(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 43320, 1364]*) (*NotebookOutlinePosition[ 44104, 1393]*) (* CellTagsIndexPosition[ 44034, 1387]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"] }], "Title"], Cell[CellGroupData[{ Cell[TextData[{ "Inspiration: Showing off ", StyleBox["Mathematica", FontSlant->"Italic"], "'s capabilities" }], "Section"], Cell["\<\ Work through the following demos and then explore any other demos that strike \ your fancy. You can also access these tutorials and demos if you go to the \ Help menu, select Help Browser, and click on the Getting Started/Demos \ button.\ \>", "Text", CellTags->"T.3"], Cell[TextData[{ ButtonBox["Mathematica as a Calculator", ButtonData:>{"Calculator", None}, ButtonStyle->"TourLink", ButtonNote->None], "\n", ButtonBox["Power Computing with Mathematica", ButtonData:>{"Power", None}, ButtonStyle->"TourLink", ButtonNote->None], "\n", ButtonBox["Accessing Algorithms in Mathematica", ButtonData:>{"Accessing", None}, ButtonStyle->"TourLink", ButtonNote->None], "\n", ButtonBox["Mathematical Knowledge in Mathematica", ButtonData:>{"Mathematical", None}, ButtonStyle->"TourLink", ButtonNote->None], "\n", ButtonBox["Handling Data", ButtonData:>{"Handling", None}, ButtonStyle->"TourLink", ButtonNote->None], "\n", ButtonBox["Visualization with Mathematica", ButtonData:>{"Visualization", None}, ButtonStyle->"TourLink", ButtonNote->None], "\n", ButtonBox["Formula Gallery", ButtonData:>{"G.1.1", None}, ButtonStyle->"DemosLink", ButtonNote->None], "\n", ButtonBox["Graphics Gallery", ButtonData:>{"G.2.1", None}, ButtonStyle->"DemosLink", ButtonNote->None], "\n", ButtonBox["Sound Gallery", ButtonData:>{"Simple Sound", None}, ButtonStyle->"DemosLink", ButtonNote->None] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Perspiration: Getting ", StyleBox["Mathematica", FontSlant->"Italic"], " to do what you want" }], "Section"], Cell[CellGroupData[{ Cell[TextData[{ "The Anatomy of a ", StyleBox["Mathematica", FontSlant->"Italic"], " Notebook" }], "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " documents, called notebooks, are composed of cells. The brackets along \ the right side of the window indicate the extent of the cells. Each cell is \ assigned a style. The cell that you are currently reading is a ", StyleBox["Text ", FontSlant->"Italic"], "cell. The cell above this one is a ", StyleBox["Subsection", FontSlant->"Italic"], " cell. Go to the ", StyleBox["Format", FontSlant->"Italic"], " menu at the top of this window and select ", StyleBox["Style", FontSlant->"Italic"], ". You will see a list of all of the cell styles that are available. In \ addition to cell styles, ", StyleBox["Mathematica", FontSlant->"Italic"], " has notebook styles, that determine how the various cell styles are to be \ displayed. You can go to the ", StyleBox["Format", FontSlant->"Italic"], " menu and select ", StyleBox["Style Sheet", FontSlant->"Italic"], " to change the style for this notebook." }], "Text"], Cell[TextData[{ StyleBox["Input", FontSlant->"Italic"], " cells contain ", StyleBox["Mathematica", FontSlant->"Italic"], " commands. When you evaluate an ", StyleBox["Input", FontSlant->"Italic"], " cell, ", StyleBox["Mathematica", FontSlant->"Italic"], " creates ", StyleBox["Output", FontSlant->"Italic"], " cells and possibly ", StyleBox["Graphics", FontSlant->"Italic"], " cells to hold the results. Click anywhere in the following ", StyleBox["Input", FontSlant->"Italic"], " cell or on the cell's bracket along the right side, and then press \ Shift-Enter to evaluate the cell. ", StyleBox["Mathematica", FontSlant->"Italic"], " will create a ", StyleBox["Graphics", FontSlant->"Italic"], " cell and an ", StyleBox["Output ", FontSlant->"Italic"], "cell." }], "Text"], Cell[BoxData[ \(Plot3D[ Cos[\(x\^2 + y\^2\)\/2], {x, \(-\[Pi]\), \ \[Pi]}, {y, \(-\[Pi]\), \ \ \[Pi]}]\)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " actually consists of two programs \[LongDash] the front end and the \ kernel. The front end displays ", StyleBox["Mathematica", FontSlant->"Italic"], " notebooks and the kernel performs calculations. The kernel is launched \ when you evaluate the first ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["Input", FontSlant->"Italic"], " cell. The kernel maintains a history of the commands that you ask it to \ evaluate, independent of the order in which your ", StyleBox["Input", FontSlant->"Italic"], " cells appear in your notebooks. That is, you can jump around in the \ notebooks and the ", StyleBox["Mathematica", FontSlant->"Italic"], " kernel will keep track of the order in which you evaluated the ", StyleBox["Input", FontSlant->"Italic"], " cells. The front end will help you keep track of the order by displaying \ ", StyleBox["In[n]:=", FontSlant->"Italic"], " before each evaluated ", StyleBox["Input", FontSlant->"Italic"], " cell, where ", StyleBox["n", FontSlant->"Italic"], " is incremented each time you evaluate a cell. The values that you have \ assigned to your variables will be kept in the kernel until you ask the \ kernel to clear them or until you quit the ", StyleBox["Mathematica", FontSlant->"Italic"], " application. If you want to resume your work later, you will have to \ save your notebook and re-evaluate the ", StyleBox["Input", FontSlant->"Italic"], " cells when you re-launch ", StyleBox["Mathematica.", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ "Cells are grouped, as indicated by the nested square brackets along the \ right side of any ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook. Cell groups can either be open or closed. This group, under \ the heading ", StyleBox["The Anatomy of a ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" Notebook", FontWeight->"Bold"], ", is open. The group just below this one, under the heading, ", StyleBox["Getting Help", FontWeight->"Bold"], ", is closed. This is indicated by the small triangle at the bottom of the \ cell bracket. You can open a closed group by double-clicking on the cell \ bracket or the triangle. You can close an open group by double-clicking on \ its cell bracket. As you finish each section of this notebook, double-click \ on the next closed section to progress to the next topic." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Getting Help", "Subsection"], Cell[TextData[{ "There are several places you can turn to for help when you are trying to \ get ", StyleBox["Mathematica", FontSlant->"Italic"], " to do what you want. Your first source for help is the ", ButtonBox["Help Browser", ButtonData:>{"Help", None}, ButtonStyle->"GettingStartedLink", ButtonNote->None], ", under the Help menu. The Help Browser has six sections:\n", ButtonBox["Getting Started/Demos", ButtonData:>{"Starting Mathematica", None}, ButtonStyle->"GettingStartedLink", ButtonNote->None], " \[LongDash] introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " with some inspiring demos,\n", ButtonBox["Built-in Functions", ButtonData:>{"Numerical Computation (Alphabetical Listing)", None}, ButtonStyle->"RefGuideLink", ButtonNote->None], " \[LongDash] syntax and examples for the thousands of functions that are \ built into ", StyleBox["Mathematica", FontSlant->"Italic"], ",\n", ButtonBox["Add-ons", ButtonData:>{"5.0.1", None}, ButtonStyle->"AddOnsLink", ButtonNote->None], " \[LongDash] syntax and examples for the many additional packages that \ extend ", StyleBox["Mathematica", FontSlant->"Italic"], "'s capabilties,\n", ButtonBox["The ", ButtonData:>{"C", None}, ButtonStyle->"MainBookLink", ButtonNote->None], StyleBox[ButtonBox["Mathematica", ButtonData:>{"C", None}, ButtonStyle->"MainBookLink", ButtonNote->None], FontSlant->"Italic"], ButtonBox[" Book", ButtonData:>{"C", None}, ButtonStyle->"MainBookLink", ButtonNote->None], " \[LongDash] in-depth description of how to use ", StyleBox["Mathematica", FontSlant->"Italic"], ", including many examples,\n", ButtonBox["Front End", ButtonData:>{"Menu Listing", None}, ButtonStyle->"OtherInformationLink", ButtonNote->None], " \[LongDash] documentation of front end options, menus, keyboard \ shortcuts, and other miscellany,\n", ButtonBox["Master Index \[LongDash]", ButtonData:>{"C", None}, ButtonStyle->"HelpLink", ButtonNote->None], " a complete index for the other five sections of the Help Browser" }], "Text"], Cell["\<\ Secondly, if you already know the name of the command that you want to use, \ you can get basic help on that command by putting a question mark before the \ command name. For example, click anywhere in the following cell and then \ press Shift-Enter. If you need more help than this, then go to the Built-In \ Functions section of the Help Browser.\ \>", "Text"], Cell[BoxData[ \(\(?ContourPlot\)\)], "Input"], Cell[TextData[{ "Finally, check out Wolfram Research's extensive Web site for additional \ assistance, inspiration, and user-contributed ", StyleBox["Mathematica", FontSlant->"Italic"], " packages. See ", ButtonBox["http://www.wolfram.com/services/", ButtonData:>{ URL[ "http://www.wolfram.com/services/"], None}, ButtonStyle->"Hyperlink"], "." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "The Anatomy of a ", StyleBox["Mathematica", FontSlant->"Italic"], " Command" }], "Subsection"], Cell[TextData[{ "In ", StyleBox["Mathematica", FontSlant->"Italic"], ",the arguments of a function are enclosed in square brackets. Built-in \ functions start with capital letters. Here is an example that uses the ", StyleBox["Factor", "MR"], " function. Evaluate this command by clicking anywhere in the following \ cell and pressing Shift-Enter. From now on, whenever you see an ", StyleBox["Input", FontSlant->"Italic"], " cell in this notebook, go ahead and evaluate it." }], "Text"], Cell[BoxData[ \(Factor[x\^99 - 1]\)], "Input"], Cell[TextData[{ "If a function has more than one argument, they are separated by commas. \ Here is the ", StyleBox["Solve", "MR"], " command with two arguments. Notice that equations have two equal signs. \ When you evaluate this command, ", StyleBox["Mathematica", FontSlant->"Italic"], " will give you a warning that it may not have found all of the solutions. \ In this case, however, it does find both solutions." }], "Text"], Cell[BoxData[ \(Solve[ 4 \[ExponentialE]\^\(k\ t\)\ - 7 \[ExponentialE]\^\(2 k\ t\) \[Equal] 1\/2, t]\)], "Input"], Cell[TextData[{ StyleBox["Exercise:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " Solve the equation ", Cell[BoxData[ \(TraditionalForm\`\(sin\^2\)(4 x) + \(cos\^2\)(5 x) = 1\)]], "." }], "Text", Background->GrayLevel[0.750011]], Cell[TextData[{ StyleBox["Exercise:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " Solve the equation ", Cell[BoxData[ \(TraditionalForm\`x\^4 - 3 x\^2 + 2 x - 5 = 0\)]], "." }], "Text", Background->GrayLevel[0.750011]], Cell["\<\ Notice that the solution to this second exercise includes some nasty \ expressions. If you want a decimal approximation to the solution, instead of \ the exact solution, then enclose the Solve equation inside of the N function. \ For example,\ \>", "Text"], Cell[BoxData[ \(N[Solve[x^3 + 1 \[Equal] 0, x]]\)], "Input"], Cell["\<\ You can assign a value to a variable by using a single equal sign.\ \>", "Text"], Cell[BoxData[{ \(k = 5\), "\[IndentingNewLine]", \(Plot[t\^2\ Sin[k\ t], {t, \(-\[Pi]\), \[Pi]}]\)}], "Input"], Cell[TextData[{ "Notice that ", StyleBox["Mathematica", FontSlant->"Italic"], " displayed the output of the \"k = 5\" command as well as the output of \ the ", StyleBox["Plot", FontFamily->"Courier"], " command. If you want to have ", StyleBox["Mathematica", FontSlant->"Italic"], " evaluate a command, but not display the output, then end the command with \ a semicolon. The output of the ", StyleBox["Plot", FontFamily->"Courier New"], " command is the word \"- Graphics -\". If you end a ", StyleBox["Plot", FontFamily->"Courier"], " command with a semicolon, it will not display the word \"- Graphics -\", \ but it will display the graph. The ", StyleBox["Input", FontSlant->"Italic"], " cell below has a semicolon after each command. Notice that it displays \ less output when you evaluate it. What should happen to the graph if you \ change the value of ", StyleBox["k", FontSlant->"Italic"], " in the following ", StyleBox["Input", FontSlant->"Italic"], " cell to a larger number? Try it and see if you get what you expected." }], "Text"], Cell[BoxData[{ \(\(k = 5;\)\), "\[IndentingNewLine]", \(\(Plot[t\^2\ Sin[k\ t], {t, \(-\[Pi]\), \[Pi]}];\)\)}], "Input"], Cell["\<\ You can temporarily substitute values for variables or for any pattern.\ \>", "Text"], Cell[BoxData[{ \(Clear[x, y, a]\), "\[IndentingNewLine]", \(\((x\^2 + 4 x\ y\^3)\) /. x \[Rule] a\^2\)}], "Input"], Cell[BoxData[{ \(Clear[x, y, a]\), "\[IndentingNewLine]", \(\((x\^2 + 4 x\ y\^3)\) /. {x \[Rule] a\^2, y \[Rule] a\^6}\)}], "Input"], Cell[TextData[{ "This does not permanently assign values to ", StyleBox["x", FontSlant->"Italic"], " or ", StyleBox["y", FontSlant->"Italic"], ". These values are only applied to the expressions above, not in future \ calculations involving ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], ". The ", StyleBox["Solve", FontFamily->"Courier"], " command gives output in this format, so that you can substitute the \ solutions into other expressions." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Creating Your Own Functions", "Subsection"], Cell[TextData[{ "You can define your own functions in ", StyleBox["Mathematica", FontSlant->"Italic"], ". Here is an example showing the concentration of the product of a \ chemical reaction. If we start the function name with a lower case letter, \ it will not interfere with any of ", StyleBox["Mathematica", FontSlant->"Italic"], "'s built-in functions, which all start with capital letters." }], "Text"], Cell[BoxData[{ \(Clear[concentration, \ t]\), "\[IndentingNewLine]", \(concentration[ t_] = \(15 \((\ \[ExponentialE]\^\( .45\ t\) - 1)\)\)\/\(6\ \ \[ExponentialE]\^\( .45\ t\) - 5\)\)}], "Input"], Cell["\<\ The function's arguments on the left side of the assignment statement must be \ followed by an underscore. Those same variables on the right side should not \ be followed by underscores. You can evaluate the function at specific \ numerical values.\ \>", "Text"], Cell[BoxData[ \(concentration[ .5]\)], "Input"], Cell[TextData[{ "You can use the function in other ", StyleBox["Mathematica", FontSlant->"Italic"], " commands, such as" }], "Text"], Cell[BoxData[ \(Solve[concentration[t] \[Equal] 1, \ t]\)], "Input"], Cell["and here is its graph.", "Text"], Cell[BoxData[ \(\(Plot[concentration[t], {t, 0, 5}];\)\)], "Input"], Cell[TextData[{ "It looks like the concentration is approaching about 2.5. We can confirm \ this by taking the limit as ", StyleBox["t", FontSlant->"Italic"], " approaches \[Infinity]." }], "Text"], Cell[BoxData[ \(Limit[concentration[t], t \[Rule] \[Infinity]]\)], "Input"], Cell["The reaction rate is the derivative.", "Text"], Cell[BoxData[ \(\(concentration'\)[t]\)], "Input"], Cell["Initially, the reaction rate is", "Text"], Cell[BoxData[ \(\(concentration'\)[0]\)], "Input"], Cell[TextData[{ StyleBox["Exercise:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " The height of a Haflinger horse is given by the logistic function, ", StyleBox["h", FontSlant->"Italic"], "(", StyleBox["t", FontSlant->"Italic"], ") = ", Cell[BoxData[ \(TraditionalForm\`59\/\(1 + .639\ \[ExponentialE]\^\(\(-1.248\)\ \ t\)\)\)]], ", ", StyleBox["t", FontSlant->"Italic"], " years after birth. Define this as a function in ", StyleBox["Mathematica", FontSlant->"Italic"], ". Find the height of this horse at birth. Find the adult height of the \ horse. Find the time at which the horse reaches half its adult height. At \ what time is the horse's growth rate at its maximum?" }], "Text", Background->GrayLevel[0.750011]], Cell[TextData[{ "The general logistic function is ", StyleBox["f", FontSlant->"Italic"], "(", StyleBox["t", FontSlant->"Italic"], ") = ", Cell[BoxData[ \(TraditionalForm\`a\/\(1 + \ b\ \[ExponentialE]\^\(\(-c\)\ t\)\)\)]], ". The height of a Burmese Mountain Dog at birth is 4 inches and its \ mature height is 24 inches. It reaches 95% of its mature height at one year \ old. Let's find a logistic function describing its height and then plot the \ function. We first define the general logistic function. " }], "Text"], Cell[BoxData[{ \(Clear[f, t, a, b, c]\), "\[IndentingNewLine]", \(f[t_] := a\/\(1 + b\ \[ExponentialE]\^\(\(-c\)\ t\)\)\)}], "Input"], Cell[TextData[{ "We have three data points, so we should be able to use Solve to find the \ values of ", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], ", and ", StyleBox["c", FontSlant->"Italic"], ". But first notice that the limiting value of f(t) is a, since ", Cell[BoxData[ \(TraditionalForm\`e\^\(\(-c\)\ t\)\)]], " approaches 0 as t approaches infinity. So, ", StyleBox["a", FontSlant->"Italic"], " must be 24 for our problem. We can either permanently assign ", StyleBox["a", FontSlant->"Italic"], " the value 24 with the command ", StyleBox["a", FontSlant->"Italic"], " = 24, or we can temporarily substitute 24 for ", StyleBox["a", FontSlant->"Italic"], " with f[t]/.a\[Rule]24. We will use the latter approach." }], "Text"], Cell[BoxData[ \(parameters = Solve[{a \[Equal] 24, f[0] \[Equal] 4, f[1] == .95*24}, \ {a, b, c}]\)], "Input"], Cell["\<\ and here is a graph with the parameters temporarily replaced by these values.\ \ \>", "Text"], Cell[BoxData[ \(\(Plot[f[t] /. parameters, {t, 0, 2}];\)\)], "Input"], Cell[TextData[{ "Notice the the original definition for ", StyleBox["f", FontSlant->"Italic"], " is unaltered, so that we can continue to work with the general logistic \ equation if we wish." }], "Text"], Cell[BoxData[ \(f[t]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Working with Lists", "Subsection"], Cell[TextData[{ "Braces are used to denote lists in ", StyleBox["Mathematica", FontSlant->"Italic"], ". The second argument to the ", StyleBox["Plot", FontFamily->"Courier"], " command is a list, in the case above, {t, 0,5}. Suppose that we want to \ plot the total population of the United States at the turn of each decade \ since the U.S. Census began in 1790. Notice that the list below is enclosed \ in braces." }], "Text"], Cell[BoxData[{ \(\(population = {3929214, 5308483, 7239881, 9633822, 12866020, 17069458, 23191876, 31443321, 38558371, 50155783, 62979766, 76303387, 91972266, 105710620, 122775046, 131669275, 151325798, 179323175, 203302031, 226545805, 248709873, 281421906};\)\), "\[IndentingNewLine]", \(\(ListPlot[population];\)\)}], "Input"], Cell[TextData[{ "As you can see, the dots are pretty small. You can add options to many ", StyleBox["Mathematica", FontSlant->"Italic"], " commands to modify their action. Options are listed after all of the \ command arguments, and they are of the form ", StyleBox["optionName", FontSlant->"Italic"], " \[Rule] ", StyleBox["optionValue", FontSlant->"Italic"], ". The available options for a command are described in the Help Browser. \ We can use the ", StyleBox["PlotStyle", FontFamily->"Courier"], " option for ", StyleBox["ListPlot", FontFamily->"Courier"], " to change the point size." }], "Text"], Cell[BoxData[ \(\(ListPlot[population, \ PlotStyle \[Rule] PointSize[0.02]];\)\)], "Input"], Cell["\<\ Or you can also list data as ordered pairs. Each ordered pair is a list of \ two numbers.\ \>", "Text"], Cell[BoxData[{ \(\(populationPairs = {{1790, 3929214}, {1800, 5308483}, \ {1810, 7239881}, \ {1820, 9633822}, \ {1830, 12866020}, \ {1840, 17069458}, \ {1850, 23191876}, \ {1860, 31443321}, \ {1870, 38558371}, \ {1880, 50155783}, \ {1890, 62979766}, \ {1900, 76303387}, \ {1910, 91972266}, \ {1920, 105710620}, \ {1930, 122775046}, \ {1940, 131669275}, \ {1950, 151325798}, \ {1960, 179323175}, {1970, 203302031}, {1980, 226545805}, {1990, 248709873}, \ {2000, 281421906}};\)\), "\n", \(\(ListPlot[populationPairs, \ PlotStyle \[Rule] PointSize[0.02]];\)\)}], "Input"], Cell["\<\ You can read the coordinates of points on the graph by clicking on the graph \ once, holding down the \[ControlKey] key, and moving your mouse over the \ graph. You can also resize the display of the graph by clicking on the graph \ once and dragging the small black boxes at the corners of the graph. Notice \ that near the middle of the twenthieth century there is a significant \ deviation from the trend. Can you explain why?\ \>", "Text"], Cell[TextData[{ "Exponential functions have constant percent growth rates. In fact, they \ are the ", StyleBox["only", FontSlant->"Italic"], " functions that have constant percent growth rate, except for boring \ constant functions, which don't grow at all.", " Let's examine the U.S. population data to see if it has a constant \ percent growth rate. We need to calculate ", Cell[BoxData[ \(TraditionalForm\`\(p\_n - p\_\(n - 1\)\)\/p\_n\)]], "\[Times]100 to get the percent growth rate at the ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["n", FontSlant->"Italic"], "th"], TraditionalForm]]], "decade. ", StyleBox["Mathematica", FontSlant->"Italic"], " can do element-wise calculations with lists. For example, we could \ measure the population in thousands by dividing by 1000." }], "Text"], Cell[BoxData[ \(population/1000. \)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " applied the operation to each element of the list. Subtraction works the \ same way. In order to calculate ", Cell[BoxData[ \(TraditionalForm\`p\_n - p\_\(n - 1\)\)]], ", we can shift the population data by one decade and then subtract this \ from the orginal population list. Although, we also have to drop one element \ from the original population list so that the two lists are the same length." }], "Text"], Cell[BoxData[{ \(\(percentGrowth = \((Drop[population, 1] - Drop[population, \(-1\)])\)/ Drop[population, 1];\)\), "\n", \(\(ListPlot[percentGrowth, PlotJoined \[Rule] True, PlotLabel \[Rule] "\"];\)\)}], \ "Input"], Cell["\<\ Does it look like the U.S. population is growing exponentially?\ \>", "Text"], Cell[TextData[{ StyleBox["Exercise:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " Using the following population data, in billions of people, determine if \ the world population is growing exponentially. " }], "Text", Background->GrayLevel[0.750011]], Cell[BoxData[ \(worldPopulationPairs = {{1900, 1.65}, {1910, 1.75}, {1920, 1.86}, {1930, 2.07}, {1940, 2.30}, {1950, 2.52}, {1960, 3.02}, {1970, 3.70}, {1980, 4.44}, {1990, 5.27}, {2000, 6.06}}\)], "Input"], Cell[BoxData[ \(worldPopulation = Last[Transpose[worldPopulationPairs]]\)], "Input"], Cell[TextData[{ "Some commands can accept single entries or lists as arguments. Examples \ are the ", StyleBox["Solve", FontFamily->"Courier"], " and ", StyleBox["Plot", FontFamily->"Courier"], " commands." }], "Text"], Cell["We can solve one equation for one unknown.", "Text"], Cell[BoxData[ \(Solve[x\^4 - 6 x\^3 + \ 2 x - 5 \[Equal] 0, x]\)], "Input"], Cell["\<\ Or we can solve a list of two equations for a list of two unknowns.\ \>", "Text"], Cell[BoxData[ \(Solve[{12 x - 6 y == 7, x\^2 - 5 y == 2}, {x, y}]\)], "Input"], Cell[TextData[{ "Notice that the answer to the second ", StyleBox["Solve", FontFamily->"Courier"], " example is a list of lists. There is a list of two solutions, and each \ solution is a list giving a value for ", StyleBox["x", FontSlant->"Italic"], " and a value for ", StyleBox["y", FontSlant->"Italic"], "." }], "Text"], Cell["Here is the plot of one function.", "Text"], Cell[BoxData[ \(\(Plot[Sin[x], {x, \(-3\) \[Pi], 3 \[Pi]}];\)\)], "Input"], Cell["Here is the plot of a list of two functions.", "Text"], Cell[BoxData[ \(\(Plot[{Sin[x], x - x\^3\/6 + x\^5\/120}, {x, \(-3\) \[Pi], 3 \[Pi]}];\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Using Palettes for Input", "Subsection"], Cell[TextData[{ "You may have noticed that this ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook uses some fancy mathematical notation. You can build up \ mathematical expressions using the palettes that are accessible under the \ File menu. Try going to File \[Rule] Palettes \[Rule] BasicInput. When \ you click on one of the buttons in the BasicInput palette, that item is \ inserted at the current location in your ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook. " }], "Text"], Cell[TextData[{ StyleBox["Exercise:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " ", "See if you can use the BasicInput palette to create the expression, ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\(1\/\@\(x\^2 + 1\)\%4\) \ \[DifferentialD]x\)]], ", in a new Input cell. Then evaluate that cell to find the value of the \ integral. You will see that ", StyleBox["Mathematica", FontSlant->"Italic"], " can evaluate some pretty difficult integrals. If you are interested in \ learning more about hypergeometric functions, visit the MathWorld Web site at \ ", ButtonBox["http://mathworld.wolfram.com/GeneralizedHypergeometricFunction.\ html", ButtonData:>{ URL[ "http://mathworld.wolfram.com/GeneralizedHypergeometricFunction.html"], None}, ButtonStyle->"Hyperlink"], ", but be prepared for some heavy reading!" }], "Text", Background->GrayLevel[0.750011]], Cell["\<\ Explore the other palettes under the File menu to see the massive collection \ of symbols and mathematical forms that are available.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Creating Graphs and Animations", "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has a variety of commands for graphing functions:\nPlot \[LongDash] graph \ a function of one variable, or graph several functions of one variable \ together.\nPlot3D \[LongDash] graph a function of two variables\nListPlot \ \[LongDash] create a 2D scatter plot of a data set\nListPlot3D \[LongDash] \ create a 3D scatter plot of a data set\nParametricPlot \[LongDash] graph a \ parametrically defined curve in a plane\nParametricPlot3D \[LongDash] graph a \ parametrically defined curve or surface in 3D\nContourPlot \[LongDash] \ display a contour plot of a function of two variables\nDensityPlot \ \[LongDash] similar to a contour plot\nListDensityPlot \[LongDash] density \ plot of a data set\nLogPlot, LogLogPlot, LogListPlot, LogLogListPlot, \ PolarPlot, ErrorListPlot, TextListPlot, BarChart, PieChart, PlotVectorField, \ ListPlotVectorField, SphericalPlot3D \[LongDash] and many more in ", StyleBox["Mathematica", FontSlant->"Italic"], "'s add-on packages." }], "Text"], Cell["\<\ In the most basic version of a plot, the first argument is the function and \ the second argument is a list of three items representating the domain.\ \>", "Text"], Cell[BoxData[ \(\(Plot[ArcTan[x], {x, \(-6\), 6}];\)\)], "Input"], Cell["\<\ You can plot several functions together by making the first argument a list \ of functions.\ \>", "Text"], Cell[BoxData[ \(\(Plot[{Cos[x], 1 - x^2/2}, {x, \(-2\) \[Pi], 2 \[Pi]}];\)\)], "Input"], Cell["You can add annotations to the graph.", "Text"], Cell[BoxData[ \(\(Plot[Log[1 - x], {x, \(-3\), 1}, AxesLabel \[Rule] {"\", "\"}, PlotLabel \[Rule] "\"];\)\)], "Input"], Cell[TextData[{ "And you can change the style of the graph.", " Don't worry about the ", StyleBox["Needs", FontFamily->"Courier New"], " statement for now. We will look at that in the next section." }], "Text"], Cell[BoxData[{ \(Needs["\"]\), "\[IndentingNewLine]", \(\(Plot[\[ExponentialE]\^\(-x\^2\), {x, \(-\[Pi]\), \[Pi]}, PlotStyle \[Rule] {{Thickness[0.02], CadmiumYellow, Dashing[{0.05, 0.05}]}}, Background \[Rule] Azure, GridLines \[Rule] Automatic];\)\)}], "Input"], Cell["\<\ The individual functions in a plot can each have their own style. The \ PlotStyle option requires a list of lists. Each list is a collection of \ options for one function in the list of functions.\ \>", "Text"], Cell[BoxData[ \(\(Plot[{\[ExponentialE]\^x, 1 + x}, {x, \(-2\), 2}, PlotStyle \[Rule] {{SeaGreen}, {Peacock, Dashing[{0.05, 0.05}]}}];\)\)], "Input"], Cell[TextData[{ StyleBox["Exercise:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " Plot the sine and cosine functions using different colors and line \ thicknesses. If you want to experiment with new colors, all available color \ names are listed in the Help Browser under Add-ons\[Rule]Standard Packages\ \[Rule]Graphics\[Rule]Colors." }], "Text", Background->GrayLevel[0.750011]], Cell[TextData[{ "You can create animations by displaying a sequence of individual graphs. \ In the following example, the ", StyleBox["Table", FontFamily->"Courier New"], " command, which is used to creating lists, creates a list of graphs \ showing the Taylor polynomial approximations to Sin[x]. We use the ", StyleBox["PlotRange", FontFamily->"Courier New"], " option in order to guarantee that the range of ", StyleBox["y", FontSlant->"Italic"], " values for all of the graphs will be the same. Double-click on any of \ the graphs to start the animation. Controls will appear at the bottom of the \ window that allow you to adjust the speed of the animation." }], "Text"], Cell[BoxData[{ \(Needs["\"]\), "\[IndentingNewLine]", \(\(Table[ Plot[{Sin[ x], \[Sum]\+\(k = 0\)\%n\(\(\((\(-1\))\)\^k\) x\^\(2 k + 1\)\)\ \/\(\((2 k + 1)\)!\)}, {x, \(-4\) \[Pi], 4 \[Pi]}, PlotRange \[Rule] {\(-1.5\), 1.5}, PlotStyle \[Rule] {{EnglishRed}, {BlueViolet}}], {n, 0, 9}];\)\)}], "Input"], Cell[TextData[{ StyleBox["Exercise:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " Create an animation showing how different choices for ", StyleBox["a", FontSlant->"Italic"], " effect the graph of ", StyleBox["y", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`x\^3 + \ a\ x\)]], ". Include both negative and positive values for ", StyleBox["a", FontSlant->"Italic"], "." }], "Text", Background->GrayLevel[0.750011]] }, Closed]], Cell[CellGroupData[{ Cell["Sounds", "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can play functions as sounds, based on the frequency and amplitude of the \ function. Here is a sum of three tones, an A note at a frequency of 220 \ cycles per second, another A note one octave higher at 440 cycles per second, \ and a C note three half-steps above a 220-frequency A note. Since doubling \ the frequency moves up an octave and the 12-half-tone chromatic scale is \ logarithmic, you multiply a frequency by ", Cell[BoxData[ \(TraditionalForm\`\(\(2\^\(x/12\)\)\(,\)\)\)]], "the tone is moved up in pitch by ", StyleBox["x", FontSlant->"Italic"], " half-tones." }], "Text"], Cell[BoxData[ \(\(Play[ Sin[220\ 2 \[Pi]\ x] + .5\ Sin[440\ 2 \[Pi]\ x] + .25\ Sin[ 220\ 2\^\(3/12\)\ 2 \[Pi]\ x], {x, 0, 1}];\)\)], "Input"], Cell[TextData[{ StyleBox["Exercise:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " Create a pleasing combination of tones by adding together Sin functions \ of various harmonious frequencies. Hint: A major cord consists of three \ notes: the root note, the third note above the root, and the fifth note above \ the root. See the ", ButtonBox["Mathematics of Musical Scales", ButtonData:>{ URL[ "http://www.wordiq.com/definition/Mathematics_of_the_Western_music_\ scale"], None}, ButtonStyle->"Hyperlink"], " for more background information." }], "Text", Background->GrayLevel[0.750011]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Extending ", StyleBox["Mathematica", FontSlant->"Italic"], "'s Capabilities with Packages" }], "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has built-in functions to deal with problems in calculus, linear algebra, \ differential equations, discrete mathematics, number theory, numerical \ analysis, statistics and other fields in the mathematical sciences. Wolfram \ Research and independent developers have created many hundreds of ", StyleBox["Mathematica", FontSlant->"Italic"], " packages to extend these built-in functions to cover a variety of \ specific applications. Some of these packages come with ", StyleBox["Mathematica", FontSlant->"Italic"], ". You can see these in the Help Browser under the ", ButtonBox["Add-ons", ButtonData:>{"Loading Packages", None}, ButtonStyle->"AddOnsLink", ButtonNote->None], " button. Additional packages are available on the Web at ", ButtonBox["MathSource", ButtonData:>{ URL[ "http://library.wolfram.com/infocenter/MathSource/AppliedMathematics/"], None}, ButtonStyle->"Hyperlink"], "." }], "Text"], Cell[TextData[{ "If you want to use a package, you must first load it into your ", StyleBox["Mathematica", FontSlant->"Italic"], " session. For example, suppose that you want to display a 3D graph and \ include a shadow of that graph on the ", StyleBox["xy", FontSlant->"Italic"], "-plane. ", StyleBox["Mathematica", FontSlant->"Italic"], " does not have a built-in function to draw the shadow. If you look in the \ Help Browser under Add-ons \[Rule] Standard Packages \[Rule] Graphics \[Rule] \ Graphics3D, you will see that there is a function called ShadowPlot3D. Here \ are the two commands that you need to load the package and evaluate the \ ShadowPlot3D command." }], "Text"], Cell[BoxData[{ \(Needs["\"]\), "\[IndentingNewLine]", \(\(ShadowPlot3D[ Cos[\@\(x\^2 + y\^2 + 1\)], \ {x, \(-4\) \[Pi], 4 \[Pi]}, {y, \(-4\) \[Pi], 4 \[Pi]}, PlotPoints \[Rule] 30];\)\)}], "Input"], Cell["\<\ In the previous section, we loaded the package Graphics`Colors` so that we \ could plot graphs using English language names for colors instead of having \ to use numerical values for the red, green, and blue (RGB) components of the \ colors. For example, the RGB values for ForestGreen are\ \>", "Text"], Cell[BoxData[{ \(Needs["\"]\), "\[IndentingNewLine]", \(ForestGreen\)}], "Input"], Cell["\<\ and here are two ways to display a circle filled with ForestGreen.\ \>", "Text"], Cell[BoxData[{ \(\(Show[Graphics[{ForestGreen, Disk[{0, 0}, 2]}], AspectRatio \[Rule] Automatic];\)\), "\[IndentingNewLine]", \(\(Show[ Graphics[{RGBColor[0.133305, 0.545106, 0.133305], Disk[{0, 0}, 2]}], AspectRatio \[Rule] Automatic];\)\)}], "Input"], Cell[TextData[{ StyleBox["Exercise:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " Use the Graphics`ImplicitPlot package to plot a graph of ", Cell[BoxData[ \(TraditionalForm\`x\^4 + y\^4 = x\^2 - y\^2\)]], " for -1 \[LessEqual] ", StyleBox["x", FontSlant->"Italic"], " \[LessEqual] 1. You will need to look in the Help Browser to determine \ how to use the ImplicitPlot function." }], "Text", Background->GrayLevel[0.750011]], Cell[TextData[{ StyleBox["Exercise:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " Use the Graphics`Polyhedra package to display a stellated icosahedron." }], "Text", Background->GrayLevel[0.750011]] }, Closed]], Cell[CellGroupData[{ Cell["Where to Go from Here", "Subsection"], Cell[TextData[{ "There are many avenues you can explore to further your experience with ", StyleBox["Mathematica", FontSlant->"Italic"], ". Some good places to start are:\n\n ", ButtonBox["The Mathematica Book", ButtonData:>{"C", None}, ButtonStyle->"MainBookLink", ButtonNote->None], ", available in the Help Browser, especially the chapter, A Practical \ Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], "\n ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", ButtonBox["Standard Packages", ButtonData:>{"5.0.1", None}, ButtonStyle->"AddOnsLink", ButtonNote->None], "\n ", ButtonBox[" Mathematica Information Center", ButtonData:>{ URL[ "http://library.wolfram.com/infocenter"], None}, ButtonStyle->"Hyperlink"], "\n ", ButtonBox["Numerous books about Mathematica", ButtonData:>{ URL[ "http://library.wolfram.com/infocenter/Books/"], None}, ButtonStyle->"Hyperlink"] }], "Text"], Cell[TextData[{ "If you are interested in any of the following subject areas, visit these \ links to get started.\n\nCalculus\n The ", StyleBox["Mathematica", FontSlant->"Italic"], " Book / Advanced Mathematics in ", StyleBox["Mathematica", FontSlant->"Italic"], " / Calculus / ", ButtonBox["Contents", ButtonData:>{"3.5", None}, ButtonStyle->"MainBookLink", ButtonNote->None], "\n The ", StyleBox["Mathematica", FontSlant->"Italic"], " Book / Advanced Mathematics in ", StyleBox["Mathematica", FontSlant->"Italic"], " / Series, Limits and Residues / ", ButtonBox["Contents", ButtonData:>{"3.6", None}, ButtonStyle->"MainBookLink", ButtonNote->None], "\n Add-ons / Standard Packages / ", ButtonBox["Calculus", ButtonData:>{"Calculus", None}, ButtonStyle->"AddOnsLink", ButtonNote->None], "\nLinear Algebra\n The ", StyleBox["Mathematica", FontSlant->"Italic"], " Book / Advanced Mathematics in ", StyleBox["Mathematica", FontSlant->"Italic"], " / Linear Algebra / ", ButtonBox["Contents", ButtonData:>{"3.7", None}, ButtonStyle->"MainBookLink", ButtonNote->None], "\n Add-ons / Standard Packages / ", ButtonBox["Linear Algebra", ButtonData:>{"LinearAlgebra", None}, ButtonStyle->"AddOnsLink", ButtonNote->None], "\nStatistics\n The ", StyleBox["Mathematica", FontSlant->"Italic"], " Book / A Practical Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " / Numerical Mathematics / ", ButtonBox["Statistics Packages", ButtonData:>{"1.6.7", None}, ButtonStyle->"MainBookLink", ButtonNote->None], "\n Add-ons / Standard Packages / ", ButtonBox["Statistics", ButtonData:>{"Statistics", None}, ButtonStyle->"AddOnsLink", ButtonNote->None], "\nDifferential Equations\n The ", StyleBox["Mathematica", FontSlant->"Italic"], " Book / A Practical Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " / Symbolic Mathematics / ", ButtonBox["Differential Equations", ButtonData:>{"1.5.9", None}, ButtonStyle->"MainBookLink", ButtonNote->None], "\n The ", StyleBox["Mathematica", FontSlant->"Italic"], " Book / Advanced Mathematics in ", StyleBox["Mathematica", FontSlant->"Italic"], " / Calculus / ", ButtonBox["Differential Equations", ButtonData:>{"3.5.10", None}, ButtonStyle->"MainBookLink", ButtonNote->None], "\nDiscrete Mathematics\n Add-ons / Standard Packages / ", ButtonBox["DiscreteMath", ButtonData:>{"DiscreteMath", None}, ButtonStyle->"AddOnsLink", ButtonNote->None], "" }], "Text"] }, Closed]] }, Open ]] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowSize->{853, 668}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, StyleDefinitions -> "PastelColor.nb" ] (******************************************************************* Cached data follows. 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