Высшая математика. Том 2

(! "$ & ϕ) – & , "$ ! A ! & + ! @- 9 + ! # ' ! < " ! # # OM. 3 " M(ρ,ϕ).

" <# 3 9 ! + # C # $ ! " # +-

+ # # ! %, & C D ' %		0 ≤ ρ < +∞ , 0 ≤ ϕ < 2 .
?'" # C ! " # $ # # 3 D '+" +!-
x = ρ cosϕ	ρ =	x2 + y2
= "#:	,	.

y = ρ sinϕ ϕ = arctg( )

x

1.6.% & ' ( 1.2

& ! ’" & > C3 3, A = 99 + 9# # # & C& @ &!. y

A& & ! " > > @ 6 6& E$:

1. ρ = 3e3ϕ 4 , − π 2 ≤ ϕ ≤ π 2.

2. ρ = 2e4ϕ 3 , − π 2 ≤ ϕ ≤ π 2.

3. ρ = 2 eϕ , − π 2 ≤ ϕ ≤ π 2.

4. ρ = 5e5ϕ 12 , − π 2 ≤ ϕ ≤ π 2.

5. ρ = 6e12ϕ 5 , − π 2 ≤ ϕ ≤ π 2.

6. ρ = 3e3ϕ 4 , 0 ≤ ϕ ≤ π 3.

7. ρ = 4e4ϕ 3 , 0 ≤ ϕ ≤ π 3.

8. ρ = 2 eϕ , 0 ≤ ϕ ≤ π 3.

9. ρ = 5e5ϕ 12 ,

0 ≤ ϕ ≤ π 3.

10.

ρ = 12e12ϕ 5 ,0 ≤ ϕ ≤ π 3.

11.

ρ = 1 − sinϕ,−π 2 ≤ ϕ ≤ −π 6.

12.

ρ

(

− cosϕ

)

,

− π ≤ ϕ ≤ −π 2.

= 2 1

)

13.

(

+ sinϕ

)

,

− π 6 ≤ ϕ ≤ 0.

14.

ρ

(

− sinϕ

,

0 ≤ ϕ ≤ π 6.

ρ = 3 1

= 4 1

)

15.

ρ = 5(1 − cosϕ ), −π 3 ≤ ϕ ≤ 0.

16.

(

, − π 2 ≤ ϕ ≤ 0.

ρ = 6 1 + sin ϕ

17.

(

− sinϕ

)

, −π 6 ≤ ϕ ≤ π 6.

18.

ρ

(

− cosϕ

)

,

− 2π 3 ≤ ϕ ≤ 0.

ρ = 7 1

= 8 1

19.

ρ = 2ϕ,

0 ≤ ϕ ≤ 3 4.

20.

ρ = 2ϕ,

0 ≤ ϕ ≤ 4 3.

21.

ρ = 2ϕ,

0 ≤ ϕ ≤ 5 12.

22.

ρ = 2ϕ,

0 ≤ ϕ ≤ 12 5.

23.

ρ = 4ϕ,

0 ≤ ϕ ≤ 3 4.

24. ρ = 3ϕ,

0 ≤ ϕ ≤ 4 3.

25.

ρ = 5ϕ,

0 ≤ ϕ ≤ 12 5.

26. ρ = 2cosϕ,

0 ≤ ϕ ≤ π 6.

27.

ρ = 8cosϕ,

0 ≤ ϕ ≤ π 4.

28.

ρ = 6cosϕ,

0 ≤ ϕ ≤ π 3.

29.

ρ = 2sinϕ,

0 ≤ ϕ ≤ π 6.

30.

ρ = 8sinϕ,

0 ≤ ϕ ≤ π 4.

31.

ρ = 6sinϕ,

0 ≤ ϕ ≤ π 3.

1.7. * + %

A # 6 +$ ! % Π " & 3 & ! # ', % > '9, . B # @, @ " # +' Oz, % ! > ' 3 ! ! & " ! % Π.

M$ M A& '-" 3 ! + &, N – ! E " E <9 3 ! % & Π, Mz – ! E " M +' Oz (+. 1.5).

11

+. 1.5. / 3 -	+. 1.6. 6 3
& ! +	& ! +
/ 3 # #	3 M D '+" 3 + ρ, ϕ z,

! = " > (ρ ϕ) < ! " # # 3 N ! % Π % ! D+ ! " 9 + , 3 + z < 3 OM z .

3 " M(ρ, ϕ, z).

% ! + E 3 > +! ! D ' +"# - 9 + + # & ! + , '" # C # 3 M E- 3$ $ + + # > 3 D '+" +! = "#:

x = ρ cosϕ,

y = sin

=

z z.

1.8., + %ϕ,ρ

@ " # <# ! ! & " > + , Oy, Oz @ ' #

! 3 # .
M$ M – A& '-", #	3 ! + &, N – 99 ! E " !-

% & Oxy, ρ – + ' M . M$ θ – & , "$ & D< !$

OM ++D Oz, ϕ – & , "$ ! A ! & ! @ 9 + -+' ! < " ! # # ON. θ ϕ D ' -" . "

! (+. 1.6).

6 3 # # 3 M D '+" 3 + ρ, θ ϕ. - 3 " M(ρ, θ, ϕ).

" <# 3 9 ! + # C # $ +6 3 # + + # # , & C D ', % 0 ≤ ρ < +∞ , 0 ≤ϕ < 2 , 0 ≤θ ≤ .

% + +6 3 9 + + # +! ! D ' +"# # + + # & ! + , '" # C # 3 M +6 3$$ + + # > 3 D '+" +! = "#

12

x = ρ sinθ cosϕ,y = ρ sinθ sinϕ, .

= ρ cosθ . z

1.9. ) / !

F& '-"$ & '$ , ! '$ $ ! "#$, # #

%"+ # E <9 ! "# 9.

F& '-"$ & '$ , ! ! & "$ +! "# &D3 #& - & 9 ! "# 9, # # & # E <9 ! "# 9.

@ " # ! "#&, % ! > ' 3 3 & M(x0,y0) ! ! & "& N = ( A, B) (+. 1.7). @ " # A& '-" & 3 & ! "#$ M(x,y) -M0M = (x − x0, y − y0 ) . # D ! ! & " + ! "# 9 MM0 N < + ' & D 9> ' @ + " @ A& &:

N M0M = (A, B) (x − x0, y − y0 ) = A(x − x0 ) + B( y − y0 ) = 0 .

<# &C $ ! 3 # C = − Ax0 − By0 .

" " @ " & Ax+By+C=0 < '+" ' . & /

!.

? @ ' " " ! "# 9 < '+" , " % + $@ 6 E < !, B, " # & ".

@ " # + # C ! > " ':

1)C=0, " " @ " & Ax+By=0 3 < ! "#&, % ! > ' 3 ! 3 (+ ' ! 3 & ' "D ' E' #& "- D).

2)B=0, " " @ " & Ax+C=0 3 < ! "#&, ! ' & + Oy

(+ ' # '$ N = ( A,0) @ '$ + Oy).

3)	A=0,	" " @ " & By+C=0 3 < ! "#&, ! ' & +
(+ ' # '$				= (0, B) @ '$ +	).
			N
4)	B=0 C=0, " " @ " & Ax=0 3 < +' Oy (+!, E" ! "-
# ! ' + Oy ! > ' 3 ! 3 ).					(+!, E" ! "-
5)	A=0 C=0, " " @ " & By=0 3 < +'
# ! ' +		! > ' 3 ! 3 ).

1.10. ' & /

@ " # ! " " ! "# 9, + ' + 6 E < !, B, " -

# & ", ! ! = # " " & @ "		x			+		y			= 1.
	−		C			−		C
			A					B

13

+. 1.7. ? @ ' " "									+. 1.8. " " ! "# 9
! "# 9 ! %									! % & >
3 # a = −	C		,	b = −			C	.
	A						B
" " @ " &		x		+	y	= 1 < '+" / !
		a
					b

% ' 0 (+. 1.8).

& 3 + $ b # D ' ! + @ # 3 3 ": DD '

3 # , " + < ! "# +">			Oy ! ( -
3&D '+" ! 3 & ).			3 & M(x0, y0) ! ' -
@ " # ! "#&, % ! > ' 3
&		= (l, m) . @ " # A& '-" & 3 &	! "#$ M(x,y)
	n

M0M = (x − x0, y − y0 ) . # D ! ' + ! "# 9 MM0 n < -

+ '

,

A , ! ! E$+ ' 9>

x − x0

=

y − y0

.

n

M0M

l

x − x0

y − y0

m

" " @ " &

=

< '+" +

l

m

/ ! (+. 1.9).

	+. 1.9. B 3 "-	+. 1.10. " "# ! "# 9
		& # 6 E < #
	" ! "# 9 ! %

$# # ! # t 3 &, % + 9 ' & $ !$ 3 + > - 3 @ " ", # x 3 t.

14

	" " @ " &	x = lt + x0 < '+" +
	/ !.	y = mt + y0
			m
	# C # 3 " " ! "# 9 m, ! #			= k . -
			l
	C # y − y0 = k(x − x0 ) . 3 # 3 b ! + $& b = y0 − kx0 .
	" " @ " &	y = kx + b < '+" / ' %

,$1 (+. 1.10.).

E' #& " k ! 3 < & $ 6 E < 9 ! "#$ $

k = tgα , α &

> & ! "# 9 +

. 6 E < b < 3 D ,

% + < '+" $ ! "#$ + Oy, ! 3 D3 ! 3 & .

1.11. 2' 1 '-% 0 0

M$ ! "# @ ' # " "#

L1 : A1x + B1 y + C1 = 0 L2 : A2 x + B2 y + C2 = 0 .

+ '

= ( A1, B1 )

= ( A2 , B2 ) , & # C ! "# # L1 L2 $

N1

N2

& & # C # ' # # E > ! "# #. ? 3 " + " @ A&-

& # <#: cosϕ =

A1 A2 + B1B2

.

A2

+ B2

A2

+ B2

1

1

2

2

# ! ! "# > L1 L2 + # '-

>

E > ! "# >, A ! ! E$+ 9>

A1

=

B1

.

N

N

2

1

A2

B2

# ! ! & " + ! "# > L1 L2 @-

' + # ' >

N1

N2

E > ! "# >, A + ' & D 9> + "-

@ A& &: A1 A2 + B1B2 = 0 .

M$ ! "# + 9# 3 # " "#

L :

x − x1

=

y − y1

L :

x − x2

=

y − y2

.

1

l1

2

l2

m2

m1

+ ' +! "# &D3 # # ! "# > L1 L2 <

= (l1, m1)

n1

= (l2 , m2 ) , @ 3 C&<#:

n2

– B& # C # ! "# #: cosϕ =

l1l2 + m1m2

.

l2

+ m2

l2

+ m2

1

1

2

2

– # ! ' + > ! "# >:

l1

=

m1

l2 m2

– # ! ! & " + > ! "# >:

l1l2 + m1m2 = 0 .

15

M$ ! ! "# + 9# " "# 3 & 6 E < :

L1 : y = k1x + b1 L2 : y = k2 x + b2 .

% α1 α2 – & > & ! "# > L1 L2 +						, ϕ – & # C
E # ! "# #, ϕ = α1 − α2 .	tgα1 − tgα2					k1 − k2
# 3 #, tgϕ = tg(α1 − α2 ) =					=		.
	1 + tgα1 tgα
			2			1 + k1k2
, C, & # C # ! "# #:	tgϕ =	k1 − k2		.
	1 + k1k2

# ! ' + : k1=k2.

# ! ! & " + ! "# > L1 L2 A " & ' @ + & # C ! "# # $, C, # < @ ": k1k2+1=0. A k1k2= −1.

			)
?$ & # C # ! "# #: 2x + 3y − 5 = 0,										3x − 2 y + 8 = 0
N + ' ! "# @ ' # " "# ! "# 9 ! %
L1 : A1x + B1 y + C1 = 0	L2 : A2 x + B2 y + C2 = 0 , & " > 6 E < 3 D ' +-
A D # ' >					= ( A1, B1 )						= ( A2 , B2 ) E > ! "# >.
				N1					N2
B& # C ! "# # L1 L2 $ & & # C # ' # # E > ! "# >.
? 3 " + " @ A& & # <#:
	cosϕ =		A1 A2			+ B1B2		.
		A2	+ B2
						A2 + B2
	1		1			2	2
B	C = − Ax0 − By0 > &< + A D									3 &, 3 " & ! >-

' ! "# , ! A3 + & # C ! "# # A &3 + ' A3 +- ">, + ' ! ' @ ! + " ! "# > 3 & # C #

# '+". A , cosϕ =	2 3 + 3	(	−2	)	=	6 − 6	= 0 , ϕ = 900 .O
	22 + 32 32 + (−2)2
					13
	1.12. 2 & + /
" > C " +			3 M(x1,y1) ! "# 9 L, A> #

@ ' " " ! "# 9 L : Ax + By + C = 0 . , + ' 3 ! "# 9:

d = Ax1 + By1 + C .

A2 + B2

& ' & 6 #& < A+ D 3 " + . ? + +& " 6 #&-A # & " ' &+ 3 ! % + – ! D - + D, =$ @ #. + E > 3 C ' ! + % 9 ! "# 9.

16

1.13. ) ! % . & !.

!, ! 0 & + ' % + %

@ " # ! + ' & ! % & α . J9 ! C " 3 < '+" "# n , ! ! & " @ E$ ! %, " 9 6 + 9 3

M0 (x0, y0, z0), % = & ! % α .

n

! ! & "$ ! %

α , < '+" # ' # # E <9 !-

%.

M$

# <

n

= (A, B,C ).

n

# " " ! % M α ,

% ! > ' 3 &

3 & M0 # < #-

'$

n

. " E' @ '# # ! %-

+. 1.11. % & ! +

# ' &

3 & M(x, , z) @ " # -

.

M0M

" A& '-" 9

3 M α

. #& 9> + "$ A&-

M0M

n

$ & D

= 0 . / + ' – @, %

3 M α .

M0M

n

+! " + >

3 E <9 ! % $ ! &=&< '+", " ' 3 M -

" '+" ! ! % D M α .

% ! 3 3

&+-

3 M,

– &+-

3-

r

r0

M0,

=

−

" " # C ! + & @ ":

(

−

)

= 0 .

M0M

r

r0

r

r0

n

/ " " < '+" " "# ! %. ? ! = # $@

$ 6 #. + '

= (x − x0 , y − y0 , z − z0 ),

= (A, B,C ),

M0M

n

A(x − x0 )+ B(y − y0 )+ C (z − z0 ) = 0 .

C, # C " " ! %, % ! > ' 3 & 3 &. # 3 #, " @ % A + + " " ! %, ! A -# ' @ $ " 9 3, % = ! %.

# #, % " " ! % < " "# 1-@ + &! " % ! 3- > x, z.

)

1: + " " ! %, % ! > ' 3 3

!(1;2;3), B(–1;0;0), C(3;0;1).

N P A + + A> " ", ! A $ ! ! &- "$ ! %. # #, % # # A& n = AB × AC .

AB = (−2, −2, −3), AC = (2, −2, −2). ?$# n :

17

i

j

k

=

−2

−2 −3

= −2

− 10

+ 8

.

n

i

j

k

2

−2

−2

" =, " 3 &, 3 " & ! > ' ! %, 3 & !, C # -

" " –2(x–1) –10(y–2)+8(z–3)=0 A x+5y–4z+1=0.

! ': x+5y–4z+1=0. O

2: + " " ! %, % ! > ' 3 3 & !(2;5;–

3) ! ! & " & ". B 3 (7;8;−1) "(9;7;4).

N ! 3 & $# ":

= ( A; B;C) = ((xC − xB );(yC − yB );(zC − zB ))= ((9 − 7);(7 − 8);(4 + 1))= (2;−1;5)

+ ', " @ % A + + " " ! %, ! A -# ' @ " 3, % = ! % !, ! + = + # & " " # <#:

A(x − x0 )+ B(y − y0 )+ C (z − z0 ) = 0

2(x − 2) − ( y − 5) + 5(z + 3) = 0 2x − 4 − y + 5 + 5z + 15 = 0

! ': 2x − y + 5z + 16 = 0 .O

1.14.% & ' ( 1.3

& ! ’" & > C3 3, A = 99 + 9# # # & C& @ &!.

! + " " ! %, % ! > ' 3 3 & ! ! ! & "-
& BC .
1.	A(1, 0, −2), B(2, −1, 3), C (0, −3, 2).
2.	A(−1, 3, 4), B(−1, 5, 0), C (2, 6, 1).
3.	A(4, −2, 0), B(1, −1, −5), C (−2, 1, −3).
4.	A(−8, 0, 7), B(−3, 2, 4), C (−1, 4, 5).
5.	A(7, −5, 1), B(5, −1, −3), C (3, 0, −4).
6.	A(−3, 5, −2), B(−4, 0, 3), C (−3, 2, 5).
7.	A(1, −1, 8), B(−4, −3, 10), C (−1, −1, 7).
8.	A(−2, 0,		−5),	B(2,	7,	−3), C (1,	10,	−1).
9.	A(1, 9, −4), B(5, 7, 1), C (3, 5, 0).
10. A(−7,		0,	3),	B(1, −5,		−4), C (2,	−3,	0).
11. A(0,		−3,	5),	B(−7,	2,	6), C (−3,	2,	4).

18

12.	A(5,	−1, 2),		B(2,	−4,		3),		C (4,		−1,		3).
13.	A(−3,	7,	2),	B(3,	5,		1), C (4,				5,	3).
14.	A(0,	−2, 8),		B(4,	3,		2), C (1,				4,	3).
15.	A(1,	−1, 5),		B(0,	7,		8), C (−1,				3,		8).
16.	A(−10, 0, 9),			B(12,		4,		11),		C (8,		5,		15).
17.	A(3,	−3, −6),		B(1,		9,	−5), C (6,					6,	−4).
18.	A(2,	1,	7), B(9, 0,			2),		C (9,		2,		3).
19.	A(−7,	1, −4),		B(8,		11, −3),				C (9,		9, −1).
20.	A(1,	0, −6),		B(−7,		2,	1), C (−9,					6,	1).
21.	A(−3,	1,	0),	B(6,	3,		3), C (9,				4,	−2).
22.	A(−4,	−2, 5),		B(3,		−3, −7),				C (9,		3, −7).
23.	A(0,	−8, 10),		B(−5, 5,				7),	C (−8,			0,		4).
24.	A(1,	−5, −2),		B(6,		−2, 1),			C (2,			−2, −2).
25.	A(0,	7, −9),		B(−1,		8,	−11), C (−4,						3, −12).
26.	A(−3,	−1, 7),		B(0,		2,		−6), C (2,				3, −5).
27.	A(5,	3, −1),		B(0,	0,		−3),		C (5, −1,				0).
28.	A(−1,	2, −2),		B(13,		14,		1),	C (14,			15,		2).
29.	A(7,	−5, 0),		B(8,	3,		−1),		C (8,		5,		1).
30.	A(−3,	6,	4),	B(8,	−3,		5), C (10,					−3, 7).
31.	A(2,	5,	−3),	B(7,	8,		−1),		C (9,		7,		4).

1.15. 3 . & !

C ! , % A& '-" " " ! = @ + &! " % - > x, , z < " "# " 9 ! %. <# &C & - #& " ! % A(x − x0 )+ B(y − y0 )+ C (z − z0 ) = 0 .

3 # D = − Ax0 − By0 − Cz0 .

C # " " & @ " Ax+By+Cz+D=0, " $ < '+" ' . &-

!, ! 3 #& !, B, " – & < # # ' @ ! %.

@ " # # ! @ ' @ " ". ?'"+&<#, " = &- < '+" ! % % + + # , " % A ' 6 E < - " " A D '+" & '.

19

1. '$ 3 D< & D D=0.

E' #& ! & " " ! % !$# < @ " Ax+Cy+Bz=0. +- ' 3 + x=0, y=0, z=0 ' "D ' " D ! %, ! > ' 3 ! 3 .

2. 6 E < ! ! 3 > > D< &- D (+. 1.12). M$, !, A=0. E' #& ! & " " ! % # < -

@ " By+Cz+D=0. # '$ ! % # < n = B j + Ck

! ! & "$ + . C, ! % ! ' + .

@ 3, " % B=0, ! % ! ' + Oy C=0 – ! % ! ' + Oz.

	+. 1.13. 6 E < !
+. 1.12. 6 E <	! 3$ '$
! ! 3$ $ &D	3 &D

# 3 #, " % " ! % 6 E < ! ! 3-$ D< & D, ! % ! ' !$ $ +.

3. B 6 E < ! ! 3$ $ '$ 3 D- D ' & D (+. 1.13). !, " % ! = D = 0, " D By + Cz = 0 !- < ! %, % ! > ' 3 ! 3 (@ ! & 1).

B # @, > &D3 ! 2, ! % ! A& ! '- D + . C, ! % ! > ' 3 +' . @ 3, ! B=D=0 ! % Ax+Cz=0 ! > ' 3 +' Oy. C=D=0 ! % ! > ' 3-+' Oz.

4. 6 E < ! ! 3 > > DD ' & D (+. 1.14). A=B=0 ! % Cz+D=0 A& ! ' D +$ Oy, C, ! ' $ ! % xOy, ! > ' 3 3 & D

				D
			0,0, −		. " "# Ax+D=0	By+D=0
				C
+. 1.14. 6 E < !			! D ' ! %, ! ' -
			# ! % # yOz xOz.
! 3 > > &D

20