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Консультация № 198090
31.03.2020, 07:51
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Здравствуйте! У меня возникли сложности с таким вопросом:
Разложить по формуле Тейлора в окрестности точки (1;1) до членов второго порядка включительно функцию f(x;y)=y^x
Разложить по формуле Тейлора в окрестности точки (1;1) до членов второго порядка включительно функцию f(x;y)=y^x
Обсуждение
05.04.2020, 04:13
общий
это ответ
Здравствуйте, mega.chepyrnukh0699@list.ru!
В общем случае ряд Тейлора для функции двух переменных f(x, y) в окрестности точки (x[sub]0[/sub], y[sub]0[/sub]) имеет вид:
где d[sup]n[/sup]f - полный дифференциал n-го порядка, определяемый выражением:
(в частности, при n=0 имеем d[sup]n[/sup]f = f). Следовательно, для разложения функции в ряд Тейлора требуется вычислить её частные производные.
В данном случае для f(x,y) = y[sup]x[/sup] имеем
частные производные третьего и последующих порядков по условию не требуются. Тогда разложение в ряд Тейлора в окрестности точки (x[sub]0[/sub], y[sub]0[/sub]) запишется как
В частности, в окрестности точки (1, 1) имеем
В общем случае ряд Тейлора для функции двух переменных f(x, y) в окрестности точки (x[sub]0[/sub], y[sub]0[/sub]) имеет вид:
где d[sup]n[/sup]f - полный дифференциал n-го порядка, определяемый выражением:
(в частности, при n=0 имеем d[sup]n[/sup]f = f). Следовательно, для разложения функции в ряд Тейлора требуется вычислить её частные производные.
В данном случае для f(x,y) = y[sup]x[/sup] имеем
частные производные третьего и последующих порядков по условию не требуются. Тогда разложение в ряд Тейлора в окрестности точки (x[sub]0[/sub], y[sub]0[/sub]) запишется как
В частности, в окрестности точки (1, 1) имеем
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