Демус В. Энциклопедический Справочник. Самые...
DESCRIPTION
Научно-популярная.TRANSCRIPT
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087.5 92
68
ISBN 978-966-14-1305-3 () ISBN 978-5-9910-1585-1 ()
Depositphotos/Oleksiy Fedorov, Iarygin Andrii, Andreus, Elnur Amikishiyev, Wolfgang Filser, K ostyantin Pankin, plrang, , 2011 , , 2011 , , 2011 , . , 2 0 1 1
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, . , .
N.N = { 1, 2, 3 ... }
. Z. .
Z = {... -3, -2 , -1, 0, 1, 2, 3 ...}
, / ^ * 0), , . : , , ( ). Q.
Q = J ...-3 , -2 ,-1 , 0, 1, - , 2, 3,... I1 4 8 6 J
, / , , .
I. /2, /, /5, , . ,
(I, ). R. . , .
, . , , 21/2, 2 2 , i ( - i ) . .
, , , , . , .
. z ( , ) z = (; ) , , , . .
z, Rez. z, Im z. : z = Rez + iIm z. : z = + iy, i
, i2 = - 1 . ,
. , ,
, .
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4
a + ib = 0 = + i 0 , , .
0 + ib . .
1) a + ib + id , = b = d.
2) a + ib c + id a + c + i(b + d).
3) a + ib c + id a c - b d + i(ad + bc). , ,
. a + ib a - b i
:(a + b i ) ( a - b i ) = a 2 - a b i + a b i - b 2i2 = 2 +b2.
: ib id = i2bd = -bd.
4) a + bi + di 0 , :
a + bi _ ( + bi)(c - di) _ a c + bd b e - a d .c + di (c + d i ) ( c - d i ) c2 + d 2 c2 + d 2
c + di = 0, , .
. z = a + ib , Rez (; >), a Im z (. 1).
, .
. z = a + ib Z (; ) (. 2).
z : |z| = 'Ja2 +b2 . z |z|
. z = a + ib, (z 0)
( z ( -
, ,. 2 ).
= argz. z = 0 .
. : 0 , , .
. b z = a + ib ( z):
(, )
-
5
a = rcos(p b = rsintp
z = a + ib = r(cos(p + isinip). . x + iy
( I I. , , -
( I 1 0 1: 1= > 1 =
0 1 - 1 0
+ , -
= *
,
=> , ..., ... ,
,
V ,
,
V , ..., ...
3 ,
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/ , ,
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{ 1 , ... , ...
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/ : , ,
, ... ,
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N
Z ,
Q ,
R ,
; ,
; ;
:
* - : - ...
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( ),
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: ... ... ... ...
! :
Jdx : ( ... ...) ... ( d) ...
: ... ...
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()
: a + b = b + a ; a - b = a + ( - b) : a x b = b * a , =
b
(): (a + b) + c = a + (b + c) : (ab)c = a(bc)
()
: ( + b) = + cb : (ab) =
= = , =
> > , >
\] || , | |
=>= > , =>
,
b , ,
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=
||
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= , + = +
> , + > +
() () () ,
n - , = ... .
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:1 . ()" = ". 3. " = "*
2.I " , > .
5. ( ' = ,\6 . " (),
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- , - .
( ), n- , , . . /.
, . n - .
n- . : .
/ = 0 ; VI = 1 . :
, : ,
;
;
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, , () .
, .
(i") * ( \ , -
V
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9
(:a , b t i c e R n n n k e N )
" ^ - 2 - 2+ = 2" ^ 2?Ta-2f b - 2f c = 2 c ( > 0, b > 0, > 0)
2n+l[db~c = 2^ i /b 2 2yjabc = \ 2>\ , (abc > 0)
2 [ [= (#0) = 2?[^ , ( > 0, b > 0) 2 ,
1 2"/2"+l/ = ---- ( b * 0) ( ; . )
4 ^ = 2" ^ , (> 0 )
2" ^ = [ " ( 2 ) = 2 7 , ( >0)
2+ 2" = (2- 1>'2"+ 2 = ( ^ ) 2\ ( R)
() () , ().
, , . . .
( ):
1. j = , > , > , ./* ___
: %/"~,
^ ^ : -= = = = = = = = -----------, ( > 0 ).
/ - *
-
10
A(yja + ylb\ A (y fa + y fb \- > 0 , > 0, .
a - by [ a - y [ b ^-Ja - y f b ^ y f a + ylb^
3. - 7=-----;= - = ----- -----== .
*1 +
yfa + yfb /2 - y/ab + yfb2, y /a - y /b yfa2 + y[ab + yfb2 , ( + ) { ) . -
- l I r i + - ^ + : -j= = = -----= ----------------------------------------------- 1------------
l fa+1/b ( ^ + Vb)(Vfl7 - ^ b + V b r ) +
b , + b 0 .
+ + + ^ + ~=-----1= = ^= ----- / ______ -----f= v = ----------------------------- . b 1 - 4 + ^ + -
, .
A ^ y j a + y / b ^ A ^ y l a + y f b }
- 4 + ^ ~ { - 1 1 + ^ ) { ^ + ~ +
, + * 0 .
^ A ^ f a - y j b ' j A ^y fa -y f l
b
b ^ + i f c b + + t f e b + ) ( & - ~
, .
4. j=---- J .yja - yjb
: ( - ) [ " ~ ' +
+ "~2 + ... + "~2 + "~' ) = " ". x = \fa, y = y[b,
{tfa -yjb^{yla"~' +yla"~2b +...+ yjab"~2 + yjb"~' | = - . :
(1"~' + yja"~2b + ... + yjab"~2 + ylb"~' V=-----= = ---------------------------------------------------- - , ( > 0 , > 0, y ja -y /b a - b
; , b , ).
5. yfa+yfb
( + ){"~ '~
- " ~ 2 + ...+ ( - 2 + (-> ')" ' ) = * + (_ l ) " ^ . = [ , y = y/b,
+ ^ ' -yla"~2b + ... + ( - l ) " 2ylab"~2 + ( - l ) ' yjb"~' | = + (-1 )" 'b . -
^ + 2 ZIb + . . .+ 2< 1 2 + 2 ) : - - = ---------------------------------------------------------------- ,
y /a+ ylb a - b b + b 0.
-
11
----------------------, > 0 , > ,
-, > , > , > ,
2 + 2?/ + .
6 . j=-----, j=---------------- j=-----j= -j= j= .
Ja+yJb + ylc y la + y lb -y jc / - / - /
, , {yfa + yfb - Vc j >
x(a + b - c - 2 Jab'j. :
{ / + y/b -y /c^{a + b - c - 2 y / a b ^
y/a+y/b+y/c ( + b - )2 - 4ai>
(a + b - c )2 - 4 a b * 0 .
7. - = -----= j= = , ab = c d .yja+yjb+yjc + yld
, , ( \/ + /b j -
-{y fc + y/d'j . :
A ^ y f a + y f b ^ - { y f c + ^ / d ^ A ^ y f a + yfb - / - y fd ^4~a+yTb + y T c + y f d = ( + )2_( + ) 2
a > 0 , b > 0 , > 0 , d > 0 , a + b ^ c + d .
8. -j= -j= -j=yja +yjb+yjc
(x + y + z ) (x2 + 2 + z 2 - x y - x z - y z ) = x* + 1 + z 3 -3 x y z . -
= /, y = yfb, z = y/c,-ro{yfa + y fb + y fc ^ \ fa 2 + yfb2 + + yfc2 - ^ fa b -y fa c - y fb c ^ j =
= a + b + c -3y/abc.
= ( + b + )2 + 3( + b + c)yfabc + 9yj(abc)2,
( + b + - /abcJ = (a + b + )3 - 27abc.
2 + y f ^ + /2 - y / a b - / - y /b c Y : ----------------------------------------------------------- ,
( + b + ) - 27abc yfa + yfb + yfc 0 , (a + b + c)3 .
/ / (). :
V a 3 I = + / - / - 2 V 2
> , > 0 2 - > 0. , 2 - (
, - , s, = s2).
-
12
b , , . log0 . :
log, b = x : ' = b ; log, 1 = 0 ( = 1 ); loga = 1 ( 1 = )
> 0, * 0. :
log, (be) = logn i) + log |c |, (be > 0 )
log,, j = log |b| - log,, |c |, ^ > 0 j lg(b ', ) = p lo g 0 |b |, (bp > 0 ) ;
: log > =m
,
lg . b = 7 logb
, , ,
1 , = - i o g a b
lg. b = lo g a b
logc
1g a
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13
-4 -3 - 2 - 1 0
-2-3-4
1 2 3 4
() ( ) ( ) , , z ( , , ) (. 3,4).
. . , OY
, OZ . .
, . .
( ) ( , () 1), . , .
i , j , k , , .
, - (. 3, 4).
(I, II, III, IV), OY, .
: ( ; ; /
, , , .
. 3
( ; , )
e y
. 4
, (. 5).
() , ( ).
. , . ,
( , , ).
: (; + 2) N . ' 5
= 0, . , . R ,
() = = , 0 < - < 2 (. 6 ). -
1 2 - / ( ) ] 2
-
14
. .
(; ) , (; 0,2
= , = 0 ,
-
15
f( .x) = g(x), / g .
() , .
, . , .
, ( ) .
: + = 0. . = = 0, ,
Vo : 0 = 0. = 0, 0, , 0 a e R : a O = - b ^ O . * 0, 0, -V = -/.
2 + + = 0, 0.'
, , 0 2 D = b2 - 4 .
D > 0 - 4 - 4 , , = -------------------- ;
2 D = 0
= ; (, , 2
), 2
D < 0 ; , :
- b \ l b 2 -4 , ~ 2
, :
- 1 \J-b2 + 4
'- ~ 2
+ + q = 0 , = 1 ( )
-
16
2 +2 + = 0 , D - 4(2 - ) D > 0:
- - \ 1 2 - - + \12 - ,= , 2 = ;
D = 0: , = - /
2 + = 0, * 0 II 0 * II 1 >
2 + = 0, < 0*1 = ~ 4 ~ ! ' *2 = yj~C/ a
2 = 0 .= 0
( )
,, 2 + + = 0, : x t + x 2 = ,
, 2 = - .
2 + + = 0 : , + 2 = -b , , 2 = .
D > 0 : 1 + + = ( - , ) ( - 2) .
D = 0 : 2 + + = ( - ,)2.
4 + 2 + = 0 2 = .
* =, > 0, 1. > 0 = log0 b. < 0 .
log = b, > 0, 1.
= .
= - 1 =0 = 1sin* = ,
- 1 = (--1) arcsina + = - / 2 + 2 = = /2 + 2
-
17
= - 1 = 0 = 1
cosx = a,|| < 1
= ia rc c o sa + 2 = + 2 = /2 + = 2
tg j = , (-; )
= arctg + 7 = - / 4 + = = /4 +
ctg = , (-; )
= arcctga + = - / 4 + = /2 + = /4 +
,
|/( ) | = , < 0, = 0, / 0 0 = 0 .
> 0, :
.
' / 0 0 = / ( ) = -
|/M | = *W (
I
:7 * 0 0 s o .
1 / ( * ) = *(*);( * 0 0 > 0 ,
1 - m = g (X)4
f / W > 0 ,
j / ( x ) = * 0 0 ;
j / ( x ) > 0 ,
| - / ( * ) = *(*)
|/(*)|=|*| f ( x ) = g 2(x)
'/ ( * )= * ( * ) . f ( x ) = -g (x ) .
, .
=
=
-
18
= = \1 )
= 2 + + = yjax2 +bx + c
- y = Pn(x) /Q m(x), () Qm(x) - - m -
{ ( ) g(x) X, 2. f ( x ) < g(x). ( < >, .)
() , .
, / ( ) < g(a).
, .
. . . X : / () < g ( x ) ,
f ( x ) < g l(x), , f ( x ) < g(x) / < 8 1 (*)> : , - , .
.
= const, > 0 ; < 0 , .
f ( x ) g(x) > g(x) + h(x) .
h (x )> 0 X, f ( x ) > g(x) f ( x ) h ( x ) > g(x )h (x ) .
h ( x )< 0 X, f ( x ) > g(x) f ( x ) h ( x ) < g(x )h (x ) .
f ( x ) > 0 , g(x) > 0 X, / () > g(x) f 2(x) > g 2(x) .
-
19
( - , )' ( - ... ( - )"* < 0 . ( < >, .)
. , 2 + + .
D = b2 - 4 .
D < 0 D > 0
II
> 0 ,2 + + > 0
(00; ) ( - ; x , ] U [ x 2;oo) bX = -----2
> 0 ,2 + + > 0
X (-; ) x e ( - c c ; x , ] U [ x 2; )
< 0 ,2 + + > 0
[ . ; 2 ] 1IIX
< 0 ,2 + + > 0
( , ; 2)
( 0 ,
/ ( ) < 2().
^ / () > () : J (p (x )> 0 , () < 0 ,
| / ( ) > 2"(); | / ( ) > 0 .
= (). R :
| |> 0 ; || = 0 |^ | | - - .
-
20
.
= sinx, = cosx = 2, s in x > cos > ( > (|| < 1) e (arcsina + 2rai, J i-a rc s in a + 27tn), V Z
sinx < ( | |< 1) ( - - arcsine + 2nn, arcsina + 2nn), V n e Z
cosx > ( | |< 1) x e (-arccos a + 2 nn, arccosa + 2nn), V n e Z
cos < (|| < 1) x e (arccosa + 2 , 2n - arccosa + 2nn), V n e Z
tg x > a x e (arctga + nn, n/2 + nn), \ / n e Z
tg x < x e ( - n / 2 + nn,arctga + nn), V n e Z
ctgx > xe (n n ,a rc c tg a + nn), V n e Z
ctgx < x e (arcctga + nn, n + nn), V n e Z
, .
: f ( x ) = , , ; , , .
, : X; ; () , ,
. ,
() .
, , , . , , : f ( x ) = .
-
21
, , , . , , / ( ) = ' , 1, = 0,5.
. ,
, .
f (x) > 0 (, ), / () .
f ' ( x ) < 0 (, ), / () .
: , , , .
.
.
/, /, ( 0) / + , , , 0 6 > 0 , | - \ < 8 |/ ( ) - / ( ) | < .
, / , : lim / ( ) = /( ) .-*
/ , .
/ ( ) (. . , ).
/ ( ) , / ( ) = , , .
/ ( ) , lim ^/(x ) lim ^/(x ).
.1. : lim ^ /(x )= lim ^/(x ).
.
-
22
/(- 0 - +0 |
f ( x ) = , .
, : f ( - x ) = f { x ) .
(. 8 ).
, : f ( - x ) = - f ( x ) .
(. 9).
, .
, (
), .
: , .
y = f ( x ) , 0, f x i}, , f ( x ) > / ( 0).
, , .
, / \ ) < 0 < 0 f (x) > 0 > 0< . . , 0 ( . 10 ( - /2 + 2; - 1), Z ).
= f ( x ) 0 , , , 0, . . 0, 0, , f i x ) < f ( x 0).
, , .
, / '( * ) > 0 < 0 / '( * ) < 0 > 0, . . 0,
. 9
-
23
= sinx 7 1
\ - /2 / \ \ /2 22 - /2 N. | 0 /2 V ! / \ . 1 /
-1
. 10
0 ( . 10 (/ 2 + 2; 1 ) , k e Z ).
, 0 .
0 () (), () .
. = f i x ) = 0 , .
. .
, 0, (, , 0). , = 0 . 0 , .
/ 0> f ' ( x Q) = 0, / " ( 0)* 0 . /% 0 )< 0 , 0 . f (xQ)> 0, 0 . / 0
/'(*)= * )= = " (*0)= , /'" (*) * . ^ ,) (0) 0, 0 .
/ ()(0 )> 0 , . , 0 .
() f i x ) ( ) (; b), , (; ),
] ; ^ (. 1 1 ).
( ) (. 12 ).1. / ' ] ; [ , / ]; b[
( / ' , / ).2. f (x) > 0 \ / ] ; b[, / .
3. / '( ) < 0 V x Jfl; b [, / .
. = / ( ) 0, 0 . 0 / , 0 /, (0; 0) / . 12
.
1
7 ^
y = f W
' 0 X
-
24
, (jc0; ,,), . / / ' ( . 13 0 (0 ; 0 ) / ( ) = 3).
: / ' ( , ) = .
:1. /"(*) 0, 0 .2. f (x0) = f m(x0 ) = ... = (0) = 0, / (| (0) # 0, 0 , 0 .
, f ( x ) < , / (. . 14; , (=; 0 ) = 1 / ).
, f ( x ) < b , / (. . 15; , , = 2 ).
, , = sinx (. 10 ).
, , (. . 14; ( - ; )
= 3 ).
. 14
= f i x ) , , .
= :1. l i m / ( x ) = ao;
X-KJ-02 . lim f ( x ) = .
>d+0
= / = 0 > (. 16).
= lim f ( x ) = a.
-
25
= * = > - (. 17).
= + :
1. h m ^ Ui-> 2. lim ( / ( ) - ) = b.
JC (
), > + ( > - ) .X 1
/ ( * ) = + - = = / 2 (. 18).2 \/
f ( x ) , , , : / ( + ) = /( ) . . .
= f ( x ) ( ) ( ; ) , (; d), (; d) = g(y) (;), = / ( ) .
>> = / ( ) . / ( ) = 0 . , , ,
. g / , /
g, = / ( ) x = g(y) , . . , = / ( ) , = g(y).
y = f ( x ) x = g (y ): f n g . , = / ( )
= g(y) . , f ( g ( y ) ) = y g ( f ( x ) ) = x.
-
26
, y = f ( x ) = g(y ) .
= . = f ( x ) , x = g(y ) ; y = f ( x ) ,
= g(y) . = = ". = = logo .
.
, (. . ; F G G F ) ; , , , , .
, , . .
, .
. ( ).
, , , , , , .
( ) ( ).
, . , ,
, , .
1- : + = , .
(. 19). 0 -/ .
/ .
-
27
= ( ).
: , (. ),
1. : D (y) = (; + 0, R ( > 0 ); = , < 0 , R ( < 0 ); = , = 0, R ( - 0 ).
= ( 1, 2, , , . . . > )
f ( x ) = a0 + alx l +a2x 1 + ... + at:x n, 0, , , 2..... .
- ,, 2, 3..... , .
0 = 0 , . , , 2, 3,..., 0, , , 2.....
, ( + 1) - ,, 2, 3 ..... - = 0 + ,, + 2 +... + , = 1 .
. = ,
( ). ,
, (. 2 0 ).1. : D ( /) = (-; + ), . .
R.2. : * 0 : () = (-; + ), . . R; = 0 : ( ) = 0 .
-
28
3. () : = , . . / ( - ) = ( - ) = - = - / ().
4. : * 0 , = 0 = 0 ; = 0 , = 0 .5. : > 0 , ; < 0 , .
= / , , 0.
(. 2 1 ).
, , .1. : D(y) = (*>; 0) U (0; + ).
7 2. : () = (-; 0) U (0 ;-).3. () :
, . . / ( - ) = = = - / ( ) .
X X4. .5. = 0 -> +.6 . = 0 , . . / ( ) >+ - 0 + / ( * ) - > - - > 0 - .
7. : < 0 > 0 , - = 0 .
. 22
= 2 + + , , , , .
, = = 0, = 2.
= 2 , . (. 22).
.
= 2 + + (. 23).
: 2
= ----- , = ------ .' 2 / 4
: D = b2 - 4.
-
29
, , ( ) -2 . , - + 4 - - 4
+ + = 0. : , = ------------; 2 = ------------ .2 2
. 23, 24.1. :
D(y) = (-; + ), . . R.2. :
() = \_~D/4 , + ), > 0 ;
(y) = ( -o o ;-D /4 a ] , < 0 .
3. () : = = 0 .
4. : D < 0 ; D = 0 = -/2; D >
- 4 12= ------------.
25. : a > 0 ( a < 0 ) (; /2 ] ;
> 0 ( < 0 ) [ - /2 ;+ ).6 . : > 0 ( < 0 ) / ( x ) min = -D /4 a .
\ , R.
s N - (. 25). N = "
- (. 26). = 1 (. 25) (.
): = . = 2 (. 25) (.
): = 2. = -1 (. 26)
(. ): = / . , . = 0 (. 25)
: = . , X, , . . 0 .
= 2, a e N , . . (. 27) = 1.1. : D ( /) = .2. : ( / ) = [ 0 ; + |\
3. () : .4. = 0.5. :
J ; ] [ 0 ; +
-
30
6 . : = 0. = 2 - 1, N , . .
(. 28): = 2"-'.1. : D ( f ) = .2. : ( / ) = .3. () : .4. = 0.5. : .
= -2 , N , . . (. 29) = i j x 2".1. : D ( f ) = R \ j 0 }.
2. : ( / ) = ]0; + |\3. () : .4. .5. = 0,
> +.6 . = 0 , . . f ( x ) > - 0 + f ( x ) > -
-> 0 - .7. : ] * ; [
] 0 ; +|\
= - (2 -1 ), N , . . (. 30): = l / 2~1.1. : D ( f ) = R \ {}.2. : ( / ) = R\{o}.
. 26
,
1 = \\ \
\ \ 11
\ 1 1
* ^ -1 0 1 *
. 27
,
-1
75 -1
. 30
-
31
3. () : .4. .5. 0 .6 . = 0 , . . / ( ) - + 0 + /(jc ) >
> 0 .7. : ]-; [ ]0; + |\
n e N : = ".1. : D ( f ) = ]0; + | .
2. : ( / ) = ] 0 ; + |\
D ( f ) ( / ) .
f ( x ) = ax, = ( = 2,7182818284590452...)(. 31).
= = ().1. : D ( /) = .2. : ( / ) = ] 0 ; + [\3. : .4. . 0 5. = 0 . 31
-> - .
= .
, , : = mjn ( Z, N ). = 1
: > 0 , = y fa " ; = 0 , = 1, , X, . . ;
< 0 , = ( > 0 ). 1*1
1, : = 1| .1. : D ( /) = .
2. : ( / ) = ]0; + |\3. : 0 < < 1
, > 1 (. 32). . 324. .5. 0 < < 1 - 0 >-.
> 1 = 0 > -.
f ( x ) = loga . > 0, * 1, > 0.
: D (y) = ]0; + o[,
() = ] ; + |\ .
-
32
. 33
1 - -
= log2 .
= i0g ' 9 = log0 ,
> 0, * 1 (. 33).
1. : D ( f ) = ]0; + [.^ 2. : ( / ) = .
3. : 0 < < 1 , > 1 .
4. = !, . . log01 = 0.5. 0 < < 1 > 1 = 0 - 0.
= \ (. 34).1. : D ( / ) = ]0 ;+ [.2. : ( / ) = .3. : .4. = 1, . . 11 = 0.5. = 0 .. 34
= sinx (. 35).1. : D ( f ) = .2. : ( / ) = ] - l ; l [ .3. () : .4. : = 2.5. : [ - /2 + 2; /2 + 2 ] ,
Z , , [ /2 + 2\ /2 + 2:], Z , .6 . : (/2 + 2-, 1),
( - /2 + 2-,-1), k e Z .7. = , Z.
= cosx (. 36).1. : D ( f ) = .
-
33
2. : ( / ) = ] - l ; l [ .3. () : .4. : = 2.5. : [2 -, + 2 J, k e Z \
: [ - + 2:;2:], Z.6 . : (2; 1), k e Z ;
( + 2;-1), k s Z .7. = /2 + , e Z .
y = tg x (. 37).
1. : D ( f ) = R \ {/2 + \ Z j .2. : ( / ) = R.3. () :
.4. : = .5. :
] - / 2 + ; /2 + [, fceZ .
6 . = , k e Z .
7. = /2 + , k e Z , , . .
lim tg(x) = -oo, lim tg(x) = +-x - > + n-Jt+0 - - + - 0
2 2
8 . , . = (2 -1 ) /2 , Z.
= ctgx (. 38).1. : D ( f ) = R \{/ | e Z j .2. : ( / ) = R.3. () :
.4. : = .5. :
[; + [, e Z.
6 . = /2 + , Z.
7. = , Z , , . . jg
lim tg(x) = +oo, lim tg(x) = -oo. - ^ -fc+O x - m - fc -0
8 . , . = , k e Z .
-
34
()
= arcsinx (. 39).1. : D ( / ) = [ - l ; l J .
2. : ( / ) = [ - /2 ; /2 ] .3. () : .4. : .5. = 0.
y = arccos* (. 40).
1. : D ( /) = [ - l ; l ] .
2. : ( / ) = [0; ] .3. () : , .4. : .
. 39 5. = 1.
= arctg (. 41).1. : D ( /) = .
2. : ( / ) = J-T t/2; /2 [ .3. () : .4. : .5. = - n / l - /2
>.6 . = 0.
-
35
arcctgx. (. 42).1. : D ( f ) = .2. : ( / ) = ]0 ; [ .3. () : , .4. : .5. = > - = 0
X > + 0 0 .6 . .
, .
- ~ = shx = ---- -----
1. : D ( /) = .2. : ( / ) = .3. () :
.4. : .6 . = 0.
= ch =
(. 43).1. : D ( /) = .
2. : ( / ) = [ l; + [.3. () :
.4. :
] - ;[ ] 0 ;+ [\5. : (0; 1)
.6 . .
shx
= th = ----- =chx
~ (. 44).
+ '1. : D ( f ) = .2. : ( / ) = ] - ! ; l [ .
. 43
-
36
3. () : .4. : .5 . : - - 1 ( > - ) = 1 ( + ).6 . = 0.
chx +~ ,
= cth x = ----- = -------- (. 44).shx -
1. : D ( f ) = R \ {}.
2. : E ( f ) = R \ ] - ] ; l [ .3. () : .4. : ] - ; 0 [ ]0 ; + [ .5 . : = - 1 ( > - ) = 1 ( > + ) .
: = 0 0.6 . .
. , . , = / ( ) , N, N ( ), = /{ ) 1, 2, . . . ,,....
, , , = + . .
( ) , ( ) : { < 2 < 3 < ...... 2 > 3 >...
, , , d, . d () . an = , + ( n - l ) d .
, , , q , . q -
-
37
. bn = b, q"~'.
, .
/ ( ) = / () = 0, (, ) , / '
, / W - / ( ) ( > (, ), - - ------= / ()
(a',b) N : = lim e > - 00.
.
.
lim = , .* ( ) ( ) , lim(a ) = lim a + lim ;
- - -
1 lim(ca ) = c lim e , ; lim (a b ) = lim a . l i m bn ; 1 - ^ = "^* , -+ .-*00 n->D -5 -0 - fo lim p
-
38
lim an = < bn Vn, < .-
0 8 ( e ) > 0 , < Ijc | < 5 / ( - ) < . : lim x
1 1 ' I - / ( ) - .
/ ( ) - , / ( - 0 ) = / ( + 0 ).
.
lim = , .-* - > / ( ) , g(x),
lim (/(x ) g(x)) = lim f ( x ) lim g ( x ) ; lim (/(x ) g(x)) = l im /(x ) l im g (x ) ;x-*a x *a x *a x *ti z *a x-*a
n x ) l im /(x )lim ------= 2------- , l im g (x )* 0 .*- g(x) lim g(x) *-
x-*a
.
, .
: / , / ,0 , , 0 , , 1 .
) -
() ( 0/0 /), - ().
(): :
Um lim () , , VI/'()
= , , , ( = 0 , ) , (), ( ) , V|/'(x) 0 ;
'(), /'() ( ). / /
, . .
: - ; 0 ; 0, , 1" , - : 0/0 /, .
/ ( ) =
-
39
:
()() = ( ): )
/ ( ) = ()\|/(), > ( > ) ( - ),
() = ------ ; V'(-) ------- , 0 /0 :() v M
v (x )-u (x )f i x ) = ---------------.' m ( x ) v ( x )
f ( x ) = ( )
-
40
* 3 * 5 ! -1 X2" sinx = x ------- h------... + ( - 1) --------------I-..., x e R ;3! 5! (2m -1)!2 4 2
cosx = l ------+ -------... + ( - 1) -------- + ..., ;2! 4! (2)!
3 25 177 629 te x = x + + ------ -------+ ----------- < < ;
3 3-5 3 *5-7 3 5 7 9 2 21 1-3 5 1-3-5 7
arcsinx = x-l------ + --------- + ------------ X ;2-3 2 -4 -5 2 -4 -6 -73 5 2*i-l
arctgx = + ... + ( I)" 1 +..., l ; l l .3 5 2 - 1 L J
1 ( 1 )2 ( 1 )3 = 1------ +-------------------- 1- ..., < < ;
1! 2 ! 3!, 2 3
= 1-1----1-----1-----1-..., 0 , * 1 ; lim ----------------= 1 . 1 *-* olx
-
41
, :
/ '( * ) = / ( * + / ( ) .-0
., ,
. .
. , , . .
, / .
(cf)' = c f
( f + g ) ' - = f '+ g '
( f - g ) ' = f ' - g '
/ g
( f g ) ' = f ' g + f g '
, ( f * j =(e*l" ' j +
/ > 0
' = 0 .
: ' = 1.
: = ; (/*) =
: (sinjc)' = cosx; (cosx)' = - s in x ; (tgx)' = ; (ctgx)' = ------ .cos x sin x
: (e*) - l1 ; (a ' j = a* ln a .
: ( ln x ) = ; (log x ) = p .x x ln a
-
42
: = (), : = ' ' ; (sin )' = cosh ; (cosu)' = - s in u - ';
(tgu) = ;(c tgu )' = -----; ( 7 ) = - ^ = ; ( ) = -;() = 1 ; ( 1 ) = ;cos sin 2yju
u ln a
, . ( ).
F / I, : F'{x) = f ( . x ) V x e l .
/ / F(x) + , .
F / , G g,
F + G f + g : F ' = f G '= g, (F+ G)'= F '+ G '= f + g. F /, ,
kF kf. (kF )' = kF' = kf. (k F ) ' = kF = kf. F(x) f ( x ) , ,
* 0 , F(kx + b) f ( k x + b) :
^ F ( f a c + b ) j = j F ' ( k x + b ) x k = f ( k x + b).
/ / .
/ ( 0 , 1 , 0) .
.
. [; b] , / , , [a; b] = = (. 45), .
/ [; fcj, F , S [; b j , . . S = F(b)-F(a).
-
43
[a; b] S (. 46):
s = ^ ( / ( * ) + / ( * , ) + + '/(* -,))
> 0 -> S (. 47). -
b J > Sn > j f ( x )d x ,
;/ , .
, f ( x ) .
1
. 47
f ( x ) (;) F(x)
, . . F'(x) = f ( x ) < < , j f ( x ) d x = F'(x) + C, < < , .
(J /(x )< ix ) = / ( ) , jF '{x)dx = F(.x) + C,
d j f ( x ) d x = f ( x ) d x ; jdF(x) = F( x) + ;
j f ( x ) d x = F(x) + C, to j f ( a x + b)dx = F(ax + b) + C, a f 0;
f ( a f ( x ) + f}g(x))dx = a j f ( x ) d x + p f g ( x ) dx , a 2 + p 2 * 0
du = d(u + C); du = - d ( a u ) ; du = - d ( u 2) ; cosu du = d{sinu) ; n ? v /
i u d u = -d (c o s u ); = d(ln|w|); f ' ( u ) - d u = d( f (u ) )
jg {x )dx = G(x) + C, jg (u)du = G{u) + , =
-
44
g(x) = g l(.x) + g 2(x), jg (x )d x = j g l(x)dx+ j g 2(x)dx
g(x) , , =
-
45
rea dx 1 f ea (ea d x \-------= ----------------- + ------ n 1
J x" n - l ^ x"-' J x -' J
je Inxdx = - e a In|jc - Ei(cx), Ei(cx)
f e e sin b xdx = ----- -(cs inbx -b co sb x )3 +bf\ea cos b x d x - ----- -(cosbx + bsinbx)J +bf ex . e^sin"-1* , . 4 r t ( r t - l ) f . n_2 ,\e sin x d x = :----- : (csm x - ncosx) + ------ \e sin xdxJ +n +n Je . cos"-1 x . . . n(n - 1) r _2 ,e cos x d x - :----- - (ccosx + Hsmx) + :------ \e cos xdx
J + n +n J
{xedx = eJ 7r
f dx = i ( 1 + erf * j t ), erf(...) aV 2 Tt 2 ^ a v 2 j
^ 2 rx ir~l ) 2n J x in
fsincxcfct = - - c o s cx J f . , sin"'1 cxcoscx - 1 . -2 ,sin cxdx = --------------------- + ------ sin cxdx, n> 0J nc n Jr , sincx xcoscxxsin cxdx = -----------------
J , . 2 coscx I x s in cx x 2coscx\x" sin cxdx = ----------- h----- ---------------------J
6 sincc 6xcos cx 3x2 sincx x3coscxf 3 . , osincx \x sincx ax = ------------- hJ c4f 4 . 24 cc\x sincxax = -------- ;J c
24 cos cx 24x sin cx 12x2 cos cx 4x 3 sin cx x 4 cos cx
------- "120 sin cx 120xcos cx 60x2 sincx 20xJ coscx 5x4 sincx
x""1 x"-3 x"~5
2 -(-1)! 4 ( -3 )! 6 -( -5 )!
- 4
Jx 5* in m fe = ^
Jx" sin cxdx = n ls in cx
- u lc o s c x , i .c-nl ( -2 ) ! - ( -4 )!
fx" sincxdx = coscx + fx_l cos cxdx, n> 0 J 1
& = (- 1 ()1 J U (2 / + 1) -(2 i + 1)!rsincx , sincx f cos cx .I------- dx = --------------- r + ------I dx
x 5 coscx
-
46
t dx 1, J-----= "ln tgJ sincx 2 dx _ coscx'sin" cx c(l - n)sin"_1 cx
r dx _ 1 f cx _ l i s i n c c c g U + 4r x d x x f cx 4! 2 .------------= - t g --------- + I" In
h + sincx 1^2 4 J r xd x x ( n ex') 2 ,---------= - c t g -------
4 sincx 1 4 2 J f sin cxdx l ( cx ----------- = X + - t g - +
J l sin c x I 4 2
n - 2 r dx-------- ^ n - 1 J sin cx
>1
7U c xsin | --------
4 2
lsinc.xsinc.xdx =sin(c, - c2)x sin(c, +c2)x
2 (Cj - c 2) 2 (c, + c2)
cos cxdx = -sincx J
cxsincx n - 1----------+-----nc n
co s cx X s in cx x c o s cxdx = - -----------
jcos" 2 cxdx, n> 0
x sincx n fx 1 sin LX fix
J
Jcos cxdx =
Jjx" cos cxdx
(E S l f ld x = In Icxl + ( - 1)'J x 1 1 t t 2i-(2i)\fc o sc x . c o s c x fs in c x ,I--------dx = --------------- ;------------------- d x , n * l* x ( n - l ) x f l - l - x
I;
(cx)1'
( - D x - ( cx n+1
X
dxcoscx
dx
Jcosn cx c ( n - l)cosn cxr dx 1 cxf------------= - t g J l + coscx 2r dx 1 cxJ------------ = ctgJl- c o s c x 2r x d x X cx 2 ,------------ = - t g + In
J l +coscx 2 r x d x x cx 2 ,------------= ctg + In
J 1 -c o sc x 2 c cos cxdx l cx------------ x tg
J l + coscx 2rcoscxdx l cxj------------ = - x ctg' I - coscx 2
. i z l f_n - 1
dx n>l
r sin(c, - c 2)x s in (c ,+ c2)x , , , ,cosc.xcosc.xax = ------ - 1 -------- ------ , c, * c,
J 1 2 2 ( c , - c 2) 2 (c,+c2) 1,1 121
-
47
tg c x - 1 2 2
seccxdx = iln lseccx + tgcxl ' 1
tg cxdx = In|coscx|
tgcxdx = - tg'c x - ftg"~2cxdx, 1 c ( n - l ) 1
dx X 1 . I . I- = + In sincx + coscx
tgcx + 1 2 2c dx
---------- = ------ 1---- ln|sm cx -coscxltg cx - 1 2 2ctg cxdx
ln|sin
c X l . i . I- = -------- in sin cx + cos cx1 2 2c 1 1
tgcxdx x 1 , 1 - I- + ln |sincx-coscx l
tgcx +
sec" cxsincx n - 2 rsec cxdx = -------------------- 1- ------ sec cxdx, n * 1
( 1) 1dx x--I7 = x_tgTsecx + 1 2
coseccxdx = --ln lco seccx + ctgcxl
cosec"- 'cxcoscx n - 2 r .cosec cxdx = -------------------------- 1--------cosec cxdx, n * l
' " 1 ( -1 )
ctgcxdx = iln |s in cx |
ctg"cxdx = ------ - ctg" 'cx - fctg" 2cxdx, n * 1c(n-l) 3c { n - 1)dx _ f tg cxdx
1 + ctgcx ^tgcx + 1dx _ rtg cxdx
1 - c tg c x ^ tg cx - 1
dx 1 .=ln
coscx sincx ~ c S, cx ,
d x 1 7 tr = tg CX + -
(coscx sincx) 2c ^ 4
_____ = _ ! _ ( s in x - c o s x _ 2( _ 2 ) f______ ^ ______(cosx + sinx)" H - l l^ c o s x + s in x )" '1 (cosx + sinx)" 2
+ Ini sir7 T r I
cos cxdxI sincx + coscx I
coscx + sincx 2 2 c
cos cxdx x 1 I I- = ---------In s in c x -co scx
co scx -s in cx 2 2c
sin cxdx
coscx + sincx 2 2clnlsi T/- Isincx + coscx
-
48
sin cxdx x 1 , | .---------------= ------------ln ls in c x -c
2 2c 1co scx -s in cx cos cxdx 1 1 ,
----- tg + Insincx(l +coscx) 4c 2 2c
cos cxdx l 2 cx 1 ,------------------= ------ ctg2 ---------- In
sincx (l-co scx ) 4c 2 2c
cxtgT
cxt67
sin cxdx cx ] 1 . + - + In 2 4 J 2 c
, CX 71
tgl T + 4cx + 2 4
-In2c
, cx t g l T + -
coscx(l +sincx) 4c sin cxdx l
coscx (l-sincx ) 4c g
sincx coscx dx = sin2 cx 2c
cos(c, + c2 )x cos(c, - c2 )xsinc,xcosc2xdx =
sin cx cos cxdx =
sin cxcos" cxdx =
sin cxcos cxdx
2 (c. + c2) 2 (ci - c , )1
-s in cx, n * 1c(n + l)
1
c(n + l)
sin""1 cxcos' cx n - 1 c(n + m) n + m
sin" excos" cxdx =sin cx cos
c(n + m)dx 1 , I |
------------= - l n tgcxsin or coscx 1
dx 1
cx m - 1 + -------
dx
jsin" 2 cxcos cxdx
jsin" cxcos -2 cxdx;
b 1sincxcos"cx ( 1) cos"-1 cx sincxcos""2 cx dx 1 r dx
sin"cxcoscx c (n -l) s in " 1 cx 's in '' cxcoscx sincxdx l
, n * 1
, n * 1
cos cx c (n -l)c o s" cxsin2 cxdx l l
---------- = sincx + - lncoscx
sin2 cxdx sincx
, * 1
cos' cx c ( n - l)cos_1 cx :in" cxdx
. cx ~4+ ~2
Jn 1 dx
n * 1
sin" 1 cx fSin" 2 cxdx+ --------------- , n * 1
J rn s r rc ( n - l )
:in cxdx
cos cx c (m - l)c o s cx
,sin" cxdx sin"-1 cx
n - m + 2 fSin cxdx---------- > mm - 1 3 cos cx
n - 1 fsin""2 cxdx -------------------------- , m * n m JJ cosmcx c(H -m )cos cx n ~ m i cosm cx
t s\n" cxdx sin-1 cx
cosmcx c (m - l)c o s m cx m
n - 1 rs in -1 c x d x~ > m * 1 1 J cos cx
; m ,n > 0
m,n> 0
-
49
, cos cxdx iI------------ ------------------ 13 s in " cx c(n l ) s in " cx
o s 2 cxdx;----------= - | COSCX + Ins in c x c ^
rco s2 cxdx
cxt g y
pCOs c x d x l { c o sc x r dx )f ; = -------- t j + f-------- 5 > * 1J sin cx - l ^ c s in cx J sin cx J /cos"cxdx cos"+1 cx - - 2 i-cos"cxdx
* sin" c x c ( m - 1)sin"'-1 cx m 1 *sin2cx 1
|.cos cxdx ei
cos" 1 cx n - l rc o s n cxdx + ------- ------- --------, m * n
c i r isin cx c(n - m)sin' cx n - m 3 sin cx
cos"-1 cx------------- : I - .sin cx
/cos" cxdx
^ s in 1 cx c(n-l)sin '"'c x m
Jsin cx tgcx dx = - ( In | sec cx + tg cx | - sin cx )
rtg "cxdx lp ;-----= -----------tg" 'cx, n I sin cx c ( n - l )
ftg"cxdx i
n - 1 fCos" 2 cxdx- f------ ----- , m 1
- 1 J s in
J cos1 cx c(n + 1) rdg"cxdx i
tg cx , - 1
ctg"*'cx, - \3 s in 2 cx c (n + l)
fc t e"cxdx ij - 1 "!----- = T77LTTtg ,"', ' * 1cos2 cx c ( l - n )
tgm (cx)p u dx = - 1 - tg"1*'1-1 (cx) - J tg ^ dx n4t lt " r t a l i - v lctg "(cx) c(m + n - l ) 3 ctg(cx)
x > 0Jlncxdx = xlncx - X
|( ln x )2dx = x(lnx)2 - 2 x ln x + 2x
j(lncx)"dx = x(lncx)" - J(lncx)"_1dx
f = ln |lnx | + y ^ ln x 1 1 t t >'!
dx x l r dxJ;(lnx)" (n - l) ( ln x ) ' Jx " ln x d x = x"'1
* ' 7 1
m + 1 (m + 1)
"" ( ln x )"
(lnx)"
m - 1
n 1
fx"(Inx)"dx = - ------ --------- fx (lnx)" 'dx, - 13 m + 1 m + 1 3
(lnx )V x (lnx)"*'X
In xdx + 1
lnx
- l
1
( r n - l ) x m-' ( m - l ) 2x
-
50
(In x)"dx
xx mdx
(lnx)"
( m - l ) x n 1 3(lnx)" 'dx
m -m + 1 r x mdx
, 1
(lnx)"
x mdx lnx
- ^ - = ln|L x ln x
dx
m + 1 r:r r+"^T Ji(n 1)(1)" n - 1 (lnx)"
= Ei((m + l)lnx), Ei(x)
lnx
x " ln xdx
= lnjl]n x |+ ( - ! ) ' i = l 1
( (w - l) '( ln x )1
i i!
* 1x(lnx)" ( 1)(1)"
xsin(lnx)rfx = (sin (lnx) - cos (lnx))
2 x
cos (In x )d x = (sin (In x) + cos (In x))
-
- d x = -a ( l - n ) x - b
( + b)" ( -1 )( -2 )( b) x2 ^ \ ( (ax + b)7
ax + b a3 2
- {1.2}
- 2b(ax + b) + b2 ln\ax + b\
(ax + b)2dx = - ^ A a x + b - 2bln|ox + b\ -
(ax + b)
- dx = ^ \ (ax + b)" a1 '
{1, 2, 3}
dx 1 .-------- = In
x(ax + b) b
dx
j d x = ln |ax + b\ +2b
ax + b
b2ax + b 2 (ax + b)
1 2b( -3 )( + )"3 (n -2 ) (a x + b)"~2 (n - l)(ax + b)
ax + b dx 1 a 1- = ------+ In
x 1 (ax + b fdx I ax
a ^ = 7 b aTCt4dx x
b2 (ax + b) ab2x b3
Jx (ax + b) bx b
ax + b
ax + b
dx
J (x + a ) 2a (x + ) 2a
1 x+ a rc tg -
3x 3 x + ra rc tg
(x + a ) 4 a (x + a ) 8 a (x + a ) 8 adx 1 , x 1 , a - x 1 1 1 1 ;----- - = arcth = In-------, x < a
x ' - a a a 2a a + xdx 1 . x 1 . x a I 1 . I;----- - = arcth= In-------, x > a
x" - a a a 2a x + adx 2 l a x + b , ,
r----------- = . = a rc tg -p 4a c - b > 0ax +bx + c y l4 a c -b 2 %l4ac-b2
-
51
dx
3ax2 +bx + c ylb2 - 4ac \Ibz - 4 a c b2 - 4 a c
- 2 < 0 * dx 2 ,2 ;----------- = -------------, 4 - = 0
+ + 2 + * j 1 | I 2 | ------------ dx = In + ftx + c ------ I-3ax +bx + 2a ' ' 2 a 3 i
2 , 2ax + b-a rc th - ln
2a x + b - \ l b 2 - 4 ac
2 ax + 4 ac
3ax2 +bx + cr m x + n , m , I 2 , I 2an - bm 2ax + b , 2J - ;-------dx = ln |ax +bx + c |H----- , arctg / , 4 a c - b >0
I:
ax +bx + 2 a m x + n
a ^ 4 a c - b 2 !4 a c - b 2
ax +bx + c 2 a m x + n
i 2 a n - b m . 2 ax + b l2dx = ln |ax +bx + c \ ------ arcth , , 4a c - b < 0
3ax +bx + c 2 a r dx 2 ax + b
a\lb2 - 4 ac J b 2 - 4 ac
4ac - b
(2n - 3)2a
, . . . , | 2 , i 2a n - b m l2dx = ln |ax +bx + c | -----;-------- 77 , 4 a c - b = 0
a(2ax + b)
r + - f---------* L _} (ax2 +bx +* (ax2 +bx + )" (n - l)(4ac - b2)(ax2 +bx + )" ' (n - l)(4ac - b 2) 1 (ax2 +bx + )"
x , bx + 2c b(2n 3) j- dx
c - b 2) h a x 2 +bx + c)"-'- d x = -
! (ax2 +bx + c)n ( n - l ) ( 4 a c - b 2)(ax2 +bx + c)" ' (n - l)(4ac - b2) J (ax2dx
= In
I
x(ax +bx + c) 2c
dx
ax2 +bx +
b dx 2c 3ax2 +bx + c
= arcsin + C, Ixl < a x \ a J
f , = arcsin(x) + C, Ixl < 1
f . - = ln (x + Va2 + x 2 j + C, |x |< a J . ^ - = ln lx + Va2 - x 2 l + C, |x |> a Va2 + x 2 ' ' \ la2 - x 2 1
Js h c x d x = - c h c x j c h c x d x = - s h c x
(sh2 c x d x = s h 2 c x ~ fc h 2 c x d x = s h 2 c x + J 4c 2 J 4c 2
fsh cx 03 cn n 3
: fsh" cxdx = ----------sh"*1 cxchcx--------- I s h " 2 cxdx, n < 0, n -1J ( + 1) n + l J
fchcxdx = shcxch""cx + - - fch"-2cxdx, n > 03 cn n 3
: fch" cxdx = ------ - shcxch"fl - + ^ fch',+2 cxdx, n < 0, n -13 c(n + l) n + 1 J
r dx 1, , cx 1, c h c x -1 1, shcx--------= - l n t h = - I n --------------= - l n ---------------
J shcx 2 c shcx chcx + 1r dx 1 - = cthcx
J sh cx f dx 2 - = - arctg e
J chcx
= i l n
ch cx - 1
chcx + 1
-
52
1 , = -th c x ch
dx chcx n - 2 t dx------=-------------;---------- ------ ;---, (I# 1sh"cx c (n - l) sh " cx n - 1 'sh"
r d x s h e * n - 2 r d x----------= ------------------------ 1--------- , 1
J c h " cx c ( n - l ) c h " cx n - l J c h " ~ c x
rc h " c x , c h cx n - 1 rcW~2 cx ,----------d x = ------------------- -- + -------- I-------------- d x ,
J s h cx c ( n - m ) s h m cx n - m J s h m cx
( ch" cx , c h "+1 cx n - m + 2 t ch " cx , : ---------- d x = -----------------------:------ 1- -------------- ------ d x , 1
J s h " c x c t m - D s h 1 c x m - 1 J s h " cx
t ch"cx , ch" 1 cx n - 1 rch" 2cx ,: -------- dx = ------------------ ------ 1-------- ------- dx, m 1
J sh "cx c ( m - l)sh cx m - l J shm cx
rshmcx , sh'"-1 cx m - 1 rshm2cx-------- dx = ---------------- ------ h------- -----------dx,
J ch"oc c ( m -n )c h " cx m - n J ch" cx
rshmcx . sh"1*1 cx m - n + 2 e sh" c x . : -------- dx = --------------------- H------------ ----- dx, 1
J ch" cx c ( - l) c h " cx n - 1 J ch" cx
rsh'"cx . sh "1 cx m - 1 rshm2cx: -------- dx = ------------------ + ------- ----- , 1
J ch" cx c ( n - l)ch cx n - 1 J ch" cx
fx shcxdx = - x c h c x - -i-shcx J
\x ch cx dx = - x sh cx - -i- ch cxJ '
j th c x d x = - lnjchcxl
Icth cx dx = i In I sh cx Ij c i i
Ith2 cxdx = x - - th o c 1
fcth2 cx dx = x - - th cx J
fth" cxdx = ------ - th"_I cx + fth"~2 cxdx, 13 c ( n - l ) 3
fcth" cxdx = ------ - cth"-1 cx + fcth-2 cxdx, n 13 c ( n - 1) J
j shbx shcxdx = ^(bsbcxchbx - cchcxshbx), b2 2
[chbxchcxdx + ^ (bshbxchcx - cshcxchbx), b2 2 1 b -cjchb xsh cxd x = -(fcshfocshcx-cchbxchcx), b2 2
fsh(ax + b)sin(a: + d)dx = a ch(ax + f>)sin(cx + d) C sh(ax + b)cos(cx + d) 1 a +c a +c
fsh(ax + b)cos(cx + d)dx = U ch(njc + b)cos(cx + d)+ sh (as + b)sin(cx + d) J " -i- -- a +
-
53
fch(ax + b)sin(cx + d)dx = a -sh (ax + b)sin(cx + d) ch(flx + b)cos(cx + d)1 a +c a +c
fch(ox + b)cos(cx + d )d x = --sh(ax + b)cos(cx + d) + -C ch (ax + b)sin(oc + d)J a +c a +c
- > 0, .
f ( x ) [ , ]. = 0 < , < 2 0, [a , fa].
I = lim / ( ! ; , -)(, - ) = f f ( x )d x(,.-. . ) - * 0 *i - l a
I / ( ) [ , b]. > 0,
[ , , 6, max(xr - ^ , ) < 5,
i=i
f ( x ) , .
, (. 45).
j f ( x ) d x ,
, = = fa }( ) . .
, : .
F / [a; fa] j f ( x ) d x = F (b)-F(a).
a
/ , [ ; ].
f ( x ) [a; fa]. f ( x )
[a ;b ] , [ , ; 2] [ ; ] .
-
54
f i x ) () [ ; ], f { x )x .g (x ) .
f { x ) , V a :
J f ( x ) d x = j f ( x ) d x .
/ ( * ) g(x) [a; b],
f ( x ) < g ( x ) V x e [ a ; b \ . : j f ( x ) d x < jg (x)dx .
.1. f ( x ) [a; b ', < f ( x ) < [; b],
(m, = const). m { b - a ) < j f ( x ) d x < ( - ) .
j f ( x ) d x < |/( ) |< & , a / + B g(x)} ix = A \ f ( x ) d x + jg (x)dx .
f ( x ) [ ;* ] ,
j f ( x ) d x = j f ( x ) d x + j f ( x )d x .
f ( x ) = 1, j f ( x ) d x = jd x = b - a , >.
. f { x ) [a ; b ],
[ ; ], j f { x ) d x = f ( c ) ( b - a ) .
. / ( * ) , g(x) , g{x) ,
f f ( x ) g ( x ) d x = f ( ) jg ( x )d x , a
-
55
- = {) < u < |3, / ( ) < < ,
, ()
< < , J f ( x ) d x = |/[()] du*
-
56
/180*0,017453, /(1 8 0 60)0,000291, / (180 60 60) 0,000005 .
, ,
. 49
, . 49 .
,
. 49 a
: sin = /
: cos = /
sinP = fc/c, ,
: t g a = /
: c tg a = b/fl
,
: seca = c/b
: cosec a = / a
, 0 90 ( 0 /2 ).
. (. 50) . 0(0; 0).
( , ).
. : . .
-
57
. (
, ). , . , .
. 51 .
. 49 .
: , . : , .
, 1.
, .
, ( . ).
0(0 )
30(/6)
45( /4)
60( /3)
90(/2)
180()
270(/2)
sin 0 1/2 4 2 /2 4 1 /2 1 0 -1
cos 1 41/2 41 1/2 0 -1 0tg a 0 1 41 0 ctg 41 1 \/4 0 0sec 1 2(41 41 2 -1 cosec 2 4i 2 1 -1
sin cos tg a ctg
I 0 < < / 2 + + + +
II / 2 < < + - - -
III < 3 7 1 :/2 - - +
IV /2 < < 2 - + - -
. 50
-
58
I :
. 51
p sinp cosp tgp ctgP/2 + cos a -s in a -c tg a - t g a
+ -s in a -c o s a tga ctg a
/2 + -co s a sina -ctg a - tg a
2 + sina cosa tg a ctg a- a -s in a cosa - tg a -ctg a
/2- a cos a sina ctg a era P
- sina -c o s a - tg a -ctg a
/ 2- -c o s a -s in a ctg a tg a
2 - -s in a cosa - tg a -ctg a
sin2 + cos2 = 1 tg a x c tg a = l, /2, n e Z
l + tg2a = l/cos2a , a n/2 + n ,n e Z1 + ctg2a = l/sin2 , , n u Z
-
59
, .
sin I----- sin Vl - sin2
= V l-s in , tg a = . ctga = ----------------- V l-s in 2a sina
cosa , ,--------2 >/l -c o s 2 a cos ai a = V l-c o s a , tg a = ----------------- , ctga = -+V 1 -c o s 2 a
tga 1 tga 1ctga = ----- , sina = , cosa =* ) Jill IA , , C u e Ur " ^ lga -y/l + tg2a l + tg2a
ctg a tg a = -------, s in a = -------------ctg a 4/l + ctg2a
1 ctg a, cosa =
t-y/l + ctg^a
sin sin = 2 sin----- -cos-------
2 2_ . + -
cos + cos = 2 cos-------cos-------2 2
. + . - cosa -co sB = -2 s in ----- - s in ----- -
2 2tg a tg P = sin a ~P^ a ,p -t n/2 + , n e Z
cos a cos p sin (aP)
ctga + ctgP = ;------, , * , n e Z
(bcosx = yJa2 + b2
bsinx = \la2 + b2
x arcsin
Va2 +b J
sinasinp = - ( c o s ( a -p ) - c o s (a + P))2
cosacosp = -(c o s (a - p) + cos(a + P))2
sin ac0sp = - (s in (a + p) + s in (a -p ))
cosasinp = i (s in (a + p) - sin(a - P))2
-
60
sin(a + - ) + sin(p + - a ) + sin(a - + ) - sin(a + + )sinasinPsiny = -----------------------------------------------------------------------------------
4-co s (a + - ) + cos(P + - a ) + cos(a - + ) - cos(a + + )
.sin a sinp cos = ---------------------------------------------------------------------------------------4
sin (a + p - ) - sin (p + X - a ) + sin (a - p + ) - sin (a + p + ) sinacosPcosy = ---------------------------------------- -----------------------------------------
cos (a + p - y) + cos(p + - a ) + cos(a - p + y) + cos(a + P + y) cosacospcosy = -------------------------------------------------------------------------------------
(, . . )
sin2a = 2sin a cos asin3a = 3sina - 4sin3 asin 4 a = cosa^4sina - 8sin3 a )
sin 5a = 16sin5 a -2 0 sin3 a + 5sin a- i f
sin(Ha) = 2',_irT sin a-I-----i i l J
cos2a = cos2 a - sin2 a = 1 - 2sin2 a = 2cos! a -1 cos3a = 4cos3a - 3 c o s a cos 4 a = 8 cos4 a - 8 c o s 2a + l cos5a = 16coss a - 2 0 c o s 3a + 5cosa
2tg a tg2a = 2
1 - tg a 3 t g a - t g 3a
tg3a =l - 3 t g a4 t g a - 4 t g 3a
tg 4a = , , l - 6 t g a + tg a
tg4a - 1 0 t g 2a + 5tg5a = tg a 2 - --------------------
5tg a - lO tg a + 1
ctg2 a -1 ctg 2a = e
2 ctg a3 c tg a -c tg 3a
ctg 3a = 2------- f l -3 c tg a
ctg4 a - 6 c t g 2 a + 1ctg 4 a = S ------ - --------
4ctg a - 4 c t g a
, /2.
. II- c o s asm = , ----------- , 0 < < 2
2 V 2
-
61
/2
- < <
1 -c o s a sm a 1 -c o s a----------- = -------------= ------------ , 0 < < 1 + cosa 1 + cosa sina
tgI =
a ll + cosa sina 1 + cosactg = J ----------- = ------------ = ------------ , 0 < a <
2 V l - c o s a 1 co sa sina
2 t g | l - t g 2! 2 tg | l - t g 2|sina = ------- ; cosa = --------- ; tg a = ; ctga = ---------
2 oc 2 a 2 ot , ; al + t g - l + tg - l - t g y g ~2
l - c o s 2 a . , 3 s in a -s in 3 a
sin a = -2 4
3 -4 c o s 2 a + cos4a . . 1 0 sin a -5 s in 3 a + sin5asin a =
cos a = -
8 16l + cos2a , 3cosa + cos3a
2 4. 3 + 4cos2a + cos4a
cos a = ---------------------------8, 10cosa + 5cos3a + cos5a . , , l - c o s 4 a
cos a = ------------------------------------ sin acos a = -------------16 8
. , . 3 s in 2 a -s in 6 a . . . 3 -4 c o s 4 a + cos8asin acos a = --------------------- sin acos a = ---------------------------
32 128
sin(a + p) = sinacos(3cosasinp cos(a + P) = cos a cos P + sin asinp
tg (a P ) = ^ ~ ^ , , */ 2 + n + * / 2 + , - * / 2 + , n e Z IT tgatgp
ctg (a P ) = ct8 a c t6P + 1| > ^ + # , - ^ n , n e Z c tg P ctg a
.
shx = -isin(ix), chx = cos(ix), thx = -itg (ix); sh(ix) = isinx, ch(ix) = cosx, th(ix) = /tgx.
-
62
shx chx ch2 sh2x = l
ch(-Jt) = shx; sh (-x) = -s h x th (-x ) = - th x ; cth (-x) = -c th x
sh(x y) = sh xch chxsh ch (x y) = ch x ch sh x sh , , . thx th v
th(x v) = ------------ i-1 thxthy
cthxcthy + 1 c th (x v )= '
e th y l cthx
sh2x = 2 ch x sh x ^thx
1 - th 2 X
ch2x = ch2 x + sh2 x = 2ch2 x - l = l + 2sh2x = ' + 1 - th 2 x
, 2thxth2x = -------
1 + th x
cth 2x = (th x + cth x)2
c h 2 x - l sh2xthx = ----------- = ------------
sh2x l + ch2xch 2x sh 2x = (sh x + ch x)2
sh3x = 4sh3 x + 3shx ch3x = 4ch3x - 3 c h x, . 3 + th2x
th3x = th x --------- l + 3th x
sh 5x = 16sh! x + 20sh3 x + 5shx ch5x = 16ch5x - 2 0 c h 3x + 5chx
, th4x + 10th2x + 5th5x = th x --------------------------
5th x + lOth x + 1
ch(x + y ) - c h ( x - y )shxsh = ----------------------------
2sh(x + y) + s h (x -y )
sh x ch = ----------------------------2
ch(x + y) + c h ( x - y ) chxch y - ----------------------------
ch(x + y) - ch(x - y)th xth y = -----------------------------
ch(jc + y) + ch(x - y)
-
63
s h * s h y = 2 s h ^ c h ^
2 2
ch * + c h , = 2 c h ^ c h ^ 2 2
c h * - c h y = 2 s h ^ s h ^ 2 2
sh (x v)thjcth_y = -------------
ch xch y
, 2 ch2x + lch x = ------------
2, , c h 2 jc - l
sh x = -----------2
. j 2thxth x = -------
1 + th x:
sh * chx
shx + chx = e'
.
, , (, , ) .
, ., , . X
X. : X, X, X .
X, : . : X, X .
. , : {a, b, } , {, , , ... } .
, . 0 , 0. .
() , . U.
, .
1. , , . . , , : = .
-
64
2. , , : ) = !
( -1 ) - - - ( - + 1) (-1)---(- + 1) \ 1-2-3
= , 0 < < , = " = 1, +,' + ... + = 2
; + c ;+1 = *,', < <
, :
( + )" =" + '-' + ' 2 2 + ... + &" = - ' ' ,1=0
, , () - ,
-
65
*
( >):
:
(, + 2 +... + a j = ^ -------'? - , , + 2 +... + = ., !2!... !
, .
,. : -
( )2 = 2 2> + 2 a2 - b 2 =(a + b)(a-b)( + b - )2 = 2 + 2 + 2 + 2 - 2 - 2
( b)3 = 3 2 + 3ab2 3 a3 b 3 = ( b)(a2 +ab + b2)
( )4 = 4 + 43 + 622 43 + 4 4 - = ( - )( + )(2 +2)
-
" - " = ( - )("~' + "~2 + "'32 +... + d2br- 3 + ~2 + ") 2" - 2 =(a + b)(a2'-' - a2"~2b + a2"~3b2 - . . . - a 2fa2"3 + 2"-2 - 2), 2" - 2" = ( + )(2"-1 - 22 + a2"V - . . . - a2b2" '3 + ab2'"2 - 2- ' ), n e N2"*1 + 2"*' = ( + )(2" - 2~' + 222 - . . . - a2b2"2 + ab2"~ - 2" ), 2"+1 + 2+| = ( + b)(a2r- a 2 'b + a2,- 2b2 - . . . - a V " 2 -nab2'" 1 - 2"), n e N
1. ( - )2" = ( - )2" , N .2. ( - )2"*' = - (6 - )2"* ', N .
, .
0 11 1 12 1 2 13 1 3 3 14 1 4 4 15 1 5 10 10 5 16 1 6 15 20 15 6 17 1 7 21 35 35 21 7 18 1 8 28 56 70 56 28 8 19 1 9 36 84 126 126 84 36 9 110 1 10 45 120 210 252 210 120 45 10
(- , ,... , ; ) .
-
66
1. ( + ) 2".2. , , .3.
; 2.
( ) (. 52).: , , , , .
(, ) :
| |,| |,| |,| |,.
, , : = .
. ( )
(') . . ,
.
:
* =/ = |
, , , ,,..., (| .
, , () 2 (. 53).
>------------------ -' . . -
. 53 , , .
( ,; 2) = + ,? ,.
, , , : + = +,
+ { + ) = ( +) + , + 0 = ,
( ) = ( ), + = ( + ),
aa + ab = ( +), = - 0 = 0.
. 52
-
67
i , j , , (. 54);
\ \> zi ; 2 < 2 < z 2 b ;
a = x ti + y j + zjc , b = x 2i + y2j + z 2k,
a = U ,;> '1; 21),
: a +b = (x, + x2; y l + y2; z, + z ,), a - b = ( x , - x 2; y , - y2;z , - z 2),
aa = (a x ,; a y ,; a z ,),
l l= V x.2 + >'i2 + z i2- A (a,;a2;fl3) ,
B(b,; b2; b3) , = (fy - \2 - 2\ - ),
= /(, - , )2 + ( - 2 )2 + (3 - , )2. ,
(. 55). : a x f r = = 0 . -
. : || : a t t : 4--
, :
; ; . ( ) , ,
, ( ) = 0 .
a, b ,c ,d . :
, .
, , . (,,) = 0 (
). ( ), ,
, 2 , = Xfi + 2
, , , b = 0 = 0. , b, .
d : d = xta + 2 + 3. |,,2, 3} d .
-
68
2, 3 , , , , , 2 2 3 .
, . :
, , , , , . .
1. (. 56).
, . ,
, :
+ = .2. (. 57).
, . , .
+ = , ()
:
^| | +| | -2| o a |x |o b |xc o s
-
69
a = x j + y tj + z,fc, b = x 2i + y 2j + z2k,to a b = xtx2 + y , y2 + z,z2, a 2 = x\ + y* + z,2.
/>\ ,2 + , 2 +z,z2
s , = . cos\a,b = --------- -----V I \\\ \ / * , +, +Z,
() , .
() :
(, ) = 0 .
/.|||. * 0 , X > 0 (. 59), , X < 0 (. 60).
= ^ ( ) 2+ ( )2 = \ 1 < + 1 =1 ( ,; 2) X
(,;2), . . (,;2) = (1;2). X, :
( + ) = +| .
.:( + ) = + .
+ + 2
,
0 \ { ,
. 59
. . . .
, 0 / \
'
,
. 60
, ,
, (. 61).
b , [a b ]
a x b , |[abj| = |a||b|sin
-
70
( ), : [ , b] = Se .
[ , b] = - [b , ] ;
[( ), ] = [ , ( )] = [ , b j ;
[( + ),] = [ , ] + [& ,] ;
[[a,b],c] + [[b,c],a] + [[c,a],b] = 0 , R1 R7
[ , ] = 0 [, [, ]] = (, ) - (, )
|[,]| +(, )2 =||2|&|2 ([, ], ) = (, [6 , ])
, , ( , , ) ( ,,)
i, j, , = {,, yv z ,}, b = {x2,y 2,z 2}=> [, ] = i j
*i , z,
*2 2 Z2
, z, 2 Z2
vl 1 , z,
*, , X,
= 1 z. 2 z2
X , Z,
X , Z ,
*1 >1
*2
(, , ),,
& :(, 6, ) = (6 ) .
,
, , .(, , ) = V^b , ,, , (, , ) = ~Vab , , ,
, , , . ( , b , V = 0 . )
(. . , ), .
e = { w r zi}> = { 2 - 2 }- C = {x 3,y 3,z ,} , =>(,,) =\ z,
*2 z2 z3
-
, , .
, , .
: , , , : , .
, .
, .
, (, : , , . .).
: .
( ) .
, .
. .
, .
, , .
, .
, .
, (, , ).
-
72
, .
, , .
.
, ( , ; ).
, , , .
, .
, ' , ', , , , ', , , , , ', ', ' ( ).
: , ( ).
: , , .
, , , .
, () , , .
, : , , ( : , , ), , , , .
, . .
. .
, .
-
73
2)
: + + = (2 + 2 >0). :
1) By + = 0 ; + = OY;
3) + By = 0 ;4) = 0 ;5) = 0 OY.
: -7 + = , 7 - .
= (, ) .
(. 1): = + , b 0, , = tga, a ,( < a < ).
(. 2): + = 1,
, 0Y.
(. 2): x co sa + y s in a - p = 0 , a ,
; .
- + +
---------------- -
. 1
i-jA2 + :- = 0 .
1
sJ a 2 + 2
; , * 0 , = 0.
, ( ,, 1) (2, 2)
- _ 2 - 1
. *, * * 2 ,*
, (0; 0) ~ = ,
.
, (. 3, ).
(. 4, end) , ,
, . . 5 . _____________________
, , . . 5 . -----------------------------------L- . 3
-
74
, , , , , . . 5 > , .
, .
, .
, .
, .
,
. 6 : 3 6 ; 4 5. 1 8; 2 7. 3 5; 4 6 . 1 7; 2 8 . 1 5; 2 6 ; 3 7; 4 8 .
+ By + = 0 A .-B D = 0Dx + Ey + F = 0
= + = y = p x + q
+ By + = 0 A D + B E =0Dx + Ey + F = 0
= + = 1 = px + q
( ,, ) (2, 2)
d = ^j(xl - x 2f + ( y l - y 2)2
___________.
. 4
-
75
(0, 0)
+ By + = 0 | 0 +0 +\
U ' + b >
+ By + = 0 Dx + Ey + F = 0
I C - F /- 1 1
1 2 + 2
= + y = px + q
+ By + = 0 Dx + Ey + F = 0
1) = + y = px + q\
- tana = ------ .
1 + 2) + By + = 0 Dx + Ey + F = 0 :
\a d + b e \
J a 2 + b 2 J d 2 + e >
aIx + bIy + cl = 0 x ,2 ~ 21 a2C, ~ a,C2a2x + b2y + c2 = 0 ,2 ~ 2, " ~ 2,
= , + , .. b2 ~ b; .. ~ , = 2 + 2 >
1J*
r 1 N1
_P*T 1
, ,
, . ,
. , ,
. , ,
.
, . . 180.
-
76
180. .
. 7
b ( ), b .
( ), b .
b 180 ( ), b .
180 ( ), .
b ( ), b .
, , .
, , .
, .
, () , (. 8).
. , , , (. 9).
, , .
( ) , , (. 10).
, (. 10).
, ( ) (. 11).
, , , :
, (. 12); , , . .
. 8
, / \ ,
V\Ja ,
/ 1\,
. 9
. 10
-
77
[) , .
. , .
() .
, , (. 13). .
. .
. ,
, .
, . 180 (. 14).
90. ---------------- 90. . 14 90
180.
, . . 15 : 1 3; 2 3; 2 4; 4 1. ,
, .
. 15 : 1 2; 3 4.
. 13
= 180
. 15 180 (. 16).
(. 17). (. 18). : , ,
.
-
78
, () (. 19); , , .
, .
: ( . 19 ) ( . 19 ),
; ; :
+ >
+ >
+ >
, .
+ + = . (. 20):
= + ; , = + ; , = +
2 = + + , .
S = \ ah = \ bhb=\ ch (-21),
S = -absinY = -acsin$ = -bcsm a,2 2 2
S = ^ { - ) { - )( - ) ( ),
_ abcS = -^ - , 5 = , -. 19
(. 22).
: a2 =b2 +2 - 2cbcosa b2 = 2 + 2 -2accosP2 = b 2 +2 -2afccosy (. 23)
_ b : ------ = -------= -------= 2, R -
sin sinp sin (. 23).
. 21 . 22
-
79
, (. 24).
= ,,,, :
= ,,; = ,,; = ,, :
1. , .
2. , .
3. , .
,
, (. 25).
ZA = ZA,; ZB = Z B ,; ZC = ZC,
. 24
~ 111ZA = ZA, ZB = ZB, ZC = ZC,
-
80
1. , .
2. , , , .
3. , .
( < 90, . 26), ( = 90, . 27), (>90, . 28).
. 28
, (. 29), (. 30) (. 31).
. 29 . 30 . 31
-
81
, (. 32). .
, , ; ;
> , > ; ;
+ > ; = /2, + = /2 .
;2 +2 = 2, , b ;
. , , , + - =, =, \ = , = - , = - , = - , ,
b; hb mh R ,r , , ;
a = bsina = bcosy,
c = b siny = cosa,
fl = c t g a = c-ctgv,
c = a tg y = a c t g a
, ; ; ; ; .
, .
; (. 31)
; ~
= ------ ;2
, ; , ,
:
; = = 1 ' = ^ - .
( , ), - , , .
_ chc _ 2 siny2 " " 2
:
m = h = l = a 4 b / 2 ,
= !/'3, = -\//,
R = 2r, S = 2 -\//4.
-
82
,
.
,
2 : 1, :
O F = - A F3
1 = -
3
= - 3
. .
:
' 22 + 22 - 2
; , ,
:
a = ^ ^ 2(mj + ) - m'j, fna,mh,mi
, , , .
, :
mb = 22 + 2 2 1
,
(. 34).D
: , , (. 34):
. 35
EF\\AC, EF = -,2
, () 1/2 (. 35).
-
83
, , : SABC = 4SEBF.
,
, (. 36).
,
= ( ) (. 37).b
(. 37) .
. .
, .
, .
, ( ).
. , :
y]ab(a + b + ){ + b~c) _
-
84
: , , ( . 38) , ; ;
:
1 1 1 : \
, , , .
., 2 Shtl = , S , ,
. , :
h = -\j4a2 - 2, ;2a j3
- ;h = -
, , , . , :
2 ^ ( - ) ( - ) { - )
, , (. 39). .
.
6. 39
-
85
.
. abc _
: R = -----; R = ---------, , , -45 2 sin
, , , S .
, , (. 40).
().
:
;
+ ,------ , , , .
:
S 1 ( - )( - )( - )
\
, ,
, .
, , (. 41).
, , (. 42). , .
. 41
ABCD , :1. : AB = CD,AD = BC.2. : ZA = /, ZB = ZD.3.
: = , = OD.4. 180 :
ZA + ZB = 180, ZB + Z C = 180,Z C + ZD = 180, ZD + ZA = 180.
5. - ' : AB = CD, AB || CD. pUc. 42
-
86
6 . . : AC2 + BD2 = 2(2 +2).
: S = ah, S = abs ina, S = - AC BD sin/.2
ABCD , ZBAD + ZBCD = , ZABC + ZADC = .
, (. 43). , .
: :
[]||[], []||[} = = CD = AD;
[A C ]x [B D ]; : AC2 +BD2 = 42; .
: = 4
a2sina, S = AC BD.2
, (. 44). , .
: : = CD, AD = BC, [AB]||[CD],
[AD]||[BC]; 90: ZBAD = ZABC = ZBCD = ZADC = /2; :
a = 2 arctg(a/b), p = 2 arctg(b/a), a + p = 180. :
= BD; .
: = 2 ( + ).
: S = ab d = ylaz +b2 S = d2 sin(a/2)cos(a/2)
-------------------------- . : R = yj(a2 +b2)/2 ( ) , . ,
D , . 44 .
-
87
, (. 45). , .
: : = = CD = DA; 90: ZBAD = ZABC = ZBCD = ZADC = /2.
: d = -\ ( )
; ( )
.: P = 4a = 8r = 2^2 R .: S = a2 = d lj2. : S = 2 = 42 = 2R2, R :
; = /2 .
: ; R = dj2 = (V2/2).
- .
, (. 46).
. ( ).
. : [ a d ]||[BC], :
(. 47), , , ( );
. :
(. 46): [iiFjIlljAD], EF = (a + b)/2.
: S = (a + b)/i/2, S = EF-h. ,
, , (. 48, ).
, / , / , (. 4 8 ,6). /
- 1, (. 48, ). . 47
-
88
, (. 48, ).
. 48
. 49
/ ' \/ / V j \/ V \
, , (. 49). . , . . , . (. 50): = /2 = //2 .
. 50
, , (. 51).
, (. 52 ).
, : = 2" 1 , ... , ,, 2,..., pt
= 2 +1 (s ). = 3, 4, 5, 6 , 8 , 10, 12, 15 16, 17, 20, 24, 32, 34, ...
= 7 ,9 ,1 1 ,1 3 , 14, 18,19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 3 3 ,... : 3, 5,17, 257,65537. ,
(. 53).
-
89
, (. 54).
, .
, . . .
, 180. . .
, .
. 0 , a R , 0 , - :
( 2 . ( 2 .; = 0 +.KCOS
-
90
, 2 , 1 2,S, S2 ,
, : ,= , S, :S , = 2.
. 56
, , , , (. 56).
, , .
, .
, ( . 56 ).
, , ( . 56 ).
, R
:
= 2Ksin.2
. 58
, (. 57). , , , .
: L R ( ) L = ;
; L = 2nR------ .
F 3601 ,
, .
. ,
. .
-
91
: ,
; , ,
, (. 58); , ,
, . ,
( . 58 '; . 59 ). ( . 60 ZAOB ).
, . , ,
(. 61, /. ). , .
: , , ; , , ;
;
, . , ,
(. 62). , ,
(. 62). , .
, .
, : /A B C = ^ / , ZADC = n - ^ Z A O C (. 63, ).
: = KD, = 2rsin(a/2) (. 63, ).
: AC -AD = AF = 2 (. 64).
: = 2 = d. : S = 2 = nd2 / 4 . ( ): / = . ( ): SK = a r 2/2.
( (5 ): / = /180. ( ): SeK = 2/360. , :
D 6. 63 . 64
-
92
, . , .
.
. .
, , , .
, , , , .
, .
, , .
, : , , ; , ,
.
, .
-
= r0 + at, 0 ( 0) = 0 (0; 0; z0) , ; = (1; ; ) . ( ): x = x0 +lt, y = y0 +mt, z = z6 +nt.
~ ~ Z~ Z0 I m
x ~ xa ~ z ~ zox , ~ xo 1 - 0 z, ~ zo
fAlx + Bly + C,z + Dl =0,s , , - [A2x + B2y + C2z + D2 = 0 * ' *
= = .2 2 2
- ---- 1 ( ^1 1 \ 1 = . , = ; ;L 1 [ 2 2 2 2 2 2 )
, = (,; ,; ,); 2 = (2; 2;2)
-
93
, .
, ,
, . .
, .
, , . , , . , ,
, .
.
.
()
. 65
Ax + By + Cz + D = , , , , D , , , . (. 65):(, JV) + D = , - M(x\y\z), N = (; ; ) (). N
' J a 2
COSp =
+-
+ 2
4 2 + 2
+ 2
/7cosy =
' ' + 2 + 2 , . D = . = 0 ( = , = 0 ) ( OY, OZ). = = ( = = 0, = = 0 ) OXY ( QXZ, OYZ).Z = 0 ; = 0 Oxz; = 0 Oyz.
-
94
+ + = 1, a = D/A\ b = - D / B ; c = D/C,
, OX, OY, OZ.
, M(xB;y 0;z0) N(A; ; )
A ( x - x 0) + B ( y - y 0) + C(.z-za); :(( ),0 = 0 .
, (: ;
( ( - , ) ,^ - ,), (, - ,) (),
- , - z " z, * 2 - * , 2 ~, Z2 -Z . * - * > - , Z3 ~ Z,
= 0 (
= 0 .
()
xco sa + 7 cosP + z c o s y -p = 0 ; :( , ) - = , , . : + By + Cz + D---- < =-\2 + 2 + 2
. ' = - 1 2 + 2 + 2
, D, D * 0, , D = 0.
, = r\ + a,f = 2 + a2t :
( - ,) , 2 = 0 ,1
( r - r 2)a,a2 = 0 .; , = ({,, m,, ,), - , - , z - z ,
/, , ,
f2 12 2
2 =(/2,m 2,n2), ; = (x 1,y , ,z 1),
= 0 .
, = ^ + at = r2 + a t :
( - ,) ( 2 - ; ) = 0 . x = xt +lt x = x2 +lt, z = z1+nt z = z2 +nt, - , - , Z - Z ,
* 2 - * l Z2 Z, /
: = , + mf = :
= 0 .
-
95
,
, .
, , . , , . , , , .
, .
, . , . , , . , , .
, ( , , , , ) 90. ,
, . ,
, , .
, .
d = *0 cosa + C0SP + zo cosY - p|>i \^xa + By0 +Cz0 +D\
-J a 2 + b 2 + c 2
5 = x0 cosa + y0 cosP + z0 cosy - p, P Ax0 +By+Cz0 +D
! 2 + 2 + 2 D, D * 0 , , D = 0.
-
96
( )
, 7 + D, = 0, 2 7 + D2 = 0, , = 2 (, ||2), D , * \ D 2. Alx + B1y + Clz + Dl = 0 A2x + B2y + C2z + D2 =0, 1 _ , _ , Dj, 2 2 D2 '
:
, , .
, .
, , .
, , .
.
,
Ax + By + Cz + Dt = 0 Ax + By + Cz + D2 = 0 :
, t e - A l
/2 + 2 + 2 ,
\ [ - (-||) = 0 ( - , ) = 0 : d = -------j:-------.
, 7 + Dl = 0, 2 7 + D2 = 0,
.2 AjX + Bty + Ctz + D, = 0 2 + 2 + C2z + D2 = 0,
-i = = ] ( - i ).
, 7 + D, = 0, 7 + D2 = 0, , = \2, D, = XD2. Alx + Bly + CIz + DI = 0 A2x + B2y + C2z + D2 =0, = bl = c l = d l
^2 ^2 ^2 ^2
( )
12 + {2 + ,2 = 0, (Np N2) = 0.
-
97
Ax + By + Cz + D, = 0 + By+ Cz + D2 = 0,
cos z 2 z i , ,
d =z2 - , 2 - ,
, /,
* 2 " * . - , /, ,
(V ',2 + ,2 +2)
? = ;+, = ^ + a ,t , , II2 11
, I, , ,
h 2 2 2 1 2 \ 2 2 Z 1
7 = rx + t = 2 +a1t, , 2 - (f2 -rI)ata2 =0.
,
h h 2
2 ~ 1 2~, Z 2 Z , , , ,/, , ,
= 0.
-
98
r = 7 t +a}t =r2 +a2t, , . (^ -^ ) , 2 #.
,
2 - 1 Z 2 " Z 1 m, , ^ 0 . d =
mod
: d =
7
- *
h
2
2 - \ 1 .
[ ,2]
z2 -z ,
,,
1 1i h+ +
m2 n2 i2
( )
cos(p/-^\ ar a2 /1/2+w,w2 +
= cos [, , 2 = rr jrr - = , )
, 0. , / + + 0.
, - 0,-70 +D = 0 . , 1 + + 0, * 0 + ByQ + Cz0 + D = 0.
, - = 0,-70 + 0. , / + + 0, 0 + 0 + Cz0 + D 0. :- - A B C \\ = = .
/
sinq>= cos1\Al + Bm + Cn\
_ _ - n r 0 +D r = r 0 - a .
:
x = x0 +ltr y = ya +mtv z = z0 +nf,, f , = -Ax0 + By0 +Cz0 +D
Al + Bm + Cn
-
99
, 0(0) - - n r + D -
=7Q+nt. : = 0 +At, = + 1. z = z0 + Ct.
, , , , (. 66). , .
, , , (. 66).
, , , .
: arccos(l/3) (70,53); /2 (90 ); Jt-arccos(l/3) (109,47); 2 -arctg () (116,56);
2 arctg(cp + l) (138,19), /5)/2 .
, , (. 67).
, , ,, , ,
, .
. ,
. .
. 67
.
-
100
: cosa = cosPcosy + sin(3sinycos cos = - cos cos + sin sin cos a , , , , , , , .
_ sina sinp sin ;-------= ------- = ------- ,
sin A sin sin , , ; , , .
, , ( ) , (. 68 ).
.
.
,
5 , : 1 = .
, , 2.
, 4 - . 68 ( ).
, , ,
, (. 69).
, , ( . 69 ABCDE, KLMNP).
, . ( . 69 ABLK, BCML, CDNM, DEPN, ).
.
.
-
101
( . 69 , BL, , DN, ).
, ( . 69 - KR).
, , ( . 69 EL).
, .
. , , , ( . 69 EBLP).
, .
: . . . .
. .
Sp = 2S,*,, + S6c,K, Sx ;S6oK ; S6oK=pl; ; I .
V = QH, V = Q,/, Q ; , Q, .
, , .
:
( ). .
:
.
. ,
, . , (
), . :
. . .
, (. 70). , , .
-
102
, , .
, , .
, , ( . 70 DBi ).
, , .
: .
, , , ; , :
, BD, = , = DB, =d,
d2 = 2 +b2 + 2. .
.
, .
: S6 = 2( + b)t , b ,
. 5 = 2{ab + + ).: V = abc, , , . ,
. : S6=P0x.h, 0 , h . : S = S6 -I- 2S0, S0 . V=S0xh. .
. : S6 = 42, . 5 = 62. V = 3.
, , , (. 71).
, . .
, (. 71 ).
-
103
, . . ,
. ,
( ).
, .
, .
: 5 . 71 :
, ;
; ,
, , . :
, ;
;
.
V = SAi, S h .
( ): Sb = .-
( ):
. , ( ). ,
.
. . ,
(, ) ( ).
. . ,
, . ,
( ).
-
104
. , , . ,
( ).
. . ,
, , , . ,
( ). . ,
, . ,
( ).
, , .
: ;
; ,
; ,
, /, ;
:
Sb = i p a = ^b2 sin , , ,
, b , .
, . .
, , .
:
V = ~(Qi + /Q, Q2 + Q2 J, h ; Q,, Q2 .
: Sc = ^(, + 2) /i6oK, ,, 2
; h6oK . .
.
-
105
,
, (. 72).
, () , ().
. S{riH = 2nRH. Su = 2nRH + 2kR2. V = R2H. . 72
, . (. 72).
, 2/ 2 + 1 j b 2 =1 (. 73).
, 2/2 - 2 / b 2 =1 (. 74) :
2 v2= ( > 0 , > 0 )
-
, 1 = 2 , > 0 ( )(. 75).
.
, , ( ) (. 76).
: , , ().
: , .
, .
, .
-
106
() .
.
.
.
271^ 1-cosy j, (
). S6oK = nRL.
SK0K = nRL + nR2. S. >= = nRL + nR2.
V = - kR2H.3
, ( ) . , , .
() , .
, . ( )
, ( ).
, . , (
). , (
). , ,
, (. 77).
. 76
-
107
S6oK = ( + )/. SIm;IH = nR2 + 2 + n(R + r)l.
V = nH(R2 +Rr + r2).3
( , , ) (. 78).
. 78: : , 6 , .
, , (. 79).
. :
. , .
. ,
. .
5 = 4nR2.
V = -n R \3
S = 2nRH ( ).
V = 3 ( - -
V = nR2H.
. 79
, . X X ' Y' , X Y = X'Y'.
, , . . , ,
, , .: ;
, ; - ; .
-
108
. 81
F'
, ', ' (. 80).
Z A.
.
, (. 81).
, .
F F', ', , . F F' (. 82).
. 82
'
. 83
'\
. 84
( ), , (. 83).
I , A e l . ' /, ' = ' X 1 (. 84).
I F F', ', /.
F F' / (. 84).
I , /, / (. 85).
. 85
. , ,
-
109
. 86
, (. 86 ).
, , .
, (. 87).
, , a W , ' , .
(; ) F '( + ; + b) F'. F, (; ) ( + ; + ), (; ), ' = (, ).
. 87
, , ', |''| = :||, , (. 88 ).
( ) = 1.
F F' , . - / 1 (. 89). F
__!____ F '' F', '' = .
. ,
, ', ', ' , ' ' '.
, , , .
, , , , .
.
. 89
-
110
* 1, , - .
D .
, (. ).
(, ). , ( ).
Ft F2, F2 F3, F, F3 (. 89).
.
( * 1) ( ), X X ', ' = (. 90).
.
. .
, 1, : . , 0, : . ( , - 1) .
-
, , .
()
, L , metre (, )
,
, , kilogram (, kg)
. , , : , ; ,
, , second (, s)
-
, I , ampere ()
, , 1 2
, 0 ,
,kelvin()
, , .
, / , candela (, cd)
, , , 1
, N
, mole (, mol)
, , . , (, , , )
-
112
(
)
(, rad)
, ( ), ( )
(, sr)
, , ( ) ,
(
)
, / . V ,
, hertz(, Hz)
, , ,
, F , newton (, N)
,
, , W , joule (, J)
,
, N , watt (, W)
, , ,
, , pascal (, )
, F , S
, O v
, lumen(, 1)
,
,
, lux (, 1)
, ,
, q, Q
, coulomb (, )
, ; ,
(), 17, .
, volt(, V)
,
-
113
, R
, ohm(, )
, , ,
,
, farad (> F)
,
,v /)
, weber(6 , Wb)
, -
, tesla (, )
, ,
, henry (, )
, , , ,
, siemens (, S)
,
( )
, becquerel (, Bq)
,
, gray (, Gy)
, ,
, sievert(, Sv)
, .
, katal (, kat)
,
-
114
,
() 1,85318
() 1,852
() 1,60934
() 185,2
914,4
304,8
25,4
(1/10 ) 2,54
(1/100 ) 254
() 2,58999 2
4046,86 2 = 0,404686
0,836127 2
929,030 2
1233,48 3
2,83168 3
0,764555 3
28,3169
16,3871 3
() 158,987 3
() 115,627 3
() 36,3687 3
() 35,2391 3
() 4,54609 3
() 4,40488 3
() 3,78541 3
() 1,1361 3
() 1,10122 3
() 0,946353 3
() 28,413 3
-
115
/
10' deca da
102 hecto h
103 kilo k
106 Mega
9 Giga G
12 Tera
15 Peta
10 Exa
21 Zetta 3 Z -
24 Yotta Y
-
10"' deci d
-2 centi
10~3 milli
micro ,
10'9 nano
12 pico
10"15 femto f
-'8 atto
-21 zepto 3 Z
-2" yocto
1 = 10241 = 210 = 1024 1 = 10242 = 2 = 1048 576 1 = 10243 = 230 = 1073741824 1 = 1024'' = 240 = 1099 511627 776 1 = 10245 = 250 = 1 125899906842624 1 = 10246 = 2 = 115292] 504606846976 1 = 10247 = 270 = 1 180591620717411303424 1 = 1024 = 20 = 1208925819614629 174706 176
-
116
, , .
( ) : , , , , , , , , , , , .
: , , . , ,
. ,
. .
, . .
, , .
q l,... qk. , ,
, . : ,
, , , , , , .
: .
.
: , .
: ( ); ( ); (- -).
, . . .
, , ().
, .
, . , : ; ( ).
, .
, .
-
117
, . *
, () .
, .
, . , . ,
.
( = 0, v = const): s = v t , , v (/), ( ().
a t 2 ( = const): s = v 0t + , s = ------- ,
2 2 0 (/), (/2), t ().
, .
( = 0, v = const): v = A s / A t , As (), At ().
( = const): v = v(1+at, v = +2as, v0 (/), J (), (/2). t ().
, As ds
v = J'm0 =
(), At (). ,
(): v = , I (), At At
().: -
s, (): = ~ >
As (), At (). , .
.
; () ( = 0), - - ( 2 ( = ): to = , = = 2nv, -
t - (), At (), (), v (-1).
( = const): Et2
) = 0+ ( ( - t 0),
-
118
-.. ( ).
, -(
= > - At
/; At (). ,
(), . (. 1).
, ^ ,
( ) ( ): F = aum.
, . -
V 2 >2 : = , = , 4
(/), (1/), - ().
-
119
. (. 2),
() ^ ). = + = * A t
(/), At (). = const
: - ^ - - (1/), - At At
1/2. -
1 : 1 = ,v
v "1. ,
, v = ,
. ,
. . 2
(, ): 7 = ^ ( w (r (2), . i- , . i- .
: / = j r 2dm = j p - r 2dV, dm = p d V dV , , d V .
, / = j r 2dV.
( ) , .
; : - mv, (), v (/).
( , , , ) . , , .
, .
L : L = r x p = r x , - , , (), v (/).
: Ll
, : L = I(bt I , .
, L (. . ).
-
120
: = 1 + ----- , I
L : L = |r| | | sin 0^, 9 , , , .
(L) ():
= = F, F (). dt
, , . , , , . F = , (), (/2).
( ) () -, , :
_ _ _ - d lM = r x F , = FI = F rs in a , M = , F (), -
(), I , 1 , (), a F .
.dl
dt , (1/), .
, (. 3).
FK = -2 ( ) - = 2 ( ), , , v .
: = 2[, ].
, (. 4).
, F .
, F .
, , . , ( ).
, :
Fml , - :
. 4
. 3
-
121
Fa6 + F ( ),
: Fu6 = , FK = 2 [ , ], ' .
, .
, , , . , . , , . .
, -
, : = -, , ; /
; ; 2.
= !]= = ^ - = / , L - dt dt
; .
, . . .
. 5: R , ,N .
, F, S '
F : = , F (), S (2).
S . 5 (): = m ( g a ) ,
(), g (/2), (/2).
( 2"! () : P = m \ g , m
(), (/2), v (/), ().
() .
: ,
, ;
, / ;
, . , . , .
-
122
: = , (