(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 19767, 658] NotebookOptionsPosition[ 17293, 571] NotebookOutlinePosition[ 17651, 587] CellTagsIndexPosition[ 17608, 584] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Lesson 3 Answers - Lists", "Section", CellChangeTimes->{{3.460065207574419*^9, 3.460065209489173*^9}, { 3.460573088458062*^9, 3.460573092485478*^9}, {3.4612688245699463`*^9, 3.461268829886034*^9}, {3.465130312004828*^9, 3.465130313034059*^9}}], Cell["1) Find a slick way to generate the following expression:", "Text", CellChangeTimes->{{3.460065222574912*^9, 3.460065255159704*^9}, { 3.4602281945121517`*^9, 3.460228197649392*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Total", "[", RowBox[{"x", "^", RowBox[{"Range", "[", RowBox[{"0", ",", "10"}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.460065261759557*^9, 3.460065293972207*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{"1", "+", "x", "+", SuperscriptBox["x", "2"], "+", SuperscriptBox["x", "3"], "+", SuperscriptBox["x", "4"], "+", SuperscriptBox["x", "5"], "+", SuperscriptBox["x", "6"], "+", SuperscriptBox["x", "7"], "+", SuperscriptBox["x", "8"], "+", SuperscriptBox["x", "9"], "+", SuperscriptBox["x", "10"]}]], "Output", CellChangeTimes->{{3.460065274603848*^9, 3.460065295608177*^9}}, FontSlant->"Italic"] }, Open ]], Cell[TextData[{ "2) Calculate ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"1", "+", FractionBox["1", "2"], "+", FractionBox["1", "3"], "+"}], "..."}], "+", FractionBox["1", "100"]}], TraditionalForm]]], "exactly by applying ", StyleBox["Total", "Input"], " to a list. (You can also calculate this using ", StyleBox["Sum", "Input"], ". See Lesson 7)" }], "Text", CellChangeTimes->{{3.460065426742648*^9, 3.460065522614431*^9}, { 3.460065583672217*^9, 3.460065641606113*^9}, {3.463581583573097*^9, 3.463581606471458*^9}, 3.463581819134109*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Total", "[", RowBox[{"1", "/", RowBox[{"Range", "[", "100", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.460065538821685*^9, 3.4600655605369673`*^9}}, FontSlant->"Italic"], Cell[BoxData[ FractionBox["14466636279520351160221518043104131447711", "2788815009188499086581352357412492142272"]], "Output", CellChangeTimes->{{3.460065557353713*^9, 3.460065561198888*^9}}, FontSlant->"Italic"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", "%", "]"}]], "Input", CellChangeTimes->{{3.460065590836577*^9, 3.460065593009156*^9}}], Cell[BoxData["5.187377517639621`"], "Output", CellChangeTimes->{3.460065594002124*^9}] }, Open ]], Cell["\<\ 3) Generate a table of the values of the sine function at 0, 0.1, 0.2, ... \ ,1.0 as below.\ \>", "Text", CellChangeTimes->{{3.460065685497843*^9, 3.460065702406146*^9}, { 3.460065753091489*^9, 3.460065793332515*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{"x", ",", RowBox[{"Sin", "[", "x", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1", ",", ".1"}], "}"}]}], "]"}], "//", "TableForm"}]], "Input", CellChangeTimes->{{3.4600657081795893`*^9, 3.460065741889655*^9}}, FontSlant->"Italic"], Cell[BoxData[ TagBox[GridBox[{ {"0.`", "0.`"}, {"0.1`", "0.09983341664682815`"}, {"0.2`", "0.19866933079506124`"}, {"0.30000000000000004`", "0.2955202066613396`"}, {"0.4`", "0.3894183423086505`"}, {"0.5`", "0.479425538604203`"}, {"0.6000000000000001`", "0.5646424733950355`"}, {"0.7000000000000001`", "0.6442176872376911`"}, {"0.8`", "0.7173560908995228`"}, {"0.9`", "0.7833269096274834`"}, {"1.`", "0.8414709848078965`"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Function[BoxForm`e$, TableForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.46006573551517*^9, 3.460065742984563*^9}}] }, Open ]], Cell[TextData[{ "4) Does ", StyleBox["Mathematica", FontSlant->"Italic"], " have a built-in function for the cross product of two vectors in ", Cell[BoxData[ FormBox[ SuperscriptBox["R", "3"], TraditionalForm]]], "?" }], "Text", CellChangeTimes->{{3.460217312162455*^9, 3.4602173755802317`*^9}}], Cell[TextData[{ "Yes, it does. ", StyleBox["Cross[u,v]", "Input"], " calculates the cross product of two vectors ", StyleBox["u", "Input"], " and ", StyleBox["v", "Input"], " in ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["R", "3"], "."}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.460573120524609*^9, 3.460573126652688*^9}, { 3.460573166182353*^9, 3.4605732255030203`*^9}}, FontSlant->"Italic"], Cell[TextData[{ "5) If ", StyleBox["M", "Input"], " and ", StyleBox["L", "Input"], " are matrices, is ", StyleBox["M+L", "Input"], " their sum and ", StyleBox["M.L", "Input"], " their product? " }], "Text", CellChangeTimes->{{3.460217724517879*^9, 3.460217729819827*^9}, { 3.460217764819642*^9, 3.460217802957287*^9}, {3.460217911730526*^9, 3.460217933696352*^9}, 3.460219626451414*^9, 3.4605732561502132`*^9}], Cell["The answer to both questions is yes. For example", "Text", CellChangeTimes->{{3.460573290415522*^9, 3.460573313024227*^9}}, FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"a", ",", "b"}], "}"}], ",", RowBox[{"{", RowBox[{"c", ",", "d"}], "}"}]}], "}"}], ".", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"A", ",", "B"}], "}"}], ",", RowBox[{"{", RowBox[{"C", ",", "D"}], "}"}]}], "}"}]}], "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.460573333378436*^9, 3.4605733679150667`*^9}}, FontSlant->"Italic"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{ RowBox[{"a", " ", "A"}], "+", RowBox[{"b", " ", "C"}]}], RowBox[{ RowBox[{"a", " ", "B"}], "+", RowBox[{"b", " ", "D"}]}]}, { RowBox[{ RowBox[{"A", " ", "c"}], "+", RowBox[{"C", " ", "d"}]}], RowBox[{ RowBox[{"B", " ", "c"}], "+", RowBox[{"d", " ", "D"}]}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{{3.460573360782331*^9, 3.460573369201696*^9}}, FontSlant->"Italic"] }, Open ]], Cell[TextData[{ "6) Create the first 11 rows of Pascal's triangle, i.e, a table that \ contains in its nth row the binomial coefficients ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"n"}, {"k"} }], ")"}], TraditionalForm]]], ", k=0,...,n, where n= 0,...,10. ", StyleBox[" ", FontWeight->"Bold"], "Display the result in tabular format, as shown below. (Hint: ", StyleBox["Mathematica", FontSlant->"Italic"], " has a function that computes binomial coeffients. Look it up!)" }], "Text", CellChangeTimes->{{3.4155334913984623`*^9, 3.41553367033363*^9}, { 3.415533748687199*^9, 3.415533872057564*^9}, {3.460227892216302*^9, 3.460227912255732*^9}, 3.460573258759729*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"Binomial", "[", RowBox[{"n", ",", "k"}], "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "0", ",", "10"}], "}"}], ",", RowBox[{"{", RowBox[{"k", ",", "0", ",", "n"}], "}"}]}], "]"}], "//", "TableForm"}]], "Input", CellChangeTimes->{{3.4605733787343903`*^9, 3.460573385728306*^9}, { 3.460573460571823*^9, 3.460573521406825*^9}}, FontSlant->"Italic"], Cell[BoxData[ InterpretationBox[GridBox[{ {"1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\ \>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"1", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\ \>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"1", "2", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\ \>", "\<\"\"\>", "\<\"\"\>"}, {"1", "3", "3", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\ \>", "\<\"\"\>"}, {"1", "4", "6", "4", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\ \>"}, {"1", "5", "10", "10", "5", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"1", "6", "15", "20", "15", "6", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"1", "7", "21", "35", "35", "21", "7", "1", "\<\"\"\>", "\<\"\"\>", "\<\"\"\>"}, {"1", "8", "28", "56", "70", "56", "28", "8", "1", "\<\"\"\>", "\<\"\"\>"}, {"1", "9", "36", "84", "126", "126", "84", "36", "9", "1", "\<\"\"\>"}, {"1", "10", "45", "120", "210", "252", "210", "120", "45", "10", "1"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[2.0999999999999996`]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], TableForm[{{1}, {1, 1}, {1, 2, 1}, {1, 3, 3, 1}, {1, 4, 6, 4, 1}, {1, 5, 10, 10, 5, 1}, {1, 6, 15, 20, 15, 6, 1}, {1, 7, 21, 35, 35, 21, 7, 1}, {1, 8, 28, 56, 70, 56, 28, 8, 1}, {1, 9, 36, 84, 126, 126, 84, 36, 9, 1}, {1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1}}]]], "Output", CellChangeTimes->{{3.460573497065933*^9, 3.460573523957732*^9}}] }, Open ]], Cell[TextData[{ "7) Can ", StyleBox["Total", "Input"], " be used to sum up a list of vectors?" }], "Text", CellChangeTimes->{{3.4611066212966747`*^9, 3.461106641681422*^9}, 3.461106714374857*^9, 3.4611845897975903`*^9}], Cell["Yes. For example,", "Text", CellChangeTimes->{{3.4611066495419483`*^9, 3.461106657808894*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Total", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "4"}], "}"}]}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.46110666172392*^9, 3.461106681397874*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"6", ",", "9"}], "}"}]], "Output", CellChangeTimes->{3.461106683364687*^9}] }, Open ]], Cell[TextData[{ "8) Suppose that ", StyleBox["M", "Input"], " is a matrix of unknown size. What expression would produce a list of the \ row sums of ", StyleBox["M", "Input"], "? A row sum is just the sum of the entries in a row of the matrix, and \ there is one row sum for each row. What expression would produce a list of \ the column sums of ", StyleBox["M", "Input"], "?" }], "Text", CellChangeTimes->{{3.460663161960495*^9, 3.460663347377853*^9}, 3.4611845940017967`*^9}], Cell["Here is a matrix:", "Text", CellChangeTimes->{{3.4606637012238493`*^9, 3.4606637082298727`*^9}}, FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"M", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "5", ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"7", ",", "8", ",", "9"}], "}"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.460663450996313*^9, 3.4606634680281467`*^9}, { 3.460663618155787*^9, 3.4606636523238173`*^9}, 3.461184326834504*^9}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "5", ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"7", ",", "8", ",", "9"}], "}"}]}], "}"}]], "Output", GeneratedCell->False, CellAutoOverwrite->False, CellChangeTimes->{ 3.460663470242649*^9, 3.4606636229826927`*^9, 3.460663653821225*^9, { 3.4611843268356037`*^9, 3.46118433099586*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "5", ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"7", ",", "8", ",", "9"}], "}"}]}], "}"}]], "Output", GeneratedCell->False, CellAutoOverwrite->False, CellChangeTimes->{3.460663470242649*^9, 3.4606636229826927`*^9, 3.460663653821225*^9, 3.4611843268356037`*^9}, FontSlant->"Italic"] }, Open ]], Cell[TextData[{ "This expression calculates the row sums of ", StyleBox["M", "Input"], "." }], "Text", CellChangeTimes->{{3.460663717974543*^9, 3.460663737520945*^9}}, FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"Total", "[", RowBox[{"M", "[", RowBox[{"[", "i", "]"}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"Length", "[", "M", "]"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.460663414954265*^9, 3.4606634350762463`*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{"{", RowBox[{"6", ",", "15", ",", "24"}], "}"}]], "Output", CellChangeTimes->{ 3.460663473354031*^9, {3.4606636304786654`*^9, 3.4606636586025248`*^9}}, FontSlant->"Italic"] }, Open ]], Cell["Here's a slicker way of getting the same result:", "Text", CellChangeTimes->{{3.461184425487892*^9, 3.461184439424872*^9}}, FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"Total", "[", "row", "]"}], ",", RowBox[{"{", RowBox[{"row", ",", "M"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.461184443630375*^9, 3.461184467576058*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{"{", RowBox[{"6", ",", "15", ",", "24"}], "}"}]], "Output", CellChangeTimes->{3.461184469531204*^9}, FontSlant->"Italic"] }, Open ]], Cell["This expression calculates the column sums.", "Text", CellChangeTimes->{{3.460663750963325*^9, 3.460663767340887*^9}}, FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"Total", "[", RowBox[{"M", "[", RowBox[{"[", RowBox[{"All", ",", "j"}], "]"}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"j", ",", RowBox[{"Length", "[", RowBox[{"M", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.460663515692171*^9, 3.460663574838943*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{"{", RowBox[{"12", ",", "15", ",", "18"}], "}"}]], "Output", CellChangeTimes->{ 3.460663577582404*^9, {3.4606636349424553`*^9, 3.460663662547944*^9}}, FontSlant->"Italic"] }, Open ]], Cell[TextData[{ "But, amazingly, ", StyleBox["Total", "Input"], " produces the same result all by itself! Why is that? See the previous \ question." }], "Text", CellChangeTimes->{{3.461184360808401*^9, 3.461184408935055*^9}, { 3.4611844876079817`*^9, 3.461184493478347*^9}, {3.461184601314886*^9, 3.461184610021743*^9}}, FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Total", "[", "M", "]"}]], "Input", CellChangeTimes->{{3.461184335554377*^9, 3.461184339780856*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{"{", RowBox[{"12", ",", "15", ",", "18"}], "}"}]], "Output", CellChangeTimes->{3.461184341458815*^9}, FontSlant->"Italic"] }, Open ]], Cell["\<\ 9) Calculate the sum of 10000 random integers between 0 and 10. How close is \ your result to 50000?\ \>", "Text", CellChangeTimes->{{3.463583288638074*^9, 3.463583349114676*^9}, { 3.4635834087045403`*^9, 3.46358341369345*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Total", "[", RowBox[{"RandomInteger", "[", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "10"}], "}"}], ",", "10000"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.463583362511965*^9, 3.4635833643749323`*^9}}, FontSlant->"Italic"], Cell[BoxData["49821"], "Output", CellChangeTimes->{{3.4635833655351048`*^9, 3.463583399085843*^9}}, FontSlant->"Italic"] }, Open ]], Cell["Pretty close in this case.", "Text", CellChangeTimes->{{3.463583369788913*^9, 3.463583379024042*^9}}, FontSlant->"Italic"] }, Open ]] }, WindowSize->{640, 750}, WindowMargins->{{Automatic, 85}, {Automatic, 20}}, ShowSelection->True, FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (April 20, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 255, 3, 67, "Section"], Cell[848, 28, 190, 2, 26, "Text"], Cell[CellGroupData[{ Cell[1063, 34, 228, 6, 27, "Input"], Cell[1294, 42, 445, 12, 30, "Output"] }, Open ]], Cell[1754, 57, 597, 18, 51, "Text"], Cell[CellGroupData[{ Cell[2376, 79, 206, 5, 27, "Input"], Cell[2585, 86, 218, 4, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2840, 95, 120, 2, 27, "Input"], Cell[2963, 99, 87, 1, 27, "Output"] }, Open ]], Cell[3065, 103, 231, 5, 26, "Text"], Cell[CellGroupData[{ Cell[3321, 112, 364, 11, 27, "Input"], Cell[3688, 125, 987, 26, 189, "Output"] }, Open ]], Cell[4690, 154, 309, 10, 29, "Text"], Cell[5002, 166, 432, 15, 30, "Text"], Cell[5437, 183, 427, 13, 26, "Text"], Cell[5867, 198, 152, 2, 26, "Text"], Cell[CellGroupData[{ Cell[6044, 204, 485, 17, 27, "Input"], Cell[6532, 223, 1003, 31, 45, "Output"] }, Open ]], Cell[7550, 257, 711, 19, 75, "Text"], Cell[CellGroupData[{ Cell[8286, 280, 452, 13, 27, "Input"], Cell[8741, 295, 1964, 41, 189, "Output"] }, Open ]], Cell[10720, 339, 227, 6, 26, "Text"], Cell[10950, 347, 101, 1, 26, "Text"], Cell[CellGroupData[{ Cell[11076, 352, 333, 10, 27, "Input"], Cell[11412, 364, 115, 3, 27, "Output"] }, Open ]], Cell[11542, 370, 489, 13, 56, "Text"], Cell[12034, 385, 125, 2, 26, "Text"], Cell[CellGroupData[{ Cell[12184, 391, 453, 12, 27, "Input"], Cell[12640, 405, 474, 14, 27, "Output"], Cell[13117, 421, 447, 13, 27, "Output"] }, Open ]], Cell[13579, 437, 193, 6, 26, "Text"], Cell[CellGroupData[{ Cell[13797, 447, 345, 10, 27, "Input"], Cell[14145, 459, 202, 5, 27, "Output"] }, Open ]], Cell[14362, 467, 152, 2, 26, "Text"], Cell[CellGroupData[{ Cell[14539, 473, 255, 7, 27, "Input"], Cell[14797, 482, 149, 4, 27, "Output"] }, Open ]], Cell[14961, 489, 147, 2, 26, "Text"], Cell[CellGroupData[{ Cell[15133, 495, 432, 13, 27, "Input"], Cell[15568, 510, 201, 5, 27, "Output"] }, Open ]], Cell[15784, 518, 351, 9, 26, "Text"], Cell[CellGroupData[{ Cell[16160, 531, 146, 3, 27, "Input"], Cell[16309, 536, 150, 4, 27, "Output"] }, Open ]], Cell[16474, 543, 240, 5, 26, "Text"], Cell[CellGroupData[{ Cell[16739, 552, 268, 7, 27, "Input"], Cell[17010, 561, 122, 2, 27, "Output"] }, Open ]], Cell[17147, 566, 130, 2, 26, "Text"] }, Open ]] } ] *) (* End of internal cache information *)