9.1.1. Расчётно-графическое задание 1
Задача 1. Найти точки локального экстремума функции и значения функции в них:
1. |
A) |
F(X) = -3(X – 10)5; |
Б) |
F(X) = -10X2E-10X; |
2. |
A) |
F(x) = -6(x + 1)5; |
Б) |
F(x) = 4x2e9x; |
3. |
A) |
F(x) = 3(x – 1)5; |
Б) |
F(x) = -6x2e-6x; |
4. |
A) |
F(x) = -4(x + 9)5; |
Б) |
F(x) = -3x2e2x; |
5. |
A) |
F(x) = 4(x + 7)5; |
Б) |
F(x) = 4x2e-9x; |
6. |
A) |
F(x) = -8(x + 8)5; |
Б) |
F(x) = -10x2e5x; |
7. |
A) |
F(x) = -8(x – 10)5; |
Б) |
F(x) = -7x2e2x; |
8. |
A) |
F(x) = -9(x – 2)5; |
Б) |
F(x) = 2x2e8x; |
9. |
A) |
F(x) = 6(x – 10)5; |
Б) |
F(x) = 5x2e-6x; |
10. |
A) |
F(x) = 9(x – 9)5; |
Б) |
F(x) = -10x2e8x; |
11. |
A) |
F(x) = -9(x – 4)5; |
Б) |
F(x) = -9x2e6x; |
12. |
A) |
F(x) = -9(x + 8)5; |
Б) |
F(x) = 3x2ex; |
13. |
A) |
F(x) = 5(x + 3)5; |
Б) |
F(x) = 3x2ex; |
14. |
A) |
F(x) = 5(x – 8)5; |
Б) |
F(x) = -8x2e7x; |
15. |
A) |
F(x) = -8(x + 7)5; |
Б) |
F(x) = 2x2e4x; |
16. |
A) |
F(x) = -7(x + 9)5; |
Б) |
F(x) = -5x2e-10x; |
17. |
A) |
F(x) = -2(x – 3)5; |
Б) |
F(x) = 9x2e-6x; |
18. |
A) |
F(x) = 5(x + 2)5; |
Б) |
F(x) = -4x2e-10x; |
19. |
A) |
F(x) = -9(x + 1)5; |
Б) |
F(x) = 8x2e2x; |
20. |
A) |
F(x) = -6(x – 5)5; |
Б) |
F(x) = 2x2e-x. |
Задача 2. Найти наибольшее и наименьшее значения функции F(X) на отрезке [A; б].
1. |
F(x) = -10x3 +15x2 + 6, |
[-0.1, 2]; |
2. |
F(x) = 3x3 –4.5x2 +9, |
[-0.1, 2.3]; |
3. |
F(x) = -4x3 +6x2 + 1, |
[-0.4, 2.5]; |
4. |
F(x) = 7x3 + 10.5x2 + 7, |
[-0.5, 2.6]; |
5. |
F(x) = -6x3 + 9x2 + 8, |
[-0.1, 2.4]; |
6. |
F(x) =-7x3 +10.5x2 + 2, |
[-0.3, 2]; |
7. |
F(x) = -3x3 + 4.5x2 + 1, |
[-0.5, 2.7]; |
8. |
F(x) = -2x3 + 3x2 + 10, |
[-0.6, 2.9]; |
9. |
F(x) = 9x3 – 13.5x2 + 9, |
[-0.5, 2.6]; |
10. |
F(x) = -9x3 + 13.5x2 + 7, |
[-0.6, 2.5]; |
11. |
F(x) = 2x3 – 3x2 + 3, |
[-0.4, 2.2]; |
12. |
F(x) = 2x3 – 3x2 + 6, |
[-0.3, 2.1]; |
13. |
F(x) = -4x3 + 6x2 + 3, |
[-0.4, 2.8]; |
14. |
F(x) = 3x3 – 4.5x2 + 9, |
[-0.3, 2.5]; |
15. |
F(x) = 5x3 – 7.5x2 + 2, |
[-0.4, 2.1]; |
16. |
F(x) = -10x3 + 15x2 + 7, |
[-0.4, 2.4]; |
17. |
F(x) = -9x3 + 13.5x2 +5, |
[-0.4, 2.8]; |
18. |
F(x) = -10x3 + 15x2 + 1, |
[-0.1, 2.3]; |
19. |
F(x) = -10x3 + 15x2 + 4, |
[-0.6, 2.9]; |
20. |
F(x) = 8x3 – 12x2 + 8, |
[-0.6, 2.9]. |
Задача 3. Найти точки перегиба, промежутки выпуклости и вогнутости графика функции F(X).
1. f(x) = 6x3-6x2+2x+6; 2. f(x) = -8x3-10x2+2x+9; 3. f(x) = -3x3+8x2-6x+1; 4. f(x) = 6x3+6x2-2x+7; 5. f(x) = 2x3-2x2-10x+8; 6. f(x) = 9x3+6x2-2x+2; 7. f(x) = 6x3+4x2-10x+1; 8. f(x) = 3x3+3x2+6x+10; 9. f(x) = 4x3-7x2-9x+9; 10. f(x) = -5x3-8x2+6x+7; |
11. f(x) = -7x3-4x2-9x+3; 12. f(x) = -3x3+4x2+7x+6; 13. f(x) = 3x3-8x2-4x+3; 14. f(x) = -7x3-8x2+3x+9; 15. f(x) = -7x3+9x2+7x+2; 16. f(x) = 2x3+2x2-4x+7; 17. f(x) = -10x3-5x2+7x+5; 18. f(x) = 2x3-3x2+6x+1; 19. f(x) = -3x3+4x2-2x+4; 20. f(x) = 8x3-2x2+4x+8. |
Задача 4. Для данной функции двух переменных Z = F(X, Y) Найти градиент функции в точке М(х0, у0) и найти производную в той же точке М по направлению вектора MN.
1. |
Z = x2 + y3 – 24x2y, |
M(-2, 3), |
N(-5, 5); |
2. |
Z = x2 + y3 – 15x2y, |
M(0, -4), |
N(2, -9); |
3. |
Z = x2 +y3 – 6x2y, |
M(-5, 4), |
N(-6, 5); |
4. |
Z = x2 + y3 – 21x2y, |
M(0, -3), |
N(0, -4); |
5. |
Z = x2 + y3 – 12x2y, |
M(-4, 2), |
N(-6, 2); |
6. |
Z = x2 + y3 – 30x2y, |
M(4, 3), |
N(6, 3); |
7. |
Z = x2 + y3 – 15x2y, |
M(0, -3), |
N(4, -1); |
8. |
Z = x2 + y3 – 15x2y, |
M(1, -4), |
N(-2, 0); |
9. |
Z = x2 + y3 – 21x2y, |
M(1, 0), |
N(-2, -1); |
10. |
Z = x2 + y3 – 27x2y, |
M(-2,0), |
N(-6, 1); |
11. |
Z = x2 + y3 – 15x2y, |
M(-4, 3), |
N(-6, 0); |
12. |
Z = x2 + y3 – 24x2y, |
M(-4, 2), |
N(-8, 1); |
13. |
Z = x2 + y3 – 9x2y, |
M(-1, -3), |
N(-6, -4); |
14. |
Z = x2 + y3 – 9x2y, |
M(3, 4), |
N(2, 6); |
15. |
Z = x2 + y3 –27x2y, |
M(2, 0), |
N(3, -5); |
16. |
Z = x2 + y3 – 15x2y, |
M(0, 4), |
N(0, 8); |
17. |
Z = x2 + y3 – 27x2y, |
M(3, -5), |
N(3, -8); |
18. |
Z = x2 + y3 – 27x2y, |
M(-1, -3), |
N(-4, -7); |
19. |
Z = x2 + y3 – 27x2y, |
M(3, 0), |
N(5, 3); |
20. |
Z = x2 + y3 – 12x2y, |
M(0, -4), |
N(-4, -9). |
Задача 5.Исследовать функцию двух переменных Z = f(x, y) на локальный экстремум.
1. z = x3 + y3 – 9xy;
2. z = x3 + y3 –18xy;
3. z = x3 + y3 – 27xy;
4. z = x3 +y3 – 3xy;
5. z = x3 + y3 – 21xy;
6. z = x3 + y3 – 30xy;
7. z = x3 + y3 – 6xy;
8. z = x3 + y3 – 17xy;
9. z = x3 + y3 – 5xy;
10. z = x3 + y3 – 13xy;
11. z = x3 + y3 – 25xy;
12. z = x3 + y3 – 16xy;
13. z = x3 + y3 – 19xy;
14. z = x3 + y3 – 4xy;
15. z = x3 + y3 – 7xy;
16. z = x3 + y3 – 20xy;
17. z = x3 + y3 – 2xy;
18. z = x3 + y3 – 29xy;
19. z = x3 + y3 – 11xy;
20. z = x3 + y3 – 12xy.
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