# Дифференциальное исчисление функций нескольких переменных. Скляренко В.А - 53 стр.

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• ## Математический анализ

28. F(x, y, z)= y
2
+ yx+ 2 yz+ 7 x
2
xz z
2
+ 5 z 5; x
0
= 0, y
0
= 1, z
0
= 1.
29. F(x, y, z)= 4 y
2
+ 14 yx + 7 yz+ x
2
+ xz z
2
+ 5; x
0
= 1, y
0
= 0, z
0
= 2.
30. F(x, y, z)= y
2
+20 yx+5 yz+2 x
2
+xzz
2
+18; x
0
= 1, y
0
= 0, z
0
= 4.
Задача 9. Найти в точке M(x
0
, y
0
)дифференциалы первого и второго
поря дков функций u = u(x, y)и v = v(x, y), за данных неявно системой
уравнений F(x, y, u,v)= 0, G( x, y, u, v)= 0, если u(x
0
, y
0
)= u
0
, v(x
0
, y
0
)=
= v
0
.
1. F = u
2
+ v + x 2, G = v
2
+ u + y 1; x
0
= 0, y
0
= 1, u
0
= 1, v
0
= 1.
2. F = u
2
+ v + x 1, G = uv + y 1; x
0
= 1, y
0
= 0, u
0
= 1, v
0
= 1.
3. F = u + v + y, G = u
2
+ v
2
+ x + 1; x
0
= 2, y
0
= 1, u
0
= 0, v
0
= 1.
4. F = u + v + x + y, G = u
2
+ xv 4; x
0
= 1, y
0
= 1, u
0
= 2, v
0
= 0.
5. F = u v x y 1, G = uv xy 1; x
0
= 1, y
0
= 1, u
0
= 1, v
0
= 0.
6. F = u
2
v
2
x, G = u
2
+ u y 3; x
0
= 0, y
0
= 1, u
0
= 1, v
0
= 1.
7. F = u
2
v y + 1, G = uv + x 1; x
0
= 0, y
0
= 1, u
0
= 1, v
0
= 1.
8. F = u v y + 2, G = u
2
+ v
2
x; x
0
= 1, y
0
= 1, u
0
= 1, v
0
= 0.
9. F = u v x + y+ 1, G = u
2
xv 1; x
0
= 1, y
0
= 1, u
0
= 2, v
0
= 3.
10. F = u + v x y, G = uv 2 xy + 2; x
0
= 1, y
0
= 1, u
0
= 0, v
0
= 2.
11. F = u
2
+ v x 2, G = v
2
u + y + 1; x
0
= 0, y
0
= 1, u
0
= 1, v
0
= 1.
12. F = u
2
v + x, G = uv
2
y x + 2; x
0
= 1, y
0
= 1, u
0
= 0, v
0
= 1.
13. F = u v + y + 1, G = u
2
v
2
+ x; x
0
= 1, y
0
= 2, u
0
= 1, v
0
= 0.
14. F = u
2
+ v 2, G = v
2
+ u + x + y; x
0
= 1, y
0
= 1, u
0
= 1, v
0
= 1.
15. F = u v + x y+ 2, G = u
2
vy 2; x
0
= 1, y
0
= 1, u
0
= 2, v
0
= 2.
16. F = u + v + x + y, G = uv + xy + 2; x
0
= 0, y
0
= 1, u
0
= 1, v
0
= 2.
17. F = u
2
+ v y, G = uv + x + y; x
0
= 1, y
0
= 1, u
0
= 1, v
0
= 0.
18. F = u + v x, G = u
2
+ v
2
y; x
0
= 1, y
0
= 1, u
0
= 0, v
0
= 1.
19. F = u + v x 2, G = uv + xy 1; x
0
= 1, y
0
= 1, u
0
= 2, v
0
= 1.
20. F = u
2
v + x, G = v
2
u y 1; x
0
= 0, y
0
= 1, u
0
= 1, v
0
= 1.
21. F = u + v + x + y 2, G = v
2
+ 2 ux 5; x
0
= 2, y
0
= 2, u
0
= 1, v
0
= 1.
53
  28. F(x, y, z) = −y2 + yx +2 yz+7 x2 − xz− z2 +5 z−5;         x0 = 0, y0 = 1, z0 = 1.
29. F(x, y, z) = −4 y2 + 14 yx + 7 yz + x2 + xz − z2 + 5;   x0 = 1, y0 = 0, z0 = −2.
30. F(x, y, z) = −y2 +20 yx+5 yz+2 x2 +xz−z2 +18;           x0 = 1, y0 = 0, z0 = −4.

Задача 9. Найти в точке M(x0 , y0 ) дифференциалы первого и второго
порядков функций u = u(x, y) и v = v(x, y), заданных неявно системой
уравнений F(x, y, u, v) = 0, G(x, y, u, v) = 0, если u(x0 , y0 ) = u0 , v(x0 , y0 ) =
= v0 .

1. F = u2 + v + x − 2, G = v2 + u + y − 1;      x0 = 0, y0 = −1, u0 = 1, v0 = 1.
2. F = u2 + v + x − 1, G = uv + y − 1;      x0 = 1, y0 = 0, u0 = −1, v0 = −1.
3. F = u + v + y, G = u2 + v2 + x + 1;      x0 = −2, y0 = −1, u0 = 0, v0 = 1.
4. F = u + v + x + y, G = u2 + xv − 4;       x0 = 1, y0 = 1, u0 = −2, v0 = 0.
5. F = u − v − x − y − 1, G = uv − xy − 1;      x0 = −1, y0 = 1, u0 = 1, v0 = 0.
6. F = u2 − v2 − x, G = u2 + u − y − 3;      x0 = 0, y0 = −1, u0 = 1, v0 = 1.
7. F = u2 − v − y + 1, G = uv + x − 1;      x0 = 0, y0 = 1, u0 = 1, v0 = 1.
8. F = u − v − y + 2, G = u2 + v2 − x;      x0 = 1, y0 = 1, u0 = −1, v0 = 0.
9. F = u − v − x + y + 1, G = u2 − xv − 1;      x0 = 1, y0 = 1, u0 = 2, v0 = 3.
10. F = u + v − x − y, G = uv − 2 xy + 2;       x0 = 1, y0 = 1, u0 = 0, v0 = 2.
11. F = u2 + v − x − 2, G = v2 − u + y + 1;      x0 = 0, y0 = −1, u0 = 1, v0 = 1.
12. F = u2 − v + x, G = uv2 − y − x + 2;       x0 = 1, y0 = 1, u0 = 0, v0 = 1.
13. F = u − v + y + 1, G = u2 − v2 + x;      x0 = −1, y0 = −2, u0 = 1, v0 = 0.
14. F = u2 + v − 2, G = v2 + u + x + y;      x0 = −1, y0 = −1, u0 = 1, v0 = 1.
15. F = u − v + x − y + 2, G = u2 − vy − 2;       x0 = −1, y0 = 1, u0 = 2, v0 = 2.
16. F = u + v + x + y, G = uv + xy + 2;       x0 = 0, y0 = −1, u0 = −1, v0 = 2.
17. F = u2 + v − y, G = uv + x + y;       x0 = −1, y0 = 1, u0 = −1, v0 = 0.
18. F = u + v − x, G = u2 + v2 − y;      x0 = 1, y0 = 1, u0 = 0, v0 = 1.
19. F = u + v − x − 2, G = uv + xy − 1;       x0 = 1, y0 = −1, u0 = 2, v0 = 1.
20. F = u2 − v + x, G = v2 − u − y − 1;      x0 = 0, y0 = −1, u0 = 1, v0 = 1.
21. F = u + v + x + y − 2, G = v2 + 2 ux − 5;      x0 = 2, y0 = −2, u0 = 1, v0 = 1.

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