{"text_format":"2","workbook":"xlsx","n_sheets":10,"sheets":["Решение уравнений1","Решение уравнений2","Решение уравнений3","Решение уравнений4","Интерполирование","Формулы прямоугольников","Формула трапеций","Формула парабол","Методы Эйлера","Методы Рунге-Кутты"],"not_rendered":{"charts":1,"images":8,"macros":0,"external_links":2,"defined_names":0,"comments":0,"pivot_tables":0},"unopened":[],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"run_cost":"one","rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"Решение уравнений1","index":0,"n_rows":41,"n_cols":9,"rows_kept":41,"cut":false,"hidden":false,"merged":[]}
["Пример решен методом дихотомии, но его можно использовать для проверки всех методов для решения уравнений"]
["Пример № 1. Уточните корни уравнения x^3+x-1=0 на [-10;10] c шагом h=1 и с точностью до 0,001"]
["x","f(x)","f'(x)=3x^2+1",null,null,"[0;1]","f'(x)"]
[-10,{"$f":"A4^3+A4-1","v":-1011},null,null,null,0,{"$f":"3*F4^2+1","v":1}]
[-9,{"$f":"A5^3+A5-1","v":-739},null,null,null,0.1,{"$f":"3*F5^2+1","v":1.03}]
[-8,-521,null,null,null,0.2,1.1200000000000001]
[-7,-351,null,null,null,0.3,1.27]
[-6,-223,null,null,null,0.4,1.48]
[-5,-131,null,null,null,0.5,1.75]
[-4,-69,null,null,null,0.6,2.08]
[-3,-31,null,null,null,0.7,2.4699999999999998]
[-2,-11,null,null,null,0.8,2.9200000000000004]
[-1,-3,null,null,null,0.9,3.43]
[0,-1,null,null,null,1,4]
[1,1,null,null,null,"знак сохраняется"]
[2,9,null,null,null,"[0;1] - отрезок изоляции "]
[3,29]
[4,67]
[5,129]
[6,221]
[7,349]
[8,519]
[9,737]
[10,1009]
["Подозрительные на отрезки изоляции: [0;1]"]
[]
["Применяем метод дихотомии к отрезку [0;1]"]
["a","b","с","f ( c )","длина [a;b]","f(a)","f(b)"]
[0,1,{"$f":"(A29+B29)/2","v":0.5},{"$f":"C29^3+C29-1","v":-0.375},{"$f":"B29-A29","v":1},{"$f":"A29^3+A29-1","v":-1},{"$f":"B29^3+B29-1","v":1},{"$f":"IF(E29<0.002,\"с - корень\")","v":false}," - так как f(с)<0, а f(b)>0, то в следующей строке b оставляем прежним, а значение a заменяем на c"]
[{"$f":"C29","v":0.5},{"$f":"B29","v":1},0.75,0.171875,0.5,-0.375,1,false," - так как f(с)>0, а f(a)<0, то в следующей строке а оставляем прежним, а значение b заменяем на c"]
[{"$f":"A30","v":0.5},{"$f":"C30","v":0.75},0.625,-0.130859375,0.25,-0.375,0.171875,false]
[{"$f":"C31","v":0.625},{"$f":"B31","v":0.75},0.6875,1.2451171875E-2,0.125,-0.130859375,0.171875,false]
[{"$f":"A32","v":0.625},{"$f":"C32","v":0.6875},0.65625,-6.1126708984375E-2,6.25E-2,-0.130859375,1.2451171875E-2,false]
[{"$f":"C33","v":0.65625},{"$f":"B33","v":0.6875},0.671875,-2.4829864501953125E-2,3.125E-2,-6.1126708984375E-2,1.2451171875E-2,false]
[{"$f":"C34","v":0.671875},{"$f":"B34","v":0.6875},0.6796875,-6.3138008117675781E-3,1.5625E-2,-2.4829864501953125E-2,1.2451171875E-2,false]
[{"$f":"C35","v":0.6796875},{"$f":"B35","v":0.6875},0.68359375,3.0373930931091309E-3,7.8125E-3,-6.3138008117675781E-3,1.2451171875E-2,false]
[{"$f":"A36","v":0.6796875},{"$f":"C36","v":0.68359375},0.681640625,-1.6460046172142029E-3,3.90625E-3,-6.3138008117675781E-3,3.0373930931091309E-3,false]
[{"$f":"C37","v":0.681640625},{"$f":"B37","v":0.68359375},0.6826171875,6.9374125450849533E-4,1.953125E-3,-1.6460046172142029E-3,3.0373930931091309E-3,"с - корень"," - длина отрезка стала меньше 2k=0,002, значит значение, записанное в ячейке, соответствующей С является корнем"]
[null,null,"корень, и так как точность k=0,001, то результат нужно округлить до 3-х знаков после запятой"]
[]
["Ответ: x=0,683"]

{"sheet":"Решение уравнений2","index":1,"n_rows":55,"n_cols":10,"rows_kept":55,"cut":false,"hidden":false,"merged":[]}
["Пример решен методом дихотомии, но его можно использовать для проверки всех методов для решения уравнений"]
["Пример № 2. Уточните корни уравнения cos2x-3x^2+1=0 на [-10;10] c шагом h=1 и с точностью до 0,001"]
["x","f(x)","f'(x)=-2sin2x-6x",null,null,"[-1;0]","f'(x)",null,"[0;1]","f'(x)"]
[-10,{"$f":"COS(2*A4)-3*A4^2+1","v":-298.59191793818661},null,null,null,-1,{"$f":"-2*SIN(2*F4)-6*F4","v":7.8185948536513639},null,0,{"$f":"-2*SIN(2*I4)-6*I4","v":0}]
[-9,{"$f":"COS(2*A5)-3*A5^2+1","v":-241.33968329175593},null,null,null,-0.9,{"$f":"-2*SIN(2*F5)-6*F5","v":7.3476952617563907},null,0.1,{"$f":"-2*SIN(2*I5)-6*I5","v":-0.99733866159012252}]
[-8,-191.95765948032337,null,null,null,-0.8,6.7991472060830107,null,0.2,-1.9788366846173013]
[-7,-145.86326278179217,null,null,null,-0.7,6.1708994599769191,null,0.3,-2.9292849467900703]
[-6,-106.1561460412675,null,null,null,-0.6,5.4640781719344522,null,0.4,-3.8347121817990457]
[-5,-74.839071529076449,null,null,null,-0.5,4.6829419696157935,null,0.5,-4.6829419696157935]
[-4,-47.145500033808617,null,null,null,-0.4,3.8347121817990457,null,0.6,-5.4640781719344522]
[-3,-25.039829713349633,null,null,null,-0.3,2.9292849467900703,null,0.7,-6.1708994599769191]
[-2,-11.653643620863612,null,null,null,-0.2,1.9788366846173013,null,0.8,-6.7991472060830107]
[-1,-2.4161468365471426,null,null,null,-0.1,0.99733866159012252,null,0.9,-7.3476952617563907]
[0,2,null,null,null,0,0,null,1,-7.8185948536513639]
[1,-2.4161468365471426,null,null,null,"знак сохраняется",null,null,"знак сохраняется"]
[2,-11.653643620863612,null,null,null,"[-1;-0,1] - отрезок изоляции ",null,null,"[0,1;1] - отрезок изоляции "]
[3,-25.039829713349633]
[4,-47.145500033808617]
[5,-74.839071529076449]
[6,-106.1561460412675]
[7,-145.86326278179217]
[8,-191.95765948032337]
[9,-241.33968329175593]
[10,-298.59191793818661]
["Подозрительные на отрезки изоляции: [-1;0], [0;1]"]
[]
["Применяем метод дихотомии к отрезку [-1;-0,1]"]
["a","b","с","f ( c )","длина [a;b]","f(a)","f(b)"]
[-1,-0.1,{"$f":"(A29+B29)/2","v":-0.55000000000000004},{"$f":"COS(2*C29)-3*C29^2+1","v":0.54609612142557706},{"$f":"B29-A29","v":0.9},{"$f":"COS(2*A29)-3*A29^2+1","v":-2.4161468365471426},{"$f":"COS(2*B29)-3*B29^2+1","v":1.9500665778412416},{"$f":"IF(E29<0.002,\"с - корень\")","v":false}," - так как f(с)>0, а f(a)<0, то в следующей строке а оставляем прежним, а значение b заменяем на c"]
[{"$f":"A29","v":-1},{"$f":"C29","v":-0.55000000000000004},-0.77500000000000002,-0.78108017219690784,0.44999999999999996,-2.4161468365471426,0.54609612142557706,{"$f":"IF(E30<0.002,\"с - корень\")","v":false}," - так как f(с)<0, а f(b)>0, то в следующей строке b оставляем прежним, а значение a заменяем на c"]
[{"$f":"C30","v":-0.77500000000000002},{"$f":"B30","v":-0.55000000000000004},-0.66250000000000009,-7.3389955853362387E-2,0.22499999999999998,-0.78108017219690784,0.54609612142557706,false]
[{"$f":"C31","v":-0.66250000000000009},{"$f":"B31","v":-0.55000000000000004},-0.60625000000000007,0.24806207296689164,0.11250000000000004,-7.3389955853362387E-2,0.54609612142557706,false]
[{"$f":"A32","v":-0.66250000000000009},{"$f":"C32","v":-0.60625000000000007},-0.63437500000000013,9.0179595339491936E-2,5.6250000000000022E-2,-7.3389955853362387E-2,0.24806207296689164,false]
[{"$f":"A33","v":-0.66250000000000009},{"$f":"C33","v":-0.63437500000000013},-0.64843750000000011,9.0950626887962294E-3,2.8124999999999956E-2,-7.3389955853362387E-2,9.0179595339491936E-2,false]
[{"$f":"A34","v":-0.66250000000000009},{"$f":"C34","v":-0.64843750000000011},-0.65546875000000004,-3.1973725719464108E-2,1.4062499999999978E-2,-7.3389955853362387E-2,9.0950626887962294E-3,false]
[{"$f":"C35","v":-0.65546875000000004},{"$f":"B35","v":-0.64843750000000011},-0.65195312500000013,-1.1395733408418041E-2,7.0312499999999334E-3,-3.1973725719464108E-2,9.0950626887962294E-3,false]
[{"$f":"C36","v":-0.65195312500000013},{"$f":"B36","v":-0.64843750000000011},-0.65019531250000018,-1.1394148814789506E-3,3.5156250000000222E-3,-1.1395733408418041E-2,9.0950626887962294E-3,false]
[{"$f":"C37","v":-0.65019531250000018},{"$f":"B37","v":-0.64843750000000011},-0.64931640625000009,3.9805566399753189E-3,1.7578125000000666E-3,-1.1394148814789506E-3,9.0950626887962294E-3,"с - корень"," - длина отрезка стала меньше 2k=0,002, значит значение, записанное в ячейке, соответствующей С является корнем"]
[null,null,"корень, и так как точность k=0,001, то результат нужно округлить до 3-х знаков после запятой"]
[]
["Применяем метод дихотомии к отрезку [0,1;1]"]
["a","b","с","f ( c )","длина [a;b]","f(a)","f(b)"]
[0.1,1,{"$f":"(A43+B43)/2","v":0.55000000000000004},{"$f":"COS(2*C43)-3*C43^2+1","v":0.54609612142557706},{"$f":"B43-A43","v":0.9},{"$f":"COS(2*A43)-3*A43^2+1","v":1.9500665778412416},{"$f":"COS(2*B43)-3*B43^2+1","v":-2.4161468365471426},{"$f":"IF(E43<0.002,\"с - корень\")","v":false}]
[{"$f":"C43","v":0.55000000000000004},{"$f":"B43","v":1},0.77500000000000002,-0.78108017219690784,0.44999999999999996,0.54609612142557706,-2.4161468365471426,false]
[{"$f":"A44","v":0.55000000000000004},{"$f":"C44","v":0.77500000000000002},0.66250000000000009,-7.3389955853362387E-2,0.22499999999999998,0.54609612142557706,-0.78108017219690784,false]
[{"$f":"A45","v":0.55000000000000004},{"$f":"C45","v":0.66250000000000009},0.60625000000000007,0.24806207296689164,0.11250000000000004,0.54609612142557706,-7.3389955853362387E-2,false]
[{"$f":"C46","v":0.60625000000000007},{"$f":"B46","v":0.66250000000000009},0.63437500000000013,9.0179595339491936E-2,5.6250000000000022E-2,0.24806207296689164,-7.3389955853362387E-2,false]
[{"$f":"C47","v":0.63437500000000013},{"$f":"B47","v":0.66250000000000009},0.64843750000000011,9.0950626887962294E-3,2.8124999999999956E-2,9.0179595339491936E-2,-7.3389955853362387E-2,false]
[{"$f":"C48","v":0.64843750000000011},{"$f":"B48","v":0.66250000000000009},0.65546875000000004,-3.1973725719464108E-2,1.4062499999999978E-2,9.0950626887962294E-3,-7.3389955853362387E-2,false]
[{"$f":"A49","v":0.64843750000000011},{"$f":"C49","v":0.65546875000000004},0.65195312500000013,-1.1395733408418041E-2,7.0312499999999334E-3,9.0950626887962294E-3,-3.1973725719464108E-2,false]
[{"$f":"A50","v":0.64843750000000011},{"$f":"C50","v":0.65195312500000013},0.65019531250000018,-1.1394148814789506E-3,3.5156250000000222E-3,9.0950626887962294E-3,-1.1395733408418041E-2,false]
[{"$f":"A51","v":0.64843750000000011},{"$f":"C51","v":0.65019531250000018},0.64931640625000009,3.9805566399753189E-3,1.7578125000000666E-3,9.0950626887962294E-3,-1.1394148814789506E-3,"с - корень"]
[null,null,"корень"]
[]
["Ответ: x=-0,649; x=0,649"]

{"sheet":"Решение уравнений3","index":2,"n_rows":63,"n_cols":9,"rows_kept":63,"cut":false,"hidden":false,"merged":[]}
["Пример решен методом хорд, но его можно использовать для проверки всех методов для решения уравнений"]
["Пример 1. Найти корни уравнения 3x^2-2x-1=0 на [-10;10] с h=1 и точностью k=0,0000001 "]
["x","f(x)","f'(x)=6x-2",{"$nulls":4},"[-1;0]","f'(x)"]
[-10,{"$f":"3*A4^2-2*A4-1","v":319},{"$nulls":5},-1,{"$f":"6*H4-2","v":-8}]
[-9,{"$f":"3*A5^2-2*A5-1","v":260},{"$nulls":5},-0.9,{"$f":"6*H5-2","v":-7.4}]
[-8,207,{"$nulls":5},-0.8,-6.8000000000000007]
[-7,160,{"$nulls":5},-0.7,-6.1999999999999993]
[-6,119,{"$nulls":5},-0.6,-5.6]
[-5,84,{"$nulls":5},-0.5,-5]
[-4,55,{"$nulls":5},-0.4,-4.4000000000000004]
[-3,32,{"$nulls":5},-0.3,-3.8]
[-2,15,{"$nulls":5},-0.2,-3.2]
[-1,4,{"$nulls":5},-0.1,-2.6]
[0,-1,{"$nulls":5},0,-2]
[1,0," - т.к. в точке х=1 f(x)=0, то x=1 - корень",{"$nulls":4},"знак сохраняется"]
[2,7,{"$nulls":5},"[-1;0] - отрезок изоляции"]
[3,20]
[4,39]
[5,64]
[6,95]
[7,132]
[8,175]
[9,224]
[10,{"$f":"3*A24^2-2*A24-1","v":279}]
["Подозрительный на отрезок изоляции: [-1;0]"]
[]
["Применяем метод хорд к отрезку [-1;0]"]
["a","b","f(a)","f(b)","x"]
[-1,0,{"$f":"3*A29^2-2*A29-1","v":4},-1,{"$f":"A29-(C29*(B29-A29))/(D29-C29)","v":-0.19999999999999996}," - после запятой должно быть знаков больше на 1, чем в точности k"]
[{"$f":"E29","v":-0.19999999999999996},{"$f":"B29","v":0},-0.4800000000000002,-1,{"$f":"A30-(C30*(B30-A30))/(D30-C30)","v":-0.38461538461538469},"действия продолжаем выполнять до того момента, пока в двух соседних ячеках у x не появятся одинаковые значения"]
[{"$f":"E30","v":-0.38461538461538469},{"$f":"B30","v":0},0.21301775147929014,-1,-0.31707317073170732]
[{"$f":"E31","v":-0.31707317073170732},{"$f":"B31","v":0},-6.4247471743010154E-2,-1,-0.33884297520661161]
[{"$f":"E32","v":-0.33884297520661161},{"$f":"B32","v":0},2.212963595382833E-2,-1,-0.33150684931506852]
[{"$f":"E33","v":-0.33150684931506852},{"$f":"B33","v":0},-7.2959279414523959E-3,-1,-0.33394327538883806]
[-0.33394327538883806,0,2.4408843099521516E-3,-1,-0.33313014324900941]
[-0.33313014324900941,0,-8.1263647866458566E-4,-1,-0.33340107712630834]
[-0.33340107712630834,0,2.709889395644538E-4,-1,-0.3333107535986452]
[-0.3333107535986452,0,-9.0317409219276001E-5,-1,-0.3333408600815147]
[-0.3333408600815147,0,3.010716268114777E-5,-1,-0.33333082443615675]
[{"$f":"E39","v":-0.33333082443615675},{"$f":"B39","v":0},-1.0035569822708368E-5,-1,-0.33333416963449042]
[{"$f":"E40","v":-0.33333416963449042},{"$f":"B40","v":0},3.3452067265127283E-6,-1,-0.33333305456651413]
[{"$f":"E41","v":-0.33333305456651413},{"$f":"B41","v":0},-1.1150670435888799E-6,-1,-0.33333342625563228]
[{"$f":"E42","v":-0.33333342625563228},{"$f":"B42","v":0},3.7168922162322815E-7,-1,-0.33333330235923658]
[{"$f":"E43","v":-0.33333330235923658},{"$f":"B43","v":0},-1.2389638415211124E-7,-1,-0.33333334365803258]
[-0.33333334365803258,0,{"$f":"3*A45^2-2*A45-1","v":4.1298797270883369E-8},-1,-0.33333332989176695]
[-0.33333332989176695,0,-1.376626546090165E-8,-1,-0.3333333344805221," - результат округлям до того количества знаков после запятой, сколько их у точности k"]
[{"$nulls":4},"корень"]
{"$rows_empty":15}
["Ответ: x=1, x=-0,3333333"]

{"sheet":"Решение уравнений4","index":3,"n_rows":16,"n_cols":2,"rows_kept":16,"cut":false,"hidden":false,"merged":[]}
["Пример решен методом хорд, но его можно использовать для проверки всех методов для решения уравнений"]
["Пример 2. Найти корни уравнения (x/2)-(ln(x)/ln(10)) на [1,3;1,4] с h=0,01 и точностью k=0,0000001 "]
["x","f(x)"]
[1.3,{"$f":"A4/2-(LN(A4)/LN(10))","v":0.5360566476931633}]
[1.31,{"$f":"A5/2-(LN(A5)/LN(10))","v":0.53772870434423581}]
[1.32,0.53942606879415012]
[1.33,0.54114835903291425]
[1.34,0.54289520163519245]
[1.35,0.54466623150499394]
[1.36,0.54646109162978251]
[1.37,0.54827943284359326]
[1.38,0.55012091359876347]
[1.39,0.55198519974590488]
[1.4,0.553871964321762]
["На данном промежутке функция не меняем знак, значит на данном отрезке корней нет"]
["Ответ: на [1,3;1,4] корней нет"]

{"sheet":"Интерполирование","index":4,"n_rows":42,"n_cols":7,"rows_kept":42,"cut":false,"hidden":false,"merged":[]}
["x","y"]
[0,1.2,null,"y=ax^5+bx^4+cx^3+dx^2+ex+f"]
[0.2,2.4]
[0.4,3.6]
[0.6,4.8]
[0.8,5.9]
[1,7.1]
[]
["x^5","x^4","x^3","x^2","x^1","x^0","y"]
[{"$f":"A2^5","v":0},{"$f":"A2^4","v":0},{"$f":"A2^3","v":0},{"$f":"A2^2","v":0},{"$f":"A2^1","v":0},1,{"$f":"B2","v":1.2}]
[{"$f":"A3^5","v":3.2000000000000019E-4},{"$f":"A3^4","v":1.6000000000000007E-3},{"$f":"A3^3","v":8.0000000000000019E-3},{"$f":"A3^2","v":4.0000000000000008E-2},{"$f":"A3^1","v":0.2},{"$f":"A3^0","v":1},{"$f":"B3","v":2.4}]
[{"$f":"A4^5","v":1.0240000000000006E-2},{"$f":"A4^4","v":2.5600000000000012E-2},{"$f":"A4^3","v":6.4000000000000015E-2},{"$f":"A4^2","v":0.16000000000000003},{"$f":"A4^1","v":0.4},{"$f":"A4^0","v":1},{"$f":"B4","v":3.6}]
[7.7759999999999996E-2,0.12959999999999999,0.216,0.36,0.6,1,4.8]
[0.32768000000000019,0.40960000000000019,0.51200000000000012,0.64000000000000012,0.8,1,5.9]
[1,1,1,1,1,1,7.1]
[]
["опр",{"$f":"MDETERM(A10:F15)","v":-1.1324620800000257E-6}]
["мобр"]
[{"$f":"MINVERSE(A10:F15)","v":-26.041666666665293},130.20833333332845,-260.41666666665924,260.41666666666055,-130.20833333333081,26.041666666666263]
[78.124999999996732,-364.58333333332189,677.08333333331609,-624.99999999998579,286.45833333332746,-52.083333333332405]
[-88.541666666663986,369.79166666665742,-614.5833333333195,510.41666666665537,-213.54166666666197,36.458333333332597]
[46.874999999999119,-160.41666666666367,222.91666666666222,-162.49999999999642,63.541666666665165,-10.41666666666643]
[-11.416666666666574,24.999999999999684,-24.999999999999535,16.666666666666295,-6.2499999999998437,0.99999999999997524]
[1,0,0,0,0,0]
[]
["a=",{"$f":"MMULT(A19:F24,G10:G15)","v":10.416666666665861}]
["b=",-23.43749999999784]
["c=",17.708333333332007]
["d=",-5.3124999999998721]
["e=",6.5249999999999773]
["f=",1.2]
[]
["y= 10 5/12x^5-23 7/16x^4+17 17/24x^3-5 5/16x^2+6 21/40x+1 1/5"]
[]
["x","y"]
[0,{"$f":"$B$26*A36^5+$B$27*A36^4+$B$28*A36^3+$B$29*A36^2+$B$30*A36+$B$31","v":1.2}]
[0.2,{"$f":"$B$26*A37^5+$B$27*A37^4+$B$28*A37^3+$B$29*A37^2+$B$30*A37+$B$31","v":2.3999999999999932}]
[0.4,3.5999999999999739]
[0.6,4.7999999999999634]
[0.8,5.9000000000000039]
[1,7.1000000000001338]
[0.1,1.8148437499999979]

{"sheet":"Формулы прямоугольников","index":5,"n_rows":34,"n_cols":9,"rows_kept":34,"cut":false,"hidden":false,"merged":[]}
["Задание: вычислить интеграл "]
["при n=20 и оценить погрешность"]
["результата"]
[]
["x","y",null,"y'",null,null,"h",{"$f":"(H7-H6)/H8","v":5.2359877559829883E-2}," - шаг интегрирования (вычисляется по формуле h=(b-a)/n)"]
[0,{"$f":"SIN(2*A6+1)","v":0.8414709848078965},"=y0",{"$f":"ABS((1/(6*$H$5))*(-11*B6+18*B7-9*B8+2*B9))","v":1.0811222042207895},null,null,"a",{"$f":"0","v":0}," - нижний предел интегрирования"]
[{"$f":"A6+$H$5","v":5.2359877559829883E-2},{"$f":"SIN(2*A7+1)","v":0.893338288440923},"=y1",0.89931017693211879,null,null,"b",{"$f":"PI()/3","v":1.0471975511965976}," - верхний предел интегрирования"]
[0.10471975511965977,0.93541799084273569,"=y2",0.70764511915219697,null,null,"n",20," - количество узлов"]
[0.15707963267948966,0.96724905798807614,"=y3",0.50822695336260326]
[0.20943951023931953,0.98848274204422126,"=y4",0.30324054671863693,null,"Инструкция по вычислению интеграла методами прямоугольников:"]
[0.26179938779914941,0.99888640232521764,"=y5",9.493177318765568E-2,null,"1. В столбце А забиваются все значения аргументов, начиная с a и далее с шагом n растягивается до b"]
[0.31415926535897931,0.99834605415192101,"=y6",0.11441709271611869,null,"2. В столбце B подставляет значения x в заданную функцию и растягиваем до конца"]
[0.36651914291880922,0.98686761769199294,"=y7",0.32251238100879537,null,"3. В ячейке С36 вычисляем интегралы по левой формуле прямоугольников: h*сумм(y0;yn-1)"]
[0.41887902047863912,0.96457685309730568,"=y8",0.52707415616505215,null,"4. В ячейке С37 вычисляем интегралы по левой формуле прямоугольников: h*сумм(y1;yn)"]
[0.47123889803846902,0.9317179826494012,"=y9",0.72586119656902959,null,"5. Для того, чтобы посчитать погрешность формул прямоугольников, необходимо воспользоваться специальной формулой:"]
[0.52359877559829893,0.88865101500906718,"=y10",0.91669554980718504,null,"6. Для нахождения максимума модуля первой производной можно воспользоваться формулой для численного"]
[0.57595865315812877,0.83584780088611355,"=y11",1.0974863947707547,null,"нахождения производной в точке с четырьмя заданными узлами: "]
[0.62831853071795862,0.77388686334425472,"=y12",1.2662529491294185,null,"Но нужно учесть, что, для того чтобы воспользоваться этой формлой, нужно найти три дополнительных значения x и y"]
[0.68067840827778847,0.70344705938135876,"=y13",1.4211461711969804,null,"(выделены желтым)"]
[0.73303828583761832,0.62530014223011943,"=y14",1.5604690184189445,null,"7. В ячейке D32 находим максимум из найденных производных"]
[0.78539816339744817,0.54030230586813977,"=y15",1.6826950405259631,null,"8. В ячейке С35 считаем погрешность"]
[0.83775804095727802,0.4493848043775428,"=y16",1.7864851036424041]
[0.89011791851710786,0.35354374893036944,"=y17",1.870702062117316]
[0.94247779607693771,0.25382919418612876,"=y18",1.9344232173300977]
[0.99483767363676756,0.15133363367321109,"=y19",1.9769504269696916]
[1.0471975511965974,4.718003020117139E-2,"=y20",1.9978177540279725]
[1.0995574287564274,-5.7490487554808831E-2]
[1.1519173063162573,-0.16153112749848061]
[1.2042771838760873,-0.26380199860671749]
[]
[null,null,"max(f'(x))=",{"$f":"MAX(D6:D26)","v":1.9978177540279725}]
["левая формула:",null,{"$f":"$H$5*SUM(B6:B25)","v":0.78968541854722585}]
["правая формула:",null,{"$f":"$H$5*SUM(B7:B26)","v":0.74809644141713738}]
["погрешность:",null,{"$f":"(((H7-H6)^2)/(2*H8))*D31","v":5.4771308049247028E-2}]

{"sheet":"Формула трапеций","index":6,"n_rows":22,"n_cols":8,"rows_kept":22,"cut":false,"hidden":false,"merged":[]}
["Задание: Вычислите определенный интеграл"]
["методом трапеций"]
["и определите погрешность результата:"]
["x","y",null,"y''"]
[{"$f":"G5","v":0},{"$f":"SIN(2*A5+1)","v":0.8414709848078965},"=y0",{"$f":"ABS((1/$G$8^2)*(2*B5-5*B6+4*B7-B8))","v":3.516530359798252},null,"a",0," - нижний предел интегрирования"]
[{"$f":"A5+$G$8","v":0.10471975511965977},{"$f":"SIN(2*A6+1)","v":0.93541799084273569},"=y1",3.9001896082564667,null,"b",{"$f":"PI()/3","v":1.0471975511965976}," - верхний предел интегрирования"]
[0.20943951023931953,0.98848274204422126,"=y2",4.1133918556477775,null,"n",10," - количество узлов"]
[0.31415926535897931,0.99834605415192101,"=y3",4.1468191407031112,null,"h",{"$f":"(G6-G5)/G7","v":0.10471975511965977}," - шаг интегрирования (вычисляется по формуле h=(b-a)/n)"]
[0.41887902047863906,0.96457685309730579,"=y4",3.9990105306638783]
[0.52359877559829882,0.8886510150090674,"=y5",3.676425971053054,null,"Инструкция по вычислению интеграла методами прямоугольников:"]
[0.62831853071795862,0.77388686334425472,"=y6",3.1931639550579791,null,"1. В столбце А забиваются все значения аргументов, начиная с a и далее с шагом n растягивается до b"]
[0.73303828583761843,0.62530014223011909,"=y7",2.5703453517263508,null,"2. В столбце B подставляем значения x в заданную функцию и растягиваем до конца"]
[0.83775804095727824,0.44938480437754241,"=y8",1.835190322638987,null,"3. В ячейке С21 вычисляем интегралы по формуле трапеций (b-a)/2n*(y0+yn+2*сумм(y1;yn-1))"]
[0.94247779607693805,0.25382919418612832,"=y9",1.0198286702317545,null,"4. Для того, чтобы посчитать погрешность формулы трапеции, необходимо воспользоваться специальной формулой:"]
[1.0471975511965979,4.7180030201170509E-2,"=y10",0.159895611254736,null,"5. Для нахождения максимума модуля первой производной можно воспользоваться формулой для численного"]
[1.1519173063162575,-0.16153112749848106,null,null,null,"нахождения второй производной в точке с четырьмя заданными узлами: "]
[1.2566370614359172,-0.36318259981410189,null,null,null,"Но нужно учесть, что, для того чтобы воспользоваться этой формлой, нужно найти три дополнительных значения x и y"]
[1.3613568165555769,-0.54896124977437832,null,null,null,"(выделены желтым)"]
[{"$nulls":5},"6. В ячейке D20 находим максимум из найденных производных"]
[null,null,"max(f''(x))=",{"$f":"MAX(D5:D15)","v":4.1468191407031112},null,"7. В ячейке С22 считаем погрешность"]
["формула трапеций",null,{"$f":"(G6-G5)/(2*G7)*(B5+B15+2*SUM(B6:B14))","v":0.76677911312290847}]
["погрешность:",null,{"$f":"(((G6-G5)^3)/(12*G7^2))*D20","v":3.9684389388828321E-3}]

{"sheet":"Формула парабол","index":7,"n_rows":34,"n_cols":11,"rows_kept":34,"cut":false,"hidden":false,"merged":[[33,0,33,1]]}
["Задание: Вычислите определенный интеграл"]
["методом парабол"]
["и определите погрешность результата:"]
[]
["a=",0,null,null,null,"a=",0]
["b=",{"$f":"PI()/3","v":1.0471975511965976},null,null,null,"b=",{"$f":"PI()/3","v":1.0471975511965976}]
["n=",10,"m=",{"$f":"B7/2","v":5},null,"n=",20,"m=",{"$f":"G7/2","v":10}]
["h=",{"$f":"(B6-B5)/B7","v":0.10471975511965977},null,null,null,"h=",{"$f":"(G6-G5)/G7","v":5.2359877559829883E-2}]
[]
["x","y",null,null,null,"x","y",null,null,null,"Метод решения интегралов методом параболпохож на предыдущие методы."]
[{"$f":"B5","v":0},{"$f":"SIN(2*A11+1)","v":0.8414709848078965},"y0",null,null,{"$f":"G5","v":0},{"$f":"SIN(2*F11+1)","v":0.8414709848078965},"y0",null,null,"Но есть несколько особенностей:"]
[{"$f":"A11+$B$8","v":0.10471975511965977},{"$f":"SIN(2*A12+1)","v":0.93541799084273569},"y1",null,null,{"$f":"F11+$G$8","v":5.2359877559829883E-2},{"$f":"SIN(2*F12+1)","v":0.893338288440923},"y1",null,null,"1. Формула применяется только для четного числа шагов"]
[{"$f":"A12+$B$8","v":0.20943951023931953},0.98848274204422126,"y2",null,null,{"$f":"F12+$G$8","v":0.10471975511965977},0.93541799084273569,"y2",null,null,"2. Нужно посчитать не только шаг (h), но и число m=n/2 (число разбиений)"]
[0.31415926535897931,0.99834605415192101,"y3",null,null,0.15707963267948966,0.96724905798807614,"y3",null,null,"2. Формула выглядит так: (б-a)/6m*(y0+yn)+2*(y2+y4+…+yn-2)+4*(y1+y3+…+yn-1)"]
[0.41887902047863906,0.96457685309730579,"y4",null,null,0.20943951023931953,0.98848274204422126,"y4"]
[0.52359877559829882,0.8886510150090674,"y5",null,null,0.26179938779914941,0.99888640232521764,"y5",null,null,"Погрешность для данного метода вычисляется без использования производной:"]
[0.62831853071795862,0.77388686334425472,"y6",null,null,0.31415926535897931,0.99834605415192101,"y6",null,null,"1. нужно посчитать интегралы с шагом n и 2n (например n=10 и n=20)"]
[0.73303828583761843,0.62530014223011909,"y7",null,null,0.36651914291880922,0.98686761769199294,"y7",null,null,"2. Вычесть одно значение из другого и потом поделить разность на 15"]
[0.83775804095727824,0.44938480437754241,"y8",null,null,0.41887902047863912,0.96457685309730568,"y8",null,null,"3. Взять модуль получившегося значения"]
[0.94247779607693805,0.25382919418612832,"y9",null,null,0.47123889803846902,0.9317179826494012,"y9"]
[{"$f":"A20+$B$8","v":1.0471975511965979},4.7180030201170509E-2,"y10",null,null,0.52359877559829893,0.88865101500906718,"y10"]
[{"$nulls":5},0.57595865315812877,0.83584780088611355,"y11"]
["Интеграл:",{"$f":"(B6-B5)/(6*D7)*((B11+B21)+2*(B13+B15+B17+B19)+4*(B12+B14+B16+B18+B20))","v":0.76960262392037115},null,null,null,0.62831853071795862,0.77388686334425472,"y12"]
[{"$nulls":5},0.68067840827778847,0.70344705938135876,"y13"]
[{"$nulls":5},{"$f":"F24+$G$8","v":0.73303828583761832},0.62530014223011943,"y14"]
[{"$nulls":5},0.78539816339744817,0.54030230586813977,"y15"]
[{"$nulls":5},0.83775804095727802,0.4493848043775428,"y16"]
[{"$nulls":5},0.89011791851710786,0.35354374893036944,"y17"]
[{"$nulls":5},0.94247779607693771,0.25382919418612876,"y18"]
[{"$nulls":5},0.99483767363676756,0.15133363367321109,"y19"]
[{"$nulls":5},1.0471975511965974,4.718003020117139E-2,"y20"]
[]
[{"$nulls":5},"Интеграл:",{"$f":"(G6-G5)/(6*I7)*((G11+G31)+2*(G13+G15+G17+G19+G21+G23+G25+G27+G29)+4*(G12+G14+G16+G18+G20+G22+G24+G26+G28+G30))","v":0.76959486893527274}]
["Погрешность:",null,{"$f":"ABS(G33-B23)/15","v":5.1699900656121405E-7}]

{"sheet":"Методы Эйлера","index":8,"n_rows":31,"n_cols":13,"rows_kept":31,"cut":false,"hidden":false,"merged":[[7,8,7,12],[8,0,8,4]]}
["a",0,"начало отрезка",{"$nulls":5},"a",0,"начало отрезка"]
["b",4,"конец отрезка",{"$nulls":5},"b",4,"конец отрезка"]
["x0",0,"нулевое x",{"$nulls":5},"x0",0,"нулевое x"]
["y0",1,"нулевое y",{"$nulls":5},"y0",1,"нулевое y"]
["h",0.2,"шаг разбиения отрезка",{"$nulls":5},"h",0.2,"шаг разбиения отрезка"]
["f(x,y)=корень(x)*y","функция",{"$nulls":6},"f(x,y)=корень(x)*y","функция"]
["i",{"$f":"(B2-B1)/B5","v":20},{"$nulls":6},"i",{"$f":"(J2-J1)/J5","v":20}]
[{"$nulls":8},"Уточненный метод Эйлера"]
["Метод Эйлера",{"$nulls":7},"i","x","y","f(x,y)","hf(x,y)"]
["i","x","y","f(x,y)","hf(x,y)",null,null,null,0,0,1,{"$f":"SQRT(J10)*K10","v":0},{"$f":"$B$5*L10","v":0}]
[0,0,1,{"$f":"SQRT(B11)*C11","v":0},{"$f":"$B$5*D11","v":0},null,null,null,1,{"$f":"J10+$J$5","v":0.2},{"$f":"K10+M10","v":1},{"$f":"SQRT(J11)*K11","v":0.44721359549995793},{"$f":"$B$5*L11","v":8.9442719099991588E-2}]
[1,{"$f":"B11+$B$5","v":0.2},{"$f":"C11+E11","v":1},{"$f":"SQRT(B12)*C12","v":0.44721359549995793},{"$f":"$B$5*D12","v":8.9442719099991588E-2},null,null,null,2,{"$f":"J11+$J$5","v":0.4},{"$f":"K10+2*M11","v":1.1788854381999831},0.74559261702352342,{"$f":"$B$5*L12","v":0.14911852340470469}]
[2,{"$f":"B12+$B$5","v":0.4},{"$f":"C12+E12","v":1.0894427190999916},{"$f":"SQRT(B13)*C13","v":0.68902407452859971},{"$f":"$B$5*D13","v":0.13780481490571994},null,null,null,3,0.60000000000000009,{"$f":"K11+2*M12","v":1.2982370468094093},1.0056100923444682,{"$f":"$B$5*L13","v":0.20112201846889366}]
[3,0.60000000000000009,1.2272475340057114,0.95062185217564821,0.19012437043512964,null,null,null,4,0.8,1.5811294751377705,1.4142051950546473,0.28284103901092944]
[4,0.8,1.417371904440841,1.2677359710912226,0.25354719421824451,null,null,null,5,1,1.8639191248312681,1.8639191248312681,0.37278382496625362]
[5,1,1.6709190986590856,1.6709190986590856,0.33418381973181716,null,null,null,6,1.2,2.3266971250702779,2.5487689997668199,0.50975379995336401]
[6,1.2,2.0051029183909028,2.1964801970442753,0.43929603940885509,null,null,null,7,1.4,2.8834267247379959,3.4117165104543199,0.68234330209086402]
[7,1.4,2.4443989577997578,2.8922518512137838,0.57845037024275681,null,null,null,8,1.5999999999999999,3.691383729252006,4.6692721208490644,0.9338544241698129]
[8,1.5999999999999999,3.0228493280425148,3.8236355600495369,0.76472711200990739,null,null,null,9,1.7999999999999998,4.7511355730776215,6.3743172670313886,1.2748634534062777]
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{"sheet":"Методы Рунге-Кутты","index":9,"n_rows":74,"n_cols":17,"rows_kept":74,"cut":false,"hidden":false,"merged":[]}
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