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Консультация № 137994
27.05.2008, 09:25
50.00 руб.
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Уважаемые эксперты! Срочно нужно решить задачу.
Исследовать функцию и построить график:
у=(2х+1)/х2.
Исследовать функцию и построить график:
у=(2х+1)/х2.
Обсуждение
Неизвестный
31.05.2008, 15:00
общий
это ответ
Здравствуйте, Юсупова М.М.!
y = (2x+1)/x²
1. Область определения.
M = (-∞, 0) ∪ (0, +∞)
2. Функция является непрерывной в области определения как частное двух непрерывных функций.
3. Так как
lim<sub>{x→0}</sub> (2x+1)/x² = ∞
то прямая <b>x=0 - вертикальная асимптота</b>.
k = lim<sub>{x→±∞}</sub> f(x)/x = lim<sub>{x→±∞}</sub> (2x+1)/x³ = 0
следовательно наклонных асимптот нет.
4. Так как
(2(-x)+1)/(-x)² = -(2x-1)/x², то функция ни чётная ни нечётная.
5. Функция не периодическая.
6. Функция y = (2x+1)/x² дифференцируемая всюду в области определения. ЕЕ производная y‘ = -2(x+1)/x³ = 0 в точке x=-1
y‘>0 при x>-1
y‘<0 при x<-1
Следовательно точка x=-1 - точка экстремума (минимума).
f(-1) = (2*(-1)+1)/(-1)² = -1
Координата минимума (-1,-1).
7. Вторая производная y‘‘ = (4x+6)/x<sup>4</sup>
y‘‘ = 0 в точке x=-3/2
y‘‘ < 0 на множестве (-∞;-3/2) функция направлена выпуклостью вверх
y‘‘ > 0 на множестве (-3/2;0) функция направлена выпуклостью вниз
8. Строим график.
<a target=_blank href=http://radikal.ru/F/i026.radikal.ru/0805/5d/05d12f364d78.gif.html><img src=http://i026.radikal.ru/0805/5d/05d12f364d78t.jpg ></a>
Good Luck!
y = (2x+1)/x²
1. Область определения.
M = (-∞, 0) ∪ (0, +∞)
2. Функция является непрерывной в области определения как частное двух непрерывных функций.
3. Так как
lim<sub>{x→0}</sub> (2x+1)/x² = ∞
то прямая <b>x=0 - вертикальная асимптота</b>.
k = lim<sub>{x→±∞}</sub> f(x)/x = lim<sub>{x→±∞}</sub> (2x+1)/x³ = 0
следовательно наклонных асимптот нет.
4. Так как
(2(-x)+1)/(-x)² = -(2x-1)/x², то функция ни чётная ни нечётная.
5. Функция не периодическая.
6. Функция y = (2x+1)/x² дифференцируемая всюду в области определения. ЕЕ производная y‘ = -2(x+1)/x³ = 0 в точке x=-1
y‘>0 при x>-1
y‘<0 при x<-1
Следовательно точка x=-1 - точка экстремума (минимума).
f(-1) = (2*(-1)+1)/(-1)² = -1
Координата минимума (-1,-1).
7. Вторая производная y‘‘ = (4x+6)/x<sup>4</sup>
y‘‘ = 0 в точке x=-3/2
y‘‘ < 0 на множестве (-∞;-3/2) функция направлена выпуклостью вверх
y‘‘ > 0 на множестве (-3/2;0) функция направлена выпуклостью вниз
8. Строим график.
<a target=_blank href=http://radikal.ru/F/i026.radikal.ru/0805/5d/05d12f364d78.gif.html><img src=http://i026.radikal.ru/0805/5d/05d12f364d78t.jpg ></a>
Good Luck!
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