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jverzani committed 2022-07-24 19:03:28 -04:00
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quarto/.gitignore
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/.quarto/
/_site/
/_book/
/_freeze/
/*/*_files/
quarto/README.md
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* author in `.jmd` files (to be run through pluto)
* run `julia make_qmd.jl` to convert to `.qmd` files
* compile with `quarto`
- The files in subdirectories are generated, they should not be edited
- The files in this main directory are quarto specific.
- `_book` and `_freeze` are conveniences
* run `quarto preview` to develop interactively (kinda slow!)
* run `quarto render` to render pages (not too bad)
# to publish
* bump the version number in `_quarto.yml`, `Project.toml`
* run `quarto publish gh-pages` to publish
* should also push project to github
* no need to push `_freeze` the repo, as files are locally rendered for now.
The files in subdirectories are generated, they should not be edited
The files in this main directory are quarto specific.
---
quarto/_freeze/ODEs/differential_equations/execute-results/html.json
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"markdown": "# The `DifferentialEquations` suite\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing OrdinaryDiffEq\nusing Plots\nusing ModelingToolkit\n```\n\n\n\n\n---\n\n\nThe [`DifferentialEquations`](https://github.com/SciML/DifferentialEquations.jl) suite of packages contains solvers for a wide range of various differential equations. This section just briefly touches touch on ordinary differential equations (ODEs), and so relies only on `OrdinaryDiffEq` part of the suite. For more detail on this type and many others covered by the suite of packages, there are many other resources, including the [documentation](https://diffeq.sciml.ai/stable/) and accompanying [tutorials](https://github.com/SciML/SciMLTutorials.jl).\n\n\n## SIR Model\n\n\nWe follow along with an introduction to the SIR model for the spread of disease by [Smith and Moore](https://www.maa.org/press/periodicals/loci/joma/the-sir-model-for-spread-of-disease-introduction). This model received a workout due to the COVID-19 pandemic.\n\n\nThe basic model breaks a population into three cohorts: The **susceptible** individuals, the **infected** individuals, and the **recovered** individuals. These add to the population size, $N$, which is fixed, but the cohort sizes vary in time. We name these cohort sizes $S(t)$, $I(t)$, and $R(t)$ and define $s(t)=S(t)/N$, $i(t) = I(t)/N$ and $r(t) = R(t)/N$ to be the respective proportions.\n\n\nThe following *assumptions* are made about these cohorts by Smith and Moore:\n\n\n> No one is added to the susceptible group, since we are ignoring births and immigration. The only way an individual leaves the susceptible group is by becoming infected.\n\n\n\nThis implies the rate of change in time of $S(t)$ depends on the current number of susceptibles, and the amount of interaction with the infected cohorts. The model *assumes* each infected person has $b$ contacts per day that are sufficient to spread the disease. Not all contacts will be with susceptible people, but if people are assumed to mix within the cohorts, then there will be on average $b \\cdot S(t)/N$ contacts with susceptible people per infected person. As each infected person is modeled identically, the time rate of change of $S(t)$ is:\n\n\n\n$$\n\\frac{dS}{dt} = - b \\cdot \\frac{S(t)}{N} \\cdot I(t) = -b \\cdot s(t) \\cdot I(t)\n$$\n\n\nIt is negative, as no one is added, only taken off. After dividing by $N$, this can also be expressed as $s'(t) = -b s(t) i(t)$.\n\n\n> assume that a fixed fraction $k$ of the infected group will recover during any given day.\n\n\n\nThis means the change in time of the recovered depends on $k$ and the number infected, giving rise to the equation\n\n\n\n$$\n\\frac{dR}{dt} = k \\cdot I(t)\n$$\n\n\nwhich can also be expressed in proportions as $r'(t) = k \\cdot i(t)$.\n\n\nFinally, from $S(t) + I(T) + R(t) = N$ we have $S'(T) + I'(t) + R'(t) = 0$ or $s'(t) + i'(t) + r'(t) = 0$.\n\n\nCombining, it is possible to express the rate of change of the infected population through:\n\n\n\n$$\n\\frac{di}{dt} = b \\cdot s(t) \\cdot i(t) - k \\cdot i(t)\n$$\n\n\nThe author's apply this model to flu statistics from Hong Kong where:\n\n\n\n$$\n\\begin{align*}\nS(0) &= 7,900,000\\\\\nI(0) &= 10\\\\\nR(0) &= 0\\\\\n\\end{align*}\n$$\n\n\nIn `Julia` we define these, `N` to model the total population, and `u0` to be the proportions.\n\n::: {.cell execution_count=4}\n``` {.julia .cell-code}\nS0, I0, R0 = 7_900_000, 10, 0\nN = S0 + I0 + R0\nu0 = [S0, I0, R0]/N # initial proportions\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n```\n3-element Vector{Float64}:\n 0.9999987341788175\n 1.2658211825048323e-6\n 0.0\n```\n:::\n:::\n\n\nAn *estimated* set of values for $k$ and $b$ are $k=1/3$, coming from the average period of infectiousness being estimated at three days and $b=1/2$, which seems low in normal times, but not for an infected person who may be feeling quite ill and staying at home. (The model for COVID would certainly have a larger $b$ value).\n\n\nOkay, the mathematical modeling is done; now we try to solve for the unknown functions using `DifferentialEquations`.\n\n\nTo warm up, if $b=0$ then $i'(t) = -k \\cdot i(t)$ describes the infected. (There is no circulation of people in this case.) The solution would be achieved through:\n\n::: {.cell hold='true' execution_count=5}\n``` {.julia .cell-code}\nk = 1/3\n\nf(u,p,t) = -k * u # solving u′(t) = - k u(t)\ntime_span = (0.0, 20.0)\n\nprob = ODEProblem(f, I0/N, time_span)\nsol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8)\n\nplot(sol)\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n![](differential_equations_files/figure-html/cell-6-output-1.svg){}\n:::\n:::\n\n\nThe `sol` object is a set of numbers with a convenient `plot` method. As may have been expected, this graph shows exponential decay.\n\n\nA few comments are in order. The problem we want to solve is\n\n\n\n$$\n\\frac{di}{dt}Line truncated
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915.328,1388 917.632,1390 919.936,1391.98 922.239,1393.95 924.543,1395.91 926.847,1397.86 929.15,1399.79 931.454,1401.71 933.757,1403.61 936.061,1405.51
938.365,1407.39 940.668,1409.26 942.972,1411.11 945.276,1412.96 947.579,1414.79 949.883,1416.61 952.187,1418.42 954.49,1420.21 956.794,1421.99 959.098,1423.77
961.401,1425.53 963.705,1427.27 966.008,1429.01 968.312,1430.73 970.616,1432.45 972.919,1434.15 975.223,1435.84 977.527,1437.52 979.83,1439.19 982.134,1440.84
984.438,1442.49 986.741,1444.13 989.045,1445.75 991.349,1447.36 993.652,1448.97 995.956,1450.56 998.259,1452.14 1000.56,1453.71 1002.87,1455.27 1005.17,1456.82
1007.47,1458.36 1009.78,1459.89 1012.08,1461.41 1014.38,1462.92 1016.69,1464.42 1018.99,1465.91 1021.3,1467.39 1023.6,1468.86 1025.9,1470.32 1028.21,1471.77
1030.51,1473.21 1032.81,1474.65 1035.12,1476.07 1037.42,1477.48 1039.72,1478.89 1042.03,1480.28 1044.33,1481.66 1046.64,1483.04 1048.94,1484.41 1051.24,1485.77
1053.55,1487.11 1055.85,1488.45 1058.15,1489.79 1060.46,1491.11 1062.76,1492.42 1065.07,1493.73 1067.37,1495.02 1069.67,1496.31 1071.98,1497.59 1074.28,1498.86
1076.58,1500.12 1078.89,1501.38 1081.19,1502.62 1083.49,1503.86 1085.8,1505.09 1088.1,1506.31 1090.41,1507.52 1092.71,1508.72 1095.01,1509.92 1097.32,1511.11
1099.62,1512.29 1101.92,1513.46 1104.23,1514.63 1106.53,1515.78 1108.83,1516.93 1111.14,1518.07 1113.44,1519.21 1115.75,1520.34 1118.05,1521.45 1120.35,1522.57
1122.66,1523.67 1124.96,1524.77 1127.26,1525.86 1129.57,1526.94 1131.87,1528.01 1134.17,1529.08 1136.48,1530.14 1138.78,1531.19 1141.09,1532.24 1143.39,1533.28
1145.69,1534.31 1148,1535.34 1150.3,1536.35 1152.6,1537.37 1154.91,1538.37 1157.21,1539.37 1159.51,1540.36 1161.82,1541.35 1164.12,1542.32 1166.43,1543.3
1168.73,1544.26 1171.03,1545.22 1173.34,1546.17 1175.64,1547.12 1177.94,1548.06 1180.25,1548.99 1182.55,1549.92 1184.85,1550.84 1187.16,1551.75 1189.46,1552.66
1191.77,1553.56 1194.07,1554.46 1196.37,1555.35 1198.68,1556.24 1200.98,1557.11 1203.28,1557.99 1205.59,1558.85 1207.89,1559.71 1210.19,1560.57 1212.5,1561.42
1214.8,1562.26 1217.11,1563.1 1219.41,1563.93 1221.71,1564.76 1224.02,1565.58 1226.32,1566.4 1228.62,1567.21 1230.93,1568.02 1233.23,1568.82 1235.53,1569.61
1237.84,1570.4 1240.14,1571.18 1242.45,1571.96 1244.75,1572.74 1247.05,1573.51 1249.36,1574.27 1251.66,1575.03 1253.96,1575.78 1256.27,1576.53 1258.57,1577.27
1260.87,1578.01 1263.18,1578.74 1265.48,1579.47 1267.79,1580.2 1270.09,1580.92 1272.39,1581.63 1274.7,1582.34 1277,1583.05 1279.3,1583.75 1281.61,1584.44
1283.91,1585.13 1286.21,1585.82 1288.52,1586.5 1290.82,1587.18 1293.13,1587.85 1295.43,1588.52 1297.73,1589.19 1300.04,1589.85 1302.34,1590.5 1304.64,1591.15
1306.95,1591.8 1309.25,1592.44 1311.55,1593.08 1313.86,1593.72 1316.16,1594.35 1318.47,1594.97 1320.77,1595.59 1323.07,1596.21 1325.38,1596.83 1327.68,1597.44
1329.98,1598.04 1332.29,1598.65 1334.59,1599.24 1336.89,1599.84 1339.2,1600.43 1341.5,1601.01 1343.81,1601.6 1346.11,1602.18 1348.41,1602.75 1350.72,1603.32
1353.02,1603.89 1355.32,1604.45 1357.63,1605.01 1359.93,1605.57 1362.23,1606.12 1364.54,1606.67 1366.84,1607.22 1369.15,1607.76 1371.45,1608.3 1373.75,1608.83
1376.06,1609.36 1378.36,1609.89 1380.66,1610.41 1382.97,1610.93 1385.27,1611.45 1387.57,1611.96 1389.88,1612.47 1392.18,1612.98 1394.49,1613.48 1396.79,1613.98
1399.09,1614.48 1401.4,1614.97 1403.7,1615.46 1406,1615.95 1408.31,1616.43 1410.61,1616.91 1412.91,1617.39 1415.22,1617.86 1417.52,1618.33 1419.83,1618.8
1422.13,1619.27 1424.43,1619.73 1426.74,1620.19 1429.04,1620.64 1431.34,1621.09 1433.65,1621.54 1435.95,1621.99 1438.25,1622.43 1440.56,1622.87 1442.86,1623.31
1445.17,1623.74 1447.47,1624.17 1449.77,1624.6 1452.08,1625.02 1454.38,1625.45 1456.68,1625.87 1458.99,1626.28 1461.29,1626.7 1463.6,1627.11 1465.9,1627.51
1468.2,1627.92 1470.51,1628.32 1472.81,1628.72 1475.11,1629.12 1477.42,1629.51 1479.72,1629.91 1482.02,1630.3 1484.33,1630.68 1486.63,1631.07 1488.94,1631.45
1491.24,1631.83 1493.54,1632.2 1495.85,1632.58 1498.15,1632.95 1500.45,1633.32 1502.76,1633.68 1505.06,1634.05 1507.36,1634.41 1509.67,1634.77 1511.97,1635.12
1514.28,1635.48 1516.58,1635.83 1518.88,1636.18 1521.19,1636.52 1523.49,1636.87 1525.79,1637.21 1528.1,1637.55 1530.4,1637.89 1532.7,1638.22 1535.01,1638.55
1537.31,1638.89 1539.62,1639.21 1541.92,1639.54 1544.22,1639.86 1546.53,1640.19 1548.83,1640.51 1551.13,1640.82 1553.44,1641.14 1555.74,1641.45 1558.04,1641.76
1560.35,1642.07 1562.65,1642.38 1564.96,1642.68 1567.26,1642.99 1569.56,1643.29 1571.87,1643.59 1574.17,1643.88 1576.47,1644.18 1578.78,1644.47 1581.08,1644.76
1583.38,1645.05 1585.69,1645.34 1587.99,1645.63 1590.3,1645.91 1592.6,1646.19 1594.9,1646.47 1597.21,1646.75 1599.51,1647.02 1601.81,1647.3 1604.12,1647.57
1606.42,1647.84 1608.72,1648.11 1611.03,1648.38 1613.33,1648.64 1615.64,1648.91 1617.94,1649.17 1620.24,1649.43 1622.55,1649.69 1624.85,1649.94 1627.15,1650.2
1629.46,1650.45 1631.76,1650.71 1634.06,1650.96 1636.37,1651.2 1638.67,1651.45 1640.98,1651.7 1643.28,1651.94 1645.58,1652.18 1647.89,1652.43 1650.19,1652.66
1652.49,1652.9 1654.8,1653.14 1657.1,1653.37 1659.4,1653.61 1661.71,1653.84 1664.01,1654.07 1666.32,1654.3 1668.62,1654.53 1670.92,1654.75 1673.23,1654.98
1675.53,1655.2 1677.83,1655.42 1680.14,1655.64 1682.44,1655.86 1684.74,1656.08 1687.05,1656.29 1689.35,1656.51 1691.66,1656.72 1693.96,1656.93 1696.26,1657.15
1698.57,1657.35 1700.87,1657.56 1703.17,1657.77 1705.48,1657.97 1707.78,1658.18 1710.08,1658.38 1712.39,1658.58 1714.69,1658.78 1717,1658.98 1719.3,1659.18
1721.6,1659.37 1723.91,1659.57 1726.21,1659.76 1728.51,1659.95 1730.82,1660.14 1733.12,1660.33 1735.42,1660.52 1737.73,1660.71 1740.03,1660.9 1742.34,1661.08
1744.64,1661.26 1746.94,1661.45 1749.25,1661.63 1751.55,1661.81 1753.85,1661.98 1756.16,1662.16 1758.46,1662.34 1760.76,1662.51 1763.07,1662.69 1765.37,1662.86
1767.68,1663.03 1769.98,1663.2 1772.28,1663.37 1774.59,1663.54 1776.89,1663.7 1779.19,1663.87 1781.5,1664.04 1783.8,1664.2 1786.1,1664.36 1788.41,1664.52
1790.71,1664.68 1793.02,1664.84 1795.32,1665 1797.62,1665.16 1799.93,1665.31 1802.23,1665.47 1804.53,1665.62 1806.84,1665.77 1809.14,1665.92 1811.44,1666.08
1813.75,1666.23 1816.05,1666.37 1818.36,1666.52 1820.66,1666.67 1822.96,1666.81 1825.27,1666.96 1827.57,1667.1 1829.87,1667.24 1832.18,1667.38 1834.48,1667.53
1836.78,1667.66 1839.09,1667.8 1841.39,1667.94 1843.7,1668.08 1846,1668.21 1848.3,1668.35 1850.61,1668.48 1852.91,1668.61 1855.21,1668.75 1857.52,1668.88
1859.82,1669.01 1862.13,1669.14 1864.43,1669.27 1866.73,1669.39 1869.04,1669.52 1871.34,1669.65 1873.64,1669.77 1875.95,1669.89 1878.25,1670.02 1880.55,1670.14
1882.86,1670.26 1885.16,1670.38 1887.47,1670.5 1889.77,1670.62 1892.07,1670.74 1894.38,1670.86 1896.68,1670.97 1898.98,1671.09 1901.29,1671.2 1903.59,1671.32
1905.89,1671.43 1908.2,1671.54 1910.5,1671.65 1912.81,1671.76 1915.11,1671.87 1917.41,1671.98 1919.72,1672.09 1922.02,1672.2 1924.32,1672.31 1926.63,1672.41
1928.93,1672.52 1931.23,1672.62 1933.54,1672.73 1935.84,1672.83 1938.15,1672.93 1940.45,1673.04 1942.75,1673.14 1945.06,1673.24 1947.36,1673.34 1949.66,1673.44
1951.97,1673.54 1954.27,1673.63 1956.57,1673.73 1958.88,1673.83 1961.18,1673.92 1963.49,1674.02 1965.79,1674.11 1968.09,1674.21 1970.4,1674.3 1972.7,1674.39
1975,1674.49 1977.31,1674.58 1979.61,1674.67 1981.91,1674.76 1984.22,1674.85 1986.52,1674.94 1988.83,1675.03 1991.13,1675.11 1993.43,1675.2 1995.74,1675.29
1998.04,1675.37 2000.34,1675.46 2002.65,1675.54 2004.95,1675.63 2007.25,1675.71 2009.56,1675.8 2011.86,1675.88 2014.17,1675.96 2016.47,1676.04 2018.77,1676.13
2021.08,1676.21 2023.38,1676.29 2025.68,1676.37 2027.99,1676.45 2030.29,1676.52 2032.59,1676.6 2034.9,1676.68 2037.2,1676.76 2039.51,1676.83 2041.81,1676.91
2044.11,1676.99 2046.42,1677.06 2048.72,1677.14 2051.02,1677.21 2053.33,1677.29 2055.63,1677.36 2057.93,1677.43 2060.24,1677.51 2062.54,1677.58 2064.85,1677.65
2067.15,1677.72 2069.45,1677.79 2071.76,1677.86 2074.06,1677.93 2076.36,1678 2078.67,1678.07 2080.97,1678.14 2083.27,1678.21 2085.58,1678.28 2087.88,1678.35
2090.19,1678.41 2092.49,1678.48 2094.79,1678.55 2097.1,1678.61 2099.4,1678.68 2101.7,1678.75 2104.01,1678.81 2106.31,1678.88 2108.61,1678.94 2110.92,1679
2113.22,1679.07 2115.53,1679.13 2117.83,1679.19 2120.13,1679.26 2122.44,1679.32 2124.74,1679.38 2127.04,1679.44 2129.35,1679.5 2131.65,1679.56 2133.95,1679.62
2136.26,1679.68 2138.56,1679.74 2140.87,1679.8 2143.17,1679.86 2145.47,1679.92 2147.78,1679.97 2150.08,1680.03 2152.38,1680.09 2154.69,1680.14 2156.99,1680.2
2159.29,1680.26 2161.6,1680.31 2163.9,1680.37 2166.21,1680.42 2168.51,1680.48 2170.81,1680.53 2173.12,1680.59 2175.42,1680.64 2177.72,1680.69 2180.03,1680.75
2182.33,1680.8 2184.63,1680.85 2186.94,1680.9 2189.24,1680.95 2191.55,1681 2193.85,1681.05 2196.15,1681.1 2198.46,1681.15 2200.76,1681.2 2203.06,1681.25
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2228.4,1681.77 2230.71,1681.82 2233.01,1681.87 2235.31,1681.91 2237.62,1681.95 2239.92,1682 2242.23,1682.04 2244.53,1682.09 2246.83,1682.13 2249.14,1682.17
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2504.84,1685.41 2507.14,1685.43 2509.45,1685.45 2511.75,1685.47 2514.06,1685.49 2516.36,1685.5 2518.66,1685.52 2520.97,1685.54 2523.27,1685.56 2525.57,1685.58
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385.478,1686.36 387.933,1686.36 390.387,1686.36 392.841,1686.36 395.295,1686.36 397.749,1686.36 400.203,1686.36 402.657,1686.36 405.111,1686.36 407.565,1686.36
410.019,1686.36 412.473,1686.36 414.927,1686.36 417.381,1686.35 419.835,1686.35 422.289,1686.35 424.743,1686.35 427.197,1686.35 429.651,1686.35 432.106,1686.35
434.56,1686.35 437.014,1686.35 439.468,1686.35 441.922,1686.35 444.376,1686.35 446.83,1686.35 449.284,1686.35 451.738,1686.35 454.192,1686.35 456.646,1686.34
459.1,1686.34 461.554,1686.34 464.008,1686.34 466.462,1686.34 468.916,1686.34 471.37,1686.34 473.824,1686.34 476.279,1686.34 478.733,1686.34 481.187,1686.34
483.641,1686.33 486.095,1686.33 488.549,1686.33 491.003,1686.33 493.457,1686.33 495.911,1686.33 498.365,1686.33 500.819,1686.33 503.273,1686.33 505.727,1686.32
508.181,1686.32 510.635,1686.32 513.089,1686.32 515.543,1686.32 517.998,1686.32 520.452,1686.32 522.906,1686.31 525.36,1686.31 527.814,1686.31 530.268,1686.31
532.722,1686.31 535.176,1686.31 537.63,1686.31 540.084,1686.3 542.538,1686.3 544.992,1686.3 547.446,1686.3 549.9,1686.3 552.354,1686.29 554.808,1686.29
557.262,1686.29 559.716,1686.29 562.171,1686.29 564.625,1686.28 567.079,1686.28 569.533,1686.28 571.987,1686.28 574.441,1686.27 576.895,1686.27 579.349,1686.27
581.803,1686.27 584.257,1686.26 586.711,1686.26 589.165,1686.26 591.619,1686.25 594.073,1686.25 596.527,1686.25 598.981,1686.24 601.435,1686.24 603.889,1686.24
606.344,1686.23 608.798,1686.23 611.252,1686.23 613.706,1686.22 616.16,1686.22 618.614,1686.21 621.068,1686.21 623.522,1686.21 625.976,1686.2 628.43,1686.2
630.884,1686.19 633.338,1686.19 635.792,1686.18 638.246,1686.18 640.7,1686.17 643.154,1686.17 645.608,1686.16 648.062,1686.16 650.517,1686.15 652.971,1686.15
655.425,1686.14 657.879,1686.14 660.333,1686.13 662.787,1686.12 665.241,1686.12 667.695,1686.11 670.149,1686.1 672.603,1686.1 675.057,1686.09 677.511,1686.08
679.965,1686.08 682.419,1686.07 684.873,1686.06 687.327,1686.05 689.781,1686.04 692.235,1686.04 694.69,1686.03 697.144,1686.02 699.598,1686.01 702.052,1686
704.506,1685.99 706.96,1685.98 709.414,1685.97 711.868,1685.96 714.322,1685.95 716.776,1685.94 719.23,1685.93 721.684,1685.92 724.138,1685.91 726.592,1685.89
729.046,1685.88 731.5,1685.87 733.954,1685.86 736.408,1685.84 738.863,1685.83 741.317,1685.82 743.771,1685.8 746.225,1685.79 748.679,1685.77 751.133,1685.76
753.587,1685.74 756.041,1685.73 758.495,1685.71 760.949,1685.69 763.403,1685.68 765.857,1685.66 768.311,1685.64 770.765,1685.62 773.219,1685.6 775.673,1685.58
778.127,1685.56 780.581,1685.54 783.036,1685.52 785.49,1685.5 787.944,1685.48 790.398,1685.45 792.852,1685.43 795.306,1685.41 797.76,1685.38 800.214,1685.36
802.668,1685.33 805.122,1685.31 807.576,1685.28 810.03,1685.25 812.484,1685.22 814.938,1685.19 817.392,1685.17 819.846,1685.13 822.3,1685.1 824.754,1685.07
827.209,1685.04 829.663,1685 832.117,1684.97 834.571,1684.93 837.025,1684.9 839.479,1684.86 841.933,1684.82 844.387,1684.78 846.841,1684.74 849.295,1684.7
851.749,1684.66 854.203,1684.62 856.657,1684.57 859.111,1684.53 861.565,1684.48 864.019,1684.44 866.473,1684.39 868.928,1684.34 871.382,1684.29 873.836,1684.23
876.29,1684.18 878.744,1684.13 881.198,1684.07 883.652,1684.01 886.106,1683.95 888.56,1683.89 891.014,1683.83 893.468,1683.77 895.922,1683.7 898.376,1683.64
900.83,1683.57 903.284,1683.5 905.738,1683.43 908.192,1683.35 910.646,1683.28 913.101,1683.2 915.555,1683.12 918.009,1683.04 920.463,1682.96 922.917,1682.87
925.371,1682.79 927.825,1682.7 930.279,1682.61 932.733,1682.51 935.187,1682.42 937.641,1682.32 940.095,1682.22 942.549,1682.11 945.003,1682.01 947.457,1681.9
949.911,1681.79 952.365,1681.68 954.819,1681.56 957.274,1681.44 959.728,1681.32 962.182,1681.2 964.636,1681.07 967.09,1680.94 969.544,1680.81 971.998,1680.67
974.452,1680.53 976.906,1680.39 979.36,1680.24 981.814,1680.09 984.268,1679.94 986.722,1679.78 989.176,1679.62 991.63,1679.46 994.084,1679.29 996.538,1679.12
998.992,1678.94 1001.45,1678.76 1003.9,1678.58 1006.35,1678.39 1008.81,1678.2 1011.26,1678 1013.72,1677.8 1016.17,1677.59 1018.62,1677.38 1021.08,1677.16
1023.53,1676.94 1025.99,1676.72 1028.44,1676.49 1030.9,1676.25 1033.35,1676.01 1035.8,1675.76 1038.26,1675.51 1040.71,1675.25 1043.17,1674.99 1045.62,1674.72
1048.07,1674.44 1050.53,1674.16 1052.98,1673.88 1055.44,1673.58 1057.89,1673.28 1060.34,1672.98 1062.8,1672.66 1065.25,1672.34 1067.71,1672.02 1070.16,1671.69
1072.61,1671.35 1075.07,1671 1077.52,1670.65 1079.98,1670.28 1082.43,1669.92 1084.88,1669.54 1087.34,1669.16 1089.79,1668.77 1092.25,1668.37 1094.7,1667.96
1097.15,1667.55 1099.61,1667.13 1102.06,1666.7 1104.52,1666.26 1106.97,1665.81 1109.42,1665.35 1111.88,1664.89 1114.33,1664.42 1116.79,1663.93 1119.24,1663.44
1121.7,1662.94 1124.15,1662.43 1126.6,1661.91 1129.06,1661.38 1131.51,1660.85 1133.97,1660.3 1136.42,1659.74 1138.87,1659.18 1141.33,1658.6 1143.78,1658.02
1146.24,1657.42 1148.69,1656.82 1151.14,1656.2 1153.6,1655.58 1156.05,1654.95 1158.51,1654.3 1160.96,1653.65 1163.41,1652.99 1165.87,1652.32 1168.32,1651.64
1170.78,1650.95 1173.23,1650.25 1175.68,1649.54 1178.14,1648.82 1180.59,1648.09 1183.05,1647.36 1185.5,1646.61 1187.95,1645.86 1190.41,1645.1 1192.86,1644.32
1195.32,1643.54 1197.77,1642.76 1200.23,1641.96 1202.68,1641.15 1205.13,1640.34 1207.59,1639.52 1210.04,1638.69 1212.5,1637.85 1214.95,1637.01 1217.4,1636.16
1219.86,1635.3 1222.31,1634.43 1224.77,1633.56 1227.22,1632.68 1229.67,1631.8 1232.13,1630.92 1234.58,1630.02 1237.04,1629.13 1239.49,1628.23 1241.94,1627.33
1244.4,1626.42 1246.85,1625.52 1249.31,1624.61 1251.76,1623.7 1254.21,1622.79 1256.67,1621.87 1259.12,1620.96 1261.58,1620.05 1264.03,1619.14 1266.48,1618.23
1268.94,1617.33 1271.39,1616.42 1273.85,1615.52 1276.3,1614.62 1278.75,1613.73 1281.21,1612.84 1283.66,1611.95 1286.12,1611.07 1288.57,1610.2 1291.03,1609.33
1293.48,1608.47 1295.93,1607.62 1298.39,1606.78 1300.84,1605.95 1303.3,1605.12 1305.75,1604.31 1308.2,1603.5 1310.66,1602.71 1313.11,1601.93 1315.57,1601.16
1318.02,1600.41 1320.47,1599.67 1322.93,1598.94 1325.38,1598.23 1327.84,1597.54 1330.29,1596.86 1332.74,1596.19 1335.2,1595.55 1337.65,1594.92 1340.11,1594.32
1342.56,1593.73 1345.01,1593.16 1347.47,1592.61 1349.92,1592.07 1352.38,1591.56 1354.83,1591.07 1357.28,1590.6 1359.74,1590.16 1362.19,1589.73 1364.65,1589.33
1367.1,1588.94 1369.55,1588.58 1372.01,1588.25 1374.46,1587.93 1376.92,1587.64 1379.37,1587.37 1381.83,1587.13 1384.28,1586.91 1386.73,1586.71 1389.19,1586.54
1391.64,1586.39 1394.1,1586.26 1396.55,1586.16 1399,1586.08 1401.46,1586.02 1403.91,1585.99 1406.37,1585.98 1408.82,1586 1411.27,1586.04 1413.73,1586.1
1416.18,1586.19 1418.64,1586.3 1421.09,1586.43 1423.54,1586.59 1426,1586.77 1428.45,1586.97 1430.91,1587.19 1433.36,1587.44 1435.81,1587.7 1438.27,1587.99
1440.72,1588.3 1443.18,1588.63 1445.63,1588.98 1448.08,1589.35 1450.54,1589.74 1452.99,1590.15 1455.45,1590.58 1457.9,1591.02 1460.35,1591.49 1462.81,1591.97
1465.26,1592.47 1467.72,1592.98 1470.17,1593.51 1472.63,1594.05 1475.08,1594.61 1477.53,1595.18 1479.99,1595.77 1482.44,1596.37 1484.9,1596.98 1487.35,1597.6
1489.8,1598.23 1492.26,1598.88 1494.71,1599.53 1497.17,1600.2 1499.62,1600.88 1502.07,1601.56 1504.53,1602.25 1506.98,1602.95 1509.44,1603.66 1511.89,1604.38
1514.34,1605.1 1516.8,1605.83 1519.25,1606.57 1521.71,1607.31 1524.16,1608.05 1526.61,1608.8 1529.07,1609.56 1531.52,1610.31 1533.98,1611.07 1536.43,1611.84
1538.88,1612.6 1541.34,1613.37 1543.79,1614.14 1546.25,1614.92 1548.7,1615.69 1551.16,1616.46 1553.61,1617.24 1556.06,1618.01 1558.52,1618.79 1560.97,1619.56
1563.43,1620.33 1565.88,1621.1 1568.33,1621.87 1570.79,1622.64 1573.24,1623.41 1575.7,1624.17 1578.15,1624.94 1580.6,1625.7 1583.06,1626.45 1585.51,1627.21
1587.97,1627.96 1590.42,1628.7 1592.87,1629.44 1595.33,1630.18 1597.78,1630.92 1600.24,1631.65 1602.69,1632.37 1605.14,1633.09 1607.6,1633.81 1610.05,1634.52
1612.51,1635.23 1614.96,1635.93 1617.41,1636.62 1619.87,1637.31 1622.32,1638 1624.78,1638.67 1627.23,1639.35 1629.68,1640.01 1632.14,1640.67 1634.59,1641.33
1637.05,1641.97 1639.5,1642.62 1641.96,1643.25 1644.41,1643.88 1646.86,1644.51 1649.32,1645.13 1651.77,1645.74 1654.23,1646.34 1656.68,1646.94 1659.13,1647.53
1661.59,1648.12 1664.04,1648.7 1666.5,1649.27 1668.95,1649.84 1671.4,1650.4 1673.86,1650.96 1676.31,1651.5 1678.77,1652.04 1681.22,1652.58 1683.67,1653.11
1686.13,1653.63 1688.58,1654.15 1691.04,1654.65 1693.49,1655.16 1695.94,1655.65 1698.4,1656.14 1700.85,1656.63 1703.31,1657.1 1705.76,1657.58 1708.21,1658.04
1710.67,1658.5 1713.12,1658.95 1715.58,1659.4 1718.03,1659.84 1720.48,1660.27 1722.94,1660.7 1725.39,1661.12 1727.85,1661.53 1730.3,1661.94 1732.76,1662.35
1735.21,1662.74 1737.66,1663.14 1740.12,1663.52 1742.57,1663.9 1745.03,1664.28 1747.48,1664.65 1749.93,1665.01 1752.39,1665.37 1754.84,1665.72 1757.3,1666.07
1759.75,1666.42 1762.2,1666.75 1764.66,1667.09 1767.11,1667.42 1769.57,1667.74 1772.02,1668.06 1774.47,1668.37 1776.93,1668.68 1779.38,1668.98 1781.84,1669.28
1784.29,1669.58 1786.74,1669.87 1789.2,1670.15 1791.65,1670.43 1794.11,1670.71 1796.56,1670.98 1799.01,1671.25 1801.47,1671.51 1803.92,1671.77 1806.38,1672.03
1808.83,1672.28 1811.28,1672.53 1813.74,1672.77 1816.19,1673.01 1818.65,1673.24 1821.1,1673.47 1823.56,1673.7 1826.01,1673.92 1828.46,1674.14 1830.92,1674.36
1833.37,1674.57 1835.83,1674.78 1838.28,1674.98 1840.73,1675.18 1843.19,1675.38 1845.64,1675.57 1848.1,1675.77 1850.55,1675.95 1853,1676.14 1855.46,1676.32
1857.91,1676.5 1860.37,1676.67 1862.82,1676.84 1865.27,1677.01 1867.73,1677.18 1870.18,1677.34 1872.64,1677.5 1875.09,1677.66 1877.54,1677.81 1880,1677.97
1882.45,1678.12 1884.91,1678.26 1887.36,1678.41 1889.81,1678.55 1892.27,1678.69 1894.72,1678.83 1897.18,1678.96 1899.63,1679.09 1902.09,1679.22 1904.54,1679.35
1906.99,1679.48 1909.45,1679.6 1911.9,1679.73 1914.36,1679.85 1916.81,1679.96 1919.26,1680.08 1921.72,1680.19 1924.17,1680.31 1926.63,1680.42 1929.08,1680.52
1931.53,1680.63 1933.99,1680.73 1936.44,1680.84 1938.9,1680.94 1941.35,1681.04 1943.8,1681.13 1946.26,1681.23 1948.71,1681.32 1951.17,1681.41 1953.62,1681.5
1956.07,1681.59 1958.53,1681.68 1960.98,1681.77 1963.44,1681.85 1965.89,1681.93 1968.34,1682.01 1970.8,1682.09 1973.25,1682.17 1975.71,1682.25 1978.16,1682.32
1980.61,1682.4 1983.07,1682.47 1985.52,1682.54 1987.98,1682.61 1990.43,1682.68 1992.89,1682.75 1995.34,1682.81 1997.79,1682.88 2000.25,1682.94 2002.7,1683
2005.16,1683.07 2007.61,1683.13 2010.06,1683.19 2012.52,1683.24 2014.97,1683.3 2017.43,1683.36 2019.88,1683.41 2022.33,1683.47 2024.79,1683.52 2027.24,1683.57
2029.7,1683.62 2032.15,1683.67 2034.6,1683.72 2037.06,1683.77 2039.51,1683.82 2041.97,1683.87 2044.42,1683.91 2046.87,1683.96 2049.33,1684 2051.78,1684.05
2054.24,1684.09 2056.69,1684.13 2059.14,1684.17 2061.6,1684.21 2064.05,1684.25 2066.51,1684.29 2068.96,1684.33 2071.41,1684.37 2073.87,1684.41 2076.32,1684.44
2078.78,1684.48 2081.23,1684.51 2083.69,1684.55 2086.14,1684.58 2088.59,1684.61 2091.05,1684.65 2093.5,1684.68 2095.96,1684.71 2098.41,1684.74 2100.86,1684.77
2103.32,1684.8 2105.77,1684.83 2108.23,1684.86 2110.68,1684.89 2113.13,1684.91 2115.59,1684.94 2118.04,1684.97 2120.5,1684.99 2122.95,1685.02 2125.4,1685.04
2127.86,1685.07 2130.31,1685.09 2132.77,1685.12 2135.22,1685.14 2137.67,1685.16 2140.13,1685.18 2142.58,1685.21 2145.04,1685.23 2147.49,1685.25 2149.94,1685.27
2152.4,1685.29 2154.85,1685.31 2157.31,1685.33 2159.76,1685.35 2162.21,1685.37 2164.67,1685.39 2167.12,1685.4 2169.58,1685.42 2172.03,1685.44 2174.49,1685.46
2176.94,1685.47 2179.39,1685.49 2181.85,1685.51 2184.3,1685.52 2186.76,1685.54 2189.21,1685.55 2191.66,1685.57 2194.12,1685.58 2196.57,1685.6 2199.03,1685.61
2201.48,1685.63 2203.93,1685.64 2206.39,1685.66 2208.84,1685.67 2211.3,1685.68 2213.75,1685.69 2216.2,1685.71 2218.66,1685.72 2221.11,1685.73 2223.57,1685.74
2226.02,1685.76 2228.47,1685.77 2230.93,1685.78 2233.38,1685.79 2235.84,1685.8 2238.29,1685.81 2240.74,1685.82 2243.2,1685.83 2245.65,1685.84 2248.11,1685.85
2250.56,1685.86 2253.02,1685.87 2255.47,1685.88 2257.92,1685.89 2260.38,1685.9 2262.83,1685.91 2265.29,1685.92 2267.74,1685.92 2270.19,1685.93 2272.65,1685.94
2275.1,1685.95 2277.56,1685.96 2280.01,1685.96 2282.46,1685.97 2284.92,1685.98 2287.37,1685.99 2289.83,1685.99 2292.28,1686 2294.73,1686.01 2297.19,1686.01
2299.64,1686.02 2302.1,1686.03 2304.55,1686.03 2307,1686.04 2309.46,1686.05 2311.91,1686.05 2314.37,1686.06 2316.82,1686.06 2319.27,1686.07 2321.73,1686.08
2324.18,1686.08 2326.64,1686.09 2329.09,1686.09 2331.54,1686.1 2334,1686.1 2336.45,1686.11 2338.91,1686.11 2341.36,1686.12 2343.82,1686.12 2346.27,1686.13
2348.72,1686.13 2351.18,1686.14 2353.63,1686.14 2356.09,1686.15 2358.54,1686.15 2360.99,1686.15 2363.45,1686.16 2365.9,1686.16 2368.36,1686.17 2370.81,1686.17
2373.26,1686.17 2375.72,1686.18 2378.17,1686.18 2380.63,1686.18 2383.08,1686.19 2385.53,1686.19 2387.99,1686.19 2390.44,1686.2 2392.9,1686.2 2395.35,1686.2
2397.8,1686.21 2400.26,1686.21 2402.71,1686.21 2405.17,1686.22 2407.62,1686.22 2410.07,1686.22 2412.53,1686.23 2414.98,1686.23 2417.44,1686.23 2419.89,1686.23
2422.34,1686.24 2424.8,1686.24 2427.25,1686.24 2429.71,1686.24 2432.16,1686.25 2434.62,1686.25 2437.07,1686.25 2439.52,1686.25 2441.98,1686.26 2444.43,1686.26
2446.89,1686.26 2449.34,1686.26 2451.79,1686.26 2454.25,1686.27 2456.7,1686.27 2459.16,1686.27 2461.61,1686.27 2464.06,1686.27 2466.52,1686.28 2468.97,1686.28
2471.43,1686.28 2473.88,1686.28 2476.33,1686.28 2478.79,1686.29 2481.24,1686.29 2483.7,1686.29 2486.15,1686.29 2488.6,1686.29 2491.06,1686.29 2493.51,1686.29
2495.97,1686.3 2498.42,1686.3 2500.87,1686.3 2503.33,1686.3 2505.78,1686.3 2508.24,1686.3 2510.69,1686.3 2513.14,1686.31 2515.6,1686.31 2518.05,1686.31
2520.51,1686.31 2522.96,1686.31 2525.42,1686.31 2527.87,1686.31 2530.32,1686.31 2532.78,1686.32 2535.23,1686.32 2537.69,1686.32 2540.14,1686.32 2542.59,1686.32
2545.05,1686.32 2547.5,1686.32 2549.96,1686.32 2552.41,1686.32 2554.86,1686.32 2557.32,1686.33 2559.77,1686.33 2562.23,1686.33 2564.68,1686.33 2567.13,1686.33
2569.59,1686.33 2572.04,1686.33 2574.5,1686.33 2576.95,1686.33 2579.4,1686.33 2581.86,1686.33 2584.31,1686.33 2586.77,1686.34 2589.22,1686.34 2591.67,1686.34
2594.13,1686.34 2596.58,1686.34 2599.04,1686.34 2601.49,1686.34 2603.95,1686.34 2606.4,1686.34 2608.85,1686.34 2611.31,1686.34 2613.76,1686.34 2616.22,1686.34
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"markdown": "# Euler's method\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing SymPy\nusing Roots\n```\n\n\n\n\n---\n\n\nThe following section takes up the task of numerically approximating solutions to differential equations. `Julia` has a huge set of state-of-the-art tools for this task starting with the [DifferentialEquations](https://github.com/SciML/DifferentialEquations.jl) package. We don't use that package in this section, focusing on simpler methods and implementations for pedagogical purposes, but any further exploration should utilize the tools provided therein. A brief introduction to the package follows in an upcoming [section](./differential_equations.html).\n\n\n---\n\nConsider the differential equation:\n\n\n\n$$\ny'(x) = y(x) \\cdot x, \\quad y(1)=1,\n$$\n\n\nwhich can be solved with `SymPy`:\n\n::: {.cell execution_count=4}\n``` {.julia .cell-code}\n@syms x, y, u()\nD = Differential(x)\nx0, y0 = 1, 1\nF(y,x) = y*x\n\ndsolve(D(u)(x) - F(u(x), x))\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n```{=html}\n<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> \n\\[\nu{\\left(x \\right)} = C_{1} e^{\\frac{x^{2}}{2}}\n\\]\n</span>\n```\n:::\n:::\n\n\nWith the given initial condition, the solution becomes:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nout = dsolve(D(u)(x) - F(u(x),x), u(x), ics=Dict(u(x0) => y0))\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```{=html}\n<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> \n\\[\nu{\\left(x \\right)} = \\frac{e^{\\frac{x^{2}}{2}}}{e^{\\frac{1}{2}}}\n\\]\n</span>\n```\n:::\n:::\n\n\nPlotting this solution over the slope field\n\n::: {.cell execution_count=6}\n``` {.julia .cell-code}\np = plot(legend=false)\nvectorfieldplot!((x,y) -> [1, F(x,y)], xlims=(0, 2.5), ylims=(0, 10))\nplot!(rhs(out), linewidth=5)\n```\n\n::: {.cell-output .cell-output-display execution_count=7}\n![](euler_files/figure-html/cell-7-output-1.svg){}\n:::\n:::\n\n\nwe see that the vectors that are drawn seem to be tangent to the graph of the solution. 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"markdown": "# ODEs\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing SymPy\n```\n\n\n\n\n---\n\n\nSome relationships are easiest to describe in terms of rates or derivatives. For example:\n\n\n * Knowing the speed of a car and how long it has been driving can summarize the car's location.\n * One of Newton's famous laws, $F=ma$, describes the force on an object of mass $m$ in terms of the acceleration. The acceleration is the derivative of velocity, which in turn is the derivative of position. So if we know the rates of change of $v(t)$ or $x(t)$, we can differentiate to find $F$.\n * Newton's law of [cooling](http://tinyurl.com/z4lmetp). This describes the temperature change in an object due to a difference in temperature with the object's surroundings. The formula being, $T'(t) = -r \\left(T(t) - T_a \\right)$, where $T(t)$ is temperature at time $t$ and $T_a$ the ambient temperature.\n * [Hooke's law](http://tinyurl.com/kbz7r8l) relates force on an object to the position on the object, through $F = k x$. This is appropriate for many systems involving springs. Combined with Newton's law $F=ma$, this leads to an equation that $x$ must satisfy: $m x''(t) = k x(t)$.\n\n\n## Motion with constant acceleration\n\n\nLet's consider the case of constant acceleration. This describes how nearby objects fall to earth, as the force due to gravity is assumed to be a constant, so the acceleration is the constant force divided by the constant mass.\n\n\nWith constant acceleration, what is the velocity?\n\n\nAs mentioned, we have $dv/dt = a$ for any velocity function $v(t)$, but in this case, the right hand side is assumed to be constant. How does this restrict the possible functions, $v(t)$, that the velocity can be?\n\n\nHere we can integrate to find that any answer must look like the following for some constant of integration:\n\n\n\n$$\nv(t) = \\int \\frac{dv}{dt} dt = \\int a dt = at + C.\n$$\n\n\nIf we are given the velocity at a fixed time, say $v(t_0) = v_0$, then we can use the definite integral to get:\n\n\n\n$$\nv(t) - v(t_0) = \\int_{t_0}^t a dt = at - a t_0.\n$$\n\n\nSolving, gives:\n\n\n\n$$\nv(t) = v_0 + a (t - t_0).\n$$\n\n\nThis expresses the velocity at time $t$ in terms of the initial velocity, the constant acceleration and the time duration.\n\n\nA natural question might be, is this the *only* possible answer? There are a few useful ways to think about this.\n\n\nFirst, suppose there were another, say $u(t)$. Then define $w(t)$ to be the difference: $w(t) = v(t) - u(t)$. We would have that $w'(t) = v'(t) - u'(t) = a - a = 0$. But from the mean value theorem, a function whose derivative is *continuously* $0$, will necessarily be a constant. So at most, $v$ and $u$ will differ by a constant, but if both are equal at $t_0$, they will be equal for all $t$.\n\n\nSecond, since the derivative of any solution is a continuous function, it is true by the fundamental theorem of calculus that it *must* satisfy the form for the antiderivative. The initial condition makes the answer unique, as the indeterminate $C$ can take only one value.\n\n\nSummarizing, we have\n\n\n> If $v(t)$ satisfies the equation: $v'(t) = a$, $v(t_0) = v_0,$ then the unique solution will be $v(t) = v_0 + a (t - t_0)$.\n\n\n\nNext, what about position? Here we know that the time derivative of position yields the velocity, so we should have that the unknown position function satisfies this equation and initial condition:\n\n\n\n$$\nx'(t) = v(t) = v_0 + a (t - t_0), \\quad x(t_0) = x_0.\n$$\n\n\nAgain, we can integrate to get an answer for any value $t$:\n\n\n\n$$\n\\begin{align*}\nx(t) - x(t_0) &= \\int_{t_0}^t \\frac{dv}{dt} dt \\\\\n&= (v_0t + \\frac{1}{2}a t^2 - at_0 t) |_{t_0}^t \\\\\n&= (v_0 - at_0)(t - t_0) + \\frac{1}{2} a (t^2 - t_0^2).\n\\end{align*}\n$$\n\n\nThere are three constants: the initial value for the independent variable, $t_0$, and the two initial values for the velocity and position, $v_0, x_0$. Assuming $t_0 = 0$, we can simplify the above to get a formula familiar from introductory physics:\n\n\n\n$$\nx(t) = x_0 + v_0 t + \\frac{1}{2} at^2.\n$$\n\n\nAgain, the mean value theorem can show that with the initial value specified this is the only possible solution.\n\n\n## First-order initial-value problems\n\n\nThe two problems just looked at can be summarized by the following. We are looking for solutions to an equation of the form (taking $y$ and $x$ as the variables, in place of $x$ and $t$):\n\n\n\n$$\ny'(x) = f(x), \\quad y(x_0) = y_0.\n$$\n\n\nThis is called an *ordinary differential equation* (ODE), as it is an equation involving the ordinary derivative of an unknown function, $y$.\n\n\nThis is called a first-order, ordinary differential equation, as there is only the first derivative involved.\n\n\nThis is called an initial-value problem, as the value at the initial point $x_0$ is specified as part of the problem.\n\n\n#### Line truncated
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quarto/_freeze/ODEs/solve/execute-results/html.json
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"markdown": "# The problem-algorithm-solve interface\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing Plots\nusing MonteCarloMeasurements\n```\n\n\n\n\n---\n\n\nThe [DifferentialEquations.jl](https://github.com/SciML) package is an entry point to a suite of `Julia` packages for numerically solving differential equations in `Julia` and other languages. A common interface is implemented that flexibly adjusts to the many different problems and algorithms covered by this suite of packages. In this section, we review a very informative [post](https://discourse.julialang.org/t/function-depending-on-the-global-variable-inside-module/64322/10) by discourse user `@genkuroki` which very nicely demonstrates the usefulness of the problem-algorithm-solve approach used with `DifferentialEquations.jl`. We slightly modify the presentation below for our needs, but suggest a perusal of the original post.\n\n\n##### Example: FreeFall\n\n\nThe motion of an object under a uniform gravitational field is of interest.\n\n\nThe parameters that govern the equation of motions are the gravitational constant, `g`; the initial height, `y0`; and the initial velocity, `v0`. The time span for which a solution is sought is `tspan`.\n\n\nA problem consists of these parameters. Typical `Julia` usage would be to create a structure to hold the parameters, which may be done as follows:\n\n::: {.cell execution_count=4}\n``` {.julia .cell-code}\nstruct Problem{G, Y0, V0, TS}\n g::G\n y0::Y0\n v0::V0\n tspan::TS\nend\n\nProblem(;g=9.80665, y0=0.0, v0=30.0, tspan=(0.0,8.0)) = Problem(g, y0, v0, tspan)\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n```\nProblem\n```\n:::\n:::\n\n\nThe above creates a type, `Problem`, *and* a default constructor with default values. (The original uses a more sophisticated setup that allows the two things above to be combined.)\n\n\nJust calling `Problem()` will create a problem suitable for the earth, passing different values for `g` would be possible for other planets.\n\n\nTo solve differential equations there are many different possible algorithms. Here is the construction of two types to indicate two algorithms:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nstruct EulerMethod{T}\n dt::T\nend\nEulerMethod(; dt=0.1) = EulerMethod(dt)\n\nstruct ExactFormula{T}\n dt::T\nend\nExactFormula(; dt=0.1) = ExactFormula(dt)\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\nExactFormula\n```\n:::\n:::\n\n\nThe above just specifies a type for dispatch –- the directions indicating what code to use to solve the problem. As seen, each specifies a size for a time step with default of `0.1`.\n\n\nA type for solutions is useful for different `show` methods or other methods. One can be created through:\n\n``` {.julia .cell-code}\nstruct Solution{Y, V, T, P<:Problem, A}\n y::Y\n v::V\n t::T\n prob::P\n alg::A\nend\n```\n\n\nThe different algorithms then can be implemented as part of a generic `solve` function. Following the post we have:\n\n::: {.cell execution_count=7}\n``` {.julia .cell-code}\nsolve(prob::Problem) = solve(prob, default_algorithm(prob))\ndefault_algorithm(prob::Problem) = EulerMethod()\n\nfunction solve(prob::Problem, alg::ExactFormula)\n g, y0, v0, tspan = prob.g, prob.y0, prob.v0, prob.tspan\n dt = alg.dt\n t0, t1 = tspan\n t = range(t0, t1 + dt/2; step = dt)\n\n y(t) = y0 + v0*(t - t0) - g*(t - t0)^2/2\n v(t) = v0 - g*(t - t0)\n\n Solution(y.(t), v.(t), t, prob, alg)\nend\n\nfunction solve(prob::Problem, alg::EulerMethod)\n g, y0, v0, tspan = prob.g, prob.y0, prob.v0, prob.tspan\n dt = alg.dt\n t0, t1 = tspan\n t = range(t0, t1 + dt/2; step = dt)\n\n n = length(t)\n y = Vector{typeof(y0)}(undef, n)\n v = Vector{typeof(v0)}(undef, n)\n y[1] = y0\n v[1] = v0\n\n for i in 1:n-1\n v[i+1] = v[i] - g*dt # F*h step of Euler\n y[i+1] = y[i] + v[i]*dt # F*h step of Euler\n end\n\n Solution(y, v, t, prob, alg)\nend\n```\n\n::: {.cell-output .cell-output-display execution_count=8}\n```\nsolve (generic function with 3 methods)\n```\n:::\n:::\n\n\nThe post has a more elegant means to unpack the parameters from the structures, but for each of the above, the parameters are unpacked, and then the corresponding algorithm employed. As of version `v1.7` of `Julia`, the syntax `(;g,y0,v0,tspan) = prob` could also be employed.\n\n\nThe exact formulas, `y(t) = y0 + v0*(t - t0) - g*(t - t0)^2/2` and `v(t) = v0 - g*(t - t0)`, follow from well-known physics formulas. Each answer is wrapped in a `Solution` type so that the answers found can be easily extracted in a uniform manner.\n\n\nFor example, plots of each can be obtained through:\n\n::: {.cell execution_count=8}\n``` {.julia .cell-code}\nearth = Problem()\nsol_euler = solve(earth)\nsol_exact = solve(earth, ExactFormula())\n\nplot(sol_euler.t, sol_euler.y;\n label=\"Euler's method (dtLine truncated
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quarto/_freeze/alternatives/plotly_plotting/execute-results/html.json
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"markdown": "# JavaScript based plotting libraries\n\n\n\nThis section uses this add-on package:\n\n``` {.julia .cell-code}\nusing PlotlyLight\n```\n\n\nTo avoid a dependence on the `CalculusWithJulia` package, we load two utility packages:\n\n``` {.julia .cell-code}\nusing PlotUtils\nusing SplitApplyCombine\n```\n\n\n---\n\n\n`Julia` has different interfaces to a few JavaScript plotting libraries, notably the [vega](https://vega.github.io/vega/) and [vega-lite](https://vega.github.io/vega-lite/) through the [VegaLite.jl](https://github.com/queryverse/VegaLite.jl) package, and [plotly](https://plotly.com/javascript/) through several interfaces: `Plots.jl`, `PlotlyJS.jl`, and `PlotlyLight.jl`. These all make web-based graphics, for display through a web browser.\n\n\nThe `Plots.jl` interface is a backend for the familiar `Plots` package, making the calling syntax familiar, as is used throughout these notes. The `plotly()` command, from `Plots`, switches to this backend.\n\n\nThe `PlotlyJS.jl` interface offers direct translation from `Julia` structures to the underlying `JSON` structures needed by plotly, and has mechanisms to call back into `Julia` from `JavaScript`. This allows complicated interfaces to be produced.\n\n\nHere we discuss `PlotlyLight` which conveniently provides the translation from `Julia` structures to the `JSON` structures needed in a light-weight package, which plots quickly, without the delays due to compilation of the more complicated interfaces. Minor modifications would be needed to adjust the examples to work with `PlotlyJS` or `PlotlyBase`. The documentation for the `JavaScript` [library](https://plotly.com/javascript/) provides numerous examples which can easily be translated. The [one-page-reference](https://plotly.com/javascript/reference/) gives specific details, and is quoted from below, at times.\n\n\nThis discussion covers the basic of graphing for calculus purposes. It does not cover, for example, the faceting common in statistical usages, or the chart types common in business and statistics uses. The `plotly` library is much more extensive than what is reviewed below.\n\n\n## Julia dictionaries to JSON\n\n\n`PlotlyLight` uses the `JavaScript` interface for the `plotly` libraries. Unlike more developed interfaces, like the one for `Python`, `PlotlyLight` only manages the translation from `Julia` structures to `JavaScript` structures and the display of the results.\n\n\nThe key to translation is the mapping for `Julia`'s dictionaries to the nested `JSON` structures needed by the `JavaScript` library.\n\n\nFor example, an introductory [example](https://plotly.com/javascript/line-and-scatter/) for a scatter plot includes this `JSON` structure:\n\n``` {.julia .cell-code}\nvar trace1 = {\n x: [1, 2, 3, 4],\n y: [10, 15, 13, 17],\n mode: 'markers',\n type: 'scatter'\n};\n```\n\n\nThe `{}` create a list, the `[]` an Array (or vector, as it does with `Julia`), the `name:` are keys. The above is simply translated via:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nConfig(x = [1,2,3,4],\n y = [10, 15, 13, 17],\n mode = \"markers\",\n type = \"scatter\"\n )\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n```\nConfig with 4 entries:\n :x => [1, 2, 3, 4]\n :y => [10, 15, 13, 17]\n :mode => \"markers\"\n :type => \"scatter\"\n```\n:::\n:::\n\n\nThe `Config` constructor (from the `EasyConfig` package loaded with `PlotlyLight`) is an interface for a dictionary whose keys are symbols, which are produced by the named arguments passed to `Config`. By nesting `Config` statements, nested `JavaScript` structures can be built up. As well, these can be built on the fly using `.` notation, as in:\n\n::: {.cell hold='true' execution_count=6}\n``` {.julia .cell-code}\ncfg = Config()\ncfg.key1.key2.key3 = \"value\"\ncfg\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\nConfig with 1 entry:\n :key1 => Config(:key2=>Config(:key3=>\"value\"))\n```\n:::\n:::\n\n\nTo produce a figure with `PlotlyLight` then is fairly straightforward: data and, optionally, a layout are created using `Config`, then passed along to the `Plot` command producing a `Plot` object which has `display` methods defined for it. This will be illustrated through the examples.\n\n\n## Scatter plot\n\n\nA basic scatter plot of points $(x,y)$ is created as follows:\n\n::: {.cell hold='true' execution_count=7}\n``` {.julia .cell-code}\nxs = 1:5\nys = rand(5)\ndata = Config(x = xs,\n y = ys,\n type=\"scatter\",\n mode=\"markers\"\n )\nPlot(data)\n```\n\n::: {.cell-output .cell-output-display execution_count=7}\n```{=html}\n<div class=\"\" style=\"height: 100vh;\" id=\"parent-of-mjddD0HOR2\">\n <div class=\"\" style=\"height: 100%;\" id=\"mjddD0HOR2\"></div>\n</div>\n<script src=\"https://cdn.plot.ly/plotly-2.11.0.min.js\"></script>\n<script>\nPlotly.newPlot(\"mjddD0HOR2\", [{\"x\":[1,2,3,4,5Line truncated
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quarto/_freeze/derivatives/curve_sketching/execute-results/html.json
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"markdown": "# Curve Sketching\n\n\n\nThis section uses the following add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing SymPy\nusing Roots\nusing Polynomials # some name clash with SymPy\n```\n\n\n\n\n---\n\n\nThe figure illustrates a means to *sketch* a sine curve - identify as many of the following values as you can:\n\n\n * asymptotic behaviour (as $x \\rightarrow \\pm \\infty$),\n * periodic behaviour,\n * vertical asymptotes,\n * the $y$ intercept,\n * any $x$ intercept(s),\n * local peaks and valleys (relative extrema).\n * concavity\n\n\nWith these, a sketch fills in between the points/lines associated with these values.\n\n::: {.cell cache='true' hold='true' execution_count=4}\n\n::: {.cell-output .cell-output-display execution_count=5}\n```{=html}\n<div class=\"d-flex justify-content-center\"> <figure> <img 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706.491,308.123 707.04,163.09 707.588,895.715 708.136,1660.98 708.685,1576.21 709.233,739.189 709.781,115.342 710.33,424.227 710.878,1309.57 711.426,1750.2
711.975,1237.9 712.523,363.548 713.071,135.647 713.62,817.058 714.168,1621.83 714.717,1621.73 715.265,816.87 715.813,135.59 716.362,363.687 716.91,1238.07
717.458,1750.21 718.007,1309.4 718.555,424.076 719.103,115.382 719.652,739.374 720.2,1576.33 720.748,1660.89 721.297,895.526 721.845,163.016 722.393,308.249
722.942,1163.71 723.49,1742.7 724.038,1377.4 724.587,489.155 725.135,102.479 725.684,663.371 726.232,1524.89 726.78,1693.35 727.329,974.441 727.877,197.37
728.425,258.417 728.974,1087.17 729.522,1727.73 730.07,1441.27 730.619,558.193 731.167,97.0553 731.715,589.741 732.264,1467.98 732.812,1718.8 733.4,985.751
733.988,172.184 734.576,339.153 735.164,1268.9 735.752,1748.84 736.339,1153.04 736.927,258.541 737.515,231.95 738.103,1107.95 738.691,1744.06 739.279,1310.66
739.867,372.997 740.455,153.968 741.043,939.232 741.63,1704.65 742.218,1451.95 742.806,510.718 743.394,108.501 743.982,769.875 744.57,1632.27 745.158,1570.96
745.746,665.891 746.334,97.4681 746.921,607.025 747.509,1530 748.097,1662.64 748.685,831.963 749.273,121.335 749.861,457.56 750.449,1402.13 751.037,1723.14
751.625,1001.92 752.213,179.094 752.8,327.791 753.388,1254.08 753.976,1749.89 754.564,1168.59 755.152,268.306 755.74,223.196 756.328,1092.09 756.916,1741.78
757.504,1324.93 758.091,385.205 758.679,148.192 759.267,923.005 759.855,1699.13 760.443,1464.34 761.031,524.854 761.619,105.947 762.207,753.961 762.795,1623.76
763.382,1580.94 763.97,681.358 764.558,98.2431 765.146,592.098 765.734,1518.84 766.322,1669.8 766.91,848.107 767.498,125.407 768.086,444.249 768.673,1388.8
769.261,1727.17 769.849,1018.06 770.437,186.291 771.025,316.658 771.613,1239.14 772.201,1750.63 772.789,1184.04 773.377,278.324 773.964,214.711 774.552,1076.17
775.14,1739.18 775.728,1339.05 776.316,397.621 776.904,142.715 777.492,906.778 778.08,1693.32 778.668,1476.53 779.255,539.144 779.843,103.707 780.431,738.114
781.019,1614.97 781.607,1590.68 782.195,696.917 782.783,99.3359 783.371,577.298 783.959,1507.45 784.547,1676.68 785.134,864.28 785.722,129.786 786.31,431.123
786.898,1375.29 787.486,1730.9 788.074,1034.16 788.662,193.771 789.25,305.759 789.838,1224.07 790.425,1751.05 791.013,1199.4 791.601,288.59 792.189,206.5
792.777,1060.19 793.365,1736.28 793.953,1353 794.541,410.239 795.129,137.538 795.716,890.558 796.304,1687.21 796.892,1488.5 797.48,553.581 798.068,101.783
798.656,722.337 799.244,1605.92 799.832,1600.15 800.42,712.565 801.007,100.746 801.595,562.632 802.183,1495.83 802.771,1683.26 803.359,880.476 803.947,134.471
804.535,418.186 805.123,1361.6 805.711,1734.31 806.298,1050.23 806.886,201.533 807.474,295.098 808.062,1208.89 808.743,1736.41 809.424,902.092 810.106,104.516
810.787,680.672 811.468,1664.74 812.149,1407.12 812.83,339.655 813.511,251.76 814.192,1290.77 814.873,1714.98 815.555,813.279 816.236,97.2028 816.917,767.118
817.598,1700.04 818.279,1332.09 818.96,280.078 819.641,307.497 820.322,1368.39 821.004,1684.36 821.685,725.75 822.366,99.4841 823.047,855.385 823.728,1726.33
824.409,1252.34 825.09,227.972 825.772,370.388 826.453,1440.85 827.134,1644.93 827.815,640.523 828.496,111.333 829.177,944.448 829.858,1743.3 830.539,1168.77
831.221,183.943 831.902,439.704 832.583,1507.32 833.264,1597.13 833.945,558.585 834.626,132.613 835.307,1033.27 835.988,1750.78 836.67,1082.36 837.351,148.502
838.032,514.639 838.713,1567.01 839.394,1541.51 840.075,480.887 840.756,163.075 841.437,1120.83 842.119,1748.66 842.8,994.117 843.481,122.06 844.162,594.325
844.843,1619.25 845.524,1478.74 846.205,408.332 846.887,202.368 847.568,1206.11 848.249,1736.96 848.93,905.059 849.611,104.924 850.292,677.837 850.973,1663.42
851.654,1409.52 852.336,341.76 853.017,250.035 853.698,1288.11 854.379,1715.84 855.06,816.221 855.741,97.2921 856.422,764.205 857.103,1699 857.785,1334.67
858.466,281.945 859.147,305.522 859.828,1365.88 860.509,1685.53 861.19,728.633 861.871,99.2537 862.552,852.427 863.234,1725.6 863.915,1255.06 864.596,229.58
865.277,368.187 865.958,1438.53 866.639,1646.38 867.32,643.313 868.002,110.786 868.683,941.48 869.364,1742.89 870.045,1171.6 870.726,185.273 871.407,437.301
872.088,1505.21 872.769,1598.85 873.451,561.25 874.132,131.755 874.813,1030.33 875.494,1750.68 876.175,1085.27 876.856,149.539 877.537,512.063 878.218,1565.14
878.9,1543.49 879.581,483.396 880.262,161.917 880.943,1117.95 881.624,1748.88 882.305,997.074 882.986,122.792 883.667,591.605 884.349,1617.64 885.03,1480.93
885.711,410.656 886.392,200.922 887.073,1203.31 887.754,1737.51 888.435,908.027 889.117,105.342 889.798,675.004 890.479,1662.08 891.16,1411.92 891.841,343.872
892.522,248.318 893.203,1285.44 893.884,1716.69 894.566,819.164 895.247,97.392 895.852,636.678 896.457,1570.23 897.063,1617.07 897.668,712.923 898.273,98.4238
898.879,616.761 899.484,1556.69 900.089,1628.49 900.694,733.629 901.3,100.005 901.905,597.048 902.51,1542.74 903.116,1639.44 903.721,754.462 904.326,102.136
904.932,577.553 905.537,1528.37 906.142,1649.91 906.748,775.407 907.353,104.813 907.958,558.288 908.563,1513.59 909.169,1659.9 909.774,796.451 910.379,108.036
910.985,539.267 911.59,1498.43 912.195,1669.4 912.801,817.581 913.406,111.802 914.011,520.503 914.617,1482.88 915.222,1678.4 915.827,838.781 916.432,116.109
917.038,502.007 917.643,1466.96 918.248,1686.9 918.854,860.038 919.459,120.955 920.064,483.792 920.67,1450.68 921.275,1694.9 921.88,881.338 922.486,126.334
923.091,465.87 923.696,1434.05 924.301,1702.38 924.907,902.666 925.512,132.245 926.117,448.253 926.723,1417.08 927.328,1709.34 927.933,924.008 928.539,138.684
929.144,430.953 929.749,1399.78 930.355,1715.78 930.96,945.35 931.565,145.645 932.17,413.981 932.776,1382.16 933.381,1721.69 933.986,966.678 934.592,153.124
935.197,397.349 935.802,1364.24 936.408,1727.07 937.013,987.978 937.618,161.117 938.224,381.068 938.829,1346.02 939.434,1731.91 940.039,1009.23 940.645,169.618
941.25,365.148 941.855,1327.53 942.461,1736.22 943.066,1030.44 943.671,178.621 944.277,349.6 944.882,1308.76 945.487,1739.99 946.093,1051.56 946.698,188.12
947.303,334.435 947.908,1289.74 948.514,1743.21 949.119,1072.61 949.724,198.11 950.33,319.662 950.935,1270.48 951.54,1745.89 952.146,1093.55 952.751,208.583
953.356,305.292 953.961,1250.98 954.567,1748.02 955.172,1114.39 955.777,219.532 956.383,291.334 956.988,1231.27 957.593,1749.6 958.199,1135.09 958.804,230.95
959.409,277.797 960.015,1211.35 960.62,1750.63 961.225,1155.66 961.83,242.83 962.436,264.69 963.041,1191.24 963.646,1751.11 964.252,1176.07 964.857,255.163
965.462,252.022 966.068,1170.96 966.673,1751.04 967.278,1196.31 967.884,267.942 968.489,239.802 969.094,1150.51 969.699,1750.42 970.305,1216.37 970.91,281.158
971.515,228.038 972.121,1129.9 972.726,1749.25 973.257,1350.07 973.788,487.234 974.318,99.7159 974.849,609.223 975.38,1461.29 975.911,1728.68 976.442,1120.4
976.972,298.404 977.503,157.219 978.034,850.483 978.565,1623.76 979.096,1635.55 979.626,873.018 980.157,165.978 980.688,283.855 981.219,1098.37 981.75,1723.14
982.28,1478.27 982.811,630.237 983.342,101.896 983.873,468.207 984.404,1330.54 984.934,1750.47 985.465,1271.03 985.996,413.944 986.527,111.935 987.058,693.655
987.588,1526.06 988.119,1703.29 988.65,1032.49 989.181,243.637 989.712,195.19 990.242,939.871 990.773,1667.29 991.304,1585.85 991.835,784.174 992.366,134.672
992.896,344.155 993.427,1184.66 993.958,1741.52 994.489,1408.74 995.02,548.466 995.55,96.8739 996.081,545.4 996.612,1405.94 997.143,1742.03 997.674,1187.92
998.204,346.617 998.735,133.649 999.266,780.78 999.797,1583.78 1000.33,1668.8 1000.86,943.316 1001.39,196.826 1001.92,241.683 1002.45,1029.07 1002.98,1702.13
1003.51,1528.41 1004.04,696.966 1004.57,112.596 1005.1,411.235 1005.64,1267.89 1006.17,1750.33 1006.7,1333.54 1007.23,471.087 1007.76,101.524 1008.29,627.019
1008.82,1475.71 1009.35,1724.03 1009.88,1101.74 1010.41,286.043 1010.94,164.606 1011.47,869.579 1012.01,1633.79 1012.54,1625.59 1013.07,853.916 1013.6,158.518
1014.13,296.155 1014.66,1117.05 1015.19,1727.87 1015.72,1463.91 1016.25,612.413 1016.78,100.009 1017.31,484.312 1017.84,1347.11 1018.37,1749.48 1018.91,1253.55
1019.44,399.003 1019.97,115.791 1020.5,712.111 1021.03,1539.03 1021.56,1696.66 1022.09,1013.47 1022.62,232.928 1023.15,204.442 1023.68,959.015 1024.21,1675.5
1024.74,1574.19 1025.28,765.335 1025.81,129.16 1026.34,357.969 1026.87,1202.76 1027.4,1744.21 1027.93,1393.09 1028.46,531.503 1028.99,97.0557 1029.52,562.529
1030.05,1421.38 1030.58,1738.98 1031.11,1169.7 1031.64,333.058 1032.18,139.509 1032.71,799.681 1033.24,1595.15 1033.77,1660.27 1034.3,924.163 1034.83,187.894
1035.36,252.692 1035.89,1048.04 1036.42,1708.42 1036.95,1515.18 1037.48,678.61 1038.01,109.097 1038.55,426.401 1039.08,1285.22 1039.61,1750.96 1040.14,1316.79
1040.67,455.182 1041.25,118.339 1041.83,803.697 1042.4,1641.7 1042.98,1569.14 1043.56,678.061 1044.14,99.0381 1044.72,566.704 1045.3,1487.71 1045.88,1693.52
1046.46,923.014 1047.03,153.776 1047.61,361.779 1048.19,1283.12 1048.77,1748.84 1049.35,1168.06 1049.93,277.641 1050.51,207.312 1051.09,1046.3 1051.66,1730.13
1052.24,1391.2 1052.82,459.515 1053.4,117.167 1053.98,798.508 1054.56,1639.08 1055.14,1572.41 1055.71,683.076 1056.29,99.4342 1056.87,561.979 1057.45,1483.85
1058.03,1695.43 1058.61,928.261 1059.19,155.704 1059.77,357.942 1060.34,1278.38 1060.92,1749.21 1061.5,1173.06 1062.08,280.928 1062.66,204.707 1063.24,1041.11
1063.82,1728.94 1064.4,1395.52 1064.97,463.866 1065.55,116.028 1066.13,793.324 1066.71,1636.43 1067.29,1575.65 1067.87,688.101 1068.45,99.8634 1069.03,557.268
1069.6,1479.98 1070.18,1697.3 1070.76,933.508 1071.34,157.663 1071.92,354.127 1072.5,1273.63 1073.08,1749.56 1073.66,1178.06 1074.23,284.241 1074.81,202.131
1075.39,1035.91 1075.97,1727.72 1076.55,1399.82 1077.13,468.236 1077.71,114.922 1078.28,788.145 1078.86,1633.75 1079.44,1578.87 1080.02,693.135 1080.6,100.326
1081.18,552.572 1081.76,1476.08 1082.34,1699.15 1082.91,938.755 1083.49,159.653 1084.07,350.335 1084.65,1268.87 1085.23,1749.87 1085.81,1183.05 1086.39,287.58
1086.97,199.584 1087.54,1030.71 1088.12,1726.46 1088.7,1404.1 1089.28,472.624 1089.86,113.848 1090.44,782.972 1091.02,1631.04 1091.6,1582.06 1092.17,698.179
1092.75,100.822 1093.33,547.891 1093.91,1472.17 1094.49,1700.97 1095.07,944.001 1095.65,161.674 1096.23,346.567 1096.8,1264.1 1097.38,1750.14 1097.96,1188.03
1098.54,290.944 1099.12,197.066 1099.7,1025.5 1100.28,1725.17 1100.85,1408.36 1101.43,477.03 1102.01,112.807 1102.59,777.804 1103.17,1628.3 1103.75,1585.23
1104.33,703.231 1104.91,101.35 1105.48,543.225 1106.06,1468.23 1106.64,1702.75 1107.22,949.247 1107.8,163.725 1108.38,342.821 1108.96,1259.31 1109.54,1750.38
1110.11,1193 1110.69,294.334 1111.27,194.577 1111.85,1020.29 1112.43,1723.85 1113.01,1412.61 1113.59,481.454 1114.17,111.798 1114.74,772.642 1115.33,1628.64
1115.91,1581.29 1116.49,691.308 1117.07,99.6517 1117.65,564.834 1118.24,1490.49 1118.82,1689.91 1119.4,907.444 1119.98,146.217 1120.57,382.131 1121.15,1312.74
1121.73,1745 1122.31,1124.74 1122.89,247.146 1123.48,237.303 1124.06,1107.83 1124.64,1742.7 1125.22,1328 1125.8,395.386 1126.39,140.473 1126.97,890.066
1127.55,1683.18 1128.13,1503.03 1128.72,580.574 1129.3,98.4097 1129.88,674.676 1130.46,1570.59 1131.04,1637.59 1131.63,789.768 1132.21,114.052 1132.79,476.714
1133.37,1412.82 1133.95,1722.27 1134.54,1008.34 1135.12,186.307 1135.7,310.016 1136.28,1220.87 1136.87,1751.15 1137.45,1221.03 1138.03,310.125 1138.61,186.234
1139.19,1008.18 1139.78,1722.22 1140.36,1412.95 1140.94,476.851 1141.52,114.019 1142.11,789.607 1142.69,1637.5 1143.27,1570.69 1143.85,674.831 1144.43,98.4196
1145.02,580.426 1145.6,1502.91 1146.18,1683.24 1146.76,890.228 1147.34,140.526 1147.93,395.261 1148.51,1327.86 1149.09,1742.72 1149.67,1107.99 1150.26,237.394
1150.84,247.053 1151.42,1124.58 1152,1744.98 1152.58,1312.89 1153.17,382.254 1153.75,146.161 1154.33,907.282 1154.91,1689.85 1155.49,1490.6 1156.08,564.981
1156.66,99.6384 1157.24,691.152 1157.82,1581.19 1158.41,1628.72 1158.99,772.802 1159.57,110.736 1160.15,491.298 1160.73,1426.6 1161.32,1717.58 1161.9,991.192
1162.48,178.678 1163.06,321.689 1163.64,1236.89 1164.23,1750.97 1164.81,1204.89 1165.39,298.715 1165.97,194.18 1166.56,1025.3 1167.14,1726.56 1167.72,1398.95
1168.3,462.458 1168.88,117.683 1169.47,806.63 1170.05,1646.06 1170.63,1559.82 1171.21,658.462 1171.79,97.5456 1172.38,596.168 1172.96,1515.09 1173.54,1676.24
1174.12,873.027 1174.71,135.174 1175.29,408.62 1175.87,1342.8 1176.45,1740.09 1177.03,1091.16 1177.62,227.939 1178.2,257.096 1178.78,1141.25 1179.36,1746.9
1179.95,1297.6 1180.53,369.357 1181.11,152.187 1181.69,924.505 1182.27,1696.19 1182.86,1477.93 1183.44,549.543 1184.02,101.225 1184.6,707.729 1185.18,1591.51
1185.77,1619.55 1186.35,755.902 1186.93,107.772 1187.51,506.07 1188.1,1440.17 1188.68,1712.55 1189.26,974.012 1189.9,133.948 1190.55,498.325 1191.19,1511.66
1191.83,1629.13 1192.48,671.661 1193.12,98.8963 1193.77,784.009 1194.41,1682.55 1195.05,1424.71 1195.7,403.559 1196.34,175.829 1196.98,1088.69 1197.63,1750.5
1198.27,1152.34 1198.91,206.093 1199.56,354.306 1200.2,1371.03 1200.85,1706.28 1201.49,848.975 1202.13,106.063 1202.78,610.103 1203.42,1592.69 1204.06,1555.89
1204.71,555.796 1205.35,117.044 1205.99,908.503 1206.64,1723.6 1207.28,1319.74 1207.93,312.592 1208.57,237.548 1209.21,1209.01 1209.86,1746 1210.5,1029.88
1211.14,152.369 1211.79,451.217 1212.43,1470.83 1213.07,1656.83 1213.72,725.653 1214.36,96.8739 1215.01,729.055 1215.65,1658.44 1216.29,1468.2 1216.94,448.348
1217.58,153.637 1218.22,1033.35 1218.87,1746.38 1219.51,1205.72 1220.15,235.6 1220.8,314.956 1221.44,1322.81 1222.09,1722.7 1222.73,905.002 1223.37,116.283
1224.02,558.936 1224.66,1558.14 1225.3,1590.62 1225.95,606.865 1226.59,106.591 1227.23,852.463 1227.88,1707.41 1228.52,1368.08 1229.17,351.772 1229.81,207.839
1230.45,1155.7 1231.1,1750.36 1231.74,1085.26 1232.38,174.343 1233.03,406.286 1233.67,1427.49 1234.31,1681.15 1234.96,780.558 1235.6,98.6588 1236.25,674.998
1236.89,1630.96 1237.53,1509.19 1238.18,495.326 1238.82,134.992 1239.46,977.507 1240.11,1738.47 1240.75,1257.8 1241.39,268.275 1242.04,278.41 1242.68,1272.76
1243.33,1735.45 1243.97,961.116 1244.61,130.221 1245.26,509.449 1245.9,1520.67 1246.54,1622.29 1247.19,659.394 1247.83,99.8998 1248.47,796.753 1249.12,1687.61
1249.76,1414.37 1250.41,393.586 1251.05,181.427 1251.69,1101.33 1252.34,1750.91 1252.98,1139.9 1253.62,199.767 1254.27,363.737 1254.91,1381.84 1255.56,1701.99
1256.2,836.124 1256.84,104.243 1257.49,622.088 1258.13,1600.21 1258.77,1547.48 1259.42,544.278 1260.06,119.977 1260.7,921.416 1261.35,1726.81 1261.99,1308.35
1262.64,303.969 1263.28,244.835 1263.92,1221.1 1264.57,1744.46 1265.21,1017.06 1265.85,147.813 1266.5,461.871 1267.14,1480.46 1267.78,1650.75 1268.43,713.141
1269.07,97.0021 1269.72,741.628 1270.36,1664.3 1271,1458.41 1271.65,437.841 1272.26,139.822 1272.87,939.762 1273.48,1717.65 1274.1,1384.35 1274.71,406.522
1275.32,153.242 1275.93,979.131 1276.55,1727.85 1277.16,1351.1 1277.77,376.377 1278.38,168.41 1279,1018.37 1279.61,1736.22 1280.22,1316.88 1280.83,347.476
1281.44,185.294 1282.06,1057.4 1282.67,1742.75 1283.28,1281.76 1283.89,319.882 1284.51,203.854 1285.12,1096.13 1285.73,1747.42 1286.34,1245.84 1286.96,293.66
1287.57,224.049 1288.18,1134.47 1288.79,1750.22 1289.41,1209.18 1290.02,268.869 1290.63,245.832 1291.24,1172.33 1291.86,1751.15 1292.47,1171.88 1293.08,245.564
1293.69,269.155 1294.31,1209.62 1294.92,1750.2 1295.53,1134.02 1296.14,223.8 1296.76,293.963 1297.37,1246.27 1297.98,1747.38 1298.59,1095.67 1299.2,203.624
1299.82,320.202 1300.43,1282.19 1301.04,1742.68 1301.65,1056.94 1302.27,185.083 1302.88,347.811 1303.49,1317.29 1304.1,1736.13 1304.72,1017.91 1305.33,168.22
1305.94,376.728 1306.55,1351.5 1307.17,1727.74 1307.78,978.664 1308.39,153.072 1309,406.887 1309.62,1384.74 1310.23,1717.52 1310.84,939.294 1311.45,139.674
1312.07,438.22 1312.68,1416.93 1313.29,1705.5 1313.9,899.889 1314.52,128.056 1315.13,470.655 1315.74,1448.01 1316.35,1691.7 1316.96,860.539 1317.58,118.244
1318.19,504.12 1318.8,1477.89 1319.41,1676.17 1320.03,821.333 1320.64,110.261 1321.25,538.537 1321.86,1506.52 1322.48,1658.92 1323.09,782.36 1323.7,104.125
1324.31,573.829 1324.93,1533.83 1325.54,1640.01 1326.15,743.708 1326.76,99.85 1327.38,609.916 1327.99,1559.75 1328.6,1619.48 1329.21,705.466 1329.83,97.4455
1330.44,646.716 1331.05,1584.23 1331.66,1597.36 1332.28,667.72 1332.89,96.9169 1333.5,684.145 1334.11,1607.21 1334.72,1573.72 1335.34,630.555 1335.95,98.2655
1336.56,722.118 1337.17,1628.65 1337.79,1548.6 1338.4,594.057 1339.01,101.488 1339.62,760.55 1340.24,1648.48 1340.85,1522.07 1341.46,558.307 1342.07,106.578
1342.69,799.353 1343.3,1666.66 1343.91,1494.17 1344.52,523.388 1345.14,113.523 1345.75,838.439 1346.36,1683.17 1346.97,1464.99 1347.59,489.377 1348.2,122.307
1348.81,877.719 1349.42,1697.95 1350.03,1434.57 1350.67,422.608 1351.31,161.198 1351.95,1041.65 1352.58,1746.01 1353.22,1219.91 1353.86,250.881 1354.49,289.463
1355.13,1277.91 1355.77,1736.6 1356.4,978.925 1357.04,139.048 1357.68,474.189 1358.31,1482.67 1358.95,1654.89 1359.59,733.051 1360.22,97.0598 1360.86,698.94
1361.5,1637.73 1362.13,1508.15 1362.77,504.167 1363.41,128.652 1364.04,943.717 1364.68,1729.28 1365.32,1309.43 1365.95,312.641 1366.59,231.015 1367.23,1186.74
1367.86,1749.19 1368.5,1076.42 1369.14,175.514 1369.77,395.039 1370.41,1406.39 1371.05,1695.66 1371.69,829.848 1372.32,104.987 1372.96,606.13 1373.6,1583.11
1374.23,1573.48 1374.87,591.654 1375.51,107.335 1376.14,845.505 1376.78,1701.19 1377.42,1393.51 1378.05,383.032 1378.69,182.349 1379.33,1091.87 1379.96,1750.12
1380.6,1171.77 1381.24,222.546 1381.87,323.355 1382.51,1323.29 1383.15,1725.54 1383.78,927.975 1384.42,124.475 1385.06,517.807 1385.69,1519.19 1386.33,1629.65
1386.97,683.832 1387.6,97.545 1388.24,748.402 1388.88,1662.13 1389.51,1470.96 1390.15,461.06 1390.79,144.153 1391.42,994.623 1392.06,1739.39 1392.7,1263.61
1393.34,279.48 1393.97,260.151 1394.61,1234.56 1395.25,1744.11 1395.88,1026.05 1396.52,155.249 1397.16,435.219 1397.79,1446.87 1398.43,1675.85 1399.07,779.4
1399.7,99.422 1400.34,653.778 1400.98,1612.65 1401.61,1540.69 1402.25,545.622 1402.89,116.965 1403.52,896.382 1404.16,1717.16 1404.8,1350.67 1405.43,345.512
1406.07,206.318 1406.71,1141.45 1407.34,1751.09 1407.98,1122.68 1408.62,196.876 1409.25,359.53 1409.89,1367.16 1410.53,1711.44 1411.16,877.01 1411.8,112.939
1412.44,562.969 1413.07,1553.45 1413.71,1601.72 1414.35,635.526 1414.99,101.17 1415.62,798.532 1416.26,1683.73 1416.9,1431.7 1417.53,419.71 1418.17,162.616
1418.81,1045.26 1419.44,1746.41 1420.08,1216.5 1420.72,248.767 1421.35,291.81 1421.99,1281.2 1422.63,1735.91 1423.26,975.284 1423.9,137.905 1424.54,477.256
1425.17,1485.36 1425.81,1653.18 1426.45,729.502 1427.08,96.9901 1427.72,702.453 1428.36,1639.57 1428.99,1505.56 1429.63,501.028 1430.27,129.662 1430.9,947.364
1431.54,1730.11 1432.08,1449.63 1432.61,587.024 1433.15,97.6348 1433.69,523.462 1434.22,1392.9 1434.76,1743.04 1435.29,1186.1 1435.83,338.897 1436.37,139.708
1436.9,809.139 1437.44,1605.79 1437.97,1647.37 1438.51,887.825 1439.05,168.357 1439.58,285.78 1440.12,1110.05 1440.65,1728.28 1441.19,1455.78 1441.73,594.345
1442.26,98.0169 1442.8,516.482 1443.33,1386.29 1443.87,1744.12 1444.41,1193.68 1444.94,344.578 1445.48,137.204 1446.01,801.223 1446.55,1601.23 1447.09,1651.22
1447.62,895.818 1448.16,171.645 1448.69,280.722 1449.23,1102.25 1449.77,1726.37 1450.3,1461.88 1450.84,601.696 1451.37,98.4762 1451.91,509.541 1452.45,1379.63
1452.98,1745.12 1453.52,1201.23 1454.05,350.314 1454.59,134.773 1455.13,793.318 1455.66,1596.61 1456.2,1654.99 1456.73,903.814 1457.27,175.004 1457.81,275.724
1458.34,1094.43 1458.88,1724.39 1459.41,1467.93 1459.95,609.078 1460.49,99.0128 1461.02,502.637 1461.56,1372.94 1462.09,1746.05 1462.63,1208.75 1463.17,356.103
1463.7,132.417 1464.24,785.425 1464.77,1591.92 1465.31,1658.7 1465.85,911.811 1466.38,178.433 1466.92,270.786 1467.45,1086.59 1467.99,1722.34 1468.53,1473.93
1469.06,616.49 1469.6,99.6265 1470.13,495.774 1470.67,1366.2 1471.21,1746.9 1471.74,1216.25 1472.28,361.945 1472.81,130.134 1473.35,777.546 1473.89,1587.17
1474.42,1662.34 1474.96,919.81 1475.49,181.931 1476.03,265.91 1476.57,1078.74 1477.1,1720.21 1477.64,1479.88 1478.18,623.93 1478.71,100.317 1479.25,488.95
1479.78,1359.42 1480.32,1747.67 1480.86,1223.72 1481.39,367.84 1481.93,127.925 1482.46,769.68 1483,1582.36 1483.54,1665.91 1484.07,927.81 1484.61,185.499
1485.14,261.095 1485.68,1070.87 1486.22,1718 1486.75,1485.78 1487.29,631.398 1487.82,101.085 1488.36,482.167 1488.9,1352.59 1489.43,1748.36 1489.97,1231.16
1490.5,373.787 1491.04,125.791 1491.58,761.828 1492.11,1577.48 1492.65,1669.42 1493.18,935.809 1493.72,189.136 1494.26,256.342 1494.79,1062.99 1495.33,1715.72
1495.86,1491.63 1496.4,638.894 1496.94,101.93 1497.47,475.426 1498.01,1345.73 1498.54,1748.98 1499.08,1238.57 1499.62,379.786 1500.15,123.732 1500.81,955.242
1501.46,1737.79 1502.12,1244.43 1502.77,248.693 1503.42,311.782 1504.08,1334.78 1504.73,1713.74 1505.39,854.5 1506.04,104.247 1506.7,639.29 1507.35,1620.74
1508.01,1509.8 1508.66,480.408 1509.31,146.534 1509.97,1031.66 1510.62,1748 1511.28,1172.43 1511.93,207.364 1512.59,365.921 1513.24,1399.5 1513.89,1687.57
1514.55,778.467 1515.2,97.5636 1515.86,712.435 1516.51,1659.03 1517.17,1453.2 1517.82,417.661 1518.48,176.018 1519.13,1107.14 1519.78,1751.14 1520.44,1098.29
1521.09,172.195 1521.75,424.857 1522.4,1460.14 1523.06,1654.83 1523.71,703.686 1524.37,97.9837 1525.02,787.398 1525.67,1691.01 1526.33,1392.06 1526.98,359.266
1527.64,211.932 1528.29,1181.06 1528.95,1747.16 1529.6,1022.66 1530.26,143.488 1530.91,488.084 1531.56,1516.16 1532.22,1615.82 1532.87,630.798 1533.53,105.504
1534.18,863.535 1534.84,1716.39 1535.49,1326.89 1536.15,305.725 1536.8,253.966 1537.45,1252.76 1538.11,1736.11 1538.76,946.183 1539.42,121.489 1540.07,555.058
1540.73,1567.1 1541.38,1570.86 1542.04,560.431 1542.69,120.059 1543.34,940.192 1544,1734.96 1544.65,1258.25 1545.31,257.498 1545.96,301.76 1546.62,1321.64
1547.27,1718.08 1547.92,869.513 1548.58,106.389 1549.23,625.202 1549.89,1612.51 1550.54,1520.33 1551.2,493.189 1551.85,141.525 1552.51,1016.71 1553.16,1746.55
1553.81,1186.75 1554.47,215 1555.12,354.902 1555.78,1387.1 1556.43,1693.23 1557.09,793.312 1557.74,98.316 1558.4,697.915 1559.05,1652.01 1559.7,1464.69
1560.36,429.649 1561.01,169.716 1561.67,1092.43 1562.32,1751.08 1562.98,1112.98 1563.63,178.596 1564.29,412.935 1564.94,1448.58 1565.59,1661.76 1566.25,718.234
1566.9,97.3403 1567.56,772.572 1568.21,1685.24 1568.87,1404.39 1569.52,370.359 1570.18,204.39 1570.83,1166.7 1571.48,1748.5 1572.14,1037.59 1572.79,148.6
1573.45,475.362 1574.1,1505.55 1574.76,1623.96 1575.41,644.925 1576.06,103.47 1576.72,848.53 1577.37,1711.94 1578.03,1339.97 1578.68,315.828 1579.34,245.25
1579.99,1238.89 1580.65,1738.84 1581.3,961.23 1581.95,125.268 1582.61,541.644 1583.26,1557.53 1583.92,1580.13 1584.52,661.198 1585.12,96.8986 1585.72,648.683
1586.32,1571.99 1586.92,1620.93 1587.51,729.469 1588.11,100.517 1588.71,582.766 1589.31,1525.48 1589.91,1656.59 1590.51,799.176 1591.11,110.212 1591.71,519.368
1592.31,1474.54 1592.91,1686.84 1593.51,869.803 1594.11,125.91 1594.71,458.954 1595.31,1419.53 1595.91,1711.47 1596.51,940.83 1597.11,147.496 1597.71,401.971
1598.31,1360.87 1598.91,1730.29 1599.5,1011.73 1600.1,174.811 1600.7,348.84 1601.3,1298.99 1601.9,1743.15 1602.5,1081.99 1603.1,207.654 1603.7,299.952
1604.3,1234.34 1604.9,1749.98 1605.5,1151.08 1606.1,245.781 1606.7,255.669 1607.3,1167.4 1607.9,1750.71 1608.5,1218.49 1609.1,288.912 1609.7,216.316
1610.3,1098.67 1610.9,1745.34 1611.49,1283.74 1612.09,336.729 1612.69,182.184 1613.29,1028.64 1613.89,1733.91 1614.49,1346.33 1615.09,388.878 1615.69,153.525
1616.29,957.846 1616.89,1716.51 1617.49,1405.8 1618.09,444.975 1618.69,130.55 1619.29,886.801 1619.89,1693.26 1620.49,1461.72 1621.09,504.607 1621.69,113.429
1622.29,816.03 1622.89,1664.34 1623.48,1513.67 1624.08,567.332 1624.68,102.288 1625.28,746.056 1625.88,1629.95 1626.48,1561.27 1627.08,632.689 1627.68,97.2099
1628.28,677.395 1628.88,1590.36 1629.48,1604.17 1630.08,700.195 1630.68,98.2313 1631.28,610.553 1631.88,1545.85 1632.48,1642.05 1633.08,769.353 1633.68,105.345
1634.28,546.024 1634.87,1496.75 1635.47,1674.64 1636.07,839.651 1636.67,118.498 1637.27,484.283 1637.87,1443.43 1638.47,1701.68 1639.07,910.572 1639.67,137.595
1640.27,425.787 1640.87,1386.27 1641.47,1722.99 1642.07,981.592 1642.67,162.493 1643.27,370.966 1643.87,1325.7 1644.47,1738.41 1645.07,1052.19 1645.67,193.009
1646.27,320.226 1646.86,1262.18 1647.46,1747.82 1648.06,1121.84 1648.66,228.918 1649.26,273.94 1649.86,1196.15 1650.46,1751.14 1651.06,1190.03 1651.66,269.955
1652.26,232.449 1652.86,1128.12 1653.46,1748.37 1654.06,1256.25 1654.66,315.818 1655.26,196.061 1655.86,1058.58 1656.46,1739.52 1657.06,1320.03 1657.66,366.168
1658.26,165.044 1658.85,988.051 1659.45,1724.64 1660.05,1380.89 1660.65,420.633 1661.21,118.316 1661.76,761.416 1662.31,1595.31 1662.87,1641.49 1663.42,845.769
1663.98,141.86 1664.53,355.74 1665.08,1236.44 1665.64,1750.53 1666.19,1294.77 1666.74,403.954 1667.3,123.383 1667.85,782.283 1668.4,1607.49 1668.96,1630.69
1669.51,824.663 1670.07,135.212 1670.62,371.352 1671.17,1255.99 1671.73,1751.08 1672.28,1275.67 1672.83,387.619 1673.39,128.979 1673.94,803.244 1674.49,1619.22
1675.05,1619.43 1675.6,803.621 1676.16,129.084 1676.71,387.328 1677.26,1275.33 1677.82,1751.09 1678.37,1256.34 1678.92,371.636 1679.48,135.097 1680.03,824.284
1680.59,1630.49 1681.14,1607.7 1681.69,782.659 1682.25,123.479 1682.8,403.658 1683.35,1294.43 1683.91,1750.55 1684.46,1236.79 1685.01,356.018 1685.57,141.736
1686.12,845.39 1686.68,1641.3 1687.23,1595.53 1687.78,761.79 1688.34,118.402 1688.89,420.33 1689.44,1313.29 1690,1749.46 1690.55,1217.04 1691.11,340.773
1691.66,148.89 1692.21,866.547 1692.77,1651.63 1693.32,1582.92 1693.87,741.028 1694.43,113.856 1694.98,437.334 1695.53,1331.89 1696.09,1747.83 1696.64,1197.09
1697.2,325.913 1697.75,156.554 1698.3,887.743 1698.86,1661.49 1699.41,1569.87 1699.96,720.386 1700.52,109.843 1701.07,454.659 1701.63,1350.22 1702.18,1745.66
1702.73,1176.96 1703.29,311.447 1703.84,164.724 1704.39,908.962 1704.95,1670.86 1705.5,1556.39 1706.05,699.879 1706.61,106.367 1707.16,472.293 1707.72,1368.27
1708.27,1742.95 1708.82,1156.67 1709.38,297.385 1709.93,173.394 1710.48,930.191 1711.04,1679.74 1711.59,1542.5 1712.15,679.519 1712.7,103.429 1713.25,490.224
1713.81,1386.03 1714.36,1739.7 1714.91,1136.22 1715.47,283.735 1716.02,182.559 1716.57,951.417 1717.13,1688.12 1717.68,1528.2 1718.24,659.32 1718.79,101.031
1719.34,508.441 1719.9,1403.49 1720.45,1735.91 1721,1115.63 1721.56,270.507 1722.11,192.212 1722.66,972.624 1723.22,1695.99 1723.77,1513.51 1724.33,639.296
1724.88,99.1764 1725.43,526.932 1725.99,1420.63 1726.54,1731.59 1727.09,1094.92 1727.65,257.709 1728.2,202.347 1728.76,993.799 1729.31,1703.36 1729.86,1498.42
1730.42,619.459 1730.97,97.8646 1731.52,545.685 1732.08,1447.01 1732.65,1720.45 1733.21,1037.73 1733.77,218.484 1734.33,246.254 1734.89,1087.7 1735.45,1732.63
1736.01,1406.78 1736.57,501.34 1737.13,103.339 1737.69,690.593 1738.25,1558.08 1738.81,1664.34 1739.38,881.804 1739.94,149.943 1740.5,347.387 1741.06,1237.09
1741.62,1750.93 1742.18,1272.01 1742.74,375.304 1743.3,137.342 1743.86,843.812 1744.42,1646.57 1744.98,1581.86 1745.55,727.379 1746.11,108.963 1746.67,469.05
1747.23,1375.34 1747.79,1739.79 1748.35,1124.86 1748.91,268.805 1749.47,199.353 1750.03,999.886 1750.59,1709.33 1751.15,1475.96 1751.71,579.955 1752.28,97.0028
1752.84,606.912 1753.4,1497.51 1753.96,1699.6 1754.52,970.558 1755.08,185.635 1755.64,287.166 1756.2,1153.26 1756.76,1744.13 1757.32,1350.41 1757.88,444.781
1758.44,114.488 1759.01,756.064 1759.57,1599.27 1760.13,1631.8 1760.69,814.597 1761.25,128.754 1761.81,397.653 1762.37,1298.47 1762.93,1749.73 1763.49,1209.68
1764.05,326.671 1764.61,160.795 1765.18,911.195 1765.74,1676.98 1766.3,1538.79 1766.86,662.532 1767.42,100.188 1767.98,526.881 1768.54,1430.35 1769.1,1725.93
1769.66,1058.78 1770.22,229.828 1770.78,234.277 1771.34,1066.78 1771.91,1727.89 1772.47,1423.9 1773.03,519.777 1773.59,100.955 1774.15,670.249 1774.71,1544.2
1775.27,1673.58 1775.83,903.076 1776.39,157.702 1776.95,332.317 1777.51,1217.29 1778.07,1750.17 1778.64,1291.21 1779.2,391.414 1779.76,131.026 1780.32,822.652
1780.88,1635.97 1781.44,1594.55 1782,748.12 1782.56,112.86 1783.12,451.424 1783.68,1357.35 1784.24,1743.03 1784.81,1145.44 1785.37,282.014 1785.93,189.33
1786.49,978.664 1787.05,1702.39 1787.61,1491.63 1788.17,599.426 1788.73,96.8983 1789.29,587.357 1789.85,1481.98 1790.41,1706.74 1790.97,991.795 1791.54,195.473
1792.1,273.794 1792.66,1132.73 1793.22,1741.09 1793.78,1368.51 1794.34,462.29 1794.9,110.386 1795.46,735.277 1796.02,1586.75 1796.58,1642.58 1797.14,835.733
1797.7,134.87 1798.27,381.408 1798.83,1279.37 1799.39,1750.7 1799.95,1229.56 1800.51,341.592 1801.07,152.843 1801.63,889.916 1802.19,1667.92 1802.75,1552.83
1803.31,682.813 1803.94,97.1237 1804.57,722.126 1805.2,1642.72 1805.83,1511.01 1806.46,519.844 1807.08,120.439 1807.71,897.589 1808.34,1713.43 1808.97,1373.44
1809.6,375.42 1810.23,180.626 1810.86,1074.26 1811.48,1747.91 1812.11,1215.24 1812.74,256.168 1813.37,274.922 1814,1244.04 1814.63,1744.59 1815.26,1043.68
1815.88,167.559 1816.51,399 1817.14,1399.14 1817.77,1703.62 1818.4,866.631 1819.03,113.658 1819.66,547.169 1820.28,1532.44 1820.91,1626.88 1821.54,692.213
1822.17,96.9399 1822.8,712.628 1823.43,1637.81 1824.06,1517.88 1824.68,528.431 1825.31,118.171 1825.94,887.786 1826.57,1710.44 1827.2,1381.64 1827.83,382.8
1828.46,176.377 1829.08,1064.61 1829.71,1746.98 1830.34,1224.4 1830.97,262.002 1831.6,268.887 1832.23,1234.98 1832.86,1745.76 1833.48,1053.38 1834.11,171.58
1834.74,391.457 1835.37,1391.08 1836,1706.84 1836.63,876.421 1837.26,115.681 1837.88,538.463 1838.51,1525.75 1839.14,1632 1839.77,701.646 1840.4,96.8724
1841.03,703.159 1841.66,1632.81 1842.28,1524.67 1842.91,537.074 1843.54,116.016 1844.17,877.989 1844.8,1707.35 1845.43,1389.78 1846.06,390.257 1846.68,172.233
1847.31,1054.93 1847.94,1745.94 1848.57,1233.52 1849.2,267.93 1849.83,262.945 1850.46,1225.86 1851.08,1746.82 1851.71,1063.06 1852.34,175.707 1852.97,383.989
1853.6,1382.95 1854.23,1709.95 1854.86,886.217 1855.48,117.818 1856.11,529.811 1856.74,1518.97 1857.37,1637.02 1858,711.11 1858.63,96.9213 1859.26,693.721
1859.88,1627.7 1860.51,1531.37 1861.14,545.771 1861.77,113.974 1862.4,868.197 1863.03,1704.14 1863.66,1397.85 1864.28,397.788 1864.91,168.195 1865.54,1045.23
1866.17,1744.78 1866.8,1242.6 1867.43,273.95 1868.06,257.096 1868.68,1216.71 1869.31,1747.77 1869.94,1072.72 1870.57,179.939 1871.2,376.597 1871.83,1374.75
1872.46,1712.95 1873.08,896.019 1873.71,120.069 1874.34,521.215 1874.97,1512.12 1875.6,1641.94 1876.23,720.604 1876.86,97.0865 1877.48,684.316 1878.11,1622.5
1878.74,1537.99 1879.37,554.522 1880,112.047 1880.63,858.414 1881.26,1700.83 1881.88,1405.86 1882.51,405.393 1883.14,164.263 1883.77,1035.52 1884.34,1723.78
1884.91,1425.34 1885.48,508.387 1886.05,104.208 1886.62,711.45 1887.19,1580.94 1887.76,1639.95 1888.33,815.686 1888.9,125.066 1889.47,420.134 1890.04,1336.85
1890.6,1744.24 1891.17,1139.68 1891.74,269.045 1892.31,206.468 1892.88,1029.15 1893.45,1722.12 1894.02,1430.43 1894.59,513.957 1895.16,103.379 1895.73,705.244
1896.3,1577.01 1896.87,1643.15 1897.44,822.062 1898.01,126.754 1898.58,415.051 1899.15,1331.27 1899.72,1745.04 1900.29,1145.88 1900.86,272.99 1901.43,203.292
1902,1022.77 1902.57,1720.41 1903.14,1435.5 1903.71,519.552 1904.27,102.599 1904.84,699.052 1905.41,1573.05 1905.98,1646.3 1906.55,828.443 1907.12,128.49
1907.69,410 1908.26,1325.66 1908.83,1745.8 1909.4,1152.06 1909.97,276.975 1910.54,200.159 1911.11,1016.39 1911.68,1718.65 1912.25,1440.54 1912.82,525.171
1913.39,101.869 1913.96,692.873 1914.53,1569.04 1915.1,1649.41 1915.67,834.831 1916.24,130.274 1916.81,404.979 1917.38,1320.03 1917.94,1746.5 1918.51,1158.23
1919.08,280.999 1919.65,197.071 1920.22,1010 1920.79,1716.84 1921.36,1445.54 1921.93,530.815 1922.5,101.188 1923.07,686.708 1923.64,1565 1924.21,1652.48
1924.78,841.224 1925.35,132.106 1925.92,399.99 1926.49,1314.38 1927.06,1747.16 1927.63,1164.39 1928.2,285.061 1928.77,194.026 1929.34,1003.6 1929.91,1714.98
1930.48,1450.52 1931.05,536.482 1931.61,100.557 1932.18,680.558 1932.75,1560.92 1933.32,1655.5 1933.89,847.622 1934.46,133.986 1935.03,395.033 1935.6,1308.7
1936.17,1747.76 1936.74,1170.54 1937.31,289.162 1937.88,191.026 1938.45,997.2 1939.02,1713.08 1939.59,1455.46 1940.16,542.172 1940.73,99.9758 1941.3,674.422
1941.87,1556.8 1942.44,1658.48 1943.01,854.025 1943.58,135.914 1944.15,390.107 1944.71,1303 1945.28,1748.32 1945.85,1176.66 1946.42,293.302 1946.99,188.069
1947.56,990.795 1948.13,1711.13 1948.7,1460.37 1949.27,547.886 1949.84,99.4444 1950.41,668.301 1950.98,1552.64 1951.55,1661.42 1952.12,860.432 1952.69,137.889
1953.26,385.214 1953.83,1297.27 1954.4,1748.83 1954.97,1182.78 1955.54,297.48 1956.11,185.158 1956.68,984.386 1957.33,1741.93 1957.98,1224.63 1958.63,238.776
1959.28,320.952 1959.94,1343.08 1960.59,1712.03 1961.24,852.745 1961.89,104.537 1962.55,633.594 1963.2,1615.31 1963.85,1519.54 1964.5,495.56 1965.15,139.381
1965.81,1006.16 1966.46,1744.9 1967.11,1204.17 1967.76,226.777 1968.41,336.117 1969.07,1361.77 1969.72,1705.12 1970.37,831.001 1971.02,101.855 1971.68,654.154
1972.33,1627.06 1972.98,1504.17 1973.63,477.019 1974.28,146.569 1974.94,1027.87 1975.59,1747.29 1976.24,1183.52 1976.89,215.265 1977.54,351.693 1978.2,1380.16
1978.85,1697.66 1979.5,809.321 1980.15,99.7474 1980.8,674.903 1981.46,1638.33 1982.11,1488.39 1982.76,458.79 1983.41,154.3 1984.07,1049.51 1984.72,1749.11
1985.37,1162.68 1986.02,204.248 1986.67,367.668 1987.33,1398.23 1987.98,1689.66 1988.63,787.722 1989.28,98.2147 1989.93,695.826 1990.59,1649.1 1991.24,1472.22
1991.89,440.886 1992.54,162.569 1993.2,1071.07 1993.85,1750.36 1994.5,1141.67 1995.15,193.733 1995.8,384.031 1996.46,1415.96 1997.11,1681.12 1997.76,766.218
1998.41,97.2584 1999.06,716.908 1999.72,1659.36 2000.37,1455.67 2001.02,423.318 2001.67,171.368 2002.33,1092.52 2002.98,1751.03 2003.63,1120.52 2004.28,183.727
2004.93,400.772 2005.59,1433.36 2006.24,1672.06 2006.89,744.824 2007.54,96.8792 2008.19,738.135 2008.85,1669.11 2009.5,1438.74 2010.15,406.101 2010.8,180.694
2011.46,1113.85 2012.11,1751.12 2012.76,1099.23 2013.41,174.239 2014.06,417.877 2014.72,1450.4 2015.37,1662.47 2016.02,723.555 2016.67,97.0773 2017.32,759.492
2017.98,1678.33 2018.63,1421.46 2019.28,389.245 2019.93,190.538 2020.58,1135.05 2021.24,1750.63 2021.89,1077.81 2022.54,165.273 2023.19,435.336 2023.85,1467.07
2024.5,1652.37 2025.15,702.425 2025.8,97.8525 2026.45,780.963 2027.11,1687.03 2027.76,1403.83 2028.41,372.762 2029.06,200.894 2029.71,1156.11 2030.37,1749.57
2031.02,1056.29 2031.67,156.837 2032.32,453.136 2032.98,1483.36 2033.63,1641.76 2034.28,681.451 2034.93,99.2044 2035.58,802.534 2036.24,1695.2 2036.89,1385.86
2037.54,356.663 2038.19,211.754 2038.84,1177 2039.5,1747.93 2040.15,1034.67 2040.75,181.604 2041.36,342.891 2041.96,1298.16 2042.56,1742.23 2043.17,1068.36
2043.77,197.267 2044.38,319.123 2044.98,1267.44 2045.58,1746.53 2046.19,1101.81 2046.79,214.164 2047.39,296.384 2048,1236.13 2048.6,1749.43 2049.21,1134.96
2049.81,232.269 2050.41,274.711 2051.02,1204.29 2051.62,1750.93 2052.23,1167.74 2052.83,251.548 2053.43,254.142 2054.04,1171.98 2054.64,1751.02 2055.24,1200.11
2055.85,271.971 2056.45,234.712 2057.06,1139.24 2057.66,1749.7 2058.26,1232.02 2058.87,293.502 2059.47,216.453 2060.07,1106.14 2060.68,1746.99 2061.28,1263.4
2061.89,316.105 2062.49,199.396 2063.09,1072.73 2063.7,1742.87 2064.3,1294.2 2064.91,339.741 2065.51,183.572 2066.11,1039.07 2066.72,1737.36 2067.32,1324.37
2067.92,364.37 2068.53,169.006 2069.13,1005.21 2069.74,1730.47 2070.34,1353.87 2070.94,389.95 2071.55,155.723 2072.15,971.21 2072.75,1722.21 2073.36,1382.63
2073.96,416.438 2074.57,143.746 2075.17,937.132 2075.77,1712.59 2076.38,1410.61 2076.98,443.789 2077.59,133.095 2078.19,903.032 2078.79,1701.63 2079.4,1437.77
2080,471.956 2080.6,123.788 2081.21,868.968 2081.81,1689.35 2082.42,1464.05 2083.02,500.891 2083.62,115.842 2084.23,834.997 2084.83,1675.77 2085.44,1489.42
2086.04,530.545 2086.64,109.269 2087.25,801.177 2087.85,1660.91 2088.45,1513.82 2089.06,560.868 2089.66,104.081 2090.27,767.567 2090.87,1644.8 2091.47,1537.22
2092.08,591.809 2092.68,100.286 2093.28,734.222 2093.89,1627.47 2094.49,1559.58 2095.1,623.314 2095.7,97.8918 2096.3,701.199 2096.91,1608.94 2097.51,1580.86
2098.12,655.33 2098.72,96.9015 2099.32,668.556 2099.93,1589.24 2100.53,1601.02 2101.13,687.802 2101.74,97.317 2102.34,636.346 2102.95,1568.41 2103.55,1620.03
2104.15,720.677 2104.76,99.1376 2105.36,604.626 2105.96,1546.49 2106.57,1637.86 2107.17,753.896 2107.78,102.36 2108.38,573.448 2108.98,1523.51 2109.59,1654.48
2110.19,787.405 2110.8,106.979 2111.4,542.866 2112,1499.51 2112.61,1669.85 2113.21,821.147 2113.81,112.987 2114.42,512.932 2115.02,1474.54 2115.63,1683.96
2116.23,855.063 2116.83,120.374 2117.44,483.697 2118.03,1437.27 2118.63,1706.69 2119.22,932.817 2119.82,147.211 2120.41,397.058 2121.01,1349.32 2121.6,1734.66
2122.2,1039.43 2122.79,190.352 2123.39,319.222 2123.99,1254.26 2124.58,1749.09 2125.18,1144.11 2125.77,245.751 2126.37,251.492 2126.96,1153.68 2127.56,1749.73
2128.15,1245.12 2128.75,312.482 2129.34,194.997 2129.94,1049.27 2130.53,1736.57 2131.13,1340.76 2131.72,389.431 2132.32,150.683 2132.91,942.76 2133.51,1709.85
2134.11,1429.44 2134.7,475.311 2135.3,119.29 2135.89,835.939 2136.49,1669.99 2137.08,1509.67 2137.68,568.688 2138.27,101.341 2138.87,730.59 2139.46,1617.66
2140.06,1580.12 2140.65,668.002 2141.25,97.1378 2141.84,628.473 2142.44,1553.75 2143.03,1639.61 2143.63,771.593 2144.23,106.75 2144.82,531.293 2145.42,1479.32
2146.01,1687.14 2146.61,877.73 2147.2,130.016 2147.8,440.675 2148.39,1395.61 2148.99,1721.92 2149.58,984.641 2150.18,166.548 2150.77,358.132 2151.37,1304.02
2151.96,1743.37 2152.56,1090.54 2153.15,215.736 2153.75,285.044 2154.35,1206.08 2154.94,1751.12 2155.54,1193.65 2156.13,276.758 2156.73,222.632 2157.32,1103.42
2157.92,1745.06 2158.51,1292.26 2159.11,348.594 2159.7,171.938 2160.3,997.775 2160.89,1725.29 2161.49,1384.72 2162.08,430.043 2162.68,133.81 2163.27,890.892
2163.87,1692.12 2164.47,1469.48 2165.06,519.746 2165.66,108.884 2166.25,784.563 2166.85,1646.12 2167.44,1545.13 2168.04,616.203 2168.63,97.5767 2169.23,680.564
2169.82,1588.06 2170.42,1610.4 2171.01,717.803 2171.61,100.078 2172.2,580.632 2172.8,1518.9 2173.39,1664.2 2173.99,822.849 2174.59,116.345 2175.18,486.437
2175.78,1439.8 2176.37,1705.63 2176.97,929.584 2177.56,146.106 2178.16,399.554 2178.75,1352.09 2179.35,1734.01 2179.94,1036.23 2180.54,188.865 2181.13,321.432
2181.73,1257.22 2182.32,1748.85 2182.92,1140.99 2183.51,243.906 2184.11,253.379 2184.71,1156.79 2185.3,1749.91 2185.9,1242.14 2186.49,310.31 2187.09,196.53
2187.68,1052.46 2188.28,1737.17 2188.87,1337.96 2189.47,386.968 2190.06,151.836 2190.66,945.992 2191.25,1710.85 2191.85,1426.87 2192.44,472.598 2193.04,120.043
2193.63,839.155 2194.21,1658.22 2194.79,1547.83 2195.37,647.738 2195.95,97.3864 2196.52,593.479 2197.1,1508 2197.68,1683.24 2198.26,897.352 2198.84,145.216
2199.41,378.975 2199.99,1302.71 2200.57,1747.04 2201.15,1149.48 2201.73,266.495 2202.3,215.874 2202.88,1061.7 2203.46,1733.22 2204.04,1380.35 2204.62,449.783
2205.19,119.558 2205.77,807.709 2206.35,1643.09 2206.93,1568.17 2207.51,677.797 2208.08,99.1096 2208.66,564.685 2209.24,1485.14 2209.82,1695.25 2210.4,929.03
2210.97,156.458 2211.55,355.549 2212.13,1274.27 2212.71,1749.59 2213.29,1179.79 2213.86,286.194 2214.44,200.025 2215.02,1030.37 2215.6,1726.07 2216.18,1406.43
2216.75,476.083 2217.33,112.781 2217.91,776.434 2218.49,1626.91 2219.07,1587.57 2219.64,708.216 2220.22,102.043 2220.8,536.419 2221.38,1461.46 2221.96,1706.13
2222.53,960.701 2223.11,168.825 2223.69,332.957 2224.27,1245.32 2224.85,1750.93 2225.42,1209.73 2226,306.83 2226.58,185.238 2227.16,998.877 2227.74,1717.74
2228.31,1431.8 2228.89,503.04 2229.47,107.193 2230.05,745.376 2230.63,1609.7 2231.2,1605.99 2231.78,738.953 2232.36,106.182 2232.94,508.721 2233.52,1436.99
2234.09,1715.86 2234.67,992.318 2235.25,182.301 2235.83,311.232 2236.4,1215.9 2236.98,1751.06 2237.56,1239.24 2238.14,328.37 2238.72,171.535 2239.29,967.277
2239.87,1708.25 2240.45,1456.44 2241.03,530.615 2241.61,102.804 2242.18,714.579 2242.76,1591.48 2243.34,1623.42 2243.92,769.96 2244.5,111.521 2245.07,481.632
2245.65,1411.76 2246.23,1724.44 2246.81,1023.83 2247.39,196.864 2247.96,290.406 2248.54,1186.04 2249.12,1749.97 2249.7,1268.29 2250.28,350.785 2250.85,158.936
2251.43,935.614 2252.01,1697.6 2252.59,1480.29 2253.17,558.766 2253.74,99.6197 2254.32,684.09 2254.9,1572.28 2255.48,1639.81 2256.06,801.194 2256.63,118.052
2257.21,455.193 2257.79,1385.82 2258.37,1731.83 2258.95,1055.21 2259.52,212.494 2260.1,270.51 2260.68,1155.81 2261.26,1747.67 2261.84,1296.84 2262.41,374.04
2262.99,147.459 2263.57,903.934 2264.15,1685.83 2264.73,1503.32 2265.3,587.454 2265.88,97.6448 2266.46,653.953 2267.04,1552.13 2267.62,1655.16 2268.21,807.969
2268.8,114.198 2269.39,491.089 2269.99,1440.35 2270.58,1706.96 2271.17,938.424 2271.76,150.837 2272.36,385.471 2272.95,1332.11 2273.54,1739.2 2274.13,1068.52
2274.73,206.782 2275.32,293.3 2275.91,1213.69 2276.5,1751.09 2277.1,1195.01 2277.69,280.636 2278.28,216.879 2278.87,1088.03 2279.47,1742.33 2280.06,1314.73
2280.65,370.556 2281.24,158.114 2281.84,958.272 2282.43,1713.13 2283.02,1424.69 2283.61,474.295 2284.21,118.474 2284.8,827.661 2285.39,1664.23 2285.98,1522.15
2286.58,589.263 2287.17,98.9479 2287.76,699.456 2288.35,1596.85 2288.95,1604.68 2289.54,712.591 2290.13,100.024 2290.72,576.858 2291.32,1512.67 2291.91,1670.21
2292.5,841.197 2293.09,121.674 2293.69,462.929 2294.28,1413.78 2294.87,1717.11 2295.46,971.871 2296.06,163.359 2296.65,360.512 2297.24,1302.67 2297.83,1744.2
2298.43,1101.35 2299.02,224.038 2299.61,272.167 2300.2,1182.11 2300.8,1750.82 2301.39,1226.4 2301.98,302.195 2302.57,200.097 2303.17,1055.09 2303.76,1736.78
2304.35,1343.9 2304.94,395.878 2305.54,146.104 2306.13,924.809 2306.72,1702.46 2307.31,1450.92 2307.91,502.749 2308.5,111.535 2309.09,794.504 2309.68,1648.7
2310.28,1544.78 2310.87,620.139 2311.46,97.2541 2312.05,667.433 2312.65,1576.84 2313.24,1623.13 2313.83,745.116 2314.42,103.617 2315.02,546.768 2315.61,1488.68
2316.2,1684.04 2316.79,874.561 2317.39,130.465 2317.98,435.524 2318.57,1386.42 2319.16,1725.96 2319.76,1005.24 2320.35,177.127 2320.94,336.477 2321.53,1272.61
2322.13,1747.86 2322.72,1133.89 2323.31,242.44 2323.9,252.1 2324.5,1150.1 2325.09,1749.19 2325.68,1257.3 2326.27,324.771 2326.87,184.502 2327.46,1021.94
2328.05,1729.91 2328.64,1372.39 2329.24,422.065 2329.83,135.368 2330.42,891.344 2331.01,1690.51 2331.61,1476.28 2332.2,531.893 2332.79,105.927 2333.38,761.558
2333.98,1631.97 2334.57,1566.38 2335.16,651.512 2335.75,96.9141 2336.35,635.829 2336.94,1555.75 2337.53,1640.45 2338.12,777.935 2338.72,108.553 2339.31,517.296
2339.9,1463.76 2340.49,1696.62 2341.09,908.006 2341.68,140.554 2342.27,408.919 2342.86,1358.29 2343.46,1733.5 2344.11,941.96 2344.75,122.661 2345.4,542.764
2346.05,1552.52 2346.7,1590.2 2347.35,597.523 2348,109.809 2348.65,881.376 2349.3,1718.88 2349.95,1327.01 2350.6,311.84 2351.25,243.482 2351.89,1227.66
2352.54,1742.2 2353.19,991.296 2353.84,136.325 2354.49,499.623 2355.14,1519.3 2355.79,1618.28 2356.44,643.475 2357.09,102.562 2357.74,832.138 2358.39,1703.81
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"markdown": "# Derivatives\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing SymPy\n```\n\n\n\n\n---\n\n\nBefore defining the derivative of a function, let's begin with two motivating examples.\n\n\n##### Example: Driving\n\n\nImagine motoring along down highway $61$ leaving Minnesota on the way to New Orleans; though lost in listening to music, still mindful of the speedometer and odometer, both prominently placed on the dashboard of the car.\n\n\nThe speedometer reads $60$ miles per hour, what is the odometer doing? Besides recording total distance traveled, it is incrementing dutifully every hour by $60$ miles. Why? Well, the well-known formula relating distance, time and rate of travel is\n\n\n\n$$\n\\text{distance} = \\text{ rate } \\times \\text{ time.}\n$$\n\n\nIf the rate is a constant $60$ miles/hour, then in one hour the distance traveled is $60$ miles.\n\n\nOf course, the odometer isn't just incrementing once per hour, it is incrementing once every $1/10$th of a mile. How much time does that take? Well, we would need to solve $1/10=60 \\cdot t$ which means $t=1/600$ hours, better known as once every $6$ seconds.\n\n\nUsing some mathematical notation, would give $x(t) = v\\cdot t$, where $x$ is position at time $t$, $v$ is the *constant* velocity and $t$ the time traveled in hours. A simple graph of the first three hours of travel would show:\n\n::: {.cell hold='true' execution_count=4}\n``` {.julia .cell-code}\nposition(t) = 60 * t\nplot(position, 0, 3)\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n![](derivatives_files/figure-html/cell-5-output-1.svg){}\n:::\n:::\n\n\nOh no, we hit traffic. In the next $30$ minutes we only traveled $15$ miles. We were so busy looking out for traffic, the speedometer was not checked. What would the average speed have been? Though in the $30$ minutes of stop-and-go traffic, the displayed speed may have varied, the *average speed* would simply be the change in distance over the change in time, or $\\Delta x / \\Delta t$. That is\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\n15/(1/2)\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\n30.0\n```\n:::\n:::\n\n\nNow suppose that after $6$ hours of travel the GPS in the car gives us a readout of distance traveled as a function of time. The graph looks like this:\n\n::: {.cell hold='true' execution_count=6}\n\n::: {.cell-output .cell-output-display execution_count=7}\n![](derivatives_files/figure-html/cell-7-output-1.svg){}\n:::\n:::\n\n\nWe can see with some effort that the slope is steady for the first three hours, is slightly less between $3$ and $3.5$ hours, then is a bit steeper for the next half hour. After that, it is flat for the about half an hour, then the slope continues on with same value as in the first $3$ hours. What does that say about our speed during our trip?\n\n\nBased on the graph, what was the average speed over the first three hours? Well, we traveled $180$ miles, and took $3$ hours:\n\n::: {.cell execution_count=7}\n``` {.julia .cell-code}\n180/3\n```\n\n::: {.cell-output .cell-output-display execution_count=8}\n```\n60.0\n```\n:::\n:::\n\n\nWhat about the next half hour? Squinting shows the amount traveled was $15$ miles ($195 - 180$) and it took $1/2$ an hour:\n\n::: {.cell execution_count=8}\n``` {.julia .cell-code}\n15/(1/2)\n```\n\n::: {.cell-output .cell-output-display execution_count=9}\n```\n30.0\n```\n:::\n:::\n\n\nAnd the half hour after that? The average speed is found from the distance traveled, $37.5$ miles, divided by the time, $1/2$ hour:\n\n::: {.cell execution_count=9}\n``` {.julia .cell-code}\n37.5 / (1/2)\n```\n\n::: {.cell-output .cell-output-display execution_count=10}\n```\n75.0\n```\n:::\n:::\n\n\nOkay, so there was some speeding involved.\n\n\nThe next half hour the car did not move. What was the average speed? Well the change in position was $0$, but the time was $1/2$ hour, so the average was $0$.\n\n\nPerhaps a graph of the speed is a bit more clear. We can do this based on the above:\n\n::: {.cell execution_count=10}\n``` {.julia .cell-code}\nfunction speed(t)\n 0 < t <= 3 ? 60 :\n t <= 3.5 ? 30 :\n t <= 4 ? 75 :\n t <= 4.5 ? 0 : 60\nend\nplot(speed, 0, 6)\n```\n\n::: {.cell-output .cell-output-display execution_count=11}\n![](derivatives_files/figure-html/cell-11-output-1.svg){}\n:::\n:::\n\n\nThe jumps, as discussed before, are artifacts of the graphing algorithm. What is interesting, is we could have derived the graph of `speed` from that of `x` by just finding the slopes of the line segments, and we could have derived the graph of `x` from that of `speed`, just using the simple formula relating distance, rate, and time.\n\n\n:::{.callout-note}\n## Note\nWe were pretty loose with some key terms. There is a distinction between \"speed\" and \"velocity\", this being the speed is the absolute value of velocity. VelLine truncated
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"markdown": "# The first and second derivatives\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing SymPy\nusing Roots\n```\n\n\n::: {.cell results='hidden' execution_count=3}\n\n::: {.cell-output .cell-output-stderr}\n```\nWARNING: redefinition of constant frontmatter. This may fail, cause incorrect answers, or produce other errors.\n```\n:::\n:::\n\n\n---\n\n\nThis section explores properties of a function, $f(x)$, that are described by properties of its first and second derivatives, $f'(x)$ and $f''(x)$. As part of the conversation two tests are discussed that characterize when a critical point is a relative maximum or minimum. (We know that any relative maximum or minimum occurs at a critical point, but it is not true that *any* critical point will be a relative maximum or minimum.)\n\n\n## Positive or increasing on an interval\n\n\nWe start with some vocabulary:\n\n\n> A function $f$ is **positive** on an interval $I$ if for any $a$ in $I$ it must be that $f(a) > 0$.\n\n\n\nOf course, we define *negative* in a parallel manner. The intermediate value theorem says a continuous function can not change from positive to negative without crossing $0$. This is not the case for functions with jumps, of course.\n\n\nNext,\n\n\n> A function, $f$, is (strictly) **increasing** on an interval $I$ if for any $a < b$ it must be that $f(a) < f(b)$.\n\n\n\nThe word strictly is related to the inclusion of the $<$ precluding the possibility of a function being flat over an interval that the $\\leq$ inequality would allow.\n\n\nA parallel definition with $a < b$ implying $f(a) > f(b)$ would be used for a *strictly decreasing* function.\n\n\nWe can try and prove these properties for a function algebraically – we'll see both are related to the zeros of some function. However, before proceeding to that it is usually helpful to get an idea of where the answer is using exploratory graphs.\n\n\nWe will use a helper function, `plotif(f, g, a, b)` that plots the function `f` over `[a,b]` coloring it red when `g` is positive (and blue otherwise). Such a function is defined for us in the accompanying `CalculusWithJulia` package, which has been previously been loaded.\n\n\nTo see where a function is positive, we simply pass the function object in for *both* `f` and `g` above. For example, let's look at where $f(x) = \\sin(x)$ is positive:\n\n::: {.cell hold='true' execution_count=4}\n``` {.julia .cell-code}\nf(x) = sin(x)\nplotif(f, f, -2pi, 2pi)\n```\n\n::: {.cell-output .cell-output-display execution_count=67}\n![](first_second_derivatives_files/figure-html/cell-5-output-1.svg){}\n:::\n:::\n\n\nLet's graph with `cos` in the masking spot and see what happens:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nplotif(sin, cos, -2pi, 2pi)\n```\n\n::: {.cell-output .cell-output-display execution_count=68}\n![](first_second_derivatives_files/figure-html/cell-6-output-1.svg){}\n:::\n:::\n\n\nMaybe surprisingly, we see that the increasing parts of the sine curve are now highlighted. Of course, the cosine is the derivative of the sine function, now we discuss that this is no coincidence.\n\n\nFor the sequel, we will use `f'` notation to find numeric derivatives, with the notation being defined in the `CalculusWithJulia` package using the `ForwardDiff` package.\n\n\n## The relationship of the derivative and increasing\n\n\nThe derivative, $f'(x)$, computes the slope of the tangent line to the graph of $f(x)$ at the point $(x,f(x))$. If the derivative is positive, the tangent line will have an increasing slope. Clearly if we see an increasing function and mentally layer on a tangent line, it will have a positive slope. Intuitively then, increasing functions and positive derivatives are related concepts. But there are some technicalities.\n\n\nSuppose $f(x)$ has a derivative on $I$ . Then\n\n\n> If $f'(x)$ is positive on an interval $I=(a,b)$, then $f(x)$ is strictly increasing on $I$.\n\n\n\nMeanwhile,\n\n\n> If a function $f(x)$ is increasing on $I$, then $f'(x) \\geq 0$.\n\n\n\nThe technicality being the equality parts. In the second statement, we have the derivative is non-negative, as we can't guarantee it is positive, even if we considered just strictly increasing functions.\n\n\nWe can see by the example of $f(x) = x^3$ that strictly increasing functions can have a zero derivative, at a point.\n\n\nThe mean value theorem provides the reasoning behind the first statement: on $I$, the slope of any secant line between $d < e$ (both in $I$) is matched by the slope of some tangent line, which by assumption will always be positive. If the secant line slope is written as $(f(e) - f(d))/(e - d)$ with $d < e$, then it is clear then that $f(e) - f(d) > 0$, or $d < e$ implies $f(d) < f(e)$.\n\n\nThe second part, follows from the secant line equation. The derivative can be written as a limit of secant-line slopes, each of which is positive. The limit of positive tLine truncated
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"markdown": "# Implicit Differentiation\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing ImplicitPlots\nusing Roots\nusing SymPy\n```\n\n\n\n\n---\n\n\n## Graphs of equations\n\n\nAn **equation** in $y$ and $x$ is an algebraic expression involving an equality with two (or more) variables. An example might be $x^2 + y^2 = 1$.\n\n\nThe **solutions** to an equation in the variables $x$ and $y$ are all points $(x,y)$ which satisfy the equation.\n\n\nThe **graph** of an equation is just the set of solutions to the equation represented in the Cartesian plane.\n\n\nWith this definition, the graph of a function $f(x)$ is just the graph of the equation $y = f(x)$. In general, graphing an equation is more complicated than graphing a function. For a function, we know for a given value of $x$ what the corresponding value of $f(x)$ is through evaluation of the function. For equations, we may have $0$, $1$ or more $y$ values for a given $x$ and even more problematic is we may have no rule to find these values.\n\n\nThere are a few options for plotting equations in `Julia`. We will use `ImplicitPlots` in this section, but note both `ImplicitEquations` and `IntervalConstraintProgramming` offer alternatives that are a bit more flexible.\n\n\nTo plot an implicit equation using `ImplicitPlots` requires expressing the relationship in terms of a function, and then plotting the equation `f(x,y) = 0`. In practice this simply requires all the terms be moved to one side of an equals sign.\n\n\nTo plot the circle of radius $2$, or the equations $x^2 + y^2 = 2^2$ we would move all terms to one side $x^2 + y^2 - 2^2 = 0$ and then express the left hand side through a function:\n\n::: {.cell execution_count=4}\n``` {.julia .cell-code}\nf(x,y) = x^2 + y^2 - 2^2\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n```\nf (generic function with 1 method)\n```\n:::\n:::\n\n\nThis function is then is passed to the `implicit_plot` function, which works with `Plots` to render the graphic:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nimplicit_plot(f)\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n![](implicit_differentiation_files/figure-html/cell-6-output-1.svg){}\n:::\n:::\n\n\n:::{.callout-note}\n## Note\nThe `f` is a function of *two* variables, used here to express one side of an equation. `Julia` makes this easy to do - just make sure two variables are in the signature of `f` when it is defined. Using functions like this, we can express our equation in the form $f(x,y) = c$ or, more generally, as $f(x,y) = g(x,y)$. The latter of which can be expressed as $h(x,y) = f(x,y) - g(x,y) = 0$. That is, only the form $f(x,y)=0$ is needed to represent an equation.\n\n:::\n\n:::{.callout-note}\n## Note\nThere are two different styles in `Julia` to add simple plot recipes. `ImplicitPlots` adds a new plotting function (`implicit_plot`); alternatively many packages add a new recipe for the generic `plot` method using new types. (For example, `SymPy` has a plot recipe for symbolic types.\n\n:::\n\nOf course, more complicated equations are possible and the steps are similar - only the function definition is more involved. For example, the [Devils curve](http://www-groups.dcs.st-and.ac.uk/~history/Curves/Devils.html) has the form\n\n\n\n$$\ny^4 - x^4 + ay^2 + bx^2 = 0\n$$\n\n\nHere we draw the curve for a particular choice of $a$ and $b$. For illustration purposes, a narrower viewing window is specified below using `xlims` and `ylims`:\n\n::: {.cell hold='true' execution_count=6}\n``` {.julia .cell-code}\na,b = -1,2\nf(x,y) = y^4 - x^4 + a*y^2 + b*x^2\nimplicit_plot(f; xlims=(-3,3), ylims=(-3,3), legend=false)\n```\n\n::: {.cell-output .cell-output-display execution_count=7}\n![](implicit_differentiation_files/figure-html/cell-7-output-1.svg){}\n:::\n:::\n\n\n## Tangent lines, implicit differentiation\n\n\nThe graph $x^2 + y^2 = 1$ has well-defined tangent lines at all points except $(-1,0)$ and $(0, 1)$ and even at these two points, we could call the vertical lines $x=-1$ and $x=1$ tangent lines. However, to recover the slope of these tangent lines would need us to express $y$ as a function of $x$ and then differentiate that function. Of course, in this example, we would need two functions: $f(x) = \\sqrt{1-x^2}$ and $g(x) = - \\sqrt{1-x^2}$ to do this completely.\n\n\nIn general though, we may not be able to solve for $y$ in terms of $x$. What then?\n\n\nThe idea is to *assume* that $y$ is representable by some function of $x$. This makes sense, moving on the curve from $(x,y)$ to some nearby point, means changing $x$ will cause some change in $y$. This assumption is only made *locally* - basically meaning a complicated graph is reduced to just a small, well-behaved, section of its graph.\n\n\nWith this assumption, asking what $dy/dx$ is has an obvious meaning - what is the slope of the tangent line to the graph at $(x,y)$. (TheLine truncated
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1029.84,1415.94 1027.42,1414.03 1025.45,1412.47 1021.92,1409.64 1021.05,1408.94 1016.66,1405.33 1016.56,1405.25 1012.26,1401.65 1011.32,1400.85 1007.87,1397.9
1006.2,1396.46 1003.47,1394.08 1001.19,1392.06 999.08,1390.18 996.291,1387.67 994.685,1386.2 991.498,1383.27 990.291,1382.15 986.809,1378.88 985.896,1378.01
982.219,1374.48 981.501,1373.79 977.724,1370.09 977.106,1369.47 973.323,1365.69 972.711,1365.07 969.012,1361.3 968.316,1360.58 964.788,1356.9 963.922,1355.99
960.649,1352.51 959.527,1351.3 956.593,1348.11 955.132,1346.51 952.616,1343.72 950.737,1341.61 948.718,1339.32 946.342,1336.6 944.895,1334.93 941.947,1331.48
941.145,1330.53 937.553,1326.24 937.468,1326.14 933.862,1321.74 933.158,1320.87 930.326,1317.35 928.763,1315.38 926.856,1312.95 924.368,1309.75 923.451,1308.56
920.109,1304.16 919.973,1303.98 916.833,1299.77 915.578,1298.06 913.618,1295.37 911.184,1291.99 910.462,1290.98 907.365,1286.58 906.789,1285.76 904.328,1282.19
902.394,1279.35 901.346,1277.8 898.42,1273.4 897.999,1272.76 895.55,1269.01 893.604,1265.98 892.733,1264.61 889.969,1260.22 889.21,1258.99 887.257,1255.82
884.815,1251.79 884.595,1251.43 881.985,1247.03 880.42,1244.35 879.424,1242.64 876.912,1238.24 876.025,1236.67 874.447,1233.85 872.03,1229.45 871.63,1228.71
869.66,1225.06 867.334,1220.66 867.235,1220.47 865.056,1216.27 862.841,1211.91 862.82,1211.87 860.63,1207.48 858.483,1203.08 858.446,1203.01 856.379,1198.69
854.317,1194.29 854.051,1193.72 852.297,1189.9 850.318,1185.5 849.656,1184.01 848.38,1181.11 846.483,1176.71 845.261,1173.83 844.625,1172.32 842.807,1167.92
841.027,1163.53 840.866,1163.12 839.287,1159.13 837.585,1154.74 836.472,1151.8 835.92,1150.35 834.294,1145.95 832.703,1141.56 832.077,1139.78 831.15,1137.16
829.634,1132.77 828.153,1128.37 827.682,1126.94 826.708,1123.98 825.298,1119.58 823.924,1115.19 823.287,1113.1 822.584,1110.79 821.279,1106.4 820.008,1102
818.892,1098.04 818.77,1097.61 817.567,1093.21 816.398,1088.82 815.261,1084.42 814.498,1081.39 814.157,1080.03 813.086,1075.63 812.048,1071.24 811.042,1066.84
810.103,1062.61 810.067,1062.45 809.125,1058.05 808.215,1053.66 807.336,1049.26 806.488,1044.87 805.708,1040.67 805.671,1040.47 804.886,1036.08 804.131,1031.68
803.407,1027.29 802.713,1022.9 802.049,1018.5 801.416,1014.11 801.313,1013.36 800.812,1009.71 800.239,1005.32 799.695,1000.92 799.181,996.526 798.696,992.131
798.241,987.737 797.815,983.342 797.418,978.947 797.05,974.552 796.918,972.842 796.711,970.157 796.401,965.763 796.12,961.368 795.867,956.973 795.643,952.578
795.447,948.183 795.28,943.788 795.141,939.394 795.03,934.999 794.948,930.604 794.893,926.209 794.867,921.814 794.869,917.419 794.898,913.025 794.956,908.63
795.041,904.235 795.155,899.84 795.296,895.445 795.464,891.051 795.661,886.656 795.885,882.261 796.137,877.866 796.416,873.471 796.723,869.076 796.918,866.511
797.058,864.682 797.42,860.287 797.81,855.892 798.228,851.497 798.673,847.102 799.145,842.707 799.645,838.313 800.173,833.918 800.728,829.523 801.311,825.128
801.313,825.113 801.922,820.733 802.56,816.338 803.226,811.944 803.92,807.549 804.641,803.154 805.39,798.759 805.708,796.958 806.167,794.364 806.972,789.97
807.805,785.575 808.665,781.18 809.554,776.785 810.103,774.15 810.47,772.39 811.416,767.995 812.389,763.601 813.39,759.206 814.419,754.811 814.498,754.483
815.477,750.416 816.564,746.021 817.678,741.626 818.821,737.232 818.892,736.965 819.994,732.837 821.195,728.442 822.425,724.047 823.287,721.034 823.684,719.652
824.972,715.258 826.29,710.863 827.635,706.468 827.682,706.319 829.012,702.073 830.418,697.678 831.853,693.283 832.077,692.609 833.319,688.889 834.814,684.494
836.338,680.099 836.472,679.721 837.895,675.704 839.481,671.309 840.866,667.538 841.096,666.914 842.745,662.52 844.422,658.125 845.261,655.963 846.132,653.73
847.872,649.335 849.643,644.94 849.656,644.909 851.448,640.545 853.283,636.151 854.051,634.339 855.151,631.756 857.05,627.361 858.446,624.182 858.982,622.966
860.947,618.571 862.841,614.403 862.944,614.177 864.976,609.782 867.039,605.387 867.235,604.974 869.138,600.992 871.269,596.597 871.63,595.862 873.436,592.202
875.635,587.808 876.025,587.039 877.871,583.413 880.14,579.018 880.42,578.483 882.447,574.623 884.787,570.228 884.815,570.176 887.165,565.833 889.21,562.106
889.578,561.439 892.029,557.044 893.604,554.253 894.515,552.649 897.04,548.254 897.999,546.603 899.602,543.859 902.201,539.465 902.394,539.142 904.841,535.07
906.789,531.867 907.518,530.675 910.235,526.28 911.184,524.763 912.992,521.885 915.578,517.818 915.788,517.49 918.627,513.096 919.973,511.034 921.506,508.701
924.368,504.392 924.425,504.306 927.39,499.911 928.763,497.897 930.396,495.516 933.158,491.531 933.443,491.121 936.537,486.727 937.553,485.299 939.675,482.332
941.947,479.188 942.857,477.937 946.085,473.542 946.342,473.195 949.361,469.147 950.737,467.32 952.683,464.752 955.132,461.552 956.052,460.358 959.469,455.963
959.527,455.889 962.937,451.568 963.922,450.333 966.455,447.173 968.316,444.874 970.024,442.778 972.711,439.51 973.644,438.384 977.106,434.238 977.315,433.989
981.042,429.594 981.501,429.058 984.824,425.199 985.896,423.966 988.661,420.804 990.291,418.958 992.555,416.409 994.685,414.032 996.506,412.015 999.08,409.185
1000.52,407.62 1003.47,404.417 1004.58,403.225 1007.87,399.725 1008.72,398.83 1012.26,395.106 1012.91,394.435 1016.66,390.56 1017.17,390.04 1021.05,386.084
1021.49,385.646 1025.45,381.678 1025.88,381.251 1029.84,377.338 1030.34,376.856 1034.24,373.065 1034.87,372.461 1038.63,368.856 1039.47,368.066 1043.03,364.709
1044.14,363.672 1047.42,360.625 1048.89,359.277 1051.82,356.6 1053.71,354.882 1056.21,352.635 1058.62,350.487 1060.61,348.727 1063.61,346.092 1065,344.876
1068.68,341.697 1069.4,341.08 1073.79,337.339 1073.84,337.303 1078.19,333.654 1079.09,332.908 1082.58,330.022 1084.43,328.513 1086.98,326.441 1089.86,324.118
1091.37,322.91 1095.39,319.723 1095.77,319.429 1100.16,315.999 1101.03,315.328 1104.56,312.618 1106.77,310.934 1108.95,309.285 1112.62,306.539 1113.35,305.997
1117.74,302.757 1118.58,302.144 1122.14,299.564 1124.66,297.749 1126.53,296.416 1130.86,293.354 1130.92,293.31 1135.32,290.252 1137.2,288.959 1139.71,287.235
1143.66,284.565 1144.11,284.26 1148.5,281.33 1150.26,280.17 1152.9,278.441 1157.01,275.775 1157.29,275.592 1161.69,272.786 1163.91,271.38 1166.08,270.019
1170.48,267.291 1170.98,266.985 1174.87,264.605 1178.21,262.591 1179.27,261.955 1183.66,259.346 1185.62,258.196 1188.06,256.775 1192.45,254.24 1193.22,253.801
1196.85,251.745 1201.02,249.406 1201.24,249.284 1205.64,246.863 1209.04,245.011 1210.03,244.476 1214.43,242.127 1217.29,240.616 1218.82,239.812 1223.22,237.533
1225.78,236.222 1227.61,235.289 1232.01,233.08 1234.53,231.827 1236.4,230.906 1240.8,228.766 1243.57,227.432 1245.19,226.659 1249.59,224.587 1252.92,223.037
1253.98,222.547 1258.37,220.542 1262.6,218.642 1262.77,218.567 1267.16,216.628 1271.56,214.72 1272.66,214.247 1275.95,212.845 1280.35,211.001 1283.13,209.853
1284.74,209.19 1289.14,207.41 1293.53,205.661 1294.05,205.458 1297.93,203.945 1302.32,202.259 1305.49,201.063 1306.72,200.603 1311.11,198.98 1315.51,197.386
1317.52,196.668 1319.9,195.823 1324.3,194.29 1328.69,192.787 1330.22,192.273 1333.09,191.315 1337.48,189.872 1341.88,188.458 1343.71,187.878 1346.27,187.075
1350.67,185.721 1355.06,184.397 1358.15,183.484 1359.46,183.101 1363.85,181.835 1368.25,180.598 1372.64,179.389 1373.76,179.089 1377.04,178.21 1381.43,177.06
1385.82,175.937 1390.22,174.843 1390.83,174.694 1394.61,173.779 1399.01,172.742 1403.4,171.734 1407.8,170.754 1409.89,170.299 1412.19,169.802 1416.59,168.878
1420.98,167.983 1425.38,167.115 1429.77,166.275 1431.78,165.904 1434.17,165.463 1438.56,164.68 1442.96,163.924 1447.35,163.195 1451.75,162.495 1456.14,161.822
1458.26,161.51 1460.54,161.176 1464.93,160.559 1469.33,159.969 1473.72,159.407 1478.12,158.873 1482.51,158.365 1486.91,157.886 1491.3,157.434 1494.6,157.115
1495.7,157.009 1500.09,156.613 1504.49,156.244 1508.88,155.902 1513.27,155.588 1517.67,155.302 1522.06,155.043 1526.46,154.812 1530.85,154.609 1535.25,154.433
1539.64,154.285 1544.04,154.165 1548.43,154.073 1552.83,154.008 1557.22,153.971 1561.62,153.963 1566.01,153.982 1570.41,154.029 1574.8,154.105 1579.2,154.209
1583.59,154.34 1587.99,154.501 1592.38,154.689 1596.78,154.906 1601.17,155.152 1605.57,155.426 1609.96,155.729 1614.36,156.061 1618.75,156.421 1623.15,156.811
1626.34,157.115 1627.54,157.23 1631.93,157.678 1636.33,158.155 1640.72,158.662 1645.12,159.198 1649.51,159.764 1653.91,160.36 1658.3,160.986 1661.81,161.51
1662.7,161.642 1667.09,162.328 1671.49,163.045 1675.88,163.792 1680.28,164.57 1684.67,165.379 1687.42,165.904 1689.07,166.219 1693.46,167.09 1697.86,167.993
1702.25,168.927 1706.65,169.893 1708.44,170.299 1711.04,170.891 1715.44,171.921 1719.83,172.984 1724.23,174.08 1726.62,174.694 1728.62,175.208 1733.02,176.37
1737.41,177.565 1741.81,178.793 1742.84,179.089 1746.2,180.055 1750.6,181.352 1754.99,182.683 1757.57,183.484 1759.38,184.049 1763.78,185.45 1768.17,186.886
1771.14,187.878 1772.57,188.358 1776.96,189.866 1781.36,191.41 1783.76,192.273 1785.75,192.99 1790.15,194.608 1794.54,196.263 1795.6,196.668 1798.94,197.956
1803.33,199.687 1806.75,201.063 1807.73,201.456 1812.12,203.265 1816.52,205.112 1817.32,205.458 1820.91,207 1825.31,208.927 1827.38,209.853 1829.7,210.896
1834.1,212.906 1836.98,214.247 1838.49,214.957 1842.89,217.051 1846.16,218.642 1847.28,219.187 1851.68,221.367 1854.98,223.037 1856.07,223.59 1860.47,225.859
1863.46,227.432 1864.86,228.172 1869.26,230.531 1871.63,231.827 1873.65,232.936 1878.05,235.389 1879.51,236.222 1882.44,237.889 1886.83,240.437 1887.14,240.616
1891.23,243.037 1894.51,245.011 1895.62,245.684 1900.02,248.384 1901.66,249.406 1904.41,251.135 1908.6,253.801 1908.81,253.938 1913.2,256.796 1915.32,258.196
1917.6,259.708 1921.87,262.591 1921.99,262.674 1926.39,265.699 1928.23,266.985 1930.78,268.781 1934.43,271.38 1935.18,271.92 1939.57,275.121 1940.46,275.775
1943.97,278.384 1946.34,280.17 1948.36,281.708 1952.07,284.565 1952.76,285.095 1957.15,288.549 1957.67,288.959 1961.55,292.07 1963.13,293.354 1965.94,295.659
1968.46,297.749 1970.34,299.318 1973.67,302.144 1974.73,303.048 1978.77,306.539 1979.13,306.851 1983.52,310.73 1983.75,310.934 1987.92,314.688 1988.62,315.328
1992.31,318.725 1993.38,319.723 1996.71,322.844 1998.05,324.118 2001.1,327.047 2002.61,328.513 2005.5,331.336 2007.08,332.908 2009.89,335.715 2011.46,337.303
2014.28,340.185 2015.75,341.697 2018.68,344.751 2019.95,346.092 2023.07,349.414 2024.07,350.487 2027.47,354.179 2028.11,354.882 2031.86,359.049 2032.07,359.277
2035.95,363.672 2036.26,364.029 2039.75,368.066 2040.65,369.124 2043.48,372.461 2045.05,374.336 2047.14,376.856 2049.44,379.67 2050.73,381.251 2053.84,385.129
2054.25,385.646 2057.7,390.04 2058.23,390.724 2061.09,394.435 2062.63,396.46 2064.41,398.83 2067.02,402.339 2067.68,403.225 2070.88,407.62 2071.42,408.371
2074.02,412.015 2075.81,414.567 2077.1,416.409 2080.12,420.804 2080.21,420.927 2083.09,425.199 2084.6,427.474 2086,429.594 2088.86,433.989 2089,434.202
2091.66,438.384 2093.39,441.139 2094.41,442.778 2097.11,447.173 2097.79,448.286 2099.76,451.568 2102.18,455.659 2102.36,455.963 2104.91,460.358 2106.58,463.284
2107.41,464.752 2109.86,469.147 2110.97,471.17 2112.27,473.542 2114.63,477.937 2115.37,479.339 2116.94,482.332 2119.21,486.727 2119.76,487.817 2121.43,491.121
2123.61,495.516 2124.16,496.634 2125.75,499.911 2127.84,504.306 2128.55,505.822 2129.89,508.701 2131.9,513.096 2132.95,515.421 2133.87,517.49 2135.8,521.885
2137.34,525.473 2137.69,526.28 2139.53,530.675 2141.34,535.07 2141.73,536.044 2143.11,539.465 2144.84,543.859 2146.13,547.2 2146.53,548.254 2148.19,552.649
2149.81,557.044 2150.52,559.034 2151.39,561.439 2152.93,565.833 2154.44,570.228 2154.92,571.657 2155.91,574.623 2157.35,579.018 2158.75,583.413 2159.31,585.229
2160.11,587.808 2161.45,592.202 2162.74,596.597 2163.71,599.96 2164,600.992 2165.23,605.387 2166.43,609.782 2167.59,614.177 2168.1,616.176 2168.72,618.571
2169.81,622.966 2170.88,627.361 2171.91,631.756 2172.5,634.36 2172.9,636.151 2173.87,640.545 2174.81,644.94 2175.71,649.335 2176.58,653.73 2176.89,655.373
2177.42,658.125 2178.23,662.52 2179.01,666.914 2179.75,671.309 2180.47,675.704 2181.16,680.099 2181.29,680.984 2181.81,684.494 2182.44,688.889 2183.03,693.283
2183.6,697.678 2184.14,702.073 2184.64,706.468 2185.12,710.863 2185.57,715.258 2185.68,716.462 2185.99,719.652 2186.38,724.047 2186.74,728.442 2187.07,732.837
2187.37,737.232 2187.65,741.626 2187.89,746.021 2188.11,750.416 2188.3,754.811 2188.46,759.206 2188.59,763.601 2188.69,767.995 2188.77,772.39 2188.82,776.785
2188.83,781.18 2188.83,785.575 2188.79,789.97 2188.73,794.364 2188.63,798.759 2188.51,803.154 2188.36,807.549 2188.19,811.944 2187.99,816.338 2187.75,820.733
2187.5,825.128 2187.21,829.523 2186.9,833.918 2186.55,838.313 2186.18,842.707 2185.79,847.102 2185.68,848.195 2185.36,851.497 2184.91,855.892 2184.43,860.287
2183.93,864.682 2183.39,869.076 2182.83,873.471 2182.24,877.866 2181.62,882.261 2181.29,884.533 2180.98,886.656 2180.3,891.051 2179.6,895.445 2178.87,899.84
2178.12,904.235 2177.33,908.63 2176.89,911.021 2176.52,913.025 2175.68,917.419 2174.81,921.814 2173.92,926.209 2173,930.604 2172.5,932.904 2172.04,934.999
2171.06,939.394 2170.06,943.788 2169.02,948.183 2168.1,951.963 2167.95,952.578 2166.86,956.973 2165.74,961.368 2164.59,965.763 2163.71,969.041 2163.41,970.157
2162.2,974.552 2160.96,978.947 2159.7,983.342 2159.31,984.644 2158.4,987.737 2157.08,992.131 2155.72,996.526 2154.92,999.083 2154.34,1000.92 2152.93,1005.32
2151.48,1009.71 2150.52,1012.58 2150.01,1014.11 2148.51,1018.5 2146.97,1022.9 2146.13,1025.28 2145.41,1027.29 2143.82,1031.68 2142.19,1036.08 2141.73,1037.3
2140.54,1040.47 2138.85,1044.87 2137.34,1048.75 2137.14,1049.26 2135.39,1053.66 2133.61,1058.05 2132.95,1059.67 2131.8,1062.45 2129.95,1066.84 2128.55,1070.13
2128.08,1071.24 2126.17,1075.63 2124.23,1080.03 2124.16,1080.2 2122.26,1084.42 2120.25,1088.82 2119.76,1089.88 2118.21,1093.21 2116.14,1097.61 2115.37,1099.22
2114.03,1102 2111.89,1106.4 2110.97,1108.26 2109.72,1110.79 2107.51,1115.19 2106.58,1117.02 2105.26,1119.58 2102.99,1123.98 2102.18,1125.51 2100.67,1128.37
2098.32,1132.77 2097.79,1133.76 2095.93,1137.16 2093.51,1141.56 2093.39,1141.77 2091.05,1145.95 2089,1149.58 2088.56,1150.35 2086.02,1154.74 2084.6,1157.17
2083.45,1159.13 2080.84,1163.53 2080.21,1164.59 2078.19,1167.92 2075.81,1171.82 2075.51,1172.32 2072.78,1176.71 2071.42,1178.88 2070.01,1181.11 2067.21,1185.5
2067.02,1185.79 2064.36,1189.9 2062.63,1192.53 2061.47,1194.29 2058.54,1198.69 2058.23,1199.14 2055.56,1203.08 2053.84,1205.6 2052.55,1207.48 2049.49,1211.87
2049.44,1211.94 2046.38,1216.27 2045.05,1218.14 2043.23,1220.66 2040.65,1224.22 2040.04,1225.06 2036.8,1229.45 2036.26,1230.18 2033.51,1233.85 2031.86,1236.03
2030.18,1238.24 2027.47,1241.77 2026.8,1242.64 2023.37,1247.03 2023.07,1247.41 2019.89,1251.43 2018.68,1252.94 2016.36,1255.82 2014.28,1258.37 2012.78,1260.22
2009.89,1263.71 2009.14,1264.61 2005.5,1268.96 2005.46,1269.01 2001.72,1273.4 2001.1,1274.12 1997.92,1277.8 1996.71,1279.19 1994.07,1282.19 1992.31,1284.18
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1717.02,1046.14 1714.6,1048.94 1714.32,1049.26 1712.19,1051.69 1710.44,1053.66 1709.77,1054.4 1707.35,1057.07 1706.46,1058.05 1704.94,1059.7 1702.52,1062.3
1702.38,1062.45 1700.1,1064.85 1698.19,1066.84 1697.68,1067.37 1695.27,1069.85 1693.9,1071.24 1692.85,1072.3 1690.43,1074.71 1689.5,1075.63 1688.02,1077.09
1685.6,1079.44 1684.98,1080.03 1683.18,1081.75 1680.76,1084.03 1680.35,1084.42 1678.35,1086.28 1675.93,1088.5 1675.59,1088.82 1673.51,1090.69 1671.1,1092.86
1670.69,1093.21 1668.68,1094.99 1666.26,1097.09 1665.66,1097.61 1663.84,1099.17 1661.43,1101.22 1660.49,1102 1659.01,1103.24 1656.59,1105.23 1655.17,1106.4
1654.18,1107.2 1651.76,1109.14 1649.68,1110.79 1649.34,1111.06 1646.92,1112.95 1644.51,1114.82 1644.03,1115.19 1642.09,1116.66 1639.67,1118.48 1638.2,1119.58
1637.26,1120.28 1634.84,1122.05 1632.42,1123.8 1632.18,1123.98 1630,1125.53 1627.59,1127.23 1625.95,1128.37 1625.17,1128.91 1622.75,1130.57 1620.34,1132.21
1619.51,1132.77 1617.92,1133.83 1615.5,1135.42 1613.08,1137 1612.84,1137.16 1610.67,1138.55 1608.25,1140.09 1605.91,1141.56 1605.83,1141.6 1603.41,1143.09
1601,1144.57 1598.7,1145.95 1598.58,1146.02 1596.16,1147.46 1593.75,1148.87 1591.33,1150.27 1591.2,1150.35 1588.91,1151.64 1586.49,1153 1584.08,1154.34
1583.36,1154.74 1581.66,1155.67 1579.24,1156.97 1576.83,1158.25 1575.15,1159.13 1574.41,1159.52 1571.99,1160.77 1569.57,1162 1567.16,1163.22 1566.53,1163.53
1564.74,1164.41 1562.32,1165.59 1559.91,1166.75 1557.49,1167.9 1557.44,1167.92 1555.07,1169.03 1552.65,1170.14 1550.24,1171.24 1547.82,1172.32 1547.81,1172.32
1545.4,1173.38 1542.99,1174.42 1540.57,1175.45 1538.15,1176.46 1537.55,1176.71 1535.73,1177.46 1533.32,1178.44 1530.9,1179.41 1528.48,1180.36 1526.54,1181.11
1526.07,1181.29 1523.65,1182.21 1521.23,1183.11 1518.81,1184 1516.4,1184.87 1514.61,1185.5 1513.98,1185.73 1511.56,1186.57 1509.15,1187.39 1506.73,1188.2
1504.31,1189 1501.89,1189.78 1501.52,1189.9 1499.48,1190.54 1497.06,1191.3 1494.64,1192.03 1492.23,1192.75 1489.81,1193.46 1487.39,1194.15 1486.89,1194.29
1484.97,1194.83 1482.56,1195.49 1480.14,1196.14 1477.72,1196.77 1475.31,1197.39 1472.89,1198 1470.47,1198.59 1470.05,1198.69 1468.05,1199.16 1465.64,1199.72
1463.22,1200.27 1460.8,1200.8 1458.39,1201.32 1455.97,1201.83 1453.55,1202.32 1451.13,1202.8 1449.64,1203.08 1448.72,1203.26 1446.3,1203.71 1443.88,1204.14
1441.47,1204.56 1439.05,1204.97 1436.63,1205.36 1434.21,1205.74 1431.8,1206.1 1429.38,1206.45 1426.96,1206.79 1424.55,1207.11 1422.13,1207.42 1421.66,1207.48
1419.71,1207.72 1417.29,1208 1414.88,1208.26 1412.46,1208.52 1410.04,1208.75 1407.63,1208.98 1405.21,1209.19 1402.79,1209.39 1400.37,1209.57 1397.96,1209.74
1395.54,1209.9 1393.12,1210.04 1390.71,1210.17 1388.29,1210.28 1385.87,1210.38 1383.45,1210.47 1381.04,1210.54 1378.62,1210.6 1376.2,1210.64 1373.79,1210.67
1371.37,1210.68 1368.95,1210.69 1366.53,1210.67 1364.12,1210.65 1361.7,1210.61 1359.28,1210.55 1356.87,1210.48 1354.45,1210.4 1352.03,1210.3 1349.61,1210.19
1347.2,1210.06 1344.78,1209.92 1342.36,1209.77 1339.95,1209.6 1337.53,1209.41 1335.11,1209.21 1332.69,1209 1330.28,1208.77 1327.86,1208.53 1325.44,1208.27
1323.03,1208 1320.61,1207.71 1318.76,1207.48 1318.19,1207.41 1315.77,1207.09 1313.36,1206.76 1310.94,1206.41 1308.52,1206.05 1306.11,1205.67 1303.69,1205.27
1301.27,1204.87 1298.85,1204.44 1296.44,1204 1294.02,1203.55 1291.64,1203.08 1291.6,1203.08 1289.19,1202.59 1286.77,1202.08 1284.35,1201.57 1281.93,1201.03
1279.52,1200.48 1277.1,1199.91 1274.68,1199.33 1272.26,1198.73 1272.11,1198.69 1269.85,1198.11 1267.43,1197.48 1265.01,1196.83 1262.6,1196.16 1260.18,1195.48
1257.76,1194.77 1256.14,1194.29 1255.34,1194.06 1252.93,1193.32 1250.51,1192.57 1248.09,1191.79 1245.68,1191.01 1243.26,1190.2 1242.38,1189.9 1240.84,1189.37
1238.42,1188.53 1236.01,1187.67 1233.59,1186.79 1231.17,1185.89 1230.16,1185.5 1228.76,1184.97 1226.34,1184.03 1223.92,1183.08 1221.5,1182.1 1219.09,1181.11
1219.09,1181.11 1216.67,1180.09 1214.25,1179.06 1211.84,1178 1209.42,1176.92 1208.95,1176.71 1207,1175.83 1204.58,1174.71 1202.17,1173.57 1199.75,1172.41
1199.56,1172.32 1197.33,1171.23 1194.92,1170.02 1192.5,1168.8 1190.8,1167.92 1190.08,1167.55 1187.66,1166.28 1185.25,1164.98 1182.83,1163.67 1182.58,1163.53
1180.41,1162.32 1178,1160.96 1175.58,1159.57 1174.83,1159.13 1173.16,1158.16 1170.74,1156.72 1168.33,1155.25 1167.49,1154.74 1165.91,1153.76 1163.49,1152.24
1161.08,1150.7 1160.52,1150.35 1158.66,1149.13 1156.24,1147.53 1153.89,1145.95 1153.82,1145.91 1151.41,1144.25 1148.99,1142.57 1147.55,1141.56 1146.57,1140.86
1144.16,1139.12 1141.74,1137.34 1141.49,1137.16 1139.32,1135.54 1136.9,1133.7 1135.69,1132.77 1134.49,1131.84 1132.07,1129.93 1130.11,1128.37 1129.65,1128
1127.24,1126.03 1124.82,1124.03 1124.76,1123.98 1122.4,1121.98 1119.98,1119.91 1119.61,1119.58 1117.57,1117.79 1115.15,1115.64 1114.64,1115.19 1112.73,1113.45
1110.32,1111.22 1109.86,1110.79 1107.9,1108.94 1105.48,1106.63 1105.25,1106.4 1103.06,1104.26 1100.79,1102 1100.65,1101.86 1098.23,1099.4 1096.49,1097.61
1095.81,1096.9 1093.4,1094.35 1092.33,1093.21 1090.98,1091.75 1088.56,1089.1 1088.3,1088.82 1086.14,1086.39 1084.41,1084.42 1083.73,1083.63 1081.31,1080.81
1080.64,1080.03 1078.89,1077.93 1077,1075.63 1076.48,1074.99 1074.06,1071.98 1073.46,1071.24 1071.64,1068.91 1070.04,1066.84 1069.22,1065.77 1066.81,1062.55
1066.73,1062.45 1064.39,1059.25 1063.52,1058.05 1061.97,1055.88 1060.41,1053.66 1059.56,1052.42 1057.39,1049.26 1057.14,1048.88 1054.72,1045.25 1054.47,1044.87
1052.3,1041.51 1051.64,1040.47 1049.89,1037.67 1048.9,1036.08 1047.47,1033.72 1046.24,1031.68 1045.05,1029.66 1043.67,1027.29 1042.64,1025.47 1041.18,1022.9
1040.22,1021.16 1038.76,1018.5 1037.8,1016.7 1036.43,1014.11 1035.38,1012.08 1034.17,1009.71 1032.97,1007.31 1031.98,1005.32 1030.55,1002.35 1029.86,1000.92
1028.13,997.201 1027.82,996.526 1025.84,992.131 1025.72,991.835 1023.94,987.737 1023.3,986.216 1022.1,983.342 1020.88,980.339 1020.32,978.947 1018.61,974.552
1018.46,974.169 1016.96,970.157 1016.05,967.634 1015.37,965.763 1013.85,961.368 1013.63,960.719 1012.38,956.973 1011.21,953.325 1010.97,952.578 1009.63,948.183
1008.8,945.359 1008.33,943.788 1007.1,939.394 1006.38,936.702 1005.92,934.999 1004.8,930.604 1003.96,927.156 1003.73,926.209 1002.72,921.814 1001.76,917.419
1001.54,916.393 1000.85,913.025 999.993,908.63 999.189,904.235 999.127,903.872 998.437,899.84 997.735,895.445 997.084,891.051 996.709,888.311 996.483,886.656
995.932,882.261 995.43,877.866 994.977,873.471 994.573,869.076 994.292,865.608 994.217,864.682 993.91,860.287 993.65,855.892 993.437,851.497 993.273,847.102
993.155,842.707 993.084,838.313 993.06,833.918 993.083,829.523 993.152,825.128 993.267,820.733 993.429,816.338 993.637,811.944 993.891,807.549 994.191,803.154
994.292,801.869 994.537,798.759 994.93,794.364 995.368,789.97 995.852,785.575 996.382,781.18 996.709,778.677 996.958,776.785 997.58,772.39 998.248,767.995
998.963,763.601 999.127,762.651 999.723,759.206 1000.53,754.811 1001.38,750.416 1001.54,749.635 1002.28,746.021 1003.23,741.626 1003.96,738.399 1004.23,737.232
1005.27,732.837 1006.36,728.442 1006.38,728.355 1007.49,724.047 1008.68,719.652 1008.8,719.227 1009.91,715.258 1011.19,710.863 1011.21,710.787 1012.52,706.468
1013.63,702.925 1013.9,702.073 1015.33,697.678 1016.05,695.529 1016.8,693.283 1018.33,688.889 1018.46,688.517 1019.91,684.494 1020.88,681.869 1021.54,680.099
1023.22,675.704 1023.3,675.504 1024.95,671.309 1025.72,669.426 1026.74,666.914 1028.13,663.575 1028.58,662.52 1030.47,658.125 1030.55,657.937 1032.41,653.73
1032.97,652.508 1034.41,649.335 1035.38,647.254 1036.47,644.94 1037.8,642.163 1038.58,640.545 1040.22,637.224 1040.75,636.151 1042.64,632.426 1042.98,631.756
1045.05,627.76 1045.26,627.361 1047.47,623.219 1047.61,622.966 1049.89,618.794 1050.01,618.571 1052.3,614.479 1052.47,614.177 1054.72,610.268 1055,609.782
1057.14,606.155 1057.59,605.387 1059.56,602.133 1060.25,600.992 1061.97,598.2 1062.97,596.597 1064.39,594.349 1065.76,592.202 1066.81,590.577 1068.61,587.808
1069.22,586.88 1071.53,583.413 1071.64,583.255 1074.06,579.705 1074.53,579.018 1076.48,576.223 1077.6,574.623 1078.89,572.803 1080.74,570.228 1081.31,569.442
1083.73,566.142 1083.95,565.833 1086.14,562.906 1087.25,561.439 1088.56,559.722 1090.62,557.044 1090.98,556.588 1093.4,553.512 1094.08,552.649 1095.81,550.486
1097.62,548.254 1098.23,547.506 1100.65,544.575 1101.24,543.859 1103.06,541.693 1104.95,539.465 1105.48,538.851 1107.9,536.057 1108.76,535.07 1110.32,533.305
1112.66,530.675 1112.73,530.59 1115.15,527.923 1116.65,526.28 1117.57,525.291 1119.98,522.698 1120.75,521.885 1122.4,520.144 1124.82,517.622 1124.95,517.49
1127.24,515.144 1129.25,513.096 1129.65,512.694 1132.07,510.283 1133.67,508.701 1134.49,507.903 1136.9,505.558 1138.21,504.306 1139.32,503.243 1141.74,500.961
1142.86,499.911 1144.16,498.711 1146.57,496.49 1147.64,495.516 1148.99,494.3 1151.41,492.138 1152.55,491.121 1153.82,490.006 1156.24,487.903 1157.61,486.727
1158.66,485.826 1161.08,483.779 1162.8,482.332 1163.49,481.756 1165.91,479.763 1168.15,477.937 1168.33,477.791 1170.74,475.852 1173.16,473.932 1173.66,473.542
1175.58,472.042 1178,470.173 1179.34,469.147 1180.41,468.33 1182.83,466.511 1185.19,464.752 1185.25,464.712 1187.66,462.943 1190.08,461.193 1191.25,460.358
1192.5,459.466 1194.92,457.763 1197.33,456.078 1197.5,455.963 1199.75,454.421 1202.17,452.782 1203.98,451.568 1204.58,451.163 1207,449.57 1209.42,447.993
1210.69,447.173 1211.84,446.439 1214.25,444.907 1216.67,443.392 1217.66,442.778 1219.09,441.899 1221.5,440.426 1223.92,438.971 1224.91,438.384 1226.34,437.536
1228.76,436.122 1231.17,434.725 1232.46,433.989 1233.59,433.347 1236.01,431.99 1238.42,430.65 1240.35,429.594 1240.84,429.327 1243.26,428.025 1245.68,426.74
1248.09,425.471 1248.62,425.199 1250.51,424.223 1252.93,422.992 1255.34,421.777 1257.31,420.804 1257.76,420.579 1260.18,419.401 1262.6,418.239 1265.01,417.093
1266.47,416.409 1267.43,415.964 1269.85,414.853 1272.26,413.759 1274.68,412.679 1276.19,412.015 1277.1,411.617 1279.52,410.572 1281.93,409.543 1284.35,408.53
1286.55,407.62 1286.77,407.531 1289.19,406.552 1291.6,405.587 1294.02,404.637 1296.44,403.703 1297.69,403.225 1298.85,402.785 1301.27,401.883 1303.69,400.996
1306.11,400.124 1308.52,399.267 1309.78,398.83 1310.94,398.426 1313.36,397.601 1315.77,396.791 1318.19,395.995 1320.61,395.214 1323.03,394.447 1323.06,394.435
1325.44,393.697 1327.86,392.962 1330.28,392.241 1332.69,391.534 1335.11,390.842 1337.53,390.164 1337.98,390.04 1339.95,389.502 1342.36,388.855 1344.78,388.221
1347.2,387.602 1349.61,386.997 1352.03,386.407 1354.45,385.83 1355.24,385.646 1356.87,385.269 1359.28,384.722 1361.7,384.189 1364.12,383.67 1366.53,383.165
1368.95,382.674 1371.37,382.197 1373.79,381.735 1376.2,381.286 1376.4,381.251 1378.62,380.852 1381.04,380.432 1383.45,380.026 1385.87,379.634 1388.29,379.255
1390.71,378.891 1393.12,378.54 1395.54,378.204 1397.96,377.881 1400.37,377.572 1402.79,377.278 1405.21,376.997 1406.48,376.856 1407.63,376.73 1410.04,376.477
1412.46,376.238 1414.88,376.013 1417.29,375.802 1419.71,375.604 1422.13,375.421 1424.55,375.252 1426.96,375.096 1429.38,374.954 1431.8,374.827 1434.21,374.713
1436.63,374.613 1439.05,374.527 1441.47,374.455 1443.88,374.397 1446.3,374.354 1448.72,374.324 1451.13,374.308 1453.55,374.307 1455.97,374.319 1458.39,374.346
1460.8,374.387 1463.22,374.442 1465.64,374.511 1468.05,374.595 1470.47,374.693 1472.89,374.805 1475.31,374.931 1477.72,375.072 1480.14,375.228 1482.56,375.398
1484.97,375.582 1487.39,375.781 1489.81,375.994 1492.23,376.223 1494.64,376.466 1497.06,376.723 1498.24,376.856 1499.48,376.996 1501.89,377.284 1504.31,377.586
1506.73,377.904 1509.15,378.236 1511.56,378.584 1513.98,378.947 1516.4,379.325 1518.81,379.718 1521.23,380.127 1523.65,380.552 1526.07,380.991 1527.44,381.251
1528.48,381.447 1530.9,381.918 1533.32,382.405 1535.73,382.908 1538.15,383.427 1540.57,383.962 1542.99,384.513 1545.4,385.081 1547.74,385.646 1547.82,385.664
1550.24,386.265 1552.65,386.882 1555.07,387.515 1557.49,388.166 1559.91,388.833 1562.32,389.517 1564.13,390.04 1564.74,390.218 1567.16,390.937 1569.57,391.673
1571.99,392.427 1574.41,393.198 1576.83,393.987 1578.17,394.435 1579.24,394.794 1581.66,395.619 1584.08,396.463 1586.49,397.325 1588.91,398.205 1590.59,398.83
1591.33,399.104 1593.75,400.022 1596.16,400.96 1598.58,401.916 1601,402.891 1601.81,403.225 1603.41,403.887 1605.83,404.903 1608.25,405.938 1610.67,406.993
1612.08,407.62 1613.08,408.069 1615.5,409.167 1617.92,410.284 1620.34,411.423 1621.57,412.015 1622.75,412.583 1625.17,413.766 1627.59,414.969 1630,416.195
1630.42,416.409 1632.42,417.444 1634.84,418.715 1637.26,420.009 1638.72,420.804 1639.67,421.327 1642.09,422.669 1644.51,424.034 1646.54,425.199 1646.92,425.422
1649.34,426.838 1651.76,428.277 1653.93,429.594 1654.18,429.74 1656.59,431.232 1659.01,432.749 1660.96,433.989 1661.43,434.291 1663.84,435.863 1666.26,437.46
1667.64,438.384 1668.68,439.085 1671.1,440.74 1673.51,442.422 1674.02,442.778 1675.93,444.135 1678.35,445.878 1680.12,447.173 1680.76,447.649 1683.18,449.455
1685.6,451.288 1685.96,451.568 1688.02,453.159 1690.43,455.059 1691.57,455.963 1692.85,456.994 1695.27,458.963 1696.95,460.358 1697.68,460.966 1700.1,463.008
1702.14,464.752 1702.52,465.083 1704.94,467.2 1707.13,469.147 1707.35,469.35 1709.77,471.546 1711.94,473.542 1712.19,473.776 1714.6,476.054 1716.58,477.937
1717.02,478.369 1719.44,480.732 1721.05,482.332 1721.86,483.137 1724.27,485.59 1725.38,486.727 1726.69,488.091 1729.11,490.638 1729.56,491.121 1731.52,493.241
1733.61,495.516 1733.94,495.889 1736.36,498.599 1737.52,499.911 1738.78,501.361 1741.19,504.177 1741.3,504.306 1743.61,507.062 1744.97,508.701 1746.03,510.003
1748.44,513.005 1748.52,513.096 1750.86,516.084 1751.95,517.49 1753.28,519.229 1755.28,521.885 1755.7,522.442 1758.11,525.735 1758.51,526.28 1760.53,529.112
1761.63,530.675 1762.95,532.57 1764.66,535.07 1765.36,536.112 1767.6,539.465 1767.78,539.745 1770.2,543.48 1770.44,543.859 1772.62,547.32 1773.2,548.254
1775.03,551.269 1775.87,552.649 1777.45,555.332 1778.45,557.044 1779.87,559.518 1780.96,561.439 1782.28,563.837 1783.38,565.833 1784.7,568.296 1785.73,570.228
1787.12,572.909 1788,574.623 1789.54,577.685 1790.2,579.018 1791.95,582.639 1792.33,583.413 1794.37,587.785 1794.38,587.808 1796.37,592.202 1796.79,593.164
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