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Je mehr da ist, desto schneller wächst es. So verhalten sich zum Beispiel Bakterienkulturen ohne Platzmangel oder radioaktive Stoffe.",[564,767,768,769,916,917,1174,1175,1219],{},"Die Lösungen sind genau die Funktionen ",[573,770,772,817],{"className":771},[576],[573,773,775],{"className":774},[580],[582,776,777],{"xmlns":584},[586,778,779,814],{},[589,780,781,783,785,787,789,791,793,795,799,801,803],{},[595,782,597],{},[599,784,607],{"stretchy":606},[595,786,610],{},[599,788,613],{"stretchy":606},[599,790,616],{},[595,792,597],{},[599,794,607],{"stretchy":606},[796,797,798],"mn",{},"0",[599,800,613],{"stretchy":606},[599,802,622],{},[592,804,805,808],{},[595,806,807],{},"e",[589,809,810,812],{},[595,811,619],{},[595,813,610],{},[632,815,816],{"encoding":634},"f(t) = f(0) \\cdot e^{kt}",[573,818,820,847,874],{"className":819,"ariaHidden":640},[639],[573,821,823,826,829,832,835,838,841,844],{"className":822},[644],[573,824],{"className":825,"style":748},[648],[573,827,597],{"className":828,"style":658},[653,657],[573,830,607],{"className":831},[700],[573,833,610],{"className":834},[653,657],[573,836,613],{"className":837},[707],[573,839],{"className":840,"style":712},[711],[573,842,616],{"className":843},[716],[573,845],{"className":846,"style":712},[711],[573,848,850,853,856,859,862,865,868,871],{"className":849},[644],[573,851],{"className":852,"style":748},[648],[573,854,597],{"className":855,"style":658},[653,657],[573,857,607],{"className":858},[700],[573,860,798],{"className":861},[653],[573,863,613],{"className":864},[707],[573,866],{"className":867,"style":734},[711],[573,869,622],{"className":870},[738],[573,872],{"className":873,"style":734},[711],[573,875,877,881],{"className":876},[644],[573,878],{"className":879,"style":880},[648],"height:0.8491em;",[573,882,884,887],{"className":883},[653],[573,885,807],{"className":886},[653,657],[573,888,890],{"className":889},[662],[573,891,893],{"className":892},[666],[573,894,896],{"className":895},[670],[573,897,899],{"className":898,"style":880},[674],[573,900,901,904],{"style":678},[573,902],{"className":903,"style":683},[682],[573,905,907],{"className":906},[687,688,689,690],[573,908,910,913],{"className":909},[653,690],[573,911,619],{"className":912,"style":730},[653,657,690],[573,914,610],{"className":915},[653,657,690],". Das prüft man durch Ableiten mit der Kettenregel: ",[573,918,920,984],{"className":919},[576],[573,921,923],{"className":922},[580],[582,924,925],{"xmlns":584},[586,926,927,981],{},[589,928,929,935,937,939,941,943,945,947,949,951,953,955,957,967,969,971,973,975,977,979],{},[592,930,931,933],{},[595,932,597],{},[599,934,603],{"mathvariant":601,"lspace":602,"rspace":602},[599,936,607],{"stretchy":606},[595,938,610],{},[599,940,613],{"stretchy":606},[599,942,616],{},[595,944,597],{},[599,946,607],{"stretchy":606},[796,948,798],{},[599,950,613],{"stretchy":606},[599,952,622],{},[595,954,619],{},[599,956,622],{},[592,958,959,961],{},[595,960,807],{},[589,962,963,965],{},[595,964,619],{},[595,966,610],{},[599,968,616],{},[595,970,619],{},[599,972,622],{},[595,974,597],{},[599,976,607],{"stretchy":606},[595,978,610],{},[599,980,613],{"stretchy":606},[632,982,983],{"encoding":634},"f'(t) = f(0) \\cdot k \\cdot e^{kt} = k \\cdot f(t)",[573,985,987,1043,1070,1088,1138,1156],{"className":986,"ariaHidden":640},[639],[573,988,990,993,1025,1028,1031,1034,1037,1040],{"className":989},[644],[573,991],{"className":992,"style":649},[648],[573,994,996,999],{"className":995},[653],[573,997,597],{"className":998,"style":658},[653,657],[573,1000,1002],{"className":1001},[662],[573,1003,1005],{"className":1004},[666],[573,1006,1008],{"className":1007},[670],[573,1009,1011],{"className":1010,"style":675},[674],[573,1012,1013,1016],{"style":678},[573,1014],{"className":1015,"style":683},[682],[573,1017,1019],{"className":1018},[687,688,689,690],[573,1020,1022],{"className":1021},[653,690],[573,1023,603],{"className":1024},[653,690],[573,1026,607],{"className":1027},[700],[573,1029,610],{"className":1030},[653,657],[573,1032,613],{"className":1033},[707],[573,1035],{"className":1036,"style":712},[711],[573,1038,616],{"className":1039},[716],[573,1041],{"className":1042,"style":712},[711],[573,1044,1046,1049,1052,1055,1058,1061,1064,1067],{"className":1045},[644],[573,1047],{"className":1048,"style":748},[648],[573,1050,597],{"className":1051,"style":658},[653,657],[573,1053,607],{"className":1054},[700],[573,1056,798],{"className":1057},[653],[573,1059,613],{"className":1060},[707],[573,1062],{"className":1063,"style":734},[711],[573,1065,622],{"className":1066},[738],[573,1068],{"className":1069,"style":734},[711],[573,1071,1073,1076,1079,1082,1085],{"className":1072},[644],[573,1074],{"className":1075,"style":726},[648],[573,1077,619],{"className":1078,"style":730},[653,657],[573,1080],{"className":1081,"style":734},[711],[573,1083,622],{"className":1084},[738],[573,1086],{"className":1087,"style":734},[711],[573,1089,1091,1094,1129,1132,1135],{"className":1090},[644],[573,1092],{"className":1093,"style":880},[648],[573,1095,1097,1100],{"className":1096},[653],[573,1098,807],{"className":1099},[653,657],[573,1101,1103],{"className":1102},[662],[573,1104,1106],{"className":1105},[666],[573,1107,1109],{"className":1108},[670],[573,1110,1112],{"className":1111,"style":880},[674],[573,1113,1114,1117],{"style":678},[573,1115],{"className":1116,"style":683},[682],[573,1118,1120],{"className":1119},[687,688,689,690],[573,1121,1123,1126],{"className":1122},[653,690],[573,1124,619],{"className":1125,"style":730},[653,657,690],[573,1127,610],{"className":1128},[653,657,690],[573,1130],{"className":1131,"style":712},[711],[573,1133,616],{"className":1134},[716],[573,1136],{"className":1137,"style":712},[711],[573,1139,1141,1144,1147,1150,1153],{"className":1140},[644],[573,1142],{"className":1143,"style":726},[648],[573,1145,619],{"className":1146,"style":730},[653,657],[573,1148],{"className":1149,"style":734},[711],[573,1151,622],{"className":1152},[738],[573,1154],{"className":1155,"style":734},[711],[573,1157,1159,1162,1165,1168,1171],{"className":1158},[644],[573,1160],{"className":1161,"style":748},[648],[573,1163,597],{"className":1164,"style":658},[653,657],[573,1166,607],{"className":1167},[700],[573,1169,610],{"className":1170},[653,657],[573,1172,613],{"className":1173},[707],". Die Gleichung legt die Funktion bis auf den Anfangswert ",[573,1176,1178,1198],{"className":1177},[576],[573,1179,1181],{"className":1180},[580],[582,1182,1183],{"xmlns":584},[586,1184,1185,1195],{},[589,1186,1187,1189,1191,1193],{},[595,1188,597],{},[599,1190,607],{"stretchy":606},[796,1192,798],{},[599,1194,613],{"stretchy":606},[632,1196,1197],{"encoding":634},"f(0)",[573,1199,1201],{"className":1200,"ariaHidden":640},[639],[573,1202,1204,1207,1210,1213,1216],{"className":1203},[644],[573,1205],{"className":1206,"style":748},[648],[573,1208,597],{"className":1209,"style":658},[653,657],[573,1211,607],{"className":1212},[700],[573,1214,798],{"className":1215},[653],[573,1217,613],{"className":1218},[707]," fest.",[1221,1222,1223,1285,1343],"ul",{},[1224,1225,1226,1280,1281,1284],"li",{},[573,1227,1229,1248],{"className":1228},[576],[573,1230,1232],{"className":1231},[580],[582,1233,1234],{"xmlns":584},[586,1235,1236,1245],{},[589,1237,1238,1240,1243],{},[595,1239,619],{},[599,1241,1242],{},">",[796,1244,798],{},[632,1246,1247],{"encoding":634},"k > 0",[573,1249,1251,1270],{"className":1250,"ariaHidden":640},[639],[573,1252,1254,1258,1261,1264,1267],{"className":1253},[644],[573,1255],{"className":1256,"style":1257},[648],"height:0.7335em;vertical-align:-0.0391em;",[573,1259,619],{"className":1260,"style":730},[653,657],[573,1262],{"className":1263,"style":712},[711],[573,1265,1242],{"className":1266},[716],[573,1268],{"className":1269,"style":712},[711],[573,1271,1273,1277],{"className":1272},[644],[573,1274],{"className":1275,"style":1276},[648],"height:0.6444em;",[573,1278,798],{"className":1279},[653],": ",[568,1282,1283],{},"exponentielles Wachstum",".",[1224,1286,1287,1280,1339,1342],{},[573,1288,1290,1309],{"className":1289},[576],[573,1291,1293],{"className":1292},[580],[582,1294,1295],{"xmlns":584},[586,1296,1297,1306],{},[589,1298,1299,1301,1304],{},[595,1300,619],{},[599,1302,1303],{},"\u003C",[796,1305,798],{},[632,1307,1308],{"encoding":634},"k \u003C 0",[573,1310,1312,1330],{"className":1311,"ariaHidden":640},[639],[573,1313,1315,1318,1321,1324,1327],{"className":1314},[644],[573,1316],{"className":1317,"style":1257},[648],[573,1319,619],{"className":1320,"style":730},[653,657],[573,1322],{"className":1323,"style":712},[711],[573,1325,1303],{"className":1326},[716],[573,1328],{"className":1329,"style":712},[711],[573,1331,1333,1336],{"className":1332},[644],[573,1334],{"className":1335,"style":1276},[648],[573,1337,798],{"className":1338},[653],[568,1340,1341],{},"exponentielle Abnahme"," (Zerfall).",[1224,1344,1345,1346,1408,1409,1549,1550,1284],{},"Pro Zeiteinheit wird der Bestand mit ",[573,1347,1349,1367],{"className":1348},[576],[573,1350,1352],{"className":1351},[580],[582,1353,1354],{"xmlns":584},[586,1355,1356,1364],{},[589,1357,1358],{},[592,1359,1360,1362],{},[595,1361,807],{},[595,1363,619],{},[632,1365,1366],{"encoding":634},"e^{k}",[573,1368,1370],{"className":1369,"ariaHidden":640},[639],[573,1371,1373,1376],{"className":1372},[644],[573,1374],{"className":1375,"style":880},[648],[573,1377,1379,1382],{"className":1378},[653],[573,1380,807],{"className":1381},[653,657],[573,1383,1385],{"className":1384},[662],[573,1386,1388],{"className":1387},[666],[573,1389,1391],{"className":1390},[670],[573,1392,1394],{"className":1393,"style":880},[674],[573,1395,1396,1399],{"style":678},[573,1397],{"className":1398,"style":683},[682],[573,1400,1402],{"className":1401},[687,688,689,690],[573,1403,1405],{"className":1404},[653,690],[573,1406,619],{"className":1407,"style":730},[653,657,690]," multipliziert. Der prozentuale Zuwachs pro Zeiteinheit ist also ",[573,1410,1412,1449],{"className":1411},[576],[573,1413,1415],{"className":1414},[580],[582,1416,1417],{"xmlns":584},[586,1418,1419,1446],{},[589,1420,1421,1423,1425,1428,1430,1432,1438,1441,1444],{},[595,1422,564],{},[599,1424,616],{},[796,1426,1427],{},"100",[599,1429,622],{},[599,1431,607],{"stretchy":606},[592,1433,1434,1436],{},[595,1435,807],{},[595,1437,619],{},[599,1439,1440],{},"−",[796,1442,1443],{},"1",[599,1445,613],{"stretchy":606},[632,1447,1448],{"encoding":634},"p = 100 \\cdot (e^k - 1)",[573,1450,1452,1471,1489,1537],{"className":1451,"ariaHidden":640},[639],[573,1453,1455,1459,1462,1465,1468],{"className":1454},[644],[573,1456],{"className":1457,"style":1458},[648],"height:0.625em;vertical-align:-0.1944em;",[573,1460,564],{"className":1461},[653,657],[573,1463],{"className":1464,"style":712},[711],[573,1466,616],{"className":1467},[716],[573,1469],{"className":1470,"style":712},[711],[573,1472,1474,1477,1480,1483,1486],{"className":1473},[644],[573,1475],{"className":1476,"style":1276},[648],[573,1478,1427],{"className":1479},[653],[573,1481],{"className":1482,"style":734},[711],[573,1484,622],{"className":1485},[738],[573,1487],{"className":1488,"style":734},[711],[573,1490,1492,1496,1499,1528,1531,1534],{"className":1491},[644],[573,1493],{"className":1494,"style":1495},[648],"height:1.0991em;vertical-align:-0.25em;",[573,1497,607],{"className":1498},[700],[573,1500,1502,1505],{"className":1501},[653],[573,1503,807],{"className":1504},[653,657],[573,1506,1508],{"className":1507},[662],[573,1509,1511],{"className":1510},[666],[573,1512,1514],{"className":1513},[670],[573,1515,1517],{"className":1516,"style":880},[674],[573,1518,1519,1522],{"style":678},[573,1520],{"className":1521,"style":683},[682],[573,1523,1525],{"className":1524},[687,688,689,690],[573,1526,619],{"className":1527,"style":730},[653,657,690],[573,1529],{"className":1530,"style":734},[711],[573,1532,1440],{"className":1533},[738],[573,1535],{"className":1536,"style":734},[711],[573,1538,1540,1543,1546],{"className":1539},[644],[573,1541],{"className":1542,"style":748},[648],[573,1544,1443],{"className":1545},[653],[573,1547,613],{"className":1548},[707],", nicht ",[573,1551,1553,1571],{"className":1552},[576],[573,1554,1556],{"className":1555},[580],[582,1557,1558],{"xmlns":584},[586,1559,1560,1568],{},[589,1561,1562,1564,1566],{},[796,1563,1427],{},[599,1565,622],{},[595,1567,619],{},[632,1569,1570],{"encoding":634},"100 \\cdot k",[573,1572,1574,1592],{"className":1573,"ariaHidden":640},[639],[573,1575,1577,1580,1583,1586,1589],{"className":1576},[644],[573,1578],{"className":1579,"style":1276},[648],[573,1581,1427],{"className":1582},[653],[573,1584],{"className":1585,"style":734},[711],[573,1587,622],{"className":1588},[738],[573,1590],{"className":1591,"style":734},[711],[573,1593,1595,1598],{"className":1594},[644],[573,1596],{"className":1597,"style":726},[648],[573,1599,619],{"className":1600,"style":730},[653,657],[559,1602,1604],{"id":1603},"beispiele","Beispiele",[564,1606,1607,1608,1758,1759,1788,1789,1817,1818,1847],{},"Eine Bakterienkultur wächst nach ",[573,1609,1611,1650],{"className":1610},[576],[573,1612,1614],{"className":1613},[580],[582,1615,1616],{"xmlns":584},[586,1617,1618,1647],{},[589,1619,1620,1626,1628,1630,1632,1634,1637,1639,1641,1643,1645],{},[592,1621,1622,1624],{},[595,1623,597],{},[599,1625,603],{"mathvariant":601,"lspace":602,"rspace":602},[599,1627,607],{"stretchy":606},[595,1629,610],{},[599,1631,613],{"stretchy":606},[599,1633,616],{},[796,1635,1636],{},"0,3",[599,1638,622],{},[595,1640,597],{},[599,1642,607],{"stretchy":606},[595,1644,610],{},[599,1646,613],{"stretchy":606},[632,1648,1649],{"encoding":634},"f'(t) = 0{,}3 \\cdot f(t)",[573,1651,1653,1709,1740],{"className":1652,"ariaHidden":640},[639],[573,1654,1656,1659,1691,1694,1697,1700,1703,1706],{"className":1655},[644],[573,1657],{"className":1658,"style":649},[648],[573,1660,1662,1665],{"className":1661},[653],[573,1663,597],{"className":1664,"style":658},[653,657],[573,1666,1668],{"className":1667},[662],[573,1669,1671],{"className":1670},[666],[573,1672,1674],{"className":1673},[670],[573,1675,1677],{"className":1676,"style":675},[674],[573,1678,1679,1682],{"style":678},[573,1680],{"className":1681,"style":683},[682],[573,1683,1685],{"className":1684},[687,688,689,690],[573,1686,1688],{"className":1687},[653,690],[573,1689,603],{"className":1690},[653,690],[573,1692,607],{"className":1693},[700],[573,1695,610],{"className":1696},[653,657],[573,1698,613],{"className":1699},[707],[573,1701],{"className":1702,"style":712},[711],[573,1704,616],{"className":1705},[716],[573,1707],{"className":1708,"style":712},[711],[573,1710,1712,1716,1719,1727,1731,1734,1737],{"className":1711},[644],[573,1713],{"className":1714,"style":1715},[648],"height:0.8389em;vertical-align:-0.1944em;",[573,1717,798],{"className":1718},[653],[573,1720,1722],{"className":1721},[653],[573,1723,1726],{"className":1724},[1725],"mpunct",",",[573,1728,1730],{"className":1729},[653],"3",[573,1732],{"className":1733,"style":734},[711],[573,1735,622],{"className":1736},[738],[573,1738],{"className":1739,"style":734},[711],[573,1741,1743,1746,1749,1752,1755],{"className":1742},[644],[573,1744],{"className":1745,"style":748},[648],[573,1747,597],{"className":1748,"style":658},[653,657],[573,1750,607],{"className":1751},[700],[573,1753,610],{"className":1754},[653,657],[573,1756,613],{"className":1757},[707]," (",[573,1760,1762,1775],{"className":1761},[576],[573,1763,1765],{"className":1764},[580],[582,1766,1767],{"xmlns":584},[586,1768,1769,1773],{},[589,1770,1771],{},[595,1772,610],{},[632,1774,610],{"encoding":634},[573,1776,1778],{"className":1777,"ariaHidden":640},[639],[573,1779,1781,1785],{"className":1780},[644],[573,1782],{"className":1783,"style":1784},[648],"height:0.6151em;",[573,1786,610],{"className":1787},[653,657]," in Stunden) und hat am Anfang ",[573,1790,1792,1805],{"className":1791},[576],[573,1793,1795],{"className":1794},[580],[582,1796,1797],{"xmlns":584},[586,1798,1799,1803],{},[589,1800,1801],{},[796,1802,1427],{},[632,1804,1427],{"encoding":634},[573,1806,1808],{"className":1807,"ariaHidden":640},[639],[573,1809,1811,1814],{"className":1810},[644],[573,1812],{"className":1813,"style":1276},[648],[573,1815,1427],{"className":1816},[653]," Bakterien. Nach ",[573,1819,1821,1835],{"className":1820},[576],[573,1822,1824],{"className":1823},[580],[582,1825,1826],{"xmlns":584},[586,1827,1828,1833],{},[589,1829,1830],{},[796,1831,1832],{},"5",[632,1834,1832],{"encoding":634},[573,1836,1838],{"className":1837,"ariaHidden":640},[639],[573,1839,1841,1844],{"className":1840},[644],[573,1842],{"className":1843,"style":1276},[648],[573,1845,1832],{"className":1846},[653]," Stunden:",[573,1849,1852],{"className":1850},[1851],"katex-display",[573,1853,1855,1913],{"className":1854},[576],[573,1856,1858],{"className":1857},[580],[582,1859,1861],{"xmlns":584,"display":1860},"block",[586,1862,1863,1910],{},[589,1864,1865,1867,1869,1871,1873,1875,1877,1879,1891,1893,1895,1897,1904,1907],{},[595,1866,597],{},[599,1868,607],{"stretchy":606},[796,1870,1832],{},[599,1872,613],{"stretchy":606},[599,1874,616],{},[796,1876,1427],{},[599,1878,622],{},[592,1880,1881,1883],{},[595,1882,807],{},[589,1884,1885,1887,1889],{},[796,1886,1636],{},[599,1888,622],{},[796,1890,1832],{},[599,1892,616],{},[796,1894,1427],{},[599,1896,622],{},[592,1898,1899,1901],{},[595,1900,807],{},[796,1902,1903],{},"1,5",[599,1905,1906],{},"≈",[796,1908,1909],{},"448,17",[632,1911,1912],{"encoding":634},"f(5) = 100 \\cdot e^{0{,}3 \\cdot 5} = 100 \\cdot e^{1{,}5} \\approx 448{,}17",[573,1914,1916,1943,1961,2025,2043,2099],{"className":1915,"ariaHidden":640},[639],[573,1917,1919,1922,1925,1928,1931,1934,1937,1940],{"className":1918},[644],[573,1920],{"className":1921,"style":748},[648],[573,1923,597],{"className":1924,"style":658},[653,657],[573,1926,607],{"className":1927},[700],[573,1929,1832],{"className":1930},[653],[573,1932,613],{"className":1933},[707],[573,1935],{"className":1936,"style":712},[711],[573,1938,616],{"className":1939},[716],[573,1941],{"className":1942,"style":712},[711],[573,1944,1946,1949,1952,1955,1958],{"className":1945},[644],[573,1947],{"className":1948,"style":1276},[648],[573,1950,1427],{"className":1951},[653],[573,1953],{"className":1954,"style":734},[711],[573,1956,622],{"className":1957},[738],[573,1959],{"className":1960,"style":734},[711],[573,1962,1964,1968,2016,2019,2022],{"className":1963},[644],[573,1965],{"className":1966,"style":1967},[648],"height:0.8641em;",[573,1969,1971,1974],{"className":1970},[653],[573,1972,807],{"className":1973},[653,657],[573,1975,1977],{"className":1976},[662],[573,1978,1980],{"className":1979},[666],[573,1981,1983],{"className":1982},[670],[573,1984,1986],{"className":1985,"style":1967},[674],[573,1987,1989,1992],{"style":1988},"top:-3.113em;margin-right:0.05em;",[573,1990],{"className":1991,"style":683},[682],[573,1993,1995],{"className":1994},[687,688,689,690],[573,1996,1998,2001,2007,2010,2013],{"className":1997},[653,690],[573,1999,798],{"className":2000},[653,690],[573,2002,2004],{"className":2003},[653,690],[573,2005,1726],{"className":2006},[1725,690],[573,2008,1730],{"className":2009},[653,690],[573,2011,622],{"className":2012},[738,690],[573,2014,1832],{"className":2015},[653,690],[573,2017],{"className":2018,"style":712},[711],[573,2020,616],{"className":2021},[716],[573,2023],{"className":2024,"style":712},[711],[573,2026,2028,2031,2034,2037,2040],{"className":2027},[644],[573,2029],{"className":2030,"style":1276},[648],[573,2032,1427],{"className":2033},[653],[573,2035],{"className":2036,"style":734},[711],[573,2038,622],{"className":2039},[738],[573,2041],{"className":2042,"style":734},[711],[573,2044,2046,2049,2090,2093,2096],{"className":2045},[644],[573,2047],{"className":2048,"style":1967},[648],[573,2050,2052,2055],{"className":2051},[653],[573,2053,807],{"className":2054},[653,657],[573,2056,2058],{"className":2057},[662],[573,2059,2061],{"className":2060},[666],[573,2062,2064],{"className":2063},[670],[573,2065,2067],{"className":2066,"style":1967},[674],[573,2068,2069,2072],{"style":1988},[573,2070],{"className":2071,"style":683},[682],[573,2073,2075],{"className":2074},[687,688,689,690],[573,2076,2078,2081,2087],{"className":2077},[653,690],[573,2079,1443],{"className":2080},[653,690],[573,2082,2084],{"className":2083},[653,690],[573,2085,1726],{"className":2086},[1725,690],[573,2088,1832],{"className":2089},[653,690],[573,2091],{"className":2092,"style":712},[711],[573,2094,1906],{"className":2095},[716],[573,2097],{"className":2098,"style":712},[711],[573,2100,2102,2105,2109,2115],{"className":2101},[644],[573,2103],{"className":2104,"style":1715},[648],[573,2106,2108],{"className":2107},[653],"448",[573,2110,2112],{"className":2111},[653],[573,2113,1726],{"className":2114},[1725],[573,2116,2118],{"className":2117},[653],"17",[564,2120,2121,2122,1280,2224,2424,2425,2589,2590,1284],{},"Die Verdopplungszeit folgt aus ",[573,2123,2125,2152],{"className":2124},[576],[573,2126,2128],{"className":2127},[580],[582,2129,2130],{"xmlns":584},[586,2131,2132,2149],{},[589,2133,2134,2144,2146],{},[592,2135,2136,2138],{},[595,2137,807],{},[589,2139,2140,2142],{},[796,2141,1636],{},[595,2143,610],{},[599,2145,616],{},[796,2147,2148],{},"2",[632,2150,2151],{"encoding":634},"e^{0{,}3t} = 2",[573,2153,2155,2215],{"className":2154,"ariaHidden":640},[639],[573,2156,2158,2162,2206,2209,2212],{"className":2157},[644],[573,2159],{"className":2160,"style":2161},[648],"height:0.8141em;",[573,2163,2165,2168],{"className":2164},[653],[573,2166,807],{"className":2167},[653,657],[573,2169,2171],{"className":2170},[662],[573,2172,2174],{"className":2173},[666],[573,2175,2177],{"className":2176},[670],[573,2178,2180],{"className":2179,"style":2161},[674],[573,2181,2182,2185],{"style":678},[573,2183],{"className":2184,"style":683},[682],[573,2186,2188],{"className":2187},[687,688,689,690],[573,2189,2191,2194,2200,2203],{"className":2190},[653,690],[573,2192,798],{"className":2193},[653,690],[573,2195,2197],{"className":2196},[653,690],[573,2198,1726],{"className":2199},[1725,690],[573,2201,1730],{"className":2202},[653,690],[573,2204,610],{"className":2205},[653,657,690],[573,2207],{"className":2208,"style":712},[711],[573,2210,616],{"className":2211},[716],[573,2213],{"className":2214,"style":712},[711],[573,2216,2218,2221],{"className":2217},[644],[573,2219],{"className":2220,"style":1276},[648],[573,2222,2148],{"className":2223},[653],[573,2225,2227,2263],{"className":2226},[576],[573,2228,2230],{"className":2229},[580],[582,2231,2232],{"xmlns":584},[586,2233,2234,2260],{},[589,2235,2236,2238,2240,2255,2257],{},[595,2237,610],{},[599,2239,616],{},[2241,2242,2243,2253],"mfrac",{},[589,2244,2245,2248,2251],{},[595,2246,2247],{},"ln",[599,2249,2250],{},"⁡",[796,2252,2148],{},[796,2254,1636],{},[599,2256,1906],{},[796,2258,2259],{},"2,31",[632,2261,2262],{"encoding":634},"t = \\frac{\\ln 2}{0{,}3} \\approx 2{,}31",[573,2264,2266,2284,2405],{"className":2265,"ariaHidden":640},[639],[573,2267,2269,2272,2275,2278,2281],{"className":2268},[644],[573,2270],{"className":2271,"style":1784},[648],[573,2273,610],{"className":2274},[653,657],[573,2276],{"className":2277,"style":712},[711],[573,2279,616],{"className":2280},[716],[573,2282],{"className":2283,"style":712},[711],[573,2285,2287,2291,2396,2399,2402],{"className":2286},[644],[573,2288],{"className":2289,"style":2290},[648],"height:1.3612em;vertical-align:-0.4811em;",[573,2292,2294,2298,2393],{"className":2293},[653],[573,2295],{"className":2296},[700,2297],"nulldelimiter",[573,2299,2301],{"className":2300},[2241],[573,2302,2305,2384],{"className":2303},[666,2304],"vlist-t2",[573,2306,2308,2379],{"className":2307},[670],[573,2309,2312,2337,2348],{"className":2310,"style":2311},[674],"height:0.8801em;",[573,2313,2315,2319],{"style":2314},"top:-2.655em;",[573,2316],{"className":2317,"style":2318},[682],"height:3em;",[573,2320,2322],{"className":2321},[687,688,689,690],[573,2323,2325,2328,2334],{"className":2324},[653,690],[573,2326,798],{"className":2327},[653,690],[573,2329,2331],{"className":2330},[653,690],[573,2332,1726],{"className":2333},[1725,690],[573,2335,1730],{"className":2336},[653,690],[573,2338,2340,2343],{"style":2339},"top:-3.23em;",[573,2341],{"className":2342,"style":2318},[682],[573,2344],{"className":2345,"style":2347},[2346],"frac-line","border-bottom-width:0.04em;",[573,2349,2351,2354],{"style":2350},"top:-3.394em;",[573,2352],{"className":2353,"style":2318},[682],[573,2355,2357],{"className":2356},[687,688,689,690],[573,2358,2360,2372,2376],{"className":2359},[653,690],[573,2361,2364,2368],{"className":2362},[2363,690],"mop",[573,2365,2367],{"className":2366},[690],"l",[573,2369,2371],{"className":2370},[690],"n",[573,2373],{"className":2374,"style":2375},[711,690],"margin-right:0.1952em;",[573,2377,2148],{"className":2378},[653,690],[573,2380,2383],{"className":2381},[2382],"vlist-s","​",[573,2385,2387],{"className":2386},[670],[573,2388,2391],{"className":2389,"style":2390},[674],"height:0.4811em;",[573,2392],{},[573,2394],{"className":2395},[707,2297],[573,2397],{"className":2398,"style":712},[711],[573,2400,1906],{"className":2401},[716],[573,2403],{"className":2404,"style":712},[711],[573,2406,2408,2411,2414,2420],{"className":2407},[644],[573,2409],{"className":2410,"style":1715},[648],[573,2412,2148],{"className":2413},[653],[573,2415,2417],{"className":2416},[653],[573,2418,1726],{"className":2419},[1725],[573,2421,2423],{"className":2422},[653],"31"," Stunden. Pro Stunde wächst die Kultur um etwa ",[573,2426,2428,2470],{"className":2427},[576],[573,2429,2431],{"className":2430},[580],[582,2432,2433],{"xmlns":584},[586,2434,2435,2467],{},[589,2436,2437,2439,2441,2443,2449,2451,2453,2455,2457,2460,2464],{},[796,2438,1427],{},[599,2440,622],{},[599,2442,607],{"stretchy":606},[592,2444,2445,2447],{},[595,2446,807],{},[796,2448,1636],{},[599,2450,1440],{},[796,2452,1443],{},[599,2454,613],{"stretchy":606},[599,2456,1906],{},[796,2458,2459],{},"35",[2461,2462,2463],"mtext",{}," ",[595,2465,2466],{"mathvariant":601},"%",[632,2468,2469],{"encoding":634},"100 \\cdot (e^{0{,}3} - 1) \\approx 35\\,\\%",[573,2471,2473,2491,2551,2572],{"className":2472,"ariaHidden":640},[639],[573,2474,2476,2479,2482,2485,2488],{"className":2475},[644],[573,2477],{"className":2478,"style":1276},[648],[573,2480,1427],{"className":2481},[653],[573,2483],{"className":2484,"style":734},[711],[573,2486,622],{"className":2487},[738],[573,2489],{"className":2490,"style":734},[711],[573,2492,2494,2498,2501,2542,2545,2548],{"className":2493},[644],[573,2495],{"className":2496,"style":2497},[648],"height:1.0641em;vertical-align:-0.25em;",[573,2499,607],{"className":2500},[700],[573,2502,2504,2507],{"className":2503},[653],[573,2505,807],{"className":2506},[653,657],[573,2508,2510],{"className":2509},[662],[573,2511,2513],{"className":2512},[666],[573,2514,2516],{"className":2515},[670],[573,2517,2519],{"className":2518,"style":2161},[674],[573,2520,2521,2524],{"style":678},[573,2522],{"className":2523,"style":683},[682],[573,2525,2527],{"className":2526},[687,688,689,690],[573,2528,2530,2533,2539],{"className":2529},[653,690],[573,2531,798],{"className":2532},[653,690],[573,2534,2536],{"className":2535},[653,690],[573,2537,1726],{"className":2538},[1725,690],[573,2540,1730],{"className":2541},[653,690],[573,2543],{"className":2544,"style":734},[711],[573,2546,1440],{"className":2547},[738],[573,2549],{"className":2550,"style":734},[711],[573,2552,2554,2557,2560,2563,2566,2569],{"className":2553},[644],[573,2555],{"className":2556,"style":748},[648],[573,2558,1443],{"className":2559},[653],[573,2561,613],{"className":2562},[707],[573,2564],{"className":2565,"style":712},[711],[573,2567,1906],{"className":2568},[716],[573,2570],{"className":2571,"style":712},[711],[573,2573,2575,2579,2582,2586],{"className":2574},[644],[573,2576],{"className":2577,"style":2578},[648],"height:0.8056em;vertical-align:-0.0556em;",[573,2580,2459],{"className":2581},[653],[573,2583],{"className":2584,"style":2585},[711],"margin-right:0.1667em;",[573,2587,2466],{"className":2588},[653],", nicht um ",[573,2591,2593,2612],{"className":2592},[576],[573,2594,2596],{"className":2595},[580],[582,2597,2598],{"xmlns":584},[586,2599,2600,2609],{},[589,2601,2602,2605,2607],{},[796,2603,2604],{},"30",[2461,2606,2463],{},[595,2608,2466],{"mathvariant":601},[632,2610,2611],{"encoding":634},"30\\,\\%",[573,2613,2615],{"className":2614,"ariaHidden":640},[639],[573,2616,2618,2621,2624,2627],{"className":2617},[644],[573,2619],{"className":2620,"style":2578},[648],[573,2622,2604],{"className":2623},[653],[573,2625],{"className":2626,"style":2585},[711],[573,2628,2466],{"className":2629},[653],[564,2631,2632,2633,1758,2784,2812,2813,2858,2859,2888,2889,1284],{},"Ein Stoff zerfällt nach ",[573,2634,2636,2677],{"className":2635},[576],[573,2637,2639],{"className":2638},[580],[582,2640,2641],{"xmlns":584},[586,2642,2643,2674],{},[589,2644,2645,2651,2653,2655,2657,2659,2661,2664,2666,2668,2670,2672],{},[592,2646,2647,2649],{},[595,2648,597],{},[599,2650,603],{"mathvariant":601,"lspace":602,"rspace":602},[599,2652,607],{"stretchy":606},[595,2654,610],{},[599,2656,613],{"stretchy":606},[599,2658,616],{},[599,2660,1440],{},[796,2662,2663],{},"0,1",[599,2665,622],{},[595,2667,597],{},[599,2669,607],{"stretchy":606},[595,2671,610],{},[599,2673,613],{"stretchy":606},[632,2675,2676],{"encoding":634},"f'(t) = -0{,}1 \\cdot f(t)",[573,2678,2680,2736,2766],{"className":2679,"ariaHidden":640},[639],[573,2681,2683,2686,2718,2721,2724,2727,2730,2733],{"className":2682},[644],[573,2684],{"className":2685,"style":649},[648],[573,2687,2689,2692],{"className":2688},[653],[573,2690,597],{"className":2691,"style":658},[653,657],[573,2693,2695],{"className":2694},[662],[573,2696,2698],{"className":2697},[666],[573,2699,2701],{"className":2700},[670],[573,2702,2704],{"className":2703,"style":675},[674],[573,2705,2706,2709],{"style":678},[573,2707],{"className":2708,"style":683},[682],[573,2710,2712],{"className":2711},[687,688,689,690],[573,2713,2715],{"className":2714},[653,690],[573,2716,603],{"className":2717},[653,690],[573,2719,607],{"className":2720},[700],[573,2722,610],{"className":2723},[653,657],[573,2725,613],{"className":2726},[707],[573,2728],{"className":2729,"style":712},[711],[573,2731,616],{"className":2732},[716],[573,2734],{"className":2735,"style":712},[711],[573,2737,2739,2742,2745,2748,2754,2757,2760,2763],{"className":2738},[644],[573,2740],{"className":2741,"style":1715},[648],[573,2743,1440],{"className":2744},[653],[573,2746,798],{"className":2747},[653],[573,2749,2751],{"className":2750},[653],[573,2752,1726],{"className":2753},[1725],[573,2755,1443],{"className":2756},[653],[573,2758],{"className":2759,"style":734},[711],[573,2761,622],{"className":2762},[738],[573,2764],{"className":2765,"style":734},[711],[573,2767,2769,2772,2775,2778,2781],{"className":2768},[644],[573,2770],{"className":2771,"style":748},[648],[573,2773,597],{"className":2774,"style":658},[653,657],[573,2776,607],{"className":2777},[700],[573,2779,610],{"className":2780},[653,657],[573,2782,613],{"className":2783},[707],[573,2785,2787,2800],{"className":2786},[576],[573,2788,2790],{"className":2789},[580],[582,2791,2792],{"xmlns":584},[586,2793,2794,2798],{},[589,2795,2796],{},[595,2797,610],{},[632,2799,610],{"encoding":634},[573,2801,2803],{"className":2802,"ariaHidden":640},[639],[573,2804,2806,2809],{"className":2805},[644],[573,2807],{"className":2808,"style":1784},[648],[573,2810,610],{"className":2811},[653,657]," in Jahren), anfangs sind ",[573,2814,2816,2836],{"className":2815},[576],[573,2817,2819],{"className":2818},[580],[582,2820,2821],{"xmlns":584},[586,2822,2823,2833],{},[589,2824,2825,2828,2830],{},[796,2826,2827],{},"50",[2461,2829,2463],{},[2461,2831,2832],{},"g",[632,2834,2835],{"encoding":634},"50\\,\\text{g}",[573,2837,2839],{"className":2838,"ariaHidden":640},[639],[573,2840,2842,2845,2848,2851],{"className":2841},[644],[573,2843],{"className":2844,"style":1715},[648],[573,2846,2827],{"className":2847},[653],[573,2849],{"className":2850,"style":2585},[711],[573,2852,2855],{"className":2853},[653,2854],"text",[573,2856,2832],{"className":2857},[653]," vorhanden. Nach ",[573,2860,2862,2876],{"className":2861},[576],[573,2863,2865],{"className":2864},[580],[582,2866,2867],{"xmlns":584},[586,2868,2869,2874],{},[589,2870,2871],{},[796,2872,2873],{},"10",[632,2875,2873],{"encoding":634},[573,2877,2879],{"className":2878,"ariaHidden":640},[639],[573,2880,2882,2885],{"className":2881},[644],[573,2883],{"className":2884,"style":1276},[648],[573,2886,2873],{"className":2887},[653]," Jahren sind es ",[573,2890,2892,2937],{"className":2891},[576],[573,2893,2895],{"className":2894},[580],[582,2896,2897],{"xmlns":584},[586,2898,2899,2934],{},[589,2900,2901,2903,2905,2907,2909,2911,2913,2915,2925,2927,2930,2932],{},[595,2902,597],{},[599,2904,607],{"stretchy":606},[796,2906,2873],{},[599,2908,613],{"stretchy":606},[599,2910,616],{},[796,2912,2827],{},[599,2914,622],{},[592,2916,2917,2919],{},[595,2918,807],{},[589,2920,2921,2923],{},[599,2922,1440],{},[796,2924,1443],{},[599,2926,1906],{},[796,2928,2929],{},"18,39",[2461,2931,2463],{},[2461,2933,2832],{},[632,2935,2936],{"encoding":634},"f(10) = 50 \\cdot e^{-1} \\approx 18{,}39\\,\\text{g}",[573,2938,2940,2967,2985,3035],{"className":2939,"ariaHidden":640},[639],[573,2941,2943,2946,2949,2952,2955,2958,2961,2964],{"className":2942},[644],[573,2944],{"className":2945,"style":748},[648],[573,2947,597],{"className":2948,"style":658},[653,657],[573,2950,607],{"className":2951},[700],[573,2953,2873],{"className":2954},[653],[573,2956,613],{"className":2957},[707],[573,2959],{"className":2960,"style":712},[711],[573,2962,616],{"className":2963},[716],[573,2965],{"className":2966,"style":712},[711],[573,2968,2970,2973,2976,2979,2982],{"className":2969},[644],[573,2971],{"className":2972,"style":1276},[648],[573,2974,2827],{"className":2975},[653],[573,2977],{"className":2978,"style":734},[711],[573,2980,622],{"className":2981},[738],[573,2983],{"className":2984,"style":734},[711],[573,2986,2988,2991,3026,3029,3032],{"className":2987},[644],[573,2989],{"className":2990,"style":2161},[648],[573,2992,2994,2997],{"className":2993},[653],[573,2995,807],{"className":2996},[653,657],[573,2998,3000],{"className":2999},[662],[573,3001,3003],{"className":3002},[666],[573,3004,3006],{"className":3005},[670],[573,3007,3009],{"className":3008,"style":2161},[674],[573,3010,3011,3014],{"style":678},[573,3012],{"className":3013,"style":683},[682],[573,3015,3017],{"className":3016},[687,688,689,690],[573,3018,3020,3023],{"className":3019},[653,690],[573,3021,1440],{"className":3022},[653,690],[573,3024,1443],{"className":3025},[653,690],[573,3027],{"className":3028,"style":712},[711],[573,3030,1906],{"className":3031},[716],[573,3033],{"className":3034,"style":712},[711],[573,3036,3038,3041,3045,3051,3055,3058],{"className":3037},[644],[573,3039],{"className":3040,"style":1715},[648],[573,3042,3044],{"className":3043},[653],"18",[573,3046,3048],{"className":3047},[653],[573,3049,1726],{"className":3050},[1725],[573,3052,3054],{"className":3053},[653],"39",[573,3056],{"className":3057,"style":2585},[711],[573,3059,3061],{"className":3060},[653,2854],[573,3062,2832],{"className":3063},[653],[564,3065,3066,3067,3095,3096,3125,3126,3155,3156,3185,3186,3299,3300,3400],{},"Aus Messwerten ",[573,3068,3070,3083],{"className":3069},[576],[573,3071,3073],{"className":3072},[580],[582,3074,3075],{"xmlns":584},[586,3076,3077,3081],{},[589,3078,3079],{},[595,3080,619],{},[632,3082,619],{"encoding":634},[573,3084,3086],{"className":3085,"ariaHidden":640},[639],[573,3087,3089,3092],{"className":3088},[644],[573,3090],{"className":3091,"style":726},[648],[573,3093,619],{"className":3094,"style":730},[653,657]," bestimmen: Eine Größe steigt in ",[573,3097,3099,3113],{"className":3098},[576],[573,3100,3102],{"className":3101},[580],[582,3103,3104],{"xmlns":584},[586,3105,3106,3111],{},[589,3107,3108],{},[796,3109,3110],{},"4",[632,3112,3110],{"encoding":634},[573,3114,3116],{"className":3115,"ariaHidden":640},[639],[573,3117,3119,3122],{"className":3118},[644],[573,3120],{"className":3121,"style":1276},[648],[573,3123,3110],{"className":3124},[653]," Tagen von ",[573,3127,3129,3143],{"className":3128},[576],[573,3130,3132],{"className":3131},[580],[582,3133,3134],{"xmlns":584},[586,3135,3136,3141],{},[589,3137,3138],{},[796,3139,3140],{},"200",[632,3142,3140],{"encoding":634},[573,3144,3146],{"className":3145,"ariaHidden":640},[639],[573,3147,3149,3152],{"className":3148},[644],[573,3150],{"className":3151,"style":1276},[648],[573,3153,3140],{"className":3154},[653]," auf ",[573,3157,3159,3173],{"className":3158},[576],[573,3160,3162],{"className":3161},[580],[582,3163,3164],{"xmlns":584},[586,3165,3166,3171],{},[589,3167,3168],{},[796,3169,3170],{},"300",[632,3172,3170],{"encoding":634},[573,3174,3176],{"className":3175,"ariaHidden":640},[639],[573,3177,3179,3182],{"className":3178},[644],[573,3180],{"className":3181,"style":1276},[648],[573,3183,3170],{"className":3184},[653],". Aus ",[573,3187,3189,3219],{"className":3188},[576],[573,3190,3192],{"className":3191},[580],[582,3193,3194],{"xmlns":584},[586,3195,3196,3216],{},[589,3197,3198,3200,3202,3212,3214],{},[796,3199,3140],{},[599,3201,622],{},[592,3203,3204,3206],{},[595,3205,807],{},[589,3207,3208,3210],{},[796,3209,3110],{},[595,3211,619],{},[599,3213,616],{},[796,3215,3170],{},[632,3217,3218],{"encoding":634},"200 \\cdot e^{4k} = 300",[573,3220,3222,3240,3290],{"className":3221,"ariaHidden":640},[639],[573,3223,3225,3228,3231,3234,3237],{"className":3224},[644],[573,3226],{"className":3227,"style":1276},[648],[573,3229,3140],{"className":3230},[653],[573,3232],{"className":3233,"style":734},[711],[573,3235,622],{"className":3236},[738],[573,3238],{"className":3239,"style":734},[711],[573,3241,3243,3246,3281,3284,3287],{"className":3242},[644],[573,3244],{"className":3245,"style":880},[648],[573,3247,3249,3252],{"className":3248},[653],[573,3250,807],{"className":3251},[653,657],[573,3253,3255],{"className":3254},[662],[573,3256,3258],{"className":3257},[666],[573,3259,3261],{"className":3260},[670],[573,3262,3264],{"className":3263,"style":880},[674],[573,3265,3266,3269],{"style":678},[573,3267],{"className":3268,"style":683},[682],[573,3270,3272],{"className":3271},[687,688,689,690],[573,3273,3275,3278],{"className":3274},[653,690],[573,3276,3110],{"className":3277},[653,690],[573,3279,619],{"className":3280,"style":730},[653,657,690],[573,3282],{"className":3283,"style":712},[711],[573,3285,616],{"className":3286},[716],[573,3288],{"className":3289,"style":712},[711],[573,3291,3293,3296],{"className":3292},[644],[573,3294],{"className":3295,"style":1276},[648],[573,3297,3170],{"className":3298},[653]," folgt ",[573,3301,3303,3329],{"className":3302},[576],[573,3304,3306],{"className":3305},[580],[582,3307,3308],{"xmlns":584},[586,3309,3310,3326],{},[589,3311,3312,3322,3324],{},[592,3313,3314,3316],{},[595,3315,807],{},[589,3317,3318,3320],{},[796,3319,3110],{},[595,3321,619],{},[599,3323,616],{},[796,3325,1903],{},[632,3327,3328],{"encoding":634},"e^{4k} = 1{,}5",[573,3330,3332,3382],{"className":3331,"ariaHidden":640},[639],[573,3333,3335,3338,3373,3376,3379],{"className":3334},[644],[573,3336],{"className":3337,"style":880},[648],[573,3339,3341,3344],{"className":3340},[653],[573,3342,807],{"className":3343},[653,657],[573,3345,3347],{"className":3346},[662],[573,3348,3350],{"className":3349},[666],[573,3351,3353],{"className":3352},[670],[573,3354,3356],{"className":3355,"style":880},[674],[573,3357,3358,3361],{"style":678},[573,3359],{"className":3360,"style":683},[682],[573,3362,3364],{"className":3363},[687,688,689,690],[573,3365,3367,3370],{"className":3366},[653,690],[573,3368,3110],{"className":3369},[653,690],[573,3371,619],{"className":3372,"style":730},[653,657,690],[573,3374],{"className":3375,"style":712},[711],[573,3377,616],{"className":3378},[716],[573,3380],{"className":3381,"style":712},[711],[573,3383,3385,3388,3391,3397],{"className":3384},[644],[573,3386],{"className":3387,"style":1715},[648],[573,3389,1443],{"className":3390},[653],[573,3392,3394],{"className":3393},[653],[573,3395,1726],{"className":3396},[1725],[573,3398,1832],{"className":3399},[653],", also",[573,3402,3404],{"className":3403},[1851],[573,3405,3407,3443],{"className":3406},[576],[573,3408,3410],{"className":3409},[580],[582,3411,3412],{"xmlns":584,"display":1860},[586,3413,3414,3440],{},[589,3415,3416,3418,3420,3432,3434,3437],{},[595,3417,619],{},[599,3419,616],{},[2241,3421,3422,3430],{},[589,3423,3424,3426,3428],{},[595,3425,2247],{},[599,3427,2250],{},[796,3429,1903],{},[796,3431,3110],{},[599,3433,1906],{},[796,3435,3436],{},"0,1014",[2461,3438,3439],{}," pro Tag",[632,3441,3442],{"encoding":634},"k = \\frac{\\ln 1{,}5}{4} \\approx 0{,}1014 \\text{ pro 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0{,}3",[573,3646,3648,3666],{"className":3647,"ariaHidden":640},[639],[573,3649,3651,3654,3657,3660,3663],{"className":3650},[644],[573,3652],{"className":3653,"style":726},[648],[573,3655,619],{"className":3656,"style":730},[653,657],[573,3658],{"className":3659,"style":712},[711],[573,3661,616],{"className":3662},[716],[573,3664],{"className":3665,"style":712},[711],[573,3667,3669,3672,3675,3681],{"className":3668},[644],[573,3670],{"className":3671,"style":1715},[648],[573,3673,798],{"className":3674},[653],[573,3676,3678],{"className":3677},[653],[573,3679,1726],{"className":3680},[1725],[573,3682,1730],{"className":3683},[653]," bedeutet nicht ",[573,3686,3688,3705],{"className":3687},[576],[573,3689,3691],{"className":3690},[580],[582,3692,3693],{"xmlns":584},[586,3694,3695,3703],{},[589,3696,3697,3699,3701],{},[796,3698,2604],{},[2461,3700,2463],{},[595,3702,2466],{"mathvariant":601},[632,3704,2611],{"encoding":634},[573,3706,3708],{"className":3707,"ariaHidden":640},[639],[573,3709,3711,3714,3717,3720],{"className":3710},[644],[573,3712],{"className":3713,"style":2578},[648],[573,3715,2604],{"className":3716},[653],[573,3718],{"className":3719,"style":2585},[711],[573,3721,2466],{"className":3722},[653]," Wachstum pro Zeiteinheit, sondern etwa ",[573,3725,3727,3745],{"className":3726},[576],[573,3728,3730],{"className":3729},[580],[582,3731,3732],{"xmlns":584},[586,3733,3734,3742],{},[589,3735,3736,3738,3740],{},[796,3737,2459],{},[2461,3739,2463],{},[595,3741,2466],{"mathvariant":601},[632,3743,3744],{"encoding":634},"35\\,\\%",[573,3746,3748],{"className":3747,"ariaHidden":640},[639],[573,3749,3751,3754,3757,3760],{"className":3750},[644],[573,3752],{"className":3753,"style":2578},[648],[573,3755,2459],{"className":3756},[653],[573,3758],{"className":3759,"style":2585},[711],[573,3761,2466],{"className":3762},[653],[1224,3764,3765,3766,3794,3795,1284],{},"Das Vorzeichen bei Abnahme vergessen: Bei Zerfall ist ",[573,3767,3769,3782],{"className":3768},[576],[573,3770,3772],{"className":3771},[580],[582,3773,3774],{"xmlns":584},[586,3775,3776,3780],{},[589,3777,3778],{},[595,3779,619],{},[632,3781,619],{"encoding":634},[573,3783,3785],{"className":3784,"ariaHidden":640},[639],[573,3786,3788,3791],{"className":3787},[644],[573,3789],{"className":3790,"style":726},[648],[573,3792,619],{"className":3793,"style":730},[653,657]," negativ, z.B. ",[573,3796,3798,3836],{"className":3797},[576],[573,3799,3801],{"className":3800},[580],[582,3802,3803],{"xmlns":584},[586,3804,3805,3833],{},[589,3806,3807,3809,3811,3813,3815,3817,3819,3821],{},[595,3808,597],{},[599,3810,607],{"stretchy":606},[595,3812,610],{},[599,3814,613],{"stretchy":606},[599,3816,616],{},[796,3818,2827],{},[599,3820,622],{},[592,3822,3823,3825],{},[595,3824,807],{},[589,3826,3827,3829,3831],{},[599,3828,1440],{},[796,3830,2663],{},[595,3832,610],{},[632,3834,3835],{"encoding":634},"f(t) = 50 \\cdot e^{-0{,}1t}",[573,3837,3839,3866,3884],{"className":3838,"ariaHidden":640},[639],[573,3840,3842,3845,3848,3851,3854,3857,3860,3863],{"className":3841},[644],[573,3843],{"className":3844,"style":748},[648],[573,3846,597],{"className":3847,"style":658},[653,657],[573,3849,607],{"className":3850},[700],[573,3852,610],{"className":3853},[653,657],[573,3855,613],{"className":3856},[707],[573,3858],{"className":3859,"style":712},[711],[573,3861,616],{"className":3862},[716],[573,3864],{"className":3865,"style":712},[711],[573,3867,3869,3872,3875,3878,3881],{"className":3868},[644],[573,3870],{"className":3871,"style":1276},[648],[573,3873,2827],{"className":3874},[653],[573,3876],{"className":3877,"style":734},[711],[573,3879,622],{"className":3880},[738],[573,3882],{"className":3883,"style":734},[711],[573,3885,3887,3890],{"className":3886},[644],[573,3888],{"className":3889,"style":2161},[648],[573,3891,3893,3896],{"className":3892},[653],[573,3894,807],{"className":3895},[653,657],[573,3897,3899],{"className":3898},[662],[573,3900,3902],{"className":3901},[666],[573,3903,3905],{"className":3904},[670],[573,3906,3908],{"className":3907,"style":2161},[674],[573,3909,3910,3913],{"style":678},[573,3911],{"className":3912,"style":683},[682],[573,3914,3916],{"className":3915},[687,688,689,690],[573,3917,3919,3922,3925,3931,3934],{"className":3918},[653,690],[573,3920,1440],{"className":3921},[653,690],[573,3923,798],{"className":3924},[653,690],[573,3926,3928],{"className":3927},[653,690],[573,3929,1726],{"className":3930},[1725,690],[573,3932,1443],{"className":3933},[653,690],[573,3935,610],{"className":3936},[653,657,690],[1224,3938,3939,3940,4039],{},"Die DGL mit ",[573,3941,3943,3971],{"className":3942},[576],[573,3944,3946],{"className":3945},[580],[582,3947,3948],{"xmlns":584},[586,3949,3950,3968],{},[589,3951,3952,3958,3960,3962,3964,3966],{},[592,3953,3954,3956],{},[595,3955,597],{},[599,3957,603],{"mathvariant":601,"lspace":602,"rspace":602},[599,3959,607],{"stretchy":606},[595,3961,610],{},[599,3963,613],{"stretchy":606},[599,3965,616],{},[595,3967,619],{},[632,3969,3970],{"encoding":634},"f'(t) = k",[573,3972,3974,4030],{"className":3973,"ariaHidden":640},[639],[573,3975,3977,3980,4012,4015,4018,4021,4024,4027],{"className":3976},[644],[573,3978],{"className":3979,"style":649},[648],[573,3981,3983,3986],{"className":3982},[653],[573,3984,597],{"className":3985,"style":658},[653,657],[573,3987,3989],{"className":3988},[662],[573,3990,3992],{"className":3991},[666],[573,3993,3995],{"className":3994},[670],[573,3996,3998],{"className":3997,"style":675},[674],[573,3999,4000,4003],{"style":678},[573,4001],{"className":4002,"style":683},[682],[573,4004,4006],{"className":4005},[687,688,689,690],[573,4007,4009],{"className":4008},[653,690],[573,4010,603],{"className":4011},[653,690],[573,4013,607],{"className":4014},[700],[573,4016,610],{"className":4017},[653,657],[573,4019,613],{"className":4020},[707],[573,4022],{"className":4023,"style":712},[711],[573,4025,616],{"className":4026},[716],[573,4028],{"className":4029,"style":712},[711],[573,4031,4033,4036],{"className":4032},[644],[573,4034],{"className":4035,"style":726},[648],[573,4037,619],{"className":4038,"style":730},[653,657]," verwechseln: Ist die Änderungsrate konstant, wächst der Bestand linear, nicht exponentiell.",[1224,4041,4042,4043,4142,4143,1549,4219,1284],{},"Beim Lösen von ",[573,4044,4046,4071],{"className":4045},[576],[573,4047,4049],{"className":4048},[580],[582,4050,4051],{"xmlns":584},[586,4052,4053,4069],{},[589,4054,4055,4065,4067],{},[592,4056,4057,4059],{},[595,4058,807],{},[589,4060,4061,4063],{},[796,4062,3110],{},[595,4064,619],{},[599,4066,616],{},[796,4068,1903],{},[632,4070,3328],{"encoding":634},[573,4072,4074,4124],{"className":4073,"ariaHidden":640},[639],[573,4075,4077,4080,4115,4118,4121],{"className":4076},[644],[573,4078],{"className":4079,"style":880},[648],[573,4081,4083,4086],{"className":4082},[653],[573,4084,807],{"className":4085},[653,657],[573,4087,4089],{"className":4088},[662],[573,4090,4092],{"className":4091},[666],[573,4093,4095],{"className":4094},[670],[573,4096,4098],{"className":4097,"style":880},[674],[573,4099,4100,4103],{"style":678},[573,4101],{"className":4102,"style":683},[682],[573,4104,4106],{"className":4105},[687,688,689,690],[573,4107,4109,4112],{"className":4108},[653,690],[573,4110,3110],{"className":4111},[653,690],[573,4113,619],{"className":4114,"style":730},[653,657,690],[573,4116],{"className":4117,"style":712},[711],[573,4119,616],{"className":4120},[716],[573,4122],{"className":4123,"style":712},[711],[573,4125,4127,4130,4133,4139],{"className":4126},[644],[573,4128],{"className":4129,"style":1715},[648],[573,4131,1443],{"className":4132},[653],[573,4134,4136],{"className":4135},[653],[573,4137,1726],{"className":4138},[1725],[573,4140,1832],{"className":4141},[653]," den Logarithmus falsch anwenden: ",[573,4144,4146,4170],{"className":4145},[576],[573,4147,4149],{"className":4148},[580],[582,4150,4151],{"xmlns":584},[586,4152,4153,4167],{},[589,4154,4155,4157,4159,4161,4163,4165],{},[796,4156,3110],{},[595,4158,619],{},[599,4160,616],{},[595,4162,2247],{},[599,4164,2250],{},[796,4166,1903],{},[632,4168,4169],{"encoding":634},"4k = \\ln 1{,}5",[573,4171,4173,4194],{"className":4172,"ariaHidden":640},[639],[573,4174,4176,4179,4182,4185,4188,4191],{"className":4175},[644],[573,4177],{"className":4178,"style":726},[648],[573,4180,3110],{"className":4181},[653],[573,4183,619],{"className":4184,"style":730},[653,657],[573,4186],{"className":4187,"style":712},[711],[573,4189,616],{"className":4190},[716],[573,4192],{"className":4193,"style":712},[711],[573,4195,4197,4201,4204,4207,4210,4216],{"className":4196},[644],[573,4198],{"className":4199,"style":4200},[648],"height:0.8889em;vertical-align:-0.1944em;",[573,4202,2247],{"className":4203},[2363],[573,4205],{"className":4206,"style":2585},[711],[573,4208,1443],{"className":4209},[653],[573,4211,4213],{"className":4212},[653],[573,4214,1726],{"className":4215},[1725],[573,4217,1832],{"className":4218},[653],[573,4220,4222,4253],{"className":4221},[576],[573,4223,4225],{"className":4224},[580],[582,4226,4227],{"xmlns":584},[586,4228,4229,4250],{},[589,4230,4231,4233,4235,4237,4239,4241,4243,4246,4248],{},[595,4232,619],{},[599,4234,616],{},[595,4236,2247],{},[599,4238,2250],{},[599,4240,607],{"stretchy":606},[796,4242,1903],{},[595,4244,4245],{"mathvariant":601},"\u002F",[796,4247,3110],{},[599,4249,613],{"stretchy":606},[632,4251,4252],{"encoding":634},"k = \\ln(1{,}5 \u002F 4)",[573,4254,4256,4274],{"className":4255,"ariaHidden":640},[639],[573,4257,4259,4262,4265,4268,4271],{"className":4258},[644],[573,4260],{"className":4261,"style":726},[648],[573,4263,619],{"className":4264,"style":730},[653,657],[573,4266],{"className":4267,"style":712},[711],[573,4269,616],{"className":4270},[716],[573,4272],{"className":4273,"style":712},[711],[573,4275,4277,4280,4283,4286,4289,4295,4299],{"className":4276},[644],[573,4278],{"className":4279,"style":748},[648],[573,4281,2247],{"className":4282},[2363],[573,4284,607],{"className":4285},[700],[573,4287,1443],{"className":4288},[653],[573,4290,4292],{"className":4291},[653],[573,4293,1726],{"className":4294},[1725],[573,4296,4298],{"className":4297},[653],"5\u002F4",[573,4300,613],{"className":4301},[707],{"title":4303,"searchDepth":4304,"depth":4304,"links":4305},"",2,[4306,4307,4308],{"id":561,"depth":4304,"text":562},{"id":1603,"depth":4304,"text":1604},{"id":3589,"depth":4304,"text":3590},[4310,4312,4315,4318,4320,4323,4326,4328,4331],{"lhs":4311,"rhs":798},"round((exp(k*(t 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mathnormal\">t\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">⇒\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">t\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" 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style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fdifferential-equations\u002Fdgl-exponentielles-wachstum",[],[4344,4345,4346,4347],"exponential-growth","exponential-decay","natural-exponential-function","bounded-growth",{"title":548,"description":4303},"de\u002Fformulas\u002Fmath\u002Fdifferential-equations\u002Fdgl-exponentielles-wachstum","Ist die Änderungsrate proportional zum Bestand, wächst oder schrumpft der Bestand exponentiell mit f(0) · e^(kt).",[4352,4353,4354,4355,4356],"differentialgleichung","exponentielles wachstum","wachstumskonstante","e-funktion","zerfall",[4358,4362,4366,4369,4372,4375],{"symbol":4359,"meaning":4360,"symbolHtml":4361},"f(t)","Bestand zum Zeitpunkt t","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>f\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>t\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f(t)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">t\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4363,"meaning":4364,"symbolHtml":4365},"f'(t)","momentane Änderungsrate des Bestands","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>f\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>t\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f'(t)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan 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