18.2. Найти асимптоты графиков функций:
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y = |
x4 |
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y = |
ln x |
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+1 |
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y = x +sin x . |
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y = (x − 2)e−1/ x . |
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18.3. Провести полное исследование и построить графики функций:
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y = |
2x2 −1 |
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б) y = x2e−x . |
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x4 |
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г) y = 3 |
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в) y = x 1− x2 . |
x2 −2x |
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д) |
y = x2 ln x . |
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Домашнее задание
18.4. Найти точки перегиба графиков функций:
а) y = |
2x −1 |
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б) y = x arctg x . |
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18.5. Найти асимптоты графика функции |
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y = x ln e + |
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18.6. Исследовать функции и построить их графики:
а) y = 1−x3x2 .
Ответы
18.4. а) |
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18.5. x = −1e ; y = x + 1e .
б) y = xe1/ x .
б) Точек перегиба нет.
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Типовой расчет № 1
Элементы линейной алгебры и аналитической геометрии
Задача 1
Исследовать систему уравнений и в случае совместности решитьее.
x1 − x2 − x3 − x4 =1, 1.1. а) 2x1 + x2 − x3 + x4 = 3,
3x1 + x4 = 4.
2x1 + x3 + 2x4 = 5, 1.2. а) x2 − x3 + x4 = 0,
2x1 + x2 +3x3 = 5.
x2 + 2x3 + 3x4 = 2, 1.3. а) x1 − x2 − x3 − 2x4 = 0,x1 + x2 + x4 = −1.
2x2 + x3 + 4x4 = 0, 1.4. а) x1 − x3 + x4 = 2,
x1 + 2x2 + 5x4 =1.
4x2 + 2x3 −3x4 = 0,
1.5. а) 3x1 −3x2 + x4 = 3,3x1 + x2 + 2x3 − 2x4 = 3.
x1 − 2x2 +3x3 = 3, 1.6. а) 2x1 + x3 − x4 =1,
x1 + 2x2 − 2x3 − x4 = −2.
б)
б)
б)
б)
б)
б)
2x1 −3x2 − x3 = 0,
x1 + x2 + x3 =1,3x1 − 2x2 =1,
x1 − 2x2 − 2x3 = −1.
2x1 −3x2 + x3 = 0,x1 + x2 + x3 =1,4x1 + 5x2 − x3 = −1,7x1 + 3x2 + x3 = 3.
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− x |
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− x |
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3x |
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− x |
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x1 |
+ 2x2 + x3 = 0, |
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2x |
− x |
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+ x |
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3x |
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x1 |
+ x3 − 2x4 = 2, |
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2x |
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2x |
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x1 |
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3x |
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4x1 − 2x3 + 5x4 = 0,
1.7.а) 3x1 + x3 − x4 = 0,x1 −3x3 + 6x4 = 0.
x1 − x3 + x4 = 0,
1.8.а) 2x1 + x3 − 2x4 = 0,3x1 + 2x2 − x4 = 0.
x1 − x2 + х3 + x4 = 0, 1.9. а) 2x1 + x2 + х3 − x4 = 0,
x1 + 2x2 − 2x4 = 0.
x1 + x2 − x4 = 0,
1.10. а) x2 + х3 + x4 = 0,
x3 − 4x4 = 0.
2x3 + 4x4 = 0,
1.11. а) x1 + х2 − x4 = 0,
2x1 + 3х2 − x3 = 0.
3x1 + х2 − x4 = 0,
1.12. а) 2x1 − х2 + x4 = 0,
x1 −3х2 = 0.
б)
б)
б)
б)
б)
б)
x1 + x3 − x4 = 7,2x1 + x2 + x4 = 6,x1 − x2 + x3 = −5,4x1 + 2x3 = 0.
2x1 + 2x2 + x3 = 5,x1 − x3 + x4 = 0,3x1 + 2x2 + x4 =1,x2 + x3 − х4 = 0.
3x1 − 2x2 − x3 =1,x1 + x2 + x3 = 0,5x2 + x3 = 7,
x1 + 3x2 = 6.
x2 + x3 − x4 = −2,x1 + x2 − x3 = 4,2x1 + х2 + x4 = 3,3x1 + 3x2 = 0.
x2 + x3 + x4 =1,
x1 − x2 + x3 − х4 = −1,x1 + 2х3 = 0,
x1 − 2x2 − 2х4 = −2.
x1 − х2 + x3 = 7,x1 + 2x2 + х4 = 5,2x2 + х3 − х4 = 0,
2x1 + x2 + х3 + х4 =1.
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2x + |
х |
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1.13. а) x1 |
+ х3 − x4 = 3, |
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3x1 + х2 + 2x3 − х4 = 3, |
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= 6. |
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х2 + х3 + x4 = 3, |
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1.14. а) |
x1 − х2 + x4 =1, |
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+ х |
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−3х |
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− 4х |
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1.15. а) |
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3x1 − 2х2 −5х3 − 4x4 = 0, |
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5x1 −8х2 −13х3 − 2х4 = 0. |
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x1 − х2 + x4 =1, |
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2 |
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1.16. а) |
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2x |
− х |
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+ х4 = |
5. |
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1.17. а) 2x1 + х2 + x3 = 0, |
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x1 − 2x2 + 4x3 = 0, |
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1.18. а) x1 + x2 − x3 = 0, |
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3x = 0. |
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б)
б)
б)
б)
б)
б)
3x3 + 4x4 = 0,
x1 + x2 + х4 = 0,2x1 + х2 − х4 = 0.
x1 − 2х2 − x3 = 0,x1 + 2x3 − х4 = 0,x1 + 3х4 = 0.
x1 + 2х2 + x3 − х4 = 4,3x1 + 2x2 − х3 − х4 = 0,2x1 − х2 + х3 + 2х4 = −1,
6x1 + 3x2 + х3 = 3.
x1 + х3 + 2x4 = 0,x1 − x3 + х4 = 0,x1 + х2 + 3х4 = 0.
2х2 + 2x3 − 4х4 =1,
3x1 + x2 − х3 − х4 = 2,х1 + x2 + х3 + х4 = −1,4x1 + 4x2 + 2х3 − 4х4 = 0.
2x1 − x3 − x4 = −3,3x1 + x2 − 2x3 = 0,x1 − x2 − x3 = −1,
6x1 − x2 − x3 −3x4 = 2.
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3x2 + x3 + 4x4 = 0, 1.19. а) x1 + x3 − x4 = 0,
2x1 + 3x4 = 0.
2x |
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1.20. а) x1 |
+ x2 − x4 =1, |
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4, |
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3x + 2x |
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x2 + x3 + 3x4 = 3, |
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1.21. а) x1 − x3 + x4 = −1, |
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4x = 2. |
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x1 − x3 − x4 = 0,
1.22.а) 2x1 − 2x2 + x4 = 0,3x1 − 2x2 + x3 = 0.
3x1 − x3 −5x4 = 5, |
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1.23. а) 2x1 − x2 + x4 =1, |
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5x1 − x2 − x3 − x4 = 6. |
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−3x |
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1.24. а) x1 − 7x3 + x4 = −1, |
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+ x |
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+ x |
+ x |
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0. |
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б)
б)
б)
б)
б)
б)
x2 + 5х3 + 2x4 = 0,− x3 + x4 =1,
x2 + 2x3 − x4 = −1,
x1 + 2x2 + 6x3 + 2x4 = 0.
x1 − x2 − x3 = 0,x1 + x2 + x3 = 0,2x1 − x2 − x3 = 0.
2x1 −3x2 + x3 = 0,x2 − x3 − x4 = 0,x1 + x2 + 2x3 = 0,
3x1 − x2 + 2x3 − x4 = 0.
3x1 + x2 + 5x3 =1,x1 − x2 − 4x4 = 5,x2 + x3 + x4 = −1,
3x1 + 2x2 + 6x3 + x4 = 9.
x1 − 2x2 + x3 = 0,2x2 + x3 + x4 = 0,
−3x1 − x2 + x3 + x4 = 0,x1 + 3x2 − x3 − x4 = 0.
3x1 + 2x2 − x3 − x4 = 0,2x1 − 2x2 + 4x3 − 2x4 = 0.
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