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Консультация № 201966
22.12.2021, 15:56
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Здравствуйте, очень срочно и важно, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Найдите площадь треугольника, образованного осью 0x, прямой y = 4 - x и касательной к графику функции y = 4 + 4x - x^2 в точке пересечения его с осью 0y
Найдите площадь треугольника, образованного осью 0x, прямой y = 4 - x и касательной к графику функции y = 4 + 4x - x^2 в точке пересечения его с осью 0y
Обсуждение
24.12.2021, 13:00
общий
это ответ
Здравствуйте, vladik_top231!
Вычислим производную функции
Вычислим значение этой производной в точке 
пересечения графика рассматриваемой функции с осью ординат: 
-- угловой коэффициент касательной к графику функции в рассматриваемой точке. Значит, уравнение касательной к графику функции в указанной точке суть 
или 
Эта касательная пересекается с осью абсцисс в точке 
поскольку при 
имеем 
откуда 
Определим координаты точки пересечения рассмотренной касательной с прямой
Приравнивая друг к другу ординаты из уравнений обеих прямых, получим 
То есть -- точка пересечения.
Определим координаты точки пересечения прямой
с осью абсцисс, то есть при 
Тогда 
То есть 
-- точка пересечения.
В результате получили треугольник с вершинами в точках
Длина основания этого треугольника составляет 
а длина высоты, опущенной на это основание (отрезка оси ординат), равна 
Следовательно, искомая площадь треугольника составляет 
(ед. площади).
Вычислим производную функции
Вычислим значение этой производной в точке пересечения графика рассматриваемой функции с осью ординат: -- угловой коэффициент касательной к графику функции в рассматриваемой точке. Значит, уравнение касательной к графику функции в указанной точке суть или Эта касательная пересекается с осью абсцисс в точке поскольку при имеем откуда Определим координаты точки пересечения рассмотренной касательной с прямой
Приравнивая друг к другу ординаты из уравнений обеих прямых, получим То есть -- точка пересечения.Определим координаты точки пересечения прямой
с осью абсцисс, то есть при Тогда То есть -- точка пересечения.В результате получили треугольник с вершинами в точках
Длина основания этого треугольника составляет а длина высоты, опущенной на это основание (отрезка оси ординат), равна Следовательно, искомая площадь треугольника составляет (ед. площади).
Об авторе:
Facta loquuntur.
Facta loquuntur.
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