数学分析学习指导书》上册（吴良森、毛羽辉、韩士安、吴畏

(, ),, :, ( 5% ),,, (CIP).. :,4.8 ISBN 7-4 - 4363 -......... - -.O7 CIP (4)63573-6454588 4 8-8 - 598-88899 http:www.hep.edu.c http:www.hep.com.c 78796 6 4.5 46 8.,

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. p q ( p, q, q ),.,,.,. =.,,. =. = +,,.,, 3.,. =. y =., > y :, > y. :,, +, y,, y + y, - y, - y. - y >,, > y,.

3,,.,,. 999 9? = = 3 3 = 999 9, = 3 333 3 999 9., 999 9. p q ( p, q, q )? p q (, ), p < q. : r, p = q + r, 9, r q -, ( : p, q, (i) p < q, =, r = p; ( ii) p = q, =, r = ; (iii) p > q,,, < 9, ( + ) q > p, q p, r = p - q.) p q = + r q, r q <. r = q + r, 9, r q -, r q = + r q, r q <, p = q + r, r = q + r, p q = + + r q. r - = q + r, 9, r q -, ( )

p q = + + + + r q,r q <, p q =.. r {,,,, q - }, ( ), p q.,, : f ( ) = f ( ) = + + + + + + +.. ( ) + = - +, >, < <, f ( ) < f ( ). + +, + + + + + + = + + +,,. + + + + +. ( + ) = C k k - k, ( 3) k = =, ( 3 ). = m, = m +, ( + ) m + m ( + ) m = C k m k m - k, k = = ( + ) ( + ) m = ( + ) C k m k m - k m k = m m = C k m k m - k + + k = k = C k m k + m - k 3

m m - = m + + C k m k m - k + + k = k = m m = m + + C k m k m - k + + k = k = C k m k + m - k + m + C k - m = m + + ( C k m + C k - m ) k m + - k + m + k = m = m + + C k m + k m + - k + m + k = m + = C k m + k m + - k, k = m k m - k + + m + C k m + C k - m = C k m +.,..35,. i 3,,,, i > -, ( i =,,, ), ( + ) ( + ) ( + ) + + + +. ( 4) = = = =, > -, (Beroulli) ( + ) + ( > - ) N +. ( 5) = ( 5). = m, ( + ) ( + )( + m ) + + + + m, i i > - ( i =,,, m ). = m +, + m + ( + ) ( + ) ( + m ) ( + m + ) ( + + + + m ) ( + m + ) >, = m, = + + + + m + m + + ( + + + m ) m + + + + + m +, m + i ( i =,,, m ). ( 5) = = = =, > -, ( + ) + ( > - ). 4 ( Cuchy ) :,,,,,,,, i = i i i i = i. ( 6) i =. 4

t, t i = 5 p,, mn + ( i + t i ) = i i = i i i = i = + t i i i = - 4 i i = i = i i i. : i = + t i, i = i, i. i = ( ) m - m = ( - ) ( m - + m - + m - 3 + + m - ). ( 7) ( ) p < ( + ) p + - p + p + (3 ) : < ( + ) p. ( 8) - + k p < p k = p + < k = ( ) ( - ) ( m - + m - + m - 3 + + m - ) = m + m - + m - + + m - - m - - m - - - m - - m = m - m. ( ) ( 7) = +, =, m = p +, k p. ( 9) ( + ) p + - p + = ( + ) p + ( + ) p - + + p, ( p + ) p < ( + ) p + - p + < ( p + ) ( + ) p, p < ( + ) p + - p + p + (3 ) =, p, ( 9 ) = m, p < p + p + < p + p. m - + m k p < mp k = p + < k = = m +, ( 8), ( ) ( m + ) p + p + < ( + ) p. k p, = ( m + ) p + - m p + p + + mp + p + ( ) 5

( m + ) p + p + < ( m + ) p + k p m + = k p. k = m k = = ( m + ) p + - m p + p + + mp + p + m p m - + k p k = m = k p. k = ( 9 ) = m +. (4 ) 6 cr + (R + ). :?, + - + c - c. + - + c = + c - c + + + c + c + + + c - c - c.. 8 p. : p, p.. p, p = u v, u, v,, v, p = u v., p, v., u v, u v ; u = pv, v u, v =,. p. 6

.,. ( -, ], ( -, ), [, + ), (, + ), ( -, + ) - +,., +, -,.. S, M S ( ) : M S ( ) S, > M ( < M ). S (), M S () ( ).. 3., ( ). : S = (, ), ( ), S. S S, S,, () ()., : S R, () () (i) S,, S ; (ii) <, S, (i) S,, S ; (ii) >, S, >, S. > -, S..( ) - ( ). <,. ( ). 4 : S, S, S ; S, S... 7

. k,. k + k, k =,,,. k S,. k + k S, =. k S,,. S S? S?,,? S ( ), S ( ). (), ( ). S ( ) S (). S = ( - ) - =,,, sup S =, if S = -. sup S =, = k,, ( - ) k -. k (i), ( - ) - ; (ii) >, k k >, - < - k, sup S =. if S = -. sup S, if S S. S sup SS, sup S ; S if SS, if S., () ()...? : S R, L, S, < L, S : S R, M >, S, > M, S, S = { - ( - ) =,, }. L, = k > L, - ( - ) = - k < L. 8

, M > M > ; SS ; M > M,. M.,. 3 S? S, S, S,. S : (i) S, > ; (ii) <, S,. S = + ( - ) N +. = k, k + k =. = k +, k + + - ( k + ) = k + k, k = + k + + k + >, k,. + = 5 S, sup S = 5, if S =., 5S, sup S = (i), (ii) = + ( - ) ; k + k + + + k +, - = ( - ) ( - k + - = >, k N + k > log k + + k + 5. if S =, k + k + k - + + k +, -, - <, k + k + + < +, k + + - + + ) 9

if S =. S = + si N +, sup S, if S. 3 = 6 k +, 6 k +,, 6 k + 5( k =,, ),. = 6 k + ( k =,,3, ) S S = + (6 k + ) 3 kn + ; = 6 k + 5( k =,, 3, ) S = - (6 k + 5 ) 3 kn+. S, S, sup S = +, if S = -. 3 S = { y y = +, }, if S, sup S. S, sup S = +. S,, y = +, S. sup S = +. M >, (M >, > M - ), + > M, S., sup S = +. if S = : (i) ys, y = + ( ) ; (ii) >,,, - < <, + < +. if S =. 4, A R, : sup( + A ) = + sup A, if( + A ) = + if A, + A = { + A}. sup( + A ) = + sup A. sup A, : (i) A, sup A ; (ii) >, A, > sup A -. : (i) A, + + sup A ; (ii) >, A, + > + sup A -. sup( + A) = + sup A.

if( + A) = + if A. 5 A, B, AB = { y A, yb}, : sup AB = sup Asup B. sup ABsup A sup B., A, sup A, yb, ysup B,, y, ysup A sup B, sup Asup B AB, y sup A sup Bsup AB. sup ABsup Asup B., > ( < ), A, > sup B -, y AB, y > ( sup A - ) ( sup B - ). sup AB y > ( sup A - ) ( sup B - ), > sup A -, y B, = sup Asup B - ( sup A + sup B)+ > sup Asup B - ( sup A + sup B + ). A, B, sup A, sup B, sup A + sup B + >, = (sup A + sup B + ). sup ABsup A sup B. (9 ) sup AB = sup Asup B. 8 >,,, : = sup{ r r, r < }, >, if{ r r, r < }, <.. >, : (i) r <, r, r ; (ii) <, r, r <, < r <. r,, (i). ( ii), < <, log <,, r, log < r <, < r <. < <.

3 : y,, y, y, y = f( ). : ( ) D M,,.( ) D, ym,,.( 3) () f, y.,.,,. R D( ) = [, ] R ( ) =,,, q, = p q ( p, qn +, p, q ),, =, (, ),. 3 : y = f( u ), ud u = g( ), E, ( fg) ( ).. E = { g( )D}E, y = f( ), D, f D f ( ), = f - ( y), yf ( D). f -. f - ( f ( D) ) = D,,. 4 ( >, ),,

= sup { r < r r }, >, if { r < r r }, < <.,. f ( ) =,,,, g( ) =, >, fg gf? y = f( u ), ud, u = g( ), E, E = { g( )D}E, E, f g E E. () f( u) = y = f ( g( ) ), E., u,, u, D = R, g( ) =, >, E = { > }, = { g( ) D}E = E, f g fg, ( ) g( u ) = u, D = { u u > }, f ( ) = g f gf. rcsi ( si,,,, E = R, E = { f( )D}E =, ) =, R?, f - ( f( ) ) =, D( f - f )? R, rcsi ( si ) =. rcsi y, - rcsi y, rcsi( si ) -,. k- < k+, - - k<, si = si ( - k), rcsi( si ) = rcsi ( si( - k) ) = - k, kz. 3

k+ 3 < k+, - < ( k + )-, si( ( k + )- ) = si, rcsi( si ) = rcsi( si[ ( k + )- ] ) = ( k + )-, kz. rcsi( si ) = - k, k- < k+, ( k + )-, k+ kz. 3 < k+, f - ( f ( ) ) =, D. f -, y f ( D), D f ( ) = y ; y[ -, ], R, si = y. -,,. rcsi( si ) =, -,, 8 kg, 4 kg, kg, kg s, 6 s, 3 s, 5 s.,,,,.,. 4 kg, G( t) t.6 s, t < 6, G( t) = 4 - t; t = 6 s,, 6 kg.6t < 9, G( t) = 6 - ( t - 6 ) ; t = 9 s,, kg.9t 5, t > 5,, 4 G( t) = G( t) = - ( t - 9 ). G( t) = 5. 4 - t, t < 6, 6 - ( t - 6 ), 6t < 9, - ( t - 9 ), 9t 5, 5, 5 < t.

-. - f ( ) = +, f ( ) = f ( f (f ( ) ) ). f f, f f, f ( ) = f ( ) = f( f ( ) ) = f ( f( f ( ) ) ) = + + + + + = + = + ; + 3. +,. +, f + ( ) = f( f ( ) ) = 3 + + + =. + ( + ) 5

y = ( ( ) ).. ( ) =,,, >, ( ) = -,,, >., { ( ) } = { - }, - 3, ( ( ) ) =. < > 4 (( ) ) = =,. 3, ( ) >, (( ) ) =,, 3,, < > 3. rcsi + rccos =,. <, = rcsi,= rccos, si =, cos =, cos = -, si = -, si(+ ) = si cos + cos si = + - - =. <, <, < + <, - <. + =, rcsi + rccos =, <. 5 D( ) R ( ) D ( R ( ) ) R ( D( ) ). D( R ( ) ). R ( ) D, D R. 6

(, ), = p, p, q, R p q q = q, D R p q = D q =., (, ), R( ) =, D( ) =, D( R ( ) ), [, ]. R ( D( ) ). D( ){, }, R ( ), R( ) = R ( ) =, R ( D( ) ), R. (5 ) y = [ ] : ( ) >, - < ( ) <, ; < -. ( ) >, - <, - <. ( ) <, - <, <, < -. 4. (), : f ( ), D M >, D, f ( ) > M., ( ). : f ( ), D,,, sup f( )if f( ) : D D = sup D f ( ) (i) D, f ( ) ; (ii) <, D, f ( ) > ( >, D, f( ) > - ).. 7

3( ),,, 3. 4, R,,. D ()? D f ( ): M, D, f ( ) > M ; D f ( ): L, D, f ( ) < L ; D f ( ): M >, D, f ( ) > M., 7 : A, B, sup( A + B) = sup A + sup B. ( 4 ), f, g D, sup{ f ( ) + g( ) }sup D D f( ) + supg( ), ( 4 ) D. ( 4 ) (4 )?? (4 )( 4 ), { f( ) + g( ) D} { f ( ) D} + { g( ) D}, ( 4 3) (4 3 )., f ( ) =, g( ) = -, D = [, ], { f( ) + g( ) D} = {}, { f( ) [, ] } + { g( ) [, ] } = [ -, ], { f( ) + g( ) D}{ f ( ) D} + { g( ) D}. { f ( ) + g( ) D} f g, ; { f ( ) D} + { g ( ) D} D. (4 3 ) (4 ),. 3 ( -, + )?. f( ) = si, =, [ k, ( k + )], kz. ( -, + ). 8 4 ()?.,,.

,, ( ). 5 I f. I I, f I?. ( 3). 6 f I? f I f I, ; f I, I, <, f( )f( ) ; f I 3, 4 I, 3 < 4, f( 3 )f( 4 ). y = l (, + ).,, y = l (, + ) : y = l (, + ) : M, > e M, l > M. L, ( < < e L ), l < L. y = l. =. f ( ) = cos, U ( ) =. M >, =, > M, U ( ), f ( ) = > M, f ( ) =. 3 D,. D f, f?, D = [, ] f ( ) =,, -,. f D f ( D), f. f,.,, 9

[, ], <,,, f ( ) =, f ( ) = -, f ( ) > f ( ), f., <,,, f( ) = -, f ( ) =, f ( ) > f ( ), f. 4, f I ( ),, f I f I. 4 : f( ), D,,, 3 D, < < 3, [ f ( ) - f ( ) ] [ f ( ) - f ( 3 ) ] >. ( 4 4) [] f ( ),,, 3 D, < < 3, (4 4 ). f ( ) - f ( ) <, f ( ) - f ( 3 ) <, [].,, 3 D, < < 3, f ( ) ( 4 4),f,, D, <, f ( ) f ( ), 3, 4 D, 3 < 4, f ( 3 ) f ( 4 ). :,, 3, 4,,, 3, < < 3, f ( ) f ( ), f ( )f ( 3 ) ( f ( )f ( ), f( )f( 3 ) ), [ f ( ) - f( ) ] [ f( ) - f( 3 ) ] ( 4 4 ). f. 3, ( 4 4 ),, 3 D, < <, f ( ) > f ( ) > f ( 3 )f ( ) < f ( ) < f ( 3 ), f. 5 : y = f( ) ( - < < + ) = = ( > ), f ( ). y = f( ) =, R, f ( + ( - ) ) = f ( + + - ) f ( - ) = f ( + ). ( 4 5) f ( - ) = f ( + ), ( 4 6) = f ( - - + ) ( 4 6) = f ( - )

= f ( + - ) = f ( - + ) ( 4.5) = f ( ), R. f ( ) ( - ). ( ) : f ( ) = + si R., R, >, f ( ) - f ( ) = - + si - si si <, ( ). = - + cos + - - si - > - - ( - ) =, si - [, + ) f [, ], [, + ) : m ( ) = m ( )M ( ), if f ( y), M ( ) = sup f ( y). y y ( ) f ( ) = cos, [, + ) ; ( ) f( ) =, [ -, + ). ( ) m( ) = ( ) m( ) = M ( ) = cos,, -, < < + ; M ( ), < +., -,, < < + ;, -,, < < +., R, : ( ) ( ) m{, } = ( + + - ) ; ( ) mi{, } = ( + - - ).

<. f g D. M ( ) = m{ f ( ), g( )}, m( ) = mi{ f ( ), g( )}, D. M ( )m( )?, M ( ) = ( f( ) + g( ) + f ( ) - g( ) ). f ( ), g( ), f ( )g( ). f ( ) - g ( ) y = u = u u = f ( ) - g( ),, M ( ). m( ). 3 f ( ) = - +, : 4f f( - ), f ( + ), f ( ) +, f + -, - +, +, - +, 5y = [ ] :, f ( ), f ( ), f( f ( ) ). + -, - +, = + +, f ( ). f ( ) = + + ( ), 5, 3. y ( 35 ) ; y = + 5, = 3, 3,, 5 () y, y.( y = [ + 5 ], > ) 6y = f( ), : ( ) y = - f ( ) ; ( ) y = f ( - ) ; (3 ) y = - f ( - ) ; ( 4) y = f ( ) ; (5 ) y = sg f ( ) ; (6 ) y = [ f ( ) + f( ) ] ; (7 ) y = [ f ( ) - f( ) ]. 7f g, : ( ) ( ) = m{ f ( ), g( ) }; ( ) ( ) = mi{ f( ), g( ) }. 6 (6 ), (7 ). 8 f, g h, f ( ) g( ) h( ), R. : f ( f( ) )g( g( ) ) h( h( ) ). R, f ( )g( ), f,

f( f ( ) )f( g( ) ) ; f ( y ) g ( y ), y = g ( ), f ( g ( ) ) g ( g ( ) ) ; f ( f ( ) )g( g( ) )., g( ) g( g( ) )h ( h( ) ). h ( ) (, h ( ) ). 9 f g (, ), 7 ( ) ( )(, ). ( ) = mi{ f ( ), g( )}., (, ), <, f( )f( ), g( )g( ), mi{ f ( ), g( )}f ( )f ( ), mi{ f ( ), g( )}g( )g( ). mi{ f ( ), g( )}mi{ f ( ), g( ) }, mi{ f ( ), g( )}. m{ f( ), g( ) }. f [ -, ] ( ). : f [, ], f [ -, ] ()., [ -, ], <, -, - [, ], - > -, f( - ) > f( - ). f, g D. : ( ) if { f ( ) + g( ) }if f( ) + sup D D D ( ) sup D f( ) + if D g( ) ; g( )sup{ f( ) + g( )}. D ( ) D, g( ) sup D 4, 7 g( ), f ( ) + g ( )f ( ) + sup D g( ). if { f ( ) + g( ) }if { f( ) + sup D D D 4. ( ) D, f ( ) + if D sup D = if f( ) + sup D D g( )f ( ) + g( ), f ( ) + if g( ) sup D g( )} g( ), { f ( ) + g( ) }, D 3

sup D f( ) + if g( )sup { f ( ) + g ( )}. D D ( ) sup g ( ) = - if { - g( ) }, 4, D D ( ). if D if D if { f ( ) + g ( )} + if { - g( ) }if D D D if { f ( ) + g( ) }if D D 3 f, g D, : ( ) if D f( )if D ( ) sup{ f ( ) g( ) }sup D D g( )if { f ( ) g( ) }; D f( ) + sup D f ( )sup D g( ). f( ), g( ). ( ) f, g D, if f ( ), if g( ). D D f ( ), if g ( ),, if D g( ) >. f, g, f ( )if g( ) f ( ) g( ), D if { f( )if g ( )}if { f( ) g( )}, D D D if f ( )if g( )if { f ( ) g( )}, D D D if { f ( ) } = if f ( ) ( > ). D D f, g, ( )., D f ( ) >, f ( ) = - ( - ), g( ) = -, D = [, ], f, g, ( ). ( ). if f ( ) =, if g( ) = - D D, if f ( ) g( ) = if D D 4 ( - ) = - 6, 4(, + ) f R, (i), ( ii), 4 ( ) f ( ) = si + ; ( ) f ( ) = - -, <, 3, >.

( ) f (, + ), f R, f ( -, ], R F( ), F ( )(, + ) f ( ). (i) f,, F ( ) = si +, >,, =, si - <., R, F ( - ) = - F ( ), F ( ). (ii) F ( ) =. ( ) ( i). (ii). F ( ) = F ( ) = si +,, - si, < 3, >, - -,, - + -, - <, 3, < - 3, >, - -,, - 3, < - 5 f R h,. : f [, + h], f R. f [, + h ], M >, [, + h], f ( ) M. R, mz, = mh +, [, + h ]. f h, R, f R. 6 f I. f( ) = f( mh + ) = f( ) M, 5

:, : M = sup I f ( ), m = if I f( ). sup f ( ) - f ( ) = M - m., I ( ), I, f( ) - f( ) M - m ; ( ) >,, I, f ( ) - f ( ) > M - m -. ( )., I, f( )M, f ( )m, f( ) - f( )M - m, f( ) - f( )M - m, f( ) - f( ) M - m. ( ).M = m, f ( ) I,. M > m, < M - m. M = sup I f ( ), >,, I, m = if I f( ), M - < f ( ), f( ) < m +, M - m - < f( ) - f( ) f ( ) - f ( ). sup f ( ) - f ( ) = M - m., I :, ( A) : m,. ( + - ) < < ( - - ). m m - < = < m. S, S, S 6

3f ( ) = si, R,. 4 f ( ) R, g ( ) R, f ( g( ) ), g( f( ) )? 5: rct + rccot = sg ( ). 6 : ( ) y = + + c; ( ) y = + + -. 7 A, B R, : ( ) A, B < ; ( ) >, A, yb, y - <. : sup A = if B.. ( ) : k -. r = + ( ) >, : < + (B) k - = + k! k = r = - r < + k! < 3. E = { r r < 7, r }, sup E, if E. 3 A, B, : if AB = if A if B. k = AB = { y A, yb}. 4 f ( )(, + ), f ( ) R, f( ) : ( ) f ( ) = e ; ( ) f( ) = l. 5 y = f( ), D. 6: y <, rct + rct y = rct + y - y. 7 A, B, S = AB, : ( ) sup S = m{sup A, sup B}; ( ) if S = mi{if A, if B}., 7

( ) - N : =, : = >, N, > N, - <. ( ) : = U( ;), U( ; ). > N. : R,, : ( ) >, NN +, N, -. ( ) U( ; ), U(, ).,. { },..,,., - N.,,. >,, ; NN + ; > N - <,.,, N., 8

(, ). 3 =, { }.: { } { - }.. - N - -, - N? =, : >, - G( ) <, G( ) < N, > N G( ) <, - <. : 9 ( ) G( ) () ; ( ) G( ) < N ; (3 ), > N, N = m{ N, N }. 4, 3, 3-3 - 3 < 9 < N = 3-3 = 3, N = 3, G( ) = 9 N = 9, N = m 3, 9. 4 q = ( q < ), G( ) = h, h = q, q - < h < N =. 5 h = ( > ), G( ) = -, - - - < N =.. - N? { } {( - ) }. : >, NN +, N, -. { } R,. - ( ) =., = 4, NN +, = m +, N, 9

- - + - 4, { }. ( ) = ( - ). =, =,, - = >. = -, =,, - ( - ) = >., = mi{ +, - }, N +, -. {( - ) }. = k = k( k + ) ( k + ),., = k = = = = k = k( k + ) ( k + ) k( k + ) - ( k + ) ( k + ) - 3 + 3-34 + + ( + ) - ( + ) ( + ) - ( + ) ( + ) - N, = 4. 5 + - = 5 3-3. 5 + - 3 - N = m 3 3 +, 4 = 3 + 4 3 (3 - ) - 5 3 4 3 ( > 4 ) = 3,, > N, 5 + - 3 - - 5 3 <.

. > 4, G( ) = 3. 3 c = (>, c > ). k = [ ] +, k = c c ( k c) k c = >,. k = [ ] +, k = c c k c = + h( h > ), N = ( k c) k = k = ( k c) k ( + h) ( - )! = ( - ) h + h, > N k 4 < c, m, c m + <. c < ( - ) h k k. k h k k <,, k. =. c c! = ( c > ). c!! <,.c >, c! = ccc m c m + c - c, c >, k = [ c] +, k - c < k, > k, 3

M =, > N, c! = ccc ( k - ) c k c ck - ( k - )! c = M, c k ( k - )!., >, N > > N = m{ N, k}, c! = ( c > ). 5 si. A R, si si - A. <, M c! M <. A, >, N >, > N, A, ( A < ). = N+ 5 7, N+ 4 4, > N,, N >, si - A > si 5 4 - A =. (7 ) 4: =, k, + k =. = : >, N, > N, - <. : >, N, > N, 7: + k - <. =, =.. =, >, N >, > N, - <. - -, =. = = =. =, =, >, N >, > N, - = - <, =. 3

= = ( - ), =, ( - ). =, =.,, ( ) { },. ( ) { }, M, N +, M. (3 ) = > ( < ), N N +, > N, > > < <. (4 ) { }{ }, N, > N,. (5 ) { }{ }, N, > N, { c } { c }, c =. c, m m + m - m - + + + ( ) =, k = m, k k + k - k - m + + +, k > m, km, m, k. ( ) =, = ( > ). (3 ) =, + + + ) = ; ) > ( =,, ), m =. log (4 ) = ( >,, k ). k 33

k (5 ) = ( c > ). c c (6 )! = ( c > ). (7 )! =. (8 ) =, >, >, =. 3.{ k }{ }, k ( ) k < k + ; ( ) k k,. { } { }{ k }. { } { }, { },.. { }{ k } k, k? { k } k k. { k } { }, k = k. k k < k + ; k k.., > { }{ k }, k -?, >, N >, > N, -. N =, >, -, N =, >, -, N = k -, k > k -, k -, { }{ k } k -. { k }, k. 34 -.

+ S = + + + + + ( > )., S = + + +, 3 + - S = + + + - = - - - +. + =, + S =, - = - - - + = ( - ). + + + + + + + + + + +, =, + + = + + +,. + = ( ( 8) ), + + + + + + =. 3 { }. : N k( < k < ), N 35

=., N + < k <, k < + < k, < + N < k + N - < < k N. =, + N =, 4 : >,, k, >, =. log =. k < log log k =, log =. = log, >, k., log log k log =., >, N >, > N, < log <, < <. =, >, N >, > N, < <. < <, log = - log, >, 36 log log = - k =. k, >,, k, c >,

, log k, 5 ( Stolz) : k c, c!,! log k c! ( ). { }, { y } N +, < +, =, y + - y =, + - : = y =. y + - y + - -,,, y., >, N ( > N ) : = y + - y -. + - >, N <. y = y - + ( - + ) ( - - ) = y - + ( - + ) ( - - - ) + ( - + ) ( - - ) = = y N + ( N + ) ( N + - N ) + + ( - + ) ( - - ) = y N + N ( N + - N ) + + - ( - - ) + ( - N ), ( y - y N - N + > ),, N + ( N + - N ) + + - ( - - ) y N - N + - N 37

y N - N +. =, N >, > N, N = m{ N yn - N, N }, > N <. y - < + =. =, =, y = + + +, y + - y = + -. 6 : + + + + =, p + p + + p = p + p + ( p ). y = p + p + + p, = p +, + >, =. =, y + - y = + -, ( + ) p ( + ) p + - p + = ( + ) p p + + C p + p + + - p +. y + - y ( + ) p = + - ( p + ) p + + = p +, p + p + + p = p + p +. (33 ) 5 { }{ },. { }. { } ( ). { }, { }., { }. = c, = ( c - ), { },, { }., { }, : =, =. 38

9,,, m m, : + + + m = m{,,, m }. = m{,,, m }, = + + + m m = m,, m =, + + + m =. 3,. e +, + = e. ( 3 ) e, l = log e. +, + +, + < e < + +. ( 3 ) + +, 9. 3 ( Cuchy ) { } : >, N,, m > N ( > N, p) - m < ( + p - < ). ( 3 3),,., ( )( 3 3).,,. 39

? : { } >, N >,, m > N, - m. :, N, N, m, m., = + + +. : = - =. 4, N,, > N, + + + + + =.. ( ) : = >, N, > N, - <. ( ) : (3 ) : (4 ) : = >, U ( ; ){ }. = { k } { }, k k =. { } { k } { }, { k }. { } >, N,, m > N, - m <. (5 ) : { } { }. (6 ) : c, ( ) : = { },. =, c =. >, N, > N, -. ( ) :

>, U( ; ) { }. (3 ) : { k } { }, { k }{ }. { k } { }, k k =, { }. { k } { }, k =, (4 ) : k { } >, N,, m > N, - m. (5 ) { } { }. + +. + +. + +. (3 ), + = + < + - = -, + + -,. < + + + = + - + +, + + < + = e, - - + + + - = e.., = e. > - 6, + = + 6 ( =,, ). { }, + - = + 6 - = 6 + - + 6 + 4

= ( + ) ( 3 - ) + 6 +. ( 3 4) < 3, = + 6 < 3,, < 3, + = + 6 < 3, { }.(3 4 ) +. >, { }, > 3, > 3, =,3,.(3 4 ),, + <,, { },. = 3, 3,. =, = + 6, = 3, 3 : ( ) + < l + = 3. < ; ( 3 5) ( ) { c } = + + + - l. + < e < + + (3 5 ); (3 5 ) ( ). ( ) ( 3 ), + < e < + +,, l + < < ( + )l +, + < l + <. ( ) { c }: ( 3 5), c + - c = = + + l - l( + ) + - l + { c }: (3 5 ), kn +, l( k + ) - l k < k, k =,, 3,,, <. 4

l ( + ) < + + +, < l( + ) - l < + + + - l. { c },, c ( Euler)., c = c = 577 5 664 9. + + 3 + + = l + c +, ( 3 6), ( 3 6 ) + + + l. 4 =, =, + = =, + = + ( =,, ),. = +, +, = 68, = - 68,, = 68., { }. + - = + - + - = - ( - - ) ( + ) ( + - ), - <, 3 - >, 4-3 <,, k - k - <, k + - k >,. { }. + = ( = 68), <, + = + = ; >, + = + < + > + = ; = > 68, k - >, k <, { }.{ k }{ k - }. + - = = + + - ( 3 7) - + + 43

= - - + = ( + + ) ( - ) +, <, + - > ; >, + - <. k <, k - >, { k } ( ), { k - } ( )., + =. k k =, k - k =. + = k -, = k, = +, = +, = = = 68. { k - }{ k }, = =.68. = 68. 5 ( Kepler) = qsi +, q, q < q <., :{ },. (3 8 ) + = qsi +, =,,,. ( 3 8) - = q si - si = q si - q - ; cos + 3 - q - q - ;, + - q - - q -. pn +, + p - + p - + p - + + + - 44

q ( q + p - + + q ) - = q ( - q p ) - q - q - q -. - =, { }. - q < q <, + p - q - q -, q - - q >, N >, > N, pn + =. + p - q - - q { }, = l. si - si l - l, si = si l., ( 3 8), l. l = si l +,, <,, { }. (39 ) 8: { }( ),? = sup{ } (if{ } ). { }., = sup{ },, >,, - <, { }, >, - < < +. : >,, >, - <, 45

, : = - (). { }, : ( ), ; m ; = sup{ }. ( + ( - ) ). =,,, < = sup{, +, }, = if{, +, }. ( ) { }, { },, m (3 ) { }{ },. (4 ) { } =. ( ) :,. sup{, +, }if{, +, }, ( ) { }, M >,, - M M, - M M, { }, { }. + = sup{ +, +, }sup{, +, } = { }. { }., m, m + m + m. (3 ),,. =, =., (4 ) [ ] =, >, N, > N - < < +. > N - +,, - + - = =. [] = =, =, =, >, N, > N - < < +, - < < +, =. : 46 (4 )

3 ( ) + 3 5 ( = 3) ; ( ) ( = ) ; e (3 ) ( + - + + ) ( = ). : ( ) q lg = ( q < ) ; ( ) = ( ) ; (3 ) =.! ( ) q = lg ( ) =. + h, h >. (3 ) >, N, > N, 3. =, : <, M =!, M! <. ( ) + + + = ( = ) ; ( ) > ( =,, ), =. ( ) =, >, N, > N, - <. + + + - - + + N - + N + - + + - - + + N - + - N. N,, > N - + + N - > N = m{ N, N } + + + - + =. <,, + + + =. = ( - ), { }, + + + =. 5. ( ) =, 47

+ + +,, ( ) + + + =,, >, =. =, ( 5 8 ( ) ) =. i > ( i =,,, ), + + + + + +, + + + =, + + + = + + + =. = =. ( ). 4: + ( ) + + = ; ( ) = ( > ) ; ( : =, i = ( i ) ) (3 ) = ; (4 ) = ;! (5 ) = e; ( : 3 ( ) = +! ) + (6 ) + 3 3 + + = ; + (7 ) = ( > ), (8 ) ( = ; - - ) = d, = d. 5: { }, { }, 48

. ( - ) =, { }, { }. 6 { } : M,, A = - + 3 - + + - - M. : { }{ A }. { A },, { A }. (), >, N, > N, p, A + p - A <, + p - + p - + + + - <. + p - = + p - + p - + + p - - + p - + + + - + p - + p - + + p - - + p - + + + - <, { } (), { }.. 7 >,>, = { },., +, + = +, =,,. : + = + =, = { }. +, { } :, + = + = + + =., = l. + = +, l = l +, l 49

l =, =. 8 > >, = - + -, = - -, =, 3,. - + - : { }{ }.. 9 { }, { } : ( ) = ( - ) ; ( ) = si ; (3 ) = + + +. ( ) =, N, = N +, m = N +, - m. ( ) =, N, = N +, m = N +, : ( ) S =, - m. =, S = m{, }, T = mi{, }, =,,. = m{, } ; ( ) T = mi{, }.. : = + + + ( A) 5 + - 4 = 5-3.,. 3 ( + + + ) = S, 4 5 ( + + + ) =. + = 3 + + 3 ( >, > ).

=. 5., : ( ) + + + = + + + = + + + =. ( ) =, if{ }sup{ }. 6 = A, : ( + C + + C k k + + C ) = A. 7 { } m + m +, = if., (B) ( ) = + + 4 + ; ( ) = i =. 3 + 3 + + i 3., y =,, y = + y - 3 =, 4 { }, 5 =, ( =, 3, ). y. =, <. { c }, c =, =, { }. 6 + - + + =. =, = + + +, i >, ( i =,,, ), 7: - + + + =. + + + + +, ( - - ) =. 5 =.

( ) f( ) = A >, M >, > M, f( ) - A <. + ( ) f( ) = A >, M >, < - M, f( ) - A <. - (3 ) f ( ) = A >, M >, > M, f ( ) - A <. () f( ) = A >, >, < - <, f( ) - A <. () + f( ) = A >, >, < < +, f( ) - A <. (3) - f( ) = A >, >, - < <, f( ) - A <. (4 ) f ( ) = A f ( ) = f ( ) = A. + -,.,,., 9,.,, :,,.. ( Weierstrss ) : = f ( ).,,, f ( ) L, L f ( ). 5

-,, f ( ).? -, -. >, ( ) >, < - <, f ( ) - A <, ;,... f ( ), - f ( ) = A ( ) f ( ) : ( - ),. ( ) f ( ) - A : f ( ) - A = ( ) -. (3 ) >, U ( ; ), ( ) M, ( ) U ( ; ). (4 ) >, = mi, M, < - <, f ( ) - A <. 3 f ( )A f ( ) A? + f( )A : >, >,, < - <, : f ( ) - A. >,, U ( ; ), f ( ) U ( A ; ). f ( )A : + >, M >, > M, f ( ) - A. : >, M, U( + ) = { > M}, f ( ) U( A ; ). 53

- : + - ( - 3 + ) = - 3. ( ) ( - ) : f ( ) = + - ( - 3 + ) = ( + ) ( - ) ( - ) ( - ) = + ( - ). ( ) f ( ) - ( - 3 ) ( ) -, ( ) : ( ) = 3 - -. f ( ) + 3 = + ( - ) + 3 = 3-5 + ( - ) = 3 - - -, (3 ) = < - <, ( ) : =, < - <, 3-3 - + < 5, - = - ( - ) > 3 4, 3 - - < 5 3 4 = 3. (4 ) f ( ) + 3 = 3 - - - 3 - <, - < 3. = mi < - <, f ( ) - ( - 3 ) <. 54 - M : -, 3, + - = -.

+ - - - = = + + ( + - ). ( + - ) -,, <, ( + - ) < - M,, M. ( + - ) = <, ( - ) 8 >, - < - 8. >, M = 8, < - M, 8 3 si. + + - - - <. si. +, si, > > M ( M > ), si.. f ( ) A, : >, M >, > + M, si. =, M >, N +, = + > M, si. + 4 + si = si + f ( ) = - [ ]. f( ) - f( ). = >,,. 55

+ f( ). <, < - <, f ( ) = - [ ] = -, + f( ) = + ( - ) =., <, - < - <, 5 f ( ) = - [ ] = - ( - ), - f ( ) = f ( ) = - ( - + ) =., f ( ).,,, (, + ), A, (, + ) f ( ) A. >, >, U ( ; ), f ( ) - A. A =, =, > ( < ),, U + ( ; ) (U + ( ; )?) f ( ) - = =. A, = A, > (< ), U ( ;), f ( ) - A = A > A =. f( ). =, + f ( ). (47 ) 7 f ( ) = A, f + + = A. f( ) = A >, M >, > M, f ( ) - A < +. y =, =, >, < y <, M 56

f y - A <, f + = A. 8: R( ) R( ) =, [, ] ( =, ). R ( ) = (, ), : q, = p q ( p, qn +, p q ),,, (, ). =. >, >, < - <, R( ) - <. R( ) =, U ( ; ), R ( ) =, R ( ) < ; U ( ; ), p q, R ( ) = q, >, p q U ( ; ), q <. >, q p q, q,, U ( ; ).,,, k, >, = mi{ -, -,, k -,, - }, U ( ; ) (, ), R( ) <., >, >, U ( ; ),, R ( ) <, R( ) =. ( ) f ( ),. ( ) f ( ) = A > ( < ), r < A ( r < - A ), U ( ), U ( ) 57

(3 ) (4 ) f ( )g( ), (5 ) h ( ) = A. f( ) > r > (f( ) < - r < ). f ( ), f. f ( ), g ( ), U ( ; ) ( ) [ f( )g( ) ] = ( ) [ f( )g( ) ] = (3 ) f ( ) g( ) = f( ) g( ). ( ) f ( ) = g( ) = A, U ( ;) f( ) f ( )h( )g( ), f ( ), g( ), f( ) g( ). ( ) f ( ) g( ). ( 3) g( ) ( g( ) ). ( 4) (, ) f ( ), f ( ) (, )? (, ) f( ),? f ( ), (, ), (, ), U ( ), f ( ) U( ), f( )., (, ),. (, )(, ); (, ). : (, ), = f( ) =, (, )., f( )(, )., f ( ) (, ) f ( ), (, ). : (, ) 58 f ( ) = q, = p q, p, q,, (, ).

(, ), { k },, k. k = p k q k, { q k }, { q k }, { k }, { k }. U ( ), K >, k > K, k U ( ). { q k }, M >, k ( k > K ), f( k ) = q k > M, f( ), f( )(, ).? ( 4) g( ), g( ) g ( ),, U ( ), U ( ), f( ), f( ) g( ) U ( f( ) ), g( ) = f( ) g( ). g ( ), g ( ) =,. g( ) = ( - ),, g( ), g( ) =. 5 5. + 5 - ( + ) ( 5 + 5 - ( + ) ) =, + 5, y = 5 + 5, y. y = 5 + 5, = y5-5 5 = + 5 - ( + ) 5 ( y5 - ) y 5 + 4 y - 5, + = y5 + 4, 5 = 5 ( y5 - ) 5( y - ) - y 5 - = 5 ( y4 + y 3 + y + y + ) ( y - ) [4 - ( y 4 + y 3 + y + y) ] ( y - ) = - 5 ( y 4 + y 3 + y + y + ) y 3 + y + 3 y + 4. 59

, y, 5 + 5 - ( + ) = y - 5 ( y 4 + y 3 + y + y + ) y 3 + y + 3 y + 4 = -. ( ) =, U ( ; ) ( ), t f ( t) = A. t f (( ) ) = A. ( 5) f ( t) = A, >, >, tu ( ;), f ( t) - A <. ( ) =, >, > ( < ), U ( ; ), ( ) - <. : >, >, U ( ; ) f(( ) ) - A <, f (( ) ) = A. ( 5), ( 5), +, -,, +, -. 3 + + + + +. + + + + + = + + + + + =,. 6 4 >,

+ =. : =, + =. >, < + ( N + ) + <. =, >, N, > N, - < > N +, [ ] + > [ ] > N, < +. - < [ ] + < [ ] < +,. 5 i m i i k + =. > ( i =,,, ), + + + + = k ( k), = k +, 4 + + + = m i i. k,, = k, + + + + = m i i. (5 ) 5 f ( ) >, f( ) = A. f ( ) = A,.. A = A >. A > f( ) - A = f ( ) - A f( ) - + f( ) - A + + A - 6

9 ( ) : ( ) f ( 3 ), f ( ), f ( ) = f ( 3 ). f ( ) = f( )? ( ) f ( 3 ), >, >, < <, f( 3 ) - A <, y = 3, < 3 y <, f ( y) - A <, < y < 3, f ( y) - A <, ( ). : f ( ) = f ( ) =, f ( ) = f( 3 ) = A.,, <,,, <,. =, - f( ), 3 f( ). + - f( ) = A ( ), f( ) = A. ( 3 ) f( ) = A + ( ), f( ) = A. ( 3 ) f( ) = A - ( ), : + - f( ) = A. ( 3 3) f( ) = A { } U + ( ), f ( ) = A. f( ) = A { } U - ( ), f ( ) = A. ( ) f( ) U + ( ), f, f ( + ) = if U + ( ) ( 3 4) ( 3 5) f ( ) ; ( 3 6) 6

f, f ( + ) = sup U + ( ) ( ) f U ( ), f ( - ) = sup U( ) - f ( ), f ( + ) = if U + ( ) f ( ). ( 3 7) f ( ). ( 3 8) f U ( ). 3 f( ) = A f ( ) = A + >, >,, U ( ;), f ( ) - f ( ) <. >, M >, > M, > M, f( ) - f( ) <. ( 3 9) (3 ) f U + ( ), f ( + ). (3 8 ), f U ( ), f( + ),,? f U ( ), f U + ( ). U - ( ), U + ( ), f ( ) f ( ), f ( ) f ( ) U + ( ),, f ( + ) = if U + ( ) f ( ). f ( ). ( ) : A, f ( ) A, f( )A >, >, U ( ; ), ( ) : (i) (ii), =, f ( ) { }, { } =, f ( ) - A. =, f ( ) f ( ) f ( ). (3 ) : f( ). (3 ) 63

f( ) >, >,,, < - <, < - <, f ( ) - f ( ). f ( ) =,,,,. f ( )., { }, = ; { }, =. f( ) = =, f ( ) = =, f( ). f( ) =,,, -,. f ( ). f ( )? >,,, U ( ), f ( ) - f ( ). >, =, >,, U + ( ; ),,, f( ) - f( ) = - ( - ) = + > >. <, =, >,, U - ( ; ),,, f ( ) - f ( ) = - - = - - > >. f( ). f ( ) ( ), f ( ) =. 3 f ( ) [, ], (, ] ( =,, ), f( ) = f( ). =. f ( )[, ], f -. f( ) = f( ) ( f - f ) ( ) = ( f - f ) ( ), 64

=,, f ( ) [, ] ( ), f -.,.,, >, { k } { }, k -. k +, f ( ), f ( k )f( + ) > f ( ). k f ( k ) = f ( ) = f ( ), k, f ( )f( + ) > f ( ),. =. 4 f ( )[, ), - f [, ). f ( ),. [] - f( ),, >, f ( ) U - ( ; ). [, - ] ( < - )f ( ), [, f [, ). - ], f ( ) m{ f ( ), f ( - ) }, [] f [, ), f [, ),, - f ( ). 5 f ( ) U ( ). : { }, U ( ),, f( ) = A, f ( ) = A. < + - < -, (3 ) :,., { } (3 )... f ( ) A, >, >, U ( ; ), f ( ) - A. =, U ( ; ), f ( ) - A ; = mi, -, U ( ; ), f( ) - A ; 65

= mi, - -, U ( ; ), f ( ) - A ; { }, < + - < -, =, f( ) - A f( ) = A. f ( ) = A. R. (55 ) 7: f R, f ( ) =, f ( ), + T f. f ( ) =, >, M >, > M +, f ( ) <. R, N +, = + T > M. f, f( ) = f( + T ) = f( ) <,, f ( ) =, f ( ), R. 83 9.. + { } U + ( ), f( ) - A,. f ( ) A, 4 si t =, rcsi + =, = - cos =. rct =, = e y ( + y) y = e. 66

+ +, + + < + + + < + = + + + + + ( < + ), = e + +, + + : f ( ) = + g( ) = + + +, < +,, < +, + N +. = e?, = e. f ( ), g( ), f ( ) = g( ) = e. + + f( ) < + < g ( ), +,, + + = e., - N - M + + = + + = e + + = e,. si - 3 - cos. 3 y = - 3,, y, 3 si - 3 - cos = y 3 = y si y - cos y + 3 si y - cos y + 3si y = y - cos y + 3 si y 67

= y si y si y cos y + 3 t ( + )t ( - ) + t,, ). t( + ) t( - ) + t = t - t - t t + t t ( - t 4 ) = ( - t t ) = 3., k+ ( k =, = ( - t 4 ) t - t t = ( - t 4 ). 3 y = + - + - = + ( ). -,, y, - + -. = y ( + y) y + 4 : ( ) si = y ( + y) y y ( + y) = e. ; ( ) si ; (3 ) si ; (4 ) si ; ( 5) si -. 68.( ) si ; ( ) ( 4 )

, ; ( 3)(5 ).. ( ) si si, =. ( ) si, si = ( si = si = ). (3 ) si, si =. (4 ) = +, si = +,, si. (5 ) y =, y, si = y si y y =. 5 B, V = kb, k, t? t, t t,,,, t, - t, t.,,., t B + k t ; kb t, t, t B + k t t, B + k t ;.,. = kt, B + k t = B + kt = B + k t = B e kt. 69

,., ( ). (58 ) 3: cos cos cos cos =. + cos cos cos cos si = si, si =. 5 : f ( ) U ( ), f. f ( ) =, +, -,,. f ( ) =.,,. ( ).. 3 f( ) = A f( ) - A. 4: f ( ), f ( ),,. f( ), f g. ( ) f ( ) g( ) =, f g, g f, 7 f ( ) = o( g( ) ) ( ). ( ) K L, U ( ) < K f ( ) g( ) L,

f( ) f g. g( ). f ( ) (3 ) g( ) f ( ) g( ) L, U ( ), f( ) = O( g( ) ) ( ). = A, f g =, f g, f ( )g ( ) ( ). :, si, t, rcsi, rct, ( 5 ) - cos. (4 ) : f, g, h U ( ), (i) f ( ) g( ) ( ). f( ) h( ) = A, g( ) h( ) = A. h( ) (ii) f( ) = B, h( ) g( ) = B. ( 5 ) (5 ), f( ) = k ( < k < + ), ( 5 3) f ( ). +, -,, +, -. 3f U ( ), f. f( ) = G >, >, U ( ;), f ( ) > G. f( ) = + G >, >, U ( ;), f ( ) > G. f( ) = - G >, >, U ( ;), f ( ) < - G. = ( ) G >, N, > N, > G > G < - G. 7

+, -,,. 4 f, g, f ( ) g( ) =, g f. : >, A >, c >, >, +, A + c log =, =. ( 5 4) + +, c A, log. 5y = f ( )y = k +, f f ( ) k = + f ( ) - = [ f ( ) - k ] ( +, - [ f ( ) - k ] ). ( 5 5) f ( ) = ( f( ) =, f ( ) = ), + - f ( ) = ( = ).,, y y - =, y y + =, y = f( ),.. t - si si 3, si, t? 3,,.,,. t = + 3 3 + o( 3 ), si = - 6 3 + o( 3 ), 7

o( 3 ) 3, t - si = 3 + o( 3 ), si, t, 3,.? f ( ), G >, >, U ( ;), f ( ) > G. f ( ), G >, >, U ( ;), f ( ) G., : f ( ) U ( ), G { }, =, f ( ) G., =, U ( ; ), f ( ) G, = mi, -, U ( ; ), f ( ) G, = mi, - -, U ( ; ), f ( ) G,. f ( ), { }, { f ( ) }., ( - ) : ( ) 3-3 + ; ( ) l ; ( 3) e - e. ( ) 3 = ; - 3 + = ( - ) ( - ), 3-3 + ( - ) ( + ) = ( - ) = 3, ( - ) ( ) y = -,, y, = ; l - = y l( + y) y = y l( + y) y =, 73

(3 ) e - e - = =. ee - - (y = - ) - = y e e y - ( z = e y - ) y = e z z l( + z ) = e z l( + z ) z = e, y l( + y) y = ue l u =., ( ) + 4 + ; ( ) + - ; ( 3) si. ( ) = 3. ( ) + - = =. (3 ) 3 + 4 + =, + + :, ( + - ) = + + + + =, =. si si = 3, ( ) f ( ) = -, ( ) f( ) = e l ( > ). 74 ( ) N +,, =,

f ( ) + = + l = - y + = - y + = -, y l y = y l + ( > ). e - ( ) N +, f ( ) = e - y ( (5 4 ) ) l y y = y = y (y = y ) e y = y y + e y =, ( > ). ( (5 4 ) ) :. 4, f( ), g( ), h( ), g( ) [ h( ) + f ( ) ] ( ) f( ) = o( g( ) ), g( )h( ), f ( ) = o( g( ) ), g( ) h( ), ( ), f ( ) g( ) h( ) + f( ) g( ) h( ) =, g( ) =, h( ) = g( ) + f ( ) g( ) = + =. g ( )[ h( ) + f ( ) ] ( ).,,. 75

5 - y =, (, > ) y =. > : (5 5 ) y = f ( ) = k k = + - = + -, y = f ( ) = - - - = + = -, - = + ( f ( ) - k ) = + = =, + ( - - ) - - + = + ( f ( ) - k ) = - =. + ( - - ) -. =, + y = f ( )y =, y = f ( ) y = -., < y = f 3 ( ) = - y = f 3 ( )y = -. - (66 ) -, y = f 4 ( ) = - -,, y = f 4 ( )y = y = y =. 7: S, { } S, + ( ). 76 S, M >, S, > M. M =, S, > M,

M = +, S, > M, M = - +, S, > M, { }, +. : (67 ) ( ) 3 - ( - [ ] ) ( = ) ; ( ) + ( [ ] + ) - = ; (3 ) ( ( + ) ( + ) - ( - ) ( - ) ) ( = + ) ; + (4 ) + ( = ) ; (5 ) - - ( = - ) ; - (6 ) + - - 3 + - 3 - = 3 ; (7 ) m - m - -, m, = m - ( 7) m,, m,. = + y,, y,. = y m - m - - m - ( + y) m - - ( + y) = - y m C m y + C m y + + y m - C y + C y + + y ( mc - C m ) y + ( mc - C m ) y + = - y ( C m y + C m y + + y m ) ( C y + C y + + y ) ( mc - C ) y + ( ) y 3 + = - y ( C m y + C m y + + y m ) ( C y + C y + + y ) = - mc - C m C m C = m -. 77

m, =, - = y, = - y m - m - - m C m y + C m y + + y m - y C m y + + y m = y ( C m y + C m y + + y m ) y = C m C m = m -. m =,.m = =,. m, m - m - : ( ) + + + - - = ; ( ) - ( - + - - ) = ; (3 ) + ( - + - - ) =. ( ) - = m - + + - - = ( - ) - ( + ) + ( - ), + + ( - ) =, + =,,, + + + - - =, =, = -, + + - + = ( ) - ( - + - - ) +. + =. ( - ) - ( + ) + ( - ) = - - + + + 78

( - ) - ( + ) + - = - - - + + +,, - =, + =, - +, = -, =., - = - - + + - 3 4 - + - - = - =, = -, =. (3 ) +, ( ) 3 4 - - + - + + ( - + - - ) ( - ) - ( + ) + - = + - + + +, - =, + =, +, =, = -,, + = + - + - + 3 4 - + + - = + =, 3 4 - + + - 79

=, = -. 3f : ( ) f ( ) f ( ) ; ( ) f( ). 4 f, f( ) >, f ( ) =.? 5 f( ) = A, ua g( u) = B, g( f ( ) ) = B? U ( )f ( ) A. 6 f ( ) = cos. ( ) { } ( ), f( ) ( ) ; ( ) { y } y ( ), f ( y ) + ( ) ; (3 ) { z } z ( ), f ( z )- ( ). 7: { }, { } : ( ) = r > ; + ( ) = s > (, =,, ). ( ) >, r - >. N, = r, >, N, r - <, > ( r - ). r - >, ( r - ) = +, { }. = +, ( ) >, s - >, + = s, N, > N, k + k > s - ( k = N +,, ). - N + > ( s - ) - N N. 8( ) : 8 = +.

( ) + 9 ( = + ) ; ( ) - = +, : ( ) ( + + + ) = + ; ( = ). ( ) > ( =,, ), = +. ( ) = +, (, > ), G >, N, > N, > G. > N, - N >, ( + + + ) + + + N > - N G + N + + + > G = G, G >, N, > N ( ) + + + > G, ( + + + ) = +. = +, e G >, N, > N, > e G, l > G, ( ) : ( ) l = +. (l + l + + l ) = +, l( ) = +, = +. l(!)!( = + ) ; ( ) ( = + ). 8

f U - ( ), : { } U - ( ) ( ), f ( ) = A, sup U - ( ) f ( - ) = sup U( ) - f( ) = A. ( ) U - ( ), f ( )A. f ( ) = A., f( )A. U - ( ), =, - >, N, > N, <, f, f ( ) f ( ), f ( ) A.,, f ( ) A. U - ( ), i, j i j, f, f ( i )f ( ) f ( j )A. j, j, f ( )f ( j ) A. ( ) >, U - ( ), A - < f( ). f ( ) = A, >, N, > N, A - < f ( ) A, = N +, A - < f( ). sup U - ( ) sup U - ( ) f( ) = A. f( ) = A. f ( - ) = f (, + )f ( ) = f ( ), + f ( ) = A. : f ( )A, (, + ). (, + ), f ( ) = f ( ) = f ( ) = = f ( ). f ( ) = A, f( + ) = A, f ( ) = f ( ) = A, f ( ) A, (, + ). 3 f (, + )f ( ) = f ( ), + : f ( )f ( ), (, + ). f( ) = f ( ) = f ( ), + >, f ( ) = f ( ) ; <, f( ) = f( ). 4 f (, + ), f (, ), ( f( + ) - f ( ) ) = A. : + f( ) = A. A =, + ( f ( + ) - f ( ) ) =, >, X, X, f( + ) - f( ) <. X, = +, [ X, X + ], N +, 8

f ( ) [ X, X + ], M >, [ X, X + ], f ( ) M. f ( + k + ) - f ( + k) <, k =,,, f( ) = f( + ) + = f( + ) - f( ) + + f( ) + = f( + ) - f( + - ) + f( + - ) - + f( + ) - f( ) + + f( ) + f( + ) - f( + - ) + + f( + ) - f( ) + + f( ) + + + M +. ( > N ), M + <, > X + N +, f( ) + =. A, F( ) = f( ) - A. f ( ) ( F( + ) - F( ) ) = [ ( f( + ) - f ( ) ) - A] =, + + <, F( ) f( ) + A, F (, ). F( ) + f ( ) + =, = A. : ( A) - - - =. f ( ) = +, : >, - - - = +. 83

3 4 : + + + - - +. - cos 3. 3 - cos 5 f( ) = A : >, >, < - <, f( ) - A <. 6: f ( ) = si U ( ),. 7, A [, ], m A A m =, f( ) =, A ( =,, ),, k A k. [, ], f ( ) =. : + 3 : 4 : 5: 84 ( ) - ( ) f f [ + (B) - - - =. f( ) = -, l = -. + ( + )( + ) - ]. + si - cos. = - f ( ) ; = f ( ).

6 f ( t)tt, P( )Q( ) : tt, P( ) = + - - + + +,, Q( ) = m m + m - m - + + +, m, ( f( t) ), > m, P( f ( t) ) + Q( f ( t) ) ( + ) ( f ( t) ), = m, 7 f ( ) ( ), >. ( ) = ( ) = f i = i = i - ; i -, =. m ( f ( t) ) m, < m. 85

f U ( ), f f( ) = f( ) f U + ( ), y = f( ) = f >, >, - f( ) - f( ) <. <, f + f ( ) = f ( ), f ( + ) = f ( ). f U - ( ), f - f f. f ( ) = f ( ), f ( - ) = f ( ). f U ( ), f, f, f. : f( - ) = f( + ) = A, f, f( )A. : f( - )f( + ). f ( + ), f ( - ).,,..,, f ( + ) - f( ), 86

. f ( ) :,,, f ( ) -. f ( ), :,, <, f ( ) - f ( ). -. f U ( ), = ( =,, ), >, - <, f ( ) - f ( )? f ( ) <,, f( ), : >, N, N, < ;, N N >, - < N, f ( ) - f ( ) < ; N >, ( = N ) >, - <, f( ) - f( ) <. N, N, N,., >, =. f ( )U( ), f ( )? f ( ), f ( ). : f ( ), f( ) >, >,, U ( ;), f ( ) f( ) - f( ) ; ( ) f( )f( ) >, >, U ( ; ), f ( ) - f ( ). ( ) D( ) =,,,. 87

R, D( ),, D( ). R ( ) = q, = p q 8 ( p, qn +, p, q ),, =, (, ). R( ) =,, (, ), ; (, ), R ( ) R ( ), (, ). ( ) : = p q, =, >, U q ( ; ), f ( ) - f ( ) = - q =, ( ), R ( ) = p q. - : ( ) y = ; ( ) y = 3. ( ) =, >, =, <, <., - < = mi, - = - + - 3,, - <, 3 - <. <, ( ) =, >, = 3, <, 3 <., - < 3-88 3 3 -,, 3 >, 3 = <, 3 - + 3 + 3 = mi, 3, - 3.

- <, 3-3 <. f ( ) = 3, =,..,, =, f( ) 3, f ( ) =, f ( ) =, >, = 3, >, U + ( ;), f( ) - f( ) = 3 > 3 =. f ( ), f ( ). <. 3. f ( ) = l =, =. l = f ( ). f ( )U ( ), =. l =, =, 3. 4. f ( ) = sg si si U ( ),,, f( ) -, f ( ), =. 89

sg u u =, si, sg si, k = k ( k =,, ).. k = k ( k =,, ), k k f ( k + ) = + k f( k - ) = - k f( k + ) = + k f ( k - ) = - k sg si = -, sg si = ; sg si =, sg si = - ; k = k ( k =,, ). =. =, >,, + <, + 3 <, =, =, + + 3 f ( ) - f ( ) = sg si + - sg si + 3 = >,, f ( ), = f ( ). 5 f ( ) (, ), e f ( ) e -, f( )(, ). f( ) (, ) (, ), f ( ), f ( + ) = f ( - ) = f ( ). e - f ( f ( ) (, ) f ( )(, ), + ),. e - f( ) (, ), f ( )(, ). (, ), f ( 9 + ) = f( ). >, f ( )f ( ),

+ f( )., f ( + ) = + f ( )f ( )., e f ( )(, ), >, +, e f ( )e f ( ), e f ( )e f( + ), f ( )f ( + ), f ( ) = f ( + ). f ( - ) = f ( ), (, ), f ( )(, ). e. (73 ) 7 f, g. g( ) = y g( ) = y f( y), f( y), >, >, < y - <, f ( y ) - g ( ) <., < - <, y, f, y f ( y), g( ) - g( ). >, >, - <, g ( ) - g ( ), g ( ) =,, g( ). ( ) f, f U ( ). ( ) f, f ( ) > (< ), r < f ( ) (- r > f( ) ), U( ), U ( ), f ( ) > r (f( ) < - r). 9

( 3) f g, fg, fg,. f g ( g( ) ) (4) f, g u, u = f ( ), g( f ( ) ) = g( f ( ) ) = g( f ( ) ) ( ) (5 ) ( ) f, f( ) =, g u =, g( f( ) ) = g( f ( ) ). (+, -, +, - ).,,. ( ) : f [, ], f [, ],... ( ) : f [, ], f ( ) f( ), (, ), f ( ) =.,. f ( ) =, f ( ) =. (3 ) : f [, ], f - [ f( ), f ( ) ] [ f ( ), f ( ) ].,,. (4 ) : f [, ], f [, ], >, = () >,, [, ], - <, f ( ) - f ( ) <. [, ], [, ],,,. ;. 9

. f (, ), f ( + ), f ( - )? f (, ) f (, ), >, ( ) >,, (, ), - <, f( ) - f( ) <., U + ( ; ), - <, f ( ) - f ( ) <. f +, f ( + ). f ( - ). + f (, )f( + ), f ( - ). [, ] F( ) : F( ) = + F( ) = f( + ), =, f ( ), (, ), f ( - ), =, f ( ) = f ( + ) = F( ), - F( ) = - f ( ) = f ( - ) = F( ), F( ) [, ], (, ) f [, ] F. F ( ) [, ], (, ), (, )F ( ) = f ( ), (, ) (, )., I,.. : ( ) ; ( ) f, f ( 4 ) ; (3 ) ; (4 ) I ci, I ci, f I, I, f I = I I. (5 ) (, ) f ( ) f ( + ), f( - )., f ( ) = si (, ), 93

si si = + - = si. 3 I f( ). f I : >, >,, I, - <, f( ) - f( ). y = si, [, + ). >, >,, - <, f ( ) - f ( ). si [, ] ( > ),, U( + ),,. = +, =, < - = + - = < + +,, - <, f ( ) - f ( )., =, >, = +, =, >, 4 - <, si - si =, si [, + ). f ( ) (, ), + f ( ) (, ). + f ( ) = -,. f ( ) = - f ( ) =, f ( ) =, f ( ) [, ] f ( )[, ] F( ), F( ) = f ( ), (, ),, =,. F( )[, ], 94 + - F( ) = f ( ) = = F( ), + F( ) = - f ( ) = = F( ).

[, ] F ( ),, [, ], F( ),. =, =, f ( ) (, ), f ( )(, ) ;, (, ), f( )(, ). 3 : (, ) f ( ), (, ). f ( )(, ),, f( + ), f( - ).f ( ) [, ] F( ), F( )[, ]. + f ( )(, ),, f ( ) - F( ), f ( ),, f ( ) [, ] + f( ), =, F( ) = - f ( ), (, ), f( ), =., F( )[, ], f ( ) (, ).. 4 f I, f I f, < < 3, f ( ) > f ( ), f ( 3 ) > f ( ), ( f ( ) < f ( ), f ( 3 ) < f ( ) ), 4 -,, y =, y = f ( ) (, f ( ) ), (, f ( ) ), f. 4-95

< 3,, f I,,, 3, < ( f ( ) - f ( ) ) ( f ( ) - f ( 3 ) ) <. f ( ) > f ( ), f( ) < f( 3 ). : [, ], [ <, < < 3, f. f ( ) < < mi{ f( ), f ( 3 )}., 3 ],, < f( ) = f ( ) =,,,. 5 f ( ) I, f ( )I : { }, { } I, ( - ) =, ( f ( ) - f( ) ) =. [] f ( )I, >, ( ) >,, I, - <, f( ) - f( ) <. I { }, { }, ( - ) =, >, N >, > N, - <,, f ( ) - f( ) <, ( f ( ) - f( ) ) =. [] I { }{ }, ( - ) =, ( f ( ) - f( ) ) =. f( )I..f( )I, >, >,,, - <, f( ) - f( ). =,, I, - <, f ( ) - f ( ), =,, I, - <, f ( ) - f ( ), =,, I, - <, f ( ) - f ( ), ( - ) =, ( f ( ) - 96 f ( ) ),.

f ( ) I. f ( ) I,, { }, { },. (8 ) 6 f [, + ), f( ). : f [, + ). + f [, + )? f [, + ). [, + ), f ( ) = B > A = f ( ), + X, > X, f ( ) < B + A, [, X] f f [, + ). [, + ), f ( ) < A, f [, + ). :. f ( ) = + + + + + =, k ( k =,,, + ). f ( ) = +, f ( ) = + - -, X >, f ( - X) <, f ( X) >. 6 f 6. f [, + ). f ( ) = A,, >, X,, + > X, f ( ) - f ( ) <. [, X + ], >, ( ) >,, [, X + ], - <, f( ) - f( ) <. ( ) = mi{, },, [, + ), - <, : f( ) - f( ) <. ( ), [, X], - <, ( ) f ( ) - f ( ) <. ( ) [, X ], ( X, + ), - <, [, X + ], ( ), f ( ) - f ( ) <. (3 ), ( X, + ), - <, f( ) - f( ) <., >, ( ) >,, [, + ), - <, f [, + ). f( ) - f( ) <, 97

3, : = sup r < { r r } ( > ), = if r < { r r } ( < < ). : = + ; ( ) =. ( 3 ) ( ) ( >, )R, - - =, + = + ( > ) ; = +, + = ( < < ). ( ) log (, + ), log + = -, log = + ( > ) ; + log + = +, log = - ( < < ). + ( ) u ( ) = >, v( ) =, u( ) v( ) =. ( 3 ) ( ) u = >, v 3 =, u v =. ( 3 3),.,?.,, f ( ) = - + - =,, 98

f ( )., - ( ). ( + ) ( ). ( + ) = e. = p q, p, q, q >, : ( + ) = + p q = + p q p q p q = = e p q. q + p q q p + p q,. (3 ) ( ),, ( p) q p ( + ) = [ ( + ) ], u( ) = ( + ), v( ) =, u( ) = ( + ), (3 ) = e, v( ) =. ( + ) = [ ( + ) ] = [ ( + ) ] = e. =, ( + ) = = e, R, ( + ) - ( > ). y = -, - = = e. log ( + y) y, 99

. - = y, = log ( + y), y. - = y = y = y = l. = e, e - 3 y log ( + y) y l( + y) l l l( + y) y y l( + y) y = l y ( + y) y =. >, 3( ), =.. = >. =. = >,, l = l. l + l + + l = l., = el l + l + + l = e 4 >, =, = e l =. =. =,, l = -. 9( ) l + l + + l = -.,

= e l 34 : 5 + l + l + + l = e =. =. ( >, > ). + =, >, =, + + + -, + - =. ( + ) + = e. = + + - = + (3 3 ) - + - u = + ( - ) + ( - ) - + - - + - - + -,. v = ( - ) + ( - ). ( + ) = e, u = + - + - = e ; - - + - = l, v ( - ) + ( - ) =

= l + l = l. + = u v = e l =. (84 ) f (, ), f ( + )f( - ). : ( ) f (, ); ( ) (, ), f ( )m{ f ( + ), f ( - ) }, f (, ). ( ), f [, ]. F( ) = f( + ), =, f( ), (, ), f( - ), =, F( )[, ]. F (, ), F( ) = f ( + ) = + F( ) = f( - ) = - f( ) = + F( ), f ( ) = - F( ). [, ], M >, F( ) M, [, ]. (, ) f ( ) = F( ), f( ) M, (, ). () F( )[, ],, [, ], F()F( ) [, ].=, F( ) = f( + ) f ( - ), f ()F () ; f ( ) m{ f ( + ), f ( - ) }, f () F(). f ( ) = F( ), f (, ). (, ), f (, ). f : >, f (, + ) ( -, ), [ +, - ] f.

f (, ), f ( + ) = f( - ) = +, f (, ). f ( + ) = f( - ) = +, (, )M > f ( ), > < -, (, + ) ( -, ), f ( ) > M. f ( ) [ +, - ],, [ +, - ], f()f ( ) [ +, - ]. f () f ( ) (, )., [ +, - ] ; (, + ) ( -, ), f ( )f ( ) < M < f ( ). f () f (, ). 3 f I, : ( ) ri f ( r) =, I f ( ) ; ( ) r, r, r < r, f ( r ) < f ( r ), f I. ( ) I, R, { r }, r =. f r I, f ( r ) =, f ( ) = f ( r ) =, f ( ). ( ), I, <, R, r, r, < r < r <. { r ( ) }, { r ( ) }, < r ( ) < r < r < r ( ) < ) ), r( =, r( =. f ( r ( ) ) < f ( r ) < f ( r ) < f ( r ( ) ) ;, f f ( ) r(, f I. ) = f ( ), f( r ( ) ) = f ( ) ; f ( )f ( r ) < f ( r ) f ( ), 4,, 3, < < 3. : + - + - (, )(, 3 ). 3 = - 3 f ( ) = ( - ) ( - 3 ) + ( - ) ( - 3 ) + 3 ( - ) ( - ). 5 f [, ], [, ], y[, ], 3

: [, ], f ( ) =. f ( y) f( ). f ( ) [, ] m = f ( ), m =, ; m >,. 6 f [, ],,,, [, ],,,, + + + =. : [, ], + f( ) = f( ) + f ( ) + + f ( ). 9, = = = =. 7 f [, + ), f( ), [, + )., = f ( ), =,,. : ( ) { }; ( ) = t, f ( t) = t ; (3 ) f ( ) <, (, + ), t =. ( ) +, { }. = f ( ), { }, ( ) + = f( ), t = + = t, f : = f ( ) = f( ) = f( t). (3 ) f ( ) <, (, + ),. t >, f( t) < t, ( )f( t) = t, t =. 8 f [, ], f ( ) = f ( ). :, [, ], f + = f( ). =, =. > F ( ) = f + - f ( ), F( ) + F + + F - =,. 9 f =,, yr, f( + y) = f ( ) + f ( y). : ( ) f R ; ( ) f ( ) = f ( ). ( ) f( + y) = f ( ) + f ( y) y = -, f ( ) = f ( ) + f ( - )., f =, 4

f ( ) =. R, f ( ) = f ( ) + f ( - ) = f ( ) + f( ), f( ) = [ f ( - ) + f ( ) ] = f ( - ) + f( ) = f( ) + f ( ) = f( ), f R. p ( ) p, f( p ) = f ( + + + ) = p f ( ).p, - p. f( p - p) = f ( p) + f ( - p) = f ( ) =, p, f ( p) = pf ( ). : q, f f( p ) = - f( - p) = - ( - p) f ( ) = pf ( ), qf q = f q = q f ( ), q + q + + q q = f( ). r = p q, ( p, q ), f ( r) = f p q = pf q = p q f( ) = r f( ). : R, f ( ) = f ( )., { r }, r f ( ) = f ( r ) = r f ( ) = f( ). =, R f,, R, f ( ) = f ( ). f. f ; R, f ( ) = f ( ) f ( ) ( ), : f( ) = f( ) ; f ( ) = f ( ) = f ( ). y = ( A) si. ( ) f ( ), g ( ) ; ( ) 5

f ( ), g( ). f( ) + g( )f ( )g( )? 3 : 4 : h - + ( >, > ). log ( + h ) + log ( - h) - log h ( > ). 5ABC, y,. 6: ( ) f ( ) = cos (, ) ; ( ) g( ) = cos (, ). 7 f ( ) [, + ), ( ) [, + ), [ f( ) - ( ) ] =, ( )[, + ). + y = : 3 : (B). + t + + + c + + + c 4 + 4 f ( )I, : ( ), I, f + ( ), I, ;. f ( ) + f ( ) f( + ( - ) )f( ) + ( - ) f ( ). 5 f ( -, + ),, f ( ) >, + f( ) = f ( ) =, f ( ). - 6 f [, ],. ( ) f (, ), [, ] 6. ;

(, ) (, f ( ) ) (, )( z, f( z ) ). ( ) R [, ]. 4-7 f [, ),. : (, ) [, ), f( ) (, ), f [ f ( ), + ). 7

( ). ( ) y f ( ) = = (3 ) : f ( ), f ( +) - f ( ) y = f ( ) + o().,,. (4 ) : f ( ),., y = =,. (5 ) : y f + ( ) = + = f ( +) - f ( ) + y f - ( ) = - = f ( +) - f ( ) -... f ( ) f + ( ) = f - ( ).,.,.( 3) ( ) : f ( ), f ( ) y = f ( ) (, f( ) ). 8

y - y y - y = - = f ( ) ( - ), f ( ) ( - ), ( f ( ) ). ( ) : f ( ),, f, f ( ) =. : f( ) =,. f ( ) = ( ) f ( ),,. 3 f [, ], f + ( ) f - ( ), k f + ( ), f - ( ), (, ), f () = k. f ( ), f ( ),,...,. 638 :,, - A = ( - ). + E, A = ( + E) ( - - E)., A, E, =. A - A E = ( + E) ( - - E) - ( - ) E = E( - ) - E E = - - E =, E =, - =, =.) 9

E, A ( ) =, =. 67,,, y y.y =, +, y + y y,, y + y = ( + ) = + C - + C - ( ) + + C ( ). y =,,, y = -. ( )= -,. f ( ), f ( ) ( f( ) )? f ( ) ( f ( ) ). f ( ) f ( ) ; ( f ( ) )f ( ),. f ( ) =, f (3 ) = = 3 = 6, ( f (3 ) )= (9 )=. f( ) =,, +, <. f + ( )f - ( ) : f ( ) =, >,, < ; =, f + ( ) = =, f - ( ) =.?,?. f ( ) = : ( +) - f + ( ) = + = + ( + ) =, (3 +) - f - ( ) = -

f + ( ) =, f - ( ). = - + = -, =. 3 f ( )? ( ) ( ). ( ). f ( ) = si, >,,, (3 ). : f - ( ) =, f + ( ) f( ) =, f + ( ) =, f - ( ) = -. f( ) = g( ) =,, -, ;, f ( ). f ( +) - f ( ) f( + ) - f ( ), -,. = =,,, -,,, f ( )., f ( ), f ( ). g( ). g( + ) - g( ) =, =, g( + ) - g( ), -,, g ( ) =., g( ), g( ). =, f ( ) [ -, ] ( > ), f ( ), f ( ) =.

f ( ), f( ) =. f ( ) - f( ) ( ) f() - f ( ) =, f ( ) =. 3 f ( ), g( )[, ], (, ), f ( ) = g( ), f - ( ) = g + ( ), h( ). h( ) = f ( ) = g( ), =, f( ),, g( ), >. h( + ) - h( ) h + ( ) = + g( + ) - f ( ) = + g( + ) - g ( ) = + = g + ( ) ; f - ( ) = g + ( ) h( ). h - ( ) = f - ( ). h + ( ) = h - ( ), 4 f ( ), P(, f ( ) ) N M, P T ( 5 - ). : P T N T PN = N T = f ( ) f ( ) f ( ) f ( ), TM = f ( )f ( ), + f ( ), PM = f ( ) + f ( )., P, t = f ( ). = t, PT = f ( ), N T = f ( ) f ( ).

5 - T PM, TM P T = t, T M = f( )f ( ). PN = N T + P T = f ( ) f ( ) = f( ) f ( ) + f ( ) + f ( ). PM = T M + P T = f ( ) f ( ) + f ( ) 5 = f( ) + f ( ). y =. y, y, 4. P(, ), y Q. P y R, ( ) =, PR R y y - = ( - ), y = -., 5 -, 3

5 -, =.PQR,, PQR. Q y q, PQ T PQ = QR, PQ = Q T + P T = + ( - q) + ( - q) = ( q + ), q = 4. 6 : f( ), f ( ), f ( )., f ( ), f( )., f ( ) = =, f ( ) =., f ( ), f ( ). f ( ) =,, -, f( ) =, f ( ) =,., f( ), f ( ) f ( ). f ( ),, U ( ), f ( ), f f ( ). f( ) =, f, f, 4

f ( ) =, f ( ) =. f ( ),, U ( ), f( ) U ( ), f ( ), f( ). f ( ) =, f ( ), f ( ), f ( ) =, f ( ) =, f ( + ) - f ( ) f ( + ) - f( ) =. f ( ) =. : ( ) (94 ) 8 f ( ) = m, f = ; ( ) m, f = ; (3 ) m, f =. =, m si,, ( m )., = ( ) m, f ( ) = m si m, f ( ) = f ( ), f =. ( ) m, m si - f ( ) = = m - si =. f ( ) =, m. f ( ) = m m - si - m - cos, (3 ) m3, f =.,, =,. 5

g( ) = g ( ) =, f ( ). f ( ) = g( ) si,,, =. f ( ) = f() - f ( ) = g() si, g() =. f R,, R, f ( + ) = f ( )f ( ).f ( ) =, R, f ( ) = f( ). f ( + ) = f ( )f ( ) = =, f( ) = f ( )., f ( ) =, f ( ) ; f ( ) =, f ( + ) = f ( )f ( ) =, =, f ( + ) = f( )f ( ), f ( ) = f( ). ( ) ( uv)= uv. ( ) ( uv)= u v + uv, ( cu)= cu ( c ). (3 ) u v u v - uv =, v v = - v v. ( ) ( c)= ( c ). ( ) ( )= - ( ). (3 ) ( )= l, (e )= e. (4 ) ( log )= l, (l )=. (5 ) ( si )= cos, ( cos )= - si, ( t )= sec, (cot )= - csc, ( sec )= sec t, (csc )= - csc cot. 6

(6 ) ( rcsi )= ( rct )= -, (rccos )= - -, +, (rccot )= - +.,. 3 d y d =, d d y d y d = d y d u d u d.,,,. 4 ( ) f ( ) = u ( ) v ( ). y = f( ),. (l f ( ) )= f ( ) f( ), f ( g( ) )( f ( g( ) ) )? f ( g( ) ) y = f ( u) u = g( ). f ( g( ) ) f ( u) u, u g( ), f ( g( ) ) = f ( u ) u = g( ). ( f ( g( ) ) )f( g( ) ),. ( f( g( ) ) )= f ( g( ) )g ( ), f ( ) = ( ) + ( ), g( ) = ( ) ( ), f( )g( ), ( ), ( ).?. ( ), ( ),, f ( ) = ( ) + ( )g( ) = ( ) ( ).., ( ) =, ( ) = - =, f ( ) = ( ) + ( ) =, =, g( ) = ( ) ( ) = - = - =. 3 f ( )U ( ), f ( + ), f ( - ). 7

f + ( ) = f ( + ), f - ( ) = f ( - )? f + ( )f ( + ). f + ( ) f, f ( + ) f ( ), f ( + ) - f( ) f + ( ) =, + f ( + ) = + f( ) =, U ( ), f ( )., +, < f ( ) =, >,, <, f ( + ) = = + f ( - ) = =. - = f + ( ), f - ( ), f - ( ) = f ( - ).., f ( + ), f ( - ). f ( ) = si,,, =, + f ( ) - f ( ) = si - cos,,, =, f ( ) = f ( ). ( ) si =. : f ( + ), f ( - ), f ( )., f ( ) =,,, =, f ( + ) = f ( - ) = ; f( ) =,, 8

- f + ( ) = + - f - ( ) = -. f ( ). =, =. f + ( ) = f ( + ), f - ( ) = f ( - ) f( ) = + + l( + + ),, : + = = + + + + + + ; u = + +, f ( ). l + + = f ( ) = f ( ) = = + +, y= + + + +. + + + + = +. - rct - + t ( > ), - + - - + sec + t 9

= = + - + + ( - ) t - + sec = ( + )cos + ( - ) si + cos. 3 y =, y., y =, l y = l. y y = l l +, y= 4 : l l + y = ( - ) ( + ) 3., y : y = ( - ) ( + ) 3, -, - ( - ) ( + ) 3, < -., - : y= ( - ) ( + ) ( 5 - ), > -, - ( - ) ( + ) (5 - ), < -. = -, ( - ) ( ) 3 - y + ( - ) = + =, y - ( - ) =, y ( - ) =.. y= 5 ( - ) ( + ) ( 5 - ), -, - ( - ) ( + ) (5 - ), < -. f( ) = + e,,, =

., =, f ( ) = f =.. + e + e + e = f + ( ) = + + + e + e = + + e =, 6 : f - ( ) = - + e =, + e -. ( ) [, + ), { }, N + ( ) f R, ff R. ( ) f ( ) = ( N + f ( ) =, =, N +,, N +,. ), f ( ) = ; N+, f, f. ( ) D( ) R,. u = D( ) D( u), D( u)d( )., D( ) =, ( DD ) ( ) = D( ) = ;, D( ) =, ( DD) ( ) = D( ) = ; ( DD) ( ), DD R, ( DD) ( ), R. ( ) 9 sh -, ch -, th -, coth -. : ( ) y = sh - ; ( ) y = ch - ;

(3 ) y = th - ; ( 4) y = coth - ; (5 ) y = th - - coth - ; ( 6) y = sh - ( t )., : ch y - sh y =, ch y = - th y, sh y = coth y -. ( ) ( ) ( 3), y = - ( ) ( y ),. ( ) = sh y,, d y d = d d y = ( sh y) = ch y,, ( )ch y = + sh y, ( ) = ch y, d y d = d d y = (3 ) = th y, ( ), ch y = (4 ) = coth y, d y d = ch y = + sh y =. + (ch y) = sh y = ch y - = - ( > ). d y d = d d y = ( th y) = ch y, - th y =, ( < ), - d y d = d d y ( th - )= -, ( < ). = (coth y) = = - sh y, - sh y ( 3 ), sh y = coth y - = -, (coth - )= - ( > ).

(5 ) y = th - - coth -, (6 ) y = sh - ( t ), y= y= - - - - =. + t sec = sec. 3 (34) = ( t), y = ( t), t, ( t) ( t), ( t). = ( ), C: = ( t), y = ( t) d y d = ( t) ( t). ( 3 ) d y d = ( ) t + ( ) ( ) - ( ) t. ( 3 ), t,, [,], ( t) + ( t), C., ( t)t. C (( t), ( t) ) ( Y - ( t) ) ( t) - ( X - ( t) ) ( t) =, ( 3 3) ( Y - ( t) ) ( t) + ( X - ( t) ) ( t) =. ( 3 4) 3 f ( ) f ( ), f ( ) f( ), f ( ), f ( ) = f ( ). f ( ) - f ( ), - 3

, f ( )., f ( - ) ( ) f ( - ), f 4 f ( ) ( ) = ( f ( - ) ( ) ( ) ( ) = (l ) ( > ), (e ) ( ) = e ; ( ) ( si ) ( ) = si + ; (3 ) ( cos ) ( ) = cos + ; ( ) ). (4 ) ( m ) ( ) = m( m - ) ( m - + ) m - ( m ) ; (5 ) ( l ) ( ) = ( - ) - ( - )! ; (6 ) + ( ) = ( - )! ( + ) +. 5u( ), v ( ), ( 3 5) ( uv) ( ) = C u ( k ) v ( - k) k =. ( 3 6) 6 ( t), ( t) [, ], = ( t), y = ( t), t[, ] y = ( t) ( - ( t) (. ( 3 7) d [ ( t) ] 3 C: d y d = ( t) ( t). y d d?? d d t d = d d t ( t) ( t). d y = d d d d y d 4 = ( t),, t y = ( t) = ( t) ( - ( t) ( [ ( t) ],. d y d, y d d

d y = d d d t d y d d d t = d ( t) d t ( t) ( t). = ( t) ( - ( t) (. [ ( t) ] 3. : ( ) ( 3 5). ( ) ( 3 6) ( 5 (4 ) ). (3 ) ( 5 (6 ) ). (4 ), ( 5 (3 ), (5 ) ). (5 ), ( 9 ). (6 ) (7 ). = t e cos t, y = e t si t, d y d. d y d t t = e si t + e t si tcos t, d d t = e t cos t - e t si tcos t, d y d = d y d t d d t r =, d y d. (3 ), = si t + si tcos t cos t - si tcos t. d y d = r ( ) t + r ( ) r ( ) - r ( ) t = = t + - t t + - t. 3 f ( t), f ( t), C: = f ( t), y = t f ( t) - f ( t), 5

d y d, d y. d d y d t = f ( t) + t f ( t) - f ( t) = tf ( t), d d t = f ( t), y = d d d y d = d d t 4 : d y d t d d t d y d d d t = tf ( t) f ( t) = = t. d d t ( t) f ( t) = f ( t). ( ) [ e si( + c) ] ( ) = e ( + ) si ( + c + ), ( ) [ e cos( + c) ] ( ) = e ( + ) cos( + c + ), si = ( ). =,. + + [e cos ( + c) ]= e cos( + c) - e si( + c) = e + + cos( + c) - si( + c) + = e + cos cos( + c) - si si( + c) = e + cos( + c + ), cos =, =. + +, = k -, [e cos ( + c) ] ( k - ) = e ( + ) k - cos[ + c + ( k - )], = k, 6 [e t cos( + c) ] ( k ) = ( [e t cos( + c) ] ( k - ) ) = [e ( + ) k - cos( + c + ( k - )) ] = ( + ) k - [ e cos( + c + ( k - )) ]

= e ( + ) k cos( + c + k), ( ). [e cos( + c) ] ( ) = e ( + ) cos( + c + ). 5 f ( ) = ( + + 3)e -, f ( ) ( ). u( ) = + + 3, v ( ) = e -, k3 u ( k ) ( ) =,, f ( ) ( ) = ( + + 3) (e - ) ( ) + C ( + + 3) (e - ) ( - ) + C ( + + 3) (e - ) ( - ) = ( - ) ( + + 3)e - + ( - ) - ( + )e - + ( - ) - ( - )e - = ( - ) e - [ - ( - ) + - 3 + 3 ]. 6 f( ),.l, m, F( ) = f ( ),, l( - ) + m( - ) +, >. f ( )( -, ]., F( ), >, < F ( ), F ( ), l, m, F( )., ; F ( ) m, F l., f - ( ), f - ( ).F( ), = f ( ). + [ l( - ) + m( - ) + ] = f( ) F ( ), F - ( ) = F + ( ), F - ( ) = f - ( ), F + ( ) = + = + F( ) - F( ) - l( - ) + m( - ) + f ( ) - f ( ) - = m, 7

m = f - ( ). F( ) F ( ) = f ( ), <, f - ( ), =, l( - ) + f - ( ), >. F =, F - ( ) = F + ( ), F - ( ) = f - ( ), F + ( ) = + = + = l, l = f - ( ), 7 + F ( ) - F ( ) - l( - ) + f - ( ) - f - ( ) - l = f - ( ), m = f - ( ), = f ( ). ( ) = ( - )! ( + ) + si[ ( + ) rccot ], (rct ) ( ) = ( - ) - ( - )! si( rccot ). ( + ) + = i + ( ) = i = ( - )! i - i - + i, - i - + i ( ) ( - i) + -, ( + i) + = ( - )! [ ( + i) + i ( + ) + cos + isi = e i,. 8 - ( - i) + ],

+ ( ) + i = + = ( - )! [ ( + i) + i ( + ) + si, cot =, - ( - i) + ]. + + i +, + = cos, = + + i = + (cos + isi ) = + e i, - i = + ( cos - isi ) = + e - i, ( + i) + = ( + ) + e i( + ) = ( + ) + [cos( + )+ isi( + )], ( - i) + = ( + ) + e - i( + ) = ( + ) + [cos( + )- isi( + )],, + ( ) = ( - )! [ isi ( + )] i ( + ) + = ( - )! ( + ) + (rct ) ( ) = si[ ( + ) rccot ], + ( - ) = ( - ) - ( - )! si( rccot ). ( + ) 3 ( 5 ) 4:. = (cos t + tsi t), y = ( si t - tcos t), ( t) ( t) = tcos t, y ( t) = tsi t, (3 4 ), [ Y - ( si t - tcos t) ] y ( t) + [ X - ( cos t + tsi t) ] ( t) =, ( t), y ( t), si ty + cos tx =,,. 4 (9 ) 9 y = rct. ( ) ( + ) y+ y= ; 9

( ) y ( ) =. y ( m ) = =, y ( m + ) = = ( - ) m ( m)! ( ).( )( ),, y ( + ) ( ), y ( + ) ( ), y ( ) ( ) ; =,. y = rcsi. ( ) ( - ) y ( + ) - ( + ) y ( + ) - y ( ) = ( ) ; ( ) y ( ) =. y ( m ) = =, y ( m + ) = = [ ( m - )!!] y= -, - y=, - y y - =, y 9. : ( - ) y- y=. f( ) = e -,,, = = f ( ) ( ) =,. f ( ) - f ( ) f ( ) = - = e - y = = e y =, y y f ( ) = 3 e -,,, =. 3

, f ( ) = - 6 3 + 4 6 e -,,, =. P 3 f ( ) ( ) = P 3 e - 3. f ( + ) f ( ) ( ) = =,,, =, P 3 ( ) - f ( ) ( ) - e - yp 3 ( y) = y e y =, y ( + ) ( ) = P 3 ( + ) e - y =,,, =, =, f ( ) ( ) =. 4 ( 5) y = f ( ) y = f ( + ) - f( ) = A + o() ( 4 ) ( A ), f, A f, d y, d y = f ( ) = f ( ) d. 3

f( ) f ( ). y = f( )I, f ( )I, I d y = f ( ) d, I. f ( ) d,, d y d. ( ) d[ u( )v( ) ] = d u( ) d v ( ) ; ( ) d[ u( ) v( ) ] = v( ) d u( ) + u ( ) d v( ) ; (3 ) d u( ) v( ) = v( ) d u ( ) - u( )d v( ) v ( ) (4 ) d( fg( ) ) = f ( u) g ( ) d, u = g( ). ; ( 4 ) y = ( fg) ( ), d y = ( fg) ( ) d, y = f ( u), d y = f ( u) d u, u d y,.. 3 f ( ), f d d.f ( ), f d y = f ( ) ( ) d. 4 ( ) :, ( ) y = f ( ) f( +)f ( ) + f ( ). ( 4 3) si ; t ; l( + ) ; e +. y y = f ( ),, = -, f ( ) f ( ), y d y f ( ) f ( ) f( ). y, y + d y + d y, ( + d ) ( y + d y) - y = d y + yd + d d y, d y yd d d y, 3

d( y) = d y + yd.,,.,,.,,,. d u, d u,, d( u )? d u u ; d u u, ( d u ) ; d( u ) u.,.? u ( ) =, d u = d ; d u = 4 d ; d( u ) = 4 3 d.? y = f( u )u, d y = f ( u )d u. u u = ( ), d u = ( ) d. y = ( f ) ( ) d y = f (( ) ) ( ) d = f ( u) d u. u ( ), d y = f ( u) d u,... u = ( ), d u = ( ) d.u, y u u = ( ), y d y = d( d y) d = d( f ( ( ) ) ( ) d ) = ( f (( ) ) ( ) ) d y = f ( u )d u. ( 4 4) = [ f (( ) ) ( ( ) ) + f (( ) ) ( ) ] d 33

= f ( ( ) ) ( ( ) d ) + f (( ) ) ( ) d = f ( u) d u + f ( u) d u. ( 4 5) (4 4 ) (4 5 ) f ( u ) d u, u. u,,. d rcsi f ( ) = rcsi - : :. f ( ) = f ( ) = - d rcsi A, - + A + A. 4 - -,,, -, - ( ). = -, - d. A - ( A > ), 8. (4 3 ) 34 f ( ) = +, =, = y, + y + ( + ) y = + = y.

A + = A ( 4 ) : + A A + = A + 4 8 = 4 3 4 3 - A A -. - 43 3 = 99 7. ( 4 8 = 99 697 56) d 3 u, y = l u, d y., d y = ( l u) d = u u d, d y = d( d y) =, u d = u u - u d. u u d y = d u u, d y = d u d u = - u d u + u d u = ud u - d u u. u = u d, d u = u d,. 4 f ( ), f ( +) - f ( ) = A + B + o( ), ( 4 6) A, B. f( )? f ( + ) - f ( ) = A + o() f ( ). f( + ) - f ( ) = A + B + o( ) f( ),. (4 6 ), f ( ). 35

=, f ( ) = (4 6 )( A = B = ).. 3 si,,, =. f() - f ( ) = 3 si = o( ), f ( ) 3 si f ( ) = 3 si - cos,,, =. f ( ) - f ( ) = 3 si - cos, =, cos, f( ) =, f( ), f ( + ) - f ( ) = f ( ) + f ( ) ( ) + o( ).! f( ) ( 4 6),. 5 T : T = l g, l (), g = 9 8 m s, 5 s, l = m? l T l = l, l = l g l gl = ( m), g = 9 8 ( m s ), l gl T 36 = 9 8 5 3 46

= 3( m). 5 (6 ) 6 m, 55(5-3),, 5, 3 mm,.,,. 5-3 y = + c + d, : ( ) y = ( c + d) c d ; ( 7 ) ( ) y ( ) = ( - ) +! c - ( c + d) + c d ( ). = : ( ) f ( ) = 3 ; ( ) f ( ) = l -. ( ) ; ( ) f + ( )f - ( ). 3 ( ),,,, ; ( ),,,,. ( ) f ( ) = =, - i = i.. 37

( ) ( ) =,,. 4:,,, = ; ( ), ( ), ; ( ), ; (3 ),. 5,, ;, : ( ) f = +, f,, () ; ( ) f = +,,, f () ; (3 ) f =, f,, () ; (4 ) f =,,, f (, ( ), ( ) =, = ). 6 ( ), f( ) = - ( ), f - ( )f + ( ). f ( )? ( f + ( ) = ( ), f - ( ) = - ( ), ( ) =, f ( ).) 7 f, : ( ) y = f (e ) e f ( ) ( y= e f ( ) ( f (e ) e + f (e ) f ( ) ) ) ; ( ) y = f ( f ( f( ) ) ) ( y= f ( f ( f ( ) ) ) f ( f( ) ) f ( ) ). 8,, y : ( ) y = ( ( ) ) + (( ) ) ( y = ( )( ) + ( )( ) ( ) y = rct ( ) ( ) ( ( ( ) ) + ( ( ) ) ( y = ( )( ) - ( ) ( ) ( ) + ( ) (3 ) y = log ( ) ( ) (, >, ) ( ) + ( )) ) ; ( ( ) + ( )) ) ; y = ( ) ( ) l ( ) - ( )( )l ( ) ( ) ( ) l ( ) 9 f ij ( ) ( i, j =,,, ), : 38.

d d f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) F ( ) : ( ) F( ) = - - 3 3 - - 3 + D( ) =, D( ) = ( - ) ( ) {,,, }, ( ) = D ( ) = ( - ) ( ) ( ) F ( ) = d d = ( - ) ( ) = k = = k = = k = f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f k ( ) f k ( ) f k ( ) f ( ) f ( ) f ( ) ; ( ) F( ) = f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) ( ) f ( ),, -,. ( )f ( ) ( ),, 3 3 6 f ( ) ( ) f ( ) ( )f ( ) ( ) k = ( - ) ( ) f ( ) ( ) f k ( k ) ( )f ( ) ( ) f ( ) ( ) f k ( k ) ( )f ( ) ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f k ( ) f k ( ) f k ( ) f ( ) f ( ) f ( ) - - 3 3 - - 3 +. 39,.

= ( ) - 3 3 - - 3 + = 3 ( + 5). F ( ) = d d + - 3 3 6 - - 3 + = 6. + - - 3 3 f ( ): ( A) ( ) f ( ) = si (cos )cos( si ) ; ( ) f ( ) = si f( ) = + e,,, = =, f =? 3 f( ) = si,,, f ( ) = c. 4, : c, +, > c. si si ( ) f, U ( )f( ) >, f ( ) > ; ( ) f [ -, ], f ( ), f ( ) =. 5 : ( ) f ( ) = - 3 + ; ( ) f( ) = - 3. 6 f ( ) =, f( ) =, = g( ), f( ) = g ( ). 7 f ( ) = si( mrcsi ), : ( ) f ( ) 4.

( ) f ( ) ( ). ( - ) f ( ) - f ( ) + m f( ) = ; f ( ): ( ) f ( ) = ( ) f ( ) = si (B) + ( + + ) ; - si - si f ( ) = l =, f =? 3 f ( ) =, f ( ) f ( ) =? 4 f( ) = + si,. =, f ( ) =,,, =. : f ( k) ( ) ( k =,,, ), f ( ) ( ) =, f ( ) ( ) =. 5 = t + t, y = 5 t + 4 t t y = f( )t =. 6. f ( )(, + ).: ( ) f( ) + + f ( )? ( ) f ( ) f ( )? + + 7 f (, ),, { }, { y }, : = y < < < y <, =,, =, f ( y ) - f( ) = f ( y - ). 4

( Rolle)f : ( ) [, ]; ( ) (, ); (3 ) f ( ) = f ( ), (, ), f ( ) =.,.. ( 8). ( Lgrge)f : ( ) [, ]; ( ) (, ), (, ), f ( ) = f( ) - f ( ) -,,. 3 f U( ), U ( ), ( ) f ; ( ) f ( ) = f ( ), f ( ). 4. 4

( ) ( ) f ( ) I, f ( ) I () f ( ) (). ( ) () f (, ), f (, ) () : (). (i) (, ), f ( )( f ( )) ; (ii) (, ) f ( ). (3 ) ( ): f I, f ( ) > ( f ( ) < ), f I f ( ) (, ),, f ( ) [, )? >, f( ) > f ( )., (, ), < < <, f ( ) (, ) f ( ) < f ( ) < f ( ), +, f( ), f ( ) = + f ( )f ( ) < f ( ), f ( ) < f( )..,? ( ), f ( ) U( ),. f ( ), f ( ) =,,, =, f ( ) U ( ), f ( ) =, f ( ) =, f ( ), f ( ) =,., f ( ),. ( ),. f ( ) U + ( ) ( U -, ( ) ), U + ( ) ( U - ( ) ), + (i) f + ( ) ( f - ( ) ) ; f ( ) - f ( ) 43

(ii) f + ( ) = f ( ) ( f - ( ) = f ( ) ). + - (3 ) f( )U( ), U ( ), f ( ), f ( )., f ( ) = f ( ) U( ), U ( ), si,,, =. f ( ) = - cos + si,. f ( ), f ( ) =. :,. f ( ) f ( ) (, ), [, ], f ( ), f( ) = f ( ), : (, ) f ( ) < f ( ) = f( )., (, ), f ( ) f ( ) = f ( ), [, ], [, ],,, < <, < <, f ( ) = f ( ) - f ( ) -, f ( ) = f ( ) - f( ), - f ( ). f ( ) [, + ), (, + ), f ( ) <. : (, + ), f ( ) > c >, (, + ) f ( ) =. f ( ) > c f ( ) > ( < < + ),? f ( ) <, >, f ( ) >, f ( ),, f () =. >, [, ],, < <, f ( ) - f ( ) = f () ( - ). f ( ) = f ( ) + f ( ) ( - ) > f ( ) + c( - ). c >, 44 f ( ) > f ( ) + c( - ) >.

>, f( ) >. f ( ) > c >, f ( ) [, + ) ; [, ],, < <,. f ( ) f ( ) >,., f( ) = rct -,, +, f ( ) = - <, f ( ) = >, f ( ),, + +, f ( ) =. 3 f ( t). f( t) = t si t, < t,, t =,,, < <, f ( ) - f ( ) - +, +, = si = f () = si - cos. + si =, si + =, cos =, + cos =. + cos +,. :, f,,. cos =. + cos + =, ; + cos 4 + =,., + +. f ( ) = l + = l +, f ( ) = l + + - + = l + - + 45

= l ( + ) - l - = + + - + < <, l y, +. f ( ) > f ( )R., + - + =, + = e f ( ). + +. 5 F( ), +,. : F( ) = F ( ) = = F ( - ) ( ) =, F ( ) ( ) >,, +, F( ) >,, +. =, : F( ) =, F( ), +, F ( ) >,, +, F( ) >,, +,., F ( ) F ( - ) ( ) >,, +, ( ) =, F ( - ) ( ) >,, +,.6., +,, F ( - ) ( t), < <, F ( - ) ( ) - F ( - ) ( ) = F ( ) ( ) ( - ) >. F ( - ) ( ) =, >, F ( - ) ( ) >. -, F ( ) >,, +.F ( ),, 6 F( ) = F( ) - F( ) = F ( ) ( - ), < <, F( ) >,, +. e > + + >. F( ) = e - - - 46, >. F ( ) = e - -, F ( ) = e -, F( ) = e >,

5 F( ) > >, F( ) = F ( ) = F ( ) =, e > + + >. 5(4 ) F( ) = e - - - F( ), e > +., >.F( ) =, e e F ( ) = e - - >, F( ) > F( ) =, >, > + + >. < + + <,.. 7 :., + f ( ) -. f ( ),, + f ( ) f ( ) = f +, f ( ) = f - -, f ( ). f ( ). f ( ), f ( ) ; f( ),, f ( ) = f ( ), f ( ). f ( ).f ( ), U - ( ), U + ( ), f( ), + -, f ( ) f ( ), f( )U - ( )U + ( ) 47

f - ( ) = f ( - ), f + ( ) = f ( + ), f ( - )f ( + ), f + ( ) f - ( ), f ( ).. 8, f ( ) = - +. f ( ) ( ) = -,. : f ( ), f ( ) = f ( ) =., f( ) =, f ( ) = f( ) =,,, < <, f () =, f ( ) = f ( ) =. ( - ) ( + ) =,. f ( ) =,. f ( ) ( ) =. f ( ) = ( - ) - ( + ) + ( - ) ( + ) - = ( - ) - ( + ) -, f ( ) = =, = ( - ). f ( ) = f ( ) = f ( - ) =,, ( ), ( ), - < ( ) < < ( ) <, f ( ) f ( ) =, f ( ) = ; f ( ) = P ( ) ( - ) - ( + ) -, P ( ), f ( ) = -. =,,.. f ( k ) ( ) =, k < k ( k) < ( k) < < ( k) k, - k. f ( k ) ( ) = P k ( ) ( - ) - k + - k,, f ( k + ) ( ) = k + ( k + ) i - < ( k + ) < ( k ) < ( k + ) < < ( k ) k k + i =, < ( k + ) k + <, f ( k + ) ( ) = P k + ( ) ( - ) - ( k + ) ( + ) - ( k + ), P k + ( )k +, f ( k + ) k = -, f ( ) ( ) =. (4 ) ( ) - ( k + ). 8. S ( ), f ( ),, f( ),, f ( ), S( ). 48

<, S ( ) = f( ) f( ) f ( ) S ( ),,,, S ( ) = S ( ) =,,,, S ( ) =. 9 S ( ) = f( ) f ( ) f ( ), = f( ) - f ( ) - ( - ) f ( ), f ( ) - f ( ) = ( - ) f ( ).. f,, f. f,. f,, f ( + ), f ( - ), f ( - )f ( )f ( + ), f ( - ) = f ( + ).( ) 4. : t > si,,., si >, f( ) = si t - >,,. f ( ) = si + si sec -, f ( ) = cos + sec + sec t - = cos - sec + t sec >, f ( ) = f ( ) =, f ( ) >,,, 5 f ( ) >,,, t > si,,. 49

f g : ( ), ; ( ), ; (3 ) f ( )g ( ); (4 ) g( )g( ),,, f ( ) f ( ) - f ( ) = g ( ) g ( ) - g( ). f g : ( ) f ( ) = g( ) = ; ( ) U ( ), g ( ) ; (3 ) f ( ) = A ( A, ), g ( ) f( ) g( ) = f ( ) g ( ) = A. ( +, -,,.) 3 ( ) + f g : f ( ) = + g( ) = ; ( ) U + ( ), g ( ); (3 ) + f ( ) = A ( A,, ), g ( ) + f( ) g( ) = + f ( ) g ( ) = A. (, -,, ). 4,,,, -. 5 u( ) = e l u ( ) u( ) >

: f ( )g( ),,, f ( ) = f ( ) - f( ) -, g () = g( ) - g( ) - g ( ), g( ) - g( ),?? f ( ) f ( ) - f ( ) = g ( ) g ( ) - g( ).., f g,.,,,, f ( ) = f( ) - f ( ) -, g ( ) = g( ) - g( ) -,.?? ( ) :,,,.. ( ) : F( ) = f( ) - f( ) - f( ) - f ( ) - F( ) = f( ) - f( ) - f ( ) - f ( ) g( ) - g( ) - ; g( ) - g( )., g ( ) =, ; f ( ) = f( ),.,,. (3 ) : 5

(i), f ( ) = f ( ) =. 8. (ii)., f () f ( ),,, f ( ), f ( ),.,. (iii), f, f g.. f, g f ( ) = g( ) =, f ( ) g( ) = f( ) - f( ) g( ) - g( ) = f () g ( ), ;. 3 5. f ( ) g ( ),,,,,, f ( ) - f ( ) g () = g( ) - g( ) f (). f ( )g ( ) g( ) g( ),,. : F( ) = f( ) g( ) - g( ) - g( ) f ( ) - f ( ), F( ) = f( ) g( ) - g( ) - g( ) f ( ) - f ( ), F( ) = f ( ) g( ) - f( ) g( ) - g( ) f ( ) + g( ) f( ) = f ( ) g( ) - g( ) f( ), F( ) = f ( ) g( ) - f ( ) g( ) - g( ) f ( ) + g( ) f ( ) = f ( ) g( ) - g( ) f( ), F( ) = F ( ) ; F ( ),,,., 5

,, F ( ) =, f ( ) g ( ) - g( ) - g ( ) f ( ) - f( ) =. f ( ), g ( ), g ( ) - g( ), g () ( f ( ) = ), f ( ) f ( ) - f ( ) = g ( ) g ( ) - g( ),.. f ( ),,,, > >,,. : f ( ) = + f ().,,, f () = + f ( ).f ( ) :. f () f ( ) - f ( ) =, - f ( ), g( ) =,, : f, g,,, ; < <, g( ) g( ), g ( ) =.,,, f () f ( ) - f ( ) =. - + f ( ) f( ) - f ( ) =, -,,,,,, f ( ) - f ( ) - = f ( ). f ( ) = + f (). 3 f ( ),,,,,, f ( ) - f + + f ( ) = - 4 f ( )..F, G 53

F( ) = f ( ) - f + + f ( ), G( ) = - 4 F( ) = G( ) = ;, F, G.,. F( ) = f( ) - f + F( ) = G( ) =, F( ) = f ( ) - f + f ( ), G( ) = F ( ) = f ( ) - f + + - 4,,,, + f ( ), G( ) = -, G ( ) = -, F( ), G( ),,,, G ( ) G ( ), F ( ), G ( ),,,, f ( F( ) - F( ) ) - f + G( ) - G( ) = - +, [, ] f ( ), +,,, f ( ) - f + - f ( ) - f, + 54 + = F( ) = f + - F + f ( ) - f + - + + f ( ) = - - F( ) = f( ) - f - f( ),, + 4 +. = f ( ), f ( )., + f ( ). F( ),, + F + - F( ) 4,,

, + -, + - = f + - - - f f( ),,, f ( ) - f + - = f ( ) 4 + f( ).. 4 : l ( ) > ; ( ) + (3 ) + l >. + e c c >, > ; ( ),, l = + + - = + =. ( ),,, + e c - = + ce c, > - - = + c e c + (3 ) + e c e c e c + e c. + = =, + e c + l l = = + + e c =. - + = = = + ( ) ( ) : >, p >,>, +.! =. c e c - =. 55

l p =, + l p. 5 - si + =, +. <,,. =,.,, f ( ) = -, g( ) = si,,,, < < <, - si = - - si - - = y < y <, - cos = y cos - y - si < si = y = si y = = - cos, <,,, y si y. y si y < =,,.,, t = -, t,,,, - si - si = t - t si t <. <. (3 ) 3. f. : 56 f ( + h) + f ( - h) - f ( ) h h = f ( ).

. : F( ) = f ( + ) + f ( - ),. f ( + h ) + f( - h) - f( ) h = F( h) - F( ) h -, 6. f. : h,, < <, f ( + h) + f( - h) - f( ) h (h > ) : F( ) = f ( + ) + f ( - ) - f( ), G( ) =, F( ) = G( ) =,, = = f ( + h ) + f ( - h) f( + h) + f ( - h ) - f ( ) h F( h) - F( ) h = F ( h ) h < < = f ( + h) - f ( - h) h., h. F ( ) = f ( + ) - f ( - ), F ( ) =,, h, f ( + h) - f ( - h) h = F ( h) - F ( ) h = F ( h ) < < = f ( + h) + f ( - h ) =,. h <... 57

3 ( Peo) ( Tylor) : f ( ), f ( ) = f ( ) + f ( ) - + f ( ) - +! + f ( ) ( ) -! + o -. ( 3.) f,,, +,,,,,, f ( ) = f ( ) + f ( ) - + f ( ) - +! + f ( ) ( ) -! + f ( + ) () + -, ( 3.) +! = + -, < <. R ( ) = f ( + ) () + - ( 3.3) +!. 3( Mcluri) (3. ) =, f ( ) = f ( ) + f ( ) + f ( ) + + f ( ) ( )!! + f ( + ) ( ) +! + < <. ( 3.4) : 58

() e = + + + +!! + < <, e + +!, -, + ; () si = - 3 3! + 5 5! + + ( - ) m - m - m -! + ( - ) m < <, cos m + m +!, -, + ; (3) cos = -! + 4 4! + + ( - ) m m m! + - m + cos m + m +!, < <, -, + ; (4) l + = - + 3 3 + + - - + - + < <, > - ; (5) ( + ) = + + -! + - - +! < <, > - ; (6) - = + + + + + < <, <. + + + - - + + +! + - - +, + - +,, : f ( + h ) = f( ) + Ah + Bh + Ch 3 + Dh 4 +, ( 3.5).,, f ( + h) = f ( ) + f ( ) h + f ( ) h + + f ( ) ( )!! h +.,.? 59

, f ( ) ; f ( ) f ( ) ( ), + f ( + ) ( ) ;., o -, ; +,., ;., ( 4, ) ;.,. : + - - ( ) > ; ( ) 3 + + + - -. ( ) o, = + l + l! + o, - = - l + l! + o, + - - ( ), + = l + o = l. o + = + = + + - + o, - = - = - + - + o, 6

+ + - - = - 4 3 + o 3., 3 + + + - - = - 4 + o( ) = - 4. f ( ) +,. f ( ) +,, f ( ) = f ( ) + f ( ) - + + f ( ) ( )! + f ( + ) f ( + ) U ( + - ) +! - + -, < <. ( ), M >, >,,, f ( + ) ( ) M. R ( ) = f ( + ) ( + - ) +! - + R ( ) - M +! -, R ( ) = o -. M +! - +,, f ( + ) ( ), R ( ) = o -,. 3 s, d, (6 - ). d s sa d + B, A, B,. r, s., d = rsi = r - 6 3 + 5,= cos, < < ; = rsi = r - 48 3 + 3 84 5,= cos,< < ; 6

Ad + B= r A + B - 6 A + 48 B 3 + s = r, A, B: A + 3 84 B 5. A + B =, 6 A + 48 B =, A = - 3, B = 8 3, s+ - d 3.. 4 : f ( ),,,,,, f ( ) - f + + f ( ) = - 4 f ( ). 6 - f ()f( ),, +,., f ( ), f ( ) f( ) = f f( ) = f + +, f ( ) - f + f + + f + + f ( )f ( ), = +,. = + : - - + f ( )! + f ( )! - -, + + f( ) = f ( ) + f ( ) - 4 f ( ) f ( ) + f ( ) f ( )., f ( ),, 6 f ( ) - f f ( ) = f ( ) + f ( ), + + f ( ) = - 4 < <,, < < +. f ( )..

3,. 5 f ( ),,, f ( ), f ( )., f ( ). f ( ), f( ) f ( ). f ( ), f ( )f ( ).,,, f ( ), f ( ) f ( ) = f( ) - f ( ) + f ( ), <! <, f( ) = f ( ) + f ( ) - + f ( )! f ( ) = f ( ) - f( ) - f ( ), f ( ), -, < <, f ( ) - + f ( ). f ( ) + + - = + - + 4, f ( ). :,,. 4() ( ) ( ) f, f, f ( ) =. f. ( ) f, U ;. (i) -, f ( ),, + f ( ), 63

f ; (ii) -, f ( ),, + f ( ), f. (3 ) f U ;, =, f ( ) =. (i) f ( ) <, f ; (ii) f ( ) >, f. (4) f U ; -,, f ( k ) ( ) = k =,,, -, f ( ) ( ), (i), f, f ( ) ( ) <, f ( ) ( ) > ; (ii), f. ( )f,, (), f, f ( ) [, ] (). (), (). f ( ),,,?. f( ) = - + si,,, =, f ( ) =. f ( ) = - + si + cos,,, =,, f ( ) cos, cos =,, f ( ),.,. f ( ) I, I. 64

f ( ) (), f( ) f ()?.f( )I,,. f ( ) f, I, f ( ) < f ( ), <. f ( ),,,, f,. < <, (i) f ( ) < f ( ), ; ( ii) =, f I, f,, U - ( ), f ( ), I.,.., f, f I. I,. ( ). e < - <,. e - < <,, f ( ) = e -, <. f ( ) = e - <, f ( ) = e - - e = - e, f ( ) =. = (),. <, f ( ) >, > f ( ) <, f ( ) =, f ( ), e - < f ( ) =, <,. e < - <,. 65

O, P, R, Q(6 - )., :. P R Q, P,, R,, Q c, d, PR = + -, RQ = d + c -, l( ) = PR + RQ 6 - = + - + d + c -.,, l( ), R. l ( ) = 6 - - + -,. - + - - c - d + c - =,, c - d + c - =, =, R l ( ), l ( ) = =,. ( ). 3 G,, F, (6-3 )., F? G - Fsi, Fcos = G - Fsi, F( ) = G cos + si. 6-3 66

F ( ) = G si - cos, < cos + si, = rct. < F ( ) <,> F () >,., F rct, F. = rct. 4 f ( ) = - ( ), ( ) =, ( ). f ( )?,? f( ) - f( ) = - ( ), ( ) >,, U( ) ( ) >., f ( ) - f ( ) U ( ),., U ( ), f ( ) - f ( ) >, f ( )., ( ) <, f( ). 5, R,,.,. r, H(6-4), H = R - r cot, S ( r) = r + r R - r cot, r, R.. S ( r) = 4r + R - r cot = Rcot + 4r - cot = 4t - t r + Rcot - cot =, r = r = Rcot cot - = R - t 6-4 < t <, r R, R, R. 67

S ( r) >, r < r, <, r > r, r (),. t, r R, r,, R. r, R, S ( r) >, S ( r), R,., S ( r ) H =, 6-5 ). sup r, R (46 ) S( r ) = R.. y = p ( y P, p Y - p = - = p p X -. Q, : Y - p = - p X -, 6-5 Y = - p, X = + p, Y = - p + p. l ( ) = PQ = p + p + p + + p - = 8 p + p + 6 p3 + p4 < < +. l ( ) = 8 p - 6 p3 - p4 3 =, 68

= p, y = p, (),. = p, y = -. p 5 ( ) f I,, I, < <, f I ( ). f + - f + - f( ) f + - f + - f,. ( ) f I < < 3 f I.. f( ) - f( ) - f( 3 ) - f( ) 3 -, f ( ) - f ( ) f( 3 ) - f( ) - 3 - ( ) f I, fi, f I (3 ) 3 f I, I,, f I. f( ) f ( ) + f ( ) -, (4 ) 4 f I, f ( ), I, f I. 3 f,, i,, i > i =,,,, i i = =, f i i i = i f( i ). i = 69

, (8 ( ) ),. 4 y = f ( ), f( ),,,, f( ) y = f( ). f,, f ( ), f ( ) =. y = f ( ) f, U ( ). U + ( )U - ( ) f ( ),, f( ) y = f ( )., f( ) y = f ( ), f ( ),,, f ( ). =,,? y = 3, f ( ) = - 9-5 3. <, f ( ) >, y = f ( ) ; >, f ( ) <, ;, (, ). I? 5 5 : f, f I. f I,.,,, y, l + yl y > l, 7 l + + y = l l = - + y l + y +. <, + y + y + y y

, > + y l + y + y l y, + y l + + y > l + yl y, l + yl y > + y l + y +. f ( ), l f ( ) f( )f ( ) - f ( ). 4, l f ( ) l f ( ) = f ( ) f ( ) l f ( ) l f ( ). = f( ) f ( ) - f ( ) f ( ) f( ) f ( ) - f ( ) f ( ) f ( ) - f ( ), f ( ),. 3 f ( ) I, : I, I, f( ).,,, = + -, f ( ) f ( ) + - f ( )m f( ), f ( ) = G, f ( ). f ( ),.d = +,,,, d, d = + f ( d) = f f ( ). +, f ( ) + f ( ) f ( ) f ( ) f( d ) - G, + G, 4 f I,, I, f, ( Lipschitz), L >,,,, 7

f ( ) - f ( ) L -., I, I, I,, c, d, < < < < c < d.,,,, <, f( ) - f ( ) - L = m f ( ) - f ( ) ( d) - f ( c) f - d - c f ( ) - f ( ) -, f ( d) - f ( c) d - c f ( ) - f ( ) L, -,,. L,,, f, : L >, f( ) - f( ) L -,,,. : I, f ( )I.,. 5 ( ), +, : ( ),. : f I : I < < 3, f( ) - f( ) f( 3 ) - f( ), - 3 - f ( ) 3 - + f( ) - 3 + f ( 3 ) -. ( ). < < < 3, 3 < <, - 3 3 + - 3 + - 3. 3 3 - + - 3 + 3 -, 7

3 - + - 3 + 3 -,. :, ( ). (53 ) 7. : ( ) f I I < < 3, = f( ) f( ) 3 f( 3 ) ( ) f >. ; f I I < < 3 f( ) - f( ) - f( 3 ) - f( ) 3 -, f ( ) 3 - + f( ) - 3 + f ( 3 ) -. 8. : ( ) i > i =,,,, + + + + + + ( ) i, i > i =,,,, i = p >, q >, p + q =. i i p i i = p q i i = q,. ( ) l = - l + + + <, l, l i = l, i = + + +. 73

- l, l + + + l + + l, + + +, + + +. ( ) p >, p = p p - p -,, +, p. f( ) = p, i = i = i q - i q i q j j = i i i = p q i q j j = i = i = i q - i p i, q i = j j = i i i = q j j = p q j j = = q j j = p, i = p p p - p q i i p i i = i =,,, p p i i = q j j = p i i = p i i = p, q - p = q, q i i =. p - = p q q. 74

6 (67) : ( ) ; ( ) : ; (3 ) : ; (4 ) () ; (5 ) ; (6 ),. ( ) f ( ),,,, f ( )f ( ), f ( ) f ( ) <, f ( ) =,. 6-6 6-6 f ( ) >, f ( ) >, f ( ) <, f ( ) >. P, f( ) P, f( ). = - < f( ) f ( ) - f ( ) - <, f ( ) <.,, = - - ) f( ) f( ) - f ( ), 75

- : = - - - - f ( - ) f ( ) - f( - ), =, 3,. ( 6.) m = if =, - f ( ) m f ( ).( 4.) 3f ( ),, f ( )f ( ), f ( ) f( ) <. ( ) :, 6-7 (i) f ( ) <, f ( ) >, f ( ) >, f ( ) <, = ; (ii) f ( ) >, f ( ) >, f( ) <, f ( ) >, = ; (iii) f ( ) >, f ( ) <, f( ) <, f( ) >, = ; (iv) f ( ) <, f ( ) <, f ( ) >, f ( ) <, =. 6-7 ( )( 4). f ( )f ( ) >,, f ( ) f ( ) <,. 76

( ) : = - - f ( - ), =,,. ( 6.) f ( - ) (3 ) : - f ( ) m, m = if, f ( ). y = si + 3 si 3. -, +.,,,. y= cos + cos 3 = cos + 4cos 3-3cos = cos cos - =, = 4,, 3.6 -. 4 6 -, 4 4 4,, 3 4 3 4 3 4, y 3 3 3 y + - + - y= - si 5-6si, =, rcsi 6 -. 5 6,- rcsi 5 6, y=, 77

6 -, rcsi 5 6 rcsi 5 6 rcsi - rcsi 5 6, 5 6 - rcsi 5 6 - rcsi 5 6, y 4 7 3 4 7 3 y - + - = rcsi 5 6,- rcsi 5 6 A, B, C, D, E (6-8 ).., y = 6-8 + e.. y= e - +, =, -.6-3. 6-3 -, - -,,, + y e 4 e y + - + 78 y= e 5 + 4,

= - 5, -, - 5, -, (, + ) 5. y = + e =, y - = + e - y = + 3. = e - + e = 3, + e + = +, + e =, - 6-9 y ( ). 6-9, B, E, C.. 3 3 + y 3-3 y = >,., y = t, = ( t), y( t) t. 3 t + t 3, y = 3 t + t 3 ; 79

t-., y, y =. d d t = 3 - t3, d y 3 t - t3 =, + t 3 d t + t 3 t = 3 ( t), t =, 3 y( t). ( t), y( t) 6-4. 6-4 t -, - -,, 3 3 3,, 3 3 3, + + - 3 4 3 4 3 4 3 3 3 3 + + + - - - y - + 3 3 3 3 3 4 3 4 3 4 y - - + + + -.t-,.. k = y = t = -, t- = y + = t- + y =, 6-3 t + t + t 3 = -, : +,. -,,. 4 f ( ),, 6 - f ( ) <, f ( ) >, f ( ) >, f ( ) >,,,, f ( ) =. = - f ( ) f ( ) - f ( ) -, 8

+ = - { }.. - f f( ) - f, =,, =. { },,, = c. + = -, f ( ), - f f ( ) - f - c f ( c) f ( ) - f( c) =, f ( c) =. f ( ),, c =, =. 5 f ( ),, + - M m -, f ( ) =, M f ( ),, m = if < < f ( )., = - - = - + f ( ) f ( ), f( ) f ( ). ( 6.3) f ( ) = f ( ) = f ( ) + f ( ) - + f ( c ) -,< c <, ( 6.4) (6.4 ) ( 6.3),, + = - - = f ( c ) f ( ) f ( ) f ( ), -. - + = - + f ( ) f ( )., f( ), = f () = f( ) + f ( ) - + f ( c ) -, < c <, 8

+ - = +.. f ( c ) f ( ) -, - M m -., - <, ( 58 ). : f( ),, +, ( ) +, f ( ) =. f ( ) = - f ( ) = - f ( ), f ( ), f ( ),.( ) :,, f () =, f ( ),. 3. f,,,, >.,, - F( ) = f ( ) f( ) f ( ), G( ) = = f ( ) - f ().,,. 4. f,,,, f ( ) = f ( ) + - f ( ) + f ( ) - - 3 f( ). F( ) = f ( ) - f ( ) - G( ) = - 3, F( ) =, G( ) =. F ( ) = f ( ) - - f ( ) + f ( ), f ( ) + f ( ) - -,, f ( ), F ( ) = - - f( ), F ( ) =, F ( ) =. 8

, F, G,, < <, F( ) G( ) F( ) - F( ) = G( ) - G( ) = F ( ). 3 -,,, < <, F ( ) = F ( ) - F ( ) 3-3 - - = F () 6 - = - f( ). F( ) G( ) = - f( ), f ( ) = f ( ) + - f ( ) + f ( ) - - 3 f( ). []. f ( ) - f( ) - L. - ( ) = f( ) - f ( ) - - f ( ) + f ( ) + - 3 L =, f ( ) + f ( ) + - 3 L, ( ) = ( ),, ( ), ( ) =, < <., ( ),, < <, ( ) =. L = f( ). 6.,,,, : ( ) f ( ) = ; f( ) = + + + ( ) + f( ) = m,,,.. ( ) l f ( ) = l + + +,, l f( ), + + + l f ( ) = l = l + l + + l + + + = l + l + + l 83

, = l. + + + =. ( ) A = m,,,, < k k =,,,, A f ( ) = A A A + A + + A < f ( )A + + + +,, f U f ( ) = A. +, >. = A, 8. h >, f U ; h +, f ( + ) ( ), : h = ; h +. f U f( + h) = f ( ) + f ( ) h + + f ( ) ( )! + f ( + ) ( + h) h +, < <. +! ; h + f( + h) = f ( ) + f ( ) h + + f ( + ) ( ) + +! h + f ( + ) ( + h) +! h h +, < <. (6 - ), f ( + ) ( ), + h, f ( + ) ( + h) - f ( + ) ( ) = f ( + ) ( + h)h, < <, f ( + ), 84 ( + h ) = f ( + ) ( ) + f ( + ) ( + h)h. f ( + h) = f ( ) + f ( ) h + + f ( ) ( ) h + f ( + ) ( ) +! +! h + f ( + ) ( + h ) h +, (6 - ) +!

(6 - ), ( 6 - ), f ( + ) ( + h) = + f ( + ) h, : f ( + ) f ( + ) = h +. ( + h)., 9. k >, k, rct - k =. f ( ) = rct - k k >. f ( ) = -, X >, > X, f( ) <. + f ( ) = < k <, >, U + - k + k - + k, ;, f ( ) >, f ( ) U + ;, >, f ( ) > f ( ) =. >, f ( ) <,,,,, f ( ) =,>., < k < rct - k =. k, F( ) = k - rct, F ( ) = k - rct - k =., < k <. + >, F( ) =, F( ) > >, rct - k =. f,, f ( ) = f ( ) =.,, f f f + + + f ( ) 4 f ( ) - f( ). -,, = f ( ) + f ( ) = f ( ) + f ( ) - - + f ( )! + f ( )! - -, < < +,, + < <. 85

f ( ) = f ( ) =,, f( ) - f ( ) - f ( ) = m f ( ), f ( ), 8 f ( ) + f ( ). f ( ) 4 f( ) - f ( ). - 3. f,,,, f ( ) M. f ( ) + f ( ) M. f,,, f ( ) =.,,, f ( ). 4. f, +, f ( ) f ( ), f ( ) =. :, + f ( ). F( ), F( ) = e - f( ),, +. F ( ) = e - f ( ) - f ( ), F( ) = e - f( )F( ) =, >. f ( ); f ( ), f ( ),. 5. f( ) f ( ) + f ( ) g( ) - f ( ) =, g( ). : f( ) = f( ) = <, f,. f ( ),, f ( ),,. f ( ), f( ) =., f () >, f( ) = f( ) =, < <., f ( ) =. f ( ) + f ( ) g( ) - f( ) =, f ( ) = f () >,,, f () =. f () =, f ( ),,. 8. : ( ) f, +, f ( ), f ( ), + + 86 f ( ) =. +

( ) f, +, f ( ) f ( ) + + f ( k) + ( ) = k =,,,. ( ), ( ) f ( ),, k > k, + X k,, > X k f ( ) - f ( ) < k. k > X k k + > k +,, k, k + f f ( k + ) - f ( k ) < k. k < k < k +, f ( k ) ( ), ( ) f ( + j) f ( + j) = f ( ) + f ( )! < k, f ( k ) =. f ( ) =. + j =,,, -, j + + f ( - ) ( ) - -! j + f ( ) ( j ) j,! j =,,, -, < j < + j. f ( ), f ( ),, f ( - ) ( ), f ( + j) - f ( ), f ( ) : f( ), f ( ) + + f ( k ) + f ( ) + + f ( ) + ( j ), ( ), f( + j) - f ( ) =, j =,,, -. ( j ), j =,,, -, ( ), k =,,, -.( ) : f ( - ) + ( ) f ( k) + f ( ) + ( ) =. ( ) =, k =,,, -. ( ), 9. f -, +.f -, +, -, +, f ( ) =.. -, +, f ( ),,, f ( ) > ( f ( ) < ), f ( ) ().f ( ), 87

, f ( ),, f ( )., -, +, f ( ) f ( ) + f ( ) -, f ( ) >, + ; f ( ) <, -, f ( )+, f -, +.f, f., -, +, f () =.. >, >. : ( A) + > + >.. f ( ),, f ( ),. : f,., f ( ) f ( )? 3. : f( ) : f ( + ) ( ), R. 4. - 3 <, 3 + + + c =. 5. f ( ), +, + :, +, f( ) = + f ( ), f ( ) =. 6. f ( ), +, : >, f ( ) = f( ) < -. + +. 7. f ( ),, f ( ) =, f ( ) =, f ( ) = f ( ) =,,, f ( ). (B). f, g R, f ( ) > g ( ), f ( ) = g ( ). : 88

>, f( ) > g( ) ; <, f( ) < g( ).. - + < e -, < <. 3. f ( ), +, >, f ( ) > k >, k, f( ) <. f( ), + f ( ) k. 4. f ( ), f ( 4 ) ( ) M. : f ( ) - f ( ) - f( ) + f( ) M - -,. 5. f ( ),,,, f ( ) = f ( ) =. : R,, f ( ) = f (). 6., f ( ) : ( ) f ( ) = ; + ( ) - k < f( ) < k k >. : + f ( ) =. 7. f ( ),, f ( ) = f ( ) =,, f ( ) 8. mi, f( ) = -, 89

,,, =,,,,, > N, = =.,,, =,., >, N >, U ;; : >, N >, m, > N, 3 m - <. S, : ( ) S ; ( ) S, (3 ) U ; S ; S, =... 4 H, ( ), H,. 5 ( ) ; ( ) ; (3 ) ; (4 ) ; ( 5) ; ( 6),, 9

.. S, S? S >, U ; S S = - +, S., - k + k, ( - ) k + -, S., = mi -, +, + k + U ; S, S. S. S, sup S if S S,?, sup S if S S?, sup SS, S. S =, N +, sup S =, S. = sup S S,, S, = ;, sup S S.. f ( ),,,, f( ). f ( ),. U f,, ;, f, H = U ;,,,...,,. f ( ),,,,, U ; M >, tu ;, f ( t) M. H = U ;, 9

, ;,, M = m M i i U k ; k H = U i ; i i H., H,,,, H, f ( ) M k M. ( ).,, M i i =,,, M..,, k, k k,. =,,, k, k k =., U ;., k, k k =. -,. 3,. =., ;,, = U -, N, > N, U ;, ; ( U ; ).,.,,, = sup,= if. =,,,.,,, k, k =, =, k k k k 9

,.. 4. S M., S, S ( S, S ),, =, M.,,, S, S,,.,,, S, S, 3, 3.,,, S, S, - =,, - - - = - - = M - -. =,,. sup S : ( ) S,,,, S.( ) >, =, > - ; S, - S. S, sup S =.., S, S, S.. 5.,,, =,,. :,,,,,, >, U ;, =. H = U ;,,,,., U i ; i i =,,, H,,. > m { i i },, U i = i, i =, i, i,,,,.,, i =, =,,.,,. (68 ) 8.. 93

.S, S, S, U,, S, ; U ; S, H = U ;,,, S. 9..,,, k, k k =, =.... 3. 4.., ( ) :,,,., ( ) : S, S,,,,,,,, S. (3 ) :.,,,,,. (4 ) : f ( ) <, f ( ) >,,, f ( ) <, f ( ) >, f ( ), 94

, f ( ) =. (4 ) f ( ) =..,. U ( ) :,, ;,, f ( ) M, M, H = U ;,,.H,, f. ( ) :, >,,, >, U ;, f( ) - f( ) <. H = U ;,,,. 3.. ( ) :, f ( ),,,, f ( ) >, k, k k f ( k k ) = f ( ),. =,, () :, f,, >, N +,,, - <, f ( ) - f( )., k,,,. 4. E, E. ( ) : f ( ),, f( ) <, f( ) >, E = f( ) >,,, k 95

= if E, f ( ) =. ( ) : E = f( t),,,, sup E =, f( ),.(4 ). f ( ),,,, f ( ).,,, >, f ( ) U ;, H = U ;,,,., H = U i ; i,. f ( ) U ik H i ; i, f ( ),, f ( ),. f,,,, f ( ) >. : c >,,, f( )c.,, f ( ) >,, >, U ;, f( ) > f ( ) H = { U. ;, },,,,,. H = U i ; i c = mi ik ik H f ( i ) >,,, i ik, U i ; i, f ( ) > f ( i ) c. 3 f,, >, f -, +, f,.,, < < <, f ( ) < f ( ).,,, >, f -, +, 96

H = { U ;, },,, H = U i ; i ik H,., U ;, U ;,, <. U ;, U ;, f U ;, f ( )f( ) ; U ;, U ;, V = U ; U ;, V, < <. f ( )f( )f ( ). k >., f ( ),. 4 : f ( ),, f,. S = f,,,. f, S,,, sup S. = sup S, S, S =,, f,. S = f,,,,, S,, = sup S. =. <, >, f ( ) -,+, >, S, = sup S, =. f,. f, >, f -, ; = sup S, f, -, f,. f ( ) 5 f ( ) I, I,. f I..f I, >,, I, - <, f ( ) - f( ). 97

,, k, k - k k, k k =.,,,,,, k, k,, ; f( k ) - f ( k ) k k, f( ), f ( ), f ( ), f ( ),, f ( k ), f( k ),,. = ;,.,, I, f( ).. (7 ) 3. : f ( ) = si, +. + si si =, + 4.. =, f ( ), +, f ( ), f ( ) <, f ( ) >,,, f ( ) =.,, f ( ),, U ;, U ;, f ( ) > (f ( ) < ), H = U ;,,., f ( ) f ( ),. 5. :, f f ( + ) f ( - )., f,,., f,, f ( + ), f ( - ). 98

3,.,,, ;,,.,. A A, A = 3 A =. ( ) A, : (i) >, N, > N, < A + ; k, k > A -, k =,,. (ii) > A, ; < A,. (iii) ( ) A = sup k k. A, : (i) >, N >, > N, > A - ; k, k < A +, k =,,. (ii) < A,, > A,. (iii) 4 ( ) = A ( ) A = if k k. = = A. ( 3.). ( 3.) 99

(3 ) ( ), y : N >, > N y, (4 ), y, y, y. ( 3.3) + y + y, ( 3.4) + y + y. ( 3.5) sup k k if k k ( ), ( ) ( ),? ( ) : sup k k if k k ; (). ( ) :, ;. (3 ) :, (3 4 ) ;, () (5, ).. ( ) sup =, if =?? ( ) sup, if, : sup =, if =. ( )., = = ; = -, if =, =., sup =, ( ) = sup, k, k k =., ;,. =.

: if, if =.. :,. ()., ( 3. ),, k, k, k k =, k k =,. 3 :, y, - y = - y, (3.4 ). 3 (iii), (3.4 ), + y + + y + y. + y - y. + y + - y = - sup k = if k = - y + y - y, yk - y k - y. + y - y =, 4, y, >, y >. y + y. y. =,. >, > N, >.

= if ( 3), y 5 k = k y = sup k k = y y. =. y, y. k, k k =., k, k k =. k k k k =,.,, k k k. k k, k k k = k k, k. (75 ) 3. :,. =. 4. : > =,, =, k, k k =,. =, k k = +,. 6. 7.9. =, >. 3

, A A = if k. A A = sup k k., 7.7 (i) >, N, > N < A + ; (ii) >, k, A - < k. (i), > N, (ii), sup k k A + ; A - < k sup i k i ; g = sup k k, A - sup i k = sup i k k k A +., A = sup k sup i A i < A +, k. = sup i sup i, k > N, k, i, >, N, > N, A - < i A - < k, < A +. 7.7, A = A = sup k k. ( 76 ) k. :. ;,, k k =.. f,, +,. + f ( ) = - f ( ) = -, f ( ) =. : f ( ) f ( ) =, f ( ), 3

F( ) : F( ) = f( ),,,, =,., F( ), M m.m = m =, f ( ),,, f,.m m, M, M,,,, m, f,. 3. f,,,, f ( ) = A. :,, f ( ) = A.. 4. f g I. ( ) I, fg I ; ( ) I, fg I. I, f, g, f, g, fg. 5. f,. :,, f( ), f,..f,, >, >,,,, - <, f( ) - f( ). =,,,, - <, f( ) - f( ).,, k,,, k k =. z =,,,,, k, k,, k k = z, f( k ) - f( k ), f ( z ),, f,. 6. f, +,, c, + f, +. + >, M >, > M, f ( ) - - c =, f( ) - - c =, 4

, > M, f ( ) - f ( ) f ( ) - - c < 3. = f ( ) - - c - f ( ) - - c + - f ( ) - - c + f ( ) - - c + - < 3 + 3 + -, - <, f ( 3 ) - f ( ) <. f ( ), M +, >, >,,, M +, - <, f ( ) - f ( ) <. = mi, 3,,,, +, - <, f( ) - f( ) <. (7 - ),, M, +,, M ;, M, M, +, - <,, M +, (7 - ). f( ), +. ( A). : = + ( - ). +. : k, k k =. 3. :, y, + y = + y. 4. f ( ),, f ( ) <, f ( ) >,,, f ( ) =, < <, f ( ) >. 5. f :,,, k < k <,, y,, f( ) - f( y) < k - y, :,, = f ( ). 5

6... : (B) ( ) rct ( - ) ; ( ) si -.. f ( ),, E E. 3. : E = f ( ) =,,. >,, y, y 4. : > =,,, = y. +. 5.. 6. + - =.,,,. 6

( ) I F ( ) = f( ), F f I. f I,. 3f I f I, f( ) d. F f, C ( ). f( ) d = F( ) + C, (.) 45 ( 9.) : f I, f I, ; (),. 5, : ( ) d = C; ( ) d = d = + C; (3 ) d = + + + C -, > ; (4 ) d = l + C ; (5 ) e d = e + C; (6 ) d = l + C < ; (7 ) cos d = si + C ; 7

(8 ) si d = - cos + C ; (9 ) sec d = t + C; () csc d = - cot + C; () sec t d = sec + C; () csc cot d = - csc + C; (3) d - = rcsi + C = - rccos + C ; (4) d + = rct + C = - rccot + C.,.,. 6: k f ( ) + k f ( ) d = k f ( ) d + k f ( ) d. (.) 7: f ( ) = g(( ) ) ( ), I. g( u ) G( u ), f( )d = g(( ) ) ( ) d =g( u) d u = G( u ) + C = G(( ) ) + C. (.3) ( ),, f ( ) F( ), g( u ), -. g( u) d u = g(( ) ) ( ) d = f ( ) d = F( ) + C = F( - ( u ) ) + C, (.4)??, F( ) f ( )I, f( )I, 8 f( ) d = F( ) + C C.

( 79 8 - ) y = F( ) y = F( ) + C., ( (. ), (.3 ), (.4 ) ),.,??,,., :f ( ). : F ( ), F( )C.,, f ( ) I F ( ) + C ( F ( ) f ( ) I ). 3? + = = -, y = y =, ;,, y =f ( ) d y= f( ) (.5). (.5) ((. ) ) ( C ), y = = y, C C = y - F( ). (.5 ) y = F( ) + C. f, y.,,,. 4 f ( ) d??, ( ). f ( ), f ( ) d, f ( ) d, 9

S ( Sum, ). ;,,.,, f ( ) d. d d f ( ) d = f ( ), d f ( ) d = f( ) d, d,,. : d f ( ) = f ( ) d = f ( ) + C, C.,,. 5? ( ),,. : ) e( ( )d = ) e( d( ) = e ( ) + C, ( ) cos ( )d = cos ( ) d( ) = si ( ) + C, ( ) si ( ) d = si ( ) d( ) = - cos ( ) + C, ( ) ( d = ( ) d( ) = ( ) + C, ( ) ( ) d = ( ) d( ) = - ( ) + C, ( ) ( ) d = d( ) = l ( ) + C, ( ). ( ), = ( t), ( t) t = - ( ). ( t), 85 7 t <, 8 < t <. (3 ), ( )( ),

., J = d + cos. + cos = - cos - cos = - cos si = csc - csc cot, J = csc d - csc cot d = - cot + csc + C. + cos = cos = sec, J = sec d = sec d = t + C., - cos, kkz.,., : t + C. - cot + csc + C = - cos si = = + C - cos si + cos si + cos + C, + C : ( ) 4 + + 4 + d ; ( ) + 5 d. ( ) 4 + + 4 + = + + = + - 4 + + d 4 + = d + d +, - d + = - - rct + C.

( ) + 5 = + 5 + + 3 + 5 + 5, + 5 d = + 3 3 + 5 + 4 5 + 5 3 3 + 7 7 + C., + = t, = t -, d = t - d t, + 5 d = t5 t - d t = t6 - t 5 d t = t7 7 - t6 6 + C = 7 + 7-3 + 6 + C. ( )( + d ), ;, ; + d,,. ( ),, C. : ( ) e d ; ( ) si cos m d ; + d ; (4 ) 5 3 (3 ) rct 3 - d. ( ) e d = - e d = - e + C. ( ) si cos m d = si + m + si - m d = - + m cos + m - - m cos - m + C. (3 ) rct + d = rct 3 + d = rct d rct = rct + C.

(4 ) 5 d 3 - = 3 3 - d 3 3 - = 3 - + 3 3 - d 3 - = 3 t + t d t t = 3 - = 3 d t t + t d t = 3 l t - t + C = 3 l 3 - - 3 - + C. ( ),. 3 : ( ) (3 ) d - - d ; ( ) d + ; -. ( ) = si t, t <, d = cos td t, >, - d = 3 si tcos t cos t d t = - cos t d t = t - si t + C = = t - si tcos t + C rcsi - - + C. ( ) = t t, t <, d = sec td t, d + = sec t t tsec t d t = csc td t = csc t - csc tcot t d t csc t - cot t d csc t - cot t =csc t - cot t = l csc t - cot t + C 3

= l + - + C = l, + + + C. d + = l + + + C, d + = d + = - + d (3 ) = - l + + = l + + + C. - - + - - =, + C - = - si t, - = - cos t, = + - si t, d = - si tcos td t. d - - =, t = - ( ) - ( ) = - si tcos t - si tcos t d t = d t = t + C = rct f( ) d = f (( t) ) ( t) d t ( t) - - + C., ( t). t, ( )( ) t <. 4 d 4 -. = si t, t <, t = rcsi, d = cos td t. 4

d = cos t 4-4si tcos t d t = csc td t = l csc t - cot t + C = l - 4 - + C. = t, t >, t =, d = - t d t. d 4 - = - t t 4 - t d t = - d t t - = - l t + 4 t - + C = - l + 4-85 8,. + C. = t, t > 4, d = - t d t. d 4 - = - d t 4 t - t = - 4 d t - = - 4 l t - 8 + t - t 4 + C t - 8 8-8 = - 4 l - 8 + 4 - + C. 4 - = t, t,, d = - td t 4 - t. d 4 - = - d t 4 - t = 4 t - - t + d t = 4 l t - t + + C 5

: + - = 4 l 4 - - 4 - + + C. d 4 - = d = t, t >, = t - t + d, - = = t 4 - -, + - 4 t +, d = 8 t d t. t + + t + 8 t d t t - 4t t + = d t t - = l t - t + = l + - - + + - + C + C.,,.,,. (88 ). : () d + si = - si cos d, : t - sec + C cos + si + C; (3) csc d = csc + csc cot d, csc + cot csc - csc cot d ; csc - cot (5) d = 4 + 4 4 d ; + (8) 3 8 - d = 4 4 - d 4 ; () cos5 d = cos4 d si ; () d si cos = sec t d, d si ; 6

(3) d e + e - = e e + d ; (6) d + >, = t t (8) 5 sec td t = sec t + sec tt t sec t + t t d t, cos t - si t d t ; - d, = si t si5 td t( ( ) ) ; (9) - 3 d, 6 = t 6 t8 - t d t = - 6 t6 + t 4 + t + + (3) + - d, + = t + + t t - d t; - t t + d t = t t + - t + + d t. t + ( 3 ) u( ) v( ), u ( ) v( )d, u( ) v ( ) d = u( ) v ( ) - u ( ) v( ) d ud v = uv - vd u. (.) uv = u v + uv., 4. 3 R ( ) = P ( ) Q m ( ) = + - + + (.) m + m - + + m. m >, R( ) ; m, R( ).,.,,. 4, 7

( ) ;. 9 94, 5. (. ) u ( )v ( ). 87 3, u = 3, v= l, u = 3, v = l -. (.) 3 l d = 3 d l - = 4 l - - 3 3 l - d = 4 l - + 3 4 4-3 3 l d, 4 3 l d = 4 l - 4 4 + C, 3 l d = 4 6 4l - + C.,, l d = l - + C., l d (u =, v= l ),.? : e - cos d, u = e -, v= cos, u= - e -, v = si, e - cos d = e - si + e - si d, (.3), e - si d. u = si, v= e -, u= cos, v = - e -, e - si d = - e - si + e - cos d. (.4) (.3 ), =,. 8

,, (.3 )(.4),,., A B,, A.,? :, u v., u = e -, v= si, u= - e -, v = - cos, e - si d = - e - cos - e - cos d. (.5) (.5 )(.3), e - cos d = e - si - e - cos - e - cos d,, e - cos d = e - si - cos + C. (.6) 3? ( ) ( ) (3 ) - = A + 3 + B + + - 3 + - + C + D + + E + F + G + 3 ; = A + B + C + D - + ; + = A 4 - + B + C - + D + E. -. : ( ), - = A + 3 + B + + C + + ( ), (3 ) + 4 - = - 3 + - + = - - + - + D + 3. - + - + = A + B + C + D - +. -, = A 4 + - + B + + C - + D -. () : 9

( ) - cos 3 d ; ( ) e 3 d. ( ) u = -, v= cos 3, u=, v = si 3, 3 - cos 3 d = 3 - si 3 - si 3 d 3 = 3 - si 3 + 9 cos 3 + C. ( ) e 3 d = d 3 e3 = 3 e 3-3 e3 d = 3 e 3-3 d 3 e3 = 3 e 3-9 e3 + 9 e3 d = 7 e3 9-6 + : P ( ) e d, P si d, P ( )cos d, P ( )., u = P ( ), v= e (si, cos ),, ;,, e ( si, cos ). () : ( ) - l d ; ( ) - rc t d ; (3 ) rcsi d. ( ) u = l, v= -, u=, v = -, - l d = - l - - d ( ) u = rct, v= -, u= - rct d = = - l - + + C. +, v = 3 3 -, 3 3 - rct - 3-3 3 + d

= 3 = 3 = 3 3-3 rct - 3-4 + d 3-3 rct - 3 3-3 rct - (3 ) rcsi d = rcsi d 3 3 3 = 3 3 rcsi - 3 3 - d, - d = d - - = - - + - d = - - - - d - - l + + C + l + + C. = - - - 3-3 + C, rcsi d = 3 3 rcsi + 3 - + 9 : - 3 + C. P ( ) l m d, P ( ) rct m d m P ( )rcsi d P ( )rccos d., u = (l ) m (rct ) m, v = P ( ),,, ( l ) m (rct ) m. m,. 3 () : ( ) e - cos d ; ( ) sec3 d ; (3 ) + d ( > ). ( ) e - cos d = e - si + e - si d

= e - si - e - cos - e - cos d, ( 6 )., e cos d = e si d = (88 5). ( ) sec3 d = sec d (t ) si + cos + e si - cos + e = sec t - sec t d + C, + C, = sec t + sec d - sec3 d, sec3 d = sec t + sec d = ( sec t + l sec + t ) + C. (3 ) + d = + - + d = + d = + + d + - + d, + + d + = ( + + l + + ) + C. = t t, + d = sec3 td t, ( ). : P ( si )e d, P (cos )e d, ( ) (3 )., I = = F( ) + I, ( ),

I = 4 () : - F( ) + C. ( ) I = cos d ; ( ) I(, m ) = (l ) m d,, m, ( ) 3 cos d (l ) d. I = d( si ) = si - - si d = si + - d( cos ) = si + - cos - ( - ) - cos d, : I = si + - cos - ( - ) I - ( : I = si + cos + C, I = si + C )., 3 cos d = I3 ( ) : = 3 si + 3 cos - 6 I = 3 si + 3 cos - 6( si + cos + C ) = ( 3-6 ) si + (3-6 )cos + C. I(, m) = ( l ) m d = (l ) m d = I(, m) =, + + + + + + (l ) m - m (l ) m - d, (l ) m - : I(, ) = d = m I(, m - ) + + + + C. (l ) d = I (, ) = 3 3 (l ) - 3 = 3 3 (l ) - 3 I(, ) 3 3 l - 3 I(, ) 3

= 3 3 (l ) - 9 3 l + 7 3 + C = 3 7 [ 9(l ) - 6l + ] + C., : I = d ( + ) = ( - ) ( + ) - + - 3 ( - ) I - I = rct + C, ( 7) ( 93 ). 5 I = 3-9 ( - - ) ( - + 5 ) d. R ( ): R ( ) = = 3-9 ( + ) ( - ) ( - + 5) A + + B - + C + D - + 5 + E + F, ( - + 5) 3-9 A ( - ) ( - + 5) + B( + ) ( - + 5) + ( C + D) ( - + 5) ( - - ) + ( E + F) ( - - ). = - : - 364 A = - 6 A = 3 ; = : 35 B = 3 B = 5 ; 5 : A + B + C = C = - ( A + B) = - = + i: ( E + F + Ei) ( - 6 + i) = - 8 + i, - E - 6 F = - 8, - E + F =, E = - 9, F = 5 4 ; = () : - 5 A + 5 B - D - F = - 9, 4 D = 3 ( 9-5 A + 5 B - F) = 3. 57 8 ;

I = 3( + ) + 5 ( - ) - 57-75 8 ( - + 5 ) - 9-75 ( - + 5 ) d. d 3 ( + ) = 3 l + + C ; d 5 ( - ) = 5 l - + C ; 57-75 - + 5 d = ( 7 ), 57( - ) - 8 ( - ) + 4 d = 57 d( - + 5) - + 5 d( - ) - 8( - ) + 4 = 57 l - + 5-9rct - 9-75 d = 9 ( - + 5 ) d( - + 5) - ( - + 5) 66 = - 9 - + 5-33 4 ( 8 ), I = 3 l + + 5 l - - - 8 = - - + 5 + + C 3 ; d( - ) [ ( - ) + 4] d( - ) ( - ) + 4 5-33 4( - + 5) - 33 8 rct - 57 l - + 5-9rct - 5-33 4( - + 5) - 33 8 rct - + C = 3 l + + 5 l - - 57 6 l - + 5 + 87 4 rct - + + C 4. ( 8) 33-5 8 ( - + 5) + C. ( ),,,. ( ),,., MATLAB. 5

syms it( ( 3 ^ - 9 )/ ( ^ - - )/ ( ^ - + 5)^ ) ; ( it), ( 5 ). Eter,, : s = / 3 log ( + ) + / 5 log( - ) - 57/ 6 log( ^ - + 5 ) + 87/ 4 t (/ - / ) - 3/ 3 ( - 44 + )/ ( ^ - + 5) log l,t rct. 5,. (89 ) 4: ( ) I = t d, =,3,, I = - t - - I -. t d = t - ( sec - ) d. ( ) I ( m, ) = cos m si d, m +, I( m, ) = cos m - si + m + = - cos m + si - m +, m =, 3,. I( m, ) = cos m - si d( si ) 3 (98 ) : + m - I( m -, ) m + = = cos m - si + - I ( m, ) + - m + I( m, - ), + ( m - ) [ I( m -, ) - I( m, ) ]. (3 ) d + 3 = 3 + - - - + d ; 6

(4 ) d + = 4 4 + + + - 4 - - + d ; (5 ) d ( - ) ( + ) = 4 - - + + - + ( + ) d ; d ( + + ) = (6 ) - = 4 ( + + ) 4 (4 + ) - 5 d ( + + ) ( + + ) d - 5 8 +. + + d, 3 R( si, cos ), si cos. R ( si, cos ) d. t = t, = rct t, d = si = R ( si, cos ) d = R + t d t, t + t, cos = - t + t, t + t, - t + t + t d t. ( 3 ) 3,,. cos5 si 4 ( ) R( - si, cos ) = - R ( si, cos ), t = cos ; ( ) R( si, - cos ) = - R ( si, cos ), t = si ; (3 ) R( - si, - cos ) = R ( si, cos ), t = t. d, ( ), t = si, d t = cos d, 7

cos5 si 4 d = ( - t ) t 4 d t = ( t - 4 - t - + ) d t,.t = t, cos5 si 4. - t d = + t t + t 5 4 + t d t = 8 ( - t ) 5 t 4 ( + t ) d t, 4, : R, d - c, t = + c + d d R (, + + c)d. +,. c + d 96. 5 R (, + + c ) d,, : + + c = t, >, t c, c >, t( - ), + + c = ( - ) ( - ). 3 ( ) ( ) ( 3). R ( u, v) u v ( ).,, R ( - u, v) = R( u, v), u : : R( u, v) = R ( u, v) ; R( - u, v) = - R ( u, v), R ( u, v) = R ( u, v) u., : 8

R ( u, v) = R ( - u, v ) = R ( - u, v ) - u = R ( u, v) u, R( u, v) u, R ( u, v ) = R ( u, v), = R ( u, v ), R( u, v) = R ( u, v) u = R ( u, v ) u., : ( ) R( - si, cos ) = - R ( si, cos ), R (si, cos ) d = R ( si, cos ) si d = - R ( - cos, cos ) d( cos ) (t = cos ) = - R ( - t, t) d t,. ( ), R ( si, - cos ) = - R( si, cos ), R ( si, cos ) d = R ( si, cos )cos d = R ( si, - si ) d( si ) (t = si ) = R ( t, - t ) d t,. (3 ) R( - u, - v) = R ( u, v), u v v u, R R R ( u, v ) = R u v v, v = R u v, v, u v, - v = - u R - v, - v = R( - u, - v) = u R v, v,, R u v, v = R u v, v, R ( si, cos ) = R (t, cos ) = R ( t, cos ) = R t, + t = R ( t ). 9

R ( - si, - cos ) = R ( si, cos ), t = t d = + t d t, R (si, cos ) d = R ( t )d = R ( t) + t d t,. 5. ( ) >, = t + + c = t +, ( 3 ) + + c = t + t +, - c - t, + + c = t + d = t - - c R (, + + c) d = R t - c. t d t. - t, t + ( t - ( t - c) - t - c) t, t - - c t d t,, ( 3 ), t.. ( ) c >,, c, = + + c = t + c, + = t + ct. ct -, + + c = ct - t + c, - t - t d = ct - - t R (, + + c) d = R ct - - t, 3 d t, ct - t + c ct - - t - t d t.

(3 ) + + c = ( - ) ( - ),, -, + + c = ( - ) ( - ) = t( - ), ( - ) = t ( - ). = t - t -, + + c = d = t t R(, + + c) d = R t - d t, - (- ) t, t - - (- ) t, t - t - t - t d t. -,,,,.. ( ) cos5 si 4 d ; ( ) cos si 4 + cos 4 d ; 3si + cos (3 ) si + 3cos d. ( ) 3, t = si, cos5 si 4 d = cos4 si 4 d( si ) = ( - t ) t 4 = ( t - 4 - t - + ) d t = - 3 t - 3 + t - + t + C = - 3si 3 + si + si + C. ( ) 3 (3 ), t = t, cos si 4 + cos 4 d = cos - si si 4 + cos 4 d = - t d(t ) t 4 + d t = - t + t 4 d t 3

= - t d t + t t t = - t + t = - d u u - + t d t u = t + t - = u + - u - d u = l u + u - + C = l t + t + t - t + + C = sec + t l sec - t + C. (3 ) ( ),. (si + 3cos ) cos - 3si ( 3si + cos ) cos si, :, 3si + cos = (cos - 3si ) + (si + 3cos ),., = - 3si + cos = ( - 3 + ) si + ( + 3 )cos, 5 3, = 3. - 3 + = 3, + 3 = 3si + cos si + 3cos d = 3 d - 5 3 d(si + 3cos ) si + 3cos = 3-5 l si + 3cos + C. 3 ( ): 3 cos si 4 + cos 4 d = ( si ) = - si d cos ( si + cos ) - si cos d

= d t - t ( t = si ) = t + - t - d t = l t + t - + C : ( ) I = - + + - d ; = si + l si - + C. ( ) I = d (, ). + - ( - ) ( - ) ( ), : I = ( - + + + - + ) d. - d = t td t ( = sec t) = ( sec t - ) d t = t t - t + C = - - rccos + C ; + d = t d t ( t - + = t) = + t - d t = + + l + - + + + C ; - d = t d t ( t + - = t) = - t + d t = - - rct - + C 3 ; 33

d = l + C 4. t = I = - + + + - - rccos - rct - + l + - + C. ( ) - -, I = ( - ) ( - ) - - d. = t - t -, d = ( - ) - t d t, ( t - ) - = - ( - ) t, - =, t - t - I = ( t - ) ( - ) t t ( - ) t - ( t - ) d t = = - d t = - t - + C - - + C. 3 : ( ) I = + + 5 d ; ( ) I = - + - 8 d. ( ) = >, c = 5 >, + + 5 = t + + 5 = t 5. + + 5 = - t, = t - 5 ( t + ), d = t + t + 5 d t, ( t + ) + + 5 = t - 5 ( t + ) - t = - ( t + t + 5 ) ( t + ) I = ( t - 5) 4( t + ) - ( t + ) t + t + 5 t + t + 5 ( t + ) d t 34.

= - ( t - 5) 4( t + ) 3 d t = - 4 ( t + ) 4-4 ( t + ) 3-4( t + ) + 6( t + ) + 6 ( t + ) 3 d t = - 4 ( t + ) - 4-4 t + + 6 ( t + ) + = - 4 = - 6 ( t + ) 3 d t t - 3 t - 4l t + - 6 t + - 8 + C ( t + ) t 8 + 3 t 4 + l t + + 4 t + + + C ( t + ) ( ) + - 8 = >, + - 8 = ( + 4) ( - ), t = - + + 5. + - 8 = t ( + 4) ( - ) = t( + 4) (t( - ) ). ( + 4 ) ( - ) = t( + 4 ), I = 4 t + - t, d = + - 8 = t - + - 8 = = + t ( - t) t d t ( - t ) = 6 t ( - t) ( t + ) ( t - ) d t t ( - t ) d t, 4 t + - t + 4 = 6 t - t, ( - t) + t = 9 d t t - - d t t + - 8 d t t - - 3 d t ( t - ) = 9 l t - - l t + - 4l t - + 3 t - + C t = ( ) ( ). + - 8 + 4. + + 5 = t + 5, I = - 8 ( 5 t - ) ( t - ) 3 d t. ( ) 96 35

, I = d ( + = t t) ( + ) + = (t t - ) sec sec t t d t = 4 sec3 td t - 3 sec td t - 4sec t. I = d ( + = 3sec t) - ( + ) - 3 3sec t = t t 3sec t - - 3t t d t = 3t t 3 - cos t - 3si t d t. (3 ) ( ) - + 4 ( + 4 ) ( - ) = t( + 4) = t., - 4 c >, + + c = ( - ) ( - ) = t( - ) R(, + + c) d =R (, ( - ) ( - ) ) d t =. = R, ( - ) ( - ) - d, ( - ),.,, - R(, + + c)d R, 3 (99 ) : ( ) d 5-3cos ( ) d + si 36 = rct t + C ; = 6 6 rct 6 t + C ; + c + d d

(3 ) d + t = l cos + si + + C ; (4 ) + - d = 7 8 rcsi - 5 (5 ) d + - + 3 4 = l + + + + C ; (6 ) - d = l + - + ( ) t = t, d t + 4 t ; ( ) t = t, d t 3 t + ; (3 ) t = t, d t ( + t) ( + t ) ; - - + - + C ; + C. (4 ) - = 5 si t, 4 ( + 5si t + 5si t) d t; (5 ) d( + ) ( + ) - ; (6 ) - + = t, - 4 t ( - t ) d t. : ( 99 ) ( ) - 3-4 d = 4 5 5 4-4 3 3-4 3 3 4 + C ; ( ) rcsi d = rcsi - 4 rcsi + 4 - + C ; (3 ) d ( = - l( + ) + C) ; + (4 ) esi si d ( = e si ( si - ) + C) ; (5 ) e d ( = e ( - ) + C) ; (6 ) d - = rccos + C ; 37

(7 ) - t d ( = l cos + si + C) ; + t (8 ) - ( - ) 3 d = l - - 3 - - + C ( - ) ; (9 ) d = t + cos 4 3 t3 + C ; () si4 d = 3 8-4 si + 3 si 4 + C ; () - 5 3-3 + 4 d = 3 l - + + - + C ; () rct( + )d ( = rct( + ) - + l + + + C) ; (3) 7 4 + d = 4 4 - l( 4 + ) + C ; (4) t + t + t d = - t + rct 3 3 (5) ( - ) d = 99 ( - ) - 99-49 ( - ) - 98 + C ; + 97 ( - ) - 9 7 (6) rcsi d = - rcsi - l + - + C ; (7) l + - d = - l + + + C ; - (8) d = t + si cos 7 5 t + C ; (9) e - + d = e + + C ; + C ; () I = v u d, u = +, v = +,. I = ( + ) [ v u + ( - ) I - ] (6 ) = sec t = t,. cos - si (7 ) cos + si d. (8 ) - + 3 ( - ) + ( - ) 3 d. () - 3 ( + ) + 3 ( - ) - ( - ) d. 38

() + = t. (3). (4) t = t, + t - (8) t = t, + t t d t. (9) e ( + ) - ( + ) d. + t + t d t. () I = v d ( u ) = v u - v - v - ud = v - ud, u ( + )d = I - + v - = I - + v - ( v - ) d u = I - + I - I -. ( v - ) d u ( A), (, y) l : ( ) e + l d ; ( ) d ( + ) ( + ) ; (3 ) 4 + d ; (4 ) d - ; (5 ) e + l d ; (6 ) (rccos ) d. -, (, ). f ( si ) = cos, f ( ). (B) : t d, t d = si cos d = - sec d (cos ) 39

= - + t d. t d, - =. 3 : ( ) 6 + 4 d ; ( ) d ( + ) ( + + ) ; (3 ) 3 d ; (4-3 ) si( l ) d ; (5 ) 3 - si 3 + cos d ; (6 ) sec4 d. 4I = d +. 4

- (, ),,. = f( i ) i ( ) i = ( T ) f ( i ) i = J ( ) i = f ( ), [, ], ( ), T { i }, f [, ], J f [, ],. f ( ) d = J, ( 3) f C [, ]f [, ], fr [, ]f [, ], R[, ] [, ]. 3 - f ( ) d = F( ) = F( ) - F( ) ( 4). 9 : f [, ], F, F ( ) = f ( ), [, ]., ( 4 ), f R[, ].? 4

f [, ],, ( 3 ) :,.,,. ( ),. ( ) : ( ) T, T, T. ( ) f ( ) = A,, f ( ) ; ( ), T, f( i ) i ( T { i } i = ).,. ( 3) ( ), T, { i } ; T, T, J. (4 ) ( 3),,,. D( ) =, [, ],, [, ], [, ].T, T (), { i }, { i }, D( i ) i i = = i =, i = D( i )i i = = i =, i =, D( )[, ]., f [, ] : >, >, T, T { i }, { j } m, T<, T<, f ( i ) i T - f ( j ) j. T ( 5), fr[, ], T, { i }, T f ( ) d. 4

(6 ), f [, ],, f ( ) d = f ( t) d t = f ( ) d=. ( 5) 3 - ( 4)? ( ) ( 4).f [, ] F, f [, ], ( 4 ). F( ) =, =, si,, F ( ) = f( ) =, =, si - cos, ; f ( ), f ( ). f [, ], F f, - ( 4 ). ( ) 9 ( 4),. ( i ) F : [, ], (, ) F ( ) = f( ). T, 9. (ii) f [, ] ( ), 9 ( ), f R [, ], ( 5), ( ) T f ( ) d, F( ) - F( ), (3) 5, 9 F( )( ). ( ). f( ) = -, [ -, ), e, [, ]. =, [ -, ] ( 3 ).- J = f ( ) d, - 43

F ( ) = -, [ -, ) e, [, ] ; F ( ) = [ -, ] =, F ( ) = f ( ) = F ( ) ; - +, [ -, ) e, [, ]. F ( ) - F ( - ) = e -, F ( ) - F ( - ) = e -. : J e -, e -?? ( ) F ( ), [ -, ] ; F ( ) F ( ), F ( ), F ( ) =. J = e -. : fr [, ], f( ) d >, [,] [, ], f ( ) >, [, ]., [,] [, ] f ( ), f ( i ) i, T, i =. T f( i ) i i = : = f ( ) d, d = -, <. ( 6) l, -., ( 6), ( 5),., T : i = + T = d =,+ i -,,+ = T= i(- ), i =,,,. i(- ) + i = ( - ) (- ) -, i =,,, ; -,, 44

= i = - - i - = ( - - ) - - - = ( - - ) = ( - ) t - t - t = ( - ) t - t l = -. l - - - 3 f R [, ], g f. gr[, ], g( )d = f ( ) d., g f, g ( ) f ( ). f( ) d = J.fR[, ], >, >, T<, { i } g( i ) i i = f ( i ) i - J < i = ; - J g( i ) i i = + f( i ) i i = - f ( i ) i i = - J < g( i ) - f ( i ) T+ i =. i -, g( i ) - f ( i ) =, i = =, g( ) - f ( ) g( ) - f ( ), T < = mi, g( ) - f( ), 45

gr [, ], g( i ) i - J < i = + =. g( ) d = J = f( ) d. :,,,.. 4 : ( + ) ( - ) = J. ( 7), f ( ) [, ], - J = f ( ) d. ( 7 ),,. J = ( + ) ( - ) = + + - I = l J = - l + i i =, I f( ) = l ( + )[, ],. i = i i -, i, i =,,, f ( ) = l( + )[, ], F( ) = ( + ) [l( + ) - ], 9 f ( )R[, ], I = I = l( + ) d = ( + ) [l( + ) - ] J = J = e I., = l -. = e I = e l 4 e = 4 e. 46

I l [, ], l( - ) [,3 ], I = l( + ) d = l d =. 5 y = - = ( 9 - )S. - =, ( - ), [, ], ( - ), (, ], S = ( - ) d + = - 3 3 + 3 3 - ( - ) d = 6 + 5 6 =. 6 f( )[, ],. : f ( ) d f : ( < < ), f ( ) + ( - ) f ( ) f (+ ( - ) ) ;. ( 8) =, 47

[ f ( ) + f ( ) ] f +. ( 9) ( 8 ),, ( 9 ) f., [, ], i i ( i =,,, ), ( 9 ) ( 8 ). f ( ) d = i = f ( i ). ( i + - i + ) =, i =,,,, f ( i ) = [ f( ) + f ( ) ] + + [ f ( ) + f( + ) ] i = f = f + + + f. i = f( i ) f + +, [, ] [, ],,. f ( ) d f + (4 ) : kd = k( - ). f ( i ) i i = = k i. i = ( - )., { i },, : ( ) 3 d = 4 48 ; ( ) e d ( = e - ) ;

(3 ) e d ( = e - e ) ; (4 ) d ( < < ) = - ( ) i 3 = i = 4 ( + ). ( ) (3 ) 4. (4 ) i = i - i [ i -, i ], i =,,,, f ( i ) i i = (6 ) : ( ) ( ) ( + 3 + + 3 ) = 4 4 ( + ) + ( + ) + +. = ; i = i - i -. i - i ( + ) = ; (3 ) + + + + + = ; 4 (4 ) ( - ) si + si + + si =. ( ) 3 d ; ( ) d (3 ) ; ( 4) + d ; ( + ) si d. 3: f R [, ], FC [, ], F ( ) = f ( ), f ( ) d = F( ) - F( ). [, ] T, F ( ) = f ( ) T. [ i -, i ] F, i ( i -, i ), F( i ) - F( i - ) = F ( i ) ( i - i - ) = f ( i ) i, i =,,,. 49

(3, 6) 3 ( ) ; 6.9 -.( ),, ( 4). 9-5

:.,, [, ],.,,. 3 s( T )f ( i ) i S ( T ) ( ) ( T ) s( T ) = m i i, S( T ) = ( T ) M i i, ( T ) f T, m i M i ( i =,,, )f T i., s( T )S ( T ) T, { i }., T, s( T ), S ( T ) ( 9-3, [ i -, i ], ; )., T, sup ( T ) = s( T ) = s, if ( T ) S ( T ) = S, f [, ], s = f ( ) d, S = f ( ) d., ( ): fr[, ] f ( ) d = f ( ) d.,, ( 3, 6). 4,.,, 5

,.35 ( ) ( ). 5 ( 9 9 4 9 6), ( )., 9 4, f [, ].,, 9 4.,.? A >, T, i i ( T ) < ; B >, >, T< T, i i ( T ) <., B A., A, T, s( T )sss ( T ) S - s S ( T ) - s( T ) = i i ( T ) <. s = S, (33 6 ), [ S ( T ) - s( T ) ] = S - s =. T -, B., A B,,,.. ( ) [, ], ((, ) )., : [, ], [, ]?., : f [, ], f [, ].. 5

( ) 9 49 6 ( 9 ),. (3 ) 3 35,.,. 3? 35, f,, f.,.,,. : u = ( ), f ( u) =, < u,, u =,. f (( ) ), [, ]. f R[, ], g( ) = e f( ). gr[, ]. [] f R [, ], M g i = sup e f( ) f( - e ), i >, f( ) M, [, ] ; g( ) e M = M, [, ]. = sup e i f( ) - f( ), i M sup f( ) - f( ), i = M f i ( i f ( ) f ( ) ). ( ) (), >, T, T, ( ) f ( T ) i i < M ; g i i M ( T ) f i i < M M =. 53

(), g( ) = e f( ) R [, ]. [] (35 ), h( u ) = e u, u = f( )[, ], g( ) = h( f( ) ) = e f ( ) R[, ]. : fr [, ], [, ] [, ], fr[,]. >, fr[, ], T, i i ( T ) <.,T T, TT, ( T ) i i i i ( T ) <., T[, ] [,]T, ( T ) i i ( T i i <. ) f R [, ]. 3 h( ) [, ], [, ] T, h( )T ( i -, i ) ( h ( i ), h ( ) ), i =,,,. : ( ) f R[, ], >, h ( ) f ( ) ( h ( ) f ( ) ), [, ], f( )d - h ( ) d - ( ) >, f R[, ]. h ( ) d < h ( ) f ( ), h ( )f ( ), [, ], h ( ) d - ( ) fr[, ], T, f ( T ) i i s( T ) f( ) d S ( T ), f( ) d - s( T ) < h h 54 f ( ) d <. ( 3) h ( ) d <, = S ( T ) - s( T ) <., S ( T ) - f( )d <. ( 4)

h ( ) = m i, h ( ) = M i, ( i -, i ), i =,,,, h ( ) d = s( T ), ( 4 ), ( 3 ). h ( ) d = S ( T ). ( ) h h,, T T, T = T S( T ) - s( T ) h ( ) d - h ( ) d <. + T, T T, T, i i = S( T ) - s( T ) f ( T ) f R[, ]. S( T ) - s( T ) <, ( ), h ( )d - h ( ) d <, ( ), fr[, ] : h h, h ( )f ( )h ( ), [, ], h ( ) d - h ( ) d <. ( )( ),. 4 : fr[, ], >, g ( ) f ( ), [, ], f ( ) d - 3( ), h, g( ) d <. h( )f ( ), [, ], f ( ) d - f [, ],, h ( ) d <. f( ) M, [, ]. h( ) ( i -, i ) m i, i =,,,, 55

< M, g( ) ( 9-4 ) : i - +, i - g ( ) = h( ) = m i ; i -, i i, i +, g( ) g ( i ) = - M. 9-4 g( )h( )f( ), [, ], h ( ) d - f ( ) d - g( )d M < ; g( )d = + h( ) d - f ( ) d - h( ) d g( ) d <. : fr[, ], g( )f ( ), g ( ) d - f ( ) d <. 5 : f [, ], f [, ], [, ]. : ( ) T S ( T ) - s( T ) < -, T i, f ( i ) < ; ( ) I = [, ] (, ), f ( I ) = sup I f ( ) - if I f ( ) < ; (3 ) I = [, ] (, ), 56

f ( I ) = sup I f( ) - if I f( ) < ; (4 ), { I } = { [, ] }, [, ] ( -, - ), f ( I ) = sup f( ) - if f ( ) < I I, { I }, I, =,, f ; (5 ) f [, ]. ( ) : f ( i ), i =,,,, S( T ) - s( T ) = f ( i ) i ( T ) i = -. ( T ) ( ) ( ), i = [ i -, i ] [, ], f ( i ) <. I : < i <, [, ] = [ i -, i ] (, ) ; i =, [, ] = [, ], < <, [, ] (, ) ; i =, [, ] = [ -, ], - < <, [, ] (, ). I = [, ] i, f ( I ) f ( i ) <. (3 ) I ( ) [, ],, f R[, ]. : [, ] T S ( T ) - s( T ) < ( - ) ; T i, f ( i ) <. ( ), I = [, ] (, ), f ( I ) f ( i ) <. (4 ) T, T =, + - + T, 57

I + I, { I }. ( - ) = ;, I, =,, ; >, NN +, N, <, [, ] U( ; ). { I }: [, ] ( -, - ), =,,, (, ), =,,. = mi{ - N, N - } >, U( ; ) U( ; ), f. f ( ) - f ( ) f ( I N ) = N <, (5 ) [, ] [, ], [, ] [, ], fr[, ], ( )( 4) f (, ). f [, ]..,.. f, [, ], f R[, ]. f( ) = si,,,, =,. 9-5 58

9-5,, =,, k =,,. k,, f,. 3 ( ) : T, ( T ) i i 3(9-3). : fr[, ], [, ] [, ], f R[, ]. (.) i i. ( T ) 3 f g [, ], f ( )g( ). : f [, ], g [, ], f ( ) d = g( ) d. 3 ;.f g c., >, T, f ( T ) i i c T k, g i i ( T ), g k k = ( T ) ik gr[, ],. <. f i i < + g kk. + k g k <, g [, ]. g( ) d = f( ) d 4 f [, ], { } [, ], ( =,, ), f R[, ]. = c =. >, >, = c. : f [, ] 59

<, f [, + ] (=?). [ +, ]f (?), [ +, ] T, i i ( T ) <. [, + ] T[, ] T, i i ( T ) 5: f, <. sup f( ) - if f ( ) = sup f ( ) - f ( ).,,. m = if f( ), M = sup f ( )., : sup f ( ) - f ( ) = M - m,, ),, f ( ) - f ( ) M - m ; ) >,,, f( ) - f( ) > M - m -. 6 (36 ) (3 ). ( ). 6 T s( T ) = s. T 3 f ( ) =, S( T ) = S.,,. f [, ]; f [, ].? s =, S =. 4 f R[, ], f( ), [, ]. f [, ]? (35 ). 5: 9 4 >, >, T < T, i i ( T ) = S ( T ) - s( T ) <. 6

( ) 6: ( )? ( ) ()? (3 ), ( ), ( )? ( ), ( ) ; (3 )( ),. 7 (, 5.) 3 (4) 6; 78. fr[, ] k f ( ) d = k f ( ) d. fgr[, ] ( f g)r[, ], [ f( )g( ) ] d = 3fgR[, ] ( fg) R[, ]. 4fR[, ] c(, ), f R[, c], fr[ c, ], f( ) d = c f( ) d f ( ) d + c 5fR[, ], f ( ) f ( ) d ; [ ] f gr[, ], f ( ) g( ), f( )d 6fR[, ] f R[, ], f ( ) d 7f [, ], [, ], g( ) d. f ( ) d. ( 3 ) g( ) d. ( 3 ) f( ) d. ( 3 3) f ( ) d = f( ) ( - ). ( 3 4) 8fg [, ], g ( ) [, ], [, ], 6

f( ) g( ) d = f () g ( ) d. ( 3 5) 3?? ( ) f, g, h = f + g( f - g), [, ], [, ]., f ( f + g) R[, ], g = ( f + g) - f, gr[, ]. ( ) f, g, h = f + g( f - g), [, ], [, ], [, ] (, ( ) ). (3 ) f gr[, ], f ( ) m >, [, ], g f R[, ]., f R[, ] ( 7 ), 3, g f = g f R[, ]. (4 ) fr[, ], f( ) m >, [, ], gr[, ], ( fg) R[, ] (, (3 )). ( 4 ),, c (3 )? f R[, c], < c, f ( ) d =, f ( ) d = - f ( ) d, ( 3 ).4 c c f( ) d = - c f ( ) d = c f ( ) d = f ( ) d + c f( ) ; f ( ) d,, ( 3 ) f ( ) d + c f( ) d. (3 ). 3. 6 f( ) =, [, ], -, [, ].

, f ( )[, ] ; f ( ), [, ], [, ].,. 4 ( 7 ) ( 8 ) : ( ) 7 ( f [, ] ) f [, ], : =. fr[, ], m = M = if [, ] sup [, ] f( ), f ( ). ( m M ), f( ) d = ( - )., mf( )M, [, ], 5, m( - ) = - md m f ( ) d - f( ) d M. f( )d, mm, f ( ) d = ( - ). Md = M ( - ), 7 f ( ), f ( ) [, ] 8( 9 ). ( ) ( 3 4 ) ( 3 5 ) (, ).(8.) (3 ).f [, ], F, F ( ) = f ( ), [, ]. F [, ], (, ), F( ) - F( ) = F ( ) ( - ) = f( ) ( - ) ;, -, F( ) - F( ) = f ( ) d., (, ), f ( ) d = f( ) ( - ), (3 4 )., f [, ], f F [, ] ( ). F 63

,. (4 ) 8 ()(3 5 ) f ( ) g( ) d f ( ) =, g( )d f () f( ) [, ], g ( ), [, ] f ( ). I = e e l d., I = - e l d = l - + C, e l d + e l d I = ( - l ) e + ( l - ) e = - e - l e + [e(l e - ) - ( - ) ] = - e + = - e. : < + d <. ( 3 6) (3 4 ), e - m( - ) f ( ) d M ( - ), ( 3 7) M m f ( )[, ],. ( 3 5), g( ), [, ], m g( )d, ( 3 8) (3 7 ). (3 8 ), 64 f ( ) = f ( ) g( ) d M -, g( ) = e +, g( ) d. ( 3 8)

e - d e - + d e - d = ( - e - e - d = ( - e - e - d. ) <, ) <,, ( 3 6) ;. : (3 6 ). e - e - + d > 5 + d 5 5 e - d f ( ) = e -, g ( ) = = 5 ( - e - 5 ) > 5 3 4 =,,? +.,, 45,? 3 : fr[, ], f ( ), f( ) d = f ( ). ( ). [, ]f, f( ) >, f ( ) d >, f( ) d = ( 7 ). f ( ). ( ) 5 ( ), f, [, ] T, f T m i, i =,,,, s( T ) =. f [, ], f ( ) d = f ( ) d = 4 ( Schwrz): f ( )cos kd s( T ) =. T + f ( ) si k d, ( 3 9) 65

f [, ], f( ) d =, k. f ( ) g( ) d ( f ( ) ) d f gr[, ] ( 37 6 ). f( )cos k d f ( ) cos k d = f ( )f ( ) cos k d f ( ) d f ( )cos = f ( )cos kd, f( ) si kd ( g( ) ) d, kd f ( ) si k d. + f ( ) si k d f ( ) d =. f,, f ( ) [, ] (?), (3 9 ). 5 f( )[, ], F ( ) = ( f( ) ), =,,., [, ], : =. F ( ) = F ( ) d, =,,.,, F ( ) [, ], > <, < N >, > N, f( - ) < f( - ) f ( - ) f ( - ) F ( - ) < F ( - f( - ) f( - ) = F ( - ) F ( - ) <. ) < - F ( ) d =, 66

< F ( ) d = F ( ). F ( ), > N - <, =, f ( ), )? [, ] [, ],? 6 : e cos d =. f ( ) = e cos,,. f ( ) = e cos - (cos - si ) =, =, = rct. m, f ( ) = f ( ) = f ( ) = e rct + = + +, = ( = =. + 9-6, f( ) =,3,5,. : f ( ) d = f ( ) d +f ( ) d,, ; f ( ) [, ] (, = rct < ), f()( ). >, = 4. rct =, N >, > N, < rct <., [, ] 67

9-6 f ( ) = e cos e rct + = M. M =, N >, > N, < M <. > m{ N, N }, < f( ) d M d = M 4 <., > N f ( ),, > N 3 = l < f ( ) d e cos - = 4 4 e cos ( - 4 ) <, - e - 4 l cos 4. > N = m{ N, N, N 3 }, < f( ) d = e cos d =. f ( ) d +f ( ) d <, 7 f [, ], f( ). : 68 ( f ( ) ) d = m [, ] f( ). (3 )

M = m [, ] > (< M ), [, ] [, ], f ( ), M ( - ) f ( ) > (M =, f ( ), (3 ) ). < M - f ( )M, [, ], ( M - ) ( f ( ) ) M, [,]. M ( - ) = M d ( f ( ) ) d ( M - ) d = ( M - ) (- ) ; M ( - ) ( f ( ) ) = M, ( M - ) (- ) M -, ( 3 ). m = mi [, ] f ( ). ( f( ) ) d ( M - ) (- ). ( f( ) ) d = M -, M. m ( f ( ) ) ( m + ), [,] [, ], ( f( ) ) d = m?,,. 8 : fr[, ], p p f ( ) si p d =, f ( ) cos pd =. (3 ) (3 )., : 9-7, p, y = f( ) si p., f, >.T [, ], i = f i i <. T i = [ i -, i ], m i = if f( ), i =,,,. i 69

9-7 f ( ) si pd = f ( ) - m i i si pd = i - i = i i- i = i = i- + i = f ( ) si pd [ f( ) - m i ] si pd m i i i- si pd, f i, i =,,,, p (cos p i - - cos p i ) p, f ( ) si pd f i i + p m i. i = T, m i, p > P i = = 4 m i, i = p > P, (3 ). p m i <. i = f ( ) si pd i = < + =, ( 3 ) ( Leesgue), ( Fourier),. 7