b) My friend wasn't waiting for me when I arrived late. 7.a) When did he have the accident? -As he was climbing up the ladder. b) When was he having the accident? -As he was climbing up the ladder. 8.a) l read a book when she sent me an SMS. b) l was reading a book when she sent me am SMS. 9.a) Did you eеat lunch when the fire alarm rang? b) Were you eating luncn when the fire alarm rang. 10.a) lt was a beautiful day The sun shone and there wasn't a cloud in the sky. b) lt was a beautiful day. The sun was shining and there wasn't a cloud in the sky.
Look at the of sentences below. Both of them are grammatically correct. Choose one sentence in each pair, a or b, that is more logical. 1.a) lt rained heavily when I ran for the bus. B) lt was raining heavily when I ran for the bus. 2.a) The rain jjust came into the station when we arrived on the platform. b) The train was just coming into the station when we arrived on the platform. 3.a) He broke his leg while riding his bike at the weekend. b) He was breaking his leg while riding his bike at the weekend. 4.a) While I was watching TV, someone knocked on the window. bjWhile I was watching TV, someone was knocked on the window. 5 a) No wonder she fell off her bike. She didnt Innk here che iwac in
6 Complete the sentences with when or while. 1) The boys were playing football ... the ball flew into the window. 2) Ronny was painting the fence ... the bucket of paint turned over. 3) Denny was putting a tent ... the girls were gathering wood for fire. 4) Rita was sleeping .. the alarm- rang. 5) Nelly was peeling potatоes her sister was cutting chicken into pieces. 6) Nigel was skiing he fell down and broke his leg. 7) They were arguing ... Mrs Elliot entered the room. 8) ... my mother was watching her favorite show on TV, father making coffee. 9) She was tying a letter some mistakes. 10) They were driving towards the village it started clock she found to snow.
Дана функция у = (х-1)²/x².
1.Область определения функции. D ∈ R : x ≈ 0.
2. Нули функции. Точки пересечения графика функции с осью ОХ.
График функции пересекает ось X при f = 0.
Значит, надо решить уравнение (х-1)²/x² = 0.
Решаем это уравнение (достаточно приравнять нулю числитель):
(х-1)² = 0, х-1 = 0, х = 1.
Точки пересечения с осью X: (1; 0).
График пересекает ось Y, когда x равняется 0.
Подставляем x = 0 в (x - 1)²/x².
Результат: (0 - 1)²/0² невыполним, значит, график не пересекает ось Оу.
3. Промежутки знакопостоянства функции.
Так как переменная в числителе и знаменателе в квадрате, то функция на всей числовой оси только положительна.
4. Симметрия графика (чётность или нечётность функции).
f(-x) = ((-x) - 1)²/((-x)²) = (x + 1)²/x² ≠ f(x) ≠ -f(-x).
Поэтому функция не чётная и не нечётная.
5. Периодичность графика. Не периодична.
6.Точки разрыва, поведение функции в окрестностях точек разрыва, вертикальные асимптоты - смотри приложение.
7. Интервалы монотонности функции, точки экстремумов, значения функции в точках экстремумов.
Первая производная: y' = (1/x²)*(2x - 2) - (2/x³)*(x - 1)²
или y' = (2x - 2)/x³.
Находим нули функции. Для этого приравниваем производную к нулю
(достаточно числитель): 2x-2 = 0
Откуда: x1 = 2/2 = 1.
(-∞ ;0) (0; 1) (1; +∞)
f'(x) > 0 f'(x) < 0 f'(x) > 0
функция возрастает функция убывает функция возрастает.
В окрестности точки x = 1 производная функции меняет знак с (-) на (+). Следовательно, точка x = 1 - точка минимума.
8. Интервалы выпуклости, точки перегиба.
Найдем точки перегибов, для этого надо решить уравнение
\frac{d^{2}}{d x^{2}} f{\left (x \right )} = 0.
(вторая производная равняется нулю),
корни полученного уравнения будут точками перегибов для указанного графика функции:
\frac{d^{2}}{d x^{2}} f{\left (x \right )} =
Вторая производная
\frac{1}{x^{2}} \left(2 - \frac{1}{x} \left(8 x - 8\right) + \frac{6}{x^{2}} \left(x - 1\right)^{2}\right) = 0
Решаем это уравнение
Корни этого ур-ния
x_{1} = \frac{3}{2}
Также нужно подсчитать пределы y'' для аргументов, стремящихся к точкам неопределённости функции:
Точки, где есть неопределённость:
x_{1} = 0.
\lim_{x \to 0^-}\left(\frac{1}{x^{2}} \left(2 - \frac{1}{x} \left(8 x - 8\right) + \frac{6}{x^{2}} \left(x - 1\right)^{2}\right)\right) = \infty.
\lim_{x \to 0^+}\left(\frac{1}{x^{2}} \left(2 - \frac{1}{x} \left(8 x - 8\right) + \frac{6}{x^{2}} \left(x - 1\right)^{2}\right)\right) = \infty.
- пределы равны, значит, пропускаем соответствующую точку.
Интервалы выпуклости и вогнутости:
Найдём интервалы, где функция выпуклая или вогнутая, для этого посмотрим, как ведет себя функция в точках перегибов:
Вогнутая на промежутках
(-oo, 3/2]
Выпуклая на промежутках
[3/2, oo)
9. Поведение функции в бесконечности. Наклонные (в частности, горизонтальные) асимптоты - смотри приложение.
10. Дополнительные точки, позволяющие более точно построить график - даны в приложении.
11. Построение графика функции по проведенному исследованию дан в приложении.
b) My friend wasn't waiting for me when I arrived late. 7.a) When did he have the accident? -As he was climbing up the ladder. b) When was he having the accident? -As he was climbing up the ladder. 8.a) l read a book when she sent me an SMS. b) l was reading a book when she sent me am SMS. 9.a) Did you eеat lunch when the fire alarm rang? b) Were you eating luncn when the fire alarm rang. 10.a) lt was a beautiful day The sun shone and there wasn't a cloud in the sky. b) lt was a beautiful day. The sun was shining and there wasn't a cloud in the sky.
Look at the of sentences below. Both of them are grammatically correct. Choose one sentence in each pair, a or b, that is more logical. 1.a) lt rained heavily when I ran for the bus. B) lt was raining heavily when I ran for the bus. 2.a) The rain jjust came into the station when we arrived on the platform. b) The train was just coming into the station when we arrived on the platform. 3.a) He broke his leg while riding his bike at the weekend. b) He was breaking his leg while riding his bike at the weekend. 4.a) While I was watching TV, someone knocked on the window. bjWhile I was watching TV, someone was knocked on the window. 5 a) No wonder she fell off her bike. She didnt Innk here che iwac in
6 Complete the sentences with when or while. 1) The boys were playing football ... the ball flew into the window. 2) Ronny was painting the fence ... the bucket of paint turned over. 3) Denny was putting a tent ... the girls were gathering wood for fire. 4) Rita was sleeping .. the alarm- rang. 5) Nelly was peeling potatоes her sister was cutting chicken into pieces. 6) Nigel was skiing he fell down and broke his leg. 7) They were arguing ... Mrs Elliot entered the room. 8) ... my mother was watching her favorite show on TV, father making coffee. 9) She was tying a letter some mistakes. 10) They were driving towards the village it started clock she found to snow.