[{"data":1,"prerenderedAt":5682},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fbogenlaenge-eines-graphen":546,"formula-related-de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fbogenlaenge-eines-graphen":5669},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":555,"category":101,"checks":5559,"conditions":5584,"conditionsHtml":5585,"deprecated":5586,"description":5552,"extension":5587,"figure":5588,"figureCaption":5589,"figureHtml":5636,"formulaId":5637,"latex":5638,"latexHtml":5639,"level":5640,"meta":5641,"navigation":5597,"path":5642,"redirectFrom":5643,"related":5644,"seo":5649,"stem":5650,"subject":577,"summary":5651,"tags":5652,"variables":5657,"__hash__":5668},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Fcalculus\u002Fbogenlaenge-eines-graphen.md","Bogenlänge eines Graphen",[550,551,552,553,554],"Bogenlänge Integral","Länge einer Kurve","Kurvenlänge berechnen","Länge eines Funktionsgraphen","Bogenlänge Funktion",{"type":556,"value":557,"toc":5551},"minimark",[558,563,697,1055,1058,1062,1232,1634,1701,1705,2089,2507,2820,3412,4161,4950,5156,5160],[559,560,562],"h2",{"id":561},"erklärung","Erklärung",[564,565,566,567,642,643,696],"p",{},"Die Formel berechnet, wie lang der Graph einer Funktion zwischen ",[568,569,572,602],"span",{"className":570},[571],"katex",[568,573,576],{"className":574},[575],"katex-mathml",[577,578,580],"math",{"xmlns":579},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[581,582,583,597],"semantics",{},[584,585,586,590,594],"mrow",{},[587,588,589],"mi",{},"x",[591,592,593],"mo",{},"=",[587,595,596],{},"a",[598,599,601],"annotation",{"encoding":600},"application\u002Fx-tex","x = a",[568,603,607,633],{"className":604,"ariaHidden":606},[605],"katex-html","true",[568,608,611,616,621,626,630],{"className":609},[610],"base",[568,612],{"className":613,"style":615},[614],"strut","height:0.4306em;",[568,617,589],{"className":618},[619,620],"mord","mathnormal",[568,622],{"className":623,"style":625},[624],"mspace","margin-right:0.2778em;",[568,627,593],{"className":628},[629],"mrel",[568,631],{"className":632,"style":625},[624],[568,634,636,639],{"className":635},[610],[568,637],{"className":638,"style":615},[614],[568,640,596],{"className":641},[619,620]," und ",[568,644,646,665],{"className":645},[571],[568,647,649],{"className":648},[575],[577,650,651],{"xmlns":579},[581,652,653,662],{},[584,654,655,657,659],{},[587,656,589],{},[591,658,593],{},[587,660,661],{},"b",[598,663,664],{"encoding":600},"x = b",[568,666,668,686],{"className":667,"ariaHidden":606},[605],[568,669,671,674,677,680,683],{"className":670},[610],[568,672],{"className":673,"style":615},[614],[568,675,589],{"className":676},[619,620],[568,678],{"className":679,"style":625},[624],[568,681,593],{"className":682},[629],[568,684],{"className":685,"style":625},[624],[568,687,689,693],{"className":688},[610],[568,690],{"className":691,"style":692},[614],"height:0.6944em;",[568,694,661],{"className":695},[619,620]," ist, so als würde man einen Faden auf die Kurve legen und danach gerade ziehen.",[564,698,699,700,919,920,949,950,1003,1004,1054],{},"Der Integrand ",[568,701,703,750],{"className":702},[571],[568,704,706],{"className":705},[575],[577,707,708],{"xmlns":579},[581,709,710,747],{},[584,711,712],{},[713,714,715],"msqrt",{},[584,716,717,721,724,728,739],{},[718,719,720],"mn",{},"1",[591,722,723],{},"+",[591,725,727],{"stretchy":726},"false","(",[729,730,731,734],"msup",{},[587,732,733],{},"f",[591,735,738],{"mathvariant":736,"lspace":737,"rspace":737},"normal","0em","′",[729,740,741,744],{},[591,742,743],{"stretchy":726},")",[718,745,746],{},"2",[598,748,749],{"encoding":600},"\\sqrt{1 + (f')^2}",[568,751,753],{"className":752,"ariaHidden":606},[605],[568,754,756,760],{"className":755},[610],[568,757],{"className":758,"style":759},[614],"height:1.24em;vertical-align:-0.305em;",[568,761,764],{"className":762},[619,763],"sqrt",[568,765,769,910],{"className":766},[767,768],"vlist-t","vlist-t2",[568,770,773,905],{"className":771},[772],"vlist-r",[568,774,778,882],{"className":775,"style":777},[776],"vlist","height:0.935em;",[568,779,783,788],{"className":780,"style":782},[781],"svg-align","top:-3.2em;",[568,784],{"className":785,"style":787},[786],"pstrut","height:3.2em;",[568,789,792,795,799,803,806,810,851],{"className":790,"style":791},[619],"padding-left:1em;",[568,793,720],{"className":794},[619],[568,796],{"className":797,"style":798},[624],"margin-right:0.2222em;",[568,800,723],{"className":801},[802],"mbin",[568,804],{"className":805,"style":798},[624],[568,807,727],{"className":808},[809],"mopen",[568,811,813,817],{"className":812},[619],[568,814,733],{"className":815,"style":816},[619,620],"margin-right:0.1076em;",[568,818,821],{"className":819},[820],"msupsub",[568,822,824],{"className":823},[767],[568,825,827],{"className":826},[772],[568,828,831],{"className":829,"style":830},[776],"height:0.6779em;",[568,832,834,838],{"style":833},"top:-2.989em;margin-right:0.05em;",[568,835],{"className":836,"style":837},[786],"height:2.7em;",[568,839,845],{"className":840},[841,842,843,844],"sizing","reset-size6","size3","mtight",[568,846,848],{"className":847},[619,844],[568,849,738],{"className":850},[619,844],[568,852,855,858],{"className":853},[854],"mclose",[568,856,743],{"className":857},[854],[568,859,861],{"className":860},[820],[568,862,864],{"className":863},[767],[568,865,867],{"className":866},[772],[568,868,871],{"className":869,"style":870},[776],"height:0.7401em;",[568,872,873,876],{"style":833},[568,874],{"className":875,"style":837},[786],[568,877,879],{"className":878},[841,842,843,844],[568,880,746],{"className":881},[619,844],[568,883,885,888],{"style":884},"top:-2.895em;",[568,886],{"className":887,"style":787},[786],[568,889,893],{"className":890,"style":892},[891],"hide-tail","min-width:1.02em;height:1.28em;",[894,895,901],"svg",{"xmlns":896,"width":897,"height":898,"viewBox":899,"preserveAspectRatio":900},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.28em","0 0 400000 1296","xMinYMin slice",[902,903],"path",{"d":904},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[568,906,909],{"className":907},[908],"vlist-s","​",[568,911,913],{"className":912},[772],[568,914,917],{"className":915,"style":916},[776],"height:0.305em;",[568,918],{}," ist immer mindestens ",[568,921,923,936],{"className":922},[571],[568,924,926],{"className":925},[575],[577,927,928],{"xmlns":579},[581,929,930,934],{},[584,931,932],{},[718,933,720],{},[598,935,720],{"encoding":600},[568,937,939],{"className":938,"ariaHidden":606},[605],[568,940,942,946],{"className":941},[610],[568,943],{"className":944,"style":945},[614],"height:0.6444em;",[568,947,720],{"className":948},[619],". Deshalb ist die Bogenlänge nie kleiner als ",[568,951,953,972],{"className":952},[571],[568,954,956],{"className":955},[575],[577,957,958],{"xmlns":579},[581,959,960,969],{},[584,961,962,964,967],{},[587,963,661],{},[591,965,966],{},"−",[587,968,596],{},[598,970,971],{"encoding":600},"b - a",[568,973,975,994],{"className":974,"ariaHidden":606},[605],[568,976,978,982,985,988,991],{"className":977},[610],[568,979],{"className":980,"style":981},[614],"height:0.7778em;vertical-align:-0.0833em;",[568,983,661],{"className":984},[619,620],[568,986],{"className":987,"style":798},[624],[568,989,966],{"className":990},[802],[568,992],{"className":993,"style":798},[624],[568,995,997,1000],{"className":996},[610],[568,998],{"className":999,"style":615},[614],[568,1001,596],{"className":1002},[619,620],", und gleich ",[568,1005,1007,1024],{"className":1006},[571],[568,1008,1010],{"className":1009},[575],[577,1011,1012],{"xmlns":579},[581,1013,1014,1022],{},[584,1015,1016,1018,1020],{},[587,1017,661],{},[591,1019,966],{},[587,1021,596],{},[598,1023,971],{"encoding":600},[568,1025,1027,1045],{"className":1026,"ariaHidden":606},[605],[568,1028,1030,1033,1036,1039,1042],{"className":1029},[610],[568,1031],{"className":1032,"style":981},[614],[568,1034,661],{"className":1035},[619,620],[568,1037],{"className":1038,"style":798},[624],[568,1040,966],{"className":1041},[802],[568,1043],{"className":1044,"style":798},[624],[568,1046,1048,1051],{"className":1047},[610],[568,1049],{"className":1050,"style":615},[614],[568,1052,596],{"className":1053},[619,620]," nur bei einer waagrechten Geraden.",[564,1056,1057],{},"Nur für wenige Funktionen lässt sich das Integral geschlossen berechnen, weil die Wurzel meist keine elementare Stammfunktion hat. Sonst rechnet man numerisch.",[559,1059,1061],{"id":1060},"herleitung","Herleitung",[564,1063,1064,1065,1101,1102,1231],{},"Man nähert den Graphen durch kurze Strecken an. Ein Stück mit der Breite ",[568,1066,1068,1085],{"className":1067},[571],[568,1069,1071],{"className":1070},[575],[577,1072,1073],{"xmlns":579},[581,1074,1075,1082],{},[584,1076,1077,1080],{},[587,1078,1079],{"mathvariant":736},"Δ",[587,1081,589],{},[598,1083,1084],{"encoding":600},"\\Delta x",[568,1086,1088],{"className":1087,"ariaHidden":606},[605],[568,1089,1091,1095,1098],{"className":1090},[610],[568,1092],{"className":1093,"style":1094},[614],"height:0.6833em;",[568,1096,1079],{"className":1097},[619],[568,1099,589],{"className":1100},[619,620]," steigt um ",[568,1103,1105,1145],{"className":1104},[571],[568,1106,1108],{"className":1107},[575],[577,1109,1110],{"xmlns":579},[581,1111,1112,1142],{},[584,1113,1114,1116,1119,1122,1128,1130,1132,1134,1138,1140],{},[587,1115,1079],{"mathvariant":736},[587,1117,1118],{},"y",[591,1120,1121],{},"≈",[729,1123,1124,1126],{},[587,1125,733],{},[591,1127,738],{"mathvariant":736,"lspace":737,"rspace":737},[591,1129,727],{"stretchy":726},[587,1131,589],{},[591,1133,743],{"stretchy":726},[1135,1136,1137],"mtext",{}," ",[587,1139,1079],{"mathvariant":736},[587,1141,589],{},[598,1143,1144],{"encoding":600},"\\Delta y \\approx f'(x)\\,\\Delta x",[568,1146,1148,1171],{"className":1147,"ariaHidden":606},[605],[568,1149,1151,1155,1158,1162,1165,1168],{"className":1150},[610],[568,1152],{"className":1153,"style":1154},[614],"height:0.8778em;vertical-align:-0.1944em;",[568,1156,1079],{"className":1157},[619],[568,1159,1118],{"className":1160,"style":1161},[619,620],"margin-right:0.0359em;",[568,1163],{"className":1164,"style":625},[624],[568,1166,1121],{"className":1167},[629],[568,1169],{"className":1170,"style":625},[624],[568,1172,1174,1178,1212,1215,1218,1221,1225,1228],{"className":1173},[610],[568,1175],{"className":1176,"style":1177},[614],"height:1.0019em;vertical-align:-0.25em;",[568,1179,1181,1184],{"className":1180},[619],[568,1182,733],{"className":1183,"style":816},[619,620],[568,1185,1187],{"className":1186},[820],[568,1188,1190],{"className":1189},[767],[568,1191,1193],{"className":1192},[772],[568,1194,1197],{"className":1195,"style":1196},[776],"height:0.7519em;",[568,1198,1200,1203],{"style":1199},"top:-3.063em;margin-right:0.05em;",[568,1201],{"className":1202,"style":837},[786],[568,1204,1206],{"className":1205},[841,842,843,844],[568,1207,1209],{"className":1208},[619,844],[568,1210,738],{"className":1211},[619,844],[568,1213,727],{"className":1214},[809],[568,1216,589],{"className":1217},[619,620],[568,1219,743],{"className":1220},[854],[568,1222],{"className":1223,"style":1224},[624],"margin-right:0.1667em;",[568,1226,1079],{"className":1227},[619],[568,1229,589],{"className":1230},[619,620],". Nach dem Satz des Pythagoras ist es",[568,1233,1236],{"className":1234},[1235],"katex-display",[568,1237,1239,1319],{"className":1238},[571],[568,1240,1242],{"className":1241},[575],[577,1243,1245],{"xmlns":579,"display":1244},"block",[581,1246,1247,1316],{},[584,1248,1249,1279,1281,1310,1312,1314],{},[713,1250,1251],{},[584,1252,1253,1255,1257,1259,1265,1267,1269,1271,1273],{},[591,1254,727],{"stretchy":726},[587,1256,1079],{"mathvariant":736},[587,1258,589],{},[729,1260,1261,1263],{},[591,1262,743],{"stretchy":726},[718,1264,746],{},[591,1266,723],{},[591,1268,727],{"stretchy":726},[587,1270,1079],{"mathvariant":736},[587,1272,1118],{},[729,1274,1275,1277],{},[591,1276,743],{"stretchy":726},[718,1278,746],{},[591,1280,1121],{},[713,1282,1283],{},[584,1284,1285,1287,1289,1292,1298,1300,1302,1304],{},[718,1286,720],{},[591,1288,723],{},[591,1290,727],{"fence":726,"stretchy":606,"minsize":1291,"maxsize":1291},"1.2em",[729,1293,1294,1296],{},[587,1295,733],{},[591,1297,738],{"mathvariant":736,"lspace":737,"rspace":737},[591,1299,727],{"stretchy":726},[587,1301,589],{},[591,1303,743],{"stretchy":726},[729,1305,1306,1308],{},[591,1307,743],{"fence":726,"stretchy":606,"minsize":1291,"maxsize":1291},[718,1309,746],{},[1135,1311,1137],{},[587,1313,1079],{"mathvariant":736},[587,1315,589],{},[598,1317,1318],{"encoding":600},"\\sqrt{(\\Delta x)^2 + (\\Delta y)^2} \\approx \\sqrt{1 + \\big(f'(x)\\big)^2}\\,\\Delta x",[568,1320,1322,1470],{"className":1321,"ariaHidden":606},[605],[568,1323,1325,1329,1461,1464,1467],{"className":1324},[610],[568,1326],{"className":1327,"style":1328},[614],"height:1.24em;vertical-align:-0.2561em;",[568,1330,1332],{"className":1331},[619,763],[568,1333,1335,1452],{"className":1334},[767,768],[568,1336,1338,1449],{"className":1337},[772],[568,1339,1342,1436],{"className":1340,"style":1341},[776],"height:0.9839em;",[568,1343,1345,1348],{"className":1344,"style":782},[781],[568,1346],{"className":1347,"style":787},[786],[568,1349,1351,1354,1357,1360,1389,1392,1395,1398,1401,1404,1407],{"className":1350,"style":791},[619],[568,1352,727],{"className":1353},[809],[568,1355,1079],{"className":1356},[619],[568,1358,589],{"className":1359},[619,620],[568,1361,1363,1366],{"className":1362},[854],[568,1364,743],{"className":1365},[854],[568,1367,1369],{"className":1368},[820],[568,1370,1372],{"className":1371},[767],[568,1373,1375],{"className":1374},[772],[568,1376,1378],{"className":1377,"style":870},[776],[568,1379,1380,1383],{"style":833},[568,1381],{"className":1382,"style":837},[786],[568,1384,1386],{"className":1385},[841,842,843,844],[568,1387,746],{"className":1388},[619,844],[568,1390],{"className":1391,"style":798},[624],[568,1393,723],{"className":1394},[802],[568,1396],{"className":1397,"style":798},[624],[568,1399,727],{"className":1400},[809],[568,1402,1079],{"className":1403},[619],[568,1405,1118],{"className":1406,"style":1161},[619,620],[568,1408,1410,1413],{"className":1409},[854],[568,1411,743],{"className":1412},[854],[568,1414,1416],{"className":1415},[820],[568,1417,1419],{"className":1418},[767],[568,1420,1422],{"className":1421},[772],[568,1423,1425],{"className":1424,"style":870},[776],[568,1426,1427,1430],{"style":833},[568,1428],{"className":1429,"style":837},[786],[568,1431,1433],{"className":1432},[841,842,843,844],[568,1434,746],{"className":1435},[619,844],[568,1437,1439,1442],{"style":1438},"top:-2.9439em;",[568,1440],{"className":1441,"style":787},[786],[568,1443,1445],{"className":1444,"style":892},[891],[894,1446,1447],{"xmlns":896,"width":897,"height":898,"viewBox":899,"preserveAspectRatio":900},[902,1448],{"d":904},[568,1450,909],{"className":1451},[908],[568,1453,1455],{"className":1454},[772],[568,1456,1459],{"className":1457,"style":1458},[776],"height:0.2561em;",[568,1460],{},[568,1462],{"className":1463,"style":625},[624],[568,1465,1121],{"className":1466},[629],[568,1468],{"className":1469,"style":625},[624],[568,1471,1473,1477,1625,1628,1631],{"className":1472},[610],[568,1474],{"className":1475,"style":1476},[614],"height:1.84em;vertical-align:-0.4541em;",[568,1478,1480],{"className":1479},[619,763],[568,1481,1483,1616],{"className":1482},[767,768],[568,1484,1486,1613],{"className":1485},[772],[568,1487,1490,1596],{"className":1488,"style":1489},[776],"height:1.3859em;",[568,1491,1494,1498],{"className":1492,"style":1493},[781],"top:-3.8em;",[568,1495],{"className":1496,"style":1497},[786],"height:3.8em;",[568,1499,1501,1504,1507,1510,1513,1521,1553,1556,1559,1562],{"className":1500,"style":791},[619],[568,1502,720],{"className":1503},[619],[568,1505],{"className":1506,"style":798},[624],[568,1508,723],{"className":1509},[802],[568,1511],{"className":1512,"style":798},[624],[568,1514,1516],{"className":1515},[619],[568,1517,727],{"className":1518},[1519,1520],"delimsizing","size1",[568,1522,1524,1527],{"className":1523},[619],[568,1525,733],{"className":1526,"style":816},[619,620],[568,1528,1530],{"className":1529},[820],[568,1531,1533],{"className":1532},[767],[568,1534,1536],{"className":1535},[772],[568,1537,1539],{"className":1538,"style":830},[776],[568,1540,1541,1544],{"style":833},[568,1542],{"className":1543,"style":837},[786],[568,1545,1547],{"className":1546},[841,842,843,844],[568,1548,1550],{"className":1549},[619,844],[568,1551,738],{"className":1552},[619,844],[568,1554,727],{"className":1555},[809],[568,1557,589],{"className":1558},[619,620],[568,1560,743],{"className":1561},[854],[568,1563,1565,1571],{"className":1564},[619],[568,1566,1568],{"className":1567},[619],[568,1569,743],{"className":1570},[1519,1520],[568,1572,1574],{"className":1573},[820],[568,1575,1577],{"className":1576},[767],[568,1578,1580],{"className":1579},[772],[568,1581,1584],{"className":1582,"style":1583},[776],"height:1.054em;",[568,1585,1587,1590],{"style":1586},"top:-3.3029em;margin-right:0.05em;",[568,1588],{"className":1589,"style":837},[786],[568,1591,1593],{"className":1592},[841,842,843,844],[568,1594,746],{"className":1595},[619,844],[568,1597,1599,1602],{"style":1598},"top:-3.3459em;",[568,1600],{"className":1601,"style":1497},[786],[568,1603,1606],{"className":1604,"style":1605},[891],"min-width:1.02em;height:1.88em;",[894,1607,1610],{"xmlns":896,"width":897,"height":1608,"viewBox":1609,"preserveAspectRatio":900},"1.88em","0 0 400000 1944",[902,1611],{"d":1612},"M983 90\nl0 -0\nc4,-6.7,10,-10,18,-10 H400000v40\nH1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7\ns-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744\nc-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30\nc26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722\nc56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5\nc53.7,-170.3,84.5,-266.8,92.5,-289.5z\nM1001 80h400000v40h-400000z",[568,1614,909],{"className":1615},[908],[568,1617,1619],{"className":1618},[772],[568,1620,1623],{"className":1621,"style":1622},[776],"height:0.4541em;",[568,1624],{},[568,1626],{"className":1627,"style":1224},[624],[568,1629,1079],{"className":1630},[619],[568,1632,589],{"className":1633},[619,620],[564,1635,1636,1637,1670,1671,1700],{},"lang. Summiert man alle Stücke und lässt ",[568,1638,1640,1655],{"className":1639},[571],[568,1641,1643],{"className":1642},[575],[577,1644,1645],{"xmlns":579},[581,1646,1647,1653],{},[584,1648,1649,1651],{},[587,1650,1079],{"mathvariant":736},[587,1652,589],{},[598,1654,1084],{"encoding":600},[568,1656,1658],{"className":1657,"ariaHidden":606},[605],[568,1659,1661,1664,1667],{"className":1660},[610],[568,1662],{"className":1663,"style":1094},[614],[568,1665,1079],{"className":1666},[619],[568,1668,589],{"className":1669},[619,620]," gegen ",[568,1672,1674,1688],{"className":1673},[571],[568,1675,1677],{"className":1676},[575],[577,1678,1679],{"xmlns":579},[581,1680,1681,1686],{},[584,1682,1683],{},[718,1684,1685],{},"0",[598,1687,1685],{"encoding":600},[568,1689,1691],{"className":1690,"ariaHidden":606},[605],[568,1692,1694,1697],{"className":1693},[610],[568,1695],{"className":1696,"style":945},[614],[568,1698,1685],{"className":1699},[619]," gehen, entsteht das Integral.",[559,1702,1704],{"id":1703},"beispiele","Beispiele",[564,1706,1707,1708,1862,1863,642,1891,1919,1920,2088],{},"Für die Gerade ",[568,1709,1711,1744],{"className":1710},[571],[568,1712,1714],{"className":1713},[575],[577,1715,1716],{"xmlns":579},[581,1717,1718,1741],{},[584,1719,1720,1722,1724,1726,1728,1730,1739],{},[587,1721,733],{},[591,1723,727],{"stretchy":726},[587,1725,589],{},[591,1727,743],{"stretchy":726},[591,1729,593],{},[1731,1732,1733,1736],"mfrac",{},[718,1734,1735],{},"3",[718,1737,1738],{},"4",[587,1740,589],{},[598,1742,1743],{"encoding":600},"f(x) = \\frac{3}{4}x",[568,1745,1747,1775],{"className":1746,"ariaHidden":606},[605],[568,1748,1750,1754,1757,1760,1763,1766,1769,1772],{"className":1749},[610],[568,1751],{"className":1752,"style":1753},[614],"height:1em;vertical-align:-0.25em;",[568,1755,733],{"className":1756,"style":816},[619,620],[568,1758,727],{"className":1759},[809],[568,1761,589],{"className":1762},[619,620],[568,1764,743],{"className":1765},[854],[568,1767],{"className":1768,"style":625},[624],[568,1770,593],{"className":1771},[629],[568,1773],{"className":1774,"style":625},[624],[568,1776,1778,1782,1859],{"className":1777},[610],[568,1779],{"className":1780,"style":1781},[614],"height:1.1901em;vertical-align:-0.345em;",[568,1783,1785,1789,1856],{"className":1784},[619],[568,1786],{"className":1787},[809,1788],"nulldelimiter",[568,1790,1792],{"className":1791},[1731],[568,1793,1795,1847],{"className":1794},[767,768],[568,1796,1798,1844],{"className":1797},[772],[568,1799,1802,1818,1829],{"className":1800,"style":1801},[776],"height:0.8451em;",[568,1803,1805,1809],{"style":1804},"top:-2.655em;",[568,1806],{"className":1807,"style":1808},[786],"height:3em;",[568,1810,1812],{"className":1811},[841,842,843,844],[568,1813,1815],{"className":1814},[619,844],[568,1816,1738],{"className":1817},[619,844],[568,1819,1821,1824],{"style":1820},"top:-3.23em;",[568,1822],{"className":1823,"style":1808},[786],[568,1825],{"className":1826,"style":1828},[1827],"frac-line","border-bottom-width:0.04em;",[568,1830,1832,1835],{"style":1831},"top:-3.394em;",[568,1833],{"className":1834,"style":1808},[786],[568,1836,1838],{"className":1837},[841,842,843,844],[568,1839,1841],{"className":1840},[619,844],[568,1842,1735],{"className":1843},[619,844],[568,1845,909],{"className":1846},[908],[568,1848,1850],{"className":1849},[772],[568,1851,1854],{"className":1852,"style":1853},[776],"height:0.345em;",[568,1855],{},[568,1857],{"className":1858},[854,1788],[568,1860,589],{"className":1861},[619,620]," zwischen ",[568,1864,1866,1879],{"className":1865},[571],[568,1867,1869],{"className":1868},[575],[577,1870,1871],{"xmlns":579},[581,1872,1873,1877],{},[584,1874,1875],{},[718,1876,1685],{},[598,1878,1685],{"encoding":600},[568,1880,1882],{"className":1881,"ariaHidden":606},[605],[568,1883,1885,1888],{"className":1884},[610],[568,1886],{"className":1887,"style":945},[614],[568,1889,1685],{"className":1890},[619],[568,1892,1894,1907],{"className":1893},[571],[568,1895,1897],{"className":1896},[575],[577,1898,1899],{"xmlns":579},[581,1900,1901,1905],{},[584,1902,1903],{},[718,1904,1738],{},[598,1906,1738],{"encoding":600},[568,1908,1910],{"className":1909,"ariaHidden":606},[605],[568,1911,1913,1916],{"className":1912},[610],[568,1914],{"className":1915,"style":945},[614],[568,1917,1738],{"className":1918},[619]," ist ",[568,1921,1923,1955],{"className":1922},[571],[568,1924,1926],{"className":1925},[575],[577,1927,1928],{"xmlns":579},[581,1929,1930,1952],{},[584,1931,1932,1938,1940,1942,1944,1946],{},[729,1933,1934,1936],{},[587,1935,733],{},[591,1937,738],{"mathvariant":736,"lspace":737,"rspace":737},[591,1939,727],{"stretchy":726},[587,1941,589],{},[591,1943,743],{"stretchy":726},[591,1945,593],{},[1731,1947,1948,1950],{},[718,1949,1735],{},[718,1951,1738],{},[598,1953,1954],{"encoding":600},"f'(x) = \\frac{3}{4}",[568,1956,1958,2014],{"className":1957,"ariaHidden":606},[605],[568,1959,1961,1964,1996,1999,2002,2005,2008,2011],{"className":1960},[610],[568,1962],{"className":1963,"style":1177},[614],[568,1965,1967,1970],{"className":1966},[619],[568,1968,733],{"className":1969,"style":816},[619,620],[568,1971,1973],{"className":1972},[820],[568,1974,1976],{"className":1975},[767],[568,1977,1979],{"className":1978},[772],[568,1980,1982],{"className":1981,"style":1196},[776],[568,1983,1984,1987],{"style":1199},[568,1985],{"className":1986,"style":837},[786],[568,1988,1990],{"className":1989},[841,842,843,844],[568,1991,1993],{"className":1992},[619,844],[568,1994,738],{"className":1995},[619,844],[568,1997,727],{"className":1998},[809],[568,2000,589],{"className":2001},[619,620],[568,2003,743],{"className":2004},[854],[568,2006],{"className":2007,"style":625},[624],[568,2009,593],{"className":2010},[629],[568,2012],{"className":2013,"style":625},[624],[568,2015,2017,2020],{"className":2016},[610],[568,2018],{"className":2019,"style":1781},[614],[568,2021,2023,2026,2085],{"className":2022},[619],[568,2024],{"className":2025},[809,1788],[568,2027,2029],{"className":2028},[1731],[568,2030,2032,2077],{"className":2031},[767,768],[568,2033,2035,2074],{"className":2034},[772],[568,2036,2038,2052,2060],{"className":2037,"style":1801},[776],[568,2039,2040,2043],{"style":1804},[568,2041],{"className":2042,"style":1808},[786],[568,2044,2046],{"className":2045},[841,842,843,844],[568,2047,2049],{"className":2048},[619,844],[568,2050,1738],{"className":2051},[619,844],[568,2053,2054,2057],{"style":1820},[568,2055],{"className":2056,"style":1808},[786],[568,2058],{"className":2059,"style":1828},[1827],[568,2061,2062,2065],{"style":1831},[568,2063],{"className":2064,"style":1808},[786],[568,2066,2068],{"className":2067},[841,842,843,844],[568,2069,2071],{"className":2070},[619,844],[568,2072,1735],{"className":2073},[619,844],[568,2075,909],{"className":2076},[908],[568,2078,2080],{"className":2079},[772],[568,2081,2083],{"className":2082,"style":1853},[776],[568,2084],{},[568,2086],{"className":2087},[854,1788],":",[568,2090,2092],{"className":2091},[1235],[568,2093,2095,2163],{"className":2094},[571],[568,2096,2098],{"className":2097},[575],[577,2099,2100],{"xmlns":579,"display":1244},[581,2101,2102,2160],{},[584,2103,2104,2107,2109,2119,2135,2137,2140,2142,2144,2151,2154,2156,2158],{},[587,2105,2106],{},"L",[591,2108,593],{},[2110,2111,2112,2115,2117],"msubsup",{},[591,2113,2114],{},"∫",[718,2116,1685],{},[718,2118,1738],{},[713,2120,2121],{},[584,2122,2123,2125,2127],{},[718,2124,720],{},[591,2126,723],{},[1731,2128,2129,2132],{},[718,2130,2131],{},"9",[718,2133,2134],{},"16",[1135,2136,1137],{},[587,2138,2139],{},"d",[587,2141,589],{},[591,2143,593],{},[1731,2145,2146,2149],{},[718,2147,2148],{},"5",[718,2150,1738],{},[591,2152,2153],{},"⋅",[718,2155,1738],{},[591,2157,593],{},[718,2159,2148],{},[598,2161,2162],{"encoding":600},"L = \\int_0^4 \\sqrt{1 + \\frac{9}{16}}\\,dx = \\frac{5}{4} \\cdot 4 = 5",[568,2164,2166,2184,2402,2480,2498],{"className":2165,"ariaHidden":606},[605],[568,2167,2169,2172,2175,2178,2181],{"className":2168},[610],[568,2170],{"className":2171,"style":1094},[614],[568,2173,2106],{"className":2174},[619,620],[568,2176],{"className":2177,"style":625},[624],[568,2179,593],{"className":2180},[629],[568,2182],{"className":2183,"style":625},[624],[568,2185,2187,2191,2250,2253,2384,2387,2390,2393,2396,2399],{"className":2186},[610],[568,2188],{"className":2189,"style":2190},[614],"height:2.5635em;vertical-align:-0.9119em;",[568,2192,2195,2201],{"className":2193},[2194],"mop",[568,2196,2114],{"className":2197,"style":2200},[2194,2198,2199],"op-symbol","large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[568,2202,2204],{"className":2203},[820],[568,2205,2207,2241],{"className":2206},[767,768],[568,2208,2210,2238],{"className":2209},[772],[568,2211,2214,2226],{"className":2212,"style":2213},[776],"height:1.564em;",[568,2215,2217,2220],{"style":2216},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[568,2218],{"className":2219,"style":837},[786],[568,2221,2223],{"className":2222},[841,842,843,844],[568,2224,1685],{"className":2225},[619,844],[568,2227,2229,2232],{"style":2228},"top:-3.8129em;margin-right:0.05em;",[568,2230],{"className":2231,"style":837},[786],[568,2233,2235],{"className":2234},[841,842,843,844],[568,2236,1738],{"className":2237},[619,844],[568,2239,909],{"className":2240},[908],[568,2242,2244],{"className":2243},[772],[568,2245,2248],{"className":2246,"style":2247},[776],"height:0.9119em;",[568,2249],{},[568,2251],{"className":2252,"style":1224},[624],[568,2254,2256],{"className":2255},[619,763],[568,2257,2259,2375],{"className":2258},[767,768],[568,2260,2262,2372],{"className":2261},[772],[568,2263,2266,2355],{"className":2264,"style":2265},[776],"height:1.6516em;",[568,2267,2270,2274],{"className":2268,"style":2269},[781],"top:-4.4em;",[568,2271],{"className":2272,"style":2273},[786],"height:4.4em;",[568,2275,2277,2280,2283,2286,2289],{"className":2276,"style":791},[619],[568,2278,720],{"className":2279},[619],[568,2281],{"className":2282,"style":798},[624],[568,2284,723],{"className":2285},[802],[568,2287],{"className":2288,"style":798},[624],[568,2290,2292,2295,2352],{"className":2291},[619],[568,2293],{"className":2294},[809,1788],[568,2296,2298],{"className":2297},[1731],[568,2299,2301,2343],{"className":2300},[767,768],[568,2302,2304,2340],{"className":2303},[772],[568,2305,2308,2320,2328],{"className":2306,"style":2307},[776],"height:1.3214em;",[568,2309,2311,2314],{"style":2310},"top:-2.314em;",[568,2312],{"className":2313,"style":1808},[786],[568,2315,2317],{"className":2316},[619],[568,2318,2134],{"className":2319},[619],[568,2321,2322,2325],{"style":1820},[568,2323],{"className":2324,"style":1808},[786],[568,2326],{"className":2327,"style":1828},[1827],[568,2329,2331,2334],{"style":2330},"top:-3.677em;",[568,2332],{"className":2333,"style":1808},[786],[568,2335,2337],{"className":2336},[619],[568,2338,2131],{"className":2339},[619],[568,2341,909],{"className":2342},[908],[568,2344,2346],{"className":2345},[772],[568,2347,2350],{"className":2348,"style":2349},[776],"height:0.686em;",[568,2351],{},[568,2353],{"className":2354},[854,1788],[568,2356,2358,2361],{"style":2357},"top:-3.6116em;",[568,2359],{"className":2360,"style":2273},[786],[568,2362,2365],{"className":2363,"style":2364},[891],"min-width:1.02em;height:2.48em;",[894,2366,2369],{"xmlns":896,"width":897,"height":2367,"viewBox":2368,"preserveAspectRatio":900},"2.48em","0 0 400000 2592",[902,2370],{"d":2371},"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z",[568,2373,909],{"className":2374},[908],[568,2376,2378],{"className":2377},[772],[568,2379,2382],{"className":2380,"style":2381},[776],"height:0.7884em;",[568,2383],{},[568,2385],{"className":2386,"style":1224},[624],[568,2388,2139],{"className":2389},[619,620],[568,2391,589],{"className":2392},[619,620],[568,2394],{"className":2395,"style":625},[624],[568,2397,593],{"className":2398},[629],[568,2400],{"className":2401,"style":625},[624],[568,2403,2405,2409,2471,2474,2477],{"className":2404},[610],[568,2406],{"className":2407,"style":2408},[614],"height:2.0074em;vertical-align:-0.686em;",[568,2410,2412,2415,2468],{"className":2411},[619],[568,2413],{"className":2414},[809,1788],[568,2416,2418],{"className":2417},[1731],[568,2419,2421,2460],{"className":2420},[767,768],[568,2422,2424,2457],{"className":2423},[772],[568,2425,2427,2438,2446],{"className":2426,"style":2307},[776],[568,2428,2429,2432],{"style":2310},[568,2430],{"className":2431,"style":1808},[786],[568,2433,2435],{"className":2434},[619],[568,2436,1738],{"className":2437},[619],[568,2439,2440,2443],{"style":1820},[568,2441],{"className":2442,"style":1808},[786],[568,2444],{"className":2445,"style":1828},[1827],[568,2447,2448,2451],{"style":2330},[568,2449],{"className":2450,"style":1808},[786],[568,2452,2454],{"className":2453},[619],[568,2455,2148],{"className":2456},[619],[568,2458,909],{"className":2459},[908],[568,2461,2463],{"className":2462},[772],[568,2464,2466],{"className":2465,"style":2349},[776],[568,2467],{},[568,2469],{"className":2470},[854,1788],[568,2472],{"className":2473,"style":798},[624],[568,2475,2153],{"className":2476},[802],[568,2478],{"className":2479,"style":798},[624],[568,2481,2483,2486,2489,2492,2495],{"className":2482},[610],[568,2484],{"className":2485,"style":945},[614],[568,2487,1738],{"className":2488},[619],[568,2490],{"className":2491,"style":625},[624],[568,2493,593],{"className":2494},[629],[568,2496],{"className":2497,"style":625},[624],[568,2499,2501,2504],{"className":2500},[610],[568,2502],{"className":2503,"style":945},[614],[568,2505,2148],{"className":2506},[619],[564,2508,2509,2510,2572,2573,1919,2634,2819],{},"Das passt zum Satz des Pythagoras: Die Strecke von ",[568,2511,2513,2536],{"className":2512},[571],[568,2514,2516],{"className":2515},[575],[577,2517,2518],{"xmlns":579},[581,2519,2520,2533],{},[584,2521,2522,2524,2526,2529,2531],{},[591,2523,727],{"stretchy":726},[718,2525,1685],{},[591,2527,2528],{},"∣",[718,2530,1685],{},[591,2532,743],{"stretchy":726},[598,2534,2535],{"encoding":600},"(0 \\mid 0)",[568,2537,2539,2560],{"className":2538,"ariaHidden":606},[605],[568,2540,2542,2545,2548,2551,2554,2557],{"className":2541},[610],[568,2543],{"className":2544,"style":1753},[614],[568,2546,727],{"className":2547},[809],[568,2549,1685],{"className":2550},[619],[568,2552],{"className":2553,"style":625},[624],[568,2555,2528],{"className":2556},[629],[568,2558],{"className":2559,"style":625},[624],[568,2561,2563,2566,2569],{"className":2562},[610],[568,2564],{"className":2565,"style":1753},[614],[568,2567,1685],{"className":2568},[619],[568,2570,743],{"className":2571},[854]," nach ",[568,2574,2576,2598],{"className":2575},[571],[568,2577,2579],{"className":2578},[575],[577,2580,2581],{"xmlns":579},[581,2582,2583,2595],{},[584,2584,2585,2587,2589,2591,2593],{},[591,2586,727],{"stretchy":726},[718,2588,1738],{},[591,2590,2528],{},[718,2592,1735],{},[591,2594,743],{"stretchy":726},[598,2596,2597],{"encoding":600},"(4 \\mid 3)",[568,2599,2601,2622],{"className":2600,"ariaHidden":606},[605],[568,2602,2604,2607,2610,2613,2616,2619],{"className":2603},[610],[568,2605],{"className":2606,"style":1753},[614],[568,2608,727],{"className":2609},[809],[568,2611,1738],{"className":2612},[619],[568,2614],{"className":2615,"style":625},[624],[568,2617,2528],{"className":2618},[629],[568,2620],{"className":2621,"style":625},[624],[568,2623,2625,2628,2631],{"className":2624},[610],[568,2626],{"className":2627,"style":1753},[614],[568,2629,1735],{"className":2630},[619],[568,2632,743],{"className":2633},[854],[568,2635,2637,2671],{"className":2636},[571],[568,2638,2640],{"className":2639},[575],[577,2641,2642],{"xmlns":579},[581,2643,2644,2668],{},[584,2645,2646,2664,2666],{},[713,2647,2648],{},[584,2649,2650,2656,2658],{},[729,2651,2652,2654],{},[718,2653,1738],{},[718,2655,746],{},[591,2657,723],{},[729,2659,2660,2662],{},[718,2661,1735],{},[718,2663,746],{},[591,2665,593],{},[718,2667,2148],{},[598,2669,2670],{"encoding":600},"\\sqrt{4^2 + 3^2} = 5",[568,2672,2674,2810],{"className":2673,"ariaHidden":606},[605],[568,2675,2677,2681,2801,2804,2807],{"className":2676},[610],[568,2678],{"className":2679,"style":2680},[614],"height:1.04em;vertical-align:-0.1266em;",[568,2682,2684],{"className":2683},[619,763],[568,2685,2687,2792],{"className":2686},[767,768],[568,2688,2690,2789],{"className":2689},[772],[568,2691,2694,2772],{"className":2692,"style":2693},[776],"height:0.9134em;",[568,2695,2698,2701],{"className":2696,"style":2697},[781],"top:-3em;",[568,2699],{"className":2700,"style":1808},[786],[568,2702,2705,2734,2737,2740,2743],{"className":2703,"style":2704},[619],"padding-left:0.833em;",[568,2706,2708,2711],{"className":2707},[619],[568,2709,1738],{"className":2710},[619],[568,2712,2714],{"className":2713},[820],[568,2715,2717],{"className":2716},[767],[568,2718,2720],{"className":2719},[772],[568,2721,2723],{"className":2722,"style":870},[776],[568,2724,2725,2728],{"style":833},[568,2726],{"className":2727,"style":837},[786],[568,2729,2731],{"className":2730},[841,842,843,844],[568,2732,746],{"className":2733},[619,844],[568,2735],{"className":2736,"style":798},[624],[568,2738,723],{"className":2739},[802],[568,2741],{"className":2742,"style":798},[624],[568,2744,2746,2749],{"className":2745},[619],[568,2747,1735],{"className":2748},[619],[568,2750,2752],{"className":2751},[820],[568,2753,2755],{"className":2754},[767],[568,2756,2758],{"className":2757},[772],[568,2759,2761],{"className":2760,"style":870},[776],[568,2762,2763,2766],{"style":833},[568,2764],{"className":2765,"style":837},[786],[568,2767,2769],{"className":2768},[841,842,843,844],[568,2770,746],{"className":2771},[619,844],[568,2773,2775,2778],{"style":2774},"top:-2.8734em;",[568,2776],{"className":2777,"style":1808},[786],[568,2779,2782],{"className":2780,"style":2781},[891],"min-width:0.853em;height:1.08em;",[894,2783,2786],{"xmlns":896,"width":897,"height":2784,"viewBox":2785,"preserveAspectRatio":900},"1.08em","0 0 400000 1080",[902,2787],{"d":2788},"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",[568,2790,909],{"className":2791},[908],[568,2793,2795],{"className":2794},[772],[568,2796,2799],{"className":2797,"style":2798},[776],"height:0.1266em;",[568,2800],{},[568,2802],{"className":2803,"style":625},[624],[568,2805,593],{"className":2806},[629],[568,2808],{"className":2809,"style":625},[624],[568,2811,2813,2816],{"className":2812},[610],[568,2814],{"className":2815,"style":945},[614],[568,2817,2148],{"className":2818},[619]," lang.",[564,2821,2822,2823,1919,3076,3225,3226,2088],{},"Für ",[568,2824,2826,2864],{"className":2825},[571],[568,2827,2829],{"className":2828},[575],[577,2830,2831],{"xmlns":579},[581,2832,2833,2861],{},[584,2834,2835,2837,2839,2841,2843,2845,2851],{},[587,2836,733],{},[591,2838,727],{"stretchy":726},[587,2840,589],{},[591,2842,743],{"stretchy":726},[591,2844,593],{},[1731,2846,2847,2849],{},[718,2848,746],{},[718,2850,1735],{},[729,2852,2853,2855],{},[587,2854,589],{},[1731,2856,2857,2859],{},[718,2858,1735],{},[718,2860,746],{},[598,2862,2863],{"encoding":600},"f(x) = \\frac{2}{3}x^{\\frac{3}{2}}",[568,2865,2867,2894],{"className":2866,"ariaHidden":606},[605],[568,2868,2870,2873,2876,2879,2882,2885,2888,2891],{"className":2869},[610],[568,2871],{"className":2872,"style":1753},[614],[568,2874,733],{"className":2875,"style":816},[619,620],[568,2877,727],{"className":2878},[809],[568,2880,589],{"className":2881},[619,620],[568,2883,743],{"className":2884},[854],[568,2886],{"className":2887,"style":625},[624],[568,2889,593],{"className":2890},[629],[568,2892],{"className":2893,"style":625},[624],[568,2895,2897,2901,2969],{"className":2896},[610],[568,2898],{"className":2899,"style":2900},[614],"height:1.299em;vertical-align:-0.345em;",[568,2902,2904,2907,2966],{"className":2903},[619],[568,2905],{"className":2906},[809,1788],[568,2908,2910],{"className":2909},[1731],[568,2911,2913,2958],{"className":2912},[767,768],[568,2914,2916,2955],{"className":2915},[772],[568,2917,2919,2933,2941],{"className":2918,"style":1801},[776],[568,2920,2921,2924],{"style":1804},[568,2922],{"className":2923,"style":1808},[786],[568,2925,2927],{"className":2926},[841,842,843,844],[568,2928,2930],{"className":2929},[619,844],[568,2931,1735],{"className":2932},[619,844],[568,2934,2935,2938],{"style":1820},[568,2936],{"className":2937,"style":1808},[786],[568,2939],{"className":2940,"style":1828},[1827],[568,2942,2943,2946],{"style":1831},[568,2944],{"className":2945,"style":1808},[786],[568,2947,2949],{"className":2948},[841,842,843,844],[568,2950,2952],{"className":2951},[619,844],[568,2953,746],{"className":2954},[619,844],[568,2956,909],{"className":2957},[908],[568,2959,2961],{"className":2960},[772],[568,2962,2964],{"className":2963,"style":1853},[776],[568,2965],{},[568,2967],{"className":2968},[854,1788],[568,2970,2972,2975],{"className":2971},[619],[568,2973,589],{"className":2974},[619,620],[568,2976,2978],{"className":2977},[820],[568,2979,2981],{"className":2980},[767],[568,2982,2984],{"className":2983},[772],[568,2985,2988],{"className":2986,"style":2987},[776],"height:0.954em;",[568,2989,2991,2994],{"style":2990},"top:-3.363em;margin-right:0.05em;",[568,2992],{"className":2993,"style":1808},[786],[568,2995,2997],{"className":2996},[841,842,843,844],[568,2998,3000],{"className":2999},[619,844],[568,3001,3003,3008,3073],{"className":3002},[619,844],[568,3004],{"className":3005},[809,1788,841,3006,3007],"reset-size3","size6",[568,3009,3011],{"className":3010},[1731],[568,3012,3014,3064],{"className":3013},[767,768],[568,3015,3017,3061],{"className":3016},[772],[568,3018,3021,3036,3046],{"className":3019,"style":3020},[776],"height:0.8443em;",[568,3022,3024,3027],{"style":3023},"top:-2.656em;",[568,3025],{"className":3026,"style":1808},[786],[568,3028,3030],{"className":3029},[841,3006,1520,844],[568,3031,3033],{"className":3032},[619,844],[568,3034,746],{"className":3035},[619,844],[568,3037,3039,3042],{"style":3038},"top:-3.2255em;",[568,3040],{"className":3041,"style":1808},[786],[568,3043],{"className":3044,"style":3045},[1827,844],"border-bottom-width:0.049em;",[568,3047,3049,3052],{"style":3048},"top:-3.384em;",[568,3050],{"className":3051,"style":1808},[786],[568,3053,3055],{"className":3054},[841,3006,1520,844],[568,3056,3058],{"className":3057},[619,844],[568,3059,1735],{"className":3060},[619,844],[568,3062,909],{"className":3063},[908],[568,3065,3067],{"className":3066},[772],[568,3068,3071],{"className":3069,"style":3070},[776],"height:0.344em;",[568,3072],{},[568,3074],{"className":3075},[854,1788,841,3006,3007],[568,3077,3079,3109],{"className":3078},[571],[568,3080,3082],{"className":3081},[575],[577,3083,3084],{"xmlns":579},[581,3085,3086,3106],{},[584,3087,3088,3094,3096,3098,3100,3102],{},[729,3089,3090,3092],{},[587,3091,733],{},[591,3093,738],{"mathvariant":736,"lspace":737,"rspace":737},[591,3095,727],{"stretchy":726},[587,3097,589],{},[591,3099,743],{"stretchy":726},[591,3101,593],{},[713,3103,3104],{},[587,3105,589],{},[598,3107,3108],{"encoding":600},"f'(x) = \\sqrt{x}",[568,3110,3112,3168],{"className":3111,"ariaHidden":606},[605],[568,3113,3115,3118,3150,3153,3156,3159,3162,3165],{"className":3114},[610],[568,3116],{"className":3117,"style":1177},[614],[568,3119,3121,3124],{"className":3120},[619],[568,3122,733],{"className":3123,"style":816},[619,620],[568,3125,3127],{"className":3126},[820],[568,3128,3130],{"className":3129},[767],[568,3131,3133],{"className":3132},[772],[568,3134,3136],{"className":3135,"style":1196},[776],[568,3137,3138,3141],{"style":1199},[568,3139],{"className":3140,"style":837},[786],[568,3142,3144],{"className":3143},[841,842,843,844],[568,3145,3147],{"className":3146},[619,844],[568,3148,738],{"className":3149},[619,844],[568,3151,727],{"className":3152},[809],[568,3154,589],{"className":3155},[619,620],[568,3157,743],{"className":3158},[854],[568,3160],{"className":3161,"style":625},[624],[568,3163,593],{"className":3164},[629],[568,3166],{"className":3167,"style":625},[624],[568,3169,3171,3175],{"className":3170},[610],[568,3172],{"className":3173,"style":3174},[614],"height:1.04em;vertical-align:-0.2397em;",[568,3176,3178],{"className":3177},[619,763],[568,3179,3181,3216],{"className":3180},[767,768],[568,3182,3184,3213],{"className":3183},[772],[568,3185,3188,3200],{"className":3186,"style":3187},[776],"height:0.8003em;",[568,3189,3191,3194],{"className":3190,"style":2697},[781],[568,3192],{"className":3193,"style":1808},[786],[568,3195,3197],{"className":3196,"style":2704},[619],[568,3198,589],{"className":3199},[619,620],[568,3201,3203,3206],{"style":3202},"top:-2.7603em;",[568,3204],{"className":3205,"style":1808},[786],[568,3207,3209],{"className":3208,"style":2781},[891],[894,3210,3211],{"xmlns":896,"width":897,"height":2784,"viewBox":2785,"preserveAspectRatio":900},[902,3212],{"d":2788},[568,3214,909],{"className":3215},[908],[568,3217,3219],{"className":3218},[772],[568,3220,3223],{"className":3221,"style":3222},[776],"height:0.2397em;",[568,3224],{},", also ",[568,3227,3229,3273],{"className":3228},[571],[568,3230,3232],{"className":3231},[575],[577,3233,3234],{"xmlns":579},[581,3235,3236,3270],{},[584,3237,3238,3240,3242,3244,3250,3252,3254,3256,3262,3264,3266,3268],{},[718,3239,720],{},[591,3241,723],{},[591,3243,727],{"stretchy":726},[729,3245,3246,3248],{},[587,3247,733],{},[591,3249,738],{"mathvariant":736,"lspace":737,"rspace":737},[591,3251,727],{"stretchy":726},[587,3253,589],{},[591,3255,743],{"stretchy":726},[729,3257,3258,3260],{},[591,3259,743],{"stretchy":726},[718,3261,746],{},[591,3263,593],{},[718,3265,720],{},[591,3267,723],{},[587,3269,589],{},[598,3271,3272],{"encoding":600},"1 + (f'(x))^2 = 1 + x",[568,3274,3276,3295,3385,3403],{"className":3275,"ariaHidden":606},[605],[568,3277,3279,3283,3286,3289,3292],{"className":3278},[610],[568,3280],{"className":3281,"style":3282},[614],"height:0.7278em;vertical-align:-0.0833em;",[568,3284,720],{"className":3285},[619],[568,3287],{"className":3288,"style":798},[624],[568,3290,723],{"className":3291},[802],[568,3293],{"className":3294,"style":798},[624],[568,3296,3298,3302,3305,3337,3340,3343,3346,3376,3379,3382],{"className":3297},[610],[568,3299],{"className":3300,"style":3301},[614],"height:1.0641em;vertical-align:-0.25em;",[568,3303,727],{"className":3304},[809],[568,3306,3308,3311],{"className":3307},[619],[568,3309,733],{"className":3310,"style":816},[619,620],[568,3312,3314],{"className":3313},[820],[568,3315,3317],{"className":3316},[767],[568,3318,3320],{"className":3319},[772],[568,3321,3323],{"className":3322,"style":1196},[776],[568,3324,3325,3328],{"style":1199},[568,3326],{"className":3327,"style":837},[786],[568,3329,3331],{"className":3330},[841,842,843,844],[568,3332,3334],{"className":3333},[619,844],[568,3335,738],{"className":3336},[619,844],[568,3338,727],{"className":3339},[809],[568,3341,589],{"className":3342},[619,620],[568,3344,743],{"className":3345},[854],[568,3347,3349,3352],{"className":3348},[854],[568,3350,743],{"className":3351},[854],[568,3353,3355],{"className":3354},[820],[568,3356,3358],{"className":3357},[767],[568,3359,3361],{"className":3360},[772],[568,3362,3365],{"className":3363,"style":3364},[776],"height:0.8141em;",[568,3366,3367,3370],{"style":1199},[568,3368],{"className":3369,"style":837},[786],[568,3371,3373],{"className":3372},[841,842,843,844],[568,3374,746],{"className":3375},[619,844],[568,3377],{"className":3378,"style":625},[624],[568,3380,593],{"className":3381},[629],[568,3383],{"className":3384,"style":625},[624],[568,3386,3388,3391,3394,3397,3400],{"className":3387},[610],[568,3389],{"className":3390,"style":3282},[614],[568,3392,720],{"className":3393},[619],[568,3395],{"className":3396,"style":798},[624],[568,3398,723],{"className":3399},[802],[568,3401],{"className":3402,"style":798},[624],[568,3404,3406,3409],{"className":3405},[610],[568,3407],{"className":3408,"style":615},[614],[568,3410,589],{"className":3411},[619,620],[568,3413,3415],{"className":3414},[1235],[568,3416,3418,3531],{"className":3417},[571],[568,3419,3421],{"className":3420},[575],[577,3422,3423],{"xmlns":579,"display":1244},[581,3424,3425,3528],{},[584,3426,3427,3429,3431,3439,3449,3451,3453,3455,3457,3495,3497,3503,3505,3508,3510,3512,3514,3516,3523,3525],{},[587,3428,2106],{},[591,3430,593],{},[2110,3432,3433,3435,3437],{},[591,3434,2114],{},[718,3436,1685],{},[718,3438,1735],{},[713,3440,3441],{},[584,3442,3443,3445,3447],{},[718,3444,720],{},[591,3446,723],{},[587,3448,589],{},[1135,3450,1137],{},[587,3452,2139],{},[587,3454,589],{},[591,3456,593],{},[2110,3458,3459,3491,3493],{},[584,3460,3461,3464,3470,3472,3474,3476,3478,3488],{},[591,3462,3463],{"fence":606},"[",[1731,3465,3466,3468],{},[718,3467,746],{},[718,3469,1735],{},[591,3471,727],{"stretchy":726},[718,3473,720],{},[591,3475,723],{},[587,3477,589],{},[729,3479,3480,3482],{},[591,3481,743],{"stretchy":726},[1731,3483,3484,3486],{},[718,3485,1735],{},[718,3487,746],{},[591,3489,3490],{"fence":606},"]",[718,3492,1685],{},[718,3494,1735],{},[591,3496,593],{},[1731,3498,3499,3501],{},[718,3500,746],{},[718,3502,1735],{},[591,3504,727],{"stretchy":726},[718,3506,3507],{},"8",[591,3509,966],{},[718,3511,720],{},[591,3513,743],{"stretchy":726},[591,3515,593],{},[1731,3517,3518,3521],{},[718,3519,3520],{},"14",[718,3522,1735],{},[591,3524,1121],{},[718,3526,3527],{},"4,67",[598,3529,3530],{"encoding":600},"L = \\int_0^3 \\sqrt{1 + x}\\,dx = \\left[\\frac{2}{3}(1 + x)^{\\frac{3}{2}}\\right]_0^3 = \\frac{2}{3}(8 - 1) = \\frac{14}{3} \\approx 4{,}67",[568,3532,3534,3552,3693,3958,4041,4062,4139],{"className":3533,"ariaHidden":606},[605],[568,3535,3537,3540,3543,3546,3549],{"className":3536},[610],[568,3538],{"className":3539,"style":1094},[614],[568,3541,2106],{"className":3542},[619,620],[568,3544],{"className":3545,"style":625},[624],[568,3547,593],{"className":3548},[629],[568,3550],{"className":3551,"style":625},[624],[568,3553,3555,3559,3610,3613,3675,3678,3681,3684,3687,3690],{"className":3554},[610],[568,3556],{"className":3557,"style":3558},[614],"height:2.476em;vertical-align:-0.9119em;",[568,3560,3562,3565],{"className":3561},[2194],[568,3563,2114],{"className":3564,"style":2200},[2194,2198,2199],[568,3566,3568],{"className":3567},[820],[568,3569,3571,3602],{"className":3570},[767,768],[568,3572,3574,3599],{"className":3573},[772],[568,3575,3577,3588],{"className":3576,"style":2213},[776],[568,3578,3579,3582],{"style":2216},[568,3580],{"className":3581,"style":837},[786],[568,3583,3585],{"className":3584},[841,842,843,844],[568,3586,1685],{"className":3587},[619,844],[568,3589,3590,3593],{"style":2228},[568,3591],{"className":3592,"style":837},[786],[568,3594,3596],{"className":3595},[841,842,843,844],[568,3597,1735],{"className":3598},[619,844],[568,3600,909],{"className":3601},[908],[568,3603,3605],{"className":3604},[772],[568,3606,3608],{"className":3607,"style":2247},[776],[568,3609],{},[568,3611],{"className":3612,"style":1224},[624],[568,3614,3616],{"className":3615},[619,763],[568,3617,3619,3666],{"className":3618},[767,768],[568,3620,3622,3663],{"className":3621},[772],[568,3623,3626,3650],{"className":3624,"style":3625},[776],"height:0.9144em;",[568,3627,3629,3632],{"className":3628,"style":2697},[781],[568,3630],{"className":3631,"style":1808},[786],[568,3633,3635,3638,3641,3644,3647],{"className":3634,"style":2704},[619],[568,3636,720],{"className":3637},[619],[568,3639],{"className":3640,"style":798},[624],[568,3642,723],{"className":3643},[802],[568,3645],{"className":3646,"style":798},[624],[568,3648,589],{"className":3649},[619,620],[568,3651,3653,3656],{"style":3652},"top:-2.8744em;",[568,3654],{"className":3655,"style":1808},[786],[568,3657,3659],{"className":3658,"style":2781},[891],[894,3660,3661],{"xmlns":896,"width":897,"height":2784,"viewBox":2785,"preserveAspectRatio":900},[902,3662],{"d":2788},[568,3664,909],{"className":3665},[908],[568,3667,3669],{"className":3668},[772],[568,3670,3673],{"className":3671,"style":3672},[776],"height:0.1256em;",[568,3674],{},[568,3676],{"className":3677,"style":1224},[624],[568,3679,2139],{"className":3680},[619,620],[568,3682,589],{"className":3683},[619,620],[568,3685],{"className":3686,"style":625},[624],[568,3688,593],{"className":3689},[629],[568,3691],{"className":3692,"style":625},[624],[568,3694,3696,3700,3949,3952,3955],{"className":3695},[610],[568,3697],{"className":3698,"style":3699},[614],"height:2.6537em;vertical-align:-0.9997em;",[568,3701,3704,3900],{"className":3702},[3703],"minner",[568,3705,3707,3715,3777,3780,3783,3786,3789,3792,3795,3894],{"className":3706},[3703],[568,3708,3712],{"className":3709,"style":3711},[809,3710],"delimcenter","top:0em;",[568,3713,3463],{"className":3714},[1519,843],[568,3716,3718,3721,3774],{"className":3717},[619],[568,3719],{"className":3720},[809,1788],[568,3722,3724],{"className":3723},[1731],[568,3725,3727,3766],{"className":3726},[767,768],[568,3728,3730,3763],{"className":3729},[772],[568,3731,3733,3744,3752],{"className":3732,"style":2307},[776],[568,3734,3735,3738],{"style":2310},[568,3736],{"className":3737,"style":1808},[786],[568,3739,3741],{"className":3740},[619],[568,3742,1735],{"className":3743},[619],[568,3745,3746,3749],{"style":1820},[568,3747],{"className":3748,"style":1808},[786],[568,3750],{"className":3751,"style":1828},[1827],[568,3753,3754,3757],{"style":2330},[568,3755],{"className":3756,"style":1808},[786],[568,3758,3760],{"className":3759},[619],[568,3761,746],{"className":3762},[619],[568,3764,909],{"className":3765},[908],[568,3767,3769],{"className":3768},[772],[568,3770,3772],{"className":3771,"style":2349},[776],[568,3773],{},[568,3775],{"className":3776},[854,1788],[568,3778,727],{"className":3779},[809],[568,3781,720],{"className":3782},[619],[568,3784],{"className":3785,"style":798},[624],[568,3787,723],{"className":3788},[802],[568,3790],{"className":3791,"style":798},[624],[568,3793,589],{"className":3794},[619,620],[568,3796,3798,3801],{"className":3797},[854],[568,3799,743],{"className":3800},[854],[568,3802,3804],{"className":3803},[820],[568,3805,3807],{"className":3806},[767],[568,3808,3810],{"className":3809},[772],[568,3811,3814],{"className":3812,"style":3813},[776],"height:1.004em;",[568,3815,3817,3820],{"style":3816},"top:-3.413em;margin-right:0.05em;",[568,3818],{"className":3819,"style":1808},[786],[568,3821,3823],{"className":3822},[841,842,843,844],[568,3824,3826],{"className":3825},[619,844],[568,3827,3829,3832,3891],{"className":3828},[619,844],[568,3830],{"className":3831},[809,1788,841,3006,3007],[568,3833,3835],{"className":3834},[1731],[568,3836,3838,3883],{"className":3837},[767,768],[568,3839,3841,3880],{"className":3840},[772],[568,3842,3844,3858,3866],{"className":3843,"style":3020},[776],[568,3845,3846,3849],{"style":3023},[568,3847],{"className":3848,"style":1808},[786],[568,3850,3852],{"className":3851},[841,3006,1520,844],[568,3853,3855],{"className":3854},[619,844],[568,3856,746],{"className":3857},[619,844],[568,3859,3860,3863],{"style":3038},[568,3861],{"className":3862,"style":1808},[786],[568,3864],{"className":3865,"style":3045},[1827,844],[568,3867,3868,3871],{"style":3048},[568,3869],{"className":3870,"style":1808},[786],[568,3872,3874],{"className":3873},[841,3006,1520,844],[568,3875,3877],{"className":3876},[619,844],[568,3878,1735],{"className":3879},[619,844],[568,3881,909],{"className":3882},[908],[568,3884,3886],{"className":3885},[772],[568,3887,3889],{"className":3888,"style":3070},[776],[568,3890],{},[568,3892],{"className":3893},[854,1788,841,3006,3007],[568,3895,3897],{"className":3896,"style":3711},[854,3710],[568,3898,3490],{"className":3899},[1519,843],[568,3901,3903],{"className":3902},[820],[568,3904,3906,3940],{"className":3905},[767,768],[568,3907,3909,3937],{"className":3908},[772],[568,3910,3913,3925],{"className":3911,"style":3912},[776],"height:1.654em;",[568,3914,3916,3919],{"style":3915},"top:-1.7003em;margin-right:0.05em;",[568,3917],{"className":3918,"style":837},[786],[568,3920,3922],{"className":3921},[841,842,843,844],[568,3923,1685],{"className":3924},[619,844],[568,3926,3928,3931],{"style":3927},"top:-3.9029em;margin-right:0.05em;",[568,3929],{"className":3930,"style":837},[786],[568,3932,3934],{"className":3933},[841,842,843,844],[568,3935,1735],{"className":3936},[619,844],[568,3938,909],{"className":3939},[908],[568,3941,3943],{"className":3942},[772],[568,3944,3947],{"className":3945,"style":3946},[776],"height:0.9997em;",[568,3948],{},[568,3950],{"className":3951,"style":625},[624],[568,3953,593],{"className":3954},[629],[568,3956],{"className":3957,"style":625},[624],[568,3959,3961,3964,4026,4029,4032,4035,4038],{"className":3960},[610],[568,3962],{"className":3963,"style":2408},[614],[568,3965,3967,3970,4023],{"className":3966},[619],[568,3968],{"className":3969},[809,1788],[568,3971,3973],{"className":3972},[1731],[568,3974,3976,4015],{"className":3975},[767,768],[568,3977,3979,4012],{"className":3978},[772],[568,3980,3982,3993,4001],{"className":3981,"style":2307},[776],[568,3983,3984,3987],{"style":2310},[568,3985],{"className":3986,"style":1808},[786],[568,3988,3990],{"className":3989},[619],[568,3991,1735],{"className":3992},[619],[568,3994,3995,3998],{"style":1820},[568,3996],{"className":3997,"style":1808},[786],[568,3999],{"className":4000,"style":1828},[1827],[568,4002,4003,4006],{"style":2330},[568,4004],{"className":4005,"style":1808},[786],[568,4007,4009],{"className":4008},[619],[568,4010,746],{"className":4011},[619],[568,4013,909],{"className":4014},[908],[568,4016,4018],{"className":4017},[772],[568,4019,4021],{"className":4020,"style":2349},[776],[568,4022],{},[568,4024],{"className":4025},[854,1788],[568,4027,727],{"className":4028},[809],[568,4030,3507],{"className":4031},[619],[568,4033],{"className":4034,"style":798},[624],[568,4036,966],{"className":4037},[802],[568,4039],{"className":4040,"style":798},[624],[568,4042,4044,4047,4050,4053,4056,4059],{"className":4043},[610],[568,4045],{"className":4046,"style":1753},[614],[568,4048,720],{"className":4049},[619],[568,4051,743],{"className":4052},[854],[568,4054],{"className":4055,"style":625},[624],[568,4057,593],{"className":4058},[629],[568,4060],{"className":4061,"style":625},[624],[568,4063,4065,4068,4130,4133,4136],{"className":4064},[610],[568,4066],{"className":4067,"style":2408},[614],[568,4069,4071,4074,4127],{"className":4070},[619],[568,4072],{"className":4073},[809,1788],[568,4075,4077],{"className":4076},[1731],[568,4078,4080,4119],{"className":4079},[767,768],[568,4081,4083,4116],{"className":4082},[772],[568,4084,4086,4097,4105],{"className":4085,"style":2307},[776],[568,4087,4088,4091],{"style":2310},[568,4089],{"className":4090,"style":1808},[786],[568,4092,4094],{"className":4093},[619],[568,4095,1735],{"className":4096},[619],[568,4098,4099,4102],{"style":1820},[568,4100],{"className":4101,"style":1808},[786],[568,4103],{"className":4104,"style":1828},[1827],[568,4106,4107,4110],{"style":2330},[568,4108],{"className":4109,"style":1808},[786],[568,4111,4113],{"className":4112},[619],[568,4114,3520],{"className":4115},[619],[568,4117,909],{"className":4118},[908],[568,4120,4122],{"className":4121},[772],[568,4123,4125],{"className":4124,"style":2349},[776],[568,4126],{},[568,4128],{"className":4129},[854,1788],[568,4131],{"className":4132,"style":625},[624],[568,4134,1121],{"className":4135},[629],[568,4137],{"className":4138,"style":625},[624],[568,4140,4142,4146,4149,4157],{"className":4141},[610],[568,4143],{"className":4144,"style":4145},[614],"height:0.8389em;vertical-align:-0.1944em;",[568,4147,1738],{"className":4148},[619],[568,4150,4152],{"className":4151},[619],[568,4153,4156],{"className":4154},[4155],"mpunct",",",[568,4158,4160],{"className":4159},[619],"67",[564,4162,4163,4164,4168,4169,4247,4248,4467,4468,4681,4682,642,4791,2088],{},"Für die ",[4165,4166,4167],"strong",{},"Kettenlinie"," ",[568,4170,4172,4202],{"className":4171},[571],[568,4173,4175],{"className":4174},[575],[577,4176,4177],{"xmlns":579},[581,4178,4179,4199],{},[584,4180,4181,4183,4185,4187,4189,4191,4194,4197],{},[587,4182,733],{},[591,4184,727],{"stretchy":726},[587,4186,589],{},[591,4188,743],{"stretchy":726},[591,4190,593],{},[587,4192,4193],{},"cosh",[591,4195,4196],{},"⁡",[587,4198,589],{},[598,4200,4201],{"encoding":600},"f(x) = \\cosh x",[568,4203,4205,4232],{"className":4204,"ariaHidden":606},[605],[568,4206,4208,4211,4214,4217,4220,4223,4226,4229],{"className":4207},[610],[568,4209],{"className":4210,"style":1753},[614],[568,4212,733],{"className":4213,"style":816},[619,620],[568,4215,727],{"className":4216},[809],[568,4218,589],{"className":4219},[619,620],[568,4221,743],{"className":4222},[854],[568,4224],{"className":4225,"style":625},[624],[568,4227,593],{"className":4228},[629],[568,4230],{"className":4231,"style":625},[624],[568,4233,4235,4238,4241,4244],{"className":4234},[610],[568,4236],{"className":4237,"style":692},[614],[568,4239,4193],{"className":4240},[2194],[568,4242],{"className":4243,"style":1224},[624],[568,4245,589],{"className":4246},[619,620]," (Kosinus hyperbolicus, ",[568,4249,4251,4296],{"className":4250},[571],[568,4252,4254],{"className":4253},[575],[577,4255,4256],{"xmlns":579},[581,4257,4258,4293],{},[584,4259,4260,4262,4264,4266,4268],{},[587,4261,4193],{},[591,4263,4196],{},[587,4265,589],{},[591,4267,593],{},[1731,4269,4270,4291],{},[584,4271,4272,4279,4281],{},[729,4273,4274,4277],{},[587,4275,4276],{},"e",[587,4278,589],{},[591,4280,723],{},[729,4282,4283,4285],{},[587,4284,4276],{},[584,4286,4287,4289],{},[591,4288,966],{},[587,4290,589],{},[718,4292,746],{},[598,4294,4295],{"encoding":600},"\\cosh x = \\frac{e^x + e^{-x}}{2}",[568,4297,4299,4323],{"className":4298,"ariaHidden":606},[605],[568,4300,4302,4305,4308,4311,4314,4317,4320],{"className":4301},[610],[568,4303],{"className":4304,"style":692},[614],[568,4306,4193],{"className":4307},[2194],[568,4309],{"className":4310,"style":1224},[624],[568,4312,589],{"className":4313},[619,620],[568,4315],{"className":4316,"style":625},[624],[568,4318,593],{"className":4319},[629],[568,4321],{"className":4322,"style":625},[624],[568,4324,4326,4330],{"className":4325},[610],[568,4327],{"className":4328,"style":4329},[614],"height:1.3324em;vertical-align:-0.345em;",[568,4331,4333,4336,4464],{"className":4332},[619],[568,4334],{"className":4335},[809,1788],[568,4337,4339],{"className":4338},[1731],[568,4340,4342,4456],{"className":4341},[767,768],[568,4343,4345,4453],{"className":4344},[772],[568,4346,4349,4363,4371],{"className":4347,"style":4348},[776],"height:0.9874em;",[568,4350,4351,4354],{"style":1804},[568,4352],{"className":4353,"style":1808},[786],[568,4355,4357],{"className":4356},[841,842,843,844],[568,4358,4360],{"className":4359},[619,844],[568,4361,746],{"className":4362},[619,844],[568,4364,4365,4368],{"style":1820},[568,4366],{"className":4367,"style":1808},[786],[568,4369],{"className":4370,"style":1828},[1827],[568,4372,4373,4376],{"style":1831},[568,4374],{"className":4375,"style":1808},[786],[568,4377,4379],{"className":4378},[841,842,843,844],[568,4380,4382,4414,4417],{"className":4381},[619,844],[568,4383,4385,4388],{"className":4384},[619,844],[568,4386,4276],{"className":4387},[619,620,844],[568,4389,4391],{"className":4390},[820],[568,4392,4394],{"className":4393},[767],[568,4395,4397],{"className":4396},[772],[568,4398,4401],{"className":4399,"style":4400},[776],"height:0.7385em;",[568,4402,4404,4408],{"style":4403},"top:-2.931em;margin-right:0.0714em;",[568,4405],{"className":4406,"style":4407},[786],"height:2.5em;",[568,4409,4411],{"className":4410},[841,3006,1520,844],[568,4412,589],{"className":4413},[619,620,844],[568,4415,723],{"className":4416},[802,844],[568,4418,4420,4423],{"className":4419},[619,844],[568,4421,4276],{"className":4422},[619,620,844],[568,4424,4426],{"className":4425},[820],[568,4427,4429],{"className":4428},[767],[568,4430,4432],{"className":4431},[772],[568,4433,4436],{"className":4434,"style":4435},[776],"height:0.8477em;",[568,4437,4438,4441],{"style":4403},[568,4439],{"className":4440,"style":4407},[786],[568,4442,4444],{"className":4443},[841,3006,1520,844],[568,4445,4447,4450],{"className":4446},[619,844],[568,4448,966],{"className":4449},[619,844],[568,4451,589],{"className":4452},[619,620,844],[568,4454,909],{"className":4455},[908],[568,4457,4459],{"className":4458},[772],[568,4460,4462],{"className":4461,"style":1853},[776],[568,4463],{},[568,4465],{"className":4466},[854,1788],") gilt mit dem Sinus hyperbolicus ",[568,4469,4471,4516],{"className":4470},[571],[568,4472,4474],{"className":4473},[575],[577,4475,4476],{"xmlns":579},[581,4477,4478,4513],{},[584,4479,4480,4483,4485,4487,4489],{},[587,4481,4482],{},"sinh",[591,4484,4196],{},[587,4486,589],{},[591,4488,593],{},[1731,4490,4491,4511],{},[584,4492,4493,4499,4501],{},[729,4494,4495,4497],{},[587,4496,4276],{},[587,4498,589],{},[591,4500,966],{},[729,4502,4503,4505],{},[587,4504,4276],{},[584,4506,4507,4509],{},[591,4508,966],{},[587,4510,589],{},[718,4512,746],{},[598,4514,4515],{"encoding":600},"\\sinh x = \\frac{e^x - e^{-x}}{2}",[568,4517,4519,4543],{"className":4518,"ariaHidden":606},[605],[568,4520,4522,4525,4528,4531,4534,4537,4540],{"className":4521},[610],[568,4523],{"className":4524,"style":692},[614],[568,4526,4482],{"className":4527},[2194],[568,4529],{"className":4530,"style":1224},[624],[568,4532,589],{"className":4533},[619,620],[568,4535],{"className":4536,"style":625},[624],[568,4538,593],{"className":4539},[629],[568,4541],{"className":4542,"style":625},[624],[568,4544,4546,4549],{"className":4545},[610],[568,4547],{"className":4548,"style":4329},[614],[568,4550,4552,4555,4678],{"className":4551},[619],[568,4553],{"className":4554},[809,1788],[568,4556,4558],{"className":4557},[1731],[568,4559,4561,4670],{"className":4560},[767,768],[568,4562,4564,4667],{"className":4563},[772],[568,4565,4567,4581,4589],{"className":4566,"style":4348},[776],[568,4568,4569,4572],{"style":1804},[568,4570],{"className":4571,"style":1808},[786],[568,4573,4575],{"className":4574},[841,842,843,844],[568,4576,4578],{"className":4577},[619,844],[568,4579,746],{"className":4580},[619,844],[568,4582,4583,4586],{"style":1820},[568,4584],{"className":4585,"style":1808},[786],[568,4587],{"className":4588,"style":1828},[1827],[568,4590,4591,4594],{"style":1831},[568,4592],{"className":4593,"style":1808},[786],[568,4595,4597],{"className":4596},[841,842,843,844],[568,4598,4600,4629,4632],{"className":4599},[619,844],[568,4601,4603,4606],{"className":4602},[619,844],[568,4604,4276],{"className":4605},[619,620,844],[568,4607,4609],{"className":4608},[820],[568,4610,4612],{"className":4611},[767],[568,4613,4615],{"className":4614},[772],[568,4616,4618],{"className":4617,"style":4400},[776],[568,4619,4620,4623],{"style":4403},[568,4621],{"className":4622,"style":4407},[786],[568,4624,4626],{"className":4625},[841,3006,1520,844],[568,4627,589],{"className":4628},[619,620,844],[568,4630,966],{"className":4631},[802,844],[568,4633,4635,4638],{"className":4634},[619,844],[568,4636,4276],{"className":4637},[619,620,844],[568,4639,4641],{"className":4640},[820],[568,4642,4644],{"className":4643},[767],[568,4645,4647],{"className":4646},[772],[568,4648,4650],{"className":4649,"style":4435},[776],[568,4651,4652,4655],{"style":4403},[568,4653],{"className":4654,"style":4407},[786],[568,4656,4658],{"className":4657},[841,3006,1520,844],[568,4659,4661,4664],{"className":4660},[619,844],[568,4662,966],{"className":4663},[619,844],[568,4665,589],{"className":4666},[619,620,844],[568,4668,909],{"className":4669},[908],[568,4671,4673],{"className":4672},[772],[568,4674,4676],{"className":4675,"style":1853},[776],[568,4677],{},[568,4679],{"className":4680},[854,1788]," die Ableitung ",[568,4683,4685,4717],{"className":4684},[571],[568,4686,4688],{"className":4687},[575],[577,4689,4690],{"xmlns":579},[581,4691,4692,4714],{},[584,4693,4694,4700,4702,4704,4706,4708,4710,4712],{},[729,4695,4696,4698],{},[587,4697,733],{},[591,4699,738],{"mathvariant":736,"lspace":737,"rspace":737},[591,4701,727],{"stretchy":726},[587,4703,589],{},[591,4705,743],{"stretchy":726},[591,4707,593],{},[587,4709,4482],{},[591,4711,4196],{},[587,4713,589],{},[598,4715,4716],{"encoding":600},"f'(x) = \\sinh x",[568,4718,4720,4776],{"className":4719,"ariaHidden":606},[605],[568,4721,4723,4726,4758,4761,4764,4767,4770,4773],{"className":4722},[610],[568,4724],{"className":4725,"style":1177},[614],[568,4727,4729,4732],{"className":4728},[619],[568,4730,733],{"className":4731,"style":816},[619,620],[568,4733,4735],{"className":4734},[820],[568,4736,4738],{"className":4737},[767],[568,4739,4741],{"className":4740},[772],[568,4742,4744],{"className":4743,"style":1196},[776],[568,4745,4746,4749],{"style":1199},[568,4747],{"className":4748,"style":837},[786],[568,4750,4752],{"className":4751},[841,842,843,844],[568,4753,4755],{"className":4754},[619,844],[568,4756,738],{"className":4757},[619,844],[568,4759,727],{"className":4760},[809],[568,4762,589],{"className":4763},[619,620],[568,4765,743],{"className":4766},[854],[568,4768],{"className":4769,"style":625},[624],[568,4771,593],{"className":4772},[629],[568,4774],{"className":4775,"style":625},[624],[568,4777,4779,4782,4785,4788],{"className":4778},[610],[568,4780],{"className":4781,"style":692},[614],[568,4783,4482],{"className":4784},[2194],[568,4786],{"className":4787,"style":1224},[624],[568,4789,589],{"className":4790},[619,620],[568,4792,4794,4836],{"className":4793},[571],[568,4795,4797],{"className":4796},[575],[577,4798,4799],{"xmlns":579},[581,4800,4801,4833],{},[584,4802,4803,4805,4807,4817,4819,4821,4831],{},[718,4804,720],{},[591,4806,723],{},[729,4808,4809,4815],{},[584,4810,4811,4813],{},[587,4812,4482],{},[591,4814,4196],{},[718,4816,746],{},[587,4818,589],{},[591,4820,593],{},[729,4822,4823,4829],{},[584,4824,4825,4827],{},[587,4826,4193],{},[591,4828,4196],{},[718,4830,746],{},[587,4832,589],{},[598,4834,4835],{"encoding":600},"1 + \\sinh^2 x = \\cosh^2 x",[568,4837,4839,4857,4909],{"className":4838,"ariaHidden":606},[605],[568,4840,4842,4845,4848,4851,4854],{"className":4841},[610],[568,4843],{"className":4844,"style":3282},[614],[568,4846,720],{"className":4847},[619],[568,4849],{"className":4850,"style":798},[624],[568,4852,723],{"className":4853},[802],[568,4855],{"className":4856,"style":798},[624],[568,4858,4860,4864,4894,4897,4900,4903,4906],{"className":4859},[610],[568,4861],{"className":4862,"style":4863},[614],"height:0.8984em;",[568,4865,4867,4870],{"className":4866},[2194],[568,4868,4482],{"className":4869},[2194],[568,4871,4873],{"className":4872},[820],[568,4874,4876],{"className":4875},[767],[568,4877,4879],{"className":4878},[772],[568,4880,4882],{"className":4881,"style":4863},[776],[568,4883,4885,4888],{"style":4884},"top:-3.1473em;margin-right:0.05em;",[568,4886],{"className":4887,"style":837},[786],[568,4889,4891],{"className":4890},[841,842,843,844],[568,4892,746],{"className":4893},[619,844],[568,4895],{"className":4896,"style":1224},[624],[568,4898,589],{"className":4899},[619,620],[568,4901],{"className":4902,"style":625},[624],[568,4904,593],{"className":4905},[629],[568,4907],{"className":4908,"style":625},[624],[568,4910,4912,4915,4944,4947],{"className":4911},[610],[568,4913],{"className":4914,"style":4863},[614],[568,4916,4918,4921],{"className":4917},[2194],[568,4919,4193],{"className":4920},[2194],[568,4922,4924],{"className":4923},[820],[568,4925,4927],{"className":4926},[767],[568,4928,4930],{"className":4929},[772],[568,4931,4933],{"className":4932,"style":4863},[776],[568,4934,4935,4938],{"style":4884},[568,4936],{"className":4937,"style":837},[786],[568,4939,4941],{"className":4940},[841,842,843,844],[568,4942,746],{"className":4943},[619,844],[568,4945],{"className":4946,"style":1224},[624],[568,4948,589],{"className":4949},[619,620],[568,4951,4953],{"className":4952},[1235],[568,4954,4956,5005],{"className":4955},[571],[568,4957,4959],{"className":4958},[575],[577,4960,4961],{"xmlns":579,"display":1244},[581,4962,4963,5002],{},[584,4964,4965,4967,4969,4977,4979,4981,4983,4985,4987,4989,4991,4993,4995,4997,4999],{},[587,4966,2106],{},[591,4968,593],{},[2110,4970,4971,4973,4975],{},[591,4972,2114],{},[718,4974,1685],{},[718,4976,720],{},[587,4978,4193],{},[591,4980,4196],{},[587,4982,589],{},[1135,4984,1137],{},[587,4986,2139],{},[587,4988,589],{},[591,4990,593],{},[587,4992,4482],{},[591,4994,4196],{},[718,4996,720],{},[591,4998,1121],{},[718,5000,5001],{},"1,1752",[598,5003,5004],{"encoding":600},"L = \\int_0^1 \\cosh x\\,dx = \\sinh 1 \\approx 1{,}1752",[568,5006,5008,5026,5113,5137],{"className":5007,"ariaHidden":606},[605],[568,5009,5011,5014,5017,5020,5023],{"className":5010},[610],[568,5012],{"className":5013,"style":1094},[614],[568,5015,2106],{"className":5016},[619,620],[568,5018],{"className":5019,"style":625},[624],[568,5021,593],{"className":5022},[629],[568,5024],{"className":5025,"style":625},[624],[568,5027,5029,5032,5083,5086,5089,5092,5095,5098,5101,5104,5107,5110],{"className":5028},[610],[568,5030],{"className":5031,"style":3558},[614],[568,5033,5035,5038],{"className":5034},[2194],[568,5036,2114],{"className":5037,"style":2200},[2194,2198,2199],[568,5039,5041],{"className":5040},[820],[568,5042,5044,5075],{"className":5043},[767,768],[568,5045,5047,5072],{"className":5046},[772],[568,5048,5050,5061],{"className":5049,"style":2213},[776],[568,5051,5052,5055],{"style":2216},[568,5053],{"className":5054,"style":837},[786],[568,5056,5058],{"className":5057},[841,842,843,844],[568,5059,1685],{"className":5060},[619,844],[568,5062,5063,5066],{"style":2228},[568,5064],{"className":5065,"style":837},[786],[568,5067,5069],{"className":5068},[841,842,843,844],[568,5070,720],{"className":5071},[619,844],[568,5073,909],{"className":5074},[908],[568,5076,5078],{"className":5077},[772],[568,5079,5081],{"className":5080,"style":2247},[776],[568,5082],{},[568,5084],{"className":5085,"style":1224},[624],[568,5087,4193],{"className":5088},[2194],[568,5090],{"className":5091,"style":1224},[624],[568,5093,589],{"className":5094},[619,620],[568,5096],{"className":5097,"style":1224},[624],[568,5099,2139],{"className":5100},[619,620],[568,5102,589],{"className":5103},[619,620],[568,5105],{"className":5106,"style":625},[624],[568,5108,593],{"className":5109},[629],[568,5111],{"className":5112,"style":625},[624],[568,5114,5116,5119,5122,5125,5128,5131,5134],{"className":5115},[610],[568,5117],{"className":5118,"style":692},[614],[568,5120,4482],{"className":5121},[2194],[568,5123],{"className":5124,"style":1224},[624],[568,5126,720],{"className":5127},[619],[568,5129],{"className":5130,"style":625},[624],[568,5132,1121],{"className":5133},[629],[568,5135],{"className":5136,"style":625},[624],[568,5138,5140,5143,5146,5152],{"className":5139},[610],[568,5141],{"className":5142,"style":4145},[614],[568,5144,720],{"className":5145},[619],[568,5147,5149],{"className":5148},[619],[568,5150,4156],{"className":5151},[4155],[568,5153,5155],{"className":5154},[619],"1752",[559,5157,5159],{"id":5158},"typische-fehler","Typische Fehler",[5161,5162,5163,5260,5548],"ul",{},[5164,5165,5166,5195,5196,5259],"li",{},[568,5167,5169,5182],{"className":5168},[571],[568,5170,5172],{"className":5171},[575],[577,5173,5174],{"xmlns":579},[581,5175,5176,5180],{},[584,5177,5178],{},[587,5179,733],{},[598,5181,733],{"encoding":600},[568,5183,5185],{"className":5184,"ariaHidden":606},[605],[568,5186,5188,5192],{"className":5187},[610],[568,5189],{"className":5190,"style":5191},[614],"height:0.8889em;vertical-align:-0.1944em;",[568,5193,733],{"className":5194,"style":816},[619,620]," statt ",[568,5197,5199,5217],{"className":5198},[571],[568,5200,5202],{"className":5201},[575],[577,5203,5204],{"xmlns":579},[581,5205,5206,5214],{},[584,5207,5208],{},[729,5209,5210,5212],{},[587,5211,733],{},[591,5213,738],{"mathvariant":736,"lspace":737,"rspace":737},[598,5215,5216],{"encoding":600},"f'",[568,5218,5220],{"className":5219,"ariaHidden":606},[605],[568,5221,5223,5227],{"className":5222},[610],[568,5224],{"className":5225,"style":5226},[614],"height:0.9463em;vertical-align:-0.1944em;",[568,5228,5230,5233],{"className":5229},[619],[568,5231,733],{"className":5232,"style":816},[619,620],[568,5234,5236],{"className":5235},[820],[568,5237,5239],{"className":5238},[767],[568,5240,5242],{"className":5241},[772],[568,5243,5245],{"className":5244,"style":1196},[776],[568,5246,5247,5250],{"style":1199},[568,5248],{"className":5249,"style":837},[786],[568,5251,5253],{"className":5252},[841,842,843,844],[568,5254,5256],{"className":5255},[619,844],[568,5257,738],{"className":5258},[619,844]," in die Formel einsetzen.",[5164,5261,5262,5263,5547],{},"Die Wurzel auseinanderziehen: ",[568,5264,5266,5313],{"className":5265},[571],[568,5267,5269],{"className":5268},[575],[577,5270,5271],{"xmlns":579},[581,5272,5273,5310],{},[584,5274,5275,5297,5300,5302,5304],{},[713,5276,5277],{},[584,5278,5279,5281,5283,5285,5291],{},[718,5280,720],{},[591,5282,723],{},[591,5284,727],{"stretchy":726},[729,5286,5287,5289],{},[587,5288,733],{},[591,5290,738],{"mathvariant":736,"lspace":737,"rspace":737},[729,5292,5293,5295],{},[591,5294,743],{"stretchy":726},[718,5296,746],{},[591,5298,5299],{"mathvariant":736},"≠",[718,5301,720],{},[591,5303,723],{},[729,5305,5306,5308],{},[587,5307,733],{},[591,5309,738],{"mathvariant":736,"lspace":737,"rspace":737},[598,5311,5312],{"encoding":600},"\\sqrt{1 + (f')^2} \\neq 1 + f'",[568,5314,5316,5491,5509],{"className":5315,"ariaHidden":606},[605],[568,5317,5319,5322,5442,5445,5488],{"className":5318},[610],[568,5320],{"className":5321,"style":759},[614],[568,5323,5325],{"className":5324},[619,763],[568,5326,5328,5434],{"className":5327},[767,768],[568,5329,5331,5431],{"className":5330},[772],[568,5332,5334,5419],{"className":5333,"style":777},[776],[568,5335,5337,5340],{"className":5336,"style":782},[781],[568,5338],{"className":5339,"style":787},[786],[568,5341,5343,5346,5349,5352,5355,5358,5390],{"className":5342,"style":791},[619],[568,5344,720],{"className":5345},[619],[568,5347],{"className":5348,"style":798},[624],[568,5350,723],{"className":5351},[802],[568,5353],{"className":5354,"style":798},[624],[568,5356,727],{"className":5357},[809],[568,5359,5361,5364],{"className":5360},[619],[568,5362,733],{"className":5363,"style":816},[619,620],[568,5365,5367],{"className":5366},[820],[568,5368,5370],{"className":5369},[767],[568,5371,5373],{"className":5372},[772],[568,5374,5376],{"className":5375,"style":830},[776],[568,5377,5378,5381],{"style":833},[568,5379],{"className":5380,"style":837},[786],[568,5382,5384],{"className":5383},[841,842,843,844],[568,5385,5387],{"className":5386},[619,844],[568,5388,738],{"className":5389},[619,844],[568,5391,5393,5396],{"className":5392},[854],[568,5394,743],{"className":5395},[854],[568,5397,5399],{"className":5398},[820],[568,5400,5402],{"className":5401},[767],[568,5403,5405],{"className":5404},[772],[568,5406,5408],{"className":5407,"style":870},[776],[568,5409,5410,5413],{"style":833},[568,5411],{"className":5412,"style":837},[786],[568,5414,5416],{"className":5415},[841,842,843,844],[568,5417,746],{"className":5418},[619,844],[568,5420,5421,5424],{"style":884},[568,5422],{"className":5423,"style":787},[786],[568,5425,5427],{"className":5426,"style":892},[891],[894,5428,5429],{"xmlns":896,"width":897,"height":898,"viewBox":899,"preserveAspectRatio":900},[902,5430],{"d":904},[568,5432,909],{"className":5433},[908],[568,5435,5437],{"className":5436},[772],[568,5438,5440],{"className":5439,"style":916},[776],[568,5441],{},[568,5443],{"className":5444,"style":625},[624],[568,5446,5448,5481,5485],{"className":5447},[629],[568,5449,5451],{"className":5450},[629],[568,5452,5455],{"className":5453},[619,5454],"vbox",[568,5456,5459],{"className":5457},[5458],"thinbox",[568,5460,5463,5466,5477],{"className":5461},[5462],"rlap",[568,5464],{"className":5465,"style":5191},[614],[568,5467,5470],{"className":5468},[5469],"inner",[568,5471,5473],{"className":5472},[619],[568,5474,5476],{"className":5475},[629],"",[568,5478],{"className":5479},[5480],"fix",[568,5482],{"className":5483},[624,5484],"nobreak",[568,5486,593],{"className":5487},[629],[568,5489],{"className":5490,"style":625},[624],[568,5492,5494,5497,5500,5503,5506],{"className":5493},[610],[568,5495],{"className":5496,"style":3282},[614],[568,5498,720],{"className":5499},[619],[568,5501],{"className":5502,"style":798},[624],[568,5504,723],{"className":5505},[802],[568,5507],{"className":5508,"style":798},[624],[568,5510,5512,5515],{"className":5511},[610],[568,5513],{"className":5514,"style":5226},[614],[568,5516,5518,5521],{"className":5517},[619],[568,5519,733],{"className":5520,"style":816},[619,620],[568,5522,5524],{"className":5523},[820],[568,5525,5527],{"className":5526},[767],[568,5528,5530],{"className":5529},[772],[568,5531,5533],{"className":5532,"style":1196},[776],[568,5534,5535,5538],{"style":1199},[568,5536],{"className":5537,"style":837},[786],[568,5539,5541],{"className":5540},[841,842,843,844],[568,5542,5544],{"className":5543},[619,844],[568,5545,738],{"className":5546},[619,844],".",[5164,5549,5550],{},"Ein Ergebnis kleiner als der direkte Abstand der Endpunkte nicht als Fehler erkennen. Die Bogenlänge ist immer mindestens so groß wie die Sehne.",{"title":5552,"searchDepth":5553,"depth":5553,"links":5554},"",2,[5555,5556,5557,5558],{"id":561,"depth":5553,"text":562},{"id":1060,"depth":5553,"text":1061},{"id":1703,"depth":5553,"text":1704},{"id":5158,"depth":5553,"text":5159},[5560,5562,5564,5567,5570,5573,5576,5579,5582],{"lhs":5561,"rhs":2148},"sqrt(1 + (3\u002F4)^2) * (4 - 0)",{"lhs":5563,"rhs":2148},"sqrt(4^2 + 3^2)",{"lhs":5565,"rhs":5566},"sqrt(1 + sqrt(x)^2)","sqrt(1 + x)",{"lhs":5568,"rhs":5569},"2\u002F3 * (1 + 3)^(3\u002F2) - 2\u002F3 * (1 + 0)^(3\u002F2)","14\u002F3",{"lhs":5571,"rhs":5572},"round(14\u002F3, 2)","4.67",{"lhs":5574,"rhs":5575},"round(sum(map(0:999, f(k) = sqrt(1 + (k + 0.5) * 3 \u002F 1000))) * 3 \u002F 1000, 4)","round(14\u002F3, 4)",{"lhs":5577,"rhs":5578},"sqrt(1 + sinh(x)^2)","cosh(x)",{"lhs":5580,"rhs":5581},"round(sinh(1) - sinh(0), 4)","1.1752",{"lhs":5583,"rhs":5581},"round(sum(map(0:999, f(k) = sqrt(1 + sinh((k + 0.5) \u002F 1000)^2))) \u002F 1000, 4)","f \\text{ stetig differenzierbar auf } [a, b],\\ a \u003C b","\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\">stetig differenzierbar auf\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">[a, 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:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\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>a\u003C\u002Fmi>\u003Cmo><\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a < 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:0.5782em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\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>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md",{"caption":5589,"x":5590,"y":5593,"unit":5596,"axes":5597,"ticks":5598,"items":5601},"Bogenlänge L des Graphen von f(x) = ⅔ x√x zwischen a = 0 und b = 3, gestrichelt die kürzere Sehne",[5591,5592],-0.5,4,[5594,5595],-0.6,4.2,58,true,[5599,5600],10,1,[5602,5610,5613,5619,5621,5624,5626,5630,5634],{"type":5603,"from":5604,"to":5606,"style":5609},"segment",[5605,5605],0,[5607,5608],3,3.4641,"dashed",{"type":5603,"from":5611,"to":5612,"style":5609},[5607,5605],[5607,5608],{"type":5614,"expr":5615,"domain":5616,"style":5618},"function","2\u002F3 * x^(3\u002F2)",[5607,5617],3.8,"muted",{"type":5614,"expr":5615,"domain":5620},[5605,5607],{"type":5622,"at":5623},"point",[5605,5605],{"type":5622,"at":5625},[5607,5608],{"type":5627,"at":5628,"text":2106},"label",[5553,5629],1.4,{"type":5627,"at":5631,"text":596},[5632,5633],0.2,-0.35,{"type":5627,"at":5635,"text":661},[5607,5633],"\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"0 0 261 278.4\" width=\"261\" height=\"278.4\" role=\"img\" aria-label=\"Bogenlänge L des Graphen von f(x) = ⅔ x√x zwischen a = 0 und b = 3, gestrichelt die kürzere Sehne\" class=\"formula-figure-svg\">\u003Cdefs>\u003Cmarker id=\"fig-arc-length-curve-arrow\" viewBox=\"0 0 10 10\" refX=\"9\" refY=\"5\" markerWidth=\"7\" markerHeight=\"7\" orient=\"auto-start-reverse\">\u003Cpath d=\"M 0 0 L 10 5 L 0 10 z\" class=\"fig-arrowhead\"\u002F>\u003C\u002Fmarker>\u003CclipPath id=\"fig-arc-length-curve-clip\">\u003Crect x=\"0\" y=\"0\" width=\"261\" height=\"278.4\"\u002F>\u003C\u002FclipPath>\u003C\u002Fdefs>\u003Cline x1=\"0\" y1=\"243.6\" x2=\"261\" y2=\"243.6\" class=\"fig-axis\" marker-end=\"url(#fig-arc-length-curve-arrow)\"\u002F>\u003Cline x1=\"29\" y1=\"278.4\" x2=\"29\" y2=\"0\" class=\"fig-axis\" marker-end=\"url(#fig-arc-length-curve-arrow)\"\u002F>\u003Cline x1=\"26\" y1=\"185.6\" x2=\"32\" y2=\"185.6\" class=\"fig-axis\"\u002F>\u003Ctext x=\"23\" y=\"178.6\" text-anchor=\"end\" dominant-baseline=\"middle\" class=\"fig-tick\">1\u003C\u002Ftext>\u003Cline x1=\"26\" y1=\"127.6\" x2=\"32\" y2=\"127.6\" class=\"fig-axis\"\u002F>\u003Ctext x=\"23\" y=\"120.6\" text-anchor=\"end\" dominant-baseline=\"middle\" class=\"fig-tick\">2\u003C\u002Ftext>\u003Cline x1=\"26\" y1=\"69.6\" x2=\"32\" y2=\"69.6\" class=\"fig-axis\"\u002F>\u003Ctext x=\"23\" y=\"62.599999999999994\" text-anchor=\"end\" dominant-baseline=\"middle\" class=\"fig-tick\">3\u003C\u002Ftext>\u003Ctext x=\"255\" y=\"231.6\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">x\u003C\u002Ftext>\u003Ctext x=\"41\" y=\"8\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">y\u003C\u002Ftext>\u003Cline x1=\"29\" y1=\"243.6\" x2=\"203\" y2=\"42.68\" class=\"fig-dashed\"\u002F>\u003Cline x1=\"203\" y1=\"243.6\" x2=\"203\" y2=\"42.68\" class=\"fig-dashed\"\u002F>\u003Cpath d=\"M 203 42.68 L 203.19 42.35 L 203.39 42.01 L 203.58 41.68 L 203.77 41.34 L 203.97 41.01 L 204.16 40.67 L 204.35 40.33 L 204.55 40 L 204.74 39.66 L 204.93 39.32 L 205.13 38.99 L 205.32 38.65 L 205.51 38.31 L 205.71 37.98 L 205.9 37.64 L 206.09 37.3 L 206.29 36.96 L 206.48 36.62 L 206.67 36.29 L 206.87 35.95 L 207.06 35.61 L 207.25 35.27 L 207.45 34.93 L 207.64 34.59 L 207.83 34.25 L 208.03 33.91 L 208.22 33.57 L 208.41 33.23 L 208.61 32.89 L 208.8 32.55 L 208.99 32.21 L 209.19 31.87 L 209.38 31.53 L 209.57 31.19 L 209.77 30.85 L 209.96 30.51 L 210.15 30.17 L 210.35 29.82 L 210.54 29.48 L 210.73 29.14 L 210.93 28.8 L 211.12 28.46 L 211.31 28.11 L 211.51 27.77 L 211.7 27.43 L 211.89 27.08 L 212.09 26.74 L 212.28 26.4 L 212.47 26.05 L 212.67 25.71 L 212.86 25.36 L 213.05 25.02 L 213.25 24.68 L 213.44 24.33 L 213.63 23.99 L 213.83 23.64 L 214.02 23.3 L 214.21 22.95 L 214.41 22.6 L 214.6 22.26 L 214.79 21.91 L 214.99 21.57 L 215.18 21.22 L 215.37 20.87 L 215.57 20.53 L 215.76 20.18 L 215.95 19.83 L 216.15 19.49 L 216.34 19.14 L 216.53 18.79 L 216.73 18.44 L 216.92 18.1 L 217.11 17.75 L 217.31 17.4 L 217.5 17.05 L 217.69 16.7 L 217.89 16.35 L 218.08 16 L 218.27 15.66 L 218.47 15.31 L 218.66 14.96 L 218.85 14.61 L 219.05 14.26 L 219.24 13.91 L 219.43 13.56 L 219.63 13.21 L 219.82 12.86 L 220.01 12.51 L 220.21 12.15 L 220.4 11.8 L 220.59 11.45 L 220.79 11.1 L 220.98 10.75 L 221.17 10.4 L 221.37 10.04 L 221.56 9.69 L 221.75 9.34 L 221.95 8.99 L 222.14 8.64 L 222.33 8.28 L 222.53 7.93 L 222.72 7.58 L 222.91 7.22 L 223.11 6.87 L 223.3 6.52 L 223.49 6.16 L 223.69 5.81 L 223.88 5.45 L 224.07 5.1 L 224.27 4.74 L 224.46 4.39 L 224.65 4.03 L 224.85 3.68 L 225.04 3.32 L 225.23 2.97 L 225.43 2.61 L 225.62 2.26 L 225.81 1.9 L 226.01 1.54 L 226.2 1.19 L 226.39 0.83 L 226.59 0.47 L 226.78 0.12 L 226.97 -0.24 L 227.17 -0.6 L 227.36 -0.95 L 227.55 -1.31 L 227.75 -1.67 L 227.94 -2.03 L 228.13 -2.39 L 228.33 -2.74 L 228.52 -3.1 L 228.71 -3.46 L 228.91 -3.82 L 229.1 -4.18 L 229.29 -4.54 L 229.49 -4.9 L 229.68 -5.26 L 229.87 -5.62 L 230.07 -5.98 L 230.26 -6.34 L 230.45 -6.7 L 230.65 -7.06 L 230.84 -7.42 L 231.03 -7.78 L 231.23 -8.14 L 231.42 -8.5 L 231.61 -8.86 L 231.81 -9.22 L 232 -9.59 L 232.19 -9.95 L 232.39 -10.31 L 232.58 -10.67 L 232.77 -11.03 L 232.97 -11.4 L 233.16 -11.76 L 233.35 -12.12 L 233.55 -12.48 L 233.74 -12.85 L 233.93 -13.21 L 234.13 -13.57 L 234.32 -13.94 L 234.51 -14.3 L 234.71 -14.67 L 234.9 -15.03 L 235.09 -15.39 L 235.29 -15.76 L 235.48 -16.12 L 235.67 -16.49 L 235.87 -16.85 L 236.06 -17.22 L 236.25 -17.58 L 236.45 -17.95 L 236.64 -18.32 L 236.83 -18.68 L 237.03 -19.05 L 237.22 -19.41 L 237.41 -19.78 L 237.61 -20.15 L 237.8 -20.51 L 237.99 -20.88 L 238.19 -21.25 L 238.38 -21.61 L 238.57 -21.98 L 238.77 -22.35 L 238.96 -22.72 L 239.15 -23.09 L 239.35 -23.45 L 239.54 -23.82 L 239.73 -24.19 L 239.93 -24.56 L 240.12 -24.93 L 240.31 -25.3 L 240.51 -25.67 L 240.7 -26.03 L 240.89 -26.4 L 241.09 -26.77 L 241.28 -27.14 L 241.47 -27.51 L 241.67 -27.88 L 241.86 -28.25 L 242.05 -28.62 L 242.25 -29 L 242.44 -29.37 L 242.63 -29.74 L 242.83 -30.11 L 243.02 -30.48 L 243.21 -30.85 L 243.41 -31.22 L 243.6 -31.59 L 243.79 -31.97 L 243.99 -32.34 L 244.18 -32.71 L 244.37 -33.08 L 244.57 -33.46 L 244.76 -33.83 L 244.95 -34.2 L 245.15 -34.57 L 245.34 -34.95 L 245.53 -35.32 L 245.73 -35.7 L 245.92 -36.07 L 246.11 -36.44 L 246.31 -36.82 L 246.5 -37.19 L 246.69 -37.57 L 246.89 -37.94 L 247.08 -38.32 L 247.27 -38.69 L 247.47 -39.07 L 247.66 -39.44 L 247.85 -39.82 L 248.05 -40.19 L 248.24 -40.57 L 248.43 -40.94 L 248.63 -41.32 L 248.82 -41.7 L 249.01 -42.07 L 249.21 -42.45 L 249.4 -42.83\" class=\"fig-muted fig-graph\" clip-path=\"url(#fig-arc-length-curve-clip)\"\u002F>\u003Cpath d=\"M 29 243.6 L 29.73 243.55 L 30.45 243.45 L 31.17 243.32 L 31.9 243.17 L 32.63 243 L 33.35 242.81 L 34.08 242.6 L 34.8 242.38 L 35.53 242.14 L 36.25 241.89 L 36.97 241.63 L 37.7 241.35 L 38.42 241.07 L 39.15 240.77 L 39.88 240.46 L 40.6 240.14 L 41.33 239.81 L 42.05 239.47 L 42.78 239.12 L 43.5 238.77 L 44.22 238.4 L 44.95 238.02 L 45.68 237.64 L 46.4 237.25 L 47.13 236.85 L 47.85 236.44 L 48.58 236.02 L 49.3 235.59 L 50.03 235.16 L 50.75 234.72 L 51.47 234.27 L 52.2 233.82 L 52.93 233.36 L 53.65 232.89 L 54.38 232.41 L 55.1 231.93 L 55.83 231.44 L 56.55 230.94 L 57.28 230.44 L 58 229.93 L 58.72 229.41 L 59.45 228.89 L 60.18 228.36 L 60.9 227.83 L 61.63 227.29 L 62.35 226.74 L 63.08 226.19 L 63.8 225.63 L 64.53 225.06 L 65.25 224.49 L 65.97 223.92 L 66.7 223.34 L 67.43 222.75 L 68.15 222.16 L 68.88 221.56 L 69.6 220.95 L 70.32 220.35 L 71.05 219.73 L 71.78 219.11 L 72.5 218.49 L 73.22 217.85 L 73.95 217.22 L 74.68 216.58 L 75.4 215.93 L 76.13 215.28 L 76.85 214.63 L 77.57 213.96 L 78.3 213.3 L 79.03 212.63 L 79.75 211.95 L 80.47 211.27 L 81.2 210.59 L 81.93 209.9 L 82.65 209.2 L 83.38 208.5 L 84.1 207.8 L 84.82 207.09 L 85.55 206.37 L 86.28 205.66 L 87 204.93 L 87.73 204.21 L 88.45 203.47 L 89.18 202.74 L 89.9 202 L 90.63 201.25 L 91.35 200.5 L 92.07 199.75 L 92.8 198.99 L 93.53 198.23 L 94.25 197.46 L 94.98 196.69 L 95.7 195.91 L 96.43 195.14 L 97.15 194.35 L 97.88 193.56 L 98.6 192.77 L 99.32 191.98 L 100.05 191.17 L 100.78 190.37 L 101.5 189.56 L 102.23 188.75 L 102.95 187.93 L 103.68 187.11 L 104.4 186.29 L 105.13 185.46 L 105.85 184.63 L 106.57 183.79 L 107.3 182.95 L 108.03 182.1 L 108.75 181.26 L 109.48 180.4 L 110.2 179.55 L 110.93 178.69 L 111.65 177.83 L 112.38 176.96 L 113.1 176.09 L 113.82 175.21 L 114.55 174.33 L 115.28 173.45 L 116 172.56 L 116.73 171.68 L 117.45 170.78 L 118.18 169.88 L 118.9 168.98 L 119.63 168.08 L 120.35 167.17 L 121.07 166.26 L 121.8 165.34 L 122.53 164.43 L 123.25 163.5 L 123.98 162.58 L 124.7 161.65 L 125.43 160.71 L 126.15 159.78 L 126.88 158.84 L 127.6 157.89 L 128.32 156.95 L 129.05 156 L 129.77 155.04 L 130.5 154.09 L 131.23 153.12 L 131.95 152.16 L 132.68 151.19 L 133.4 150.22 L 134.13 149.25 L 134.85 148.27 L 135.57 147.29 L 136.3 146.3 L 137.02 145.32 L 137.75 144.33 L 138.48 143.33 L 139.2 142.33 L 139.93 141.33 L 140.65 140.33 L 141.38 139.32 L 142.1 138.31 L 142.82 137.3 L 143.55 136.28 L 144.27 135.26 L 145 134.23 L 145.73 133.21 L 146.45 132.18 L 147.18 131.14 L 147.9 130.11 L 148.63 129.07 L 149.35 128.03 L 150.07 126.98 L 150.8 125.93 L 151.52 124.88 L 152.25 123.82 L 152.98 122.76 L 153.7 121.7 L 154.43 120.64 L 155.15 119.57 L 155.88 118.5 L 156.6 117.43 L 157.32 116.35 L 158.05 115.27 L 158.77 114.19 L 159.5 113.1 L 160.23 112.01 L 160.95 110.92 L 161.68 109.82 L 162.4 108.73 L 163.13 107.62 L 163.85 106.52 L 164.58 105.41 L 165.3 104.3 L 166.02 103.19 L 166.75 102.08 L 167.48 100.96 L 168.2 99.83 L 168.93 98.71 L 169.65 97.58 L 170.38 96.45 L 171.1 95.32 L 171.83 94.18 L 172.55 93.04 L 173.27 91.9 L 174 90.76 L 174.73 89.61 L 175.45 88.46 L 176.18 87.3 L 176.9 86.15 L 177.63 84.99 L 178.35 83.83 L 179.08 82.66 L 179.8 81.49 L 180.52 80.32 L 181.25 79.15 L 181.98 77.98 L 182.7 76.8 L 183.43 75.61 L 184.15 74.43 L 184.88 73.24 L 185.6 72.05 L 186.33 70.86 L 187.05 69.67 L 187.77 68.47 L 188.5 67.27 L 189.23 66.06 L 189.95 64.86 L 190.68 63.65 L 191.4 62.44 L 192.13 61.22 L 192.85 60 L 193.58 58.78 L 194.3 57.56 L 195.02 56.34 L 195.75 55.11 L 196.48 53.88 L 197.2 52.64 L 197.93 51.41 L 198.65 50.17 L 199.38 48.93 L 200.1 47.68 L 200.83 46.44 L 201.55 45.19 L 202.27 43.94 L 203 42.68\" class=\"fig-main fig-graph\" clip-path=\"url(#fig-arc-length-curve-clip)\"\u002F>\u003Ccircle cx=\"29\" cy=\"243.6\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ccircle cx=\"203\" cy=\"42.68\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"145\" y=\"162.4\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">L\u003C\u002Ftext>\u003Ctext x=\"40.6\" y=\"263.9\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">a\u003C\u002Ftext>\u003Ctext x=\"203\" y=\"263.9\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">b\u003C\u002Ftext>\u003C\u002Fsvg>","arc-length-curve","L = \\int_a^b \\sqrt{1 + \\big(f'(x)\\big)^2}\\,dx","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi>L\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmsubsup>\u003Cmo>∫\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>b\u003C\u002Fmi>\u003C\u002Fmsubsup>\u003Cmsqrt>\u003Cmrow>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>+\u003C\u002Fmo>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">(\u003C\u002Fmo>\u003Cmsup>\u003Cmi>f\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmsup>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">)\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003C\u002Fmsqrt>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi>d\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">L = \\int_a^b \\sqrt{1 + \\big(f'(x)\\big)^2}\\,dx\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:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\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>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:2.511em;vertical-align:-0.9119em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop op-symbol large-op\" style=\"margin-right:0.4445em;position:relative;top:-0.0011em;\">∫\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.599em;\">\u003Cspan style=\"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">a\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.8129em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">b\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9119em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord sqrt\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3859em;\">\u003Cspan class=\"svg-align\" style=\"top:-3.8em;\">\u003Cspan class=\"pstrut\" style=\"height:3.8em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"padding-left:1em;\">\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6779em;\">\u003Cspan style=\"top:-2.989em;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 mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.054em;\">\u003Cspan style=\"top:-3.3029em;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\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.3459em;\">\u003Cspan class=\"pstrut\" style=\"height:3.8em;\">\u003C\u002Fspan>\u003Cspan class=\"hide-tail\" style=\"min-width:1.02em;height:1.88em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"400em\" height=\"1.88em\" viewBox=\"0 0 400000 1944\" preserveAspectRatio=\"xMinYMin slice\">\u003Cpath d=\"M983 90\nl0 -0\nc4,-6.7,10,-10,18,-10 H400000v40\nH1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7\ns-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744\nc-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30\nc26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722\nc56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5\nc53.7,-170.3,84.5,-266.8,92.5,-289.5z\nM1001 80h400000v40h-400000z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.4541em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},"\u002Fde\u002Fformulas\u002Fmath\u002Fcalculus\u002Fbogenlaenge-eines-graphen",[],[5645,5646,5647,5648],"pythagorean-theorem","distance-two-points","arc-length","fundamental-theorem",{"title":548,"description":5552},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fbogenlaenge-eines-graphen","Die Länge eines Funktionsgraphen ist das Integral über die Wurzel aus eins plus Ableitung zum Quadrat.",[5653,5654,5655,5656],"bogenlänge","kurvenlänge","integral","ableitung",[5658,5661,5664],{"symbol":2106,"meaning":5659,"symbolHtml":5660},"Länge des Graphen zwischen x = a und x = b","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>L\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">L\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:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":5216,"meaning":5662,"symbolHtml":5663},"Ableitung der Funktion f","\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>\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:0.9463em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;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 mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":5665,"meaning":5666,"symbolHtml":5667},"a, b","Grenzen des Abschnitts","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>b\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a, 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=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","bxYMM7h2XTOsZt2BXWOkelIxtmk71WaYrFeNEkV7jps",[5670,5673,5676,5679],{"stem":5671,"title":5672,"category":101},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fhauptsatz-der-differential-und-integralrechnung","Hauptsatz der Differential- und Integralrechnung",{"stem":5674,"title":5675,"category":208},"de\u002Fformulas\u002Fmath\u002Ffunctions\u002Fabstand-zweier-punkte","Abstand zweier Punkte",{"stem":5677,"title":5678,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fkreisbogen","Länge eines Kreisbogens",{"stem":5680,"title":5681,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fsatz-des-pythagoras","Satz des Pythagoras",1793824281306]