Контрольные работы 1

Задание 6. Дана функция z = f (x; y;t). x = x(t), y = y(t)

Найти полную производную dz

x

dt

x

00. z = arcsin

+t

01. z =

+t

2

y

y

x = cost

=

e

t

x

2

= ln t

y =t

y

02. z = ln sin(

y x) +3t

03. z = ex−2 y +

t

2

x =sin t

x =3t

=t5

t2 +9

y

y =

05. z = x2 ln y +t4

04. z = x2 ln y + 43 t

=sin(1 −t

2

)

2

+1

x

x =t

= cos2 (t −5)

y

y =5cost2

1

06. z =

x2 + y2 + 2t

07. z =

+ 3 t

x2 − y2

x = ln t

x = et

1

2

y

=t

−

1

t

y

=

t

08. z = ln arctg

x

+ 4t

x

+t3

y

09. z = arcsin

1

− y

x = 2t −et 2

=ty

t

x

=t

−1

=1 (1−t)

y

y

t

11. z = x

y − tg(1− x) + t

10. z =3

x −arccos

y

+t

2

x

= tg

t

x

2

= cos3 t

y

x =

−t2

1

13. z =

1

−arc tgxy3 +t

=1

− x

4

y

x

x

x = (1−t)(1+t)

12. z = arcsin

+t

1− y

1−e2t

2

t

x = ctg

y =1−t4

81

14. z = x −12 y + 3 t

x = ln(1−t)

y =t3 +1t

16. z = x2 + y2 − xy + 2t

x =3t2 −et3y = 22

t

18. z = e4x−3 y +t

			2
x =3t
y = 4t +ln(1−t)
20. z =3			x +sin xy +3t
		3
x =
		t5
					4t	2	)
y = ln(1+
22. z =			x		+t2
		x − y

x =3cost

y =t −sin 4t

24. z = x2 ln y + 3 t

x = 5t

y = 5t

26. z = arcsin(x − y) +t3

x =3t

y = 4t3 −1t

15. z =

2 − x

+t4

x2 − y

4t

x = e

=3 −t5

y

17. z =3x +

y

−

x

+

t

y

1

x

= 4t +

t

=1−sin

2

t

y

19. z = ln(x − y2 ) + 2

t

= ctg

t

x

=1−sin2 t

y

21. z = x y +t5

x

= ctgt

=

7

y

t

23. z = cos

x

+t

1

− y2

= e

7−2t

x

=t2 ln t

y

25. z = arctgxy + 2t

t

x = e

y =t5 +7

27. z = arccos x y +t

1−t2

x = e

y =sin5t

82

	28. z = x2 y −	y	+ t
		x

x =5t2y = 1−7 t

30. z = ln(ex − y2 )

x = ln ty =5 −t

32. z =			5x		−	3t
			x − y
			3
x =			t
	=5t
y
34. z = x − y				+t4
			x
x = 2 (1−t)
	= 43 t
y

36. z = ln2 (x − y2 ) +t3

x = cost

y = 13t

38. z = arcsin 1−y x +t

x = cos5t

y = 12t

40. z = x −ln y2 +3t2

	t
x = 2t −cos

y = e1−4t

29. z = e2( x−y)				+t2
	cos y
x = cos5t
				1
			t −
y = 4				t
		x
31. z = arcsin			+t
		y
x =3	t
	1
y =ty
	t

33. z = arcsin(x − y2 ) +t3

x =		3t
		4
y	=
			5 −t
35. z =			x ln			y	+ 2t
						x
				2	t
x =sin
	=		3 t
y

37. z = arccos(2 − x2 y) + t

x = e4ty = 5t

39. z = tg2 (x − y) −t2x =sin(1 −4t)

y = e2t

41. z = x2 − y12 − 3 t

		1
	x =
		2t
		1−t
	y =

83

	42. z = xy −	y	+ 4t
		x

x = ln(1−4t)

= cos2 t

y

44. z =

x +

y

+t2

1− x

= tg

2

t

x

y =3 −t

46. z = tg(x − y)2 +t

x

=5 −3t

4

y

=

t

=

5e

2t

x

=

cost

y

50. z =

x − y

+5t2

x + y

x

=3t3 −5

1

y

=

2

t

tg

52. z = 4x3 −cos xy −t

=

3

t

x

= arcsin5t

y

54. z =

sin(x − y) − 3 t

= e

5−4t

x

= tgt2

y

56. z = tg3(x − y) −t

7t

x = e

+sin xy +t

48. z = x y

= 7 −8t2

y

43. z = x y −cos xy +t

	x =t2 +5
					1
	y =
				sin t
45. z =5 − x3 y −5t3
	x =3 −7t
					1
	y =
				sin t
47. z = e5x−y +							t
			2		(1−t)
x = ln
y =3t +7
49. z = x2 y − x							y −t2
		2
x =		t

y = arctg(1−t)

51. z =3x2 − xy +cos y2 +t

x = arcsin 2t
		1
y	=
			3		t
53. z =			xy − x − y + 4t3
x = e4t
			5
y	=
			t2
55. z = x2 − xy + y +t
	=	4		3			t
x
	= ln t2
y
57. z = ln2 (x − y2 ) +3t4
							5
x		=
								t
		=					5t
y				3

84

58. z = 3 cos(x2 − y) −t3

x = 2 −3t

= e4t

y

60. z = x −cos

x

y

4

−3t

x = e

=8 3 t

y

62. z =

1

+

t

x −3y

x = tg25t

1

y

=

t

64. z =

1−ln(x − y) +3 t

x =

3e4t

1

y

=

t3

66. z =

x2 −4x2

5 −cos x

x =

4 −3t

1

y

=

3

t

68. z =

1

+3t2

x − y

2

t

x =sin

= ln3

t

y

70. z = sin xy

−t2

1− y

x =1ty = e2t

59. z =

x − y

+t2

x + y

= cos

2

(1−t)

x

= 4t

y

61. z = xy −3

y x2 −t2

7

x

=5t −

1

−8t

= ln t

2

y

63. z =sin2 (x −

y ) +t3

x =3t2

1

y

=

4 −t

65. z =

5 − x

+ 3 t

y − x

2

(1

−t)

x =sin

y =5t7

67. z

=

x + x

y −

1

t

x = e5t

1

y

=

2

−t

1
69. z =		+5t
	sin (x − y)

x = e5t −4

= cos2 3

y t

71. z = tg(x −4 y) − t

x = ln t2

y = arcsin(1−t)

85

72. z = ctg2 (x −2 y) +3t

x = e4t

y =3 −5t

74. z = x2 − xy +3 y + 2t

x =3 −t2
				1
y	=
				t
	4
76. z =				4		−t2
		sin(x − y)

x =5 ty = 12t

78. z = arccos1−x y −3t2

x = e5t

	−t2
y = 4

80.z = cos(x1− y2 ) + 2t

x =35ty = 31t

82. z =		x − y				+t2
		ln(xy)
				2
x = 4 −t
	=5cos2 t
y
84. z =		x − y		2			+5t
		x + y
			5t
x = e
	=1−t2
y

73. z = arcsin

x

+ 3 t

y

x =3 −5t2

1

y =

t

x

75. z = 2arctg

+t

y

2

x =1−3t

= cos2 t

y

77. z =

5 − x

+t

tgxy

1

x =

t

=sin

2

t

y

79. z =

1−arctg

x

+ 3 t

y

x = 2 −ln t

=sin2 t

y

81.z = 4 − x2 +6t

x− y2

		4
	x =
		t2
	y = arctg t

83. z = arcsin(x − y) − 3 t
					4
		5t		3	−		t
x =
	= e7t
y
85. z = ex2				− x			y −t
x = ln2 t
				1
y	=
			cost

86

86. z =			x − y2							−t3
		cos(x − y)
x = 2t −cost
	= e4t
y
88. z =		cos(3x2 − y) + 5 t6
x =sin 2t
	= e4t
y
90. z =				x2 + 4 y2							+t2
						x − y
						sin5t
x =
			=3 −4t
y
92. z =				tgxy					−4t
				x − y
x =						sin5t
						1
y			=
					t2

94. z = ln2 tg(1 − xy) +t

x =1−1 t

y = cos1 t

96. z = 3 tg2 xy −t3

x = 2t −4

y = 1

cost

98. Z = x2 + y2 −t x2 − y2

x = arcsin(1−t)

y = 12t

87. z = arctg(x − y2 ) + 3 t

			1
x =				t
					t
y	= 2
89. z =							1				−t2
		sin2 (x − y)
						2		t
x = cos
	=1−e5t
y
91. z =					x				+ 2t
			1		− y2

x = arctgln t

y = 12t

93. z = x − xy +3t

x =3 −4t2y = 1t

e

95. z = arccos1 −x y +t2

		5t +7
x = e
y =sin2 t
97. z =	x	+ y	2	+ 2t
		tgx

x = 1t2

y =3t +cos2

99. z = x2 −14 y2 +6t2

x =sin3(2 −4t)

y = 1t

87

Задание 7. Найти уравнение касательной плоскости и нормали

к поверхности в точке М(x0 ; y0 ; z0 ) , если уравнение поверхности задано явно z=f(x,y) либо неявно F (x; y; z) = 0 .

00. x2 + 2 y2 −3z2 + xy − yz −2xy + 21 = 0,			M (1;−2;3)
01. x2 + 2 yz − y2 + z −6 = 0,			M (1;1;2)
02. x2 − y2 −2z2 − xy − z +11 = 0,			M (1;1;2)
03.	3x2 + y2 − z2 + xy −2zy + xz −7 = 0,		M (1;1;−1)
04.	2x2 −3y2 + z2 − yx + 2xz + 28 = 0,		M (1;3;1)
05.	3x2 −2 y2 + 4z2 − xy + yz −8xz −37 = 0,		M (0;1;3)
06.	4x2 − y2 +3z2 − xy − yz + 2xz − 24 = 0,		M (0;1;3)
07. x2 −4 y2 + 2z2 − xy + yz −3xz −3 = 0,			M (1;0;2)
08. x3 + y3 + z3 + xyz −6 = 0,			M (1;2;−1)
09.	2x2 −3y2 − z2 + 2xy + 2 yz − xz −38 = 0,		M (4;1;−1)
10. 2x2 + y2 − z2 − xy −8yz +5xz − 44 = 0,			M (3;−1;1)
11. 3x2 − y2 + 4z2 − xy +3yz −3xz −46 = 0,			M (4;2;−1)
12. 3x2 + 2 y2 + z2 + xy − yz + 2xz − 21 = 0,			M (0;3;−1)
13. 4 + x2 + y2 + z2 = x + y + z,			M (2;3;6)
14. ez−y − xz + y2 −3 = 0,			M (1;0;−2)
15. ln(x2 + y + z) − yz + x3 −5 = 0,			M (1;2;−2)
16.	x + y2 + z2 + y2 − z2 −4 = 0,		M (−2;3;3)
17. x2 y2 z2 −4 = 0,			M (1;2;−1)
18. ln(x2 − z) + y2 z − xy −1 = 0,			M (2;1;3)
19. e8 + xyz −ey = 0,			M (0;1;1)
20.	x2 + y2 + z2 − y2 + 2 = 0,		M (2;3;6)
21. x2 −ln(x2 − y2 + z) − z2 +7 = 0,		M (3;2;−4)
22.	x2 + y2 + z2 + xy −2xy = 0,	M (2;1;3)
23. exy−y2 z + z −2x = 0,		M (2;0;3)
24.	x2 + yz −3x − x2 y +3 = 0,	M (2;1;3)

88

	25.	x3 + yz − x − x2 y +1 = 0,					M (2;1;3)
	26. x3 yz2 −2 y2 = 0,						M (1;2;2)
	27. exz − x3 y + y −3z = 0,						M (0;2;1)
	28. arctg(x2 y) − y3z + xz2 − 4 = 0,						M (1;0;2)
	29. 5 −		x3 + y2 + 4z + z2 y −3z = 0,				M (2;2;1)
	30. ln(x3 + y2 − z) −5xy + z2 −6 = 0,						M (1;2;4)
	31. xyz −arcsin				y	+ x3z −8z = 0,	M (2;0;1)
			x		x
	32. arctg			+ y3z −3y2 x −3 = 0,			M (0;1;3)
			z2
	33.	z2 y + 2x3 + y3x −2z −1 = 0,					M (2;1;3)
	34. ez2 y + xz2 − x2z +5 = 0,						M (3;0;2)

35. arcsin( y2 − z) + x3z2 − y3x −8 = 0, M (2;0;1)

36. ln(1 + x3 − y2 + z2 ) − z3 y +3 = 0,		M (2;3;1)
37. ex2 y − x3z + z4 −4 y −77 = 0,		M (0;1;3)
38. 3x2 z − y3 +5xy2 −5x −4 = 0,		M (1;2;−1)
39. 2xy − z2 y2 + x2 y +1 = 0,		M (2;1;3)
40. x2 + y2 −3z2 + xyz = 0,		M (1;1;1)
41.	5x2 y − z2 x + y3z −3 = 0,	M (1;1;2)
42. xy + x2 − y3 + z3x −3 = 0,		M (2;1;−1)
43. z3 − y3 + x2 y −5 = 0,		M (2;−1;2)
44.	5x2 −7 y2 + z2 − xyz = 0,	M (1;1;2)
45.	3x4 z + 2x2 y3 − yz2 −7 = 0,	M (1;2;3)
46. 6x2 y2 +5z2 x + 4 y2 zx3 −2 = 0,		M (2;1;−1)
47. 2xy +3y2 z −5xz2 −3x = 0,		M (3;2;1)
48.	5x3 −8y2 z2 +8z3x2 = 0,	M (2;3;1)
49.	3xyz − x2 y − y2 +1 = 0,	M (3;1;1)
50. z = x2 + y2 − xy +8y −3x,		M (3;1;6)
51. z = x2 −2 y2 +8xy + x −3,		M (1;3;5)

89

52. z =3x2 − y2 +9xy −5y −5,

M (1;2;4)

53. z =5x2 −4 y2 −3xy −5,

M (2;1;5)

54. z = 4x2 −3y2 + 2xy +7x +3,

M (1;3;7)

55. z =5x2 + 2 y2 − xy + y −7x +8,

M (2;−1;1)

56. z = x2 − y2 −5xy + y −4x −6,

M (3;−1;4)

57. z =9x3 −3x2 y −2 y2 x +8xy −7,

M (1;2;4)

58. z = xy − y2 + x −7 y +9,

M (1;2;−6)

59. z =3x2 −4 y2 + xy − y +7x,

M (3;4;−8)

60. z = 2x2 −3y2 − xy + y +5x,

M (2;2;4)

61. z = 2x2 − y2 − xy +9 y +7,

M (2;−1;7)

62. z =3x2 −2 y2 −4xy −8x −4 y,

M (1;−1;1)

63. z = x2 + y2 −7xy −8y − x,

M (1;−2;2)

64. z =3x2 −2 y2 −6xy −5x −9,

M (1;−2;3)

65. z = xy +ln(x2 + y2 −1),

M (1;1;1)

66. z = x2 + y2 − xy −2 y,

M (3;2;3)

67. z = ln(3y2 −2x2 ),

M (1;1;0)

68. z = ln(5x2 −4 y2 ),

M (1;1;0)

69. z = arc tg

y

2

,

M (1;3;

π

)

9x

4

70. z = ln(x2 + y2 −12) + xy,

M (2;3;6)

71. z = arccos

4x

,

M (1;4;0)

y

72. z = arctg xy ,

M (2;1; π)

2

4

73. z = arcsin

2x2

,

M (1;2;

π

)

y

4

74. z = ln(4 y2 −3x2 ) + x2 y

M (−1;1;1)

75. z = ex+2 y + xy2

M (−2;1;−1)

76. z =

xy

2

xy

M (3;3;6)

3

−

90