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Dann legt man im Punkt ",[573,766,768,806],{"className":767},[576],[573,769,771],{"className":770},[580],[582,772,773],{"xmlns":584},[586,774,775,803],{},[589,776,777,779,786,789,791,793,799,801],{},[596,778,599],{"stretchy":598},[689,780,781,783],{},[592,782,602],{},[592,784,785],{},"n",[596,787,788],{},"∣",[592,790,594],{},[596,792,599],{"stretchy":598},[689,794,795,797],{},[592,796,602],{},[592,798,785],{},[596,800,605],{"stretchy":598},[596,802,605],{"stretchy":598},[614,804,805],{"encoding":616},"(x_n \\mid f(x_n))",[573,807,809,868],{"className":808,"ariaHidden":622},[621],[573,810,812,815,818,859,862,865],{"className":811},[626],[573,813],{"className":814,"style":631},[630],[573,816,599],{"className":817},[641],[573,819,821,824],{"className":820},[635],[573,822,602],{"className":823},[635,636],[573,825,827],{"className":826},[717],[573,828,830,851],{"className":829},[721,722],[573,831,833,848],{"className":832},[726],[573,834,837],{"className":835,"style":836},[730],"height:0.1514em;",[573,838,839,842],{"style":734},[573,840],{"className":841,"style":739},[738],[573,843,845],{"className":844},[743,744,745,746],[573,846,785],{"className":847},[635,636,746],[573,849,754],{"className":850},[753],[573,852,854],{"className":853},[726],[573,855,857],{"className":856,"style":761},[730],[573,858],{},[573,860],{"className":861,"style":653},[652],[573,863,788],{"className":864},[657],[573,866],{"className":867,"style":653},[652],[573,869,871,874,877,880,920],{"className":870},[626],[573,872],{"className":873,"style":631},[630],[573,875,594],{"className":876,"style":637},[635,636],[573,878,599],{"className":879},[641],[573,881,883,886],{"className":882},[635],[573,884,602],{"className":885},[635,636],[573,887,889],{"className":888},[717],[573,890,892,912],{"className":891},[721,722],[573,893,895,909],{"className":894},[726],[573,896,898],{"className":897,"style":836},[730],[573,899,900,903],{"style":734},[573,901],{"className":902,"style":739},[738],[573,904,906],{"className":905},[743,744,745,746],[573,907,785],{"className":908},[635,636,746],[573,910,754],{"className":911},[753],[573,913,915],{"className":914},[726],[573,916,918],{"className":917,"style":761},[730],[573,919],{},[573,921,923],{"className":922},[648],"))"," die Tangente an den Graphen und nimmt deren Nullstelle als nächste Näherung ",[573,926,928,954],{"className":927},[576],[573,929,931],{"className":930},[580],[582,932,933],{"xmlns":584},[586,934,935,951],{},[589,936,937],{},[689,938,939,941],{},[592,940,602],{},[589,942,943,945,948],{},[592,944,785],{},[596,946,947],{},"+",[610,949,950],{},"1",[614,952,953],{"encoding":616},"x_{n+1}",[573,955,957],{"className":956,"ariaHidden":622},[621],[573,958,960,964],{"className":959},[626],[573,961],{"className":962,"style":963},[630],"height:0.6389em;vertical-align:-0.2083em;",[573,965,967,970],{"className":966},[635],[573,968,602],{"className":969},[635,636],[573,971,973],{"className":972},[717],[573,974,976,1006],{"className":975},[721,722],[573,977,979,1003],{"className":978},[726],[573,980,982],{"className":981,"style":731},[730],[573,983,984,987],{"style":734},[573,985],{"className":986,"style":739},[738],[573,988,990],{"className":989},[743,744,745,746],[573,991,993,996,1000],{"className":992},[635,746],[573,994,785],{"className":995},[635,636,746],[573,997,947],{"className":998},[999,746],"mbin",[573,1001,950],{"className":1002},[635,746],[573,1004,754],{"className":1005},[753],[573,1007,1009],{"className":1008},[726],[573,1010,1013],{"className":1011,"style":1012},[730],"height:0.2083em;",[573,1014],{},". Das wiederholt man, bis sich die Werte in der gewünschten Genauigkeit nicht mehr ändern.",[565,1017,1018,1019,1340,1341,1523],{},"Die Formel ist die Nullstelle der Tangente ",[573,1020,1022,1094],{"className":1021},[576],[573,1023,1025],{"className":1024},[580],[582,1026,1027],{"xmlns":584},[586,1028,1029,1091],{},[589,1030,1031,1034,1036,1038,1040,1042,1044,1046,1052,1054,1056,1066,1068,1074,1076,1078,1080,1083,1089],{},[592,1032,1033],{},"t",[596,1035,599],{"stretchy":598},[592,1037,602],{},[596,1039,605],{"stretchy":598},[596,1041,608],{},[592,1043,594],{},[596,1045,599],{"stretchy":598},[689,1047,1048,1050],{},[592,1049,602],{},[592,1051,785],{},[596,1053,605],{"stretchy":598},[596,1055,947],{},[1057,1058,1059,1061],"msup",{},[592,1060,594],{},[596,1062,1065],{"mathvariant":1063,"lspace":1064,"rspace":1064},"normal","0em","′",[596,1067,599],{"stretchy":598},[689,1069,1070,1072],{},[592,1071,602],{},[592,1073,785],{},[596,1075,605],{"stretchy":598},[596,1077,599],{"stretchy":598},[592,1079,602],{},[596,1081,1082],{},"−",[689,1084,1085,1087],{},[592,1086,602],{},[592,1088,785],{},[596,1090,605],{"stretchy":598},[614,1092,1093],{"encoding":616},"t(x) = f(x_n) + f'(x_n)(x - x_n)",[573,1095,1097,1124,1189,1291],{"className":1096,"ariaHidden":622},[621],[573,1098,1100,1103,1106,1109,1112,1115,1118,1121],{"className":1099},[626],[573,1101],{"className":1102,"style":631},[630],[573,1104,1033],{"className":1105},[635,636],[573,1107,599],{"className":1108},[641],[573,1110,602],{"className":1111},[635,636],[573,1113,605],{"className":1114},[648],[573,1116],{"className":1117,"style":653},[652],[573,1119,608],{"className":1120},[657],[573,1122],{"className":1123,"style":653},[652],[573,1125,1127,1130,1133,1136,1176,1179,1183,1186],{"className":1126},[626],[573,1128],{"className":1129,"style":631},[630],[573,1131,594],{"className":1132,"style":637},[635,636],[573,1134,599],{"className":1135},[641],[573,1137,1139,1142],{"className":1138},[635],[573,1140,602],{"className":1141},[635,636],[573,1143,1145],{"className":1144},[717],[573,1146,1148,1168],{"className":1147},[721,722],[573,1149,1151,1165],{"className":1150},[726],[573,1152,1154],{"className":1153,"style":836},[730],[573,1155,1156,1159],{"style":734},[573,1157],{"className":1158,"style":739},[738],[573,1160,1162],{"className":1161},[743,744,745,746],[573,1163,785],{"className":1164},[635,636,746],[573,1166,754],{"className":1167},[753],[573,1169,1171],{"className":1170},[726],[573,1172,1174],{"className":1173,"style":761},[730],[573,1175],{},[573,1177,605],{"className":1178},[648],[573,1180],{"className":1181,"style":1182},[652],"margin-right:0.2222em;",[573,1184,947],{"className":1185},[999],[573,1187],{"className":1188,"style":1182},[652],[573,1190,1192,1196,1230,1233,1273,1276,1279,1282,1285,1288],{"className":1191},[626],[573,1193],{"className":1194,"style":1195},[630],"height:1.0019em;vertical-align:-0.25em;",[573,1197,1199,1202],{"className":1198},[635],[573,1200,594],{"className":1201,"style":637},[635,636],[573,1203,1205],{"className":1204},[717],[573,1206,1208],{"className":1207},[721],[573,1209,1211],{"className":1210},[726],[573,1212,1215],{"className":1213,"style":1214},[730],"height:0.7519em;",[573,1216,1218,1221],{"style":1217},"top:-3.063em;margin-right:0.05em;",[573,1219],{"className":1220,"style":739},[738],[573,1222,1224],{"className":1223},[743,744,745,746],[573,1225,1227],{"className":1226},[635,746],[573,1228,1065],{"className":1229},[635,746],[573,1231,599],{"className":1232},[641],[573,1234,1236,1239],{"className":1235},[635],[573,1237,602],{"className":1238},[635,636],[573,1240,1242],{"className":1241},[717],[573,1243,1245,1265],{"className":1244},[721,722],[573,1246,1248,1262],{"className":1247},[726],[573,1249,1251],{"className":1250,"style":836},[730],[573,1252,1253,1256],{"style":734},[573,1254],{"className":1255,"style":739},[738],[573,1257,1259],{"className":1258},[743,744,745,746],[573,1260,785],{"className":1261},[635,636,746],[573,1263,754],{"className":1264},[753],[573,1266,1268],{"className":1267},[726],[573,1269,1271],{"className":1270,"style":761},[730],[573,1272],{},[573,1274,605],{"className":1275},[648],[573,1277,599],{"className":1278},[641],[573,1280,602],{"className":1281},[635,636],[573,1283],{"className":1284,"style":1182},[652],[573,1286,1082],{"className":1287},[999],[573,1289],{"className":1290,"style":1182},[652],[573,1292,1294,1297,1337],{"className":1293},[626],[573,1295],{"className":1296,"style":631},[630],[573,1298,1300,1303],{"className":1299},[635],[573,1301,602],{"className":1302},[635,636],[573,1304,1306],{"className":1305},[717],[573,1307,1309,1329],{"className":1308},[721,722],[573,1310,1312,1326],{"className":1311},[726],[573,1313,1315],{"className":1314,"style":836},[730],[573,1316,1317,1320],{"style":734},[573,1318],{"className":1319,"style":739},[738],[573,1321,1323],{"className":1322},[743,744,745,746],[573,1324,785],{"className":1325},[635,636,746],[573,1327,754],{"className":1328},[753],[573,1330,1332],{"className":1331},[726],[573,1333,1335],{"className":1334,"style":761},[730],[573,1336],{},[573,1338,605],{"className":1339},[648],". Deshalb muss ",[573,1342,1344,1377],{"className":1343},[576],[573,1345,1347],{"className":1346},[580],[582,1348,1349],{"xmlns":584},[586,1350,1351,1374],{},[589,1352,1353,1359,1361,1367,1369,1372],{},[1057,1354,1355,1357],{},[592,1356,594],{},[596,1358,1065],{"mathvariant":1063,"lspace":1064,"rspace":1064},[596,1360,599],{"stretchy":598},[689,1362,1363,1365],{},[592,1364,602],{},[592,1366,785],{},[596,1368,605],{"stretchy":598},[596,1370,1371],{"mathvariant":1063},"≠",[610,1373,612],{},[614,1375,1376],{"encoding":616},"f'(x_n) \\neq 0",[573,1378,1380,1514],{"className":1379,"ariaHidden":622},[621],[573,1381,1383,1386,1418,1421,1461,1464,1467,1511],{"className":1382},[626],[573,1384],{"className":1385,"style":1195},[630],[573,1387,1389,1392],{"className":1388},[635],[573,1390,594],{"className":1391,"style":637},[635,636],[573,1393,1395],{"className":1394},[717],[573,1396,1398],{"className":1397},[721],[573,1399,1401],{"className":1400},[726],[573,1402,1404],{"className":1403,"style":1214},[730],[573,1405,1406,1409],{"style":1217},[573,1407],{"className":1408,"style":739},[738],[573,1410,1412],{"className":1411},[743,744,745,746],[573,1413,1415],{"className":1414},[635,746],[573,1416,1065],{"className":1417},[635,746],[573,1419,599],{"className":1420},[641],[573,1422,1424,1427],{"className":1423},[635],[573,1425,602],{"className":1426},[635,636],[573,1428,1430],{"className":1429},[717],[573,1431,1433,1453],{"className":1432},[721,722],[573,1434,1436,1450],{"className":1435},[726],[573,1437,1439],{"className":1438,"style":836},[730],[573,1440,1441,1444],{"style":734},[573,1442],{"className":1443,"style":739},[738],[573,1445,1447],{"className":1446},[743,744,745,746],[573,1448,785],{"className":1449},[635,636,746],[573,1451,754],{"className":1452},[753],[573,1454,1456],{"className":1455},[726],[573,1457,1459],{"className":1458,"style":761},[730],[573,1460],{},[573,1462,605],{"className":1463},[648],[573,1465],{"className":1466,"style":653},[652],[573,1468,1470,1504,1508],{"className":1469},[657],[573,1471,1473],{"className":1472},[657],[573,1474,1477],{"className":1475},[635,1476],"vbox",[573,1478,1481],{"className":1479},[1480],"thinbox",[573,1482,1485,1489,1500],{"className":1483},[1484],"rlap",[573,1486],{"className":1487,"style":1488},[630],"height:0.8889em;vertical-align:-0.1944em;",[573,1490,1493],{"className":1491},[1492],"inner",[573,1494,1496],{"className":1495},[635],[573,1497,1499],{"className":1498},[657],"",[573,1501],{"className":1502},[1503],"fix",[573,1505],{"className":1506},[652,1507],"nobreak",[573,1509,608],{"className":1510},[657],[573,1512],{"className":1513,"style":653},[652],[573,1515,1517,1520],{"className":1516},[626],[573,1518],{"className":1519,"style":667},[630],[573,1521,612],{"className":1522},[635]," sein, sonst verläuft die Tangente waagerecht und hat keine Nullstelle.",[565,1525,1526,1527,1530],{},"Liegt der Startwert nah genug an einer einfachen Nullstelle, konvergiert das Verfahren sehr schnell: Die Anzahl der richtigen Stellen verdoppelt sich ungefähr in jedem Schritt (",[569,1528,1529],{},"quadratische Konvergenz","). Bei schlechtem Startwert kann es aber zu einer anderen Nullstelle laufen, pendeln oder ganz versagen.",[560,1532,1534],{"id":1533},"beispiele","Beispiele",[565,1536,1537,1633,1634,1754,1755,1859,1860,1952],{},[573,1538,1540,1558],{"className":1539},[576],[573,1541,1543],{"className":1542},[580],[582,1544,1545],{"xmlns":584},[586,1546,1547,1555],{},[589,1548,1549],{},[1550,1551,1552],"msqrt",{},[610,1553,1554],{},"2",[614,1556,1557],{"encoding":616},"\\sqrt{2}",[573,1559,1561],{"className":1560,"ariaHidden":622},[621],[573,1562,1564,1568],{"className":1563},[626],[573,1565],{"className":1566,"style":1567},[630],"height:1.04em;vertical-align:-0.1328em;",[573,1569,1572],{"className":1570},[635,1571],"sqrt",[573,1573,1575,1624],{"className":1574},[721,722],[573,1576,1578,1621],{"className":1577},[726],[573,1579,1582,1598],{"className":1580,"style":1581},[730],"height:0.9072em;",[573,1583,1587,1591],{"className":1584,"style":1586},[1585],"svg-align","top:-3em;",[573,1588],{"className":1589,"style":1590},[738],"height:3em;",[573,1592,1595],{"className":1593,"style":1594},[635],"padding-left:0.833em;",[573,1596,1554],{"className":1597},[635],[573,1599,1601,1604],{"style":1600},"top:-2.8672em;",[573,1602],{"className":1603,"style":1590},[738],[573,1605,1609],{"className":1606,"style":1608},[1607],"hide-tail","min-width:0.853em;height:1.08em;",[1610,1611,1617],"svg",{"xmlns":1612,"width":1613,"height":1614,"viewBox":1615,"preserveAspectRatio":1616},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[1618,1619],"path",{"d":1620},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[573,1622,754],{"className":1623},[753],[573,1625,1627],{"className":1626},[726],[573,1628,1631],{"className":1629,"style":1630},[730],"height:0.1328em;",[573,1632],{}," als Nullstelle von ",[573,1635,1637,1669],{"className":1636},[576],[573,1638,1640],{"className":1639},[580],[582,1641,1642],{"xmlns":584},[586,1643,1644,1666],{},[589,1645,1646,1648,1650,1652,1654,1656,1662,1664],{},[592,1647,594],{},[596,1649,599],{"stretchy":598},[592,1651,602],{},[596,1653,605],{"stretchy":598},[596,1655,608],{},[1057,1657,1658,1660],{},[592,1659,602],{},[610,1661,1554],{},[596,1663,1082],{},[610,1665,1554],{},[614,1667,1668],{"encoding":616},"f(x) = x^2 - 2",[573,1670,1672,1699,1745],{"className":1671,"ariaHidden":622},[621],[573,1673,1675,1678,1681,1684,1687,1690,1693,1696],{"className":1674},[626],[573,1676],{"className":1677,"style":631},[630],[573,1679,594],{"className":1680,"style":637},[635,636],[573,1682,599],{"className":1683},[641],[573,1685,602],{"className":1686},[635,636],[573,1688,605],{"className":1689},[648],[573,1691],{"className":1692,"style":653},[652],[573,1694,608],{"className":1695},[657],[573,1697],{"className":1698,"style":653},[652],[573,1700,1702,1706,1736,1739,1742],{"className":1701},[626],[573,1703],{"className":1704,"style":1705},[630],"height:0.8974em;vertical-align:-0.0833em;",[573,1707,1709,1712],{"className":1708},[635],[573,1710,602],{"className":1711},[635,636],[573,1713,1715],{"className":1714},[717],[573,1716,1718],{"className":1717},[721],[573,1719,1721],{"className":1720},[726],[573,1722,1725],{"className":1723,"style":1724},[730],"height:0.8141em;",[573,1726,1727,1730],{"style":1217},[573,1728],{"className":1729,"style":739},[738],[573,1731,1733],{"className":1732},[743,744,745,746],[573,1734,1554],{"className":1735},[635,746],[573,1737],{"className":1738,"style":1182},[652],[573,1740,1082],{"className":1741},[999],[573,1743],{"className":1744,"style":1182},[652],[573,1746,1748,1751],{"className":1747},[626],[573,1749],{"className":1750,"style":667},[630],[573,1752,1554],{"className":1753},[635]," mit ",[573,1756,1758,1788],{"className":1757},[576],[573,1759,1761],{"className":1760},[580],[582,1762,1763],{"xmlns":584},[586,1764,1765,1785],{},[589,1766,1767,1773,1775,1777,1779,1781,1783],{},[1057,1768,1769,1771],{},[592,1770,594],{},[596,1772,1065],{"mathvariant":1063,"lspace":1064,"rspace":1064},[596,1774,599],{"stretchy":598},[592,1776,602],{},[596,1778,605],{"stretchy":598},[596,1780,608],{},[610,1782,1554],{},[592,1784,602],{},[614,1786,1787],{"encoding":616},"f'(x) = 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genaue Wert ist ",[573,3039,3041,3062],{"className":3040},[576],[573,3042,3044],{"className":3043},[580],[582,3045,3046],{"xmlns":584},[586,3047,3048,3059],{},[589,3049,3050,3054,3056],{},[1550,3051,3052],{},[610,3053,1554],{},[596,3055,2072],{},[610,3057,3058],{},"1,414214",[614,3060,3061],{"encoding":616},"\\sqrt{2} \\approx 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Für ",[573,3149,3151,3184],{"className":3150},[576],[573,3152,3154],{"className":3153},[580],[582,3155,3156],{"xmlns":584},[586,3157,3158,3181],{},[589,3159,3160,3162,3164,3166,3168,3170,3176,3178],{},[592,3161,594],{},[596,3163,599],{"stretchy":598},[592,3165,602],{},[596,3167,605],{"stretchy":598},[596,3169,608],{},[1057,3171,3172,3174],{},[592,3173,602],{},[610,3175,1554],{},[596,3177,1082],{},[592,3179,3180],{},"a",[614,3182,3183],{"encoding":616},"f(x) = x^2 - 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vereinfacht sich der Schritt zu ",[573,3270,3272,3328],{"className":3271},[576],[573,3273,3275],{"className":3274},[580],[582,3276,3277],{"xmlns":584},[586,3278,3279,3325],{},[589,3280,3281,3293,3295,3301],{},[689,3282,3283,3285],{},[592,3284,602],{},[589,3286,3287,3289,3291],{},[592,3288,785],{},[596,3290,947],{},[610,3292,950],{},[596,3294,608],{},[2003,3296,3297,3299],{},[610,3298,950],{},[610,3300,1554],{},[589,3302,3303,3305,3311,3313,3323],{},[596,3304,599],{"fence":622},[689,3306,3307,3309],{},[592,3308,602],{},[592,3310,785],{},[596,3312,947],{},[2003,3314,3315,3317],{},[592,3316,3180],{},[689,3318,3319,3321],{},[592,3320,602],{},[592,3322,785],{},[596,3324,605],{"fence":622},[614,3326,3327],{"encoding":616},"x_{n+1} = \\frac{1}{2}\\left(x_n + 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dem ",[569,3662,555],{},".",[565,3665,3666,3667,1754,3808,1859,3964,4065,4066,1859,4152,4271],{},"Nullstelle von ",[573,3668,3670,3706],{"className":3669},[576],[573,3671,3673],{"className":3672},[580],[582,3674,3675],{"xmlns":584},[586,3676,3677,3703],{},[589,3678,3679,3681,3683,3685,3687,3689,3695,3697,3699,3701],{},[592,3680,594],{},[596,3682,599],{"stretchy":598},[592,3684,602],{},[596,3686,605],{"stretchy":598},[596,3688,608],{},[1057,3690,3691,3693],{},[592,3692,602],{},[610,3694,2059],{},[596,3696,1082],{},[592,3698,602],{},[596,3700,1082],{},[610,3702,950],{},[614,3704,3705],{"encoding":616},"f(x) = x^3 - x - 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1",[573,3850,3852,3908,3955],{"className":3851,"ariaHidden":622},[621],[573,3853,3855,3858,3890,3893,3896,3899,3902,3905],{"className":3854},[626],[573,3856],{"className":3857,"style":1195},[630],[573,3859,3861,3864],{"className":3860},[635],[573,3862,594],{"className":3863,"style":637},[635,636],[573,3865,3867],{"className":3866},[717],[573,3868,3870],{"className":3869},[721],[573,3871,3873],{"className":3872},[726],[573,3874,3876],{"className":3875,"style":1214},[730],[573,3877,3878,3881],{"style":1217},[573,3879],{"className":3880,"style":739},[738],[573,3882,3884],{"className":3883},[743,744,745,746],[573,3885,3887],{"className":3886},[635,746],[573,3888,1065],{"className":3889},[635,746],[573,3891,599],{"className":3892},[641],[573,3894,602],{"className":3895},[635,636],[573,3897,605],{"className":3898},[648],[573,3900],{"className":3901,"style":653},[652],[573,3903,608],{"className":3904},[657],[573,3906],{"className":3907,"style":653},[652],[573,3909,3911,3914,3917,3946,3949,3952],{"className":3910},[626],[573,3912],{"className":3913,"style":1705},[630],[573,3915,2059],{"className":3916},[635],[573,3918,3920,3923],{"className":3919},[635],[573,3921,602],{"className":3922},[635,636],[573,3924,3926],{"className":3925},[717],[573,3927,3929],{"className":3928},[721],[573,3930,3932],{"className":3931},[726],[573,3933,3935],{"className":3934,"style":1724},[730],[573,3936,3937,3940],{"style":1217},[573,3938],{"className":3939,"style":739},[738],[573,3941,3943],{"className":3942},[743,744,745,746],[573,3944,1554],{"className":3945},[635,746],[573,3947],{"className":3948,"style":1182},[652],[573,3950,1082],{"className":3951},[999],[573,3953],{"className":3954,"style":1182},[652],[573,3956,3958,3961],{"className":3957},[626],[573,3959],{"className":3960,"style":667},[630],[573,3962,950],{"className":3963},[635],[573,3965,3967,3989],{"className":3966},[576],[573,3968,3970],{"className":3969},[580],[582,3971,3972],{"xmlns":584},[586,3973,3974,3986],{},[589,3975,3976,3982,3984],{},[689,3977,3978,3980],{},[592,3979,602],{},[610,3981,612],{},[596,3983,608],{},[610,3985,2019],{},[614,3987,3988],{"encoding":616},"x_0 = 1{,}5",[573,3990,3992,4047],{"className":3991,"ariaHidden":622},[621],[573,3993,3995,3998,4038,4041,4044],{"className":3994},[626],[573,3996],{"className":3997,"style":707},[630],[573,3999,4001,4004],{"className":4000},[635],[573,4002,602],{"className":4003},[635,636],[573,4005,4007],{"className":4006},[717],[573,4008,4010,4030],{"className":4009},[721,722],[573,4011,4013,4027],{"className":4012},[726],[573,4014,4016],{"className":4015,"style":731},[730],[573,4017,4018,4021],{"style":734},[573,4019],{"className":4020,"style":739},[738],[573,4022,4024],{"className":4023},[743,744,745,746],[573,4025,612],{"className":4026},[635,746],[573,4028,754],{"className":4029},[753],[573,4031,4033],{"className":4032},[726],[573,4034,4036],{"className":4035,"style":761},[730],[573,4037],{},[573,4039],{"className":4040,"style":653},[652],[573,4042,608],{"className":4043},[657],[573,4045],{"className":4046,"style":653},[652],[573,4048,4050,4053,4056,4062],{"className":4049},[626],[573,4051],{"className":4052,"style":3133},[630],[573,4054,950],{"className":4055},[635],[573,4057,4059],{"className":4058},[635],[573,4060,2496],{"className":4061},[2495],[573,4063,2500],{"className":4064},[635],": Es ist ",[573,4067,4069,4094],{"className":4068},[576],[573,4070,4072],{"className":4071},[580],[582,4073,4074],{"xmlns":584},[586,4075,4076,4091],{},[589,4077,4078,4080,4082,4084,4086,4088],{},[592,4079,594],{},[596,4081,599],{"stretchy":598},[610,4083,2019],{},[596,4085,605],{"stretchy":598},[596,4087,608],{},[610,4089,4090],{},"0,875",[614,4092,4093],{"encoding":616},"f(1{,}5) = 0{,}875",[573,4095,4097,4133],{"className":4096,"ariaHidden":622},[621],[573,4098,4100,4103,4106,4109,4112,4118,4121,4124,4127,4130],{"className":4099},[626],[573,4101],{"className":4102,"style":631},[630],[573,4104,594],{"className":4105,"style":637},[635,636],[573,4107,599],{"className":4108},[641],[573,4110,950],{"className":4111},[635],[573,4113,4115],{"className":4114},[635],[573,4116,2496],{"className":4117},[2495],[573,4119,2500],{"className":4120},[635],[573,4122,605],{"className":4123},[648],[573,4125],{"className":4126,"style":653},[652],[573,4128,608],{"className":4129},[657],[573,4131],{"className":4132,"style":653},[652],[573,4134,4136,4139,4142,4148],{"className":4135},[626],[573,4137],{"className":4138,"style":3133},[630],[573,4140,612],{"className":4141},[635],[573,4143,4145],{"className":4144},[635],[573,4146,2496],{"className":4147},[2495],[573,4149,4151],{"className":4150},[635],"875",[573,4153,4155,4184],{"className":4154},[576],[573,4156,4158],{"className":4157},[580],[582,4159,4160],{"xmlns":584},[586,4161,4162,4181],{},[589,4163,4164,4170,4172,4174,4176,4178],{},[1057,4165,4166,4168],{},[592,4167,594],{},[596,4169,1065],{"mathvariant":1063,"lspace":1064,"rspace":1064},[596,4171,599],{"stretchy":598},[610,4173,2019],{},[596,4175,605],{"stretchy":598},[596,4177,608],{},[610,4179,4180],{},"5,75",[614,4182,4183],{"encoding":616},"f'(1{,}5) = 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also",[573,4273,4275],{"className":4274},[1956],[573,4276,4278,4313],{"className":4277},[576],[573,4279,4281],{"className":4280},[580],[582,4282,4283],{"xmlns":584,"display":1965},[586,4284,4285,4310],{},[589,4286,4287,4293,4295,4297,4299,4305,4307],{},[689,4288,4289,4291],{},[592,4290,602],{},[610,4292,950],{},[596,4294,608],{},[610,4296,2019],{},[596,4298,1082],{},[2003,4300,4301,4303],{},[610,4302,4090],{},[610,4304,4180],{},[596,4306,2072],{},[610,4308,4309],{},"1,3478",[614,4311,4312],{"encoding":616},"x_1 = 1{,}5 - \\frac{0{,}875}{5{,}75} \\approx 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läuft von der Nullstelle weg.",[5025,5347,5348,1859,5376,5439],{},[573,5349,5351,5364],{"className":5350},[576],[573,5352,5354],{"className":5353},[580],[582,5355,5356],{"xmlns":584},[586,5357,5358,5362],{},[589,5359,5360],{},[592,5361,594],{},[614,5363,594],{"encoding":616},[573,5365,5367],{"className":5366,"ariaHidden":622},[621],[573,5368,5370,5373],{"className":5369},[626],[573,5371],{"className":5372,"style":1488},[630],[573,5374,594],{"className":5375,"style":637},[635,636],[573,5377,5379,5397],{"className":5378},[576],[573,5380,5382],{"className":5381},[580],[582,5383,5384],{"xmlns":584},[586,5385,5386,5394],{},[589,5387,5388],{},[1057,5389,5390,5392],{},[592,5391,594],{},[596,5393,1065],{"mathvariant":1063,"lspace":1064,"rspace":1064},[614,5395,5396],{"encoding":616},"f'",[573,5398,5400],{"className":5399,"ariaHidden":622},[621],[573,5401,5403,5407],{"className":5402},[626],[573,5404],{"className":5405,"style":5406},[630],"height:0.9463em;vertical-align:-0.1944em;",[573,5408,5410,5413],{"className":5409},[635],[573,5411,594],{"className":5412,"style":637},[635,636],[573,5414,5416],{"className":5415},[717],[573,5417,5419],{"className":5418},[721],[573,5420,5422],{"className":5421},[726],[573,5423,5425],{"className":5424,"style":1214},[730],[573,5426,5427,5430],{"style":1217},[573,5428],{"className":5429,"style":739},[738],[573,5431,5433],{"className":5432},[743,744,745,746],[573,5434,5436],{"className":5435},[635,746],[573,5437,1065],{"className":5438},[635,746]," im Bruch vertauschen.",[5025,5441,5442,5443,5583,5584,5612,5613,5683],{},"Einen Startwert wählen, an dem ",[573,5444,5446,5478],{"className":5445},[576],[573,5447,5449],{"className":5448},[580],[582,5450,5451],{"xmlns":584},[586,5452,5453,5475],{},[589,5454,5455,5461,5463,5469,5471,5473],{},[1057,5456,5457,5459],{},[592,5458,594],{},[596,5460,1065],{"mathvariant":1063,"lspace":1064,"rspace":1064},[596,5462,599],{"stretchy":598},[689,5464,5465,5467],{},[592,5466,602],{},[610,5468,612],{},[596,5470,605],{"stretchy":598},[596,5472,608],{},[610,5474,612],{},[614,5476,5477],{"encoding":616},"f'(x_0) = 0",[573,5479,5481,5574],{"className":5480,"ariaHidden":622},[621],[573,5482,5484,5487,5519,5522,5562,5565,5568,5571],{"className":5483},[626],[573,5485],{"className":5486,"style":1195},[630],[573,5488,5490,5493],{"className":5489},[635],[573,5491,594],{"className":5492,"style":637},[635,636],[573,5494,5496],{"className":5495},[717],[573,5497,5499],{"className":5498},[721],[573,5500,5502],{"className":5501},[726],[573,5503,5505],{"className":5504,"style":1214},[730],[573,5506,5507,5510],{"style":1217},[573,5508],{"className":5509,"style":739},[738],[573,5511,5513],{"className":5512},[743,744,745,746],[573,5514,5516],{"className":5515},[635,746],[573,5517,1065],{"className":5518},[635,746],[573,5520,599],{"className":5521},[641],[573,5523,5525,5528],{"className":5524},[635],[573,5526,602],{"className":5527},[635,636],[573,5529,5531],{"className":5530},[717],[573,5532,5534,5554],{"className":5533},[721,722],[573,5535,5537,5551],{"className":5536},[726],[573,5538,5540],{"className":5539,"style":731},[730],[573,5541,5542,5545],{"style":734},[573,5543],{"className":5544,"style":739},[738],[573,5546,5548],{"className":5547},[743,744,745,746],[573,5549,612],{"className":5550},[635,746],[573,5552,754],{"className":5553},[753],[573,5555,5557],{"className":5556},[726],[573,5558,5560],{"className":5559,"style":761},[730],[573,5561],{},[573,5563,605],{"className":5564},[648],[573,5566],{"className":5567,"style":653},[652],[573,5569,608],{"className":5570},[657],[573,5572],{"className":5573,"style":653},[652],[573,5575,5577,5580],{"className":5576},[626],[573,5578],{"className":5579,"style":667},[630],[573,5581,612],{"className":5582},[635]," oder fast ",[573,5585,5587,5600],{"className":5586},[576],[573,5588,5590],{"className":5589},[580],[582,5591,5592],{"xmlns":584},[586,5593,5594,5598],{},[589,5595,5596],{},[610,5597,612],{},[614,5599,612],{"encoding":616},[573,5601,5603],{"className":5602,"ariaHidden":622},[621],[573,5604,5606,5609],{"className":5605},[626],[573,5607],{"className":5608,"style":667},[630],[573,5610,612],{"className":5611},[635]," ist. Die Tangente ist dann (fast) waagerecht, ",[573,5614,5616,5634],{"className":5615},[576],[573,5617,5619],{"className":5618},[580],[582,5620,5621],{"xmlns":584},[586,5622,5623,5631],{},[589,5624,5625],{},[689,5626,5627,5629],{},[592,5628,602],{},[610,5630,950],{},[614,5632,5633],{"encoding":616},"x_1",[573,5635,5637],{"className":5636,"ariaHidden":622},[621],[573,5638,5640,5643],{"className":5639},[626],[573,5641],{"className":5642,"style":707},[630],[573,5644,5646,5649],{"className":5645},[635],[573,5647,602],{"className":5648},[635,636],[573,5650,5652],{"className":5651},[717],[573,5653,5655,5675],{"className":5654},[721,722],[573,5656,5658,5672],{"className":5657},[726],[573,5659,5661],{"className":5660,"style":731},[730],[573,5662,5663,5666],{"style":734},[573,5664],{"className":5665,"style":739},[738],[573,5667,5669],{"className":5668},[743,744,745,746],[573,5670,950],{"className":5671},[635,746],[573,5673,754],{"className":5674},[753],[573,5676,5678],{"className":5677},[726],[573,5679,5681],{"className":5680,"style":761},[730],[573,5682],{}," liegt weit entfernt.",[5025,5685,5686,5687,5794],{},"Aus wenigen Schritten auf Konvergenz schließen, ohne ",[573,5688,5690,5718],{"className":5689},[576],[573,5691,5693],{"className":5692},[580],[582,5694,5695],{"xmlns":584},[586,5696,5697,5715],{},[589,5698,5699,5701,5703,5709,5711,5713],{},[592,5700,594],{},[596,5702,599],{"stretchy":598},[689,5704,5705,5707],{},[592,5706,602],{},[592,5708,785],{},[596,5710,605],{"stretchy":598},[596,5712,2072],{},[610,5714,612],{},[614,5716,5717],{"encoding":616},"f(x_n) \\approx 0",[573,5719,5721,5785],{"className":5720,"ariaHidden":622},[621],[573,5722,5724,5727,5730,5733,5773,5776,5779,5782],{"className":5723},[626],[573,5725],{"className":5726,"style":631},[630],[573,5728,594],{"className":5729,"style":637},[635,636],[573,5731,599],{"className":5732},[641],[573,5734,5736,5739],{"className":5735},[635],[573,5737,602],{"className":5738},[635,636],[573,5740,5742],{"className":5741},[717],[573,5743,5745,5765],{"className":5744},[721,722],[573,5746,5748,5762],{"className":5747},[726],[573,5749,5751],{"className":5750,"style":836},[730],[573,5752,5753,5756],{"style":734},[573,5754],{"className":5755,"style":739},[738],[573,5757,5759],{"className":5758},[743,744,745,746],[573,5760,785],{"className":5761},[635,636,746],[573,5763,754],{"className":5764},[753],[573,5766,5768],{"className":5767},[726],[573,5769,5771],{"className":5770,"style":761},[730],[573,5772],{},[573,5774,605],{"className":5775},[648],[573,5777],{"className":5778,"style":653},[652],[573,5780,2072],{"className":5781},[657],[573,5783],{"className":5784,"style":653},[652],[573,5786,5788,5791],{"className":5787},[626],[573,5789],{"className":5790,"style":667},[630],[573,5792,612],{"className":5793},[635]," zu prüfen.",{"title":5796,"searchDepth":5797,"depth":5797,"links":5798},"",2,[5799,5800,5801],{"id":562,"depth":5797,"text":563},{"id":1533,"depth":5797,"text":1534},{"id":5019,"depth":5797,"text":5020},[5803,5806,5809,5811,5814,5817,5820,5823,5826,5829,5832,5834],{"lhs":5804,"rhs":5805},"x - (x^2 - a)\u002F(2*x)","(x + a\u002Fx)\u002F2",{"lhs":5807,"rhs":5808},"1 - (1^2 - 2)\u002F(2*1)","1.5",{"lhs":5810,"rhs":2852},"1.5 - (1.5^2 - 2)\u002F(2*1.5)",{"lhs":5812,"rhs":5813},"17\u002F12 - ((17\u002F12)^2 - 2)\u002F(2*17\u002F12)","577\u002F408",{"lhs":5815,"rhs":5816},"round(17\u002F12, 6)","1.416667",{"lhs":5818,"rhs":5819},"round(577\u002F408, 6)","1.414216",{"lhs":5821,"rhs":5822},"round(sqrt(2), 6)","1.414214",{"lhs":5824,"rhs":5825},"1.5^3 - 1.5 - 1","0.875",{"lhs":5827,"rhs":5828},"3*1.5^2 - 1","5.75",{"lhs":5830,"rhs":5831},"round(1.5 - 0.875\u002F5.75, 4)","1.3478",{"lhs":5833,"rhs":950},"0 - (0^3 - 2*0 + 2)\u002F(3*0^2 - 2)",{"lhs":5835,"rhs":612},"1 - (1^3 - 2*1 + 2)\u002F(3*1^2 - 2)","f \\text{ differenzierbar},\\ f'(x_n) \\neq 0","\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 style=\"white-space: nowrap\">\u003Cspan class=\"katex\">differenzierbar\u003C\u002Fspan>\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath 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