Введение в математику

2n

1)

an =

;

n + 1

3n + 2

2)

an =

;

5n2 1

3)

an =

( 1)n+1

;

4n3 7

3n;

4)

a

=

2n 1

n

7n + 2

5)

an =

3 5n

;

3

6)

an =

( 1)n

.

32n 1

1)

lim

5n 1

= 5;

n!1 n 2

2)

lim

3n

=

3

;

4n + 1

n!1

4

3)

lim

2n2 + 1

= 2.

n!1 n2 + 2

1)

1

;

3n + 2

1.1 A [ B = f 2; 0; 1; 2; 3; 4; 5; 7; 9g,

A \ B = f3; 7g, An B = f 2; 0; 1; 4; 9g,

Bn A = f2; 5g.

1.2 A = f 1; 3; 5; 15g, B = f2; 3; 5; 7g, C = f2; 4; 6; 8g,

A [ B = f 1; 2; 3; 5; 7; 15g,

A [ C = f 1; 2; 3; 4; 5; 6; 8; 15g,

B \ C = f2g,

(A [ C) \ B = f2; 3; 5g.

1.3 A = f 7; 0; 7; 14; 21; 28; 35g,

B = f1; 3; 5; 7; 9; 11; 13g,

A [ B = f 7; 0; 1; 3; 5; 7; 9; 11; 13; 14; 21; 28; 35g,

A \ B = f7g, Bn A = f1; 3; 5; 9; 11; 13g.

1.4 A [ C = [ 1; 2), A \ B = [ 1; 0),

(A [ B) \ C = [0; 1].

1.9

A = f2; 4g,

B = (8; +1),

C = f2; 1; 0; 1; 2g5,

A \ C = f2g,

C = (1; 11] [ [3; +1),

A [ B =

4; 1; 16

; 2; 3 ,

B \ C =

f3g,

A \ C = ?.

A [ B = f2; 4g [

(8; +1)

,

B \ C = ?.

1.10

A = 4; 1

,

B = f1; 2; 3g

,

5

6

1.11 81 покупатель. 1.12 8

человек.

2.1

1)

z1 + z2 = 5 + 4i,

z1 z2 = 9 + 2i,

z1 z2 = 17 19i,

z1

=

11

23

2) z1 + z2 = 2 6i,

z1 z2 = 8 2i,

z1 z2 = 23 14i,

i;

z2

50

50

z1

=

7

26

213

19

17 14i;

96

153

i.

2.2

1)

+

i;

2)

3)

i;

z2

29

29

85

85

29

29

4)

10

10i. 2.3

1)

cos 0 + i sin 0;

2)

p2

cos 4

+ i sin

4

;

193

29

+ i sin 23

;

3

3

cos 6

+ i sin 6

.

3)

4

cos

23

4)

2

2.4

1)

r = 1,

; 2)

p

,

; 3)

15,

5

; 4)

7,

2

' =

' =

3

r = 2

' =

3 .

' = 2

r = 2 2

4

r = 2

p

p

p

2.5

.

2.6

1)

;

2)

3

i

. 2.7 1)

3

i

2

+

!0 = 2

+

2,

16 + 16 3 i

64

2

p

!1 =

3

i

i; 2) !0 = 1 + i, !1 = 1 + i, !2 = 1 i, !3 = 1 i;

+

, !2

=

2

2

3)

!0 = p4 2 cos

8

+ i sin 8

,

!1 = p4 2

cos

78

+ i sin

78

;

cos

,

4)

!0 = p3 2

cos 9

+ i sin 9

,

!1 = p3 2

79

+ i sin

79

!2 = p3

2

cos

5

+ i sin

5

;

9

9

2.8 1) x1 = 2 3i, x2 = 2 + 3i; 2) x1 =

3

i

,

x2 =

3

+

i

;

2

2

2

2

2.9

1) f(z) = g(z)(z3 + 2) 8z + 10;

2)

f(z) = g(z)(z2 4z + 1) 5z;

3)

f(z) = g(z)(2z2 z + 1) 4;

4)

f(z) = g(z)(z2 + 5) + 15;

5)

f(z) = g(z)(z3 3z2 + 8z 18) + 50z + 11;

6) f(z) = g(z)(2z2 + 6z + 9).

2.10

1) 3,

ò.

ê.

f(z) = (z 2)3(z + 5);

2) 2, ò. ê.

f(z) = (z + 3)2(z2 4);

3)

простой корень, т. к.

f(z) = (z + 1)(z2 5z + 6);

4)

простой

êî-

ðåíü,

ò.

x

ê.

f(z) = (z 5)(z3 2z2 1).

2.11

1)

(x + 1)(x + 4);

2)

x

;

3)

x

x

; 4) x

x); 5)

(2x + 1)(4x + 3);

(

9)(

16)

(

3)(3 + 2)

( + 3)(8

2

2(x + 1)

5;

6)2(x + 1)(5 x;

).

5)

(x + 3) 20

x 2

4

2

3

5

2

2.12 1)

;

2)

;

3)

;

4)3 3(x 1)2

71(x 7)2 7;

6)12 (x 2)2;

7)

7 5(x 2)2

8)(x + 4)2 11.

2.13

1)

z2 = 3 i,

z3 = 3 + i;

2)

z2 = 2 i,

z3 = 2 + i;

3) z2 = 1 2i,

z3 = 2 i,

z4 = 2 + i;

4)z2

= 1 + i, z3

= 6.

2.14 1)

(z 3)(z + 3)(z 3i)(z + 3i)

; 2)

(z + 2)(z 1

p

p

;

3 i)

3 i)(z 1 +

3)

(z + 4)(z + 3i)(z 3i);

4)

(z 1)(z 1 3i)(z 1 + 3i).

2.15 1) (x 1)(x 2)(x 3);

2) (x2 + 4)(x 3)(x + 3).

3.1 1) (1; +1); 2)

1; 13

[ 13; +1 ;

3) (1; 3) [ ( 3; 3) [

2

2

[(3; +1);

2; +1 ; 5)

6) (1; +1);

7) [ 3; 1];

4)

( 5; 2] [ [3; +1);

1

8)

4;

2

[

2; +1 ;

9)

(1; 3) [ (3; +1); 10)

(1; 2) [ ( 2; +1).

1

1

1

2

t

p

3.2

,

f(3) =

,

. 3.3

2.

11

f(1) = 0

f(t + 1) = t2 + 2t + 3

S(h) = h 25 h

6)

нечетная; 7) нечетная; 8) общего вида. 3.5 1) периодическая, T = ;

2)

периодическая, T = 3 ;

3)

периодическая, T = 2 ;

4)

непериодическая;

5)

непериодическая;

6) периодическая, T = 2 . 3.6

1)

f(g(x)) = 32px+1,

g(f(x)) = p3

p

3

(

( )) = 5 sin

8 sin

(

(

32x + 1

f(f(x)) = 32 32x ,

g(g(x)) =

3 p3

+ 1;

,

x + 1

2)

f(g(x)) = sin(5x2

8x),

g f x

2 x

x,

f f x)) = sin(sin x),

g(g(x)) = 5(5x2 8x)2 8(5x2 8x) = 125x4 400x3 + 280x2 + 64x.

f 1(x) =

x + 2

f 1(x) =

1

1);

3.7

1)

;

2)

log7(x

5

2

3)

f 1(x) = tg x + 3;

4)

f 1(x) =

x2 + 4

;

5)

f 1(x) =

3x 2

;

5

7

x

2

1

6)

f 1(1x) = arcsin1

2.

3.8

1)

;

4

3

3

x 2

x1 3

2)

;

3)

+

;

3(x 2)

3(x + 1)

4(x 1)

2(x + 1)2

4(x + 1)

4)

1

4

1

2x

;

6)

13 3x

+

3

;

x + 1

(x + 2)2 ;

5)

x x2 + 4

1)

4(x2 + 3) 4(x

1

5

29

8

8x + 2

7)

1 +

+

;

8)

x +

;

3x

x 2

3(x 3)

3(x + 1)

3(x2 x + 1)

9)

4x + 7

4

+

1

10) x 2 +

1

1

+

16

;

9(x2 + 2)

9(x 4)

(x 4)2 ;

6(x 1)

2(x + 1)

3(x + 2)

11)

1

8

+

1

+

1

;

12)

2

4

2

;

3(x + 2)

6(x 1)

2(x + 1)

x 2

x2

x

x + 1

x

x2

3

2

1

13)

+

;

14)

+

.

3.12

1)

a1 = 1,

x2

x + 1

x2 + 1

x

3

x + 2

x

4

n+1

5

3n + 5

1

2

8

; 3) a1 =

,

a2 =

, an+1 =

; 2) a1 =

, a2 =

, an+1

=

3

n + 2

4

19

5n2 + 10n + 4

3

a

=

1

, a

=

( 1)n+2

; 4) a

=

1

, a

=

3

6

=

729

,

2

n+1

1

729

2

16

25

4n3 + 12n2 + 12n 3

16777216

a

=

2n + 1

3n+3; 5) a

=

2

, a

=

7

, a

=

2 5n

; 6) a

=

1

,

7n + 9

3

3

8

n+1

( 1)n+1

1

2

n+1

3

1

a2 =

1

,

an+1 =

.

3.13

1)

an

=

ln(2n)

;

2) an

=

3n 1

;

32n+2 1

n + 3

80

( 1)n+1

2