(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.1' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 67024, 1426] NotebookOptionsPosition[ 64744, 1384] NotebookOutlinePosition[ 65090, 1399] CellTagsIndexPosition[ 65047, 1396] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{ RowBox[{"Clear", "[", "x", "]"}], ";"}]], "Input", CellChangeTimes->{{3.7419465833682613`*^9, 3.741946598923657*^9}, { 3.741947645171728*^9, 3.7419476715354958`*^9}, 3.741947713392439*^9, { 3.741947773594726*^9, 3.7419478061640244`*^9}, 3.742029354991497*^9, { 3.742029690872781*^9, 3.7420296937754583`*^9}},ExpressionUUID->"b4d86fd4-3e22-44df-a47b-\ e4aa9d092ed5"], Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{ RowBox[{ "Below", " ", "is", " ", "our", " ", "main", " ", "differential", " ", "equation"}], ",", " ", RowBox[{"where", " ", "instead", " ", "of", " ", RowBox[{"2", "/", "3"}], " ", "one", " ", "can", " ", "use", " ", "what", " ", "value", " ", "of", " ", "k", " ", "is", " ", "desired"}]}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Maineqn", "=", RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "x", "]"}], "\[Equal]", RowBox[{ RowBox[{ RowBox[{"y", "[", "x", "]"}], "/", "x"}], "-", RowBox[{ RowBox[{"(", RowBox[{"2", "/", "3"}], ")"}], "*", RowBox[{"Sqrt", "[", RowBox[{"1", "+", RowBox[{ RowBox[{ RowBox[{"y", "[", "x", "]"}], "^", "2"}], "/", RowBox[{"x", "^", "2"}]}]}], "]"}]}]}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"f", "[", RowBox[{"x_", ",", "y_"}], "]"}], ":=", RowBox[{ RowBox[{"y", "/", "x"}], "-", RowBox[{ RowBox[{"(", RowBox[{"2", "/", "3"}], ")"}], "*", RowBox[{"Sqrt", "[", RowBox[{"1", "+", RowBox[{ RowBox[{"y", "^", "2"}], "/", RowBox[{"x", "^", "2"}]}]}], "]"}]}]}]}], ";"}]}]}]], "Input", CellChangeTimes->{{3.741946657811358*^9, 3.74194678346246*^9}, { 3.741946906221222*^9, 3.741946906636107*^9}, {3.741947148718479*^9, 3.741947148994878*^9}, 3.741947524559182*^9, 3.742029376408154*^9},ExpressionUUID->"7307d5ef-b80e-4a52-ad83-\ c8b954a0017a"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.741946795725917*^9, 3.741946805224758*^9}, 3.7419469194733667`*^9, 3.742030317382806*^9},ExpressionUUID->"0d7efb9d-b236-4e85-a64e-\ 2b77164c10d0"], Cell[BoxData[ RowBox[{"(*", RowBox[{ RowBox[{"Let", "'"}], "s", " ", "first", " ", "check", " ", "the", " ", "exact", " ", "solution", " ", "with", " ", "DSolve"}], "*)"}]], "Input", CellChangeTimes->{{3.7419468529619303`*^9, 3.741946852965678*^9}, { 3.741946945597341*^9, 3.741947000619359*^9}, {3.741947086622533*^9, 3.741947105405962*^9}, {3.7419471765453157`*^9, 3.741947178445395*^9}, { 3.7419472223523397`*^9, 3.741947249426116*^9}, 3.741947328229033*^9},ExpressionUUID->"e1228e49-22e1-4981-98b6-\ 1d5844854e2a"], Cell[BoxData[""], "Input", CellChangeTimes->{{3.741947310529869*^9, 3.7419473118215847`*^9}, 3.7420297064661818`*^9},ExpressionUUID->"d374c0b1-c175-4ea7-a5e4-\ 45e5448f502a"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"exact", "=", RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "x", "]"}], "\[Equal]", RowBox[{"f", "[", RowBox[{"x", ",", RowBox[{"y", "[", "x", "]"}]}], "]"}]}], ",", RowBox[{ RowBox[{"y", "[", "5", "]"}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"y", "[", "x", "]"}], ",", "x"}], "]"}]}]], "Input", CellChangeTimes->{{3.741947244531969*^9, 3.74194726428927*^9}, { 3.741947395863879*^9, 3.741947400905704*^9}, 3.7419474358834543`*^9},ExpressionUUID->"c0cf3758-1d43-4b53-afb7-\ d4086f6b9ecc"], Cell[BoxData[ TemplateBox[{ "Solve","ifun", "\"Inverse functions are being used by \ \\!\\(\\*RowBox[{\\\"Solve\\\"}]\\), so some solutions may not be found; use \ Reduce for complete solution information.\"",2,7,1,23909534328258000124, "Local"}, "MessageTemplate"]], "Message", "MSG", CellChangeTimes->{ 3.742029639574135*^9, {3.742029696751771*^9, 3.74202972146881*^9}, 3.742029753195159*^9, 3.742030952407955*^9, 3.7461831678831487`*^9},ExpressionUUID->"b9baecee-e819-4a7c-999a-\ 1d1bdddd9754"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"y", "[", "x", "]"}], "\[Rule]", RowBox[{"x", " ", RowBox[{"Sinh", "[", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"(", RowBox[{ RowBox[{"2", " ", RowBox[{"Log", "[", "5", "]"}]}], "-", RowBox[{"2", " ", RowBox[{"Log", "[", "x", "]"}]}]}], ")"}]}], "]"}]}]}], "}"}], "}"}]], "Output", CellChangeTimes->{ 3.741948179532836*^9, 3.741948221186596*^9, 3.742029639584148*^9, { 3.742029696836755*^9, 3.742029721529213*^9}, 3.742029753278264*^9, 3.742030952570828*^9, 3.7461831679021497`*^9},ExpressionUUID->"1f2f8c61-1987-4499-b34b-\ 3b3f8c67138e"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{"(*", RowBox[{ RowBox[{"Let", "'"}], "s", " ", "plot", " ", "the", " ", "exact", " ", "solution"}], "*)"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"MainPath", "[", "x_", "]"}], "=", RowBox[{ RowBox[{"{", RowBox[{"x", ",", RowBox[{"y", "[", "x", "]"}]}], "}"}], "/.", "exact"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"ExactPlot", "=", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"MainPath", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", " ", "0", ",", "5"}], "}"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "6"}], "}"}]}], "}"}]}]}], "]"}]}]}]}]], "Input", CellChangeTimes->{{3.7419480622061787`*^9, 3.741948110869615*^9}, { 3.741948144974875*^9, 3.7419481759234667`*^9}, {3.7419482167921667`*^9, 3.741948250754737*^9}, {3.741948309469771*^9, 3.741948335243108*^9}, { 3.741948577139997*^9, 3.7419485791667023`*^9}, {3.742029718083053*^9, 3.742029718277068*^9}},ExpressionUUID->"244477e1-7aa6-4349-a008-\ 7b99fde4d850"], Cell[BoxData[ GraphicsBox[{{}, {}, TagBox[ {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[ 1.], FaceForm[Opacity[0.3]], LineBox[CompressedData[" 1:eJwt13k8VP0XB3BLVEIYKS1oj2yVItTnRo+UUgiF7FF5UmmzpUKWUsmSQpu0 omQpW0WksoRIKiH7vszMnRn77/v8Xuafeb1n7nzPud977zlnFjseMz0owMfH 58LPx/ffu+OozJdgWWn8/JsxZ6TsJXRybHK75PfjiKTDpFPUxykfBl9KbxFD vRTJVy7tHPjhAd4XsZN88hVT33tBYMMQn7ZCFUo61ym1up6H5m6lkjCPb1PH +8PyY3zhhthqsJcMBfSOXoLYeZPZoQk1U78PxZH13RtTIr7D3+Qzs10/DIst arrPHaudWu86Tk8sqDiu+2NqvQiMxlt8dUytwx5Lnc6+w1Fo3a74vNHq59T6 0RAZsz9dPPZzav3buFRfXpCg9BsNl6QsO1/FgmfIvLQg6/dUvHhovJGfLUPV T8W7h7wXY3W+W/5MxUtA9vQuV2PpRvidsI/tv/8Qcbsaap5daJyKnwj5An8Z he7GqfiPkGpzISLUqmkq/jO0yDLybs79C9H7mjO7Op9hstc5dffWv1P5PEex k2i83PG/U/kk42ntonClj3+n8nmJNgMRKyGX5ql80vB+rrSzbmgLoic8WO66 6XgZKWSqldQylV86FgTa6OeUtUzll4FP5pKdduKtU/llIv2D1NnosNap/LIx 57RvM3W2DddCPE79CcvG+qchGiei29BkEaJSNZyNR5syT25Lb0MQK/1eVnUO nlaUrxvoa8M35VmBwcF5yJ7FTD5k144j97J3Lh94j86Yat0nGh1oeN3hL6qa jwzdzKzlxh2YcdfDLsg5H11acvUurh2w/jdknndVPowWiqzZcLsD/DMzLjsk FcDEuc5JaaQDu/RmHVO3K0Sal7eSeVonzijeNHoeXQibL0kCAZ87cV9i8apl ZYVQj7f8GNDQCWbDhr9zNxZB48VtOX6RLtzycTSblPqIdeFqAZO2XWjNyNb8 WlyMAIvIi3f5u+G78hDfv6olMHs0+XuDSg+yLn9+td2hBJ9zJo992NQDum+V 08qoEtScyr623rgH7hldH5uHSxBo8PRGtXsP7Le4Xdn/sRSNDT+dOC96sNXa XWabTTlODk9/KbG6F0HbtGUk/6mEaHTyMUXpPizz6u4/b1sJo50FbxYt6cOH 57Gf+s9Wwl537TU+9T5Mio16lj2vxO7RPWERRn3wrMn9HSxRhUMpXhq9F/tw xGHTg4nfVVh0SsojpqcPxj6Ucq9HNfq3LEjPTe9H+5+Du35fqYZ6zDnVK+/7 4UddcS9JrMaAhFv7rtJ+vBCqTX1aWw2f16vcU5r7MTvcbf1B7Rrcf2Xr80Ni AJWJNzc38H9H2Z28wJ4jAzD92runMrwWniaCQaazBmG++PbpjBc/UeYZWiL7 dBCfYw3j8r/8xMyNtn6yKYPQlubll7X+xPXwD7rSaYNQmG4p2jb/Fx4vuMId yR1Ed6/0wznBv5A0Uq9xrWIQflnhlWfsfmPZy2TRb/Qgnu4OWb1R4g+M29q9 xjcPYeJd7mCDVhP0t9XxLfs4hLxfn3nn/mmCJr+beM3nIXhzvvMtMm3Cl9At QhfLhsBRGZSwdmuCovLqvKrqIfTHL1vz404Tik0jkoybh9DoffVEhcBf6HRe sc+eGEL+Bjvm+7K/8FP1dXZdz4T/CwH2ffsWdAil2L6LZSLl+14/Y/cWaO6d Wz95h4m6scczxnxa8O62ZsDmB0yoGRktsoxpQfW2g/vTnzBR3xFpIPa1BdKa 7GM+GUxoLl5+20unFd2ZbpeDy5nojdy22VSmDebro1LvjzNh4RUWIljWjv1N m5PMLFiY0JLVPPGzHYG1dXTbPhYe8x61NbS3wy9kt+gZaxa4Z9/r5fB3wFjt 6sPr9izcOsMcO67ZAQeN6mdPj7Dw6+S+Ew0JHfD+8fXiST8WbN2XWWV7dWKV keO2s4ksHHTKW31sZTeuS705G9zNgpS/XwW9rhsZ6rPET/ey8P4+ddKX6sYB tSMzHPpZkG0oygnd3424tYtVNJgslO8r3554pRse7QtNPg2zsN64wfXnQDe6 en7VPJjBhtBGvkT9rB5EWgXLqS5nI83yw7aSoh48uMMe71rBht2ZwJ49VT0Q bptnm7iKjez0Getsu3vQfPHHQ2llNo6qSn7wXNiLzXPVbFrWsvF96ZKmlAu9 2L9h93ptio1H4lsXztveBzfrUoHq/WzkaxnppVj0YRqlNnrRmo1fjqaH9Jz7 0HxS0VjtABvib+wz/j1Pjq+9Ov2SPRuedud2FmT2YUZP9N+FrmzsSH3te2RJ P9wtVEYnTrLRb6L4J2+0H5yKH9URYWzM8FUXNJ05gMev3KrFr7Gx5LGmYofM AMrHJLaHXmfDcuSf05JrBzDZ9P62VwTJJ8FRzOXQAO7s+9q+7RYbEay4TbO/ D0CkqO5P3EM24nZMr1C3HESh0vVLK7LZULr+qD74wCC2FL/NsclhI6dav7vB aRD5iodTbuSS87G5IHT1+CCUJr3LuW/ZmHdsWKcrdBA3pHfNyPzARlRkz9MH eYOYa8tpZZWyEVZf4S+1ZAhOafr8wX/YWLjYPfzwqiFEKm6noxvYSDooejdf dQj8B4fePGxko6TfMNtdZwhvslWScv+S8+X/MFCydwh7qvZa/mljI3B5hk1A 8BDUc2s0OvvY8HW/tYHdO4Q/DnIO8uNk/ybNLFazhvBlz6nAaRMkv/DZZxyH h3B09Vh4J3HGq6DMSiEmBN3MXFP4aPSyT2mkyDGxWLpTWmkaDRsfk7Uue5h4 NDevckCEhk6IiGpdOnnOSo0evJxLY4Zs8a7ZOUwMLTF67TGPRu2zi0cN8pmI 57Mv0JCl4VHKS84oY0Lv3J7czPk0not3rL7RxoTkqnbDpEU05kcXKu6Yy8LF 5hS5g0tpjDzwXZ7nzcIO+3X6Mqo0qT9WQd4XWVAOaI0pJhas0OrQCmGh+ABV cFqNhugY+2lmNAu9jjmRleo05C2OKr9MZaHp3ZC+5zoaW0Vs1yW0s6DiMn1t uBaNzwq6UfZ9LMzutBNS3Uhjl+Z8Wo7NQhhNZZQQWzjXvo7jZyMhfNtjPh0a h94Za0cvZOOB2WS77SYaVz2wJcSU7LPO2w1tW2hIhS5KMCD3+WhrH+u4Ho2b 90YFhMh97NioFDlKfL/sTdHFo2zMFbHzEttKI22FuqFPCBu1zLSBpQZkv34p 7HZ/x0aqOeP3gh00FPQFDuxVohEW93GvtgmNlTN9tO6R86Kvas6LJ1atYDG6 N9CIP6/eN0asa91Wcl6ffE79Lcs1pbHv5GetJBsaascs3VT30gh/eE1a4DqN vYapzh0WNGIOTx/cGU2jhCPipG1J467ahdKYOBreLT5nw4iTc0/4qzyl0SL3 tkVlH9mv6r2D+wrIPl7zrjq4n8ak4IKylywac9pX4rk1DeGSyCcjwzQ+pf18 308sGi4a8A8fB5yEcau1JI/5Cyc2/hbl4N81yX9fE2/Q+PtEeAUHLUdnaWUe oLFpZH+AiTIHg7PfRDKJt+Z/s41fy0FZyOAsNVsapjuL5qwFB4LWc80eEbs7 Pwk4sI+D6nYP/WA7GqeVFOye2nLg2Vt2+S2x7+AtbZYzB/7VRYNM4su+l4dC TnCwsitC1tqexuOoo3YZlzlQ+f2uTcGBRopVu/ZkOAei3+QV9xBnKNjJ7Ijh 4NChpVf9iD8k7ylvfMiB8RXJrDrihqJ1OrPecnC03Ou+vyO5DhIVln8KOfgq q2/4gni/zZFTL0s4cOQN9dQR17Dup5jVceC3b/Oq1U40SpeIK8SzONBy2nCu kDjE/Zmu+wgHHXHfRDuIDXK27qf4uZCwK4md6Uzim/hEtIpzkblFMN2IOOtC 5zQVJS5c9or9KSQ+UxaweFKdi4dGI66NxOvmyW+u0uTCL+Y5a5j4xUvzs6f/ 4WIaJ1Bc+SCNf0cHI7ft5CLX2eCuPrHitrBUWTMu4puG1ayIExs+dL215+KI s5NVILGjkq1wuCsXB+e9ZMUQy58ZXuLozgVj7bzw58T1BVHQOM2FpkCXWh5x rJi6jbAvF0mJbtVlxPv2l3rW+XMRwdftXU8s88gl+nkoF9GfU1b0EN/QvfPV OIYLM8fVYUIuNIxDtHoU7nLxKeqsgSSxaE31dFYiFxtoFeGFxCXyx5Z9TOKC vTamdDlxiJvIlpg0LmRKRm+qEhu8eXTgcDYXiYKJhzYQCwpu8dbJ56KW/kFt Is43rr8p9okL3cBGeX1iv9iz6Y3lXIzZCQgbEuu2S1W+quGCNz2EZUQ8vOZF b8BvLqS/vOk0Jn59bvtMi2Yu9r6ra9tDfOpL6/JVXVwsrlrTY0K8ds4FvZEB LtyyZw7/5wH7BXZlHC46xx/P/s/Jya997o5zsZm9W2038RGeya3j03jQLNix byfxqq19GXqzeBjc9+fyf/m0XQ+pkpbiQfz8rk//5Zvwe2l/+zwe+oqbxDYT 2698L5Itz0NZZL2tJrHcSauVV1bwICwblKNGXP+O1j+gwoOu1ZLFK4ljRW7Y q2nwcGE1J2IRsaWF8jl+HR5urDGQYBBLJ3y6Xb2FB/5mi9vTib/1Ob5+ZMiD S8C/qqPkeoVvnPh2djcPISqlFX3ExpduD2y34CHqy+tzjcSiVRqiCw/woCd1 UquS+MvCylX9TjyIRunz5RNvzRB2jDjBQ8I11ax4Ymrn+/CtnjwsOacx7TKx buvZ9xw/HhyTFrmeIdaQ7lpoHcYDuzXaYyexevIDI7FIHmxvJizYQKy81cr7 /W2yfs3VejniZadKfyx9wsODM8OP+8j9ryAaKFybwsOl9ZzUauKFibrrQzJ4 YGqUVGcRS9ekRPQW8HCqvd/5ArGQRviuzHoe7Euac6cRC5QZ+rq28KDsZBrZ Qp7PCSf+JNluHsoFAy4VEHMiPWb4cXkY3Sab603cztpbaCA1jIqUFyeayfPf fEWMyZs3jMajectziBuWFiskyQ/jmYQbJ5y41kzTb7bKMFpn3urXIS5Ol91Y t20Y5m8fUsH/1R+jb66XjYdBRSk3WRK/a7l8U9d8GOFByndXEr9mjLHuOw5D Tsz7TNF/9e1kw4vD54ahm3rnLovUv+B1D5eNpg1jnqH85EpSPwNLrc1Ssodh wxZ+0E3q7QUnaX+7/GFcd+A4JxN7Rl5qLCwfhvu7UjMV4kMs19iwzmHsjHaq Xkbqt2G6soSc3AiO3nK/3E36QfHI+WybZSOIoVvPJRL/o1ftGKc0gn0/9NMP EOtVeWXO0xxBu9Xh1nLST3QHiq0Ye0agZiB58BHpN2qrHR7N8B+BpouH0Bpz Uq88Moy3hYxgzsknqxtJ/1LOmc67dG0Et6M9boYRr9r+Yodg3AjWKZ/UbjOj scR1dGA8fQQyTan5V0n/k3kYrc1qG0GOnFt37m4aUd2dLWt6R3Dutv9RW2LG Wt2rx5kjWFP8UZOPWKKgubFvYgQ/Tp5I2GJMQ6RR7VLn3FHEtXyuyd1JY3x+ ScWf7aM49imw8up2Gq0RE86fU0ax5oCM1jPS32XtT+2+ljGKNNF48dXExird G/fmjkKn8JxkMpkPsj59F2/6PIoaD5k7SRTp62PJWdyWUdzsaai8v5nGehdr 0ZXzxyAxcaHGSZtG0Mas9EtBY9CzkYzatYZGrrDqXaOrY7hjcM8kk8wBg9UP QySjxjDN1nbLQmIr9+sH7jwYg2bp0qwOMh+pPnSZnpk3hhTJ1OmnlWn8EJ1j 3cocg9+ee+aHV5L9azohoG83DoFrJg3ZC2k02fi7Rh8cR1O9N0OaOOZnRFmH 2zjUhCVjjy4g/b06/WaY5zhSN4jGy5H5rfUjrVR7YxwvndlnPMn8dy/Jy+xw 0TgUh4Uez5Ii+3/2YuJ1xQkcbgmoShWmMU38hkE9awL6Il1Hxsl86iFR1Js5 MoGxTXsuLyBukuJGXOefxIX3Su1avWzkzT3QqDd7kvy//2Z2opuNU4sVPZ8p TQJxsrN+trPRplHw7IzDJLz3xooEkvm42HpIVLJiEjF/lptur2SjqOZp1xJz PirauL7n1ks2zJ+nsfYr81Ou4suXjrqwsfa4lIkCzU99u3NXz1+WjZbeD9++ fhWgjkulBO7OJXPj/i9PRGMFKbknz4TkrFhINvbK3+kwjXIpOjHtFJOJnSNe 46c2ClHNvW557Z5MaN2svKosKEw9eXWHqzo5BF7nWb2yP8LUuxUfQn09hhBc +FnbPWM6VesUl/K6cRCO144F3fOdQfV1yZ7asXsQkhKunCzzmVRa/50IE/MB iPlnZDsoiFCb931kgNGPHz5J+aIdIlSs3bFZwiW9OK3M6rn3bhblVBexYsS8 B/bl4Uc+BItS2ie8d23s64L88vNi5Q5iVOT1TUU3LDuhdLRQ0FVFnDrno3/J vKId0za92iRIi1OFP+z5XPTakFY4EBdcPps6biJesy6oBRaLPjwuN5KgigTt l03//RebMjJuGGVLUO/a/9apjjdi1mzfuyXLJCn19jNe2dr1mHTo3ysZKkkF yWTdVHtQhwMSoYvKaEnqt13PfI/7NWh+/fW6sbUUZf1LfnJBWSWMF3ct/fZO ikrrvPjJvPsLlnWsLvmixKBWHMqNkjhYAIVHsvNDrzMovx3VqeGhaQjb/Ire cYNBVZx7lqhDpYH7w7BKNJJBmX6y7mvkvMJXEc+Q8JsM6vfpZD9p51fwPV7L ibnDoHTKHJ6rbE5FnU5k9eMkBhWuKNpMNSch/Jvo1aJPDGpbUJ6cdGICxtwS DwV9YVCj6foh+bMT4Cqku9WwlEEpKhen2vk8wGYtt9HSrwwq/8SBLWdM76P3 zpfD1d8Z1C5h5/YZfPEwPBxk0NzCoKz0+y2NXkUgQ0BuSWIbg8rRDnoyFnQD 8vGZ4wc7GJSZ/3hopE04OF9bM7q6GVTBWk/ZVvEwJGroLx0aYlD7zt60bn3j D4mvvybSWAwqSt/xxH2HC/Bx9fh1imZQ0mamAbtlfGEa++AGj8egEu6Naqrn euDduo1Hc0YYlFhVTfK0+f9CsbzS0HeMQSUxe8a+5zkiyuXQss0TDKooX+b7 07em4Pv/S5rim3r9D+GfYt8= "]]}, Annotation[#, "Charting`Private`Tag$2968#1"]& ]}, Axes->{True, True}, AxesLabel->{ FormBox["\"x(t) - miles\"", TraditionalForm], FormBox["\"y(t) - miles\"", TraditionalForm]}, AxesOrigin->{0, 0}, DisplayFunction->Identity, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], Method->{"ScalingFunctions" -> None}, PlotRange->{{0, 6}, {0, 6}}, PlotRangeClipping->True, PlotRangePadding->{{0, 0}, {0, 0}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{ 3.741948337807897*^9, 3.741948579984496*^9, 3.742029639810872*^9, { 3.742029696950314*^9, 3.7420297224474688`*^9}, 3.742029753385043*^9, 3.742030952734963*^9, 3.7461831680331573`*^9},ExpressionUUID->"071893cf-7496-459c-bb99-\ ee303f7a7e9e"] }, Open ]], Cell[BoxData[ RowBox[{"(*", RowBox[{"Now", " ", RowBox[{"let", "'"}], "s", " ", "look", " ", "at", " ", "the", " ", "Midpoint", " ", "Method"}], "*)"}]], "Input", CellChangeTimes->{{3.7419474450169697`*^9, 3.741947511790153*^9}, { 3.742029449501624*^9, 3.742029462924355*^9}},ExpressionUUID->"c60e0ceb-d05e-441b-b53d-\ 030aab65f6df"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\n", RowBox[{ RowBox[{ RowBox[{"solm1", "=", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "x", "]"}], "\[Equal]", RowBox[{ RowBox[{ RowBox[{"y", "[", "x", "]"}], "/", "x"}], "-", RowBox[{"1", "*", RowBox[{"Sqrt", "[", RowBox[{"1", "+", RowBox[{ RowBox[{ RowBox[{"y", "[", "x", "]"}], "^", "2"}], "/", RowBox[{"x", "^", "2"}]}]}], "]"}]}]}]}], ",", RowBox[{ RowBox[{"y", "[", "5", "]"}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"y", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "5", ",", ".00001"}], "}"}], ",", RowBox[{"Method", "\[Rule]", "\"\\""}], ",", RowBox[{"\"\\"", "\[Rule]", ".01"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"solm2", "=", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "x", "]"}], "\[Equal]", RowBox[{ RowBox[{ RowBox[{"y", "[", "x", "]"}], "/", "x"}], "-", RowBox[{ RowBox[{"(", RowBox[{"2", "/", "3"}], ")"}], "*", RowBox[{"Sqrt", "[", RowBox[{"1", "+", RowBox[{ RowBox[{ RowBox[{"y", "[", "x", "]"}], "^", "2"}], "/", RowBox[{"x", "^", "2"}]}]}], "]"}]}]}]}], ",", RowBox[{ RowBox[{"y", "[", "5", "]"}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"y", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "5", ",", ".00001"}], "}"}], ",", RowBox[{"Method", "\[Rule]", "\"\\""}], ",", RowBox[{"\"\\"", "\[Rule]", ".01"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"solm3", "=", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "x", "]"}], "\[Equal]", RowBox[{ RowBox[{ RowBox[{"y", "[", "x", "]"}], "/", "x"}], "-", RowBox[{ RowBox[{"(", RowBox[{"1", "/", "2"}], ")"}], "*", RowBox[{"Sqrt", "[", RowBox[{"1", "+", RowBox[{ RowBox[{ RowBox[{"y", "[", "x", "]"}], "^", "2"}], "/", RowBox[{"x", "^", "2"}]}]}], "]"}]}]}]}], ",", RowBox[{ RowBox[{"y", "[", "5", "]"}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"y", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "5", ",", ".00001"}], "}"}], ",", RowBox[{"Method", "\[Rule]", "\"\\""}], ",", RowBox[{"\"\\"", "\[Rule]", ".01"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"pathm1", "[", "x_", "]"}], ":=", RowBox[{ RowBox[{"{", RowBox[{"x", ",", RowBox[{"y", "[", "x", "]"}]}], "}"}], "/.", "solm1"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"pathm2", "[", "x_", "]"}], ":=", RowBox[{ RowBox[{"{", RowBox[{"x", ",", RowBox[{"y", "[", "x", "]"}]}], "}"}], "/.", "solm2"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"pathm3", "[", "x_", "]"}], ":=", RowBox[{ RowBox[{"{", RowBox[{"x", ",", RowBox[{"y", "[", "x", "]"}]}], "}"}], "/.", "solm3"}]}], ";"}], "\n", RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"pathm1", "[", "x", "]"}], ",", RowBox[{"pathm2", "[", "x", "]"}], ",", RowBox[{"pathm3", "[", "x", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", " ", "0", ",", "5"}], "}"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", RowBox[{"PlotLegends", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\"", ",", "\"\\""}], "}"}]}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "6"}], "}"}]}], "}"}]}]}], "]"}], "\n", "\n"}]}]], "Input", CellChangeTimes->{{3.741948551126305*^9, 3.741948566798057*^9}, { 3.7420295189236507`*^9, 3.742029621870241*^9}, {3.7420297465683403`*^9, 3.74202974990718*^9}},ExpressionUUID->"56a9eb33-28ee-496f-be1e-\ 2dfdb9f4c53f"], Cell[BoxData[ TemplateBox[{GraphicsBox[{{{}, {}, TagBox[{ Directive[ Opacity[1.], AbsoluteThickness[1.6], FaceForm[ Opacity[0.3]], RGBColor[0.368417, 0.506779, 0.709798]], LineBox[CompressedData[" 1:eJw91Xk01PsbwPGZ+Y4tYvi6XbkSpSuh062uQnkepRJatJCrxZotpW7ZUz+V JSq0XJnQoq5GKpq6tsqWNBjLVLI2IeGSsczY+X1/53f0OedzPuf1z/t8/njO ebRdju1yZ9BotDDq/u+dPbsuOmZOyjHRNG9/fvdCB+hz8joz8MNecHflOfXO Hw6C6Mf8DP4PR0PZH34myT98EwLUjKVWUG69oGLflZUEFTnsw9qUw23LBzs3 3gL5gYY4Fcq8rlXLOjxS4SyrBYZlmfgo5oJNf/1deHHgocxz2dneQ6gt5Zoa UFa4vUauu+shSE2666nLzvY5MOJR/FlGdrb/CLa+8XvQJjPbfwI/dW1nX5eZ 7WdDnbOP1oD0bD8XREVj4WFSTLwcdeJkS2wu7PNVd3WlLLSLMqwdy4UKjSK6 JeWIoWepOYI8oGnGB7Mo1xnIn4+MLICm0mqHFCYTvVNzbZb0vwZWmGdFJsHE bRvkj604VAK2RxSd2HQm+uvdsOZcL4GndoFygZRvs7SX6lSWwGlJr98eyoOt Rl9+Ni6FhN7OIAXKiSEuu2dU3kBzVdafQTQmdnBz1/DLyiDDzILYMkNgqK4n 7chyHqj+7RLwdJLAnIvlWVudeTBfJrEwkrK4b6mr7jUe3EgUVh+kfJTb/aZt jAeFJSmoQNnJ3CfG4U0F7HMP8HabINDC8ei8LfurIFuvqVlunMCILSbzlDfV QI99wK96IwTqBPV8P3OwBqInp/XHJAQWc5Lefg+ogT6/30fKKc/MnQis5NRA xbq7rR6UA9/nN0WyasEzZvHAXTGB3s7r70w31cL6fdKLFIcJ3B6CBr0nBKBv a5+cIyKws8V9W1OMABLWtHqGUw7DmKO8NAGU+Y9MWFN+LPXxafpHARQYfAhs 7SdQKc7nd3eT9/A+u6eSRrkm7YZZK/0D2Mp0NZn0EbiL37uzJu4j/MUPPnu+ m8C92jdPcR83wMSrQDPfLwSWJ1myC981AN09WHYxZRPV0cLKjgaoExtd+SQk UEvGXuGreiNgYW3dBso9var3fopsBMPdOitVP1P/y4mr8T/UBAcsRds4zQSm 74jSN6Zm+cPSvM/36gmcfpUval0rBEFkRWJFJYEFjeWjpzcJgcNRifGnHCz5 QFuwSwh3fUXm2pQlhiKWo48QfK+4Tp2qIPD7LZ3f6pOFcNmljqPBI/Bz8KXj 1YwvsNdyccvBtwQWGh0afF35BY5oqF98UURg+GPG8G2ndmCb3HLIfk6gXVBs FFHZCfetYv/xTiTQ3bVA/5huD4ytXivYGUDgfUULDbWtfZD0Xi/izB4C2VYy 1SvsRfDURiPcdRk1X0cTjYZ7B0D97dQK0TQDx++ELikIHgJrntuEXw0DtTYy DuxZJgYFB+7u1JsMbC1dZSr/UgJ6uvZlB10YaMGVdkk4PgqS0/x8+iIGWj4z YGlqjoOd7ZNg61Y6diRMu5VnTsD97Mv0X9h0XCo8zth4aAq6o9uz6rbRkakY v7l5aBqOXDBqjqXTsfR9eveivTTcYbXZ1I1Dw72c7CEHAzr+oR2y3HwnDVf6 qdhqiemojt1FBqIZaO8truPzGZi4IzNFbekMCB3e/a2QRKD/obKwfrdpeLQ9 qNDGmYlV4sK4suQpsBkPmjppLIVL3B1vDxZMwtobNZcMCGkUBnvadbVNwGhX wIbKFmlcF+tnntI3DpEl5SZHuTKof3VDUe7QGLhcPhaRGiqLQVEBatGMMVBm eUhy9sqhk/7OedrDIzA3nJvrrDUHpW83vjzcL4H6kIxChW9zcJOgMkWqSwyn DIb+TX0lj7TMcKXwomFwqorzLo5UwK3mPn9GPxiChUvOzK1ynos2mvC60XsQ lvmWEB6Gihij6TyQZToAzPVZ6wmxIhaLkx1T5oggu6SfHVmlhAMeI0/4vn1g t6D4QZU1CzUELw/rt/fAei433jqXhUrp4dbzWF0grxSawtNRxozqoRbxmq8w 4/x9j3K0Mj6tOmfqf7YNDrCiF1SKlVF0zaVTd6wV2l7wr2x3VMGb3FPfLxk1 wHbt7sV1r1RQ2vOj3ZPjAtD5ps97t4xE1Xz13f+o8UDr/nz16CsknjNlb3rE z4dYsyyxVTyJ95rPfkp3yoeRestahaskrp5+Hc0ezAP+nMCouBskajZYbXGd lwehfh8lfyWTeFmQsDz0QA58Mr0qeJBBom7Dv5mCbi7E1SlcKn1LovvB859X SjJg0ifNM+IdiY5SWexnP2eAh9Q6C8sKEuML4q5pGXPAbK3PRAWfRBsrr3Wv Q9KhN/mdl+ADiZOTYW5KtPtg6RWxua2dRKfBQAOJ3C3gMjQXpX0l8cnDVaZn rJJg4a3nU+7fSNx0bU+J8GIiSPgd3O4eEufHl4Oh/HVIW71x8cAAiTobhurq yVhg8Runs4dI9FLnRRxPiIIQjxONJ8Ukynn4GhawLsCupDvxo6Mktv9nxYNy tVB4tcrYN2+cxOCbkHaOPAV6VTWWoZMk5iyvVjL/5QhcO+ypYzZNYodMisVp jf3w/22uirN7/b9U0KFk "]]}, Annotation[#, "Charting`Private`Tag$3123#1"]& ], TagBox[{ Directive[ Opacity[1.], AbsoluteThickness[1.6], FaceForm[ Opacity[0.3]], RGBColor[0.880722, 0.611041, 0.142051]], LineBox[CompressedData[" 1:eJw12Hk8Ff33AHBLKDtXiyWEUrIlT9acsaaI7Hsha4tKiqIeUUJkp4hkqUSR qCwV8ZBdUalEi33nztyV6/vp9fs1/8zr/cfcOZ9z53POmdnifcrWl4ONjU2Y nY3tz/nvcbij/12E0jTo1brXTcq4QFjs14783X8dCCnUldJG/b++AE7yO7gY Vn8dB/x7hreanv3r28B2yiDUtO6vSyDV75jAovnM/7sGNmyPKlrnNgsRigFs J1Tb4dZqd35v9DxYhWPKM8F9sPHOhN/a0gVw2HL7XNWTL7Dfzrx9OWwRWK/r Foa0f8CSsnokE5Yg6gkHnu/5Gx580DDN/r0EjhcSYjk7xyA+cWi06RIZfI/W 7zylOAUWAdOX77LIIBp1uYfYPQWUoJ/WTmw4vMnHzkZgU7D5LoMqwIGD+FBz bZzLFOh7uXaHcuHQ5dy1v+jGFJyqVLq9lx+Hf6yG/L/MT8Fy9cT7FHEcuHTY ioxfToNdWem2w7txKBY0kdq0fxbYSwdIKUdxaNC2MHrsOAsfbhtKyPji8NXb NsDIZxZqFbk+lPrhIPjCs+rEv7NweyzDsTEQh7Ajlywbq2dB3qWydvAUDgcq nkcck5uDOYeLWz+G4zBns+N7PXMOvD946nan4rA2Qp3Tdt08pAaLj2qk4yB3 X2vH+IZ50PzBMM/MwMGJYXpORGMesid37XW5heIp8BbwC5iHS5tNh3pycUgl 5+wV+jgP26wKdSIf4pBzgKdH3WkB2KnOvPP1OCglFQ9e91gARRa7y4bXONT2 GU8NHV2AU6E5Kfpv0HrcI7kSTy+AM+1Y47VGHDadoutNxi2Abh3vFr4WHNLT ph/eq1+Aq1GN/L+6cUgY7IkSlVuELPYn9MZhHKS2BCUHbl+EFb/+iZYfOJT6 8uc1qC6CPMvkXftPHNrnzGuC9BbhLr3Huec3Wi/72/l2+0XgfCzZ0jGOw9Wt Ve7R1xehzYi1kjKPQ0TQrT34zCI8Eks1c2Ch/K3aOe4kL8L2XLadmqsovmSh 8970RYjMDhAWZSOg6mlMdS/XEvgGjgy1sxMwg4doPpZegrtJrUnqXAS4h9to +B1ague1m4938hGgF8urOvBsCeyDNzrt20TAWvGWg0K1S3A/7cwLbnECPpVc OWnWsAQbPjdI/4cc3EErq+pcApoOiwMkCXgkOL4zZXQJvB6oKypKEyCR0bTj wEYy5Hx3WNsmTwDjXsTW+otkKMwrj7BXJeDyS9eYi1fIIMlnVsWhRgBnj/a4 diwZSrpfdZUj8y/jD6szyJA3YfqAexcBMo4nlcsryMA0C2De302ACe/h3QVj ZFhzP8SjUpuAd7L66Z6zZLCRbOux0iHgoJYEIY2TgUf6veIUsqPPp+c57DiY uCzEbdYjIOC1lW6GFA5CMZKbQ/cSkBgMhrG2OJx8WlhFGBIgGre5wMwFh5fp X/IijQjIvMvk4PJE+0ovI5jXmID8zhfNV07ikOGd+lnChIDKberm4bE4nFmy y1M1Q/n6KmsdhJ6rY6Sz9nIHCJA15vCwVyLgurS4kL4NAYrrwrXvqhNg4VVS kous2kMmTe0hYKEjwJmFrO822v4vus/XZj1yvS0BzmffaZe6EyBHz5dXsycg ufCmGEcSAVJ8Q9njjgRkBfIsWGYQ8Iz8MEnXiYA8tciOrBwCtla9yklALqs7 E6XykICUgBsMFWeUrz77BedGAq7abRvwdSFglVOys5xMwLH3PQaP3Ajgbk97 wKATMJRy5c0cMn8yf7QpGwXyootdNVAcElIsnW/8FNBYb//zOfIezZ8PuLdR wNjpmVa1BwF7GS7RNsoUEGEeTltCNmn4cPiOBgVyz93iUztMgK1l83oNQD7w xrYYOcjnQbSHMwX+/S5hfP0IAeeUZI88PEyB1ZGQ+FfIEQu3dMk+FPiv68TC EnJ8RPxi7BkKJI3pirt5EnA//eSRqngKXPvhNyrrRcBj1zHd1WQKnPvSuv0Q cpXskQ0HsijQf6Yj4TLy27JDXcOFFAi5VfNiAHmoebce3ysKNP7akh/lTcAd 4R6n700U+Llret8TZBf3YyHl7RQ4IZo9PYDcT85/bDdAAafQMcWdRwnokBOU vUOmAOPGUEQTcmxQiX4QgwJlrRf5x5HNak1cMHYqzCUEZ6/zQfe3CU8dEaRC a0hppQXyy8iJNSpKVLBMrR5sQj7fGb1lVZ0K7xIK/IeRd2+SMXivRQXSN1sy HflJuUPoOVMqxIOKoLIvASeYC2n7LKnAqJjLNUbesS+hQtyOClvVC9RckYuG 3k6+8qTCeA2f61Vkb6XD3Mn+VDhw2omchSxzni7nHUSF84kNSY+QBxvTQfMc FTw9U9TqkbMF1N25I6hA4RLr60R2dukIG4iiwj7f1IuDyBuK/TIexVFhUs1x 2zRyin5ut1UWFVg9fTe4/AiwitWels1D5pI1E0Hm7+/jIRdRoSHwE5cUcrvM KYX/Sqng8wg6tiLHHuc1zKqkgoNpUaYqstmLYo/AGioUXLEM2IPMyWl4Ua+B CtGhkdhe5AarwUyBVioErIuVMUa+nB36bLiLClITj7jMkfXHRHuf9lNBIF2d bIFM3/VkJvobFeScPSeskJ9f2r/O8RcV2GyujB5CDmkb2bp9kgol/l+nbJA1 1kcaMeapMGxbSfvjeU/JI50UKrjkWAv9cVnZ8/C8FZTPDJqqNfIxms2t02to EOxLdrJE3m4yW2XERwNr9pj4P/GMJsW+FxOlQbMSpeVPvAXf5OfGNtHgn/Nx AgbInopveGtkaGC679phLWTps66KN7bRoK5HtVYNefA1YeyhQoNWsQ5ZReRs 3hRPNU0aGE/lpW5GdnJUvsSuR0N9cE6IhCxW0Hq7z5AGHA9Xb/Egf5j1fl5s ToOSg+tVmej/StZhfQi1poEU82zPLLLVtdvz+x1poJVz5NIwMv97TX4pDxoY TEhq9yK3SfVunztKg5CTM6tvkE2quL1Tz9DgxunPL+4gY5Zvkk3CaDDk/Z0z Hll/JPQN5TINCtNb/M4ja4pNSrkl0ODRJ4NgS2T1snsWAmk0sE+wkNyDrGzi evHNbRp0de0ZlEZWCOn4LP+ABl5hBfdn0fMvy3+V+9NjlA+9uxV9yFJF+v/E VtHgKpzte4ks1v84daYRXU/L8olE5tJMPlg9SAP/uRt1a5A5Os0j/H+j9d5g pv5G+5N1lL1UfIoGo2rK1xqRKWnBay9TUT4vva29iDxGtm8yE6XDZdzpzC+0 /3/dEFiibaLDthrfrbXIQ/ItsqUydKjyFKMkI3+y07ospEKHwKOGc3rILc/E dQb20aFr+0Hs+p/6Y/HBP96KDrlrPg07Ib/+HZ+p70CHr8yPuYrIz0nL5Hxv OnAnyJ9v/lPfzg49CbxEh27DfXlkVP+u7y5UYFbSQZy9jaWI6ufVDje7xzV0 8KusyJ9C9TbyqFjUkQY61Cjk+5Qhh6VdG27qogNfwjk7FeQAsn92wgT6vVCB PgVUv82fKQtLSzPgZtGm+CnUD1oY/9a4KzCgiTvpUhGyqVGfd44SA8TmZis9 kI3eX6jepMUA3nDSSBfqJ/rzLa6kQwzgC6/zKUb9Rm2nV/HaKAZ0NEtx7XJA 9Sq4ympfLAPiPtnsHEb9S7mWh3btJgNoLKnMBOTt+58c4MxhQN7dzbqjdqjP +TPnV56heALdGhJR/9tQmKFLHmXAmb71U3XWBKRPTfzeNYNsr3LyMDJJQz/x 9BIDfGJOa7EhCzf+Gp5lMSANpAoMrQjgHVa7NrGRCYf+O9tfZ0nAikR7z/f9 TBD/rNabuJ+AkVSWz7vHTCjvaNQqQf1e3DPE+mYVE7qv7BPciWylMqVjX8eE ZYMdImVoPnjZ+lHwxzsmsJiNd0oxAhKWy15SfzOhJSm+N9+AgH/83PgVJZbB eEC5/6guATE6L59di1kGpfuv0g6ieaaOWzXPInEZorMsbKrRHLDQVxgrkr4M zGpeQylk16Akj9x7y1Cwq/fFOJqXVAv9eKrrlyE7w53nnDIBn/nXu40sLYOg kKVDoCLK348zHMZHVoDWzPpeI0XAD/co/wzfFTCX2E4SQ876kto5fnwFgixe 3z6J5jXuvmeZCWErsCe+JkdaAq3/P0LpU8oKfOy/dz5sIwF3Sy/YBTavgG/h s2I+UZT/0CtFSTtYsI1L430FNwFrBFPMBsks6Fe4dWxlFodg4eaZagYLKJ6r cZLIP0SpqUnsq1A39WVUewaH+o0ew0ZCqxBHvWJ3ZgqHkC07wkqUViGQ8Y73 yxgOo5qNJee9VtFcv5/3KpqfW9wW+UV6VuGb6Ceb/b04NPc/nJRzYMOUSxOn b5Xj4PCokuyizI7t6BuQY6L3DY3TojayBDt2s9neKAq9v/yeefuhu5sDE1v0 vWpdR4YfLm0P+LM5sQ1KvlzSrmQos7rQYOm1BpPB1daELC2BJePCSogOF9a6 XqV+LGwJtDN7E5U5ubG2ci+q6uoi0CZCjTq/c2NNRnFxEcGLcL3pnW5QFQ8W dMv38fPhBfC+eSrmbsRaTJhn/OwB6wUQEfanvHRYhxWGG6TaOMyDQFRVjZcs L6bVfZwEpDn4HF7awD/Oi0nlSvJxt8/AOWXy9N3XfNgmG9NtDIdp8OxKPvb2 Oj8mWa5xUGd2EmS2/ivQ5SWAWbKYTSlOE6B0sonTX0UQW87luebQMwZr9j7d y0kIYpLhSmx+RqNQ2TSfc71LCDN5/71vd8xvcNz89n6XhTDGPqypwPPtJ+yt qkqxqBHG7pEzB1RXhoFPKCKvXUEE02zdc6FGdxBWvebsReJEsJy2K5lq9wbA Qzhucychgpl6v5EIzu+HX8+7k6zcRDETxRWWZGcvWG2ZlP/wWhQrELBqdZhq A4Xxne1tSiRM6OPtdGHfRpAtFpeISyJhm+srKpLjKiHB4ClxIIWEvbh4vUgP qwTqZ/P3/GkkzN9ad3aY8hS6ecNikzNJ2AbphMtiPk8h4vQnSlYuCWt6aPxI xaACBvTS+u6XkjCR0oGf2K9SSP7An9jcSsJencuWFisqgOXjRQExbSSsvFou tkGoAPy59E3MO0iYn979iiPh98BA+zizo5uE/VMJhudt82Emty2w7yMJCyPt GlvLdgfMA2PMfv0mYR+DOp0snqZCFYe0XNEoCWPd9HuwHJMCMneqV3zHSdjX keG4NPdkoHSPVE1OkbCMXFPxEcEEKNI0ll9cJGHcT0LcRl5EgXD3V1YlmYQV t5mcyfeKhHD/4K8hBAkbWJSJtt4QAbbZ91JoNBIWuO6rlnpdMLzerXOylkHC UpILytZInIAdXb3mEcskTJ/Uvvyx3hvS/QIUDFgkTDiJ6H/4yhb+72uHGPb3 u8f/AIn6jvg= "]]}, Annotation[#, "Charting`Private`Tag$3123#2"]& ], TagBox[{ Directive[ Opacity[1.], AbsoluteThickness[1.6], FaceForm[ Opacity[0.3]], RGBColor[0.560181, 0.691569, 0.194885]], LineBox[CompressedData[" 1:eJw12Hk4lO33APCxR7aMIilLi7WFIl7LeZJ6pZKyZamkJEJeKcmSJWtkC0VZ oig7Q5HKllK20E5kp5nHMmbMht/t+n2bf+b6XNfMfT/nXs4516PgdPmEMzeB QJDiIhBWvv9+LOqyk63s+0Gv1uHlpJwtvDulIn7Z/a9dQUkU0wkL/Gs/oJ/3 jcvI+utoOHKpyCF2+K/vw8OBzcTLbr/+56fgFzshYOAz8D/XgJvE6DbRukEI ULpIcN/xAV7y3v9n6N0QmPlj6mTvHigPt9pzNmIErBTuXyWVfAcBjZ+42dox WHr9cuaXziD8ZklEcjLGIbSEez7bcRhcNxcoOw5NgLVfbBRP2xiQTw+K922Y AudzdWqXlaYgSDNIwUH5DzwWNZaVPkSBXsqxSAEuMtTrHDYqtqaAQnDAHg9e MvxwOnHR6DwF8g9pJncJkEH0uSPJ/SYFQrJqLiSIkuH6mcAjDVUUuKL4wmt+ AxlMy6oD3BRxcBq9cd5Mmwz4cZX+OjYO3I6ml464kiHDVKBzl80MrNv+dCH8 PRlU4x/3RZ6agROG1oOqH8lQ27N/6te5GTiWdOxnezua3yGYL85rBjoizwiK 9pBB+jJTbzJ6BqKPBpv59pPhbvKfgpy6GYhyVWygzJIhtq8zVEJxFmhi3W7u 6ykQ4HlPe548Cwp+yuMNThQwXbawVqPOwkSM/sAQiks6QeyaE3MWPMsfiHG7 UIBUHlHVxTcHSUUuMnqXKECe99lTvGkOtrtpDaZdoYCD/3HNC+ZzYHZD2oMT RgG9KKEd3yrngLsQi7+SS4FV61uOitXOgVzoVTmHxxT48jTE42D9HLSSVWaN 8ing/ZFRRGqbgwfkvXarCynwTHRcLXF0DnY9pplGVlBAJqVJxVSKCs/+cxxX a6AAKydga90NKlTu2Sum2k+BoBd2ETdCqKBgqy1HGKAAT6fOuE4UFbovear3 DlJAmDNfUJVCBT1vLuzaCAXkrD3US8uo0JfJtnr0hwLGQqd3PxqjwphleHw6 kwJx3rAv6sQ81KddnBCRxEEieuOjg7bzUFXMOpu3FofULDY3n+M86KyLHteR wiG77XlziMc8xDx1WO8gg0PFtl0m/lHz8Gy7qNpteRy+/JA/5vl6Hs3vnBmr joP8fu5Tlqo00CDNXlTYj4OSoL9O1i4aqD2veHTOGIcdnVTilDYN0v65opd3 AAd9+9EPN/fT4Ind4m15ExxOXnmvU+hAAz4xj+d8R3FIyL0jyR1Pg4UNlgWh 1jgs82xoK6XSwAdLcTa9iAP/h+R8FpMGrAzRa7auOAgnCIcdINAh4Z0u7YIb DjKyS7o/hekg6hGxL9AdB+09v/P5t9Fhf4eIXJoXDp7n88NOnaTDo71BVsHX cfjVvFtv9Ss6XK+r7FeLxOGBeKdNfxMdXpuc15KNwsHWwc2n9AMdBqbSWoSi ceilZhdbfKMDLeRV2lAMDh8VReUfUOlgbePIG3gHhxfBE7zbVRdAdlVD8KkU HK61hSks71qAxIm1FRqpOOyWljP8tHcBthinbeVNw6Gk1Mr36oEFOKls8Dn3 Hg55vxonXzkuwLaPn5w7MnBI1H/YYZa2AMyJn/VdOTiYRen8kc9cAIfmWovo Ryj+3h4Bat4CCJRKamC5OERdEtqXVrEAw8772gvycAhK960caF8AP5FH953z cXBjHL/nxcuA74Uxc3FFOCgbU0hGqxngXmbpqlaMw2h81CdJCQYcHEjSfIfs qPRGqEYOueH0a2YJDjbW6oFcegyIKX//3rQcB2MSv1PSfwx4Vh0q+bAKB+zI mwTj6wxwTC67vbEa7feI7xt6EAM6fhq5P0DeIzkpax/LgMj/7C6lPMdhi8/H r5vzGTAoWurhU4POk/At/i/FDJj2ttefQJbN09eKIjFgsWnG164WB8ne4iRy AwNi7ZL3/vMSB749CUer+hjA5/x5ZKwO5aE2kwCXYQbI6E1sM3uFw9I5rsL1 Uwzou2DzvRKZnuy9KmiBARY3lhNuvMZhjGrZdFCCCZclbW1ob3BoqVyv++1f JjzW8srMbsSh8XC3S4wZE5L34dLTyK+HY1L1rZjwlRoopN+EQzWRQ812YoLx 72uWn5CfXPlV4hrIhATFffGTzTg8Wp3Wv+EWE+a86tvV3uKQmXtMuCOGCdu0 70a4I6f21Ltq3mPCK/bni5PIkbtzt7ArmKBAW2J1teBw66O9RXENE3rVHaUE 3uEQfE4y9Ew9E66dOlSuh3w9OXygqZ0JjbMmZ7KRfdQMRa/1MuGks4pPF7JX E11f+ScT7lO3rlpGvkh1SY+dYILiIm+pzXsczt+WbzWcZgJeXtoYguy4+fvC DI0JT+j8Ns+QT1qYWlnxsKBuyyGeBWTLP9y3BIRYYH+DtWFDKw7mYS8rasRZ IKk9Wm2AbFKpLr5pEwt+6836BCG3sG7WOGxhgfOazgfpyAeMepwyVFlQ+nTa rwrZ6JNflfReFmQZaluNI9dLt5+2MWDBiB3LYgkZHOVXpe5H4wcaCEp+QOdp usWOaM6CXab+9/SQa7VleE9Ys+D+W4krR5F1gzyKExxYQBpz4juN/OJtvXWn EwuwkppjHsjaIpIEUVcW3KGeP+GPTLJ0eXrkMgto5ypEopA1H9SeuH2VBfE3 p0OSkcuGRTit/iiebNfSh8g71c4+XhXKgtMsj9QnyCXeJLN/o1gQlHJobwmy eq0AI/wOC6rNDiWRkJ9x2ec032WBdF9Nfg2y8qESU54MFpi0j/u+Qn6SwDW/ L4cFN6PleOqRt36zfBiczwLPw8WHGpDz5AoOvilmga5S/+EVK7qwpxcr0fqt Zguu/D67xOy+fi0L8t6cuLUynhw9x8i/ngVW/TtfrsyXaUD7U9PCglff+0tW nkc23CSF0caCz7wFZ1aeN70tw3BvD/p/xPf2lXjWS06PX/3Oglt2r5gr8d6z N0okDbCAcTJzeGU91uWm/EMdZcFqrXe3V9br7tTEsAaZBdTroXMr60nU1I/z mmNBb5iMtCdykl+8dimDBZ+Sv3NW9kO8YWiAssSC6XrBHDPkO6u0o9X52JC2 MM9rgCxiHq15aTUbVJZHlFWRhQZ2hk9IscGKwiwnIEdtC9uhtIkNYrnlUlPo fAh4fvnqvIUNM53Jht3IvIsBqsO72NCYxu7KRA417upR2MsGr24JgzBkQuzm QEcDNrxp9710AXlR5kNn/yE2tBQdEVdBDnDa6CdrzoaiVUcjVyGznnop2luz wUmqrmEMnX+6jtTVb05siM7gC81CnrE+L9Pjz4YH79brcCN7Zj5vWhPKhk6Z 891f0X2jjAp5mEex4YzMSc0i5Cmf8jftd9lwf87G+DjySNLS+ffFbNhLcnsR i+77ekefY3dIbJhv4hazRDbbPqVr+ZINGqpe9PXIL959Fh18z4Zyvh1+OShf xHKKXiwMs2FD+aBNFso3Da2Kua+m2OAR1mloj0xPvRcXNsuG1D7LmrXIjhq3 nMSW2eC2I9gtAuUrrQv2wkoyHIg0+dFijfKb2+5PdLI8B9LTli0EkLMJ//6u UOKAaJ2Xb3UD2o8MzWpDLQ50VsWFiiP/6hR0tDbnwM4SocpKlD8jdF9Uhkdw gEfW4F01yscv+XdkHo7jgEg3Y9AceaYnN2rNXQ5YBJukTqL8becZf+phDgec Sqd71yHvyL0gUFXHgeatLoWOKP9/FV5rPzLHge4btbwVqL4I/4g58IzJAT7B fUtayEb5hF1ehEW4muJ77wUJh6J9ZF6OyCKMfdEqq6lE+fJaUwlRZRFspKuv FqJ6pTz4H/f+M4vQ3aedeRTVt0GHUJcU50X45b8j5i2qh2nfk9rGLy2CispZ E31k/p7K1NjrizCe56O/rRDt11ua6pfEReDr3itELsAhq9DPwrV5EVwap7wB 1dt1viF58SpLEJP9VVkgHYcOaqLg0M4l6Dof0mt7H4dwr0eee7SXQN56rV0h qvfzrk06342W4M/WKaop6ge6HfjaFR2WoF635WLAXRzijKJpVXeWoHbbSEwB 6i94RRMP9lGX4LCGqIlSKA7e4s3kKtYS6JPkxY1DUDwSC0nxXMsgp+EVeSYY hzqpUwNGYsuw7+LIraQgVB8UVK4/VV2GREu8ceIGqvd7Gp5eO7sMc7o3oq2v oHxtPyu8pnMZ2oFfRfccqtent1ROfV6GtIk+R3UnVM8cbWyb+5bBYzHHfNNZ HG46v3rsO7UMueO8tkunUf/03FhumpuAKbqHS5baoXrRIP7CahMB69vY4tVz HIfm3oJJRSsCVm8ZHaMCqF/jdL29eZKAFXQNjTANUL3azMzpsydgTpKjQq36 OGz0PmSX6kTA+gN6a8/9g/opsakPgl4ErCydXBajhfbTVK1oJoaAPR2SUs1S Q/fhTbHn63oCRuxK/TO8DgerZxVUW3UubNuFkl2TkxRwV1EWCdzJhQ2uukOK nKDArYKH27I1uTAuvms6W8cpUPEkynZMhwsznBbZdwr11+K5p994H+DCItco 2jX8okBbhtDt26e5MFmzlk+2PajfjnNSfJWAxo/u5m5/SQFNL4nj8jQuLGwz 39230RQYJjd2d3RwY1ljaw13i1Ng0LY1XzidB6sNbB4VjSdDkZlf/ZGzvNid y4+6TnKT4QjLb9FHlw8bK1va5Gv3B3RSu+LUefgxwXO6KpzSKWBM+Bq19fNj 9kZ5sgUTkxDZ9P4fT5IAxl/23Kly3SQ43bkckRWwCou741gadnAC1oi70F9Y CWJaQZb9+TbjIBJKqjkrL4Q1W8sM1LiPwVf/wnrhcSEsJKq1IsR/FK6qU/9k vV6NeVn0Xf/iOgKO7QlujZHCGF2jzP3umWGQ23pTpP2sCGZyU3befPcQqHo0 8bhsF8VUcqKIxuK/gdeg3ICHJorZvnXXM+4egIqm6YzIdjFstmXdlTj9frDe 2Pik/bA41vTDIVP/zQ8wIJESD9eIY3QvJaeIha+wWiwg88OWNdjjRz8ODSp9 huWzuOWa6DVYhYpcnsG1bjglHr2xjbYGuzD7mr+S0gFD1R3xZvYSWJCrW032 rlYwU5jc3P1aAvNIoW0gXm6CLeNqH1pViViZtk3mHqlakH+8XiY6nogFqW4Z 1+sohFjDcpppIhELDzYtMXAshIWvJp+Ek4lY9K4+V625Z9AhdD0qIZWILVQy 6/nXPYMAry/0tIdE7OcFeW6NUwXwTS+550khEWtU6nDJm8yDhG7huOZ3RKz8 5vPpdfQM4FzKuxjRSsR0NBq106UywIVP39jkIxHLP1ueKKibDoY6l9gfO4iY eWJ8VJH/PSA/bHXt+UzEhDT9sngIKWDiGnFwaJiIrdVw1poWjAUS9ybFvFEi Vn1AP8jPNAbkHlQtOo8Tsbtfd6/vi4kCescIaXIKxcctZK66Ohzy9uzfPDtL xJQXX679QgwA8Y4fSxVUIvY1PMHBK+k6+Lt4//ChEbGj93P/eyl+FU6k5yQy GEQMjxzTfi/tDq9363rUsoiYcd2/V8OIzqDS3mUSwCFiAWIzVGyDA9y9cHGL 4RIRa7dNNQiUPQL//3ZFEvv7nuX/ABNU2fo= "]]}, Annotation[#, "Charting`Private`Tag$3123#3"]& ]}}, { DisplayFunction -> Identity, Axes -> {True, True}, AxesLabel -> { FormBox["\"x(t) - miles\"", TraditionalForm], FormBox["\"y(t) - miles\"", TraditionalForm]}, AxesOrigin -> {0, 0}, DisplayFunction :> Identity, FrameLabel -> {{None, None}, {None, None}}, FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, GridLinesStyle -> Directive[ GrayLevel[0.5, 0.4]], Method -> {"ScalingFunctions" -> None}, PlotRange -> {{0, 6}, {0, 6}}, PlotRangeClipping -> True, PlotRangePadding -> {{0, 0}, {0, 0}}, Ticks -> {Automatic, Automatic}}], FormBox[ FormBox[ TemplateBox[{"\"k=1\"", "\"k=2/3\"", "\"k=1/2\""}, "LineLegend", DisplayFunction -> (FormBox[ StyleBox[ StyleBox[ PaneBox[ TagBox[ GridBox[{{ TagBox[ GridBox[{{ GraphicsBox[{{ Directive[ PointSize[0.5], EdgeForm[None], Opacity[1.], AbsoluteThickness[1.6], FaceForm[ Opacity[0.3]], RGBColor[0.368417, 0.506779, 0.709798]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ PointSize[0.5], EdgeForm[None], Opacity[1.], AbsoluteThickness[1.6], FaceForm[ Opacity[0.3]], RGBColor[0.368417, 0.506779, 0.709798]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #}, { GraphicsBox[{{ Directive[ PointSize[0.5], EdgeForm[None], Opacity[1.], AbsoluteThickness[1.6], FaceForm[ Opacity[0.3]], RGBColor[0.880722, 0.611041, 0.142051]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ PointSize[0.5], EdgeForm[None], Opacity[1.], AbsoluteThickness[1.6], FaceForm[ Opacity[0.3]], RGBColor[0.880722, 0.611041, 0.142051]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #2}, { GraphicsBox[{{ Directive[ PointSize[0.5], EdgeForm[None], Opacity[1.], AbsoluteThickness[1.6], FaceForm[ Opacity[0.3]], RGBColor[0.560181, 0.691569, 0.194885]], { LineBox[{{0, 10}, {20, 10}}]}}, { Directive[ PointSize[0.5], EdgeForm[None], Opacity[1.], AbsoluteThickness[1.6], FaceForm[ Opacity[0.3]], RGBColor[0.560181, 0.691569, 0.194885]], {}}}, AspectRatio -> Full, ImageSize -> {20, 10}, PlotRangePadding -> None, ImagePadding -> Automatic, BaselinePosition -> (Scaled[0.1] -> Baseline)], #3}}, GridBoxAlignment -> { "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, AutoDelete -> False, GridBoxDividers -> { "Columns" -> {{False}}, "Rows" -> {{False}}}, GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}}, GridBoxSpacings -> { "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> False, GridBoxItemSize -> { "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], "Grid"], Alignment -> Left, AppearanceElements -> None, ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { FontFamily -> "Arial"}, Background -> Automatic, StripOnInput -> False], TraditionalForm]& ), InterpretationFunction :> (RowBox[{"LineLegend", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Directive", "[", RowBox[{ RowBox[{"EdgeForm", "[", "None", "]"}], ",", RowBox[{"Opacity", "[", "1.`", "]"}], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",", RowBox[{"FaceForm", "[", RowBox[{"Opacity", "[", "0.3`", "]"}], "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.368417, 0.506779, 0.709798], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.24561133333333335`, 0.3378526666666667, 0.4731986666666667], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], "RGBColor[0.368417, 0.506779, 0.709798]"], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.368417, 0.506779, 0.709798]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.368417, 0.506779, 0.709798], Editable -> False, Selectable -> False]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"EdgeForm", "[", "None", "]"}], ",", RowBox[{"Opacity", "[", "1.`", "]"}], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",", RowBox[{"FaceForm", "[", RowBox[{"Opacity", "[", "0.3`", "]"}], "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.880722, 0.611041, 0.142051], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.587148, 0.40736066666666665`, 0.09470066666666668], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], "RGBColor[0.880722, 0.611041, 0.142051]"], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.880722, 0.611041, 0.142051]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.880722, 0.611041, 0.142051], Editable -> False, Selectable -> False]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"EdgeForm", "[", "None", "]"}], ",", RowBox[{"Opacity", "[", "1.`", "]"}], ",", RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",", RowBox[{"FaceForm", "[", RowBox[{"Opacity", "[", "0.3`", "]"}], "]"}], ",", InterpretationBox[ ButtonBox[ TooltipBox[ GraphicsBox[{{ GrayLevel[0], RectangleBox[{0, 0}]}, { GrayLevel[0], RectangleBox[{1, -1}]}, { RGBColor[0.560181, 0.691569, 0.194885], RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle -> "ColorSwatchGraphics", AspectRatio -> 1, Frame -> True, FrameStyle -> RGBColor[ 0.37345400000000006`, 0.461046, 0.12992333333333334`], FrameTicks -> None, PlotRangePadding -> None, ImageSize -> Dynamic[{ Automatic, 1.35 CurrentValue["FontCapHeight"]/ AbsoluteCurrentValue[Magnification]}]], "RGBColor[0.560181, 0.691569, 0.194885]"], Appearance -> None, BaseStyle -> {}, BaselinePosition -> Baseline, DefaultBaseStyle -> {}, ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]}, If[ Not[ AbsoluteCurrentValue["Deployed"]], SelectionMove[Typeset`box$, All, Expression]; FrontEnd`Private`$ColorSelectorInitialAlpha = 1; FrontEnd`Private`$ColorSelectorInitialColor = RGBColor[0.560181, 0.691569, 0.194885]; FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; MathLink`CallFrontEnd[ FrontEnd`AttachCell[Typeset`box$, FrontEndResource["RGBColorValueSelector"], { 0, {Left, Bottom}}, {Left, Top}, "ClosingActions" -> { "SelectionDeparture", "ParentChanged", "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> Automatic, Method -> "Preemptive"], RGBColor[0.560181, 0.691569, 0.194885], Editable -> False, Selectable -> False]}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{#, ",", #2, ",", #3}], "}"}], ",", RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", RowBox[{"LabelStyle", "\[Rule]", RowBox[{"{", "}"}]}], ",", RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}]}], "]"}]& ), Editable -> True], TraditionalForm], TraditionalForm]}, "Legended", DisplayFunction->(GridBox[{{ TagBox[ ItemBox[ PaneBox[ TagBox[#, "SkipImageSizeLevel"], Alignment -> {Center, Baseline}, BaselinePosition -> Baseline], DefaultBaseStyle -> "Labeled"], "SkipImageSizeLevel"], ItemBox[#2, DefaultBaseStyle -> "LabeledLabel"]}}, GridBoxAlignment -> {"Columns" -> {{Center}}, "Rows" -> {{Center}}}, AutoDelete -> False, GridBoxItemSize -> Automatic, BaselinePosition -> {1, 1}]& ), Editable->True, InterpretationFunction->(RowBox[{"Legended", "[", RowBox[{#, ",", RowBox[{"Placed", "[", RowBox[{#2, ",", "After"}], "]"}]}], "]"}]& )]], "Output", CellChangeTimes->{ 3.742029641279893*^9, {3.7420296979870253`*^9, 3.742029723326322*^9}, 3.742029754864409*^9, 3.742030953979516*^9, 3.746183168446181*^9},ExpressionUUID->"324e29c7-1ea5-431f-9bbe-\ e69841da08bb"] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"(*", RowBox[{ RowBox[{"Let", "'"}], "s", " ", "write", " ", "a", " ", "quick", " ", "script", " ", "to", " ", "compare", " ", "this", " ", "method", " ", "to", " ", "the", " ", "exact", " ", "solution"}], "*)"}], "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"MidpointMethod", "[", RowBox[{"f_", ",", RowBox[{"{", RowBox[{"x_", ",", "x0_", ",", "xn_"}], "}"}], ",", RowBox[{"{", RowBox[{"y_", ",", "y0_"}], "}"}], ",", "steps_"}], "]"}], ":=", RowBox[{"Block", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"xold", "=", "x0"}], ",", RowBox[{"yold", "=", "y0"}], ",", RowBox[{"sollist", "=", RowBox[{"{", RowBox[{"{", RowBox[{"x0", ",", "y0"}], "}"}], "}"}]}], ",", "h"}], "}"}], ",", RowBox[{ RowBox[{"h", "=", RowBox[{"N", "[", RowBox[{ RowBox[{"(", RowBox[{"xn", "-", "x0"}], ")"}], "/", "steps"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"xnew", "=", RowBox[{"xold", "+", "h"}]}], ";", "\[IndentingNewLine]", RowBox[{"fold", "=", RowBox[{"f", "/.", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "xold"}], ",", RowBox[{"y", "\[Rule]", "yold"}]}], "}"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"ynew", "=", RowBox[{"yold", "+", RowBox[{"h", "*", RowBox[{"(", RowBox[{"f", "/.", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", RowBox[{"xold", "+", RowBox[{"h", "/", "2"}]}]}], ",", RowBox[{"y", "\[Rule]", RowBox[{"yold", "+", RowBox[{ RowBox[{"(", RowBox[{"h", "/", "2"}], ")"}], "*", "fold"}]}]}]}], "}"}]}], ")"}]}]}]}], ";", "\[IndentingNewLine]", RowBox[{"sollist", "=", RowBox[{"Append", "[", RowBox[{"sollist", ",", RowBox[{"{", RowBox[{"xnew", ",", "ynew"}], "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"xold", "=", "xnew"}], ";", "\[IndentingNewLine]", RowBox[{"yold", "=", "ynew"}]}], ",", RowBox[{"{", "steps", "}"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", "sollist", "]"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]"}]}]], "Input", CellChangeTimes->{{3.7419485948974857`*^9, 3.741948604178299*^9}, { 3.74202948482906*^9, 3.7420294878652983`*^9}, {3.7420300492300453`*^9, 3.7420300633040113`*^9}, {3.742030116205058*^9, 3.7420301457243547`*^9}, { 3.7420303001250877`*^9, 3.742030329554277*^9}, {3.742030365195013*^9, 3.742030384068185*^9}, {3.742030538160252*^9, 3.742030541495446*^9}, { 3.742030813685093*^9, 3.7420308852623463`*^9}},ExpressionUUID->"5d1650d8-bd31-4828-99ef-\ 3bb75776288c"], Cell[BoxData[ RowBox[{ RowBox[{"MidpointApproxSol", "=", RowBox[{"MidpointMethod", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"x", ",", "y"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "5", ",", "0.1"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "0"}], "}"}], ",", "10"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.742030891445306*^9, 3.742030912848215*^9}},ExpressionUUID->"60c017ae-eb7f-4e6d-9b59-\ 5f9f51b2c4f1"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"MatrixForm", "[", "MidpointApproxSol", "]"}], ";"}], "\[IndentingNewLine]", RowBox[{"MidpointApproxPlot", "=", RowBox[{"ListPlot", "[", "MidpointApproxSol", "]"}]}]}], "Input", CellChangeTimes->{ 3.742030363693204*^9, {3.742030915837304*^9, 3.742030936022913*^9}},ExpressionUUID->"3c9ad949-eddd-481f-9ea1-\ 03d356217526"], Cell[BoxData[ GraphicsBox[{{}, {{}, {RGBColor[0.368417, 0.506779, 0.709798], PointSize[0.012833333333333334`], AbsoluteThickness[1.6], PointBox[{{5., 0.}, {4.51, 0.3100279228263873}, {4.02, 0.5860705830518481}, {3.5299999999999994`, 0.8256901937655429}, { 3.039999999999999, 1.0257268078852053`}, {2.549999999999999, 1.181895167749849}, {2.0599999999999987`, 1.2880151951199759`}, { 1.5699999999999987`, 1.334351027390282}, {1.0799999999999987`, 1.3033093486991645`}, {0.5899999999999987, 1.154310938223652}, { 0.0999999999999987, 0.7179188376768568}}]}, {}}, {}, {}, {}, {}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0, 0}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{"CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0, 5.}, {0, 1.334351027390282}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.02], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{3.74203095532618*^9, 3.7461831685291853`*^9},ExpressionUUID->"3523447f-3344-424b-9215-\ b1ce63fcdce1"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"MidpointApproxPlot", ",", "ExactPlot"}], "]"}]], "Input", CellChangeTimes->{{3.741948584426202*^9, 3.741948589150589*^9}, { 3.742029611910121*^9, 3.742029614395967*^9}, {3.7420309407332067`*^9, 3.742030945480933*^9}},ExpressionUUID->"db645d55-c53e-4e64-8875-\ 228210801cd9"], Cell[BoxData[ GraphicsBox[{{{}, {{}, {RGBColor[0.368417, 0.506779, 0.709798], PointSize[ 0.012833333333333334`], AbsoluteThickness[1.6], PointBox[{{5., 0.}, {4.51, 0.3100279228263873}, {4.02, 0.5860705830518481}, {3.5299999999999994`, 0.8256901937655429}, { 3.039999999999999, 1.0257268078852053`}, {2.549999999999999, 1.181895167749849}, {2.0599999999999987`, 1.2880151951199759`}, { 1.5699999999999987`, 1.334351027390282}, {1.0799999999999987`, 1.3033093486991645`}, {0.5899999999999987, 1.154310938223652}, { 0.0999999999999987, 0.7179188376768568}}]}, {}}, {}, {}, {}, {}}, {{}, {}, TagBox[ {RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[ 1.], FaceForm[Opacity[0.3]], LineBox[CompressedData[" 1:eJwt13k8VP0XB3BLVEIYKS1oj2yVItTnRo+UUgiF7FF5UmmzpUKWUsmSQpu0 omQpW0WksoRIKiH7vszMnRn77/v8Xuafeb1n7nzPud977zlnFjseMz0owMfH 58LPx/ffu+OozJdgWWn8/JsxZ6TsJXRybHK75PfjiKTDpFPUxykfBl9KbxFD vRTJVy7tHPjhAd4XsZN88hVT33tBYMMQn7ZCFUo61ym1up6H5m6lkjCPb1PH +8PyY3zhhthqsJcMBfSOXoLYeZPZoQk1U78PxZH13RtTIr7D3+Qzs10/DIst arrPHaudWu86Tk8sqDiu+2NqvQiMxlt8dUytwx5Lnc6+w1Fo3a74vNHq59T6 0RAZsz9dPPZzav3buFRfXpCg9BsNl6QsO1/FgmfIvLQg6/dUvHhovJGfLUPV T8W7h7wXY3W+W/5MxUtA9vQuV2PpRvidsI/tv/8Qcbsaap5daJyKnwj5An8Z he7GqfiPkGpzISLUqmkq/jO0yDLybs79C9H7mjO7Op9hstc5dffWv1P5PEex k2i83PG/U/kk42ntonClj3+n8nmJNgMRKyGX5ql80vB+rrSzbmgLoic8WO66 6XgZKWSqldQylV86FgTa6OeUtUzll4FP5pKdduKtU/llIv2D1NnosNap/LIx 57RvM3W2DddCPE79CcvG+qchGiei29BkEaJSNZyNR5syT25Lb0MQK/1eVnUO nlaUrxvoa8M35VmBwcF5yJ7FTD5k144j97J3Lh94j86Yat0nGh1oeN3hL6qa jwzdzKzlxh2YcdfDLsg5H11acvUurh2w/jdknndVPowWiqzZcLsD/DMzLjsk FcDEuc5JaaQDu/RmHVO3K0Sal7eSeVonzijeNHoeXQibL0kCAZ87cV9i8apl ZYVQj7f8GNDQCWbDhr9zNxZB48VtOX6RLtzycTSblPqIdeFqAZO2XWjNyNb8 WlyMAIvIi3f5u+G78hDfv6olMHs0+XuDSg+yLn9+td2hBJ9zJo992NQDum+V 08qoEtScyr623rgH7hldH5uHSxBo8PRGtXsP7Le4Xdn/sRSNDT+dOC96sNXa XWabTTlODk9/KbG6F0HbtGUk/6mEaHTyMUXpPizz6u4/b1sJo50FbxYt6cOH 57Gf+s9Wwl537TU+9T5Mio16lj2vxO7RPWERRn3wrMn9HSxRhUMpXhq9F/tw xGHTg4nfVVh0SsojpqcPxj6Ucq9HNfq3LEjPTe9H+5+Du35fqYZ6zDnVK+/7 4UddcS9JrMaAhFv7rtJ+vBCqTX1aWw2f16vcU5r7MTvcbf1B7Rrcf2Xr80Ni AJWJNzc38H9H2Z28wJ4jAzD92runMrwWniaCQaazBmG++PbpjBc/UeYZWiL7 dBCfYw3j8r/8xMyNtn6yKYPQlubll7X+xPXwD7rSaYNQmG4p2jb/Fx4vuMId yR1Ed6/0wznBv5A0Uq9xrWIQflnhlWfsfmPZy2TRb/Qgnu4OWb1R4g+M29q9 xjcPYeJd7mCDVhP0t9XxLfs4hLxfn3nn/mmCJr+beM3nIXhzvvMtMm3Cl9At QhfLhsBRGZSwdmuCovLqvKrqIfTHL1vz404Tik0jkoybh9DoffVEhcBf6HRe sc+eGEL+Bjvm+7K/8FP1dXZdz4T/CwH2ffsWdAil2L6LZSLl+14/Y/cWaO6d Wz95h4m6scczxnxa8O62ZsDmB0yoGRktsoxpQfW2g/vTnzBR3xFpIPa1BdKa 7GM+GUxoLl5+20unFd2ZbpeDy5nojdy22VSmDebro1LvjzNh4RUWIljWjv1N m5PMLFiY0JLVPPGzHYG1dXTbPhYe8x61NbS3wy9kt+gZaxa4Z9/r5fB3wFjt 6sPr9izcOsMcO67ZAQeN6mdPj7Dw6+S+Ew0JHfD+8fXiST8WbN2XWWV7dWKV keO2s4ksHHTKW31sZTeuS705G9zNgpS/XwW9rhsZ6rPET/ey8P4+ddKX6sYB tSMzHPpZkG0oygnd3424tYtVNJgslO8r3554pRse7QtNPg2zsN64wfXnQDe6 en7VPJjBhtBGvkT9rB5EWgXLqS5nI83yw7aSoh48uMMe71rBht2ZwJ49VT0Q bptnm7iKjez0Getsu3vQfPHHQ2llNo6qSn7wXNiLzXPVbFrWsvF96ZKmlAu9 2L9h93ptio1H4lsXztveBzfrUoHq/WzkaxnppVj0YRqlNnrRmo1fjqaH9Jz7 0HxS0VjtABvib+wz/j1Pjq+9Ov2SPRuedud2FmT2YUZP9N+FrmzsSH3te2RJ P9wtVEYnTrLRb6L4J2+0H5yKH9URYWzM8FUXNJ05gMev3KrFr7Gx5LGmYofM AMrHJLaHXmfDcuSf05JrBzDZ9P62VwTJJ8FRzOXQAO7s+9q+7RYbEay4TbO/ D0CkqO5P3EM24nZMr1C3HESh0vVLK7LZULr+qD74wCC2FL/NsclhI6dav7vB aRD5iodTbuSS87G5IHT1+CCUJr3LuW/ZmHdsWKcrdBA3pHfNyPzARlRkz9MH eYOYa8tpZZWyEVZf4S+1ZAhOafr8wX/YWLjYPfzwqiFEKm6noxvYSDooejdf dQj8B4fePGxko6TfMNtdZwhvslWScv+S8+X/MFCydwh7qvZa/mljI3B5hk1A 8BDUc2s0OvvY8HW/tYHdO4Q/DnIO8uNk/ybNLFazhvBlz6nAaRMkv/DZZxyH h3B09Vh4J3HGq6DMSiEmBN3MXFP4aPSyT2mkyDGxWLpTWmkaDRsfk7Uue5h4 NDevckCEhk6IiGpdOnnOSo0evJxLY4Zs8a7ZOUwMLTF67TGPRu2zi0cN8pmI 57Mv0JCl4VHKS84oY0Lv3J7czPk0not3rL7RxoTkqnbDpEU05kcXKu6Yy8LF 5hS5g0tpjDzwXZ7nzcIO+3X6Mqo0qT9WQd4XWVAOaI0pJhas0OrQCmGh+ABV cFqNhugY+2lmNAu9jjmRleo05C2OKr9MZaHp3ZC+5zoaW0Vs1yW0s6DiMn1t uBaNzwq6UfZ9LMzutBNS3Uhjl+Z8Wo7NQhhNZZQQWzjXvo7jZyMhfNtjPh0a h94Za0cvZOOB2WS77SYaVz2wJcSU7LPO2w1tW2hIhS5KMCD3+WhrH+u4Ho2b 90YFhMh97NioFDlKfL/sTdHFo2zMFbHzEttKI22FuqFPCBu1zLSBpQZkv34p 7HZ/x0aqOeP3gh00FPQFDuxVohEW93GvtgmNlTN9tO6R86Kvas6LJ1atYDG6 N9CIP6/eN0asa91Wcl6ffE79Lcs1pbHv5GetJBsaascs3VT30gh/eE1a4DqN vYapzh0WNGIOTx/cGU2jhCPipG1J467ahdKYOBreLT5nw4iTc0/4qzyl0SL3 tkVlH9mv6r2D+wrIPl7zrjq4n8ak4IKylywac9pX4rk1DeGSyCcjwzQ+pf18 308sGi4a8A8fB5yEcau1JI/5Cyc2/hbl4N81yX9fE2/Q+PtEeAUHLUdnaWUe oLFpZH+AiTIHg7PfRDKJt+Z/s41fy0FZyOAsNVsapjuL5qwFB4LWc80eEbs7 Pwk4sI+D6nYP/WA7GqeVFOye2nLg2Vt2+S2x7+AtbZYzB/7VRYNM4su+l4dC TnCwsitC1tqexuOoo3YZlzlQ+f2uTcGBRopVu/ZkOAei3+QV9xBnKNjJ7Ijh 4NChpVf9iD8k7ylvfMiB8RXJrDrihqJ1OrPecnC03Ou+vyO5DhIVln8KOfgq q2/4gni/zZFTL0s4cOQN9dQR17Dup5jVceC3b/Oq1U40SpeIK8SzONBy2nCu kDjE/Zmu+wgHHXHfRDuIDXK27qf4uZCwK4md6Uzim/hEtIpzkblFMN2IOOtC 5zQVJS5c9or9KSQ+UxaweFKdi4dGI66NxOvmyW+u0uTCL+Y5a5j4xUvzs6f/ 4WIaJ1Bc+SCNf0cHI7ft5CLX2eCuPrHitrBUWTMu4puG1ayIExs+dL215+KI s5NVILGjkq1wuCsXB+e9ZMUQy58ZXuLozgVj7bzw58T1BVHQOM2FpkCXWh5x rJi6jbAvF0mJbtVlxPv2l3rW+XMRwdftXU8s88gl+nkoF9GfU1b0EN/QvfPV OIYLM8fVYUIuNIxDtHoU7nLxKeqsgSSxaE31dFYiFxtoFeGFxCXyx5Z9TOKC vTamdDlxiJvIlpg0LmRKRm+qEhu8eXTgcDYXiYKJhzYQCwpu8dbJ56KW/kFt Is43rr8p9okL3cBGeX1iv9iz6Y3lXIzZCQgbEuu2S1W+quGCNz2EZUQ8vOZF b8BvLqS/vOk0Jn59bvtMi2Yu9r6ra9tDfOpL6/JVXVwsrlrTY0K8ds4FvZEB LtyyZw7/5wH7BXZlHC46xx/P/s/Jya997o5zsZm9W2038RGeya3j03jQLNix byfxqq19GXqzeBjc9+fyf/m0XQ+pkpbiQfz8rk//5Zvwe2l/+zwe+oqbxDYT 2698L5Itz0NZZL2tJrHcSauVV1bwICwblKNGXP+O1j+gwoOu1ZLFK4ljRW7Y q2nwcGE1J2IRsaWF8jl+HR5urDGQYBBLJ3y6Xb2FB/5mi9vTib/1Ob5+ZMiD S8C/qqPkeoVvnPh2djcPISqlFX3ExpduD2y34CHqy+tzjcSiVRqiCw/woCd1 UquS+MvCylX9TjyIRunz5RNvzRB2jDjBQ8I11ax4Ymrn+/CtnjwsOacx7TKx buvZ9xw/HhyTFrmeIdaQ7lpoHcYDuzXaYyexevIDI7FIHmxvJizYQKy81cr7 /W2yfs3VejniZadKfyx9wsODM8OP+8j9ryAaKFybwsOl9ZzUauKFibrrQzJ4 YGqUVGcRS9ekRPQW8HCqvd/5ArGQRviuzHoe7Euac6cRC5QZ+rq28KDsZBrZ Qp7PCSf+JNluHsoFAy4VEHMiPWb4cXkY3Sab603cztpbaCA1jIqUFyeayfPf fEWMyZs3jMajectziBuWFiskyQ/jmYQbJ5y41kzTb7bKMFpn3urXIS5Ol91Y t20Y5m8fUsH/1R+jb66XjYdBRSk3WRK/a7l8U9d8GOFByndXEr9mjLHuOw5D Tsz7TNF/9e1kw4vD54ahm3rnLovUv+B1D5eNpg1jnqH85EpSPwNLrc1Ssodh wxZ+0E3q7QUnaX+7/GFcd+A4JxN7Rl5qLCwfhvu7UjMV4kMs19iwzmHsjHaq Xkbqt2G6soSc3AiO3nK/3E36QfHI+WybZSOIoVvPJRL/o1ftGKc0gn0/9NMP EOtVeWXO0xxBu9Xh1nLST3QHiq0Ye0agZiB58BHpN2qrHR7N8B+BpouH0Bpz Uq88Moy3hYxgzsknqxtJ/1LOmc67dG0Et6M9boYRr9r+Yodg3AjWKZ/UbjOj scR1dGA8fQQyTan5V0n/k3kYrc1qG0GOnFt37m4aUd2dLWt6R3Dutv9RW2LG Wt2rx5kjWFP8UZOPWKKgubFvYgQ/Tp5I2GJMQ6RR7VLn3FHEtXyuyd1JY3x+ ScWf7aM49imw8up2Gq0RE86fU0ax5oCM1jPS32XtT+2+ljGKNNF48dXExird G/fmjkKn8JxkMpkPsj59F2/6PIoaD5k7SRTp62PJWdyWUdzsaai8v5nGehdr 0ZXzxyAxcaHGSZtG0Mas9EtBY9CzkYzatYZGrrDqXaOrY7hjcM8kk8wBg9UP QySjxjDN1nbLQmIr9+sH7jwYg2bp0qwOMh+pPnSZnpk3hhTJ1OmnlWn8EJ1j 3cocg9+ee+aHV5L9azohoG83DoFrJg3ZC2k02fi7Rh8cR1O9N0OaOOZnRFmH 2zjUhCVjjy4g/b06/WaY5zhSN4jGy5H5rfUjrVR7YxwvndlnPMn8dy/Jy+xw 0TgUh4Uez5Ii+3/2YuJ1xQkcbgmoShWmMU38hkE9awL6Il1Hxsl86iFR1Js5 MoGxTXsuLyBukuJGXOefxIX3Su1avWzkzT3QqDd7kvy//2Z2opuNU4sVPZ8p TQJxsrN+trPRplHw7IzDJLz3xooEkvm42HpIVLJiEjF/lptur2SjqOZp1xJz PirauL7n1ks2zJ+nsfYr81Ou4suXjrqwsfa4lIkCzU99u3NXz1+WjZbeD9++ fhWgjkulBO7OJXPj/i9PRGMFKbknz4TkrFhINvbK3+kwjXIpOjHtFJOJnSNe 46c2ClHNvW557Z5MaN2svKosKEw9eXWHqzo5BF7nWb2yP8LUuxUfQn09hhBc +FnbPWM6VesUl/K6cRCO144F3fOdQfV1yZ7asXsQkhKunCzzmVRa/50IE/MB iPlnZDsoiFCb931kgNGPHz5J+aIdIlSs3bFZwiW9OK3M6rn3bhblVBexYsS8 B/bl4Uc+BItS2ie8d23s64L88vNi5Q5iVOT1TUU3LDuhdLRQ0FVFnDrno3/J vKId0za92iRIi1OFP+z5XPTakFY4EBdcPps6biJesy6oBRaLPjwuN5KgigTt l03//RebMjJuGGVLUO/a/9apjjdi1mzfuyXLJCn19jNe2dr1mHTo3ysZKkkF yWTdVHtQhwMSoYvKaEnqt13PfI/7NWh+/fW6sbUUZf1LfnJBWSWMF3ct/fZO ikrrvPjJvPsLlnWsLvmixKBWHMqNkjhYAIVHsvNDrzMovx3VqeGhaQjb/Ire cYNBVZx7lqhDpYH7w7BKNJJBmX6y7mvkvMJXEc+Q8JsM6vfpZD9p51fwPV7L ibnDoHTKHJ6rbE5FnU5k9eMkBhWuKNpMNSch/Jvo1aJPDGpbUJ6cdGICxtwS DwV9YVCj6foh+bMT4Cqku9WwlEEpKhen2vk8wGYtt9HSrwwq/8SBLWdM76P3 zpfD1d8Z1C5h5/YZfPEwPBxk0NzCoKz0+y2NXkUgQ0BuSWIbg8rRDnoyFnQD 8vGZ4wc7GJSZ/3hopE04OF9bM7q6GVTBWk/ZVvEwJGroLx0aYlD7zt60bn3j D4mvvybSWAwqSt/xxH2HC/Bx9fh1imZQ0mamAbtlfGEa++AGj8egEu6Naqrn euDduo1Hc0YYlFhVTfK0+f9CsbzS0HeMQSUxe8a+5zkiyuXQss0TDKooX+b7 07em4Pv/S5rim3r9D+GfYt8= "]]}, Annotation[#, "Charting`Private`Tag$2968#1"]& ]}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->{True, True}, AxesLabel->{None, None}, AxesOrigin->{0, 0}, DisplayFunction->Identity, Frame->{{False, False}, {False, False}}, FrameLabel->{{None, None}, {None, None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, Method->{"CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Identity[#]& )[ Part[#, 1]], (Identity[#]& )[ Part[#, 2]]}& )}}, PlotRange->{{0, 5.}, {0, 1.334351027390282}}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.02], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Output", CellChangeTimes->{3.74203095559859*^9, 3.746183168543186*^9},ExpressionUUID->"dfa9666e-229f-4f96-b305-\ 51cf7fd5cf24"] }, Open ]], Cell[BoxData[ RowBox[{"(*", RowBox[{"It", " ", "does", " ", "pretty", " ", RowBox[{"well", "!"}]}], "*)"}]], "Input", CellChangeTimes->{{3.742030981861587*^9, 3.7420309857335987`*^9}},ExpressionUUID->"35845abf-45ae-44a5-8a00-\ fd420132640a"] }, WindowSize->{875, 717}, WindowMargins->{{Automatic, 15}, {Automatic, 0}}, FrontEndVersion->"11.2 for Microsoft Windows (64-bit) (September 10, 2017)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[558, 20, 405, 8, 28, "Input",ExpressionUUID->"b4d86fd4-3e22-44df-a47b-e4aa9d092ed5"], Cell[966, 30, 1656, 48, 67, "Input",ExpressionUUID->"7307d5ef-b80e-4a52-ad83-c8b954a0017a"], Cell[2625, 80, 203, 4, 28, "Input",ExpressionUUID->"0d7efb9d-b236-4e85-a64e-2b77164c10d0"], Cell[2831, 86, 544, 10, 28, "Input",ExpressionUUID->"e1228e49-22e1-4981-98b6-1d5844854e2a"], Cell[3378, 98, 179, 3, 28, "Input",ExpressionUUID->"d374c0b1-c175-4ea7-a5e4-45e5448f502a"], Cell[CellGroupData[{ Cell[3582, 105, 656, 18, 28, "Input",ExpressionUUID->"c0cf3758-1d43-4b53-afb7-d4086f6b9ecc"], Cell[4241, 125, 519, 12, 21, "Message",ExpressionUUID->"b9baecee-e819-4a7c-999a-1d1bdddd9754"], Cell[4763, 139, 713, 21, 48, "Output",ExpressionUUID->"1f2f8c61-1987-4499-b34b-3b3f8c67138e"] }, Open ]], Cell[CellGroupData[{ Cell[5513, 165, 1494, 38, 105, "Input",ExpressionUUID->"244477e1-7aa6-4349-a008-7b99fde4d850"], Cell[7010, 205, 8222, 145, 329, "Output",ExpressionUUID->"071893cf-7496-459c-bb99-ee303f7a7e9e"] }, Open ]], Cell[15247, 353, 351, 8, 28, "Input",ExpressionUUID->"c60e0ceb-d05e-441b-b53d-030aab65f6df"], Cell[CellGroupData[{ Cell[15623, 365, 4980, 140, 295, "Input",ExpressionUUID->"56a9eb33-28ee-496f-be1e-2dfdb9f4c53f"], Cell[20606, 507, 28530, 532, 331, "Output",ExpressionUUID->"324e29c7-1ea5-431f-9bbe-e69841da08bb"] }, Open ]], Cell[49151, 1042, 3285, 80, 257, "Input",ExpressionUUID->"5d1650d8-bd31-4828-99ef-3bb75776288c"], Cell[52439, 1124, 473, 13, 28, "Input",ExpressionUUID->"60c017ae-eb7f-4e6d-9b59-5f9f51b2c4f1"], Cell[CellGroupData[{ Cell[52937, 1141, 372, 9, 48, "Input",ExpressionUUID->"3c9ad949-eddd-481f-9ea1-03d356217526"], Cell[53312, 1152, 1691, 42, 250, "Output",ExpressionUUID->"3523447f-3344-424b-9215-b1ce63fcdce1"] }, Open ]], Cell[CellGroupData[{ Cell[55040, 1199, 328, 6, 28, "Input",ExpressionUUID->"db645d55-c53e-4e64-8875-228210801cd9"], Cell[55371, 1207, 9101, 166, 250, "Output",ExpressionUUID->"dfa9666e-229f-4f96-b305-51cf7fd5cf24"] }, Open ]], Cell[64487, 1376, 253, 6, 28, "Input",ExpressionUUID->"35845abf-45ae-44a5-8a00-fd420132640a"] } ] *)