(************** Content-type: application/mathematica ************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 416101, 12844]*) (*NotebookOutlinePosition[ 416868, 12870]*) (* CellTagsIndexPosition[ 416824, 12866]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["\tCynthia Grantz", FontWeight->"Bold"], ", through her insightful comments and suggestions arising from computer \ lab sessions teaching, spared no time or efforts in making this presentation \ as student friendly as possible.\n\t", StyleBox["Hristo Kojouharov", FontSize->14, FontWeight->"Bold"], StyleBox[", made excellent website arrangements for this tutorial by \ incorporating various degrees of flexibility for its use.", FontSize->14], "\n\t\t\t\t\t\t\t\t\t\t", StyleBox["Andrzej Korzeniowski", FontSize->16, FontWeight->"Plain", FontSlant->"Italic"] }], "Subsubtitle", FontSize->14, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[TextData[{ StyleBox["Double click on the outer bracket of the cell holding a topic ", "Subsubtitle", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["(toggle switch for open - close)\n", "Subsubtitle", FontColor->GrayLevel[0.100008]], StyleBox["Modify the expressions in the input cells (BOLD FACE) and execute\ \n\n\t\t\t\t\t\t\t ", "Subsubtitle", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[" Creativity is everything", "Subsubtitle", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["\t", FontSize->14] }], "Subtitle", TextAlignment->Left, TextJustification->0, Background->GrayLevel[0.900008]], Cell[CellGroupData[{ Cell[TextData[{ " ", StyleBox[" ", "Subtitle", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["Basic Calculation and Graphing ", "Subtitle", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", "Subtitle", FontSize->16, FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", "Subtitle", FontSize->16, FontVariations->{"CompatibilityType"->0}], StyleBox[" ", "Subtitle", FontSize->16] }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox[" Mathematica", FontSlant->"Italic"], " " }], "Subtitle", CellMargins->{{Inherited, -22}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0], Cell["\<\ Dr. Andrzej Korzeniowski Department of Mathematics, UTA\ \>", "Subsubtitle"] }, Open ]], Cell[CellGroupData[{ Cell["Lesson 1", "Subtitle"], Cell[CellGroupData[{ Cell["Preliminaries", "Section", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[TextData[{ StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Bold"], " operations are based on the concept of a list {a, b, ..., c} where a, \ b,... are symbols or numbers. Use built-in palettes : BasicInput (shown as \ default next to the notebook) found in Files\[Rule]Palettes\[Rule]BasicInput. \ To put an expression of choice in a given place click the cursor in that \ place and then click the desired item from palette. Use Format\[Rule] Show \ ToolBar to see in the left-upper corner the attribute of the cell containing \ the active cursor. Input cells hold commands that execute mathematical \ operations while Output", StyleBox[" ", "Program"], "cells", StyleBox[" ", "Program"], "store the answer. Other layout options are: Text, Title, Section, etc. \ (word processing only and not to be executed). Expressions are stored in \ cells displayed as blue brackets on the right-hand side. Cells can be \ formatted or operated on by being selected (clicking on cell bracket to \ highlight it in black) then going to Cell (main menu) and using pull-down \ menus. Most useful initially are: Cell Grouping (first choose Manual \ Grouping \[Rule]Ungroup Cells) so each executable expression is entered in a \ separate single Input cell for easy corrections, Divide Cell (put cursor in a \ desired place in the cell), Merge Cell (highlight cells to be merged). Use \ the online HELP in the main menu to familiarize yourself with many other \ features. To delete a cell highlight the cell and press Delete key. Use copy, \ cut, paste, undo, from Edit to avoid unnecessary typing. " }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ StyleBox["MULTIPLICATION", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[".", FontVariations->{"Underline"->True}], " ", "'"}], "*", "'", " ", "in", " ", "multiplication", " ", "of", " ", "symbols", " ", "x", "*", "y", " ", "may", " ", "be", " ", "substituted", " ", "by", " ", "a", " ", "blank"}], ",", " ", \(i . e . \), ",", \(x* y = \(x\ y . \ \ \ If\ first\ factor\ is\ a\ number\ then\ 5 x = \(5\ x = 5*x\)\)\)}], TraditionalForm]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(2 + 2\)], "Input", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[BoxData[ \(4\)], "Output"] }, Open ]], Cell[BoxData[ FormBox[ RowBox[{\( (*\ comment\ by\ notebook\ user\ *) \), " ", RowBox[{"is", " ", "ignored", " ", StyleBox[ RowBox[{"by", StyleBox["Mathematica", FontSlant->"Italic"]}]], " ", "while", " ", "executing", " ", "the", " ", "cell", " ", "containing", " ", \(\(comment\)\(.\)\)}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 106}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Factor[\ 24 - 50\ x + 35\ x\^2 - 10\ x\^3 + x\^4]\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\)], "Input", CellMargins->{{Inherited, 101}, {Inherited, Inherited}}], Cell[BoxData[ \(\((\(-4\) + x)\)\ \((\(-3\) + x)\)\ \((\(-2\) + x)\)\ \((\(-1\) + x)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Together[1\/x + x\^2 - 1]\)], "Input"], Cell[BoxData[ \(\(1 - x + x\^3\)\/x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Apart[\((x - 1)\)\ \((x - 2)\)\ \((x - 3)\)\ \((x - 4)\)]\)], "Input"], Cell[BoxData[ \(24 - 50\ x + 35\ x\^2 - 10\ x\^3 + x\^4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(Expand[\((a + b)\)\^5]\)\)\)], "Input"], Cell[BoxData[ \(a\^5 + 5\ a\^4\ b + 10\ a\^3\ b\^2 + 10\ a\^2\ b\^3 + 5\ a\ b\^4 + b\^5\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\(Simplify[ 1\/\(x - 1\) - 1\/\(x + 1\)]\)\(\ \ \ \ \ \)\)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\( (*\ \(\(Shortcuts\)\(:\)\)\ // Factor, \(\(...\) \(\(,\)\(\ \)\(\(//\)\(Simplify\ or\ FullSimplify\)\ \)\)\)\ *) \)\)\)], "Input", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[BoxData[ \(2\/\(\(-1\) + x\^2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\(5!\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\( (*\ \ n\ \(factorial : \ \ n\) = \(1*2\ * ... \) \((n - 1)\)* n\ \ *) \)\)\)], "Input"], Cell[BoxData[ \(120\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Sum]\+\(k = 1\)\%n k\^5\)], "Input", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[BoxData[ \(1\/12\ n\^2\ \((1 + n)\)\^2\ \((\(-1\) + 2\ n + 2\ n\^2)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Integral]Sin[x]\ Tan[x] \[DifferentialD]x\)], "Input", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[BoxData[ \(\(-Log[Cos[x\/2] - Sin[x\/2]]\) + Log[Cos[x\/2] + Sin[x\/2]] - Sin[x]\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["TYPING TEXT.", FontWeight->"Bold", FontVariations->{"Underline"->True}], " To type a text highlight a cell and choose Text from the ToolBar. \n\n\ ", StyleBox["NUMERICAL EVALUATION.", FontWeight->"Bold", FontVariations->{"Underline"->True}], " To evaluate an expression Expr numerically use N[Expr] or Expr//N. To \ evaluate with n digit of accuracy use N[Expr,n]" }], "Text", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[Pi]\)\(//\)\(\(N\)\(\ \ \ \ \)\)\( (*\ shortcut\ for\ N[\[Pi]], \ i . e . \ to\ evaluate\ numerically\ *) \)\)\)], "Input", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[BoxData[ \(3.141592653589793`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\(N[\[Pi], 100]\)\(\ \ \ \ \ \ \ \ \ \ \ \ \)\)\( (*\ with\ accuray\ of\ 100\ decimal\ places\ *) \)\)\)], "Input", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[BoxData[ \(3.1415926535897932384626433832795028841971693993751058209749445923078164\ 0628620899862803482534211706798214808651`100\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["FUNCTIONS", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[".", FontVariations->{"Underline"->True}], " All elementary functions ", StyleBox["MUST", FontColor->RGBColor[1, 0, 0]], " begin with a", StyleBox[" CAPITAL LETTER\n", FontColor->RGBColor[1, 0, 0]], "followed by ", StyleBox["BRACKET [x]", FontColor->RGBColor[1, 0, 0]], " in place of parenthesis (x). List of basic functions: \n", StyleBox["Sin[x], Cos[x],Tan[x], Cot[x], ArcSin[x], ArcTan[x], Log[x]", FontColor->RGBColor[0, 0, 1]], "(=ln x = natural logarithm), ", StyleBox["Log [b, x]", FontColor->RGBColor[0, 0, 1]], " = ", Cell[BoxData[ \(TraditionalForm\`log\_\(\(\ \)\(b\)\)\)]], "x), ", Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^x\)], FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], "(exponential function base natural number \[ExponentialE] = 2.71828182, \ where ", StyleBox["\[ExponentialE] must be taken from the BasicInput palette (do ", FontColor->RGBColor[1, 0, 0]], StyleBox["NOT", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], StyleBox[" use 'e' from the keyboard", FontColor->RGBColor[1, 0, 0]], "), \n", StyleBox["Abs[x] ", FontColor->RGBColor[0, 0, 1]], "(= |x| = absolute value of x ).\n\n", StyleBox["BRACKETS ", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" [ . ]", FontWeight->"Bold"], " Used 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", FontVariations->{"Underline"->True}], "\nMost equations (e.g., polynomials of degree 5 or higher) cannot be \ solved symbolically. \nOne uses in those cases the built-in functions to \ obtain approximate numerical solution." }], "Text", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(NSolve[Sin[x] == 1, x]\)\(\ \ \)\)\)], "Input", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[BoxData[ \(Solve::"ifun" \(\(:\)\(\ \)\) "Inverse functions are being used by \!\(Solve\), so some solutions may \ not be found."\)], "Message"], Cell[BoxData[ \({{x \[Rule] 1.5707963267948966`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\(NRoots[x\^6 - 1 == x, x]\)\(\ \ \ \)\)\( (*\ polynomial\ equations\ only\ *) \)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\)], "Input", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[BoxData[ \(x == \(-0.778089598678601`\) || x == \(-0.6293724284703148`\) - 0.7357559529997764`\ \[ImaginaryI] || x == \(-0.6293724284703148`\) + 0.7357559529997764`\ \[ImaginaryI] || x == \(\(0.4510551586088556`\)\(\[InvisibleSpace]\)\) - 1.002364571587165`\ \[ImaginaryI] || x == \(\(0.4510551586088556`\)\(\[InvisibleSpace]\)\) + 1.002364571587165`\ \[ImaginaryI] || x == 1.1347241384015194`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\(NSolve[Log[x] + Sin[x] == 1, x]\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\( (*\ unable\ to\ solve\ *) \)\)\)], "Input", CellMargins->{{Inherited, 174}, {Inherited, Inherited}}], Cell[BoxData[ \(Solve::"tdep" \(\(:\)\(\ \)\) "The equations appear to involve the variables to be solved for in an \ essentially non-algebraic way."\)], "Message"], Cell[BoxData[ \(NSolve[Log[x] + Sin[x] == 1, x]\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["GRAPH TRACING.", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" ", FontWeight->"Bold"], "One can change the position and size of the graph by using the mouse: \ Click on the graph+hold+drag \n(= moving), click+hold corners+drag (= \ sizing). 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Cell[CellGroupData[{ Cell[BoxData[ \(r[h_] = Expand[\(-x\^4\) + \((h + x)\)\^4]/h // Simplify\)], "Input"], Cell[BoxData[ \(h\^3 + 4\ h\^2\ x + 6\ h\ x\^2 + 4\ x\^3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[r[h], h -> 0]\)], "Input"], Cell[BoxData[ \(4\ x\^3\)], "Output"] }, Open ]], Cell["\<\ To evaluate f [x] starting at x = a, ending at x = b with increment dx, use \ Table[{x, f [x]},{x, a, b, dx}] // TableForm.\ \>", "Text"], Cell[BoxData[ \(f[x_] := Sin[x]\/x\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Table[{x, f[x]}, {x, .5, .05, \(- .05\)}]\)\(//\)\(TableForm\)\(\n\)\)\)], \ "Input"], Cell[BoxData[ TagBox[GridBox[{ {"0.5`", "0.958851077208406`"}, {"0.45`", "0.9665900758027339`"}, {"0.4`", "0.9735458557716263`"}, {"0.35`", "0.9797080213012895`"}, {"0.3`", "0.9850673555377986`"}, {"0.25`", "0.9896158370180917`"}, {"0.19999999999999996`", "0.993346653975306`"}, {"0.14999999999999997`", "0.9962542164906615`"}, {"0.09999999999999998`", 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NSolve[\(f'\)[x] == 0, x]\)], "Input"], Cell[BoxData[ \({\(-0.651509027960396647`\) - 0.446475062141267908`\ I, \(-0.651509027960396647`\) + 0.446475062141267908`\ I, \(\(0.204709445032148718`\)\(\ \[InvisibleSpace]\)\) - 0.806958808211415501`\ I, \(\(0.204709445032148718`\)\(\ \[InvisibleSpace]\)\) + 0.806958808211415501`\ I, 0.90265202905377464`, 2.56237570823129257`}\)], "Output"] }, Open ]], Cell["Pick the real solutions only (use cut & paste )", "Text"], Cell[BoxData[ \(\(critnum = {0.902652, 2.56238};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(f[critnum]\)\(\ \ \ \ \)\( (*\ evaluate\ f\ at\ critical\ numbers\ *) \)\)\)], "Input"], Cell[BoxData[ \({5.1840951156744115`, \(-105.931126580864054`\)}\)], "Output"] }, Open ]], Cell["\<\ To verify that f [x] achieves a local max at x = .902652 with max = \ 5.1841 and local min at x = 2.56238 with min = f [2.56238] = -105.931 one can utilize the second derivative test as \ follows:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(f'\)'\)[critnum]\)], "Input"], Cell[BoxData[ \({\(-34.5799382448453673`\), 759.58969064097257`}\)], "Output"] }, Open ]], Cell["\<\ which shows that f '' is negative (as required) at x = .902652 and f '' is \ positive (as required) at x = 2.56238.\ \>", "Text"], Cell["\<\ Another way to find the min or max of f [x] is to use the built-in numerical \ algorithms as follows (provided that some initial approximation to the critical numbers is established by plotting \ f [x ] ).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindMinimum[f[x], {x, 2.5}]\)], "Input"], Cell[BoxData[ \({\(-105.931126587859636`\), {x \[Rule] 2.56237570836894512`}}\)], "Output"] }, Open ]], Cell[BoxData[{ FormBox[ RowBox[{\(To\ find\ the\ max\ one\ applies\ FindMinimum[ . ]\ to\ the\ \ function, \ \(-f\ [x]\)\ , \ based\ on\ the\ \(fact : \ max {f\ [x]\ }\) = \(-min\) {\(-f\ [x]\)\ }\), " "}], TraditionalForm], "\n", FormBox[ RowBox[{"(", RowBox[{\(seeks\ the\ minimum\ along\ the\ graph\ of\ f\ [ x]\ reflected\ across\ the\ x\), "-", RowBox[{ "axis", " ", "and", " ", "after", " ", "finding", " ", "the", " ", "minimum", " ", "the", " ", "graph", " ", "is", " ", "reflected", "\n", FormBox[\(\(\(back\ to\ the\ graph\ of\ f\ [ x]\)\(\ \)\()\)\)\(.\)\), "TraditionalForm"]}]}]}], TraditionalForm]}], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindMinimum[\(-f[x]\), {x, 1}]\)], "Input"], Cell[BoxData[ \({\(-5.1840951156744266`\), {x \[Rule] 0.902652030867278476`}}\)], "Output"] }, Open ]], Cell[TextData[{ "so taking ", Cell[BoxData[ \(TraditionalForm\`-\)]], "(", Cell[BoxData[ \(TraditionalForm\`-\)]], "5.1841) = 5.1841 gives the correct maximum attained at x = 0.902652." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Absolute Extrema on [a,b]", "Section"], Cell["\<\ Continuous function on a closed bounded interval attains its absolute min and \ absolute max, quite often at the endpoints of the interval [a,b]. 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-3 and around 10. To find the positive solution take the initial guess as 9:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{\(x0 = 9\), ";", "\n", RowBox[{\(f[x_]\), ":=", RowBox[{ FormBox[\(x\^4\), "TraditionalForm"], "-", RowBox[{"10", FormBox[\(x\^3\), "TraditionalForm"]}], "-", FormBox[\(x\^2\), "TraditionalForm"], "+", \(3 x\), " ", "-", " ", "50"}]}]}], "\n", \(\(f'\)[x]\)}], "Input"], Cell[BoxData[ \(3 - 2\ x - 30\ x\^2 + 4\ x\^3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x1 = x0 - f[x0]\/\(f'\)[x0]\)], "Input"], Cell[BoxData[ \(5072\/471\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(x1 // N\)], "Input"], Cell[BoxData[ \(10.7685774946921442`\)], "Output"] }, Open ]], Cell["\<\ To streamline computations define a new function g[x] which evaluates \ numerically the first 25 digits of approximation sequence:\ \>", "Text"], Cell[BoxData[ \(g[x_] := N[x - f[x]\/\(f'\)[x], 25]\)], "Input"], Cell["Then", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x1 = g[x0]\)], "Input"], Cell[BoxData[ \(10.76857749469214437367303609341825902336`25\)], "Output"] }, Open ]], Cell["\<\ To compute n iterations starting at x0 one uses the NestList[g, x0, n] as \ follows:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(NestList[g, x0, 6]\)\(//\)\(TableForm\)\(\n\)\)\)], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ {"9"}, {"10.76857749469214437367303609341825902336`25"}, {"10.216955306206155083503611723443462405`24.1064"}, {"10.12058299258635588182831319643859182795`23.2099"}, {"10.117806239312983280218683208819809`22.3144"}, {"10.11780397413295562844228212398523`21.4188"}, {"10.117803974131449032521296575722705`20.5233"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], TableForm[ {9, 10.76857749469214437367304`25, 10.21695530620615508350361`24.1064, 10.1205829925863558818283`23.2099, 10.117806239312983280219`22.3144, 10.11780397413295562844`21.4188, 10.1178039741314490325`20.5233}]]], "Output"] }, Open ]], Cell[TextData[{ " Compare the above results to built-in functions such as NRoots[expr[x] = \ = 0, x], NSolve[ expr[x] = = 0], FindRoot[expr[x] = = 0,{x, ", Cell[BoxData[ \(TraditionalForm\`x\_0\)]], "}],\n where ", Cell[BoxData[ \(TraditionalForm\`x\_\(\(0\)\(\ \ \)\)\)]], "is the initial approximation chosen in the vicinity of the exact \ solution." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NRoots", "[", RowBox[{ RowBox[{ RowBox[{ FormBox[\(x\^4\), "TraditionalForm"], "-", RowBox[{"10", FormBox[\(x\^3\), "TraditionalForm"]}], "-", FormBox[\(x\^2\), "TraditionalForm"], "+", \(3 x\), " ", "-", " ", "50"}], "==", "0"}], ",", "x"}], "]"}]], "Input"], Cell[BoxData[ \(x == \(-1.70506413927755069`\) || x == \(\(0.79363008257305081`\)\(\[InvisibleSpace]\)\) - 1.50613717733980889`\ I || x == \(\(0.79363008257305081`\)\(\[InvisibleSpace]\)\) + 1.50613717733980889`\ I || x == 10.1178039741314496`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"NSolve", "[", RowBox[{ RowBox[{ RowBox[{ FormBox[\(x\^4\), "TraditionalForm"], "-", RowBox[{"10", FormBox[\(x\^3\), "TraditionalForm"]}], "-", FormBox[\(x\^2\), "TraditionalForm"], "+", \(3 x\), " ", "-", " ", "50"}], "==", "0"}], ",", "x"}], "]"}]], "Input"], Cell[BoxData[ \({{x \[Rule] \(-1.70506413927755069`\)}, {x \[Rule] \ \(\(0.79363008257305081`\)\(\[InvisibleSpace]\)\) - 1.50613717733980889`\ I}, {x \[Rule] \(\(0.79363008257305081`\)\(\ \[InvisibleSpace]\)\) + 1.50613717733980889`\ I}, {x \[Rule] 10.1178039741314496`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{ FormBox[\(x\^4\), "TraditionalForm"], "-", RowBox[{"10", FormBox[\(x\^3\), "TraditionalForm"]}], "-", FormBox[\(x\^2\), "TraditionalForm"], "+", \(3 x\), " ", "-", " ", "50"}], "==", "0"}], ",", \({x, 9}\)}], "]"}]], "Input"], Cell[BoxData[ \({x \[Rule] 10.117803974132955`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Options[FindRoot]\)], "Input"], Cell[BoxData[ \({AccuracyGoal \[Rule] Automatic, Compiled \[Rule] True, DampingFactor \[Rule] 1, Jacobian \[Rule] Automatic, MaxIterations \[Rule] 15, WorkingPrecision \[Rule] 16}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{ FormBox[\(x\^4\), "TraditionalForm"], "-", RowBox[{"10", FormBox[\(x\^3\), "TraditionalForm"]}], "-", FormBox[\(x\^2\), "TraditionalForm"], "+", \(3 x\), " ", "-", " ", "50"}], "==", "0"}], ",", \({x, 9}\), ",", \(AccuracyGoal \[Rule] 25\), ",", \(WorkingPrecision \[Rule] 25\)}], "]"}]], "Input"], Cell[BoxData[ \({x \[Rule] 10.11780397413144903252129657601282799502`25}\)], "Output"] }, Open ]], Cell["\<\ As one can see the 4-th iteration in Newton's scheme agrees up to the first \ 7 digits with the exact solution, while the 6-th iteration provides the exact first 21 digits! Numerical \ analysts, \"Go to dust!\"\ \>", "Text"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ " Sigma ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(\[Sum]\)\)\)]], " , Riemann Sums\n Definite Integrals" }], "Subtitle", TextAlignment->Left, TextJustification->0], Cell[TextData[{ " ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Subtitle", TextAlignment->Left, TextJustification->0], Cell["\<\ Dr. Andrzej Korzeniowski Department of Mathematics, UTA\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Lesson 5 ", "Subtitle"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " built-in symbolic summation is a great tool for finding Riemann sums of \ elementary functions. \nBelow are several examples." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Sum]\+\(i = 1\)\%\[Infinity] 1\/i\^12\)], "Input"], Cell[BoxData[ \(\(691\ \[Pi]\^12\)\/638512875\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Sum]\+\(i = 1\)\%n i\^12\)], 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