Высшая математика. Том 1

' ! ! !:

	(x + 2)− 1	(3 − x)− 1			(x + 1) (2 − x)
	(2 − x)− 1	(x + 3)− 1	≤ 0		(1 − x) (x + 2)	≤ 0
	x + 2 > 0				x > −2
	3 − x > 0				x < 3

3 .

" !:

x (−2;−1] (1;2].5

$ . 1.25. $ ' -

& Maxima

log. Maxima &

& !. ! ! !

		logb x =	loga x	=	ln x	. .
			logab		ln b

&' loga xk = k loga x loga xy = loga x + loga y

& logcontract.

1.8. ' ' -' 9 1.4

2 ! ’ , &

!.

$ ’ :

	1.	log2 x + log2 (x + 1) < log2 (2x + 6) ;	2.	log3 (x2 − 5x + 6) < 0 ;
	3.	log 1 (log4 (x2 − 5)) > 0 ;	4.	logx	x + 3	> 1;
					x + 1
		3

41

5. log

x

4 x + 5 < −1 ;

6 − 5 x

7. log2 x−5 (5x − 2) ≥ 1;

9. log

x+2

(4 + 7x − 2x2 )≤ 2 ;

$ ’ :

10.

3log5 2 + 2 − x = log5 (3x − 52− x );

12.

log1− x 3 − log1− x 2 − 0,5 = 0 ;

14.

log3 (x + 6) logx 3 = 2 ;

lg x + 5

16.

x

3

= 10lg x +1 ;

18.

logx (2x2 − 4x + 3) = 2 ;

20.

lg8 − lg (x − 5)

= −1;

lg

x + 7 − lg 2

22.

4 x

− 9 2 x −1 + 2 = 0 ;

1

− 0,041− x = 0 ;

24.

5 0, 2

2 x

26.

22 x+2 x2 −2

− 5 2x−1+ x2 −2 = 6 .

2! :

1		4
27. 81	log5 3	+ 27log9 36 + 3	log7 9	;

29. (8114 − 12 log9 4 + 25log125 8 ) 49log7 2 .

6.	log 1 log 1				x + 4		< 0	;
					2 x − 3
		3	(	2				3x−1	(		)
8.	log	3x−1		2x2	)	≥ log				− 3x2
					+ x −1				11x − 6
		x+2						x+2

11.	log3 x9 − 4log9 3x = 1;
13.	lg5+ lg(x +10) =1−lg(2x −1) + lg(21x − 20);

15.	(				(								)		(	x − 36	))	= lg 2 ;
	0,5 lg					x2 − 55x + 90 − lg
17.	lg (5 − x)− 1								3	lg (35 − x3 )= 0 ;
				x − 5
19.	log2							+ log2 (x2 − 25)= 0 ;
				x + 5									= lg(27 − 31 x );
21.	1 +	1					lg3 + lg 2
				2x
23.	log	2				4 x + log							x 2	= 8 ;
		0 ,5									2		8

25. lg(lg x) + lg(lg x3 − 2) = 0 ;

28. 36log6 5 + 101−lg 2 − 3log9 36 ;

30. ' & y = 1 − log8 (x2 − 4x + 3). 31. 3& log 3 6 a , loga 27 = b .

1.9. ? '

< ’ . # ! ! ,

& , & ,

& , , , &, . -, & ! , &

, ! , -

42

.

– , !'

.

% – ’ & -

& & ! &.

2 – '

( ) ( ) . + '

! , ! -

! .

0, , ! , ' -

!; ! & ,

. ( ! & . % -

! .

% , ’ !

, & . 3

– , &

. . , & ! -? " . 2 !-

. "

& ! , & . 7 , !

! & !. # -

(& ) (& ) ! & !, ! ` -

. 7 ! ! , -

, & , ! & (

). 3, ! , & !, -

. 8 -

& .

# ` & ! &-; ! -

; , , ` ;

.

', ! !-

& & ! : -

& , ( , ) ! , &

( ) & , & !.

'

# 1. $ ’ :

! ! . # ! 6 !, B, !

! " , ! 12 . # ! , ! ! B

8 ! , ! – 9 . "

43

! & !.

4, v1, v2 ( / ) – ! !, -

! ! S. ' & ! .

3 " ! ! 8

, BC = 8v1 . ! "!

9 , AC = 9v2 .

$ . 1.26. $ !

' :

AC + 12 = BC 9v2 + 12 = 8v1 .

" , ! ! -

: t =

AC

=

9v2 , t

=

BC

= 8v1 .

2

1

v1

v1

v2

v2

9v2

= 8v1 .

# 6 , : t

+ 6 = t

2

+ 6

1

v1

v2

v2

' & :

= a ’ :

v

1

9a + 6 = 8 9a2 + 6a − 8 = 0 a = 2

, a

2

= − 4 .

a

1

3

3

. a – , & -

. 3 :

v

=

2

2

3

v1

= 6

.

v1

= 4

= 8v1

v2

9v2 − 12

", S = AC + BC = 9v2 + 8v1 = 84 . " !: S = 84, v1 = 6, v2 = 4 .5

# 2. $ ’ : &, ! , !-

& 12 . 7 ! ! &, ! & 10 !, . ' ! &-

&?

4 , x ( 3) – &’ & t ( .) – ! & -

&. % ! & ! & t + 10 .

' ! &: N

2

=

x

, N

=

x

. ' 12 !

t

1

t + 10

&

!

N1 + N2

! &:

x

x

x =

+

12 . 2 x ! -

t + 10

t

: t (t + 10)

= (2t + 10) 12 . $ ! -

! . 3 :

44

t2 − 14t − 120 = 0 t1 = 20, t2 = −6 .

3 ! ,

!. " !: t = 20 .5

# 3. $ ’ : " ,

. 6 & 9 , , 6 &, , & . 2

& , , 4,5 -

3 50 ?

4 , x , x , x – , !-

. ":

x

− x

= 9,

x

− x

= 6 . " : C

=

yi

, C

–

i

xi

i

, yi – i- , xi – i- .

'

y

= y

= y

= y . 3 C =

y

, "

=

y

, " =

y

–

x

. 2 , '

:

4,5"

=

3"

+ 0,5"

4,5y

= 3y + 0,5 y .

" : = − 9,

= − 6 = − 15.

2 y ! :

4,5 =

3

+

0,5

2 − 58,5

+ 607,5 = 0 , !

≠ 0; 9; 15 .

− 9

− 15

) : = 45,

= 13,5 .

$ & !:

1.= 45 , = 36, = 30 .

2.= 13,5, = 4,5, = −1,5. ! ,

! .

" !: 45 , 36 30 . 5

# 4. ' ` y= 2–58,5 +607,5 !- ! !.

4' solve ! Maxima.

" !: 1 = 45, 2 = 13,5. 5

45

-:

1., ! .

2.7 ’ ?

3., ! ?

4.7 !?

5.7 ` ?

6.7 ! ?

7.?

8., ?

-

, ,

’

;

.

46

2.

,

' -

,

.

m×n m·n , -

m n . -

. , :

		2	− 1		0			2	1	0				1
				3				−3	2	0		,
						,							(4),	2 .
	1		π		1				3
							1,5			−1				3

! ,

, .

" m×n :

a11	a12	a1n
A = a21	a22	a2n .
a	a	a
m1	m2	mn

#, $ , . %-

aij:

, – . , a23 –

2- 3- .

&, $ - , -

. ' (1 -

500 «)», 250 «!» 100 «*»; (2 100 «)», 500 «!» 200 «*»; (3 -

250 «)», 250 «!» 500

«*». + . , - 3×3:

	500	100	250
A = 250		500	250	.
	100	200	500

. aij , $ j a i-

. , a32 =200 , $ (2 200 -

«!» . /

47

2.2.

0$ , -

, -

. " $ –

, – 1.

), ,

. " .

1 , $

.

), A = (a11 a12 a1n ) ,

- ( ), , , -

- .

), ,

(0), 0. ,	0).
0 = (0 0 0), 0 = (0
0	0

! , $ -

.

1	3	−1
0	2	−2
	1
4		3

2 , , $ -

, , .

a11		a12	a13
	0	a22	a23
	0	0	a
			33

2 , , , $

, , .

,	a11		0	0			(1	0). +, $
		0	a22	0
		0	0	a			0	0
				33

.

! , -

, E. ,

1	0		0
3- E = 0		1	0	.
		0	1
0
2.3.		! " " ! " !

" ! . ! A B , $ -

-

48

aij = bij. , $	a11
	A = a	21
, a11 = b11, a12 = b12, a21 = b21

a12		B =
a
22

a22 = b22.

b

11

b21

b

12 , A=B

b22

# ! $". 1 A, $ m n

. 3 B n m

, $ A ( , -

A ). - , $

		a11	a12	a1n			a11	a21	am1
	A =	a21	a22	a2n		, B =	a12	a22	am2		.
		a	a	a			a	a	a
		m1	m2	mn			n1	2n	mn

. B A, -

A B .

. , – -

. ), A, AT.

+' A :

AijT = Aji .

- :

a)( T)T =

b)( +B)T = T+BT

c)( · )T = BT· T

d)T = .

+ .

, A = (2			2	7
	0 3), AT = 0			−2 ./
7	−2 1		3	1

%" . A B

, . .

, $ A B A

B, $ . . , -

A B C, $ :

a11

a12

a13

b11

b12

b13

a11

+ b11

a12

+ b12

a13

+ b13

A + B = a

a

a

23

+

21

b b

= a

+ b a

+ b

a

+ b

,

21

22

22

23

21

21

22

22

23

23

					+ b
c			= a			.
	ij			ij	ij

49

& 1. + :
1	2 2		3 3		5
, 2	4	+ 4	0	= 6	4 ./
3	5	1	5	4	10
& 2. + :
,(1	−1 0)+ (−1			3 ) – , $ . /
2	3	1	4	−2

& 3. + :

,(1 2 3) + (0 −1 1) = (1 1 4)./

4 , $ "

A+B=B+A (A+B)+C=A+(B+C) .

& ' ! . ! $ A

k A . . -

, A k , $ -

a11

a12

k a11

k a12

= k a

.

kA = k a

a

= k a

k a

c

21

22

21

22

ij

ij

a31

a32 k a31

k a32

! - a b A B :

(αβ )

A = α (β A) ,

α ( A + B)

= α A + α B ,

(α + β )

A = α A + β A .

& 1. + :

1 3 0 −2

−6 0

,−2

−1 2 4

=

2

−4

−8 . /

2

−1 3

−4 2

−6

& 2. + 2A–B, $ A = (1

−1

1

), B = ( 0

−2

−3).

2

0

−2

−1

4

1

,2A − B = (2 −2 2 )− ( 0 −2 −3)= (

2 0 5 )./

4 0

−4

−1 4 1

5

−4

−5

& 3. + C= – 3A+4B, $ A =

2

1

1 1).

0

−1 , B = (−1

−1

−2

2

−2 3

, – A B ./

& 4. 2 5 -

((1 (2) 3, 4, 5 -

50