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Консультация № 198089
31.03.2020, 07:30
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
функцию f (x,y)= x^3-5x^2-xy+y^2+10x+5y-4 разложить по формуле тейлора в окрестности точки (2;-1)
функцию f (x,y)= x^3-5x^2-xy+y^2+10x+5y-4 разложить по формуле тейлора в окрестности точки (2;-1)
Обсуждение
05.04.2020, 03:56
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 все её частные производные, отличные от нуля.
В данном случае для f(x,y) = x[sup]3[/sup] - 5x[sup]2[/sup] - xy + y[sup]2[/sup] + 10x + 5y - 4 имеем
все остальные частные производные равны нулю. Тогда разложение в ряд Тейлора в окрестности точки (x[sub]0[/sub], y[sub]0[/sub]) запишется как
В частности, в окрестности точки (2, -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 все её частные производные, отличные от нуля.
В данном случае для f(x,y) = x[sup]3[/sup] - 5x[sup]2[/sup] - xy + y[sup]2[/sup] + 10x + 5y - 4 имеем
все остальные частные производные равны нулю. Тогда разложение в ряд Тейлора в окрестности точки (x[sub]0[/sub], y[sub]0[/sub]) запишется как
В частности, в окрестности точки (2, -1) имеем
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