Контрольные работы 1

Продолжение прил. 2

20		21		22			23
a) y = (arcsin x)2	a) y =	x 5x	a) y =		x −2		a) y = (3 − x2 )e−2 x
					x2 −5
x = arccost,	x =t cost,		x = arctgt,				x = (sin t)−1	,
б)	б)	=t sin t	б)	=t(1+t2 )			б)
y = ln(1−t2 )	y		y				y = ctgt

24	25
a) y =sin3 x	a) y = (1+ x)2 cos3x
x = arctgt,	x = tgt,
б)	б)
y =sin2 t	y = (1+t2 )2

28						29
a) y =sin4 x					a) y = cos4 x
	=	ln	4	x,	x =sin t,
x					б)
б)						y = ln(1	+t2 )
y = cos3 t

			26				27
	a) y = ln tgx					a) y = ln		x −1
								x +1
		=	te	−t	,	x = arcsin t,
	x					б)
	б)						y = (1+t3 )2
	y = ln(1+t2 )

	30				31
a) y = arcsin(2sin x)					a) y = ln3 1+ x2
	=	t	2	,	x = ln(1 −cost),
x					б)
б)						y	=t3
y = cos3 t

32

33

34

35

a) y = arccos(3cos x)

a) y =

1+ x4

a) y =

2x −3

a) y = xsin x

x +

5

2

x =sin t,

x

= arctgt,

x = 4cos

2t,

x =t

,

б)

4

б)

=t

5

б)

б)

t3

−t

y =t

y

y =3sin t

y =

3

36

37

38

39

a) y = x ln x

a) y = xe−4 x

a) y =

x +3

a) y = x2 ln x

x2 −4

x =t

−sin t,

2

,

= e

2t

,

2

t,

x =t

x

x = cos

б)

−cost

б)

б)

= e3t

б)

y =1

y =t +t3

y

y =t3

71

					Продолжение табл. 2
40	41		42		43
a) y = x3 ln x	a) y = x3ex		a) y = x2 sin x		a) y = x3 4x
x = arctgt,	x = arctgt,		x = arcsin t,		x =t	−sin t,
б)	б)	+t2 )	б)	−t2	б)	−cos.
y = ln(1 +t2 )	y = ln(1		y = 1		y =1

44

45

46

47

a) y = ex2

a) y = (1+ x)2 cos3x

a) y = ln tgx

a) y = ln

x −1

x +1

=

2cos

3

t,

x = cos2t,

x = ln(1

+t

3

),

x = arc tgt,

x

б)

б)

1

б)

3 t

y =sin2 t

y =t

б)

y =

t2

y = 2sin

2

48		49		50	51
a) y = e2 x	a) y = e−x cos x		a) y = (1− x2 )cos x		a) y = x3 ln x
x = e−t ,	x = ln t,		x =sin3t,		x =3t −t3		,
	б)		б)
б)		y =t5		y = cos2 t	б)	2 t
y =t3					y = 4sin

52

53

54

55

a) y =

1+ x

a) y =

1 + x

a) y = ln

1

a) y = x arcsin5xsin5x

x

x2 −5

1

− x

x =t −sin t,

x

=

e

2t

,

x

=3cos2 t,

x =t +ln cost,

б)

б)

б)

y =t5 +3t

y = cos3t

б)

=

y =t −lnsin t

y

2sin

3

t

56					57
a) y = xtg2x					a) y = x2arctgx
x = ln t,					x =5cost,
	1		1
б)					б)
y =	2	t +	t		y = 2sin3 t

	58				59
	a) y = ectgx				a) y = xe−x2
			t		x =t −sin t,
	x = cos			,
			2
	б)				б)
		−sin t			y = arctgt
	y =1

72

Продолжение табл. 2

60

61

62

63

a) y = x2 cos x

a) y = e−3t cos 2t

a) y = xe x

a) y = arctg(x2 )

x = ln t,

x =t +3sin t,

x = arcsin t,

б)

x = 4cos2 t,

б)

б)

−cos 2t

б)

=t3

=3sin 2t.

y =t3

y = 2

y

y

64

65

66

67

a) y = ln(1+ 2x4 )

a) y = (1+ x)2 cos3x

a) y = ln tgx

a) y = ln

x −1

x +1

x = e

5t

,

x = ln t ,

x = (sin t)

−1

,

x = ln(t +

2),

б)

б)

t

б)

б)

y = cos3t

y =

y

= ctgt

y =t3 −4t

1

−t

68			69			70			71
a) y = ctg3 x			a) y = ln(1+ x)			a) y =	x2 −4x		a) y =	x −2
							x +5			x +3
		2	x = e	3t	,	x =t +ln x,			x = ln(3t + 2),
x = 1	−t ,
б)			б)			б)			б)	t
y = arccost			y =sin 2t			y =t −cos4t			y =3

72	73				74			75
a) y = (x −2)e1 x	a) y =		2x		a) y = tg3 x			a) y = lnsin 2x
			x −3
x =1+cos 4t,	x = arcsin t,				x = cos	2	t,		−t	,
								x = e
б)	б)		1−t2		б)			б)
y =t +sin t	y =				y =sin3t			y = e−t sin t

76

77

78

79

a) y = x2 sin x

a) y = x2arctgx

a) y = ectgx

a) y = xe−x2

x = ln(2t +7),

x =sin t +t cost,

x = arcsin t,

x

=

t cost

,

б)

б)

б)

− t

y =t

y

= ln(1

−t2 )

б)

y =t3

3

y =sin3t

73

Продолжение табл. 2

80

81

82

83

a) y =

x + 4

a) y = (1− x2 )e−x

a) y =

sin3 x

a) y = x2 cos3x

x2 −9

x = arcsin t,

=

x =

(sin t)

−1

,

=

−2t

x

arctg2t,

te ,

x

б)

б)

x + 2

б)

б)

y = ln(1+t)

y =

y = ln(1+t2 )

y = 1+t3 .

84

a) y =	3x −1
	x + 4

x =3cos 2t,

б) y = 4sin t

85

a) y = x2 sin 4x

x = t ,

б)

y = t3 −t3

	86			87
	a) y =	x2 −4		a) y = x4 ln x
		x +3
	x =t −sin t,				t	,
				x = e
	б)			б)
	y =t −cost			y = e2t

88

89

90

91

a) y = ln tgx

a) y = ln

x + 4

a) y = x ln 2x a) y = xe5 x

x −4

=

t

2

+1,

=sin t ,

2

,

x = cos

2

t,

x

x

x =t

б)

=

б)

=

2

б)

y =sin3 t

б)

y = ln(1

+ 2t)

y

cos2t

y

t

92				93	94		95
a) y = e−x cos3x		a) y = (1− x2 )sin 2x			a) y = etgx		a) y = xarccos4x
x = ln(1	+t3 ),	x = arc tgt,			x = cos 2t,		x = ln t,
б)		б)	y = t	2	б)		б)
y = t					y =3	sin t	y =5	t
			2

96

97

98

99

a) y = (arcsin3x)2

a) y =sin3 4x

a) y = ectgx

a) y = x e−x2

x

=

ln(2t

+

7),

x = ln cost,

x = arccost,

x

=

arctgt,

б)

б)

t

y = e2t

б)

=

−

б)

y =t3 −

y

1

t

3

y = ln(1 + x)

74

Задание 3 Найти наибольшее и наименьшее значения функции y = f (x)

на отрезке [a;b]

	00.	f (x) = x3	−12x +7,[0;3]
				3				π
	02.	f (x) = cos x +				x, 0;
				2
								2
	04.	f (x) = x3	−3x +1,[0,5;2]
	06.	f (x) =sin x3 +			3		x,[0;π]
					2

08.f (x) =3 − 2x2 ,[−1;3]

10.f (x) = x4 −2x2 +5,[−2;2]

12. f (x) = x3 −6x2 +9x −1,[−1;2] 14. f (x) =11−+ xx +− xx22 ,[0;1]

16.f (x) =sin 2x − x, −π,0

2

18.f (x) = x + x84 ,[−2;−1].

20.f (x) = 2x3 +3x2 −12x +1,[−1;2]

22. f (x) =sin x +	1	π
	2	x,	2	;π

24. f (x) = xx2+−35 ,[−2,5;2]

26.f (x) = cos 2x + x, 0; π

2

28.f (x) =sin x + cos x, 0; π

3

01 f (x) = x5 − 53 x3 + 2, [0;2]

03 f (x) =3x4 −16x3 + 2, [−3;1]

05 f (x) = x4 + 4x, [−2;2]

07 f (x) =108x − x4 , [−1;4]

09 f (x) = x −sin x, [−π;π]

11 f (x) = x5 −5x4 +5x3 +1, [−1;2] 13 f (x) = 100 − x2 , [−6;8]

15f (x) = 2x2 +6 , [0;4]

x+1

17f (x) = 2tgx −tg2 x, 0; π

3

19 f (x) = x22x++21 , [−4;0]

21 f (x) = ln2 x − 2ln x, 1;e7

23 f (x) = 2sin x + cos 2x, [0;π]

	25 f (x) =	x −3	, [4;6]
		x + 4

27f (x) = tgx −2x, 0; π

3

29 f (x) = x3 −12x +3, [0;3]

30.

f (x) =

x +5

,

[

4;6

]

31

f (x) = ln x −

x

, 1;3

x −3

2

[ ]

32.

π

;

π

33

f (x) = (x −1)e−x , [1;3]

f (x) = ctgx + 2x,

6

3

75

	34.	f (x) = x3 −3x2 −9x +1,[−2,0]
		f (x) = cos x +	x			π
	36.			,	0;
			2
						2

38.f (x) = ln x −2x, 1 ,1

4

40.f (x) = xe−2x ,[0;1]

42. f (x) = xx2+−59 ,[−4;4]

44. f (x) = arcsin x − 2x3 , 14 , 34

46.f (x) = 2ln x −3x, 1 ;1

2

48. f (x) = xx −+25 ,[3;5]

50.f (x) = 0,5cos 2x +sin x, 0; π

2

52.	f (x) =		x	+		3	,	[		−5;−1
						x
	3										[				]
	f (x) =32x − x4														]
54.											, 1;3
56.	f (x) = x3 −3x2 −9x +1,[−2,0]
58.	f (x) = ln x					−	1			ln		3					3
							3							x, 1;e
	f (x) =sin x +								x					π
60.											,				;π
								2						2
62.	f (x) =		x		+	2	, 1;6						]
						x
	8							[

64.f (x) = ctgx + x, π, 3π

8 4

	66. f (x) =	x −5	,[6;7]
		x +7

35f (x) = tgx + ctg2 x, π; π

6 3

37 f (x) = arctg − 2x , [0;2]

39 f (x) = e2x − 2x, [−1;1]

41f (x) = 2ln x −3x, 1 ;1

2

43f (x) = x2 −16 , [−3;0]

x+5

45 f (x) = x + x84 , [1;3]

47 f (x) = (x + 2)e−x , [−1;2]

49 f (x) = x3 x+ 2 , [−1,1;0]

51f (x) =3ln x −4x, 1 ;1

2

53 f (x) = 2x2x ++43 , [−2;2]

55 f (x) = 15 x5 − 43 x3 +1, [−1;3] 57 f (x) = x3 −12x, [1;3]

59 f (x) = 2ln x − 5x , [8;12] 61 f (x) = x2 − x3, [−1;0,5]

63 f (x) = x2e−x , [1;3]

65 f (x) = ex + e−x , [−1;2]

		3	π
	67 f (x) =		x + cos x,		;π
		2		2

76

68.f (x) = cos2 x +sin x, 0; π

4

70.f (x) = 2sin x + 0,5cos 2x, 0; π .

2

72. f (x) = x3 −3x2 +7,[−1;1].

74.f (x) = tgx −4x, π, 3 π .

6 8

76.f (x) = x5 −5x3 +10x + 2,[0;1,1].

78.f (x) = x3 −3x2 +3x + 2,[−2,2].

80.f (x) = 2x2 −ln x,[1;e].

82.f (x) = xx+−14 ,[2;3].

84.

f (x) = arctgx +

x

,[1;3].

5

]

[

86.

f (x) = x3 −12x +3, 1,3 .

88.

f (x) =

x −

4

,[5;7].

x +

3

]

f (x) =

8x2 − x4

[

90.

+ 2,

−1;3 .

f (x) =

9 − x2 ,

[

]

92.

−2;1 .

94.

f (x) = x3 −27x + 4,[0;4].

96.

f (x) =

x + 2

,[4;5].

x −3

98.f (x) = − 3x +sin 2x, 0, π .

2

69 f (x) = e3x −3x, [−1;1]

71 f (x) =sin 2x − x, [0;π]

73 f (x) = arctgx − 4x , [1;2]

75 f (x) = x3 −12x +3, [0;3]

77 f (x) =3ln x − 2x, [1;2]

79f (x) = 3x −sin 2x, 0; π

2

81 f (x) = ctgx +	4		π	;	π
	3	x,	6
					2

83 f (x) = xx2+−23 , [−1,5;0]

85f (x) = cos2 x +sin x, π;π

3

87f (x) = 3x +cos2x, − π; π

2 2

89f (x) = x +cos2 x, 0; π

2

91 f (x) = (x − 2)e−x , [1;4]

93 f (x) = arctgx − 4x , [−2;0] 95 f (x) = 4ln x − x, [3;5]

97 f (x) = xx2−+0,52 , [−3;0] 99 f (x) = x3 −3x2 +5, [1;3]

77

Задание 4. Исследовать данные функции и построить их графики

					(−1)i+ j
x2		−(i	+ j)x +i j
А) f (x) =						;
			i + j	+1
		x +
			2

Б) f (x) = (x + j)(−1)i e(−1) j (i+1) x ,

где i и j – последняя и предпоследняя цифры номера зачетной книжки.

Задание 5. Дана функцияZ = f (x, y) . Найти частные производныеz′x, z′y, z′′xy .

00. z = ln xy

02. z=ln(y2 −4x)+8

04.z = x + y

x− y

06. z = ln x −ln sin y

08.z = 2cos2 xy + xy

10.z = x arctgy

12.z = tg xy

	14. z =		1
		x2 − y2
	16. z = x arccos y
	18. z = arctg			xy
	20. z = x2 arctgxy
	22. z = x		y +		cos x
	24. z = xe		y + 2
	26. z =5		xy

01. z = x + y +	x − y
03. z=2x2 −3xy +	x − y

05. z = x2 +xy2

07. z = arctg(x − y2 )

09.z = x y −3y cos x

11.z = esin xy

y

13. z = 4tg x

15. z = y x y

17. z =3sin2 x cos y

19.z = x2 y −sin2 x

21.z = y 3x

23. z = arcsin xy

	25. z =	3xy −		x
				y
	27. z =	3
		x y

78

28.z = ctg2 (x −3y2 )

30.z = ln(x −3y2 )

32.z = ln2 (x +7 y)

	34. z =	y
		x +	8
	36. z =	ln xy
	38. z =	x +		y
		x − y

40. z =3sin2 xy −cos y

ctg	y
42. z = e	x

44. z = ln sin x +3 y

46. z = x arccos y

48.z = xy2

50.z=cos(xy2 )/(1− y)

52.z = 1xy

			arcsin	x
	54. z=2			y
	56. z=	x+3y
		y2 − x

	58. z=ln2		y
			1 − x
	60. z=		3
		x2 + 4 y2
	62. z=lntg y
			x
	64. z = ln(sin xy)
				y
	66. z= arcsin (1 − x2 )
	68. z =3xy2 + x y

29. z =	x
	x2 + y2

31.z = sin x

x+5y

33.z = ax+3 y2

35. z =	x +	3 y
	7
37. z =	cos(x2 + y)

39.z =5x+ y

41.z = ln(x +ln y)

43.z = x 3y

45.z = arctg xy

47.z = x + 3 y +1/ cos xy

	49. z =		sin x
			x − y
	51. z=x/(y −					x )
	53. z = 3x2			y +cos2 y
	55. z=		1
		x + y2

57.z=arctg x − y

x+ y

59. z=ecos2 ( x−5 y)
61. z= arcsin	y2
	x
63. z=x y
65. z = cosex2 −y
67. z= 3 sin2 (3x −			y )
69. z = x y +cos2			y

79

70. z=tg3 xy5

72. z =1− x2 y

1− xy

74. z = arctg			x	+	sin x
			y
76. z = 3 ctg2 (1− xy)
78. z =	ln xy
80. z = 2arctg( x−					y )
82. z = 3 x2 − y2
84. z = x ysin 3x
86. z =	1
	ln xy

88.z = esin2 (3−xy)

90.z =3arctg(x2 −4xy +5y)

92. z = 3 tg2 ln xx

94. z = xcos2 y − x ey 96. z = ln2 xy

x

98. z = 2x +3 y

71. z=xy3 −3x2 y2 + 2 y4

73. z =sin2 x +cos2			y
75. z = y x y
77. z = arccos		xy
		− y2
1

79. z = y2 1− tgxy

81.z = arcsin y2

x2

	83. z = ln(x +						y		)
					2x
	85. z = cos x sin2 xy
	87. z =	cos xy2					−ex
			4
	89. z =		1
		ln(x −					y )
	91. z = 2arcsin3 ( x−3 y)
	93. z =5arctg(							x +5y)
	95. z = arctg					y
			(1− x)
	97. z =3x2 +			x			−cos2 y
				y
	99. z =3x2 + tg2								xy −7

80