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Liter pro Minute oder Euro pro Stück.",[810,877,878,879,882],{},"Das ",[569,880,881],{},"Vorzeichen"," sagt, ob die Größe im Mittel zu- oder abnimmt.",[810,884,885,886,889,890,798],{},"Sie sagt nichts darüber, wie die Änderung innerhalb des Intervalls verläuft. Dafür braucht man die ",[569,887,888],{},"momentane Änderungsrate"," ",[574,891,893,921],{"className":892},[577],[574,894,896],{"className":895},[581],[583,897,898],{"xmlns":585},[587,899,900,918],{},[590,901,902,912,914,916],{},[903,904,905,907],"msup",{},[593,906,595],{},[597,908,911],{"mathvariant":909,"lspace":910,"rspace":910},"normal","0em","′",[597,913,600],{"stretchy":599},[593,915,860],{},[597,917,606],{"stretchy":599},[620,919,920],{"encoding":622},"f'(x)",[574,922,924],{"className":923,"ariaHidden":628},[627],[574,925,927,931,975,978,981],{"className":926},[632],[574,928],{"className":929,"style":930},[636],"height:1.0019em;vertical-align:-0.25em;",[574,932,934,937],{"className":933},[641],[574,935,595],{"className":936,"style":643},[641,642],[574,938,941],{"className":939},[940],"msupsub",[574,942,945],{"className":943},[944],"vlist-t",[574,946,949],{"className":947},[948],"vlist-r",[574,950,954],{"className":951,"style":953},[952],"vlist","height:0.7519em;",[574,955,957,962],{"style":956},"top:-3.063em;margin-right:0.05em;",[574,958],{"className":959,"style":961},[960],"pstrut","height:2.7em;",[574,963,969],{"className":964},[965,966,967,968],"sizing","reset-size6","size3","mtight",[574,970,972],{"className":971},[641,968],[574,973,911],{"className":974},[641,968],[574,976,600],{"className":977},[647],[574,979,860],{"className":980},[641,642],[574,982,606],{"className":983},[654],[565,985,986,987,1015,1016,1266,1267,1323],{},"Ist ",[574,988,990,1003],{"className":989},[577],[574,991,993],{"className":992},[581],[583,994,995],{"xmlns":585},[587,996,997,1001],{},[590,998,999],{},[593,1000,595],{},[620,1002,595],{"encoding":622},[574,1004,1006],{"className":1005,"ariaHidden":628},[627],[574,1007,1009,1012],{"className":1008},[632],[574,1010],{"className":1011,"style":841},[636],[574,1013,595],{"className":1014,"style":643},[641,642]," stetig differenzierbar, gilt ",[574,1017,1019,1084],{"className":1018},[577],[574,1020,1022],{"className":1021},[581],[583,1023,1024],{"xmlns":585},[587,1025,1026,1081],{},[590,1027,1028,1030,1032,1034,1036,1038,1040,1042,1044,1046,1049,1059,1065,1067,1070,1072,1076,1079],{},[593,1029,595],{},[597,1031,600],{"stretchy":599},[593,1033,603],{},[597,1035,606],{"stretchy":599},[597,1037,609],{},[593,1039,595],{},[597,1041,600],{"stretchy":599},[593,1043,616],{},[597,1045,606],{"stretchy":599},[597,1047,1048],{},"=",[1050,1051,1052,1055,1057],"msubsup",{},[597,1053,1054],{},"∫",[593,1056,616],{},[593,1058,603],{},[903,1060,1061,1063],{},[593,1062,595],{},[597,1064,911],{"mathvariant":909,"lspace":910,"rspace":910},[597,1066,600],{"stretchy":599},[593,1068,1069],{},"t",[597,1071,606],{"stretchy":599},[1073,1074,1075],"mtext",{}," ",[593,1077,1078],{},"d",[593,1080,1069],{},[620,1082,1083],{"encoding":622},"f(b) - f(a) = \\int_a^b f'(t)\\,dt",[574,1085,1087,1114,1143],{"className":1086,"ariaHidden":628},[627],[574,1088,1090,1093,1096,1099,1102,1105,1108,1111],{"className":1089},[632],[574,1091],{"className":1092,"style":637},[636],[574,1094,595],{"className":1095,"style":643},[641,642],[574,1097,600],{"className":1098},[647],[574,1100,603],{"className":1101},[641,642],[574,1103,606],{"className":1104},[654],[574,1106],{"className":1107,"style":659},[658],[574,1109,609],{"className":1110},[663],[574,1112],{"className":1113,"style":659},[658],[574,1115,1117,1120,1123,1126,1129,1132,1136,1140],{"className":1116},[632],[574,1118],{"className":1119,"style":637},[636],[574,1121,595],{"className":1122,"style":643},[641,642],[574,1124,600],{"className":1125},[647],[574,1127,616],{"className":1128},[641,642],[574,1130,606],{"className":1131},[654],[574,1133],{"className":1134,"style":1135},[658],"margin-right:0.2778em;",[574,1137,1048],{"className":1138},[1139],"mrel",[574,1141],{"className":1142,"style":1135},[658],[574,1144,1146,1150,1212,1216,1248,1251,1254,1257,1260,1263],{"className":1145},[632],[574,1147],{"className":1148,"style":1149},[636],"height:1.3998em;vertical-align:-0.3558em;",[574,1151,1154,1160],{"className":1152},[1153],"mop",[574,1155,1054],{"className":1156,"style":1159},[1153,1157,1158],"op-symbol","small-op","margin-right:0.1945em;position:relative;top:-0.0006em;",[574,1161,1163],{"className":1162},[940],[574,1164,1167,1203],{"className":1165},[944,1166],"vlist-t2",[574,1168,1170,1198],{"className":1169},[948],[574,1171,1174,1186],{"className":1172,"style":1173},[952],"height:1.044em;",[574,1175,1177,1180],{"style":1176},"top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;",[574,1178],{"className":1179,"style":961},[960],[574,1181,1183],{"className":1182},[965,966,967,968],[574,1184,616],{"className":1185},[641,642,968],[574,1187,1189,1192],{"style":1188},"top:-3.2579em;margin-right:0.05em;",[574,1190],{"className":1191,"style":961},[960],[574,1193,1195],{"className":1194},[965,966,967,968],[574,1196,603],{"className":1197},[641,642,968],[574,1199,1202],{"className":1200},[1201],"vlist-s","​",[574,1204,1206],{"className":1205},[948],[574,1207,1210],{"className":1208,"style":1209},[952],"height:0.3558em;",[574,1211],{},[574,1213],{"className":1214,"style":1215},[658],"margin-right:0.1667em;",[574,1217,1219,1222],{"className":1218},[641],[574,1220,595],{"className":1221,"style":643},[641,642],[574,1223,1225],{"className":1224},[940],[574,1226,1228],{"className":1227},[944],[574,1229,1231],{"className":1230},[948],[574,1232,1234],{"className":1233,"style":953},[952],[574,1235,1236,1239],{"style":956},[574,1237],{"className":1238,"style":961},[960],[574,1240,1242],{"className":1241},[965,966,967,968],[574,1243,1245],{"className":1244},[641,968],[574,1246,911],{"className":1247},[641,968],[574,1249,600],{"className":1250},[647],[574,1252,1069],{"className":1253},[641,642],[574,1255,606],{"className":1256},[654],[574,1258],{"className":1259,"style":1215},[658],[574,1261,1078],{"className":1262},[641,642],[574,1264,1069],{"className":1265},[641,642],". Die mittlere Änderungsrate ist also der Mittelwert der momentanen Änderungsrate auf ",[574,1268,1270,1295],{"className":1269},[577],[574,1271,1273],{"className":1272},[581],[583,1274,1275],{"xmlns":585},[587,1276,1277,1292],{},[590,1278,1279,1282,1284,1287,1289],{},[597,1280,1281],{"stretchy":599},"[",[593,1283,616],{},[597,1285,1286],{"separator":628},",",[593,1288,603],{},[597,1290,1291],{"stretchy":599},"]",[620,1293,1294],{"encoding":622},"[a, b]",[574,1296,1298],{"className":1297,"ariaHidden":628},[627],[574,1299,1301,1304,1307,1310,1314,1317,1320],{"className":1300},[632],[574,1302],{"className":1303,"style":637},[636],[574,1305,1281],{"className":1306},[647],[574,1308,616],{"className":1309},[641,642],[574,1311,1286],{"className":1312},[1313],"mpunct",[574,1315],{"className":1316,"style":1215},[658],[574,1318,603],{"className":1319},[641,642],[574,1321,1291],{"className":1322},[654],". Nach dem Mittelwertsatz wird sie an mindestens einer Stelle im Intervall auch als momentane Rate angenommen.",[560,1325,1327],{"id":1326},"beispiele","Beispiele",[565,1329,1330,1331,1362,1363,1392,1393,1422],{},"In einem Tank sind zu Beginn ",[574,1332,1334,1349],{"className":1333},[577],[574,1335,1337],{"className":1336},[581],[583,1338,1339],{"xmlns":585},[587,1340,1341,1347],{},[590,1342,1343],{},[1344,1345,1346],"mn",{},"120",[620,1348,1346],{"encoding":622},[574,1350,1352],{"className":1351,"ariaHidden":628},[627],[574,1353,1355,1359],{"className":1354},[632],[574,1356],{"className":1357,"style":1358},[636],"height:0.6444em;",[574,1360,1346],{"className":1361},[641]," Liter, nach ",[574,1364,1366,1380],{"className":1365},[577],[574,1367,1369],{"className":1368},[581],[583,1370,1371],{"xmlns":585},[587,1372,1373,1378],{},[590,1374,1375],{},[1344,1376,1377],{},"15",[620,1379,1377],{"encoding":622},[574,1381,1383],{"className":1382,"ariaHidden":628},[627],[574,1384,1386,1389],{"className":1385},[632],[574,1387],{"className":1388,"style":1358},[636],[574,1390,1377],{"className":1391},[641]," Minuten ",[574,1394,1396,1410],{"className":1395},[577],[574,1397,1399],{"className":1398},[581],[583,1400,1401],{"xmlns":585},[587,1402,1403,1408],{},[590,1404,1405],{},[1344,1406,1407],{},"300",[620,1409,1407],{"encoding":622},[574,1411,1413],{"className":1412,"ariaHidden":628},[627],[574,1414,1416,1419],{"className":1415},[632],[574,1417],{"className":1418,"style":1358},[636],[574,1420,1407],{"className":1421},[641]," Liter:",[574,1424,1427],{"className":1425},[1426],"katex-display",[574,1428,1430,1505],{"className":1429},[577],[574,1431,1433],{"className":1432},[581],[583,1434,1436],{"xmlns":585,"display":1435},"block",[587,1437,1438,1502],{},[590,1439,1440,1449,1451,1489,1491,1494,1496],{},[1441,1442,1443,1446],"mover",{"accent":628},[593,1444,1445],{},"m",[597,1447,1448],{},"ˉ",[597,1450,1048],{},[1452,1453,1454,1471],"mfrac",{},[590,1455,1456,1458,1460,1463,1465,1467,1469],{},[1344,1457,1407],{},[1073,1459,1075],{},[1073,1461,1462],{},"L",[597,1464,609],{},[1344,1466,1346],{},[1073,1468,1075],{},[1073,1470,1462],{},[590,1472,1473,1475,1477,1480,1482,1485,1487],{},[1344,1474,1377],{},[1073,1476,1075],{},[1073,1478,1479],{},"min",[597,1481,609],{},[1344,1483,1484],{},"0",[1073,1486,1075],{},[1073,1488,1479],{},[597,1490,1048],{},[1344,1492,1493],{},"12",[1073,1495,1075],{},[1452,1497,1498,1500],{},[1073,1499,1462],{},[1073,1501,1479],{},[620,1503,1504],{"encoding":622},"\\bar{m} = \\frac{300\\,\\text{L} - 120\\,\\text{L}}{15\\,\\text{min} - 0\\,\\text{min}} = 12\\,\\frac{\\text{L}}{\\text{min}}",[574,1506,1508,1560,1707],{"className":1507,"ariaHidden":628},[627],[574,1509,1511,1515,1551,1554,1557],{"className":1510},[632],[574,1512],{"className":1513,"style":1514},[636],"height:0.5678em;",[574,1516,1519],{"className":1517},[641,1518],"accent",[574,1520,1522],{"className":1521},[944],[574,1523,1525],{"className":1524},[948],[574,1526,1528,1538],{"className":1527,"style":1514},[952],[574,1529,1531,1535],{"style":1530},"top:-3em;",[574,1532],{"className":1533,"style":1534},[960],"height:3em;",[574,1536,1445],{"className":1537},[641,642],[574,1539,1540,1543],{"style":1530},[574,1541],{"className":1542,"style":1534},[960],[574,1544,1548],{"className":1545,"style":1547},[1546],"accent-body","left:-0.25em;",[574,1549,1448],{"className":1550},[641],[574,1552],{"className":1553,"style":1135},[658],[574,1555,1048],{"className":1556},[1139],[574,1558],{"className":1559,"style":1135},[658],[574,1561,1563,1567,1698,1701,1704],{"className":1562},[632],[574,1564],{"className":1565,"style":1566},[636],"height:2.1297em;vertical-align:-0.7693em;",[574,1568,1570,1574,1695],{"className":1569},[641],[574,1571],{"className":1572},[647,1573],"nulldelimiter",[574,1575,1577],{"className":1576},[1452],[574,1578,1580,1686],{"className":1579},[944,1166],[574,1581,1583,1683],{"className":1582},[948],[574,1584,1587,1630,1641],{"className":1585,"style":1586},[952],"height:1.3603em;",[574,1588,1590,1593],{"style":1589},"top:-2.314em;",[574,1591],{"className":1592,"style":1534},[960],[574,1594,1596,1599,1602,1609,1612,1615,1618,1621,1624],{"className":1595},[641],[574,1597,1377],{"className":1598},[641],[574,1600],{"className":1601,"style":1215},[658],[574,1603,1606],{"className":1604},[641,1605],"text",[574,1607,1479],{"className":1608},[641],[574,1610],{"className":1611,"style":659},[658],[574,1613,609],{"className":1614},[663],[574,1616],{"className":1617,"style":659},[658],[574,1619,1484],{"className":1620},[641],[574,1622],{"className":1623,"style":1215},[658],[574,1625,1627],{"className":1626},[641,1605],[574,1628,1479],{"className":1629},[641],[574,1631,1633,1636],{"style":1632},"top:-3.23em;",[574,1634],{"className":1635,"style":1534},[960],[574,1637],{"className":1638,"style":1640},[1639],"frac-line","border-bottom-width:0.04em;",[574,1642,1644,1647],{"style":1643},"top:-3.677em;",[574,1645],{"className":1646,"style":1534},[960],[574,1648,1650,1653,1656,1662,1665,1668,1671,1674,1677],{"className":1649},[641],[574,1651,1407],{"className":1652},[641],[574,1654],{"className":1655,"style":1215},[658],[574,1657,1659],{"className":1658},[641,1605],[574,1660,1462],{"className":1661},[641],[574,1663],{"className":1664,"style":659},[658],[574,1666,609],{"className":1667},[663],[574,1669],{"className":1670,"style":659},[658],[574,1672,1346],{"className":1673},[641],[574,1675],{"className":1676,"style":1215},[658],[574,1678,1680],{"className":1679},[641,1605],[574,1681,1462],{"className":1682},[641],[574,1684,1202],{"className":1685},[1201],[574,1687,1689],{"className":1688},[948],[574,1690,1693],{"className":1691,"style":1692},[952],"height:0.7693em;",[574,1694],{},[574,1696],{"className":1697},[654,1573],[574,1699],{"className":1700,"style":1135},[658],[574,1702,1048],{"className":1703},[1139],[574,1705],{"className":1706,"style":1135},[658],[574,1708,1710,1714,1717,1720],{"className":1709},[632],[574,1711],{"className":1712,"style":1713},[636],"height:2.0463em;vertical-align:-0.686em;",[574,1715,1493],{"className":1716},[641],[574,1718],{"className":1719,"style":1215},[658],[574,1721,1723,1726,1786],{"className":1722},[641],[574,1724],{"className":1725},[647,1573],[574,1727,1729],{"className":1728},[1452],[574,1730,1732,1777],{"className":1731},[944,1166],[574,1733,1735,1774],{"className":1734},[948],[574,1736,1738,1752,1760],{"className":1737,"style":1586},[952],[574,1739,1740,1743],{"style":1589},[574,1741],{"className":1742,"style":1534},[960],[574,1744,1746],{"className":1745},[641],[574,1747,1749],{"className":1748},[641,1605],[574,1750,1479],{"className":1751},[641],[574,1753,1754,1757],{"style":1632},[574,1755],{"className":1756,"style":1534},[960],[574,1758],{"className":1759,"style":1640},[1639],[574,1761,1762,1765],{"style":1643},[574,1763],{"className":1764,"style":1534},[960],[574,1766,1768],{"className":1767},[641],[574,1769,1771],{"className":1770},[641,1605],[574,1772,1462],{"className":1773},[641],[574,1775,1202],{"className":1776},[1201],[574,1778,1780],{"className":1779},[948],[574,1781,1784],{"className":1782,"style":1783},[952],"height:0.686em;",[574,1785],{},[574,1787],{"className":1788},[654,1573],[565,1790,1791,1792,1820],{},"Im Mittel fließen ",[574,1793,1795,1808],{"className":1794},[577],[574,1796,1798],{"className":1797},[581],[583,1799,1800],{"xmlns":585},[587,1801,1802,1806],{},[590,1803,1804],{},[1344,1805,1493],{},[620,1807,1493],{"encoding":622},[574,1809,1811],{"className":1810,"ariaHidden":628},[627],[574,1812,1814,1817],{"className":1813},[632],[574,1815],{"className":1816,"style":1358},[636],[574,1818,1493],{"className":1819},[641]," Liter pro Minute zu, auch wenn der Zufluss zwischendurch schwankt.",[565,1822,1823,1824,1963,1964,1993,1994,2022,2023,2052],{},"Die Höhe einer Pflanze ist ",[574,1825,1827,1865],{"className":1826},[577],[574,1828,1830],{"className":1829},[581],[583,1831,1832],{"xmlns":585},[587,1833,1834,1862],{},[590,1835,1836,1839,1841,1843,1845,1847,1850,1857,1860],{},[593,1837,1838],{},"h",[597,1840,600],{"stretchy":599},[593,1842,1069],{},[597,1844,606],{"stretchy":599},[597,1846,1048],{},[1344,1848,1849],{},"0,5",[903,1851,1852,1854],{},[593,1853,1069],{},[1344,1855,1856],{},"2",[597,1858,1859],{},"+",[1344,1861,1856],{},[620,1863,1864],{"encoding":622},"h(t) = 0{,}5t^2 + 2",[574,1866,1868,1895,1954],{"className":1867,"ariaHidden":628},[627],[574,1869,1871,1874,1877,1880,1883,1886,1889,1892],{"className":1870},[632],[574,1872],{"className":1873,"style":637},[636],[574,1875,1838],{"className":1876},[641,642],[574,1878,600],{"className":1879},[647],[574,1881,1069],{"className":1882},[641,642],[574,1884,606],{"className":1885},[654],[574,1887],{"className":1888,"style":1135},[658],[574,1890,1048],{"className":1891},[1139],[574,1893],{"className":1894,"style":1135},[658],[574,1896,1898,1902,1905,1911,1915,1945,1948,1951],{"className":1897},[632],[574,1899],{"className":1900,"style":1901},[636],"height:1.0085em;vertical-align:-0.1944em;",[574,1903,1484],{"className":1904},[641],[574,1906,1908],{"className":1907},[641],[574,1909,1286],{"className":1910},[1313],[574,1912,1914],{"className":1913},[641],"5",[574,1916,1918,1921],{"className":1917},[641],[574,1919,1069],{"className":1920},[641,642],[574,1922,1924],{"className":1923},[940],[574,1925,1927],{"className":1926},[944],[574,1928,1930],{"className":1929},[948],[574,1931,1934],{"className":1932,"style":1933},[952],"height:0.8141em;",[574,1935,1936,1939],{"style":956},[574,1937],{"className":1938,"style":961},[960],[574,1940,1942],{"className":1941},[965,966,967,968],[574,1943,1856],{"className":1944},[641,968],[574,1946],{"className":1947,"style":659},[658],[574,1949,1859],{"className":1950},[663],[574,1952],{"className":1953,"style":659},[658],[574,1955,1957,1960],{"className":1956},[632],[574,1958],{"className":1959,"style":1358},[636],[574,1961,1856],{"className":1962},[641]," (in cm, ",[574,1965,1967,1980],{"className":1966},[577],[574,1968,1970],{"className":1969},[581],[583,1971,1972],{"xmlns":585},[587,1973,1974,1978],{},[590,1975,1976],{},[593,1977,1069],{},[620,1979,1069],{"encoding":622},[574,1981,1983],{"className":1982,"ariaHidden":628},[627],[574,1984,1986,1990],{"className":1985},[632],[574,1987],{"className":1988,"style":1989},[636],"height:0.6151em;",[574,1991,1069],{"className":1992},[641,642]," in Wochen). Von Woche ",[574,1995,1997,2010],{"className":1996},[577],[574,1998,2000],{"className":1999},[581],[583,2001,2002],{"xmlns":585},[587,2003,2004,2008],{},[590,2005,2006],{},[1344,2007,1856],{},[620,2009,1856],{"encoding":622},[574,2011,2013],{"className":2012,"ariaHidden":628},[627],[574,2014,2016,2019],{"className":2015},[632],[574,2017],{"className":2018,"style":1358},[636],[574,2020,1856],{"className":2021},[641]," bis Woche ",[574,2024,2026,2040],{"className":2025},[577],[574,2027,2029],{"className":2028},[581],[583,2030,2031],{"xmlns":585},[587,2032,2033,2038],{},[590,2034,2035],{},[1344,2036,2037],{},"6",[620,2039,2037],{"encoding":622},[574,2041,2043],{"className":2042,"ariaHidden":628},[627],[574,2044,2046,2049],{"className":2045},[632],[574,2047],{"className":2048,"style":1358},[636],[574,2050,2037],{"className":2051},[641],":",[574,2054,2056],{"className":2055},[1426],[574,2057,2059,2156],{"className":2058},[577],[574,2060,2062],{"className":2061},[581],[583,2063,2064],{"xmlns":585,"display":1435},[587,2065,2066,2153],{},[590,2067,2068,2091,2093,2123,2125,2139,2141,2143,2145],{},[2069,2070,2071,2077],"msub",{},[1441,2072,2073,2075],{"accent":628},[593,2074,1445],{},[597,2076,1448],{},[590,2078,2079,2081,2083,2085,2087,2089],{},[597,2080,1281],{"stretchy":599},[1344,2082,1856],{},[597,2084,1286],{"separator":628},[1073,2086,1075],{},[1344,2088,2037],{},[597,2090,1291],{"stretchy":599},[597,2092,1048],{},[1452,2094,2095,2115],{},[590,2096,2097,2099,2101,2103,2105,2107,2109,2111,2113],{},[593,2098,1838],{},[597,2100,600],{"stretchy":599},[1344,2102,2037],{},[597,2104,606],{"stretchy":599},[597,2106,609],{},[593,2108,1838],{},[597,2110,600],{"stretchy":599},[1344,2112,1856],{},[597,2114,606],{"stretchy":599},[590,2116,2117,2119,2121],{},[1344,2118,2037],{},[597,2120,609],{},[1344,2122,1856],{},[597,2124,1048],{},[1452,2126,2127,2137],{},[590,2128,2129,2132,2134],{},[1344,2130,2131],{},"20",[597,2133,609],{},[1344,2135,2136],{},"4",[1344,2138,2136],{},[597,2140,1048],{},[1344,2142,2136],{},[1073,2144,1075],{},[1452,2146,2147,2150],{},[1073,2148,2149],{},"cm",[1073,2151,2152],{},"Woche",[620,2154,2155],{"encoding":622},"\\bar{m}_{[2,\\,6]} = \\frac{h(6) - h(2)}{6 - 2} = \\frac{20 - 4}{4} = 4\\,\\frac{\\text{cm}}{\\text{Woche}}",[574,2157,2159,2265,2386,2477],{"className":2158,"ariaHidden":628},[627],[574,2160,2162,2166,2256,2259,2262],{"className":2161},[632],[574,2163],{"className":2164,"style":2165},[636],"height:0.923em;vertical-align:-0.3552em;",[574,2167,2169,2200],{"className":2168},[641],[574,2170,2172],{"className":2171},[641,1518],[574,2173,2175],{"className":2174},[944],[574,2176,2178],{"className":2177},[948],[574,2179,2181,2189],{"className":2180,"style":1514},[952],[574,2182,2183,2186],{"style":1530},[574,2184],{"className":2185,"style":1534},[960],[574,2187,1445],{"className":2188},[641,642],[574,2190,2191,2194],{"style":1530},[574,2192],{"className":2193,"style":1534},[960],[574,2195,2197],{"className":2196,"style":1547},[1546],[574,2198,1448],{"className":2199},[641],[574,2201,2203],{"className":2202},[940],[574,2204,2206,2247],{"className":2205},[944,1166],[574,2207,2209,2244],{"className":2208},[948],[574,2210,2213],{"className":2211,"style":2212},[952],"height:0.3448em;",[574,2214,2216,2219],{"style":2215},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[574,2217],{"className":2218,"style":961},[960],[574,2220,2222],{"className":2221},[965,966,967,968],[574,2223,2225,2228,2231,2234,2238,2241],{"className":2224},[641,968],[574,2226,1281],{"className":2227},[647,968],[574,2229,1856],{"className":2230},[641,968],[574,2232,1286],{"className":2233},[1313,968],[574,2235],{"className":2236,"style":2237},[658,968],"margin-right:0.1952em;",[574,2239,2037],{"className":2240},[641,968],[574,2242,1291],{"className":2243},[654,968],[574,2245,1202],{"className":2246},[1201],[574,2248,2250],{"className":2249},[948],[574,2251,2254],{"className":2252,"style":2253},[952],"height:0.3552em;",[574,2255],{},[574,2257],{"className":2258,"style":1135},[658],[574,2260,1048],{"className":2261},[1139],[574,2263],{"className":2264,"style":1135},[658],[574,2266,2268,2272,2377,2380,2383],{"className":2267},[632],[574,2269],{"className":2270,"style":2271},[636],"height:2.1963em;vertical-align:-0.7693em;",[574,2273,2275,2278,2374],{"className":2274},[641],[574,2276],{"className":2277},[647,1573],[574,2279,2281],{"className":2280},[1452],[574,2282,2284,2366],{"className":2283},[944,1166],[574,2285,2287,2363],{"className":2286},[948],[574,2288,2291,2314,2322],{"className":2289,"style":2290},[952],"height:1.427em;",[574,2292,2293,2296],{"style":1589},[574,2294],{"className":2295,"style":1534},[960],[574,2297,2299,2302,2305,2308,2311],{"className":2298},[641],[574,2300,2037],{"className":2301},[641],[574,2303],{"className":2304,"style":659},[658],[574,2306,609],{"className":2307},[663],[574,2309],{"className":2310,"style":659},[658],[574,2312,1856],{"className":2313},[641],[574,2315,2316,2319],{"style":1632},[574,2317],{"className":2318,"style":1534},[960],[574,2320],{"className":2321,"style":1640},[1639],[574,2323,2324,2327],{"style":1643},[574,2325],{"className":2326,"style":1534},[960],[574,2328,2330,2333,2336,2339,2342,2345,2348,2351,2354,2357,2360],{"className":2329},[641],[574,2331,1838],{"className":2332},[641,642],[574,2334,600],{"className":2335},[647],[574,2337,2037],{"className":2338},[641],[574,2340,606],{"className":2341},[654],[574,2343],{"className":2344,"style":659},[658],[574,2346,609],{"className":2347},[663],[574,2349],{"className":2350,"style":659},[658],[574,2352,1838],{"className":2353},[641,642],[574,2355,600],{"className":2356},[647],[574,2358,1856],{"className":2359},[641],[574,2361,606],{"className":2362},[654],[574,2364,1202],{"className":2365},[1201],[574,2367,2369],{"className":2368},[948],[574,2370,2372],{"className":2371,"style":1692},[952],[574,2373],{},[574,2375],{"className":2376},[654,1573],[574,2378],{"className":2379,"style":1135},[658],[574,2381,1048],{"className":2382},[1139],[574,2384],{"className":2385,"style":1135},[658],[574,2387,2389,2393,2468,2471,2474],{"className":2388},[632],[574,2390],{"className":2391,"style":2392},[636],"height:2.0074em;vertical-align:-0.686em;",[574,2394,2396,2399,2465],{"className":2395},[641],[574,2397],{"className":2398},[647,1573],[574,2400,2402],{"className":2401},[1452],[574,2403,2405,2457],{"className":2404},[944,1166],[574,2406,2408,2454],{"className":2407},[948],[574,2409,2412,2423,2431],{"className":2410,"style":2411},[952],"height:1.3214em;",[574,2413,2414,2417],{"style":1589},[574,2415],{"className":2416,"style":1534},[960],[574,2418,2420],{"className":2419},[641],[574,2421,2136],{"className":2422},[641],[574,2424,2425,2428],{"style":1632},[574,2426],{"className":2427,"style":1534},[960],[574,2429],{"className":2430,"style":1640},[1639],[574,2432,2433,2436],{"style":1643},[574,2434],{"className":2435,"style":1534},[960],[574,2437,2439,2442,2445,2448,2451],{"className":2438},[641],[574,2440,2131],{"className":2441},[641],[574,2443],{"className":2444,"style":659},[658],[574,2446,609],{"className":2447},[663],[574,2449],{"className":2450,"style":659},[658],[574,2452,2136],{"className":2453},[641],[574,2455,1202],{"className":2456},[1201],[574,2458,2460],{"className":2459},[948],[574,2461,2463],{"className":2462,"style":1783},[952],[574,2464],{},[574,2466],{"className":2467},[654,1573],[574,2469],{"className":2470,"style":1135},[658],[574,2472,1048],{"className":2473},[1139],[574,2475],{"className":2476,"style":1135},[658],[574,2478,2480,2484,2487,2490],{"className":2479},[632],[574,2481],{"className":2482,"style":2483},[636],"height:1.7936em;vertical-align:-0.686em;",[574,2485,2136],{"className":2486},[641],[574,2488],{"className":2489,"style":1215},[658],[574,2491,2493,2496,2556],{"className":2492},[641],[574,2494],{"className":2495},[647,1573],[574,2497,2499],{"className":2498},[1452],[574,2500,2502,2548],{"className":2501},[944,1166],[574,2503,2505,2545],{"className":2504},[948],[574,2506,2509,2523,2531],{"className":2507,"style":2508},[952],"height:1.1076em;",[574,2510,2511,2514],{"style":1589},[574,2512],{"className":2513,"style":1534},[960],[574,2515,2517],{"className":2516},[641],[574,2518,2520],{"className":2519},[641,1605],[574,2521,2152],{"className":2522},[641],[574,2524,2525,2528],{"style":1632},[574,2526],{"className":2527,"style":1534},[960],[574,2529],{"className":2530,"style":1640},[1639],[574,2532,2533,2536],{"style":1643},[574,2534],{"className":2535,"style":1534},[960],[574,2537,2539],{"className":2538},[641],[574,2540,2542],{"className":2541},[641,1605],[574,2543,2149],{"className":2544},[641],[574,2546,1202],{"className":2547},[1201],[574,2549,2551],{"className":2550},[948],[574,2552,2554],{"className":2553,"style":1783},[952],[574,2555],{},[574,2557],{"className":2558},[654,1573],[565,2560,2561,2562,2661,2662,2713,2714,2742,2743,2920],{},"Die momentane Wachstumsrate ",[574,2563,2565,2593],{"className":2564},[577],[574,2566,2568],{"className":2567},[581],[583,2569,2570],{"xmlns":585},[587,2571,2572,2590],{},[590,2573,2574,2580,2582,2584,2586,2588],{},[903,2575,2576,2578],{},[593,2577,1838],{},[597,2579,911],{"mathvariant":909,"lspace":910,"rspace":910},[597,2581,600],{"stretchy":599},[593,2583,1069],{},[597,2585,606],{"stretchy":599},[597,2587,1048],{},[593,2589,1069],{},[620,2591,2592],{"encoding":622},"h'(t) = t",[574,2594,2596,2652],{"className":2595,"ariaHidden":628},[627],[574,2597,2599,2602,2634,2637,2640,2643,2646,2649],{"className":2598},[632],[574,2600],{"className":2601,"style":930},[636],[574,2603,2605,2608],{"className":2604},[641],[574,2606,1838],{"className":2607},[641,642],[574,2609,2611],{"className":2610},[940],[574,2612,2614],{"className":2613},[944],[574,2615,2617],{"className":2616},[948],[574,2618,2620],{"className":2619,"style":953},[952],[574,2621,2622,2625],{"style":956},[574,2623],{"className":2624,"style":961},[960],[574,2626,2628],{"className":2627},[965,966,967,968],[574,2629,2631],{"className":2630},[641,968],[574,2632,911],{"className":2633},[641,968],[574,2635,600],{"className":2636},[647],[574,2638,1069],{"className":2639},[641,642],[574,2641,606],{"className":2642},[654],[574,2644],{"className":2645,"style":1135},[658],[574,2647,1048],{"className":2648},[1139],[574,2650],{"className":2651,"style":1135},[658],[574,2653,2655,2658],{"className":2654},[632],[574,2656],{"className":2657,"style":1989},[636],[574,2659,1069],{"className":2660},[641,642]," ist bei ",[574,2663,2665,2683],{"className":2664},[577],[574,2666,2668],{"className":2667},[581],[583,2669,2670],{"xmlns":585},[587,2671,2672,2680],{},[590,2673,2674,2676,2678],{},[593,2675,1069],{},[597,2677,1048],{},[1344,2679,2136],{},[620,2681,2682],{"encoding":622},"t = 4",[574,2684,2686,2704],{"className":2685,"ariaHidden":628},[627],[574,2687,2689,2692,2695,2698,2701],{"className":2688},[632],[574,2690],{"className":2691,"style":1989},[636],[574,2693,1069],{"className":2694},[641,642],[574,2696],{"className":2697,"style":1135},[658],[574,2699,1048],{"className":2700},[1139],[574,2702],{"className":2703,"style":1135},[658],[574,2705,2707,2710],{"className":2706},[632],[574,2708],{"className":2709,"style":1358},[636],[574,2711,2136],{"className":2712},[641]," ebenfalls ",[574,2715,2717,2730],{"className":2716},[577],[574,2718,2720],{"className":2719},[581],[583,2721,2722],{"xmlns":585},[587,2723,2724,2728],{},[590,2725,2726],{},[1344,2727,2136],{},[620,2729,2136],{"encoding":622},[574,2731,2733],{"className":2732,"ariaHidden":628},[627],[574,2734,2736,2739],{"className":2735},[632],[574,2737],{"className":2738,"style":1358},[636],[574,2740,2136],{"className":2741},[641],". Die Gesamtänderung ergibt sich auch als ",[574,2744,2746,2788],{"className":2745},[577],[574,2747,2749],{"className":2748},[581],[583,2750,2751],{"xmlns":585},[587,2752,2753,2785],{},[590,2754,2755,2763,2765,2767,2769,2771,2773,2776,2778,2780,2782],{},[1050,2756,2757,2759,2761],{},[597,2758,1054],{},[1344,2760,1856],{},[1344,2762,2037],{},[593,2764,1069],{},[1073,2766,1075],{},[593,2768,1078],{},[593,2770,1069],{},[597,2772,1048],{},[1344,2774,2775],{},"18",[597,2777,609],{},[1344,2779,1856],{},[597,2781,1048],{},[1344,2783,2784],{},"16",[620,2786,2787],{"encoding":622},"\\int_2^6 t\\,dt = 18 - 2 = 16",[574,2789,2791,2874,2893,2911],{"className":2790,"ariaHidden":628},[627],[574,2792,2794,2798,2850,2853,2856,2859,2862,2865,2868,2871],{"className":2793},[632],[574,2795],{"className":2796,"style":2797},[636],"height:1.3648em;vertical-align:-0.3558em;",[574,2799,2801,2804],{"className":2800},[1153],[574,2802,1054],{"className":2803,"style":1159},[1153,1157,1158],[574,2805,2807],{"className":2806},[940],[574,2808,2810,2842],{"className":2809},[944,1166],[574,2811,2813,2839],{"className":2812},[948],[574,2814,2817,2828],{"className":2815,"style":2816},[952],"height:1.009em;",[574,2818,2819,2822],{"style":1176},[574,2820],{"className":2821,"style":961},[960],[574,2823,2825],{"className":2824},[965,966,967,968],[574,2826,1856],{"className":2827},[641,968],[574,2829,2830,2833],{"style":1188},[574,2831],{"className":2832,"style":961},[960],[574,2834,2836],{"className":2835},[965,966,967,968],[574,2837,2037],{"className":2838},[641,968],[574,2840,1202],{"className":2841},[1201],[574,2843,2845],{"className":2844},[948],[574,2846,2848],{"className":2847,"style":1209},[952],[574,2849],{},[574,2851],{"className":2852,"style":1215},[658],[574,2854,1069],{"className":2855},[641,642],[574,2857],{"className":2858,"style":1215},[658],[574,2860,1078],{"className":2861},[641,642],[574,2863,1069],{"className":2864},[641,642],[574,2866],{"className":2867,"style":1135},[658],[574,2869,1048],{"className":2870},[1139],[574,2872],{"className":2873,"style":1135},[658],[574,2875,2877,2881,2884,2887,2890],{"className":2876},[632],[574,2878],{"className":2879,"style":2880},[636],"height:0.7278em;vertical-align:-0.0833em;",[574,2882,2775],{"className":2883},[641],[574,2885],{"className":2886,"style":659},[658],[574,2888,609],{"className":2889},[663],[574,2891],{"className":2892,"style":659},[658],[574,2894,2896,2899,2902,2905,2908],{"className":2895},[632],[574,2897],{"className":2898,"style":1358},[636],[574,2900,1856],{"className":2901},[641],[574,2903],{"className":2904,"style":1135},[658],[574,2906,1048],{"className":2907},[1139],[574,2909],{"className":2910,"style":1135},[658],[574,2912,2914,2917],{"className":2913},[632],[574,2915],{"className":2916,"style":1358},[636],[574,2918,2784],{"className":2919},[641]," cm.",[565,2922,2923,2924,2952,2953,3096],{},"Die Temperatur fällt in ",[574,2925,2927,2940],{"className":2926},[577],[574,2928,2930],{"className":2929},[581],[583,2931,2932],{"xmlns":585},[587,2933,2934,2938],{},[590,2935,2936],{},[1344,2937,2136],{},[620,2939,2136],{"encoding":622},[574,2941,2943],{"className":2942,"ariaHidden":628},[627],[574,2944,2946,2949],{"className":2945},[632],[574,2947],{"className":2948,"style":1358},[636],[574,2950,2136],{"className":2951},[641]," Stunden von 18 °C auf 6 °C. Die mittlere Änderungsrate ist ",[574,2954,2956,2987],{"className":2955},[577],[574,2957,2959],{"className":2958},[581],[583,2960,2961],{"xmlns":585},[587,2962,2963,2984],{},[590,2964,2965,2977,2979,2981],{},[1452,2966,2967,2975],{},[590,2968,2969,2971,2973],{},[1344,2970,2037],{},[597,2972,609],{},[1344,2974,2775],{},[1344,2976,2136],{},[597,2978,1048],{},[597,2980,609],{},[1344,2982,2983],{},"3",[620,2985,2986],{"encoding":622},"\\frac{6 - 18}{4} = -3",[574,2988,2990,3084],{"className":2989,"ariaHidden":628},[627],[574,2991,2993,2997,3075,3078,3081],{"className":2992},[632],[574,2994],{"className":2995,"style":2996},[636],"height:1.1901em;vertical-align:-0.345em;",[574,2998,3000,3003,3072],{"className":2999},[641],[574,3001],{"className":3002},[647,1573],[574,3004,3006],{"className":3005},[1452],[574,3007,3009,3063],{"className":3008},[944,1166],[574,3010,3012,3060],{"className":3011},[948],[574,3013,3016,3031,3039],{"className":3014,"style":3015},[952],"height:0.8451em;",[574,3017,3019,3022],{"style":3018},"top:-2.655em;",[574,3020],{"className":3021,"style":1534},[960],[574,3023,3025],{"className":3024},[965,966,967,968],[574,3026,3028],{"className":3027},[641,968],[574,3029,2136],{"className":3030},[641,968],[574,3032,3033,3036],{"style":1632},[574,3034],{"className":3035,"style":1534},[960],[574,3037],{"className":3038,"style":1640},[1639],[574,3040,3042,3045],{"style":3041},"top:-3.394em;",[574,3043],{"className":3044,"style":1534},[960],[574,3046,3048],{"className":3047},[965,966,967,968],[574,3049,3051,3054,3057],{"className":3050},[641,968],[574,3052,2037],{"className":3053},[641,968],[574,3055,609],{"className":3056},[663,968],[574,3058,2775],{"className":3059},[641,968],[574,3061,1202],{"className":3062},[1201],[574,3064,3066],{"className":3065},[948],[574,3067,3070],{"className":3068,"style":3069},[952],"height:0.345em;",[574,3071],{},[574,3073],{"className":3074},[654,1573],[574,3076],{"className":3077,"style":1135},[658],[574,3079,1048],{"className":3080},[1139],[574,3082],{"className":3083,"style":1135},[658],[574,3085,3087,3090,3093],{"className":3086},[632],[574,3088],{"className":3089,"style":2880},[636],[574,3091,609],{"className":3092},[641],[574,3094,2983],{"className":3095},[641]," Grad pro Stunde, das Minus zeigt die Abnahme.",[560,3098,3100],{"id":3099},"typische-fehler","Typische Fehler",[807,3102,3103,3187,3190,3271],{},[810,3104,3105,3106,3186],{},"Nur die Gesamtänderung 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angeben und nicht durch die Intervalllänge teilen.",[810,3188,3189],{},"Die Einheit weglassen oder falsch bilden, z.B. „Liter\" statt „Liter pro Minute\".",[810,3191,3192,3193,3270],{},"Mittlere und momentane Änderungsrate verwechseln. „Wie schnell wächst die Pflanze in Woche 4?\" fragt nach 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„im Durchschnitt zwischen Woche 2 und 6\" nach dem Differenzenquotienten.",[810,3272,3273],{},"Oben und unten in verschiedener Reihenfolge subtrahieren. Das dreht das Vorzeichen um.",{"title":3275,"searchDepth":3276,"depth":3276,"links":3277},"",2,[3278,3279,3280],{"id":562,"depth":3276,"text":563},{"id":1326,"depth":3276,"text":1327},{"id":3099,"depth":3276,"text":3100},[3282,3285,3287,3289,3291,3293,3295,3297],{"lhs":3283,"rhs":3284},"(fb - fa)\u002F(b - a)","(fa - fb)\u002F(a - b)",{"lhs":3286,"rhs":1493},"(300 - 120)\u002F(15 - 0)",{"lhs":3288,"rhs":2131},"0.5*6^2 + 2",{"lhs":3290,"rhs":2136},"0.5*2^2 + 2",{"lhs":3292,"rhs":2136},"(20 - 4)\u002F(6 - 2)",{"lhs":3294,"rhs":2136},"round((0.5*(4+1e-5)^2 - 0.5*(4-1e-5)^2)\u002F(2e-5), 6)",{"lhs":3296,"rhs":2784},"6^2\u002F2 - 2^2\u002F2",{"lhs":3298,"rhs":3299},"(6 - 18)\u002F4","-3","a \\neq b,\\ f \\text{ an den Stellen } a \\text{ und } b \\text{ definiert}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\">≠\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a \\neq b,\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:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">\u003Cspan class=\"mrel\">\u003Cspan class=\"mord katex-vbox\">\u003Cspan class=\"katex-thinbox\">\u003Cspan class=\"rlap\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-inner\">\u003Cspan class=\"mord\">\u003Cspan class=\"mrel\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-fix\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace nobreak\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">f\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:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">an den Stellen\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a\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:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan 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