeight->"Bold"] }], "Text", CellTags->"Prepend and Append\t"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "8. Algebraic Expressions ", ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Section", CellTags->"Algebraic Expressions"], Cell[TextData[{ "Grouping Symbols.", StyleBox[" In typing algebraic expressions that have grouping symbols, use \ only parentheses ( ). Do not use square brackets [ ] and do not use set \ braces { }.\n", FontWeight->"Plain"] }], "Text", FontWeight->"Bold", CellTags->"Grouping Symbols"], Cell[TextData[{ StyleBox["More Built-in Functions.", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" has several built-in functions that can be used with algebraic \ expressions. For example,", FontWeight->"Plain"] }], "Text", CellTags->"More Built-in Functions"], Cell[TextData[{ StyleBox["PRACTICE 39. ", FontWeight->"Bold"], " ", ButtonBox["(back to contents) ", ButtonData:>"Contents", ButtonStyle->"Hyperlink"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]], StyleBox["\n1. Expand. This function expands out products and positive \ powers in an expression. ", FontWeight->"Plain"] }], "Text", PageBreakBelow->False], Cell[BoxData[ \(Clear[x]\)], "Input", CellLabel->"In[18]:=", CellFrame->1], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\(\((x\^2 + 3)\)\^3\) \((2 x - 5)\)]\)], "Input", CellLabel->"In[19]:="], Cell[BoxData[ \(\(-135\) + 54\ x - 135\ x\^2 + 54\ x\^3 - 45\ x\^4 + 18\ x\^5 - 5\ x\^6 + 2\ x\^7\)], "Output", CellLabel->"Out[19]="] }, Open ]], Cell["\<\ 2. Factor. This function factors a polynomial over the integers.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[9\ x\^4 - 24\ x\^3 + 43\ x\^2 - 72\ x + 48]\)], "Input", CellLabel->"In[20]:="], Cell[BoxData[ \(\((\(-4\) + 3\ x)\)\^2\ \((3 + x\^2)\)\)], "Output", CellLabel->"Out[20]="] }, Open ]], Cell["\<\ 3. Together. This function combines the terms of an expression in a sum \ over a common denominator and cancels factors in the result.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Together[2\/x + 1\/\(x - 1\) - x\/\(x + 1\)]\)], "Input", CellLabel->"In[21]:="], Cell[BoxData[ \(\(\(-2\) + x + 4\ x\^2 - x\^3\)\/\(\((\(-1\) + x)\)\ x\ \((1 + x)\)\)\)], "Output", CellLabel->"Out[21]="] }, Open ]], Cell["\<\ 4. Apart. This function rewrites a rational expression as a sum of terms \ with minimal denominators.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Apart[\(\(-x\^3\) + 4\ x\^2 - x + 3\)\/\(x\^4 + x\^2\)]\)], "Input", CellLabel->"In[22]:="], Cell[BoxData[ \(3\/x\^2 - 1\/x + 1\/\(1 + x\^2\)\)], "Output", CellLabel->"Out[22]="] }, Open ]], Cell["\<\ 5. Simplify. This function performs a sequence of algebraic transformations \ and returns the simplest form it finds.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Tan[ArcCos[x]]]\)], "Input", CellLabel->"In[23]:="], Cell[BoxData[ \(\@\(1 - x\^2\)\/x\)], "Output", CellLabel->"Out[23]="] }, Open ]], Cell[TextData[{ StyleBox["END PRACTICE 39. ", FontWeight->"Bold"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text", PageBreakAbove->False], Cell[TextData[{ StyleBox["PRACTICE 40", FontWeight->"Bold"], ".", StyleBox[" Expand the expression ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(\((2\ t\^2 + 1)\)\^3\) \((t - 5)\)\^2\)]], ". ", StyleBox["Then factor the resulting polynomial. Did you get back the \ original expression? ", FontWeight->"Plain"], StyleBox["END PRACTICE 40.", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["Replacement Rules", FontWeight->"Bold"], ".", StyleBox[" Replacing the variables in an algebraic expression with \ specific values is called a ", FontWeight->"Plain"], StyleBox["replacement rule", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". You must first indicate the algebraic expression; then type \ /. (note the period) ; then indicate the variable in the expression that \ will be replaced; then retrieve the arrow \[Rule] from the BasicInput palette \ by clicking on the arrow; then indicate the value that will replace the \ variable. So ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[\(\(w + 3\)\/w\^2\), FontFamily->"System"], TraditionalForm]]], "/. w\[Rule]5 ", StyleBox["will tell ", FontWeight->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" to replace the ", FontWeight->"Plain"], StyleBox["w", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["'s in the expression, ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(w + 3\)\/w\^2\)]], ", ", StyleBox["with 5. ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Replacement Rules"], Cell[TextData[StyleBox["PRACTICE 41.", FontWeight->"Bold"]], "Text", PageBreakBelow->False], Cell[CellGroupData[{ Cell[BoxData[ \(\(w + 3\)\/w\^2 /. w \[Rule] 5\)], "Input", CellLabel->"In[24]:="], Cell[BoxData[ \(8\/25\)], "Output", CellLabel->"Out[24]="] }, Open ]], Cell["You can also replace a variable with any expression.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(a = \@\(x\^2 - 1\)\)], "Input", CellLabel->"In[25]:="], Cell[BoxData[ \(\@\(\(-1\) + x\^2\)\)], "Output", CellLabel->"Out[25]="] }, Open ]], Cell[BoxData[ \(Clear[b]\)], "Input", CellLabel->"In[26]:=", CellFrame->1], Cell[CellGroupData[{ Cell[BoxData[ \(a /. x \[Rule] b - 2\)], "Input", CellLabel->"In[27]:="], Cell[BoxData[ \(\@\(\(-1\) + \((\(-2\) + b)\)\^2\)\)], "Output", CellLabel->"Out[27]="] }, Open ]], Cell["\<\ You may apply a replacement rule to more than one expression by putting the \ expressions in a list using { }.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({2\ x\^2, x - 3} /. x \[Rule] 10\)], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \({200, 7}\)], "Output", CellLabel->"Out[28]="] }, Open ]], Cell["\<\ You may apply more than one replacement rule by putting the rules in a list \ using { }.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\@\(2\ x\ y\^2\) /. {x \[Rule] 4, y \[Rule] 5}\)], "Input", CellLabel->"In[29]:="], Cell[BoxData[ \(10\ \@2\)], "Output", CellLabel->"Out[29]="] }, Open ]], Cell[TextData[{ StyleBox["END PRACTICE 41. ", FontWeight->"Bold"], ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Text", PageBreakAbove->False], Cell[TextData[{ StyleBox["PRACTICE 42", FontWeight->"Bold"], StyleBox["."], StyleBox[" In the expression ", FontWeight->"Plain"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\(\(-b\) + \@\(b\^2 - 4\ a\ c\)\)\/\(2\ a\)\)]]], StyleBox[", "], StyleBox["replace the a, b, and c with -12, 4, and 7, respectively.\n", FontWeight->"Plain"], StyleBox["END PRACTICE 42.", FontWeight->"Bold"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "9. Review of Grouping Symbols ", " ", ButtonBox["(back to contents)", ButtonData:>"Contents", ButtonStyle->"Hyperlink"] }], "Section", CellTags->"Review of Grouping Symbols"], Cell[TextData[{ "In your mathematical past, you probably have used the grouping symbols ( \ ), [ ], and { } interchangeably. But you have seen in this manual that ", StyleBox["Mathematica", FontSlant->"Italic"], " has very specific uses of each of these symbols within input cells. Let \ us review those uses within input cells.\n\n", StyleBox["Parentheses ( ) are used to indicate multiplication or as \ grouping symbols in algebraic expressions.\n\nSquare brackets [ ] are used \ for the inputs after a ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" built-in function.\n\nSet braces { } are used to indicate a \ list. ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellFrame->3] }, Open ]], Cell[CellGroupData[{ Cell["10. Solving Equations", "Section", CellTags->"Solving Equations"], Cell[TextData[{ StyleBox["Solve and FindRoot. ", FontWeight->"Bold"], StyleBox["Mathematica", FontSlant->"Italic"], " has the ability to solve certain equations algebraically, which is one of \ the reasons for calling it a ", StyleBox["computer algebra system", FontSlant->"Italic"], StyleBox[". ", FontWeight->"Bold", FontSlant->"Italic"], "It can also solve equations numerically provided you give it an estimate \ for a solution. As a first example we will have it solve the equation ", Cell[BoxData[ \(TraditionalForm\`3 x + 5\ = \(-4\)\)]], ". Here is how to do this. ", Cell[BoxData[ FormBox[ ButtonBox[\((index)\), ButtonData:>"Index", ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text", CellTags->"Solve and FindRoot"], Cell[TextData[StyleBox["PRACTICE 43.", FontWeight->"Bold"]], "Text", PageBreakBelow->False], Cell[CellGroupData[{ Cell[BoxData[