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Консультация № 108231
06.11.2007, 11:30
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Добрый день, Пожалуйста, проверьте решение задания на исследование ф-ии, заранее благодарна.
Исследовать ф-ю и построить ее график
Y=(x^3+16)/x
Исследование:
1. Область определения от +бескон, до 0 и от 0 до +бескон
2. Имеет разрыв в точке х=0. при остальных значениях аргумента непрерывна.
3. Вертикальная асимптота имеет х=0. Находим наклонные асимтоты. Ур-ие к-ых y=kx+b
K=Lim[+-00]f(x)/x, b= Lim[+-00][f(x) – kx]
K=Lim[+-00](x^3+16)/x=x+16/x^2=16
B=x^2+15/x=15
Y=16x+15
4. Lim[+-00] (x^3+16)/x=+-00
5. Функция ни четная ни нечетная, непериодическая.
6. График пересекает оси в точках х=0 и у=0
7. найдем интервалы монотонности и точки экстремумов
производная равна (2x^3-16)/x^2 . Определим критические точки. Производная превращается в 0 при х1=2, х2=0.
Интервалы возрастания ф-ии (-2:0) и (0:+00)
Интервалы убывания (-00;-2)
6. Определим интервалы выпуклости/вогнутости:
Вторая производная равна (2x^4+32x)/x^4. Определим крит. точки.
Это х1=0 и х2=16^1/3
Исследовать ф-ю и построить ее график
Y=(x^3+16)/x
Исследование:
1. Область определения от +бескон, до 0 и от 0 до +бескон
2. Имеет разрыв в точке х=0. при остальных значениях аргумента непрерывна.
3. Вертикальная асимптота имеет х=0. Находим наклонные асимтоты. Ур-ие к-ых y=kx+b
K=Lim[+-00]f(x)/x, b= Lim[+-00][f(x) – kx]
K=Lim[+-00](x^3+16)/x=x+16/x^2=16
B=x^2+15/x=15
Y=16x+15
4. Lim[+-00] (x^3+16)/x=+-00
5. Функция ни четная ни нечетная, непериодическая.
6. График пересекает оси в точках х=0 и у=0
7. найдем интервалы монотонности и точки экстремумов
производная равна (2x^3-16)/x^2 . Определим критические точки. Производная превращается в 0 при х1=2, х2=0.
Интервалы возрастания ф-ии (-2:0) и (0:+00)
Интервалы убывания (-00;-2)
6. Определим интервалы выпуклости/вогнутости:
Вторая производная равна (2x^4+32x)/x^4. Определим крит. точки.
Это х1=0 и х2=16^1/3
Обсуждение
Неизвестный
06.11.2007, 17:58
общий
это ответ
Здравствуйте, Дроздова Елена Владимировна!
6. График пересекает ось х в точке y=0 => x = - (16)^(1/3)
ось y не пересекает, ибо х не может быть равно 0
7. y‘ = {(3x^2)x - (x^3+16)}/(x^2) = (2x^3 - 16)/(x^2)
=> y‘ = 0 при х=2
Интервалы возрастания ф-ии (2;00)
Интервалы убывания (-00;2)
8. y‘‘ = {(6x^2)(x^2) - (2x^3 - 16)(2x)}/(x^4) = (6x^4 - 2x^4 + 32x)/(x^4) = (2x^3 + 32)/(x^3)
y‘‘(2) = 6
x = 2 - точка минимума функции
6. График пересекает ось х в точке y=0 => x = - (16)^(1/3)
ось y не пересекает, ибо х не может быть равно 0
7. y‘ = {(3x^2)x - (x^3+16)}/(x^2) = (2x^3 - 16)/(x^2)
=> y‘ = 0 при х=2
Интервалы возрастания ф-ии (2;00)
Интервалы убывания (-00;2)
8. y‘‘ = {(6x^2)(x^2) - (2x^3 - 16)(2x)}/(x^4) = (6x^4 - 2x^4 + 32x)/(x^4) = (2x^3 + 32)/(x^3)
y‘‘(2) = 6
x = 2 - точка минимума функции
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