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Консультация № 172476
22.09.2009, 13:13
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Добрый день, уважаемые! Помогите решить пример по дифференциальному исчислению функций нескольких переменных...
Даны функция z=3 x^2 y^2 + 5 x y^2, точка A(1;1) и вектор a(2;1) . Найти: 1) grad z в точке А; 2) производную в точке А по направлению вектора а.
Спасибо.
Даны функция z=3 x^2 y^2 + 5 x y^2, точка A(1;1) и вектор a(2;1) . Найти: 1) grad z в точке А; 2) производную в точке А по направлению вектора а.
Спасибо.
Обсуждение
Неизвестный
22.09.2009, 16:23
общий
это ответ
Здравствуйте, Попов Антон Андреевич.
1) Находим частные производные данной функции:
∂z/∂x = 6xy2 + 5y2
∂z/∂y = 6x2y +10xy
Вычисляем значение этих производных в точке А(1;1):
∂z/∂x = 6*1*12 + 5*12 = 11
∂z/∂y = 6*1*2*1 +10*1*1 = 16.
Окончательно получаем: grad z(A) = (11;16).
2) Находим единичный вектор а0, совпадающий с направлением вектора а:
а0 = a/|a| = (2/√5;1/√5), где |a| = √(22 + 12) = √5.
Тогда ∂z/∂a = 11 * 2/√5 + 16 * 1/√5 = 38/√5.
1) Находим частные производные данной функции:
∂z/∂x = 6xy2 + 5y2
∂z/∂y = 6x2y +10xy
Вычисляем значение этих производных в точке А(1;1):
∂z/∂x = 6*1*12 + 5*12 = 11
∂z/∂y = 6*1*2*1 +10*1*1 = 16.
Окончательно получаем: grad z(A) = (11;16).
2) Находим единичный вектор а0, совпадающий с направлением вектора а:
а0 = a/|a| = (2/√5;1/√5), где |a| = √(22 + 12) = √5.
Тогда ∂z/∂a = 11 * 2/√5 + 16 * 1/√5 = 38/√5.
Форма ответа
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