(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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[ 195644, 8060]*) (*NotebookOutlinePosition[ 196495, 8090]*) (* CellTagsIndexPosition[ 196451, 8086]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[" Math 2224\n Intro Lab"], "Title", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSize->28], Cell[CellGroupData[{ Cell[TextData["Introduction - Basics"], "Section", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Monaco"], Cell[TextData[{ StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[ " is a powerful software package which combines numerical computations, \ symbolic manipulation, graphics, and text.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Arithmetic Symbols and Calculation", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}]], "Subsection", Evaluatable->False, AspectRatioFixed->True, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], Cell[TextData[{ StyleBox[ " 1) Addition is represented by +.\n 2) Subtraction is represented by \ -.\n 3) Division is represented by /.\n 4) Exponentiation is represented \ by ^.\n 5) Multiplication can be represented in different \n ways:\n \ e.g., 7 times 4 may be 7*4 or 7 4 .(i.e.,* or space)\n 7 \ times x may be 7*x or 7x (it is not necessary to \n \ leave a space between \n \ the number and letter)\n x times y ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["must", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"Underline"->True}], StyleBox[" be represented as x*y or x y. \n (", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[ " would take the more conventional \"xy\"\n as a new \ variable.)\n Of course parentheses may be used for grouping to indicate \n \ order of evaluation. 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We will use \"==\" when writing equations to\ \n be solved or graphed.\n \n 4) != means not equal\n \n \ 5) <,<=, =>, > are less than, less than or equal to, etc.\n \n 6) To \ make an arrow used in certain commands use the minus sign \n followed by \ the greater than sign: ->", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Text and Titles"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Now let's look at typing text like the sentence you are now reading. To \ get out of \"input\" mode and into \"text\" mode which allows you to put in \ your name, comments, titles, etc.:\n \nOn Mac machines: Choose Style from the \ menu, then CellStyle,\n then Text. Now choose a font by \ returning to the \n Style menu and choosing Font.\nOn DOS \ machines: You must start typing first to create a cell. 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Functions \ (such as the trigonometric functions) \n and commands also begin with \ capital letters, are spelled out \n unless there is a standard \ abbreviation, and arguements are \n enclosed in square brackets.\n\n Some \ exceptions are the general exponential and logarithmic \n functions. For \ example, e", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["2x+y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" could be represented as either\n ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["E^(2x+y)", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" or as ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Exp[2x+y]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ ". 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", CellFrame->True, Evaluatable->False, PageBreakAbove->False, PageBreakBelow->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Mathematica", CellFrame->True, Evaluatable->False, PageBreakAbove->False, PageBreakBelow->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[ " will algebraically simplify an \n expression, but keep in mind, what \ is \"simplest form\"\n sometimes depends on what you are going to do with \ the \n expression. 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The form is:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["\n ", CellFrame->True, AspectRatioFixed->True, FontWeight->"Plain"], StyleBox[" D[function,variable]", CellFrame->True, AspectRatioFixed->True], StyleBox[ " to find the derivative of a \n function with respect to a specific \ variable. 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If this takes too \n long or if ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[ " is unable to find an antiderivative, \n you may use a command for \ numerical integration:\n \n ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["NIntegrate[expression,{variable,a,b}]", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["\n \n Realize that the", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox[" NIntegrate", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " command may return a decimal \n approximation rather than an exact \ answer.\n ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{ Cell["Integrate[x Sin[x],x]", "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ -(x*Cos[x]) + Sin[x]\ \>", "\<\ -(x Cos[x]) + Sin[x]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["integ=Integrate[(7+2x^2)/((1+x^2)*(9+4x^2)),{x,0,1}]", "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ Pi/4 - ArcTan[2/3]/3\ \>", "\<\ 2 ArcTan[-] Pi 3 -- - --------- 4 3\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["N[integ]", "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 0.5893972955482591259\ \>", "\<\ 0.589397\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["NIntegrate[(7+2x^2)/((1+x^2)*(9+4x^2)),{x,0,1}]", "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 0.5893972955482569202\ \>", "\<\ 0.589397\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Exercise 2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSize->18]], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" 1. Define the following functions in ", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Mathematica", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[ " using proper\n syntax: (Warning: Remember to clear f, g, and h \ first.)\n 1) f(x) = x", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["2", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" - 2x - 24\n 2) g(x) = cos(tan(x", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["2", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["))\n 3) h(s,t) = 3s", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["2", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["-2t", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["3", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[ "\n 2. 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Neither graph shows the \n \ same scale on the x-axis as on the y-axis. Also on the first \n graph we \ might rather have the x-axis marked off in multiples \n of \[Pi] than in \ integers. On the second graph notice that the \n axes do not cross at the \ origin, but at the point (0,1) and \n the y-values ranging from .65 to 1 \ do not allow us to see \n the whole curve from x = -1 to x = 1. These and \ other style- \n related effects are all things we can change with the \ options \n listed in the table on the following page.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ " Option Usage\n \n Axes->False \ Causes no axes to be drawn \n AspectRatio->Automatic Gives \ the same scale on the x and y\n axes\n \ AxesOrigin->{a,b} Makes the axes cross at the point\n \ (a,b). To make axes cross at the\n \ origin, let a = b = 0.\n AxesLabel->{\"__\",\"__\"} Labels the \ horizontal and verical\n axes by what is \ contained within\n the quotation marks.\n \ PlotLabel->\"________\" Labels the graph by title contained\n \ within quotation marks\n Gridlines->Automatic Puts \ in gridlines on graph\n Ticks->{{a", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["1", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",...,a", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["m", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["},{b", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["1", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[",...,b", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["n", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[ "}} \n Puts specified values on x and y\n \ axes respectively\n PlotRange->{c,d} \ Specifies the range of function \n values to \ be displayed on graph\n PlotRange->All Attempts to show entire \ graph\n PlotStyle->Dashing[{L}] Causes graph to be drawn in dashes\n \ with the dashes and spaces L times\n \ the width of the plot, e.g. L may\n \ be .05.\n PlotStyle->RGBColor[r,s,t] Causes the graph to be drawn in \n \ color. [1,0,0] gives red,[0,1,0] \n \ green,and [0,0,1] blue. Using \n \ numbers between 0 and 1 for r,s,\n and t gives \ other mixtures of\n these colors.\n \ PlotStyle->GrayLevel[m] Causes the graph to be drawn in a\n \ shade of gray. \"m\" must be \n \ between 0 and 1 with 0 being\n black and 1 \ being white.\n DisplayFunction->Identity Prevents the graph from being \ shown\n although it may be shown later by\n \ adding the option:\n \ DisplayFunction->$DisplayFunction \n Framed->True Draws a box \ around graph \n PlotPoints->n Uses n plot points to draw the \ \n graph instead of the default \n \ 25. \n PlotStyle->Thickness[n] Changes \ thickness of curve.", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"] }], "Text", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" It is ", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["not necessary to memorize", Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"Underline"->True}], StyleBox[ " these options. 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of y = 5 sin x", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" \n from x = -", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["\[Pi]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" to x = ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["\[Pi]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ ". 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Make \ the curve gray level (.3) and \n make the scale the same on both the x- \ and y-axes.\n\n 3) Use the ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Show", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command to plot the two functions on the same\n axes.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction - Solving Equations", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Monaco"]], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" \n We will now look at three other basic commands in ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[" - \n the ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Solve", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command, the ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["NSolve", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command, and the ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["FindRoots", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" \n command. ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[ " can find exact solutions for a polynomial \n equation of degree four \ or less. In certain cases ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[ " \n can solve higher degree polynomial equations and other \n \ algebraic equations. If not, the ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["NSolve", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command will give \n ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["approximate", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"Underline"->True}], StyleBox[" solutions to these equations. We will use the \n ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["FindRoot", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " command to approximate roots of equations involving \n non-algebraic \ functions, such as trigonometric functions.\n ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], Cell[TextData[{ StyleBox[" ", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Solve[LHS==RHS]", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " Can be used if the equation \n \ involves only one variable.\n ", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Solve[LHS==RHS,var]", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" Tells ", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Mathematica", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[ " what variable \n to solve for.\n \ The ", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Solve", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " command is used primarily to solve for real or \n complex roots \ of polynomials up to the 4th degree. Note\n that in the example \ below, the ", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["N[ ]", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" with the ", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Solve", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " \n command inside the brackets causes us to get decimal \n \ representations of the solutions.", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"] }], "Text", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Solve[x^2 + 7 x + 4 == 0]", "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {{x -> (-7 - 33^(1/2))/2}, {x -> (-7 + 33^(1/2))/2}}\ \>", "\<\ -7 - Sqrt[33] -7 + Sqrt[33] {{x -> -------------}, {x -> -------------}} 2 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["N[Solve[x^2 + 7 x + 4 == 0]]", "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {{x -> -6.37228132326901433}, {x -> -0.6277186767309856702}}\ \>", "\<\ {{x -> -6.37228}, {x -> -0.627719}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Solve[a x^2 + b x + c == 0,x]", "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{x -> (-b - (b^2 - 4*a*c)^(1/2))/(2*a)}, {x -> (-b + (b^2 - 4*a*c)^(1/2))/(2*a)}}\ \>", "\<\ 2 2 -b - Sqrt[b - 4 a c] -b + Sqrt[b - 4 a c] {{x -> ---------------------}, {x -> ---------------------}} 2 a 2 a\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Solve[x^3 + x + 10 == 0]", "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{x -> -2}, {x -> 1 - 2*I}, {x -> 1 + 2*I}}\ \>", "\<\ {{x -> -2}, {x -> 1 - 2 I}, {x -> 1 + 2 I}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["Solve[x^4 - 1 == 0]", "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{x -> -1}, {x -> -I}, {x -> I}, {x -> 1}}\ \>", "\<\ {{x -> -1}, {x -> -I}, {x -> I}, {x -> 1}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ StyleBox[" Mathematica", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[ " can also solve a system of two or more equations \n simultaneously. The \ format is:\n ", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Solve[{equation1,equation2}] ", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"] }], "Text", CellFrame->True, Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Suppose for example we need to find the point(s) of intersection of the \ parabola y", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[ " = 4x and the line 4x - 3y = 4. The solutions will be given as ordered \ pairs. ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{ Cell["Solve[{y^2==4x,4x-3y==4}]", "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{x -> 1/4, y -> -1}, {x -> 4, y -> 4}}\ \>", "\<\ 1 {{x -> -, y -> -1}, {x -> 4, y -> 4}} 4\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell["\<\ Now let's try to solve an eighth degree polynomial equation.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{ Cell["Solve[x^8 - 2x + 1 == 0]", "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {{x -> 1}, ToRules[Roots[x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 == 1, x]]}\ \>", "\<\ 2 3 4 5 6 7 {{x -> 1}, ToRules[Roots[x + x + x + x + x + x + x == 1, x]]}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ StyleBox[" Notice what happened when we tried to use the ", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Solve", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" command \n here. ", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Mathematica", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[ " did find one root, but couldn't find the \n others. Among the set of \ complex numbers, that equation has \n ", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["eight", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontVariations->{"Underline"->True}], StyleBox[" roots. Let's investigate the command ", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["NSolve", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " to obtain \n decimal approximations for all eight roots.\n \n The form \ for this command is\n ", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["NSolve[lhs==rhs,variable]", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["\n .", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"] }], "Text", CellFrame->True, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"], Cell[CellGroupData[{ Cell["\<\ NSolve[x^8 - 2x + 1 == 0,x]\ \>", "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {{x -> -1.053186583070133583 - 0.4835575290388050698*I}, {x -> -1.053186583070133583 + 0.4835575290388050698*I}, {x -> -0.3089096008009623204 - 1.090040293978604644*I}, {x -> -0.3089096008009623204 + 1.090040293978604644*I}, {x -> 0.5020170551781655117}, {x -> 0.6110876562820131472 - 0.8843359538908265619*I}, {x -> 0.6110876562820131472 + 0.8843359538908265619*I}, {x -> 1.}}\ \>", "\<\ {{x -> -1.05319 - 0.483558 I}, {x -> -1.05319 + 0.483558 I}, {x -> -0.30891 - 1.09004 I}, {x -> -0.30891 + 1.09004 I}, {x -> 0.502017}, {x -> 0.611088 - 0.884336 I}, {x -> 0.611088 + 0.884336 I}, {x -> 1.}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[StyleBox[" \n What about a non-algebraic equation?", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier"]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Solve[5 Cos[x]==x]", "Input", AspectRatioFixed->True], Cell["\<\ Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way.\ \>", "Message", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ Solve[5*Cos[x] == x]\ \>", "\<\ Solve[5 Cos[x] == x]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell["NSolve[5 Cos[x]==x]", "Input", AspectRatioFixed->True], Cell["\<\ Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way.\ \>", "Message", Evaluatable->False, AspectRatioFixed->True], Cell[OutputFormData["\<\ NSolve[5*Cos[x] == x]\ \>", "\<\ NSolve[5 Cos[x] == x]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ StyleBox[" Look at the messages ", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Mathematica", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontSlant->"Italic"], StyleBox[ " gave us when we tried to \n solve the equation 5 Cos x = x with the ", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["Solve", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" and ", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["NSolve", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" \n commands. In this situation, we must use the ", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["FindRoot", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" \n command. ", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier"], StyleBox["FindRoot", CellFrame->True, Evaluatable->False, PageBreakWithin->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " uses a process similar to Newton's \n Method which requires an \ initial estimate for a root. A \n graph of the function can help with \ the estimate. 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