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1. Sets and operations over them
1. If A = {x: -4 < x < 1}, B = {x: 0 < x < 4}, find AB=
a) {x: -4 < x < 4}; b) {x: 0 < x < 1}; c) {x: -4 < x 0};
d) {x: 1 x < 4}; e) {x: -1 < x < 4};q
2. If A = {x: -4 < x < 1}, B = {x: 0 < x < 4}, find AB=
a) {x: -4 < x < 4}; b) {x: 0 < x < 1}; c) {x: -4 < x 0};
d) {x: 1 x < 4}; e) {x: -1 < x < 4};
3. If A = {x: -4 < x < 1}, B = {x: 0 < x < 4}, find A\B=
a) {x: -4 < x < 4}; b) {x: 0 < x < 1}; c) {x: -4 < x 0};
d) {x: 1 x < 4}; e) {x: -1 < x < 4};
4. If A = {x: -4 < x < 1}, B = {x: 0 < x < 4}, find B\A=
a) {x: -4 < x < 4}; b) {x: 0 < x < 1}; c) {x: -4 < x 0};
d) {x: 1 x < 4}; e) {x: -1 < x < 4};
5. If A = {x: x2 - x - 2 > 0}, B = {x: 6x – x2 0}, find AB=
a) {x: -1 < x 6}; b) {x: 0 x < 2}; c) {x: -1 < x < 0};
d) {x: 2 x 6}; e) {x: -1 < x < 6};
6. If A = {x: x2 - x - 2 > 0}, B = {x: 6x – x2 0}, find AB=
a) {x: -1 < x 6}; b) {x: 0 x < 2}; c) {x: -1 < x < 0};
d) {x: 2 x 6}; e) {x: -1 < x < 6};
7. If A = {x: x2 - x - 2 > 0}, B = {x: 6x – x2 0}, find A\B=
a) {x: -1 < x 6}; b) {x: 0 x < 2}; c) {x: -1 < x < 0};
d) {x: 2 x 6}; e) {x: -1 < x < 6};
8. If A = {x: x2 - x - 2 > 0}, B = {x: 6x – x2 0}, find B\A=
a) {x: -1 < x 6}; b) {x: 0 x < 2}; c) {x: -1 < x < 0};
d) {x: 2 x 6}; e) {x: -1 < x < 6};
9. If A = {x: sin x = 0}, B = {x: cos x/2 =0}, find AB=
a) A; b) B; c) {x: x =2n, nZ}; d) ; e) {x: x = n, nZ };
10. If A = {x: sin x = 0}, B = {x: cos x/2 =0}, find AB=
a) A; b) B; c) {x: x =2n, nZ}; d) ; e) {x: x = n, nZ };
11. If A = {x: sin x = 0}, B = {x: cos x/2 =0}, find A\B=
a) A; b) B; c) {x: x =2n, nZ}; d) ; e) {x: x = n, nZ };
12. If A = {x: sin x = 0}, B = {x: cos x/2 =0}, find B\A=
a) A; b) B; c) {x: x =2n, nZ}; d) ; e) {x: x = n, nZ };
13. If A = {(x, y): x2 + y2 1}, B = {(x, y): |x|+|y| 1}, find AB=
a) A; b) B; c) {x: -1<x <1}; d) ; e) {x: x = n, nZ };
14. If A = {(x, y): x2 + y2 1}, B = {(x, y): |x|+|y| 1}, find AB=
a) A; b) B; c) {x: -1<x <1}; d) ; e) {x: x = n, nZ };
15. If A = {(x, y): x2 + y2 1}, B = {(x, y): |x|+|y| 1}, find A\B=
a) A; b) B; c) {x: -1<x <1}; d) ; e) {x: x = n, nZ };
16. If A = {(x, y): max{| x|, |y|} 1}, B = {(x, y): |x|+|y| 1}, find AB=
a) A; b) B; c) {x: -1<x <1}; d) ; e) {x: x = n, nZ };
17. If A = {(x, y): max{| x|, |y|} 1}, B = {(x, y): |x|+|y| 1}, find AB=
a) A; b) B; c) {x: -1<x <1}; d) ; e) {x: x = n, nZ };
18. If A = {(x, y): max{| x|, |y|} 1}, B = {(x, y): |x|+|y| 1}, find B\A=
a) A; b) B; c) {x: x =2n, nZ}; d) ; e) {x: x = n, nZ };
2. Functions and their types
19. Rule f: EF is called as a function, if
a) xD(f) E: y!F, y = f(x); b) xD(f) E: y!F, y = f(x);
c) xD(f) E: yF, y = f(x); d) xD(f) E: yF, y = f(x);
e) yR(f) F: !xE, y = f(x);
20. Function f: EF is surjection, if
a) f(x1) = f(x2) x1=x2; b) xD(f) E: y!F, y = f(x);
c) xD(f) E: yF, y = f(x); d) xD(f) E: yF, y = f(x);
e) yF: xD(f), y = f(x);
21. Function f: EF is injection, if
a) f(x1) = f(x2) x1=x2, x1, x2D(f); b) xD(f) E: y!F, y = f(x);
c) xD(f) E: yF, y = f(x); d) xD(f) E: yF, y = f(x);
e) yF: xD(f), y = f(x);
22. Function f: EF is bijection, if
a) f(x1) = f(x2) x1=x2, x1, x2D(f); b) xD(f) E: y!F, y = f(x);
c) xD(f) E: yF, y = f(x); d) xD(f) E: yF, y = f(x);
e) yF: x!D(f), y = f(x);
23. Function f(x) = 3 sin x/2, f: [0, 1][0, 3], is
a) injection; b) surjection; c) bijection; d) composition; e) inverse
24. Function f(x) = tg x/4, f: [0, 1][0, 3], is
a) injection; b) surjection; c) bijection; d) composition; e) inverse
25. Function f(x) = 3x, f: [0, 1][0, 3], is
a) injection; b) surjection; c) bijection; d) composition; e) inverse
26. Function f(x) = 12(x – ½)2, f: [0, 1][0, 3], is
a) injection; b) surjection; c) bijection; d) composition; e) inverse
27. Function f(x) = 3 – (16/3) (x - ¼)2, f: [0, 1][0, 3], is
a) injection; b) surjection; c) bijection; d) composition; e) inverse
28. Function f(x) = 2|x+2| - 3, f: [0, 1][0, 3], is
a) injection; b) surjection; c) bijection; d) composition; e) inverse
29. Function f(x) = cos x/2, f: [-1, 1][0, 1], is
a) injection; b) surjection; c) bijection; d) composition; e) inverse
30. Function f(x) = - x2 +1, f: [-1, 1][0, 1], is
a) infection; b) surjection; c) bijection; d) composition; e) inverse
31. Function f(x) = |x|, f: [-1, 1][0, 1], is
a) injection; b) surjection; c) bijection; d) composition; e) inverse
32. Function f(x) = (x+1)/2, f: [-1, 1][0, 1], is
a) injection; b) surjection; c) bijection; d) composition; e) inverse
33. Function f(x) = (x+1)/3, f: [-1, 1][0, 1], is
a) injection; b) surjection; c) bijection; d) composition; e) inverse
34. Function f(x) = 2x-1, f: [-1, 1][0, 1], is
a) infection; b) surjection; c) bijection; d) composition; e) inverse
3. Domain and range of a function
35. Find domain D(f1) if f1=lg (16 – x2)
a) (-4; 4); b) {xR: x/2+2n, nZ}; c) {x(-4; 4): x/2};
d) [-4, 4]; e) (-; ).
36. Find domain D(f1) if f1=1/ (1 – sin x)
a) (-4; 4); b) {xR: x/2+2n, nZ}; c) { xR: x/2};
d) [-4, 4]; e) (-; ).
37. Find domain D(f1+f2) if f1=lg (16 – x2) and f2=1/ (1 – sin x)
a) (-4; 4); b) {xR: x/2+2n, nZ}; c) {x(-4; 4): x/2};
d) [-4, 4]; e) (-; ).
38. Find domain D(1/f1) if f1=x2- x + 1
a) (-1; 1); b) {xR: x2, x-2}; c) {xR: x1, x-1};
d) [-2, 2]; e) (-; ).
39. Find domain D(1/f1) if f1=|x| - 2;
a) (-1; 1); b) {xR: x2, x-2}; c) {xR: x1, x-1};
d) [-2, 2]; e) (-; ).
40. Find domain D(f1) if f1=5x- 2x + 1
a) (-5; 5); b) {xR: x2, x-2}; c) {xR: x5, x-5};
d) [-2, 2]; e) (-; ).
41. Find range R(f), if f(x) = 2x – 5, x[-2; 2],
a) (-9; -1); b) [-9;9]; c) (-;-9)(0)(9; +);
d) [-9, +); e) (-; ).
42. Find range R(f), if f(x) = |x – 1|, x[0; 5],
a) (-4; 0); b) [0;4]; c) (-;-4)(0)(4; +);
d) [-4, +); e) (-; ).
43. Find range R(f), if f(x) = x +sign x, xR,
a) (-1; 1); b) [0; 1]; c) (-;-1)(0)(1; +);
d) [-1, +); e) (-; ).
44. Find range R(f), if f(x) = x2 + 2x – 3, xR,
a) (-4; -1); b) [-4; 4]; c) (-;-4)(4; +);
d) [-4, +); e) (-; ).
45. Find range R(f), if f(x) = x +1/x, x[0; +],
a) (-2; 2); b) [0; 2]; c) (-;-2)(2; +);
d) [2, +); e) (-; ).
4. Composition and inverse function
46. f(x)=4/x-5, g(x)=2x+7. Find h(x)=(gоf)(x)=g(f(x))
a) 2/x2+7/5; b) 8/x-3; c) 10; d) 8x-35; e) 2/x2-5/7
47. g(х)=х2 and f(х)=х+1, then g(f(х))=(gоf)(х)=
a) (х+1)2; b) х+1; c) х2+1; d) х2; e) (х-1)2
48. g(х)=х2 and f(х)=х+1, then f(g(х))= (fоg)(х) =
a) х2+1 b) (х+1)2; c) х+1 d) х2; e) (х-1)2
49. f(x)=x5, g(x)=x+5. Find h(x)=(gоf)(x)=g(f(x))
a) (х+5)5, xR; b) х5+5, xR; c) х+5, xR; d) х5, xR; e) (х-5)5, xR.
50. f(x)=x5, g(x)=x+5, then f(g(х))= (fоg)(х) =
a) (х+5)5, xR; b) х5+5, xR; c) х+5, xR; d) х5, xR; e) (х-5)5, xR.
51. For the function f(x) = tg x, 3/2 < x < 5/2, find inverse function:
a) f -1(x) = 2 + arctg x, xR; b) f -1(x) = arctg x, xR; c) f -1(x) = arcctg x, xR;
a) f -1(x) = 2 + arcctg x, xR; a) f -1(x) = ctg x, xR;
52. , then =
a) x/4 -0,2; b) 4/(x+5); c) 4x+5; d) x/(5x-4); e) 5/x – 4.
53. f(x) =5x, then =
a) x; b) 1/(5x); c) 1/x; d)1/(1+5x); e) (1+5x).
54. f(x) =2x - 1, then =
a) (x+1)/2; b) 1/(2x - 1); c)|x|1/2 sign x; d)1/(1 - 2x); e) (1+2x).
55. f(x) =(x – 1)1/2, then =
a) (x+1)2; b)1/(1- x)1/2; c)|x|1/2 sign x; d)1/(x - 1)1/2; e) (1+x).
5. Inf and sup of a set. Bounded sets
56. sup X = a
a) b)
c) d)
e)
57. inf X = a
a) b)
c) d)
e)
58. Let X={½ n/(2n + 1)}, nN. Find inf X=
a) 1; b) 0; c) -1; d) -; e) ½
59. Let X={½ n/(2n + 1)}, nN. Find sup X=
a) 1; b) 0; c) -1; d) ; e) ½
60. Let X={1/n}, nN. Find sup X=
a) 1; b) 0; c) -1; d) ; e) ½
61. Let X={1/n}, nN. Find inf X=
a) 1; b) 0; c) -1; d) ; e) ½
62. Let X={1+(-1)n/n}, nN. Find sup X=
a) 3/2; b) 0; c) -1; d) ; e) ½
63. Let X={1+(-1)n/n}, nN. Find inf X=
a) 1; b) 0; c) -1; d) ; e) ½
64. Set X is a set bounded below, if
a) b)
c) d)
e)
65. Set X is a set bounded above, if
a) b)
c) d)
e)
66. Set X is a bounded set, if
a) b)
c) d)
e)
67. If XY, then
a) sup X sup Y and inf X inf Y; b) sup X sup Y and inf X inf Y
c) sup X sup Y and inf X inf Y; d) sup X sup Y and inf X inf Y;
e) sup X inf Y and inf X sup Y.
6. Bounded and unbounded functions. Inf and sup of a function
68. Function f is bounded, if
a) C=const>0, xD(f): |f(x)| C; b) C=const>0, xD(f): |f(x)| C;
c) C=const>0, xD(f): |f(x)| C; c) sup f = inf f;
e) inf f sup f.
69. Which of the following functions is unbounded?
a) y = x2 – x – 1, x[-1; 5]; b) y = 1/(x – 10), x[0; 5]; c) y = x3 /(x4 + 1), xR;
d) y = (x2 –1)/ |x3 – 1|, xR; e) y = xx, x[0; +];
70. Which of the following functions is bounded?
a) y = 0,4x, xR; b) y = log0,1 x, x[1; +); c) y = x3 /(x4 + 1), xR;
d) y = logx 2, x[1; +); e) y = xx, x[0; +];
71. Functions and are considered on (-; ). Bounded below function is
A) both of them; B) neither of them; С ) only ; D) only .
72. Functions and are considered on (-; ). Bounded above function is
A) both of them; B) neither of them; С ) only ; D) only .
73. For the function f(x) = 2-|x+2| find inf f, if it exists:
a) 1; b) 0; c) -; d) -1; e) 4.
74. For the function f(x) = 1 - 21/(x-1) find inf f, if it exists:
a) 1; b) 0; c) -; d) -1; e) 4.
75. For the function f(x) = 8 - 2x+1- 4x find inf f, if it exists:
a) 1; b) 0; c) -; d) -1; e) 4.
76. For the function f(x) = lg (x2 + x - 2) find inf f, if it exists:
a) 1; b) 0; c) -; d) -1; e) 4.
77. For the function f(x) = log0,1 (4x - 3 - x2) find inf f, if it exists:
a) 1; b) 0; c) -; d) -1; e) 4.
78. For the function f(x) = 2-|x+2| find sup f, if it exists:
a) 1; b) 0; c) -; d) -1; e) 4.
79. For the functn f(x) = 1 - 21/(x-1) find sup f, if it exists:
a) 1; b) 0; c) -; d) -1; e) 4.
80. For the function f(x) = 8 - 2x+1- 4x find sup f, if it exists:
a) 1; b) 0; c) -; d) 8; e) 4.
81. For the function f(x) = lg (x2 + x - 2) find sup f, if it exists:
a) 1; b) 0; c) +; d) -1; e) 4.
82. For the function f(x) = log0,1 (4x - 3 - x2) find sup f, if it exists:
a) 1; b) 0; c) +; d) -1; e) 4.
7. Sequence. The greatest and least member of the sequence
83. - neighborhood of a point a
a) {x: |x - | < a}; b) {x: |x – a| < }; c) {x: |x - | > a};
d) {x: |x – | a}; e) {x: |x – a| };
84. Define which of the numbers a,b is a member of the sequence {xn}, if a=1215, b=12555, xn=532n - 3, nN,
a) a; b) b; c) both of them; d) neither of them
85. Define which of the numbers a,b is a member of the sequence {xn}, if a=6, b=8, xn=(n2+32n)½ - n, nN,
a) a; b) b; c) both of them; d) neither of them
86. Define which of the numbers a,b is a member of the sequence {xn}, if a=6, b=11, xn=(n2+11)/(n+1), nN,
a) a; b) b; c) both of them; d) neither of them
87. Define which of the numbers a,b is a member of the sequence {xn}, if a=248, b=2050, xn=2n -n, nN,
a) a; b) b; c) both of them; d) neither of them
88. Find the greatest member of the sequence {21/(3n2 – 14n - 17)}:
a) x6=3; b) x3 = 1/6; c) x3=5/64; d) x6 = 1/6; e) x2 = 6;
89. Find the greatest member of the sequence {n/(n2 + 9)}:
a) x6=3; b) x3 = 1/6; c) x3=5/64; d) x6 = 6/45; e) x2 = 6;
90. Find the greatest member of the sequence {2-n - 34-n}:
a) x6=3; b) x3 = 1/6; c) x3=5/64; d) x2 = 1/16; e) x6 = 1/6;
91. Find the least member of the sequence {(2n - 5)/(2n - 11)}:
a) x5=-5; b) x4 = - 4; c) x6=7; d) x9 = 23/29; e) x5 = -10;
92. Find the least member of the sequence {n + 5/n}:
a) x4=-9; b) x2 = 4,5; c) x5=-1; d) x1 = 6; e) x4 = 5,2;
93. Find the least member of the sequence {1,4n/n}:
a) x4=19; b) x2 = 4,5; c) x3=1,43/3; d) x4=1,44/3; e) x1=1,4/3;
94. Domain of a sequence {xn} is
a) the set of natural numbers – N; b) the set of integers – Z;
c) the set of real numbers – R; d) the set of positive real numbers – R+;
e) (-; +);
95. {xn} is a Cauchy sequence
a) >0: N = N(): |xn - x|<, n>N; b) >0: N = N(): |xn - xm|<, n, m>N;
c) >0: N = N(): |xn - x|<, n>N; c) >0: N = N(): |xn - xm|<, n, m>N;
d) >0: N = N(): |xn - a|>, n>N;
8. Bounded and unbounded sequence
96. {xn} is a bounded sequence
a) C=const>0: |xn| C, nN; b) C=const<0: |xn| C, nN;
c) C>0: |xn| C, nN; d) C>0: |xn| C, nN; e) C>0: |xn| C, nN;
97. Let {xn}= {n/3n} and {yn}= {n2/2n}. Point out bounded sequence:
a) only {xn}; b) only {yn}; c) both of them; d) neither of them.
98. Let {xn}= {n/3n} and {yn}= {5n- 4n}. Point out bounded sequence:
a) only {xn}; b) only {yn}; c) both of them; d) neither of them.
99. Let {xn}= {n/3n} and {yn}= {5n- 4n}. Point out unbounded sequence:
a) only {xn}; b) only {yn}; c) both of them; d) neither of them.
100. Let {xn}= {n/3n} and {yn}= {5n- 4n}. Point out bounded below sequence:
a) only {xn}; b) only {yn}; c) both of them; d) neither of them.
101. Which of the following statements is correct?
a) Any convergent sequence is bounded; b) Any bounded sequence converges;
c) Any divergent sequence is bounded; d) Any divergent sequence is bounded below;
e) If {xn} converges to 0 and {yn} is unbounded, then {xnyn}converges to 0;
102. Which of the following statements is correct?
a) If {xn} converges to 0 and {yn} is unbounded, then {xnyn}converges to 0;
b) If {xn} converges to 0 and {yn} is bounded, then {xnyn}converges to 0;
c) Any bounded sequence converges; d) Any divergent sequence is bounded below;
e) If {xn} diverges and {yn} is bounded, then {xnyn}converges;
9. Monotone sequence
103. Sequence {xn} is strictly increasing, if
a) xn < xm as n<m, n, mN; b) xn xm as n<m, n, mN; c) xn < xm as n>m, n, mN;
d) xn xm as n>m, n, mN; e) xn > xm as n<m, n, mN.
104. Sequence {xn} is increasing, if
a) xn < xm as n<m, n, mN; b) xn xm as n<m, n, mN; c) xn < xm as n>m, n, mN;
d) xn xm as n>m, n, mN; e) xn > xm as n<m, n, mN;
105. Sequence {xn} is strictly decreasing, if
a) xn < xm as n<m, n, mN; b) xn xm as n<m, n, mN; c) xn < xm as n>m, n, mN;
d) xn xm as n>m, n, mN; e) xn > xm as n>m, n, mN;
106. Sequence {xn} is decreasing, if
a) xn < xm as n<m, n, mN; b) xn xm as n<m, n, mN; c) xn < xm as n>m, n, mN;
d) xn xm as n>m, n, mN; e) xn > xm as n<m, n, mN;
107. Let {xn} be an increasing real sequence. Which of the following statements is true?
a) {xn} is bounded above, then xn converges as n;
b) {xn} is bounded above, then xn diverges as n;
c) {xn} is bounded sequence, then xn diverges as n;
d) {xn} is bounded below, then xn converges as n;
e) {xn} is unbounded above, then xn converges as n;
108. Let {xn} be a decreasing real sequence. Which of the following statements is true?
a) {xn} is bounded below, then xn converges as n;
b) {xn} is bounded below, then xn diverges as n;
c) {xn} is bounded sequence, then xn diverges as n;
d) {xn} is bounded above, then xn converges as n;
e) {xn} is unbounded below, then xn converges as n;
109. Suppose that {xn} satisfies |xn+1/xn|a as n. Then
a) a<1, then xn converges to 1 as n; b) a<1, then xn converges to 0 as n;
c) a<1, then xn diverges as n; d) a>1, then xn converges to 0 as n;
e) a>1, then xn converges to 1 as n;
110. Let {xn}= {(n+1)/(2n-1)} and {yn}= {n3 - 6n 2}. Point out monotone sequence:
a) only {xn}; b) only {yn}; c) both of them; d) neither of them.
111. Let {xn}= {sin n} and {yn}= {n3 - 6n 2}. Point out monotone sequence:
a) only {xn}; b) only {yn}; c) both of them; d) neither of them.
112. Let {xn}= {sin n} and {yn}= {(-1)n}. Point out monotone sequence:
a) only {xn}; b) only {yn}; c) both of them; d) neither of them.
113. Let {xn}= {(n+1)/(2n-1)} and {yn}= {n3 - 6n 2}. Point out increasing sequence:
a) only {xn}; b) only {yn}; c) both of them; d) neither of them.
114. Let {xn}= {(n+1)/(2n-1)} and {yn}= {n3 - 6n 2}. Point out decreasing sequence:
a) only {xn}; b) only {yn}; c) both of them; d) neither of them.
10. Converging sequence
115. =
a) 1/9; b) 1 c) 0; d) 1/3; e) 9.
116. =
a) 3; b) 0; c) 1; d) 2/3; e) .
117. =
a) ; b) 0; c) 1; d) -1; e) 2.
118. =
a) 1; b) 0; c) -1; d) ; e) -.sdez otveta netu pr.1/2
119. =
a) -7/4; b) 0; c) ; d) ¾. E) - sdez otveta netu pr.-5/4
120. =
a) 1; b) 0; c) ; d) -1; e) -.
121. =
а) ; в) ; с)-5; d)0; e) .
122. =
а) ; в) 10; с) 1; d) 0; e) -10.
123. =
а) ; в) 10; с) 1; d)0; e) -10.
124. =
а) ; в) 2; с) 1; d) 0; e) -10
125. =
а) ; в) e3; с) 1; d) 0; e) –e3
126. =
а) ; в) 2; с) 1; d) 0; e) -10
127.
a) 3; b) 1; c) 2/3; d) ; c) 9.
128.
a) 3; b) 1; c) 0; d) ; c) 2.
129.
a) 3/2; b) 1; c) 2/3; d) ; c) 0.
130. What does mean?
A) B)
C) D)
E)
131.
A) ;
B) ;
C) ;
D) ;
E) .
11. Limit of a function
141. = а)1; в) ; с) ; d)2; e) .
142. =
A) 4 ; B) 3 ; C) 1; D) –1; E) 0,4.
143. =
A) 2 ; B) 0,6 ; C) 1,2; D) 1,4; E) 1,6.
144. =
A) 1,5; B) 3; C) 2; D) 0,5; E) 1.
145. = A) ; B) ; C) 10 ; D ) 0,1 ; E) 1.
146. = A) e-10 ; B) 1; C) e10 ; D) ; E) .
147. = А) 0,6; B) 3; C) 2; D) ; Е) –1.
148. = А) 2; B) -; C) -; D) ; E) -2.
149. = А) ; B) ; C) - ; D) -4; E) 1.
150. = А) 5; B) ; C) 3; D); E) .
151. = А) e; В) e3; C) 1; D) 2; E) 4.
152. = A) 0; B) 1; C) -1; D) 2; E) 3.
153. = A) B) 1; C) 0; D) 5; E)
154. = A) –2; B) 0; C) 5; D) ; E)
155. = A) ; B) 1; C) 0; D) 2; E) -1.
156. = A) 0; B) ; C) 3; D) 1; E) 5.
157. = A) 0 B) 1 C) D) E) 2
158. = A) ln5 B) - C) D) ln2 E) 0
159. = A) B) -2 C)-25/2 D) -5/1 E) 25/2.
160. = A) ln2 B) C) e D) -1 E) 1
161. = a) 7/3; b) ; c) 1/5; d) 0; e) 7/5.
162. = a) -2; b) 2; c) 0; d) ; e) e.
163. = a) 1; b) 3/10; c) 0; d) ; e) e2
164. = a) -1; b) 0; c) ; d) -; e) 1
165. = a) 2; b) 22 c) 0; d) ; e) -;
166. = a) 1/3; b) 0; c) 3; d) ; e) -;
167. = a) e-1; b) e; c) 0; d) ; e) -;
168. = a) 2; b) 1; c) 0; d) ; e) e;
169. = a) -1/2; b) 1/2; c) 1; d) 0; e) 3.
170. = a) m; b) 1; c) 0; d) –m; e) .
171. = a) 12/5; b) 1; c) 0; d) ; e) 8.
172.
A) ;
B) ;
C) ;
D) ;
E) .
173.
A) ;
B) ;
C) ;
D) ;
E) .
174. (x) is called as an infinitesimal function as xx0, if
a)
b)
c)
d)
e)
175. Infinitesimal functions (x) and (x) are equivalent as xx0, if
a) b) c)
d) e)
12. Continuity and uniformly continuity of a function
176. Function f is continuous at a point a, if
A) B) and
C) ant and their equal.
D) E)
177. Let function f be defined and continuous on [a,b]. If f(a)>0 and f(b)<0 at the same time, then
a) (c(a, b)) : f(c)=0; b) (c(a, b)) : f’(c)=0; c) (c(a, b)) : f(c)=0;
d) (c(a, b)) : f’(c)=0; e) ( [f(a), f(b)] (x [a, b]) (f(x) = ).
178. Let f be a continuous function on [a, b]R. Suppose that f(a) < f(b). Then
a) (c(a, b)) : f(c)=0; b) (c(a, b)) : f’(c)=0;
c) ( [f(a), f(b)] (x [a, b]) (f(x) = );
d) ( [f(a), f(b)] (x [a, b]) (f(x) = );
e) ( [f(a), f(b)] (x [a, b]) (f(x) = );
179. Let f be continuous on [a, b]. Then
a) m, M = const such that m f(x) M when a x b.
b) (c(a, b)) : f(c)=0; c) b) (c(a, b)) : f’(c)=0;
d) ( [f(a), f(b)] (x [a, b]) (f(x) = );
e) ( [f(a), f(b)] (x [a, b]) (f(x) = );
180. Let f be a function defined on a set A R. f is uniformly continuous on A if
a) ( > 0) ( > 0) (x1, x2A) [(|x1 - x2|< ) => (|f(x1) - f(x2)|> .
b) ( > 0) ( > 0) (x1, x2A) [(|x1 - x2|< ) => (|f(x1) - f(x2)|< .
c) ( > 0) ( > 0) (xA) [(|x – x0|< ) => (|f(x) - f(x0)|< .
d) ( > 0) ( > 0) (xA) [(|x – x0|< ) => (|f(x) - f(x0)|> .
13. Differentiable functions
181. f is continuous on [a; b] and differentiable on (a; b).
Which of the following statements is correct:
І. .
ІІ. ?
A) both of them ; B) neither of them; С) only І; D) only ІІ.
182. f is continuous on [a; b] and differentiable on (a; b).
Which of the following statements is correct:
І.
ІІ.
A) both of them; B) neither of them; С ) only І; D) only ІІ.
183. Graph of is tangent to axes Ox at a point А (1; 0), then =
A) -2 ; B) 1 ; C) -1 ; D) 2; E) 0.
184. Graph of is tangent to axes Ox at a point А (1; 0), then=
A) 2 ; B) –1; C) 1 ; D) -4 ; E) 0.
13. Derivative of a function
185. ,
A) ; B) ; C) ; D) ; E) .
186. ,
А) ; B) ; C) ; D) ; E) .
187. =
A) ; B); C); D) ; E) .
188. ,
A) ; B) ; C) ; D) ; E) .
189. ,
A) 5/2; B) ; C ); D) ; E) .
190.
а) ; в) ; с) ;
d) ; e) .
191. y= cos4x; y'=
А) 4cos3x; B) 4sin3x; C) cos3xsinx; D) -4cos3xsinx; E) sin4x.
192. Find derivative of at the point
А) ; В) ; С) ; D) ; E) .
193. Find derivative of at х= 0
А) 0; В) -3; С) ; D) -6; E) 3.
194. , =
A) ; B) ; C) ;
D) ; E) .
195. Find , if
A) 1 B) 1/3 C) 2 D) 0 E) 3
196. Find differential of
A) B) C) D) E)
197. , =
A) B) C) D) 3 E)
198. y’=
a) 3/(2+3x); b) 1/(2+3x); c)3x/(2+3x); d) 3x/(2+3x)2; e) 3/(2+3x)2
199. , y’=
a) 1/sin x; b) sin x; c) 1/cos x; d) cos x; e) 1/tg x.
200. , f’(x)=
a) –e-x/cos2e-x; b) –ex/cos2e-x; c) e-x/cos2e-x; d) ex/cos2e-x; e) –e-x/sin2e-x;
201. , f’(x) =
a) –e-x, b) e-x, c) ex, d) –ex, e) e2x.
202. , =
A) B) C) - D) - E)
14. Basic theorems about theorems
203. Find a point which satisfies Lagrange’s Theorem
a) 2; b) 2,5; c) 3; d) 4; e) 4,5.
204. Using L’Hospital’s rule, find
A) 1; B) ; C) 1; D) ; E) 0.
205. If; ; a<b, then
15. Monotone functions
206. Function f is increasing, if
a) f(x) f(y) as x < y; b) f(x) f(y) as x > y; c) f(x) < f(y) as x < y;
d) f(x) > f(y) as x < y; e) f(x) > f(y) as x > y;
207. Function f is decreasing, if
a) f(x) f(y) as x < y; b) f(x) f(y) as x > y; c) f(x) < f(y) as x < y;
d) f(x) > f(y) as x < y; e) f(x) > f(y) as x > y;
208. Functions and are considered on R. Monotone increasing function is
A) both of them; B) neither of them; С ) only ; D) only .
209. Functions and are considered on R. Monotone increasing function is
A) both of them; B) neither of them; С ) only ; D) only .
210. Define increasing domain of
а) ; в) ; с) ; d) ;
e) .
211. Define increasing domain of y=2x2 - ln x:
a) (0; 0,5); b) (0,5; +); c) (-; 0); d) (0; +); e) (-; 0,5)
212. Define decreasing domain of y=2x2-ln x:
a) (0; 0,5); b) (0,5; +); c) (-; 0); d) (0; +); e) (-; 0,5)
213. Define decreasing domain of y=x2e-x
a) (0; 2); b) (-; 0)(2; +); c) (-; 0); d) (2; +); e) (0; e-2)
214. y=x2e-x is
a) increasing on (0, 2); b) decreasing on (0, 2); c) increasing on (-, 2);
d) c) increasing on (-, 0); e) increasing on (2, +);
16. Max and min of a function
215. y=x2e-x
a) ymin = 0, ymax=4e-2; b) ymax = 0, ymin=4e-2; c) ymin = 1, ymax=4e-2;
d) ymax = 1, ymin=4e-2; e) ymin = 0, ymax=2;
216. y=2x2 - ln x
a) ymin = 0, ymax=0,5; b) ymax doesn’t exist, ymin=0,5;
c) ymin = ½ +ln 2, ymax doesn’t exist; d) ymax = ½ +ln 2, ymin doesn’t exist; e) ymin = 0, ymax=2;
217. y=(x+1)3
a) ymin = 0, ymax=2; b) ymin and ymax doesn’t exist; c) ymin = 0, ymax doesn’t exist;
d) ymin doesn’t exist, ymax=2; e) ymin = -1, ymax=2;
218. y= x/(x2-6x-16)
a) ymin = -2, ymax=8; b) ymin and ymax doesn’t exist; c) ymin = -2, ymax doesn’t exist;
d) ymin doesn’t exist, ymax=8; e) ymin = 0, ymax=2;
219. y = x4 - 2x2 - 5
a) ymin = -1, ymax=0; b) ymin and ymax doesn’t exist; c) ymin = -6, ymax doesn’t exist;
d) ymin doesn’t exist, ymax=0; e) ymin = -6, ymax=-5;
220. y = x – ln (1+x)
a) ymin = -1, ymax=0; b) ymin and ymax doesn’t exist; c) ymin = 0, ymax doesn’t exist;
d) ymin doesn’t exist, ymax=0; e) ymin = -6, ymax=-5;
17. Inflection points
221. Find inflection point of
a) it does not exist; b) M1(-1; e-1/2) and M2(1; e-1/2); c) M(1; e-1/2);
d) M1(-1; - ½ ) and M2(1; ½ ); e) M(-1; e-1/2)
222. Find inflection point of f(x) = ln (1+x2)
a) it does not exist; b) M1(-1; -ln 2) and M2(1; ln2); c) M(1; ln 2);
d) M1(1; ln 2) and M2(-1; ln 2); e) M(-1; -ln 2)
223. Find inflection point of f(x) = arctg x - x
a) it does not exist; b) M1(0; 2) and M2(/2; ); c) O(0; 0);
d) O(0; 0) and M2(/2; ); e) M(/2; )
224. Find inflection point of
a) it does not exist; b) M1(-3; 0) ; c) O(0; 0);
d) M1(-3; 0) and O(0; 0 ); e) M(-1; 1)
225. Find inflection point of f(x) = x3 / (3 - x2):
a) it does not exist; b) M1(1; 1/2) and M2(2; -8); c) O(0; 0);
d) O(0; 0) and M2(1; ½ ); e) M(2; -8)
226. is
a) convex on (-1; 1); b) concave on (-1; 1); c) concave on (-; -1);
d) convex on (1; +); e) convex on (-;-1)(1; +)
227. is
a) concave on (1; +); b) concave on (-1; 1); c) convex on (-; -1);
d) convex on (1; +); e) concave on (-;-1)(1; +)
228. f(x) = ln (1+x2)
a) concave on (1; +); b) convex on (-1; 1); c) convex on (-; -1);
d) concave on (1; +); e) convex on (-;-1)(1; +)
229. f(x) = ln (1+x2)
a) concave on (1; +); b) concave on (-1; 1); c) convex on (-; -1);
d) convex on (1; +); e) concave on (-;-1)(1; +)
230. y = arctg x – x
a) concave on (-; 0); b) convex on (-; 0) and concave on (0; +);
c) convex on (0; +); d) convex on (-; -1);
e) concave on (-; 0) and convex on (0; +);
18. Asymptotes
231. Sloping asymptote of y=x3/(x2-1) is
a) y = x; b) x = 1; c) y = 0; d) y = x+1; e) doesn’t exist;
232. Vertical asymptote of y=x3/(x2-1) is
a) y = x; b) x = 1; c) y = 0; d) y = x+1; e) doesn’t exist;
233. Horizontal asymptote of y=x3/(x2-1) is
a) y = x; b) x = 1; c) y = 0; d) y = x+1; e) doesn’t exist;
234. Sloping asymptote of y=x2/(x2-1)1/2 is
a) y = x; b) x = 1; c) y = 0; d) y = x+1; e) doesn’t exist;
235. Vertical asymptote of y=x2/(x2-1)1/2 is
a) y = x; b) x = 1; c) y = 0; d) y = x+1; e) doesn’t exist;
236. Sloping asymptote of y=x3/(2(x+1)2) is
a) y = x; b) x = 1; c) y = ½ x + 1; d) x = - 1; e) doesn’t exist;
237. Vertical asymptote of y=x2/(x2-1)1/2 is
a) y = x; b) x = 1; c) y = ½ x + 1; d) x = - 1; e) doesn’t exist;
238. Sloping asymptote of is
a) y = x; b) x = 1; c) y = 0; d) y = x+1; e) doesn’t exist;
239. Horizontal asymptote of is
a) y = x; b) x = 1; c) y = 0; d) y = x+1; e) doesn’t exist;
240. Asymptotes of y = x3 / (3 - x2) are
a) y = x, y = 3 and x = 3; b) y = x and y = 3; c) y = x and x =3;
d) y = -x and x = 3; e) don’t exist;
Let M be a set and M’ be complement of M, then
MM’=
MM’=M
MM’=M’
MM’
If X=A(B\C), Y=(AB)\(AC), then
XY
X=Y
XY
X=Y’
If X=(AB)\C, Y=(A\C)(B\C), then
XY
XY
X=Y’
X=Y
If X=(AB)\C, Y=(A\C)(B\C), then
X=Y
XY
XY
X=Y’
A\(B\C)=(A\B) C if and only if
AC
AC
A=C
A=B
A\(A\B)=
AB
AB
A=B
A=B’
(AB)’=
A’B’
A’\B’
A’B’
A\B’
(AB)’=
A’B’
A’B’
A’\B’
A\B’
Let A and B be any subsets of the set U. Then (A\B)’=
AB
A’B
A’B
AB
Let A and B be any subsets of the set U. Then (AB’) (A’B)=
A’B
AB
AB
A’B
Let A and B be any subsets of the set U. Then (AB)(A’B’)=
A’B
AB
AB
A’B
Equalities AB=B and AB=A are correct if and only if
AB
A=B
AB
A=B’