1. Математический анализ (850924), страница 5
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8.1. 8 x0, f (x0 ) < 1 Ty = 1limx!0 Tx y = f (x) 1 M0(x0 f (x0)), x = x0J 1 y = f (x0 ):78 " * . 4 S *1 , * , t, ..S = S (t): t0 S0 = S (t0) , t0 + Tt { S1 = S (t0 + Tt) . = , Tt 1 TS = S (t0 + Tt) ; S (t0). 4 %TS . ( % * Tt Tt: % TTSt Tt ! 0 * t0:TS :V (t0) = tlim!0 Tt2 ", V (t) S (t) t:V (t) = S 0(t): :1) y = x2 Ty = (x + Tx)2 ; x2 = 2xTx + (Tx)2:22xTx+(Tx)20(x ) = xlim= 2x + xlim!0!0 Tx = 2x:Tx2) y = sin x Ty = sin(x + Tx) ; sin x = 2 sin T2x cos(x + T2x ):Tx cos(x + Tx )2sin22 =(sin x)0 = xlim!02sin T2xTx= xlim!0 cos(x + 2 ) xlim!0 Tx = cos x279 sin T2xlim= 1:x!0 Tx2 " & f (x) x0: 8.4. 8 1 % TTfx Tx ! +0 (..
Tx ! 0J Tx > 0), + & f () 0 " f 0 (x0 + 0):(f (x0 ) + Tx) ; f (x0 ) :f 0 (x0 + 0) = xlim!+0Tx 8.5. 8 1 %Tf Tx ! ;0 (.. Tx ! 0J Tx < 0), + Tx & f () 0 " f 0 (x0 ; 0):(f (x0 ) + Tx) ; f (x0 ) :f 0 (x0 ; 0) = xlim!;0Tx 8.2.
? , " 1 0 " , " 1 + " " .* + " 8.1. ./ f (x) 0, # ) . 1 0, ..Ty = lim f (x0 + Tx) ; f (x0 ) :f 0 (x0) = xlim!0 Tx x!0Tx2 * &, e " & Ty = f 0 (x ) + (Tx)0Tx80. (Tx) { " & Tx ! 0, ..lim (Tx) = 0:= ,x!0Ty = f 0 (x0)Tx + (Tx)Tx:6 ,0lim Ty = xlim!0 (f (x0 )Tx + (Tx)Tx) == f 0 (x0) xlim!0 Tx + xlim!0 (Tx)Tx = 0:> + , & f (x) 0. " , - , ..
0 & +. 8.1. P = jxj = 0. *, + .6 y0 (+0)jTxj + 0 = lim Tx = 1:y0 (+0) = xlimx!+0 Tx!+0 Tx>,jTxj ; 0 = lim jTxj = lim ;Tx = ;1:y0 (;0) = xlim!;0 Txx!;0 Txx!;0 Tx2 ", & y = jxj = 0 1 y0 (;0) = ;1 1 y0 (+0) = 1, y0 (;0) 6= y0 (+0) , , & y = jxj = 0.x!0! " , " 8.2. 1 ./ u = u(x) v = v(x) -# 0 .
2 ) # ,81 , :v(x0) 6= 0, ,(u + v)0 = u0 + v0 J ) (uv )0 = uv 0 + u0 v J u !0 u0 v ; uv0)v = v2 :- . ).4 & F (x) = u(x) + v(x). 8 1 0 0 + Tx :)TF = u(0 + Tx) + v(0 + Tx) ; (u(x0 ) + v(x0)) == u(0 + Tx) ; u(x0 ) + v(0 + Tx) ; v(x0) = Tu + Tv:F = , % TTxTF = Tu + Tv Tx Tx TxTF = lim Tu + lim Tv = u0(x ) + v0 (x )lim00x!0 Tx x!0 Txx!0 Tx..0(u + v) = u0(x0 ) + v0 (x0):x = x0- . ). " G(x) = u(x)v(x). 2TG = u(0 + Tx)v(0 + Tx) ; u(x0 )v(x0) == u(0 +Tx)v(0 +Tx)+ u(x0 )v(x0 +Tx) ; u(x0 )v(x0 +Tx) ; u(x0 )v(x0 ) == Lu(x0 + Tx) ; u(x0)]v(x0 + Tx) + Lv(x0 + Tx) ; v(x0 )]u(x0):= ,TG = lim ( u(x0 + Tx) ; u(x0 ) v(x + Tx)+G0(x0 ) = xlim0!0 Tx x!0Tx82Tu lim v(x + Tx)++u(x0 ) v(x0 + TTxx) ; v(x0) ) = xlim!0 Tx x!0 0Tv = u0(x )v(x ) + v0 (x )u(x ):+u(x0 ) xlim0000!0 Tx< " 1 0: & v() 0, + , ..lim v(x0 + Tx) = v(x0 ):x!02 " &(uv)0 = u0 v + uv0 :- .
). " () = uv((xx)) " , v(x0) 6= 0. 2 & v(x) 0, 0 ( - " 1 0), , v(x0) 6= 0, &, 1 U (x0) 0, , *1x U (x0), v(x) 6= 0.2 = 0 + Tx, *1x U (x0) :+ Tx) ; u(x0 ) =TT = T (x0 + Tx) ; T (x0 ) = uv((xx0 +0 Tx) v (x0 ); u(x0)v(x0 + Tx) == u(x0 + Txv)(vx(x+0) Tx)v(x0 )0; u(x0)v(x0 ) ; v(x0 + Tx)u(x0 ) == u(x0 + Tx)v(x0 ) + u(x0v)(vx(x+0) Tx)v(x0 )0; v(x0 + Tx)u(x0 ) + u(x0)v(x0 ) == u(x0 + Tx)v(x0 ) ; u(x0v)(vx(x+0) Tx)v(x0 )0; u(x0 )(v(x0 + Tx) ; v(x0)) == v(x0 )(u(x0 + Tx) ;vu((xx0+)) Tx)v(x0 )0832 ",; u(x0)Tv := v(vx(0x)T+u Tx)v(x )00Tu ; u(x ) Tv u !0 v(x)00TT0(x0 ) = limTxTx ==T=limx!0 Tx x!0 v (x0 + Tx)v (x0 )v x = x0Tu ; u(x ) lim Tvv(x0) xlim0 x!0!0 TxTx :=v2(x0 ), * ")Tu = u0 (x )J lim Tv = v0 (x )limv(x+Tx)=v(x)=60lim0 x!0000x!0x!0 TxTx+0 (x0)v(x0 ) ; v0 (x0)u(x0 )u0 (0) =v2(x0)..
& ), v(x0) 6= 0, . u !0 u0 v ; uv0v = v2 : 8.3. 8 y = Cu(x), C { , 0y = Cu0(x):y0 = (Cu)0 = C 0u + Cu0 = Cu0 C 0 = 0: 2 ",# - - # ." & 1) & y = x, x > 0, =6 0:Ty = (x0 + Tx) ; x0 Jx0 (1 + Tx x ) ; 10=lim=y0 (x0 ) = xlimx!0!0TxTx(1 + Txx ) ; 1(1 + Tx x ) ; 100= x0 ;1 xlim= x0 ;1:= x0 xlimTx xTx!0!0x0 0x084(x0 + Tx) ; x0= ,(x )0 = x;1 , x;1 .2) & y = ax, a > 0, a 6= 1:x+x ; axx ; 1aax0xx(a ) = xlim!0 Tx = xlim!0 a Tx = a ln a:3) & y = ln x:x + Txlnln(x + Tx) ; ln x = limx =y0 = (ln x)0 = xlimx!0!0TxTx!Txln 1 + xln(1 + Txx ) 1ln(1 + Txx ) 1= x:= xlim= xlim!0 Tx x = x xlim!0 Tx!0Txxx4) & y = loga x:xJy = loga x = lnln a!0y = (loga x)0 = ln x 0 = 1 (ln x)0 = 1 :ln aln ax ln a5) & y = cos x:Tx sin 2x + Tx;2sincos(x + Tx) ; cos x = lim22 =y0 = (cos x)0 = xlim!0x!0TxTxsin T2x Tx != ; xlim!0 Tx sin x + 2 = ; sin x:26) & y = tg x:!0y = (tg x)0 = sin x 0 = (sin x)0 cos x ; (cos x)0 sin x =cos xcos2 x2 x + sin2 x1 :cos==cos2cos2 x857) & y = ctg x:!0y = (ctg x)0 = cos x 0 = sin x(cos x)0 ;2 cos x(sin x)0 =sin xsin x2 x + sin2 xcos= ; sin12 x := ; sin2 x 8.4.
$ e &, *:(c1f1 + c2f2 + ::: + cnfn)0 = c1f10 + c2f20 + ::: + cnfn0 (f1f2f3 :::fn)0 = f10 f2f3:::fn + f1f20 f3:::fn + ::: + f1f2:::fn;1fn0 :$-+ ,&+1. # *: f (x) * x0 (c. 8.1).2. * * !* *: y = f (x) *M0(x0 f (x0)) (c. 8.2).3. ;5 * * !* *: y = x2 + 2x *M0(1 3):4. *% " 6# # * (c. 8.1).5. ; 0 f 0(x0 ; 0) 0 f 0(x0 + 0) #, :1) f (x) = jx + 1j x0 = ;12) f (x) = j ln xj x0 = 1:6. )*%, *: f (x) #0 * x = 0 f (0) = 0,f (x) : f 0(0) = xlim!0 x7. ; f 0(a), f (x) = (x ; a) '(x), ! *: '(x) * = .8. ; f 0(0), f (x) = x(x + 1) : : : (x + 1998):9. + *: f () #0 * x0, *: g(x) # * x0. 4 % *# 6 # *:F (x) = f (x) + g(x) * x0?863.
C* ** f 0(x) = 2x + 2, f 0(1) = 4. , * * M0(1 3) y ; 3 = 4(x ; 1) :: y ; 4x + 1 = 0:5. ?# 8.4 8.5 :1) f 0(;1 ; 0) = ;1 f 0(;1 + 0) = 1D2) f 0(1 ; 0) = ;1 f 0(1 + 0) = 1:6. + 0 # *: = f (x) * x0 = 0 (. 8.1) f (Fx) ; f (0) = lim f (x) f 0(0) = limx!0 xx!0Fx* ** Fx = x ; 0 = x.f (x) ; f (a)(x ; a)'(x)7. f 0(a) = xlim!a x ; a = xlim!a x ; a = xlim!a '(x) = '(a):x(x + 1)(x + 2) : : : (x + 1998) = 1 2 : : : 1998 = 1998!8. f 0(0) = xlim!0x9. <*: F (x) = f (x) + g(x) # * 0.
)*# " !. + F (x) #0 * 0, !*: g(x) = F (x) ; f (x) "# #0, ** # *:,06 #0 * 0 (. 8.2), 2 %0, *: g(x) #. + *# 5%.87 9446 456( 9.1. P f (x), 0, && 0, 1 Tf = f (x0 + Tx) ; f (x0 ) = ATx + (Tx)Tx(9:1) A { , 1 T, (Tx) { " & T ! 0: 9.2. $* ATx Tf:<, A 6= 0 + , & (Tx)Tx & % % ATx.(Tx)Tx = 1 lim (Tx) = 0:limx!0 ATxA x!0 9.3. O 1 Tf , ..* ATx, ../ ./ f (x) x0 dy :dy = ATx:(9:2) 9.1. 2 A 6= 0 A(Tx)Tx { & " % % ATx, f (x0 + Tx) f (x0 ) + dy:(9:3)* 9.1. , # ./ f (x) # ../ x0 , , # ) .
1 )Tf = f 0 (x0 )Tx + (Tx)TxJ xlim!0 (Tx) = 0:883 .6 & f (x) && x0,Tf = ATx + (Tx)TxJ xlim!0 (Tx) = 0:Tf = A + lim (Tx) = Alimx!0 Txx!0, , f 0 (x0) 1 f 0 (x0) = A: . x0 , ..Tf = f 0 (x ):lim0x!0 Tx * &, " & :Tf = f 0 (x ) + (Tx)J lim (Tx) = 0:0x!0TxC ,Tf = f 0 (x0 )Tx + (Tx)TxJ xlim(9:4)!0 (Tx) = 0.. & f (x) && 0. , A = f 0 (x0). 9.2. ) 9.1, df = ATx = f 0 (x0)TxJ :: df = f 0 (x0 )Tx: 9.3.
?&& dx 1 Tx, ..dx Tx: & df = f 0 (x0) dx:* 9.2. ./ f (x) ../ 0, # ) .898 & f (x) && 0, 1 Tf Tf = f 0 (x0)Tx + (Tx)TxJ xlim!0 (Tx) = 0, ,0lim Lf (x0 + Tx) ; f (x0)] = xlimx!0!0 f (x0)Tx + xlim!0 (Tx)Tx = 0Jlim f (x0 + Tx) = f (x0 )x!0..
& f (x) 0. 9.4. 4 - , & " ( 1 ) , , " " && 0:.) ; , y = f (x) f (x) { && 0 ( . 9.1).. 9.1 + M0(x0 f (x0 ))., , +&& 0K f 0 (x0) tg = f 0 (x0), { * 0K x. " 0 Tx = 0L, 1 Ty = LM1, LK = ML tg JLK = f 0 (x0)Tx dy:902 " " 1 9.4. ?&& & f (x) 0 1, y = f (x) " 0 " x0 + Tx:( , " 9.3. ./ f (x) g(x) ../# 0, ./ C1f (x) + C2g(x), C1 C2 { # ,f (x)g(x) ../# 0, , , g(x0) 6= 0,f (x) ../ , ) ./0g(x)d(C1f (x) + C2g(x)) = C1df + C2dgd(fg) = gdf + f dg(9:5) f ! gdf ; f dgd g =g2 :? & (9.5) &&dy = f 0 (x0)dx:? ,) d(C1 f (x) + C2g(x)) = (C1f (x) + C2g(x))0 dx =x = x0= C1f 0 (x0)dx + C2g0(x0)dx = C1df + C2dg:") d(fg) = (fg)0 dx = (f 0 (x0)g(x0) + g0(x0)f (x0 ))dx =x = x0= g(x0)f 0 (x0 )dx + f (x0 )g0 (x0)dx = g(x0)df + f (x0 )dg: f !0 0 (x0)g(x0) ; g0(x0)f (x0 )ff) d g = g dx =dx =g2(x0)x = x0g(x0 )f 0 (x0 )dx ; f (x0 )g 0 (x0 )dx g (x0 )df ; f (x0 )dg=:=g2(x0)g2(x0)91.
9.1. ?* ( ), & = 2&& 0 = 3.f (3) = 32 = 9J x = 3 + TxJTf = (3 + Tx)2 ; 9 = 9 + 6Tx + Tx2 ; 9 = 6Tx + TxTx:$ % = 6, (Tx) = Tx ! 0 Tx ! 0. = ,& y = x2 && x0. 9.2. D * && && &?. " && &&.6: 1) y1 = jxj y2 = jxj { && & 0, y = jxj jxj = x2 { && 0 &.2) y1 = jxj + 1 { && & 0, y2 = jxj{ * && & 0. ) y = (jxj + 1) jxj = x2 + jxj && 0&. && "* & (9.3) 9.1:f (x0 + Tx) f (x0 ) + dy = f (x0 ) + f 0 (x0)dx * dx Tpx: 9.1.
$ 15 8. $ % 4ppf (x) = xJ x0 = 16J Tx = ;0 2J f (x0 ) = 16 = 2J1:f 0 (x) = p1 3 J f 0 (x0) = 324 x= ,q1 (;0 2) = 2 ; 1 = 1 159 :15 8 f (15 8) 2 + 32160 160444492" 9.4. ./ u(x) ../ 0, y = f (u) ../ u0 = u(x0 ), - ./ F (x) = f (u(x)) ../ x0 , 00F (x0) = (f (u(x))) = f 0 (u0)u0 (x0 ):(9:6)x = x0./? = 0 1 Tx 6=0.
2 1 & u = u(x)J Tu = u(x0 +Tx) ; u(x0), , 1 & y =f (u)J Ty = f (u0 +Tu) ; f (u0 ). 2 & f (u) && u0 = u(x0 ), 1 Tf ( .& (9.4)):Tf = f 0 (u0)Tu + (Tu)Tu(9:7). (Tu) { " & Tu ! 0:? , " " (Tu) Tu =0. 6 " , " & (9.7) " " Tu , + & (Tu) Tu = 0J (0) = 0. 2 &(9.7) " Tu .= , & u = u(x) && 0, ,Tu = u0 (x0)Tx + (Tx)TxJ xlim!0 (Tx) = 0 Tu ! 0 Tx ! 0, Tu ! 0 (Tu) ! 0, Tx ! 0.2 ",Tf = f 0 (u0)Lu0 (x0)Tx + (Tx)Tx] + (Tu)Lu0 (x0)Tx + (Tx)Tx] == f 0 (u0)u0 (x0)Tx + L (Tx)f 0 (u0) + (Tu)u0 (x0) + (Tx)(Tu)]Tx == f 0 (u0)u0 (x0)Tx + (Tx)Tx:93< (Tx) = ( (Tx)f 0 (u0) + (Tu)u0 (x0 ) + (Tx)(Tu))J00lim (Tx) = xlimx!0!0 (Tx)f (u0) + xlim!0 (Tu)u (x0)++ xlim!0 (Tx)(Tu)) = 0:= , & F (x) && 0 F 0 (x0) = (f (u(x)))0 = f 0 (u0)u0 (x0).x = x02 . 9.2.
6 1 &:a) y = cos3 xJ y0 = 3 cos2 x(cos x)0 = 3 cos2 x(; sin x)J0px)px)10tg(1=tg(1=pJ y =5ln 5 tg x =") y = 50 10 1 1 !px)p111x)tg(1=tg(1=A@ln 5 cos2 p1 px = 5ln 5 cos2 p1 ; 2 p 3 J=5xxx) y = sin2(x3 + 3)J y0 = 2 sin(x3 + 3)(sin(x3 + 3))0 == 2 sin(x3 + 3) cos(x3 + 3)(x3 + 3)0 = 2 sin(x3 + 3) cos(x3 + 3)3x2 == 3x2 sin 2(x3 + 3):, (") 9.5.8 & u = u(x) && 0, & u = f (u) && 1 u0 = u(x0), * & = f (u()) && 0, d(f (u(x)) = f 0 (u0)u0 (x0)dx:? 9.4 9.2, (9:8)df (u(x)) = ddx (f (u(x)))dx = f 0 (u0 )u0(x0 )dx:94 9.5. 2 u0(x0 )dx = du, & (9.8) * df (u(x)) = f 0 (u0)du:(9:9) & , && & * & * ( * &o), & , & &.( && .= " , (..
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