(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.2' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 100514, 2570] NotebookOptionsPosition[ 84415, 2313] NotebookOutlinePosition[ 88763, 2394] CellTagsIndexPosition[ 88684, 2389] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", "FirstSlide", CellTags-> "SlideShowHeader",ExpressionUUID->"ae861ed8-5fa1-4088-84fd-2651cc7b368a"], Cell["Introdu\[CCedilla]\[ATilde]o \[AGrave] F\[IAcute]sica Computacional I \ (2023.2)", "Title", CellChangeTimes->{ 3.559948400406288*^9, {3.699269571766136*^9, 3.699269573578824*^9}, { 3.7099080105038958`*^9, 3.7099080108133383`*^9}, {3.727189496068931*^9, 3.7271894962748613`*^9}, {3.83932299294512*^9, 3.839323000605739*^9}, { 3.873367528987236*^9, 3.873367529147398*^9}, {3.9043713310631914`*^9, 3.904371331375814*^9}},ExpressionUUID->"92508663-47f1-4deb-9615-\ d2c271e8e453"], Cell["\<\ Aula 6: gr\[AAcute]ficos; zeros e extremos de fun\[CCedilla]\[OTilde]es; \ ajustes de dados\ \>", "Subtitle", CellChangeTimes->{{3.699269582406044*^9, 3.699269584285553*^9}, { 3.8393232649484053`*^9, 3.839323284264048*^9}, 3.840031268565436*^9, { 3.8407028843722677`*^9, 3.840702924713172*^9}, {3.840714600599773*^9, 3.8407146281949797`*^9}, {3.841157909295815*^9, 3.841157911477744*^9}, { 3.841480259563218*^9, 3.8414802737602463`*^9}, {3.8415945589945517`*^9, 3.841594560546528*^9}, {3.842262188714292*^9, 3.8422622084045753`*^9}, { 3.842288735025601*^9, 3.8422887376233883`*^9}, {3.842288769619337*^9, 3.8422887810036163`*^9}},ExpressionUUID->"14f8f51b-58e9-4033-aea1-\ fd611ac95bb9"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"ee3ea2a1-973b-4e9a-a704-0d8a3c30d1be"], Cell[CellGroupData[{ Cell["Gr\[AAcute]ficos mais sofisticados", "Section", CellChangeTimes->{ 3.699269664116198*^9, 3.70454748091461*^9, {3.704547516075143*^9, 3.7045475190812902`*^9}, {3.706294276733098*^9, 3.706294304716371*^9}, { 3.839323324298633*^9, 3.839323331059084*^9}, 3.840031288455207*^9, { 3.840714636688951*^9, 3.840714639748958*^9}, {3.841057130911109*^9, 3.84105713513932*^9}, {3.841480515514193*^9, 3.84148051572683*^9}, { 3.84226221732572*^9, 3.842262223457004*^9}},ExpressionUUID->"7da6da63-6378-4f26-b58d-\ d42a7250afa4"], Cell[TextData[{ "J\[AAcute] temos trabalhado com gr\[AAcute]ficos, mas agora vamos explorar \ como utilizar fun\[CCedilla]\[OTilde]es auxiliares e as ", ButtonBox["op\[CCedilla]\[OTilde]es", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/guide/PlottingOptions.html"], None}, ButtonNote-> "https://reference.wolfram.com/language/guide/PlottingOptions.html"], " da fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Plot", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Plot.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Plot.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " (e de outras fun\[CCedilla]\[OTilde]es a ela relacionadas) para melhor\ \[AAcute]-los." }], "Text", CellChangeTimes->{{3.6992696952616243`*^9, 3.6992696957779827`*^9}, { 3.704547416597644*^9, 3.704547418388081*^9}, {3.8393233348613234`*^9, 3.839323366214238*^9}, {3.839323576559081*^9, 3.839323576563138*^9}, 3.840031301280365*^9, {3.840714647849332*^9, 3.840714700814382*^9}, { 3.841480523939232*^9, 3.841480544976879*^9}, {3.841480739096553*^9, 3.841480739636643*^9}, {3.8422622643967133`*^9, 3.8422623532037497`*^9}, { 3.842262439308573*^9, 3.8422624573684273`*^9}, {3.8422649573788548`*^9, 3.8422649619920673`*^9}, {3.842358288105185*^9, 3.8423582908266573`*^9}, { 3.8733676215584908`*^9, 3.873367621563629*^9}, {3.904371368360536*^9, 3.904371368597521*^9}},ExpressionUUID->"3b730a40-c305-44bd-95b8-\ df5440115120"] }, Open ]] }, Open ]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"0d6bbe14-c226-415e-b1fc-fcf294444606"], Cell[TextData[{ "Vamos primeiro voltar ao problema do per\[IAcute]odo de um p\[EHat]ndulo de \ comprimento ", Cell[BoxData[ FormBox["L", TraditionalForm]],ExpressionUUID-> "e483227e-0e42-421d-850b-feeab8c0e4b5"], " abandonado de um \[AHat]ngulo inicial ", Cell[BoxData[ FormBox[ SubscriptBox["\[Theta]", "0"], TraditionalForm]],ExpressionUUID-> "b2ad2e74-fee3-4900-b327-e32df0f8cc58"], " com a vertical, per\[IAcute]odo que \[EAcute] dado por ", Cell[BoxData[ FormBox[ RowBox[{"T", "=", RowBox[{ SqrtBox[ FractionBox[ RowBox[{"8", "L"}], "g"]], RowBox[{ SubsuperscriptBox["\[Integral]", "0", SubscriptBox["\[Theta]", "0"]], FractionBox[ RowBox[{"d", " ", "\[Theta]"}], SqrtBox[ RowBox[{"cos\[Theta]", "-", SubscriptBox["cos\[Theta]", "0"]}]]]}]}]}], TraditionalForm]], ExpressionUUID->"2b52e853-2bb0-49b3-a308-235e586fa070"], ". O Mathematica \[OpenCurlyDoubleQuote]resolve\[CloseCurlyDoubleQuote] a \ integral em termos de integrais el\[IAcute]pticas:" }], "Text", CellChangeTimes->{{3.840715029587531*^9, 3.840715134031026*^9}, { 3.8414805564409933`*^9, 3.8414806111499968`*^9}, {3.8414806778318357`*^9, 3.841480677842259*^9}, {3.841686628790584*^9, 3.841686633357884*^9}, { 3.841770413921526*^9, 3.841770418522381*^9}, {3.842262528840704*^9, 3.842262547649932*^9}, {3.8422634217918177`*^9, 3.8422634252103033`*^9}, { 3.8422635300506496`*^9, 3.842263615755658*^9}, {3.842263652004801*^9, 3.8422637009962873`*^9}, {3.8422637455911694`*^9, 3.8422637862137833`*^9}, { 3.842263822305868*^9, 3.8422638367376823`*^9}, {3.8422641608299427`*^9, 3.8422641636529922`*^9}, {3.8422649720254*^9, 3.842264973116967*^9}, { 3.842358334218719*^9, 3.842358338459055*^9}},ExpressionUUID->"06361fc3-d0e5-4646-a4fa-\ 73129b0aa494"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"Cos", "[", "\[Theta]", "]"}], "-", RowBox[{"Cos", "[", "\[Theta]0", "]"}]}], "]"}]}], ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", ",", "\[Theta]0"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"{", RowBox[{"0", "<", "\[Theta]0", "<", "\[Pi]"}], "}"}]}]}], "]"}], " "}]], "Input", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.84148068767109*^9, 3.841480759190398*^9}, { 3.841482273972699*^9, 3.841482274767726*^9}, 3.84148233141354*^9, { 3.842263846423929*^9, 3.8422639438448563`*^9}, 3.842356876974916*^9}, CellLabel->"",ExpressionUUID->"cd8d50d6-4119-457c-96b0-28e6134ac339"], Cell["Note a demora at\[EAcute] a sa\[IAcute]da do c\[AAcute]lculo anterior. \ ", "ItemParagraph", CellChangeTimes->{ 3.842358414735612*^9},ExpressionUUID->"e8afa592-94bf-423a-91b3-\ 7baec477a2d0"], Cell[TextData[{ "Quando tra\[CCedilla]a um gr\[AAcute]fico de uma fun\[CCedilla]\[ATilde]o \ ", Cell[BoxData[ FormBox["f", TraditionalForm]],ExpressionUUID-> "f00033ca-cd7f-4fa3-9d5c-68ab5cc59358"], ", o Mathematica calcula \[OpenCurlyDoubleQuote]do zero\ \[CloseCurlyDoubleQuote] o valor da fun\[CCedilla]\[ATilde]o em cada ponto, o \ que pode resultar em um grande tempo de c\[AAcute]lculo para obter o gr\ \[AAcute]fico completo. Nesses casos, \[EAcute] conveniente invocar a fun\ \[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Evaluate", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Evaluate.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Evaluate.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " em combina\[CCedilla]\[ATilde]o com a fun\[CCedilla]\[ATilde]o ", StyleBox["Plot", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", o que for\[CCedilla]a o c\[AAcute]lculo do argumento de ", StyleBox[ButtonBox["Evaluate", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Evaluate.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Evaluate.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " antes de que se trace o gr\[AAcute]fico:" }], "Text", CellChangeTimes->{{3.842264241402898*^9, 3.842264475639538*^9}, { 3.842264527960717*^9, 3.8422645279660597`*^9}, {3.842264578487348*^9, 3.842264625501144*^9}, 3.8423584127896748`*^9, {3.904371383911541*^9, 3.9043714074648113`*^9}},ExpressionUUID->"bae55d84-2b1d-4be9-957f-\ aeb20aea8b74"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", RowBox[{"Integrate", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"Cos", "[", "\[Theta]", "]"}], "-", RowBox[{"Cos", "[", "\[Theta]0", "]"}]}], "]"}]}], ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", ",", "\[Theta]0"}], "}"}], ",", RowBox[{"Assumptions", "\[Rule]", RowBox[{"{", RowBox[{"0", "<", "\[Theta]0", "<", "\[Pi]"}], "}"}]}]}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"\[Theta]0", ",", "0", ",", RowBox[{"\[Pi]", "-", "0.01"}]}], "}"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8422645363539133`*^9, 3.842264570658684*^9}, 3.842356879550829*^9},ExpressionUUID->"2aa21682-2a41-4b26-b045-\ 55d87ea52ccb"], Cell[TextData[{ StyleBox["N\[ATilde]o tente em casa", FontWeight->"Bold", FontVariations->{"Underline"->True}], " executar a entrada anterior sem a fun\[CCedilla]\[ATilde]o ", StyleBox["Evaluate", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", a menos que seu computador tenha uma mem\[OAcute]ria gigantesca. " }], "ItemParagraph", CellChangeTimes->{{3.842264734267601*^9, 3.842264821308092*^9}},ExpressionUUID->"30deaacb-0eb3-4f1e-9797-\ a97bfe9c9492"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"e87c0300-a443-4a7a-a867-2dd6904b1c93"], Cell[TextData[{ "As op\[CCedilla]\[OTilde]es da fun\[CCedilla]\[ATilde]o ", StyleBox["Plot", FontFamily->"Source Code Pro", FontWeight->"Regular"], " controlam, entre outras coisas, aspectos como cores das curvas, espessuras \ de linhas, tipos e tamanhos de fontes das legendas, a presen\[CCedilla]a de \ uma moldura no gr\[AAcute]fico etc. Para especificar uma op\[CCedilla]\ \[ATilde]o, \[EAcute] preciso invoc\[AAcute]-la como uma regra de transforma\ \[CCedilla]\[ATilde]o do tipo ", StyleBox["Nome", FontFamily->"Source Code Pro", FontWeight->"Regular"], Cell[BoxData[ FormBox["\[Rule]", TraditionalForm]], FontFamily->"Source Code Pro", FontWeight->"Regular",ExpressionUUID-> "031e2309-fd4d-4cf9-bf87-e046de770ef4"], StyleBox["Valor", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", conforme os exemplos a seguir." }], "Text", CellChangeTimes->{{3.840714805233673*^9, 3.840714848494001*^9}, { 3.840714990245832*^9, 3.8407150050952263`*^9}, {3.8407162178028297`*^9, 3.840716217806488*^9}, {3.841482364491879*^9, 3.841482404491866*^9}, { 3.841482474444706*^9, 3.8414824881249037`*^9}, {3.842264986830921*^9, 3.842265156413743*^9}},ExpressionUUID->"bc8716a3-ec22-4f95-8297-\ dc736158740b"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"32be91fb-777d-400b-86fa-9a049cede895"], Cell[TextData[{ "O potencial de Lennard-Jones fornece uma descri\[CCedilla]\[ATilde]o razo\ \[AAcute]vel para a energia de intera\[CCedilla]\[ATilde]o entre dois \ \[AAcute]tomos de um g\[AAcute]s nobre cujos centros est\[ATilde]o separados \ por uma dist\[AHat]ncia ", Cell[BoxData[ FormBox["r", TraditionalForm]],ExpressionUUID-> "1fb65fd1-400c-4497-b407-79d6b981eac2"], ". Qualitativamente, sua forma funcional \[EAcute]" }], "Text", CellChangeTimes->{{3.840716256216364*^9, 3.8407162792437897`*^9}, { 3.84071639267841*^9, 3.8407164385558443`*^9}, {3.8414829295109167`*^9, 3.841482967164357*^9}, {3.841483033822137*^9, 3.841483128332952*^9}, { 3.842265331220846*^9, 3.842265438646007*^9}, {3.842361161853416*^9, 3.842361166296517*^9}},ExpressionUUID->"0d223cbc-4c03-401e-905d-\ 973bce4d584c"], Cell[BoxData[ RowBox[{ RowBox[{"uLJ", "[", "r_", "]"}], ":=", RowBox[{ RowBox[{"(", RowBox[{"1", "/", RowBox[{"r", "^", "12"}]}], ")"}], "-", RowBox[{"(", RowBox[{"1", "/", RowBox[{"r", "^", "6"}]}], ")"}], " "}]}]], "Input", CellChangeTimes->{{3.840716285257362*^9, 3.84071631971697*^9}, 3.841216684087008*^9, {3.841483133227302*^9, 3.84148314343582*^9}, { 3.842265418789784*^9, 3.8422654649579268`*^9}, 3.842356975755855*^9},ExpressionUUID->"2869d96c-abc0-4783-9fa1-\ d4b178ad7213"], Cell[TextData[{ "Vamos tentar tra\[CCedilla]ar seu gr\[AAcute]fico, mas j\[AAcute] \ utilizando as op\[CCedilla]\[OTilde]es ", StyleBox[ButtonBox["ImageSize", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/ImageSize.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/ImageSize.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], Cell[BoxData[ FormBox["\[Rule]", TraditionalForm]], FontFamily->"Source Code Pro", FontWeight->"Regular",ExpressionUUID-> "f341d7a6-a53b-426f-a8ab-27615336080d"], StyleBox["Large", FontFamily->"Source Code Pro", FontWeight->"Regular"], " e ", StyleBox[ButtonBox["LabelStyle", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/LabelStyle.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/LabelStyle.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], Cell[BoxData[ FormBox["\[Rule]", TraditionalForm]], FontFamily->"Source Code Pro", FontWeight->"Regular",ExpressionUUID-> "efd0b19f-a946-424c-b0b0-6df364c13a60"], StyleBox["Large", FontFamily->"Source Code Pro", FontWeight->"Regular"], " para aumentar o tamanho do gr\[AAcute]fico e das legendas: " }], "Text", CellChangeTimes->{{3.841483261737115*^9, 3.841483334649732*^9}, { 3.8422654847014227`*^9, 3.842265491665125*^9}, {3.842265588865736*^9, 3.842265686370936*^9}, {3.842265808513692*^9, 3.842265886028368*^9}, { 3.842265969929212*^9, 3.842265979457273*^9}, {3.842266381986641*^9, 3.842266488241345*^9}, {3.873367836300592*^9, 3.873367861325642*^9}},ExpressionUUID->"c8424edf-67f3-412d-847d-\ 224e5d4cb7e5"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"uLJ", "[", "r", "]"}], ",", RowBox[{"{", RowBox[{"r", ",", "0.001", ",", "3.0"}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", "Large"}], ",", RowBox[{"LabelStyle", "\[Rule]", " ", "Large"}]}], "]"}]], "Input", CellChangeTimes->{{3.841483272950675*^9, 3.841483273798868*^9}, { 3.841483370614747*^9, 3.8414833880909643`*^9}, {3.841483784461116*^9, 3.841483790659606*^9}, {3.842265494647156*^9, 3.84226550454707*^9}, { 3.8422656570495234`*^9, 3.8422656609030113`*^9}, {3.842265712471738*^9, 3.8422657781499157`*^9}, {3.84226589452752*^9, 3.842265898158119*^9}, { 3.8422661348859987`*^9, 3.842266206327133*^9}, {3.842266270008697*^9, 3.842266324373652*^9}, {3.8422663560496273`*^9, 3.842266358416554*^9}, { 3.842266493142488*^9, 3.842266496005539*^9}, {3.842356928650559*^9, 3.842356930808023*^9}}, CellLabel->"",ExpressionUUID->"b54a9736-b826-496c-a3e8-02f579064f9c"], Cell[TextData[{ "Olhando muito atentamente, podemos suspeitar que h\[AAcute] algum detalhe \ fino do gr\[AAcute]fico em torno da abscissa igual a 1. Para verificar isso, \ calibramos o intervalo de valores das ordenadas com a op\[CCedilla]\[ATilde]o \ ", StyleBox[ButtonBox["PlotRange", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/PlotRange.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/PlotRange.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{{3.842265542212652*^9, 3.842265560875412*^9}, { 3.8422659944882717`*^9, 3.842265996065256*^9}, {3.842266597456753*^9, 3.842266655090495*^9}, {3.87336781098508*^9, 3.8733678109935617`*^9}},ExpressionUUID->"384746ca-07e8-4df8-9894-\ deb0e39cd9ac"], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"uLJ", "[", "r", "]"}], ",", RowBox[{"{", RowBox[{"r", ",", "0.001", ",", "3.0"}], "}"}], ",", RowBox[{"ImageSize", "\[Rule]", "Large"}], ",", RowBox[{"LabelStyle", "\[Rule]", "Large"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "0.3"}], ",", "0.6"}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.842266713885373*^9, 3.8422667220183973`*^9}, { 3.842356981503415*^9, 3.842356982131687*^9}}, CellLabel->"",ExpressionUUID->"44f5558f-0468-4814-a5f2-3bdfe351744c"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"dbbc9628-10eb-4897-b200-5b6460fd6b1a"], Cell[TextData[{ "O Mathematica tra\[CCedilla]a gr\[AAcute]ficos com n\[UAcute]mero padr\ \[ATilde]o de pontos igual a 50. Em certos casos, esse n\[UAcute]mero pode \ ser insuficiente para reproduzir corretamente o comportamento da fun\ \[CCedilla]\[ATilde]o. Por exemplo, considere o seguinte gr\[AAcute]fico de \ batimentos, em que aproveitamos para ilustrar como padronizar op\[CCedilla]\ \[OTilde]es com o uso da fun\[CCedilla]\[ATilde]o ", StyleBox["Evaluate", FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{{3.840717295113817*^9, 3.84071729734042*^9}, { 3.840717343837861*^9, 3.840717386227171*^9}, {3.840717481133095*^9, 3.8407174814466467`*^9}, {3.8407177165619297`*^9, 3.840717758197268*^9}, { 3.840720388376389*^9, 3.840720390647537*^9}, {3.84148341793322*^9, 3.8414834621562843`*^9}, {3.842266820276202*^9, 3.842266877549309*^9}, { 3.842266950000952*^9, 3.842266973962446*^9}, {3.842267271404274*^9, 3.842267305366654*^9}, {3.8422673843647757`*^9, 3.84226738903314*^9}, { 3.873367921899115*^9, 3.873367926927175*^9}},ExpressionUUID->"651cfb64-d385-4f6a-b0f5-\ 32fba8fd40c5"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"optam", "=", RowBox[{"{", RowBox[{ RowBox[{"ImageSize", "\[Rule]", "Large"}], ",", RowBox[{"LabelStyle", "\[Rule]", "Large"}]}], "}"}]}], ";", RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"Cos", "[", RowBox[{"90", "*", "t"}], "]"}], "+", RowBox[{"Cos", "[", RowBox[{"91", "*", "t"}], "]"}]}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "\[Pi]"}], ",", "\[Pi]"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.8422671791828737`*^9, 3.842267190167901*^9}, { 3.842267324380275*^9, 3.842267324538506*^9}, {3.8423361428665743`*^9, 3.842336142869729*^9}, {3.8423570814599543`*^9, 3.842357082065295*^9}}, CellLabel->"",ExpressionUUID->"39597d20-ec56-4f2f-9e26-572d181afc23"], Cell[TextData[{ "As falhas no gr\[AAcute]fico resultam de uma amostragem insuficiente da fun\ \[CCedilla]\[ATilde]o, que pode ser corrigida com o uso da op\[CCedilla]\ \[ATilde]o ", StyleBox[ButtonBox["PlotPoints", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/PlotPoints.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/PlotPoints.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ". Note que agora tamb\[EAcute]m vamos associar o gr\[AAcute]fico a uma vari\ \[AAcute]vel: " }], "Text", CellChangeTimes->{{3.842267349726112*^9, 3.842267410307064*^9}, { 3.8422676642451*^9, 3.842267684302994*^9}, {3.873367988450076*^9, 3.87336798845914*^9}},ExpressionUUID->"62d58f3c-00ad-4b54-8fb8-\ 362086697220"], Cell[BoxData[ RowBox[{"grafico1", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"Cos", "[", RowBox[{"90", "*", "t"}], "]"}], "+", RowBox[{"Cos", "[", RowBox[{"91", "*", "t"}], "]"}]}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "\[Pi]"}], ",", "\[Pi]"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"PlotPoints", "\[Rule]", " ", "120"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.842267424765024*^9, 3.842267436386292*^9}, { 3.842267641861429*^9, 3.8422676442649193`*^9}, 3.842336142879973*^9},ExpressionUUID->"f3774ca7-c5c2-42bb-a91f-\ 933433e0148b"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"f4abf8ae-e5a5-4155-bfef-c2f83419c5de"], Cell[TextData[{ "Com o aux\[IAcute]lio da fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Show", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Show.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Show.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", podemos acrescentar (algumas) op\[CCedilla]\[OTilde]es a um \ gr\[AAcute]fico j\[AAcute] tra\[CCedilla]ado. Por exemplo, vamos rotular os \ eixos do gr\[AAcute]fico anterior com a op\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["AxesLabel", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/AxesLabel.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/AxesLabel.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{{3.840717900027441*^9, 3.8407179161894913`*^9}, { 3.84148401568287*^9, 3.841484146115486*^9}, {3.8414842239886303`*^9, 3.841484224551149*^9}, {3.841743085839407*^9, 3.841743086232205*^9}, { 3.842267602380474*^9, 3.842267606744514*^9}, {3.84226772061803*^9, 3.842267764583261*^9}, {3.842267810965003*^9, 3.8422678718199244`*^9}, { 3.8422882325645227`*^9, 3.842288234217061*^9}, {3.842357175747822*^9, 3.842357181380335*^9}, {3.873368035898404*^9, 3.8733680359025087`*^9}},ExpressionUUID->"c0e271b6-f898-4758-97f5-\ de10218251d3"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Show", "[", RowBox[{"grafico1", ",", RowBox[{"AxesLabel", "\[Rule]", " ", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"t\",FontSlant->\"Italic\"]\) (s)\>\"", ",", "\"\<\!\(\*StyleBox[\"y\",FontSlant->\"Italic\"]\) (\!\(\*StyleBox[\"A\ \",FontSlant->\"Italic\"]\))\>\""}], "}"}]}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.842267847827688*^9, 3.842267886467093*^9}, { 3.842273448994131*^9, 3.842273454879149*^9}, 3.8423571897640533`*^9},ExpressionUUID->"4335c73f-d836-4591-af53-\ 077629a4ad7a"], Cell[TextData[{ "Podemos remover os eixos e incluir uma moldura com as \ op\[CCedilla]\[OTilde]es ", StyleBox[ButtonBox["Axes", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Axes.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Axes.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], Cell[BoxData[ FormBox["\[Rule]", TraditionalForm]], FontFamily->"Source Code Pro", FontWeight->"Regular",ExpressionUUID-> "d3859b14-9d10-493c-9626-5f0c039eb365"], StyleBox["False", FontFamily->"Source Code Pro", FontWeight->"Regular"], " e ", StyleBox[ButtonBox["Frame", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Frame.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Frame.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], Cell[BoxData[ FormBox["\[Rule]", TraditionalForm]], FontFamily->"Source Code Pro", FontWeight->"Regular",ExpressionUUID-> "3b864087-0083-4e37-8895-a3905fb831c9"], StyleBox["True", FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{{3.8422735297128563`*^9, 3.8422736199596987`*^9}, { 3.8422736712382936`*^9, 3.842273704875064*^9}, 3.842357212284483*^9, { 3.842361187558885*^9, 3.8423611898205*^9}, {3.8733680774695253`*^9, 3.873368087571919*^9}},ExpressionUUID->"9c6512a8-e856-4905-91f8-\ 7c43dff710cd"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Show", "[", RowBox[{"grafico1", ",", RowBox[{"Axes", "\[Rule]", " ", "False"}], ",", RowBox[{"Frame", "\[Rule]", " ", "True"}], ",", RowBox[{"FrameLabel", "\[Rule]", " ", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"t\",FontSlant->\"Italic\"]\) (s)\>\"", ",", "\"\<\!\(\*StyleBox[\"y\",FontSlant->\"Italic\"]\) (\!\(\*StyleBox[\"A\ \",FontSlant->\"Italic\"]\))\>\""}], "}"}]}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.842273659954158*^9, 3.842273667480547*^9}, { 3.842273709556601*^9, 3.8422737353641977`*^9}, 3.842357215263414*^9, { 3.84236119398525*^9, 3.842361195245453*^9}},ExpressionUUID->"de3fe268-a631-424a-b41f-\ 903e41b44f28"], Cell[TextData[{ "Note a substitui\[CCedilla]\[ATilde]o da op\[CCedilla]\[ATilde]o ", StyleBox["AxesLabel", FontFamily->"Source Code Pro", FontWeight->"Regular"], " por ", StyleBox[ButtonBox["FrameLabel", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/FrameLabel.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/FrameLabel.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], "." }], "ItemParagraph", CellChangeTimes->{{3.8422737949776382`*^9, 3.842273846918026*^9}, { 3.873368136010127*^9, 3.873368136014377*^9}},ExpressionUUID->"e85b79e0-bfa8-455f-a6eb-\ 68f74036f4fc"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"ccbea514-18b8-498d-b295-7981bb17390b"], Cell["\<\ Para tra\[CCedilla]ar o gr\[AAcute]fico de uma fun\[CCedilla]\[ATilde]o, n\ \[ATilde]o pode haver vari\[AAcute]veis a que n\[ATilde]o tenham sido atribu\ \[IAcute]dos valores. Em problemas f\[IAcute]sicos, uma forma de atribuir \ valores \[EAcute] adotar unidades convenientes ou, o que \[EAcute] \ equivalente, mudar para vari\[AAcute]veis adimensionais. Vamos ilustrar essa \ estrat\[EAcute]gia com um problema de mec\[AHat]nica estat\[IAcute]stica. \ \>", "Text", CellChangeTimes->{{3.8407263528327703`*^9, 3.840726386414*^9}, { 3.840726458714198*^9, 3.840726482804037*^9}, {3.8414919756375732`*^9, 3.841492059474931*^9}, {3.841492206713057*^9, 3.841492278824851*^9}, { 3.841492387323824*^9, 3.841492387570656*^9}, {3.842274111103971*^9, 3.842274197627184*^9}, {3.842274231569303*^9, 3.8422743192437897`*^9}},ExpressionUUID->"90a8a30a-154b-41f4-a84e-\ 4ed101c61e2f"] }, Open ]], Cell[TextData[{ "Um sistema \[EAcute] composto de part\[IAcute]culas n\[ATilde]o \ interagentes, cada uma das quais pode ter energia igual a 0 ou a ", Cell[BoxData[ FormBox["\[Epsilon]", TraditionalForm]],ExpressionUUID-> "689122e2-9e12-405e-96f4-fddd6a201d41"], ", com esse \[UAcute]ltimo valor duplamente degenerado. Se o sistema \ \[EAcute] mantido a uma temperatura fixa ", Cell[BoxData[ FormBox["T", TraditionalForm]],ExpressionUUID-> "8c3da0ff-6f2d-4ba4-905e-e5d711a1fb65"], ", suas propriedades termodin\[AHat]micas s\[ATilde]o obtidas da fun\ \[CCedilla]\[ATilde]o de parti\[CCedilla]\[ATilde]o can\[OHat]nica ", Cell[BoxData[ FormBox[ RowBox[{"Z", "=", RowBox[{"1", "+", RowBox[{"2", SuperscriptBox["e", RowBox[{ RowBox[{"-", "\[Epsilon]"}], "/", RowBox[{"(", RowBox[{"k", " ", "T"}], ")"}]}]]}]}]}], TraditionalForm]], ExpressionUUID->"d860756a-8a5f-4dce-93f5-a767f0737252"], ", sendo ", Cell[BoxData[ FormBox["k", TraditionalForm]],ExpressionUUID-> "db12d519-f1af-4831-af82-ee16624ba1e9"], " a constante de Boltzmann. A energia m\[EAcute]dia por part\[IAcute]cula e \ o calor espec\[IAcute]fico s\[ATilde]o dados por\n", Cell[BoxData[ FormBox[ RowBox[{"E", "=", RowBox[{"k", " ", SuperscriptBox["T", "2"], FractionBox[ RowBox[{ RowBox[{"\[PartialD]", "ln"}], " ", "Z"}], RowBox[{"\[PartialD]", "T"}]], " "}]}], TraditionalForm]], ExpressionUUID->"74f906bf-b4b8-4656-8465-f06c0a289330"], " e ", Cell[BoxData[ FormBox[ RowBox[{"c", "=", FractionBox[ RowBox[{"\[PartialD]", "E"}], RowBox[{"\[PartialD]", "T"}]]}], TraditionalForm]],ExpressionUUID-> "9d065ac8-fc86-486a-a6b3-bcd9c25b5057"], ".\nVamos fazer gr\[AAcute]ficos dessas duas grandezas utilizando vari\ \[AAcute]veis adimensionais, ou seja, vamos tra\[CCedilla]ar ", Cell[BoxData[ FormBox[ RowBox[{"E", "/", "\[Epsilon]"}], TraditionalForm]],ExpressionUUID-> "166e0103-ecd7-454e-84ed-29681b28b0fe"], " e ", Cell[BoxData[ FormBox[ RowBox[{"c", "/", "k"}], TraditionalForm]],ExpressionUUID-> "b5aa161c-c338-4eb5-9579-b35d02354b84"], " em fun\[CCedilla]\[ATilde]o da \[OpenCurlyDoubleQuote]temperatura reduzida\ \[CloseCurlyDoubleQuote] ", Cell[BoxData[ FormBox[ RowBox[{"k", " ", RowBox[{"T", "/", "\[Epsilon]"}]}], TraditionalForm]],ExpressionUUID-> "bdbc4ad8-0f26-4444-999c-a87d41297074"], ":" }], "Text", CellChangeTimes->{{3.842274332617735*^9, 3.842274612430127*^9}, { 3.8422746519994707`*^9, 3.8422748095066757`*^9}, {3.842275024800037*^9, 3.842275075019041*^9}, 3.8423572736873503`*^9},ExpressionUUID->"e83d8843-e633-4d87-b2f9-\ f8b705cf1f08"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"Z", "=", RowBox[{"1", "+", RowBox[{"2", "*", RowBox[{"Exp", "[", RowBox[{ RowBox[{"-", "\[Epsilon]"}], "/", RowBox[{"(", RowBox[{"k", "*", "T"}], ")"}]}], "]"}]}]}]}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "Fun\[CCedilla]\[ATilde]o", " ", "de", " ", "parti\[CCedilla]\[ATilde]o"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"energia", "=", RowBox[{ RowBox[{"k", "*", RowBox[{"T", "^", "2"}], "*", RowBox[{"D", "[", RowBox[{ RowBox[{"Log", "[", "Z", "]"}], ",", "T"}], "]"}]}], "//", "Simplify"}]}], " ", RowBox[{"(*", " ", RowBox[{"Energia", " ", "por", " ", "part\[IAcute]cula"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{"calesp", "=", RowBox[{"D", "[", RowBox[{"energia", ",", "T"}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"Calor", " ", "espec\[IAcute]fico"}], " ", "*)"}]}]}], "Input", CellChangeTimes->{{3.842274819403225*^9, 3.842274899958036*^9}, { 3.842274977530328*^9, 3.842274995495248*^9}},ExpressionUUID->"eb4b9007-d970-4094-b64f-\ 3f4a479c63c0"], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"energia", "/", "\[Epsilon]"}], "/.", RowBox[{"{", RowBox[{"T", "\[Rule]", " ", RowBox[{"t", "*", RowBox[{"\[Epsilon]", "/", "k"}]}]}], "}"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0.01", ",", "4"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\\"Italic\"]\))\>\"", ",", "\"\<\!\(\*StyleBox[\"E\",FontSlant->\"Italic\"]\) \ (\[Epsilon])\>\""}], "}"}]}], ",", RowBox[{"Evaluate", "[", "optam", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.8422749264681396`*^9, 3.842274967459096*^9}, { 3.8422751707856617`*^9, 3.842275171106803*^9}, 3.8423361428885612`*^9},ExpressionUUID->"54fa9e22-ffc4-402e-b4b2-\ 0c795e4442b4"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"calesp", "/", "k"}], "/.", RowBox[{"{", RowBox[{"T", "\[Rule]", " ", RowBox[{"t", "*", RowBox[{"\[Epsilon]", "/", "k"}]}]}], "}"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0.01", ",", "4"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"T\",FontSlant->\"Italic\"]\) \ (\[Epsilon]/\!\(\*StyleBox[\"k\",FontSlant->\"Italic\"]\))\>\"", ",", "\"\<\!\(\*StyleBox[\"C\",FontSlant->\"Italic\"]\) \ (\!\(\*StyleBox[\"k\",FontSlant->\"Italic\"]\))\>\""}], "}"}]}], ",", RowBox[{"Evaluate", "[", "optam", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.842275116972571*^9, 3.842275164673429*^9}, 3.842336142897174*^9},ExpressionUUID->"eedd4c13-20a9-4420-962f-\ e1431f681832"], Cell[TextData[{ "Uma estrat\[EAcute]gia tamb\[EAcute]m v\[AAcute]lida, mas talvez \ fisicamente menos rica, \[EAcute] utilizar as regras de transforma\[CCedilla]\ \[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{"\[Epsilon]", "\[Rule]", " ", "1"}], TraditionalForm]], ExpressionUUID->"9cb95190-5391-4428-9e3f-ccbda086249c"], " e ", Cell[BoxData[ FormBox[ RowBox[{"k", "\[Rule]", " ", "1"}], TraditionalForm]],ExpressionUUID-> "6e3890e4-ae13-4c69-833f-3326802bfaa6"], ". 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A probabilidade de observar o el\[EAcute]tron em uma dist\ \[AHat]ncia contida no intervalo de largura ", Cell[BoxData[ FormBox[ StyleBox["dr", FontSlant->"Italic"], TraditionalForm]],ExpressionUUID-> "826d603a-3527-4b90-98ea-ce49aea53370"], " em torno de ", Cell[BoxData[ FormBox["r", TraditionalForm]],ExpressionUUID-> "b1801253-bab5-46f3-a85c-136feae5e025"], " \[EAcute] ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"[", RowBox[{"r", " ", RowBox[{ SubscriptBox["R", RowBox[{"n", ",", "l"}]], "(", "r", ")"}]}], "]"}], "2"], StyleBox["dr", FontSlant->"Italic"]}], TraditionalForm]],ExpressionUUID-> "41c6c4fe-c688-43f8-984d-00b736c81572"], ". 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Al\[EAcute]m disso, podemos rotular cada gr\[AAcute]fico, com a op\ \[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["PlotLegends", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/PlotLegends.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/PlotLegends.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ". Essas op\[CCedilla]\[OTilde]es n\[ATilde]o podem ser acrescentadas com a \ fun\[CCedilla]\[ATilde]o ", StyleBox["Show", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", de modo que temos que invocar novamente ", StyleBox["Plot", FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{{3.842288299243579*^9, 3.842288457778346*^9}, { 3.842336909995206*^9, 3.842336923967842*^9}, {3.842357549168662*^9, 3.8423575566737013`*^9}, {3.873372963109527*^9, 3.873372983132862*^9}, { 3.87337301810216*^9, 3.8733730181073303`*^9}},ExpressionUUID->"0ccd70fe-d277-441e-bf6a-\ 006c53efdbf0"], Cell[BoxData[ RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"r", "*", RowBox[{"Rnl", "[", RowBox[{"n", ",", "0", ",", "r"}], "]"}]}], ")"}], "^", "2"}], "/.", RowBox[{"a", "\[Rule]", " ", "1"}]}], ",", RowBox[{"{", RowBox[{"n", ",", "3"}], "}"}]}], "]"}], "]"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "25"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "0.55"}], "}"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"r\",FontSlant->\"Italic\"]\) \ (\!\(\*StyleBox[\"a\",FontSlant->\"Italic\"]\))\>\"", ",", "\"\<\!\(\*StyleBox[\"P\",FontSlant->\"Italic\"]\)(\!\(\*StyleBox[\"r\"\ ,FontSlant->\"Italic\"]\)) \ (1/\!\(\*StyleBox[\"a\",FontSlant->\"Italic\"]\))\>\""}], "}"}]}], ",", " ", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"Filling", "\[Rule]", " ", "Axis"}], ",", RowBox[{"PlotLegends", "\[Rule]", " ", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"n\",FontSlant->\"Italic\"]\)=1\>\"", ",", "\"\<\!\(\*StyleBox[\"n\",FontSlant->\"Italic\"]\)=2\>\"", ",", "\"\<\!\(\*StyleBox[\"n\",FontSlant->\"Italic\"]\)=3\>\""}], "}"}]}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8422885535187817`*^9, 3.842288567595677*^9}, { 3.8422888298711233`*^9, 3.842288832522523*^9}, 3.842336142907177*^9},ExpressionUUID->"f4a7311d-a0a6-41f1-bfeb-\ 744654151b7f"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.842288037625621*^9, 3.842288059640642*^9}, { 3.842288124445612*^9, 3.842288137932507*^9}, {3.8422881798382187`*^9, 3.8422881814693947`*^9}, 3.8422882901785583`*^9, {3.84228851966891*^9, 3.842288534764145*^9}},ExpressionUUID->"cb9bec5b-6160-4fe6-8b4e-\ 75f2b04d0738"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.842276366384461*^9, 3.842276367404133*^9}},ExpressionUUID->"7bf13b9b-a2c3-44e9-9b4d-\ 1639e6d49609"], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"916aafb6-db56-46e9-abc4-ac7b00e07c2a"], Cell["Zeros e extremos de fun\[CCedilla]\[OTilde]es", "Section", CellChangeTimes->{{3.8410460233083143`*^9, 3.8410460412733793`*^9}, { 3.8414947218276873`*^9, 3.841494862183041*^9}, {3.8414950184999313`*^9, 3.8414950759148483`*^9}, {3.841512344981151*^9, 3.841512400542829*^9}, { 3.84159445169516*^9, 3.8415944912259007`*^9}, {3.842290177474586*^9, 3.842290192109984*^9}},ExpressionUUID->"6026ccad-e8e5-4659-852a-\ 29f271e84752"], Cell[TextData[{ "J\[AAcute] discutimos a solu\[CCedilla]\[ATilde]o num\[EAcute]rica de equa\ \[CCedilla]\[OTilde]es alg\[EAcute]bricas usando a fun\[CCedilla]\[ATilde]o \ ", StyleBox["NSolve", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", que no entanto funciona quase que apenas para equa\[CCedilla]\[OTilde]es \ polinomiais. Em casos mais complicados, devemos utilizar a fun\[CCedilla]\ \[ATilde]o ", StyleBox[ButtonBox["FindRoot", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/FindRoot.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/FindRoot.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", que encontra uma solu\[CCedilla]\[ATilde]o por vez, a partir de um \ palpite inicial sobre essa solu\[CCedilla]\[ATilde]o. " }], "Text", CellChangeTimes->{{3.841495092716528*^9, 3.841495136864716*^9}, { 3.841495220002638*^9, 3.84149522517741*^9}, {3.8415122501164703`*^9, 3.841512255616736*^9}, {3.842336998635304*^9, 3.842337088961656*^9}, { 3.8423371323026047`*^9, 3.842337132306622*^9}, {3.8423576053399067`*^9, 3.842357609130612*^9}},ExpressionUUID->"2432d87d-accc-41bf-bf07-\ a9da126b9fae"] }, Open ]], Cell["O exemplo a seguir mostra uma fun\[CCedilla]\[ATilde]o que tem dois \ zeros: ", "Text", CellChangeTimes->{{3.8423371614805603`*^9, 3.842337175856915*^9}, { 3.8423576200879383`*^9, 3.842357620358099*^9}},ExpressionUUID->"1563b0cc-a456-4ba5-9ae6-\ 1cc66a106766"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], ":=", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "-", RowBox[{"x", "^", "2"}]}]}], " "}], "\[IndentingNewLine]", RowBox[{"grafico2", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"f", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"x\",FontSlant->\"Italic\"]\)\>\"", ",", "\"\<\!\(\*StyleBox[\"f\",FontSlant->\"Italic\"]\)(\!\(\*StyleBox[\"x\"\ ,FontSlant->\"Italic\"]\))\>\""}], "}"}]}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.842337229882998*^9, 3.842337252279893*^9}, { 3.842337321935504*^9, 3.842337338998343*^9}, {3.842337499877337*^9, 3.8423375031238537`*^9}, 3.842357623916103*^9},ExpressionUUID->"72da1838-de88-49d1-b0e7-\ 5dc000d68eca"], Cell[TextData[{ "Por inspe\[CCedilla]\[ATilde]o visual, um dos zeros \[EAcute] \ pr\[OAcute]ximo de ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "0"}], TraditionalForm]],ExpressionUUID-> "5dea56e6-fa7b-4d82-8dd6-8c511a913734"], " e o outro, de ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "0.9"}], TraditionalForm]],ExpressionUUID-> "a982b018-2745-4d9e-bd40-b7411cafc9c8"], ". Vamos determinar valores precisos desses zeros:" }], "Text", CellChangeTimes->{{3.842337299743902*^9, 3.842337318204073*^9}, { 3.842337352664077*^9, 3.8423374084976273`*^9}},ExpressionUUID->"64121dbe-c23e-402a-94ae-\ ede4dfae1589"], Cell[BoxData[{ RowBox[{ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", "x", "]"}], "\[Equal]", "0"}], ",", RowBox[{"{", RowBox[{"x", ",", "0.1"}], "}"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ "Com", " ", "palpite", " ", "inicial", " ", "igual", " ", "a", " ", "0.1"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", "x", "]"}], "\[Equal]", "0"}], ",", RowBox[{"{", RowBox[{"x", ",", "0.9"}], "}"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ "Com", " ", "palpite", " ", "inicial", " ", "igual", " ", "a", " ", "0.9"}], " ", "*)"}]}]}], "Input", CellChangeTimes->{{3.842337411822101*^9, 3.842337427578808*^9}, { 3.842337461393134*^9, 3.842337461699977*^9}, {3.8423613293229218`*^9, 3.842361355084159*^9}},ExpressionUUID->"943e8c02-669d-47c4-ae98-\ c08822131d58"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"e743604b-1c12-44a3-bfe6-87bd824714a7"], Cell["\<\ Uma maneira pr\[AAcute]tica de estimar as posi\[CCedilla]\[OTilde]es dos \ zeros \[EAcute] utilizar a op\[CCedilla]\[ATilde]o \[OpenCurlyDoubleQuote]Get \ Coordinates\[CloseCurlyDoubleQuote], que surge ao clicarmos sobre o gr\ \[AAcute]fico com o bot\[ATilde]o direito do mouse. Ap\[OAcute]s selecion\ \[AAcute]-la, podemos clicar com o bot\[ATilde]o esquerdo sobre posi\ \[CCedilla]\[OTilde]es do gr\[AAcute]fico pr\[OAcute]ximas dos zeros, \ produzindo pontos coloridos a cada clique. Ao final, utilizar Ctrl-C (ou \ Cmd-C) copia as coordenadas desses pontos, que podem ser coladas a uma lista \ de pares ordenados: \ \>", "Text", CellChangeTimes->{{3.841668041823781*^9, 3.841668100347762*^9}, { 3.841668156976202*^9, 3.841668156982239*^9}, {3.841668293038082*^9, 3.841668359311798*^9}, {3.841668440698966*^9, 3.841668454016659*^9}, 3.841743913306546*^9, {3.842337558428185*^9, 3.842337748126307*^9}, { 3.842337781020978*^9, 3.842337783608797*^9}, {3.84233784314482*^9, 3.842337846609462*^9}},ExpressionUUID->"0555e125-aa6a-41fa-8d14-\ 3d01e83ab6ea"], Cell[BoxData[ RowBox[{ RowBox[{"Show", "[", "grafico2", "]"}], " "}]], "Input", CellChangeTimes->{{3.841668363947023*^9, 3.84166838311263*^9}, { 3.8416684587710238`*^9, 3.841668460923037*^9}, {3.8416685389059353`*^9, 3.841668572910033*^9}, {3.842337593438224*^9, 3.842337599446104*^9}, 3.8423576925532227`*^9},ExpressionUUID->"98a83e91-e521-4e25-8f55-\ f2185dd134af"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"lista", "=", " ", RowBox[{"(*", " ", RowBox[{ "Cole", " ", "ao", " ", "lado", " ", "a", " ", "lista", " ", "de", " ", "pares", " ", "ordenados"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.842337793988741*^9, 3.842337809416453*^9}, { 3.842337855813051*^9, 3.8423378591644917`*^9}, {3.842337913874633*^9, 3.8423379280151253`*^9}},ExpressionUUID->"22c481aa-ad4c-4f29-ae78-\ cbcaf440a7ef"], Cell[TextData[{ "Em seguida, podemos construir uma tabela de solu\[CCedilla]\[OTilde]es \ selecionando como palpite para a fun\[CCedilla]\[ATilde]o ", StyleBox["FindRoot", FontFamily->"Source Code Pro", FontWeight->"Regular"], " a coordenada ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "7cc2a8a7-af72-468f-99c3-88d393f9c2bf"], " de cada par ordenado:" }], "Text", CellChangeTimes->{{3.842337818170128*^9, 3.842337886265854*^9}},ExpressionUUID->"8099600f-16ce-46c1-ad63-\ 5af2b6589c85"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"f", "[", "x", "]"}], "\[Equal]", "0"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"lista", "[", RowBox[{"[", RowBox[{"i", ",", "1"}], "]"}], "]"}]}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"Length", "[", "lista", "]"}]}], "}"}]}], "]"}]], "Input",Express\ ionUUID->"ef9fb4b3-bb5f-4410-a0a6-25c7209f3070"], Cell[TextData[{ "A fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Length", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Length.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Length.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " fornece o n\[UAcute]mero de elementos de uma lista." }], "ItemParagraph", CellChangeTimes->{{3.842337943878347*^9, 3.842337958457657*^9}, { 3.842338042990859*^9, 3.842338042995832*^9}},ExpressionUUID->"90f4a627-768c-4129-bd9d-\ 964be8cc0a9c"] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"819adaeb-b750-4384-8bd8-23b33ce04733"], Cell[TextData[{ "Para determinar extremos de fun\[CCedilla]\[OTilde]es, o Mathematica disp\ \[OTilde]e das fun\[CCedilla]\[OTilde]es ", StyleBox[ButtonBox["FindMinimum", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/FindMinimum.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/FindMinimum.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " e ", StyleBox[ButtonBox["FindMaximum", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/FindMaximum.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/FindMaximum.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", cuja sintaxe \[EAcute] bastante semelhante \[AGrave] da fun\[CCedilla]\ \[ATilde]o FindRoot. " }], "Text", CellChangeTimes->{{3.841069824495468*^9, 3.8410698723684464`*^9}, { 3.841126648040525*^9, 3.841126703885021*^9}, {3.841126787731592*^9, 3.841126789520269*^9}, {3.841594762980836*^9, 3.84159480454916*^9}, { 3.841595555500298*^9, 3.8415956047536497`*^9}, {3.842338126477908*^9, 3.842338156323935*^9}, {3.84233821168635*^9, 3.842338291763804*^9}, 3.8423577385192966`*^9},ExpressionUUID->"6f2af07a-0a3d-478a-8393-\ 33ce571b5160"], Cell[TextData[{ "Vamos ilustrar o uso dessas duas fun\[CCedilla]\[OTilde]es com o problema \ de um oscilador harm\[OHat]nico sujeito a uma for\[CCedilla]a de \ resist\[EHat]ncia do ar proporcional ao quadrado da velocidade. A equa\ \[CCedilla]\[ATilde]o de movimento \[EAcute] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ OverscriptBox["x", ".."], "+", RowBox[{"2", "\[Gamma]", " ", OverscriptBox["x", "."], RowBox[{"\[LeftBracketingBar]", OverscriptBox["x", "."], "\[RightBracketingBar]"}]}], "+", RowBox[{ SubsuperscriptBox["\[Omega]", "0", "2"], "x"}]}], "=", "0"}], TraditionalForm]],ExpressionUUID->"dad4f69e-baae-4f14-886c-bb3c470f983e"], ", e vamos supor que o oscilador parte do repouso com uma deforma\[CCedilla]\ \[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "0"], "=", "A"}], TraditionalForm]],ExpressionUUID-> "e5fd7046-ccad-416b-9abd-09149c65cca8"], ". A solu\[CCedilla]\[ATilde]o num\[EAcute]rica da equa\[CCedilla]\[ATilde]o \ de movimento com ", Cell[BoxData[ FormBox[ RowBox[{"\[Gamma]", "=", "0.2"}], TraditionalForm]],ExpressionUUID-> "4cc53997-e007-4322-812e-c1c79b0fd882"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Omega]", "0"], "=", "2"}], TraditionalForm]], ExpressionUUID->"e04abd8a-804b-407c-82a4-866fff35c2fb"], " e ", Cell[BoxData[ FormBox[ RowBox[{"A", "=", "1"}], TraditionalForm]],ExpressionUUID-> "e98de893-ec8a-4dc7-af6e-30b6535a47ea"], " (unidades arbitr\[AAcute]rias) \[EAcute] obtida abaixo:" }], "Text", CellChangeTimes->{{3.841595611965324*^9, 3.841595705598823*^9}, { 3.841595757427567*^9, 3.841595774068248*^9}, {3.841745923183269*^9, 3.8417459407774963`*^9}, {3.842338367486302*^9, 3.8423384351760197`*^9}, { 3.8423385007756233`*^9, 3.842338517360704*^9}, {3.842338553496283*^9, 3.842338749546898*^9}, {3.84235775114145*^9, 3.842357756534642*^9}, { 3.842361428345615*^9, 3.842361434016693*^9}, 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m\[IAcute]nimos e outra para localizar os m\[AAcute]ximos:\ \>", "Text", CellChangeTimes->{{3.842343120560115*^9, 3.842343147739512*^9}},ExpressionUUID->"0ef4201c-533c-477c-800d-\ d25bfac15efa"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"listamin", "="}], RowBox[{"(*", " ", RowBox[{ "Cole", " ", "ao", " ", "lado", " ", "a", " ", "lista", " ", "de", " ", "pares", " ", "ordenados", " ", "para", " ", "os", " ", "m\[IAcute]nimos"}], " ", "*)"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"listamax", "="}], " ", RowBox[{"(*", " ", RowBox[{ "Cole", " ", "ao", " ", "lado", " ", "a", " ", "lista", " ", "de", " ", "pares", " ", "ordenados", " ", "para", " ", "os", " ", "m\[AAcute]ximos"}], " ", "*)"}], ";"}]}], "Input", CellChangeTimes->{{3.84233880219973*^9, 3.842338802602943*^9}, { 3.842343020498365*^9, 3.8423430278490267`*^9}, {3.84234315533366*^9, 3.842343224937683*^9}, {3.842343362480514*^9, 3.842343367575862*^9}},ExpressionUUID->"aa3dd76e-7efd-4c59-b8d2-\ 27756320f4ab"], Cell[TextData[{ "Agora utilizamos ", StyleBox["FindMinimum", FontFamily->"Source Code Pro", FontWeight->"Regular"], " e ", StyleBox["FindMaximum", FontFamily->"Source Code Pro", FontWeight->"Regular"], " para produzir tabelas com as posi\[CCedilla]\[OTilde]es precisas dos \ extremos:" }], "Text", CellChangeTimes->{{3.842343250001955*^9, 3.842343284538402*^9}},ExpressionUUID->"615a9009-c8ab-40b5-8f86-\ 9aa7e333324a"], Cell[BoxData[{ RowBox[{"minimos", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"FindMinimum", "[", RowBox[{ RowBox[{"pos", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"listamin", "[", RowBox[{"[", RowBox[{"i", ",", "1"}], "]"}], "]"}]}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"Length", "[", "listamin", "]"}]}], "}"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"maximos", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"FindMaximum", "[", RowBox[{ RowBox[{"pos", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"listamax", "[", 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Agora vamos explorar outras \ capacidades do Mathematica para lidar com esse tipo de gr\[AAcute]fico." }], "Text", CellChangeTimes->{{3.8423438752382507`*^9, 3.842343918918458*^9}},ExpressionUUID->"929a17da-e7d6-494e-99a6-\ 5ff2b32f4fda"], Cell["\<\ Suponha que tenhamos registrado a posi\[CCedilla]\[ATilde]o de um carro (em \ metros) para diversos instantes de tempo (em segundos) e constru\[IAcute]do a \ lista e o gr\[AAcute]fico abaixo:\ \>", "Text", CellChangeTimes->{{3.842343932711841*^9, 3.842343997432673*^9}},ExpressionUUID->"f7ef4db5-1505-40ec-b099-\ 074d13022453"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"dados", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "1.5"}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", "6.0"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "13.5"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "24"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "37.5"}], "}"}], ",", RowBox[{"{", RowBox[{"6", ",", "54"}], "}"}], ",", RowBox[{"{", RowBox[{"7", ",", "73.5"}], "}"}], ",", RowBox[{"{", RowBox[{"8", ",", "96"}], "}"}], ",", RowBox[{"{", RowBox[{"9", ",", "121.5"}], "}"}]}], "}"}]}], ";"}], " "}], "\[IndentingNewLine]", RowBox[{"grafico3a", "=", RowBox[{"ListPlot", "[", RowBox[{"dados", ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"PointSize", "[", "0.025", "]"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"t\",FontSlant->\"Italic\"]\) (s)\>\"", ",", "\"\<\!\(\*StyleBox[\"d\",FontSlant->\"Italic\"]\) (m)\>\""}], "}"}]}], ",", RowBox[{"Evaluate", "[", "optam", "]"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.842344015366826*^9, 3.842344027092675*^9}, { 3.842344075792233*^9, 3.842344085623711*^9}, 3.8423578979813013`*^9},ExpressionUUID->"cd335b7e-9800-4a55-b9b1-\ 155427f29078"], Cell[TextData[{ "Note o uso da op\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["PlotStyle", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/PlotStyle.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/PlotStyle.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], Cell[BoxData[ FormBox["\[Rule]", TraditionalForm]], FontFamily->"Source Code Pro", FontWeight->"Regular",ExpressionUUID-> "baf80042-42c3-4fc3-9632-82edd19c30df"], StyleBox[ButtonBox["PointSize", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/PointSize.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/PointSize.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " para controlar o tamanho do ponto utilizado no gr\[AAcute]fico. " }], "ItemParagraph", CellChangeTimes->{{3.8423441539445667`*^9, 3.84234422247896*^9}, { 3.873373921169052*^9, 3.873373930961144*^9}},ExpressionUUID->"04bbcb29-1f50-4a55-8c7c-\ 3909e402d4c8"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"8819130e-2ada-4052-a082-eb0337583a7c"], Cell[TextData[{ "Agora vamos utilizar a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Fit", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Fit.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Fit.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " para produzir um ajuste c\[UAcute]bico dos dados, por m\[IAcute]nimos \ quadrados. Invocamos a fun\[CCedilla]\[ATilde]o com os seguintes argumentos: \ (i) a lista que cont\[EAcute]m os dados a ajustar; (ii) uma lista indicando \ as fun\[CCedilla]\[OTilde]es ", Cell[BoxData[ FormBox["1", TraditionalForm]],ExpressionUUID-> "74e2a500-6ec2-4f1f-9483-fbbf9644f1bb"], ", ", Cell[BoxData[ FormBox["t", TraditionalForm]],ExpressionUUID-> "e7dce63f-2cba-4087-aad4-f062b132971d"], ", ", Cell[BoxData[ FormBox[ SuperscriptBox["t", "2"], TraditionalForm]],ExpressionUUID-> "756dd832-a1ec-4c4b-afe9-e64e831cd637"], " e ", Cell[BoxData[ FormBox[ SuperscriptBox["t", "3"], TraditionalForm]],ExpressionUUID-> "4efeb95a-5e52-40db-b77f-be375a0b1866"], "; (iii) a vari\[AAcute]vel independente. " }], "Text", CellChangeTimes->{{3.8423442297137947`*^9, 3.8423444226514807`*^9}, { 3.842344575147196*^9, 3.84234457619247*^9}, {3.8423446111815443`*^9, 3.8423446378994017`*^9}, {3.842344810329218*^9, 3.842344814143237*^9}},ExpressionUUID->"0fe17f98-89e7-4053-adf9-\ ca1b704fe85f"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"ajuste", "=", RowBox[{"Fit", "[", RowBox[{"dados", ",", RowBox[{"{", RowBox[{"1", ",", "t", ",", RowBox[{"t", "^", "2"}], ",", RowBox[{"t", "^", "3"}]}], "}"}], ",", "t"}], "]"}]}], "\[IndentingNewLine]", RowBox[{"Chop", "[", "ajuste", "]"}]}], "Input", CellChangeTimes->{{3.8423448470511513`*^9, 3.8423448575033092`*^9}},ExpressionUUID->"f8b6e26a-cd13-46f8-843f-\ 1f4e6af22ed4"], Cell[TextData[{ "A fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Chop", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Chop.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Chop.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " elimina, por padr\[ATilde]o, os termos num\[EAcute]ricos de uma express\ \[ATilde]o que sejam menores do que ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "10"}]], TraditionalForm]],ExpressionUUID-> "5c15de43-b9ab-4aa7-b8c0-bf3ff643b6bc"], ". Ajustes com outras fun\[CCedilla]\[OTilde]es podem ser feitos alterando o \ argumento (ii) de ", StyleBox["Fit", FontFamily->"Source Code Pro", FontWeight->"Regular"], "." }], "ItemParagraph", CellChangeTimes->{{3.84234487182246*^9, 3.842344900948168*^9}, { 3.842344931959352*^9, 3.842344938748457*^9}, {3.8423579720469627`*^9, 3.842357994715579*^9}},ExpressionUUID->"3baa4ee6-b91f-42f9-a3a3-\ 499eef01145e"], Cell[TextData[{ "Podemos produzir um gr\[AAcute]fico da fun\[CCedilla]\[ATilde]o de ajuste e \ combin\[AAcute]-lo com o gr\[AAcute]fico dos dados usando a fun\[CCedilla]\ \[ATilde]o ", StyleBox["Show", FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{{3.842345199446851*^9, 3.8423452295401*^9}},ExpressionUUID->"0d9cd8c2-b312-4f5b-bffb-7e426098a737"], Cell[BoxData[{ RowBox[{ RowBox[{"grafico3b", "=", RowBox[{"Plot", "[", RowBox[{"ajuste", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "15"}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", "Orange"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Show", "[", RowBox[{"grafico3a", ",", "grafico3b"}], "]"}]}], "Input", CellChangeTimes->{{3.842345267457519*^9, 3.842345293565106*^9}},ExpressionUUID->"d78428d4-d600-4ed8-998c-\ 030f7ec0e056"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"bf3623c1-381c-4d74-bccc-d2feab1bf6cc"], Cell[TextData[{ "Se quisermos tra\[CCedilla]ar os dados utilizando linhas em vez de pontos, \ invocamos a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["ListLinePlot", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/ListLinePlot.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/ListLinePlot.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{{3.8423458183472567`*^9, 3.8423458581151743`*^9}},ExpressionUUID->"7bfec97d-f52f-4a9a-833d-\ a7b01c22de97"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ListLinePlot", "[", RowBox[{"dados", ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"t\",FontSlant->\"Italic\"]\) (s)\>\"", ",", "\"\<\!\(\*StyleBox[\"d\",FontSlant->\"Italic\"]\) (m)\>\""}], "}"}]}], ",", RowBox[{"Evaluate", "[", "optam", "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.842345914929584*^9, 3.842345921612193*^9}}, CellLabel->"",ExpressionUUID->"4569168e-c766-4811-afff-ef4569bed6ce"], Cell[TextData[{ "O padr\[ATilde]o \[EAcute] unir os pontos por segmentos de reta. Para obter \ uma curva mais suave, existe a op\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["InterpolationOrder", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/InterpolationOrder.html"], None}, ButtonNote-> "https://reference.wolfram.com/language/ref/InterpolationOrder.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ". Vamos interpolar os pontos com c\[UAcute]bicas: " }], "Text", CellChangeTimes->{{3.842345979031719*^9, 3.842346106302842*^9}, { 3.873374155574421*^9, 3.87337415558125*^9}},ExpressionUUID->"3e3c108a-5279-4136-9356-\ 39fa096636d9"], Cell[BoxData[ RowBox[{"ListLinePlot", "[", RowBox[{"dados", ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"t\",FontSlant->\"Italic\"]\) (s)\>\"", ",", "\"\<\!\(\*StyleBox[\"d\",FontSlant->\"Italic\"]\) (m)\>\""}], "}"}]}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"InterpolationOrder", "\[Rule]", "3"}]}], "]"}]], "Input", CellChangeTimes->{{3.842346133170863*^9, 3.8423461380861883`*^9}}, CellLabel->"",ExpressionUUID->"bbd49a86-8191-4677-8d68-1f6888e6d79d"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"dc3647df-f278-4a6d-8cf5-9ad9865172fe"], Cell["Gr\[AAcute]ficos param\[EAcute]tricos", "Section", CellChangeTimes->{{3.842347875496059*^9, 3.8423478784538527`*^9}},ExpressionUUID->"a07768e1-4740-465e-990a-\ 09dec919ddab"], Cell[TextData[{ "Para tra\[CCedilla]armos gr\[AAcute]ficos bidimensionais de ", Cell[BoxData[ FormBox[ RowBox[{"x", "(", "t", ")"}], TraditionalForm]],ExpressionUUID-> "e1988722-7089-4f62-8142-2e3a0abc3314"], " contra ", Cell[BoxData[ FormBox[ RowBox[{"y", "(", "t", ")"}], TraditionalForm]],ExpressionUUID-> "a9449387-63bd-4a6d-9a92-ff4732df973f"], ", \[AGrave] medida que mudam os valores da vari\[AAcute]vel independente ", Cell[BoxData[ FormBox["t", TraditionalForm]],ExpressionUUID-> "7be70b49-e306-486f-8b3e-828b8f0243ac"], ", invocamos a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["ParametricPlot", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/ParametricPlot.html"], None}, ButtonNote-> "https://reference.wolfram.com/language/ref/ParametricPlot.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ". Vamos ilustr\[AAcute]-la com o exemplo das figuras de Lissajous, \ produzidas a partir de ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"x", "(", "t", ")"}], "=", RowBox[{ SubscriptBox["A", "x"], RowBox[{"cos", "(", RowBox[{ RowBox[{ SubscriptBox["\[Omega]", "x"], "t"}], "+", SubscriptBox["\[Phi]", "x"]}], ")"}]}]}], TraditionalForm]], ExpressionUUID->"94e93e47-2686-4d43-b7b5-baa72e59b28c"], " e ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"y", "(", "t", ")"}], "=", RowBox[{ SubscriptBox["A", "y"], RowBox[{"cos", "(", RowBox[{ RowBox[{ SubscriptBox["\[Omega]", "y"], "t"}], "+", SubscriptBox["\[Phi]", "y"]}], ")"}]}]}], TraditionalForm]], ExpressionUUID->"7fb8a7dc-224e-42e1-8d4d-f0d01f2d9d70"], ". Isso pode corresponder, por exemplo, aos campos el\[EAcute]tricos de duas \ ondas eletromagn\[EAcute]ticas de polariza\[CCedilla]\[OTilde]es \ perpendiculares que se combinam em um ponto." }], "Text", CellChangeTimes->{{3.842347931296171*^9, 3.842348158500369*^9}, { 3.842348198992408*^9, 3.842348217949791*^9}, {3.842350046767194*^9, 3.842350082698326*^9}, {3.8423508673052473`*^9, 3.842350867311578*^9}, { 3.842358071420203*^9, 3.842358091252952*^9}},ExpressionUUID->"f293c507-4dee-4689-bc09-\ bbfce2e26b5e"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"fx", "[", "t_", "]"}], ":=", RowBox[{"ax", "*", RowBox[{"Cos", "[", RowBox[{ RowBox[{"\[Omega]x", "*", "t"}], "+", "\[Phi]x"}], "]"}]}]}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{"fy", "[", "t_", "]"}], ":=", RowBox[{"ay", "*", RowBox[{"Cos", "[", RowBox[{ RowBox[{"\[Omega]y", "*", "t"}], "+", "\[Phi]y"}], "]"}]}]}]}], "Input", CellChangeTimes->{{3.8423482203128157`*^9, 3.842348282454698*^9}, 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