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Консультация № 139342
06.06.2008, 17:04
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Здравствуйте эксперты. Составить уравнение касательной и нормали к графику функции y=f(x) в точке с абсциссой X нулевое.
y=(корень4-2x^2); Xнулевое=1
y=(корень4-2x^2); Xнулевое=1
Обсуждение
08.06.2008, 14:55
общий
это ответ
Здравствуйте, SETXAOS!
Решение.
Находим значение функции в точке x0:
f(x0) = f(1) = sqrt (4 - 2*1^2) = sqrt (4 - 2) = sqrt 2.
Находим производную функции:
f‘(x) = 1/(2*sqrt (4 - 2*x^2)).
Находим значений производной в точке x0 = 1:
f‘(x0) = f‘(1) = 1/(2*sqrt (4 - 2*1^2)) = 1/(2*sqrt 2) = (sqrt 2)/4.
Находим уравнение касательной к графику функции f(x) в точке x0 = 1:
y - f(x0) = (f‘(x0))*(x - x0),
y - sqrt 2 = ((sqrt 2)/4)*(x - 1),
y = ((sqrt 2)/4)*x - ((sqrt 2)/4) + sqrt 2,
y = ((sqrt 2)/4)*x + (3/4)*sqrt 2.
Находим уравнение нормали к графику функции f(x) в точке x0 = 1:
y - f(x0) = (-1/f‘(x0))*(x - x0),
y - sqrt 2 = (-1/(sqrt 2)/4)*(x - 1),
y - sqrt 2 = (-4/(sqrt 2))*(x - 1),
y - sqrt 2 = (-2*sqrt 2)*x + 2*sqrt 2,
y = (-2*sqrt 2)*x + 3*sqrt 2.
Ответ: y = ((sqrt 2)/4)*x + (3/4)*sqrt 2 - уравнение касательной; y = (-2*sqrt 2)*x + 3*sqrt 2 - уравнение нормали.
С уважением.
Решение.
Находим значение функции в точке x0:
f(x0) = f(1) = sqrt (4 - 2*1^2) = sqrt (4 - 2) = sqrt 2.
Находим производную функции:
f‘(x) = 1/(2*sqrt (4 - 2*x^2)).
Находим значений производной в точке x0 = 1:
f‘(x0) = f‘(1) = 1/(2*sqrt (4 - 2*1^2)) = 1/(2*sqrt 2) = (sqrt 2)/4.
Находим уравнение касательной к графику функции f(x) в точке x0 = 1:
y - f(x0) = (f‘(x0))*(x - x0),
y - sqrt 2 = ((sqrt 2)/4)*(x - 1),
y = ((sqrt 2)/4)*x - ((sqrt 2)/4) + sqrt 2,
y = ((sqrt 2)/4)*x + (3/4)*sqrt 2.
Находим уравнение нормали к графику функции f(x) в точке x0 = 1:
y - f(x0) = (-1/f‘(x0))*(x - x0),
y - sqrt 2 = (-1/(sqrt 2)/4)*(x - 1),
y - sqrt 2 = (-4/(sqrt 2))*(x - 1),
y - sqrt 2 = (-2*sqrt 2)*x + 2*sqrt 2,
y = (-2*sqrt 2)*x + 3*sqrt 2.
Ответ: y = ((sqrt 2)/4)*x + (3/4)*sqrt 2 - уравнение касательной; y = (-2*sqrt 2)*x + 3*sqrt 2 - уравнение нормали.
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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