1. .2. .3. . .4. .5. .6. . .7. ,, .8. .9. .10. . .11. . .12.
.13. .14. . .15. .16. .17. .18. .19. .20. . .21. .22. .23. .24.
.25. . , .26. .27. .28. . .29. , , .30. , , .31. .32. . .33. .34. ,
.35. , .36. .37. , .38. , .39. .40. .41. .42. .43. .44. . .45. (
).46. ( ).47. . .48. .49. . .50. .51. . .52. -, . -. , . .53. .54.
.55. , .56. .57. . . .58. . . . .59. . . .60. , . , .61. , , .62.
.63. . . .64. , .65. .66. . 67. . .68. . .69. . .70. .71. . .72. .
.73. .
1. .1. n . .1: - ({xn} - ): 2: : 2 , . {n} I {n} .
3: : {xn} - , {n} . , {xn, n} {xn} - {n}
: .: {n}, {n}-. {n} {n} {n} . {n}-, {n} 3 {n, n} .4: 1) {xn} ,
{1/xn} - .2) {n} , n=0 .: {xn}
2. -
: . , , .
- ;k - . -1) - .2) - () , +,-.
1) . . .- . , - :
, - .
=0,1,2.......
=1,2,3...., ,
, - - , , , () - . .
2) - .
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3. . .
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: , .
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: - .
,
, , lim , .
- . : 1. . - - .
. . , , 4. .. .( lim, ) ( ):
; , , a=b:
a -(2) {Xnk}a, aR => 3 lim Xn=a k n .. , .1 , .: , - 2. ,
. : , .. ,, 3: , . : - . , , 9. . ( : () {xn} (), ) {xn}, xn : xn
=(1+1/n)n., . - , . : (1) xn+1 : -, , xn=2 ., {xn} , . . ,10. .: 1:
f(x) , , . 1: ; . : - - .: ( )( ). :1) . , ; , 2) - ; ; ., .: -
f(x) . ., ! f(x)- . , . , .11. . -. - 1( ): . - f(x) , - , . 1' , :
. . !1 , , . R, , - . : 2: . , . ! , . . (), . : , - . : : . . . ,
, . - , . , .1 12. . 1: f(x) .0 , (0) , - f XO(x0). 1()=>2: ( )
- f 0 , : , =0 0a-a/2=a/2>0 a ..: 0 .14. . . f(), g(), , 0 .(0),
(), (0)f() =()g(). - f() g() 0, >0 | ()|c (0) - : |f()| c| g()|,
(0)f()=O(g()), 02.- f , - g 0 0, 1, 2>0 (0)1| ()|c21| g()| |f()|
c2| g()|f , g.. f=(g), g=O(f), 0/3.- f - g 0, ()-=0. . f=(g), 04. -
f g 0, =1, f~ g, 05. f=(g), 0 =0, f g.5+ f=(gn), 0, n N, =0, f - n
g.5++ 0 0, 0 , 1) f=(g), 0 =02) f~ g, 0 =13) =k,k0 f=(g), 0. - - f
i g -f={xn}, g={yn}. - .1) ., . n=O(yn), n2) n=O(yn), n=O(n)
n3)n~yn, n4) n=o(yn), n, .15. . : , - 0 , , 0 , a 0 O(x0):x1,x2 X
O(x0)(|-| 0, - ). , :y'= -1; ()'' = ( - 1)-2; (3) = ( - 1)(-2)-3
,... : ()(n) = ( - 1)( - 2)... ( - n + 1)-n. . = m, m , (m)(m) =
m!, (xm)(n) =0 n>m. , n- m- n>m . n- 2- -. , 1- 2- - (uv)' =
u'v' ( ) n- (u v)(n) = u(n)v(n) n- 2- - uv. : (1). , - (1): (u+v)n,
u i v . , - u i v u(0) i v(0) ( - ). - . n=1 - (uv)' = u'v + uv',
2- -. , -(1) n, n+1. , n - (1) . - ' , - , : (2)( , 1=). : . -,
(2): . - (1) (n+1). - . 35. , .. 2 1- , .:d2yd(dy) dy=y'xdx
d2y=d(y'xdx)=y''xdxdx=y''x(dx)2=y''dx2., d2y=y''xdx2
y''x=d2y/dx2dy=y'xdx2 y'x=dy/dx 1- .. n- (n-1)- ; :dny=d(dn-1y).-:
dny=y(n)xdxn(1). n=1 n=2 - (1) . , - (n-1), , dn-1y=y(n-1)xdxn-1.,
dny, , - (1) . - (1) : (1') , n>1 - (1) (1') , , .36. .- y=f(x)
max(min) .0, *(0) f(x))f(x0) y(x0)=f(x)-f(x0))0. : - f(x) .0, ,
=0.. (0), x0 0 f ' (x0)>=0 (1) x>x0, 0 min i . t , [, ], -
(t) : (5), (6). (t) : (7). Q(x), - (t). , - (t) [, ] . - (7) , -
f(x), , - (t) [,] . , () = () = 0. (5) t = (6), : , (4) =0. = 0 -
(7). , - [, ] . [, ] . , : . (7), (9). , -(9), 2-, . , (10) - (10)
t = (8), : C(11) (6), :. . . - . . : - f(x) (n-1) . n . . Rn+1(x) -
f(x) (3) , Rn+1(x) : , . (6) n . = - f(x) , . =, : , (13) (12).
(13) , . - Rn+1(x) (-)n (n-1) . , (13) - - (-)n e (n-1) . , ,
-(14),(n-1) , - : (13), (15) : (13) =0, - =0, (14). (12) . , - :
42. . .( ). - f(x) . n+1 (n - ). , - , p - . , -: (1) (2). (1) - (
. ), Rn+1(x) . (2), . , (2), . . (, ) n, , : . (3) Rn+1(x) (4) , ,
Rn+1(x) - (2). - , . , > . t , [, ], - (t) : (5), (6). (t) :
(7). Q(x), - (t). , - (t) [, ] . - (7) , - f(x), , - (t) [,] . , ()
= () = 0. (5) t = (6), : , (4) =0. = 0 - (7). , - [, ] . [, ] . , :
. (7), (9). , -(9), 2-, . , (10) - (10) t = (8), : C(11) (6), :. .
. - . - . - (2). . , 0 0,0- ( 2m-1)(0)f(0) f(0)(-0)+.+ f( 2n-1)(0)
*(-0)( n-1)+ f( n)(0)*(-0)( n)+ (n-1)! n! +(-). (0)= f( n)(0) *(
n)+(( n) n1f( n)(0)0 (( n))=(f( n)(0)*( n)=(( n)* (f( n!n)(0)*( n)
n! ( 0 1)m=2m ( 2m)>0 f(
2m)(0)>((0.1f1(x)+2f2(x)dx=11(x)dx+2f2(x)dx.53. . .: 1
f(x)dx=f(u)du,u=u(x) f(x)dx=F(x)+Cf(u)du=F(u)+C. : u=ax+,0
f(ax+)d(ax+)=F(ax+)+C f(ax+)dx=1/aF(ax+) x
2x-1dx=d(2x-1)=uxdx=1/u2x-1(uxdx)=1/u 2x-1*d(2x-1)=1/u*(2x-1)(
2/3)+C=1/6(2x-1)( 2/3)+C.:F(x),u(t) ().f(x) F(x).(t) (t) tf((t))
(t) F((t)) t, f(x)( =(t))=f((t))(t)dt. : f(x)x (t)t y(t)xf((t)t
f((t))(t) F((t)) (d/dt)F((t))=F((t)-(t)=f((t)-(t)..
deff((t))-(d)d(t)=F((t))+C F(x)- F(x) deff(x)dx=F(x)+C,x=(t)
f(x)dx) =)=F((t))+C/ : - =(t) .f(x)dx=f((t))(t)dt.54.
.(Pn(x))/(Qm(x)) , .:Qm(x)=x( m)+1x( m-1)++( m-1)x+( m) (R
i=1,n)=(x-1)( k1)(x-s) ( ks)(x+p1x+q1) ( 1)(x+p( r)*+q( r)( 2)
k1++k2+2L1++2L2=m. :1) (-) ( k) qm(x) k 1- 2- :
A1/(x-)+A2/(x-)++Ak/(x-) ( k). 2) (+px+q)( l) L 3,4 :
(B1=C1)/(x+px+q)+(B2x+C2)/(x+px+q)++(Blx+Cl)/(x+px+q) ( l) i,j,j
i=1 k; j=1 L . - -= : (Adx)/(x-)=Aln|x-|+C; (Adx)/(x-)( k)=A(x-)(
k+1)/(-k+1)+C55. , .R(sinx,cosx)dx, R( , )- - . . :t=tg/2 x=2arctgt
sin=(ctg(x/2))/(1+tgx/2)=2t/(1+t) cosx=(1-t)/(1+t) dx=(2dt)/(1+t)
R((2t/(1+t)*((1-t)/(1+t)))*(2dt)/(1+t)/ :1)R(cosx)sinxdx=-R(t)dt
t=cosx dt=-sinxdx. 2)R(sinx)cosx t=sinx. 3)R(tgx)dx=R(t)dt/(|t|)
4)R(sinx,cosx)dx, sinx,cosx- t=tgx 5)sin( n)xcos( m)xdx> :) .
m=2k+1 t=sinx dt=cosxdx t( n)(1-t)( k)dt. )m I n- , . )m i n-,
t=tgx.6)cosm( x)cosnxdx .56. .1)R(x,x( 1/n),x( 1/m)x( 1/1)dx :x=t(
N),N=(n,m,s), dx=Nt( N-1)dt ;R(t( n),t( N/nt ( N/s)Nt(
N-1)dt.2)R(x1((ax+b)/(cx=d)) ( 1/n) ((ax+b)/(cx+d)) ( 1/1))dx
(ax+b)/(cx+d)=t( N) N=(n,m,S) NN3) . . .-x( m)(a+bx( n))( p)dx ,
a,b,m,n,p R. : . m,n,p Q 3 : )p Z;)(m+1)/n Z; )((m+1)/n)+ Z57. . .
. . . n . , , . . . , , , , f(x) ( ) [a,b]; f(x) ( ) [a,b]. - , .
f(x), [a,b] R [xi,xi + 1] SR =f(i)(xi + 1 xi),i - - . [xi,xi + 1] ,
f(x) , [a,b] . . , , .58. . - . . . mk=inf f(x), x[xk; xk+1],
MK=sup f(x), x[xk; xk+1]. mk, MK R, k=0,n-1()
S*(T)=mkxk(S*(T)=Mkxk.c .1) () S*(T) S(T) S*(T)2) (),(/) S*(T)
S*(T/) S*(T) S*(T/) S*(T/) S*(T)3) S*(T)= inf f() {k} S*(T)= sup
f(T) {k} 4) () . : *= sup S*(T)I*= inf S*(T) S*(T) * I* S*(T),
(T)5) : = I* ; = I*. . , - , , - .-)=01: f(x) [a,b]k(f)xk=0, k(f)-
- [xk; xk+1], k(f)=sup(f(x)-f(x/))= supf(x)-inf f(x), x, x/ [xk;
xk+1]: f(x) [a,b], ==. :1) =( -) - : >0 ()() ()< ()- <
S(T) < I + inf sup k.- S(T) I + 0- 20 () , / [a,b]| - / |<
()(| f(x) f(/ )|0 [a,b) , /, // [,b) ( |dx|
