Высшая математика. Том 1

$ ’ :

21. 3 sin2 2x + 7 cos 2x − 3 = 0 ; 23. sin (15 + x)+ sin (45 − x)= 1;

25. cos 4x + 2sin2 x = 0 ; 27. sin 9x = 2sin 3x ;

29. sin3 x cos x − sin x cos3 x = 14 ;

, :

31. tg20° tg40° tg80° = 3 .

22. sin x cos x cos2x cos8x = 14sin12x ;

24. sin 3x = 2cos	π	− x	;
	2

26. tgx + tg2x + tg3x = 0 ;

28. cos(20 + x)+ cos(100 − x)= 12 ;

30. ctg2 x = 1 + sin x .

1 + cos x

1.3. #

# ( . planum – ! metros – ),

, , !-

.

( . 1.4) ! !: 2 α + β + γ =1800 ;

% : /sinα=b/sinβ=c/sinγ=2R, R – ! ; % : c2 = 2 + b2 – 2ab cos γ ;

	a + b		tg	α + β
% :		=			2			;
	a − b		tg	α −		β
					2
1 α:	la = (2bc cosα 2)/(b + c);
# 3 S =				p( p − a )( p − b )( p − c ) , – ! !,

p = ½ (a+b+c);

# S = bh/2, h – ! b; # 2 : S = ½·ab sin α;

# S .=( 3 a2)/4;

$ . 1.4. %

$ . 1.5. )

21

# ! S=abc/4R; # ! S=pr.

# ( . 1.5): S= πR2.

# : S =(πR2α)/3600.

# : & ( . 1.6): S = (a+b)/2 h.

# : " # ( . 1.7): S=ah;

2! !:

d12+d22=2(a2+b2). # #; ( . 1.7): S=ah= a2·sinα= ½·d1d2.

2! &: d1=2a·sinα/2; d2=2a·cosα/2; d12+d22=4a2.

$ . 1.6. % !	$ . 1.7. #

'

# 1. 2 a, b mc , !

c. ' .

4 & ! " !-

!" #. # KC = 2 mc . ' -

	!: 4m2 + c2	= 2b2 + 2a2 . '
	c
$ . 1.8. #	: c2 = 2b2 + 2a2 − 4mc .
	" !: c = 2a2 + 2b2	− 4m2	.5
!" #		c

# 2. " ! 3 : !M = 6 , $ = 4 , $" = 1 . 6 $ ! ! !$" !. ' .

4 , AMB = α . '

AMB :

AB2 = AM 2 + MB2 − 2AM BM cosα = 52 − 48cosα .

. BMC :

BC2 = BM 2 + MC2 − 2BM MC cosα = 17 − 8cosα .

" AB BC ,

, !

$ . 1.9. ) - :

22

0 = 35 − 40cosα cosα = 35 = 7 .
% ,				40	8
AB2 = 52 − 48cosα = 52 − 48 7					= 52 − 42 = 10,						sinα = 1 − cos2 α .
% AMB ,				8
				! ,				,
! :
				2R =		AB	=			10			.
											7	2
						sinα		1
										−	8
R = 1	10		=	10 64 =	2 5 4 16			=		32		" !: R =		32	.5
	64 −	49			3 5 4					3				3
2				15 4
	64
# 3.		( R α ! .
' .

$ . 1.10. 2 -

R

: AB = AC = R , BAC = α .

': OH = r .

4 3 ! -

!, AO = R − r . $ !%& ( !,

OHA = 900 ):	HAO =	α
		2

!%& !%$ ( O&=%$=r, !&=!$ , !

, !O – !).

3, '

, ( , :

sin

α

=

OH

=

r

(sin

α

(R − r) = r) (R sin

α

− r sin

α

= r)

2

R − r

2

2

2

AO

α

(R sin

α = r sin α

+ r)

" !: r =

Rsin

2

.5

α

2

2

1 + sin

2

# 4. 3 ! 5 40 . ' ,' .

4 , ABCD – !, – !", # – -

BD.

% # EF !. ' , )# – -

ABD, – , ( '' '

23

, -
' ,
! , EK =	50 = 25 .
	2
.,	EP = 40 = 20 , -
	2
) ∆ ! ".		$ . 1.11. % ! ABCD
3, PK = EK − EP = 25 − 20 = 5 .

" !: 5 . 5

1.4. ' ' -' 9 1.2

2 ! ’ , &

!.

':

1.2 B ABC, AC = 4 , BC = 5 , cos A = 92 ;

2.&, &

1:2, ! & 12 ;

3.# , ! ! , !

9 3 2, ! ;

4.# ABD , ABC ! C

1350, AC = 6 , BD = 2 ;

5.2 A ABC , AC = 4 , BC = 7 , AB = 5 ;

6.AB ABC , BC = 9 , cosC = 73 ,

sin A = 23 ;

7." , 600,

3 27 ;

8.! , !, -

4 6 , 600, !

450;

9." , , !

6, 25π 2, ! 3 ;

10.2 ! ,

! 20 , ! – 4 ;

11.&, ! & 18 2, -

300;

12. " M H CH !

ABC , AC = 7 CM = 7 , ! !-

24

AB ;

13." ! ! ! 5 ,

' & & !, !

, 2 7 ;

14.# & 1200, !

10 ;

15.# ! 6 , O1 O2 – ,

O1O2 = 6 2 ;

16.# , ! & !,

! 14 , 12 16 ;

17.# ! !

a b, ! ! ;

18.3 & ! & , 17 , ! 15 ;

19.& ,

, ! , ! &

12 8 , ! ;

20.AB AC ABC, BC = 8 , -

, ! AC BC, ! 6,4 4 .

21.& !, -

10 , – 8 , ! 300, !,

& !, ! ! ! ;

22." ! ! !, -

! ! ! 1:2, ! 2 32;

23.# ! ! , !-

24 , ! 24 2;

24.$ , ! S,

& ;

25.# ! , -15 , , ! , 3 ;

26.! ! ,

! 1 ;

27.2 a ! ", b c, ! &-

;

28.& ! , ! -

b c;

29.# & !, ! 3 ,

! ! π/4 2 ;

30.# & ABCD, , ! &

ABC ABD, ! R r;

31.# ! R r,

d.

25

1.5. " ;$

! ! !

, ! . # -

, – ! . . ,

, , -

, ! , – ’ ; ! -

, .
, !, 3x–6=0 , =2
’ ; x2 + y2 = 25 , =3, y=4

. 2 ! & $ -

, ! . $ & $, ' & $. 7 ' ,

, , & & .

’ ! !. , !, -

ax²+bx+c=0 ! ’ :

x = −b± b2 − 4ac
1,2	2a	.

# , D=b²– 4ac & , x1≠x2; D

, x1=x2, D , ’ . ) , ’ ',

:

x1+x2 = –b/a; x1 x2 = c/a.

x²+px+q=0 ! ’

:

x = −	p	±	p2	− q , & x1+x2 = –p; x1 x2 = q.
1,2	2		4

7 ! ' ! ! ,& :

a paq = a p+q ;

a pb p = (ab)p ;

(a p )q = a pq ;

a0=1; a1=a;

p a p b = p ab ;

p a =b bp =a;

a p

= a p−q

a− p =

1

a p

a p

;

;

=

;

aq

a p

b p

b

p g a = pg a ;

pk agk = p ag ;

g

p ag = a p ;

1

p a .

a p = p a ;

p a

=

b

p b

2 ! 1.

0 &

:

26

(a±b)2=a2±2ab+b2; (a±b)3=a3±3a2b+3ab2±b3; a2– b2=(a+b)(a–b);

a3± b3=(a±b)(a2 ab+b2); (a+b)3=a3+b3+3ab(a+b); (a–b)3=a3–b3–3ab(a–b).

8 – ! & -

, , & . ! -

<, . %, 2 > 1 1 < 2 ! : 2 & 1, & 1 2. + !-

(!, < b < ). 1 , b

& ( ) , & , !: ≥ b (b 9 ) -

: « & & b» («b & »); : «

b» («b & »). ' ! ≠ b , b ,

, &. ( ! -

.

, & ! . %, -

!, & (&

& ) . % &

! . 3, &

! ,

(& > <, < >).

'! < " < D ! ! + " < + D ! – D < – ", &

(! < " < D) ! ,

(! < D > ") – ! . 7 !, , " D !-

, ! < " < D ! AC < BD A/D < /", &-

( ! ) ! !-

, – ! .

,, , ! -

, & . %, x2 – 4x + 3 > 0 ! = 4 ! = 2. -

! ! ! ’ , & ! -

, & , , &

& !.

%, ! ! x2 – 4x + 3 > 0 : ( – 1)( – 3) > 0,

, & , !-

	: < 1, > 3,			’ .
	$ & &-
	, & !			P(x)	≥ 0, ( P(x) Q(x)– &) &
				Q(x)
	, .
	7 x1, , xn – ’ & P(x) Q(x),
	! ,	P(x)		(& P(x) Q(x)) -
		Q(x)

(−∞, x1 ), (x1, x2 ), , (xn−1, xn ), (xn , +∞).

27

* , & P( x)

Q( x)

, &- -

P( x) .

Q( x)

# ’ ! af(x)>(<) aϕ(x) :

f(x) >(<)ϕ(x)

& : a>1, ; a<1, -

.

" ! Maxima , !

!. / expand -

. , ! ! :

/ divide

& :

/ gcd & &, factor & :

# &- !-

! =. , !, x 5/z:

28

/ ratsimp & :

" assume (to assume - !), ! & , :

/ forget (to forget - &) &,

! assume:

				'
# 1. $ ’ f (x) =						(x − 1)(x + 2)2			≥ 0 .
							(x + 3)
4	#	'
x1 = −3, x2	= −2, x3 = 1.	"	f (x)
(−∞, −3), (−3, −2), (−2,1), (1, +∞ ). " &- (−∞, −3),
! x = −4 , ,					f (−4):
		f (−4) =	(−4 − 1) (−4 + 2)2		=		−5 4	= 20 > 0 .
			(−4 + 3)				−1

' ! , f (x) > 0 ! x (−∞, −3).

# ! x=–3 ,

, &

. 3, f (x) f (x) < 0 x (−3, −2).

# ! x=–2 f (x) , -

( +2) ! !, , , ( +2)2 &-

’ .

29

$ . 1.12. $ ’ -

" !: (−∞, −3) {−2} [1, +∞).5

# 2. $ ’

# !

x=+1 f (x) .

% & & ! '

= –2 .

2	<	3	− 3 .
3x + 2		2x + 5

4 , ! ! – . % & !' ,

! ’ . # , ! ,

! .

2(2x + 5)− 3(3x + 2)

+ 3(3x + 2)(2x + 5)

< 0

(3x + 2)(2x + 5)

18x

2 + 52x + 34

18(x + 1)(x + 17

19

)

+ 2)(2x + 5)

< 0

< 0

+ 2)(2x + 5)

(3x

(3x

a x2 + b x + c = a(x − x )(x − x ) = 18(x + 1)(x + 17

9

)

1

2

x =

−52 ± 522 − 4 18 34

= −13 ± 132 − 9 17 = −13 ± 4

1,2

2

18

9

9

x = −17

9

; x = −1

1

2

$ ! !-

!,

!:

$ . 1.13. $ ’

5

17

2

, −

−1,

−

x −

2

9

3

.5

# 3.

$ ’ :

3 − x

−

x + 2

= 5.

41 : ; -’-

$ 4 !:

3

− x ≥ 0

x ≤ 3

−2 ≤ x

≤ 3 (3 ').

1)

+ 2 ≥ 0

x

x ≥ −2

$ . 1.14. 3& -

! !

( ! !

3 − x − (x + 2) = 5 x = −2 .

* 3 ', -

’ .

30