1. Математический анализ (850924), страница 14
Текст из файла (страница 14)
( t.< ! ! S = S (t), t : a = S (t): C:S (t) = (t4 ; 3t2 + 2t) = (4t3 ; 6t + 2) = 12t2 ; 6: >0000000000000002250 !"#!$# $ %&#'# (&1) yIV (1) y(x) = x6 ; 4x3 + 4#2) y(n) y(x) = eax#3) y(n) y(x) = sin2 x#4) 8 9 : S = t2 ; 4t +1: 9.1) 360#2) aneax#3) 2n 1 sin(2x + (n ; 1)=2)# 4) S (t) = 2t ; 4 S (t) = 2:;000||||| f (x) ! x0 5 n- !4! . 1 !1!(x0) (x;x )2+: : :+ f (n) (x0) (x;x )n+R (x)f (x) = f (x0 )+ f (1!x0) (x;x0 )+ f 2!00n+1n! ! (5 (n +1) !53) 5 ! 1! ! f (x) ! Rn+1(x)5 5 !.! f (n) (x) 5 x0 5! 000Rn+1 (x) = oG(x ; x0)n] x ! x0(.. ! ! x ! x0 52 4 (x ; x0)n ).! 9 f (n+1) (x) x0 5 ! I(n+1) ( )fRn+1(x) = (n + 1)! (x ; x0)n+1 = x0 + J(x ; x0) 0 < J < 1:226 ! " A! 1! ! ! f (x) x0 (!! ).
< , !f (x0) = 0 (x0 { f (x)), ! ! x0 ! 53 ! f (x) x0: 1 5 ! Rn+1(x) ! x !3 x0 !4 ! ! 4 !45 !5 ! 1!, 5 !4, 54 4 f (x) 4 ! 1!./ n ! ! ! ! 54 4, 5 ! f (x0 ) ! 9 !, !4, .. !f (x0) = : : : = f (m 1) (x0) = 0 f (m) (x0) 6= 0 !4 n = m (m)f (x) f (x0 ) + f m(!x0) (x ; x0)m:/ 9 4 # !5 !.
9.7. C! f (x) = 2ex+1 ; x2 ; 4x x0 = ;1:< <5! f (x) 53 x0 = ;1:f (;1) = 5* f (x) = 2ex+1 ; 2x ; 4*f (;1) = 0* f (x) = 2ex+1 ; 2*f (;1) = 0* f (x) = 2ex+1*f (;1) = 2:. ! ! !.<! x0 = ;1( > 5 x > ;123f (x) 5 + 3! (x + 1) < 5 x < ;1:22700;0000000000001 f (;1) = 0 ! ! (;1 5) !!! x ! !53 # ! ! , ! { ! , !! , (;1 5)! (. 9.1). >0;. 9.1 9.8.
C! f (x) = sin2(x ; 1) ; x2 + 2x x0 = 1:< <5! f (x) 53 x0 = 1:f (1) = 1*f (x) = 2 sin (x ; 1) cos (x ; 1) ; 2x + 2 = sin 2(x ; 1) ; 2x + 2*f (1) = 0* f (x) = 2 cos 2(x ; 1) ; 2*f (1) = 0* f (x) = ;4 sin 2(x ; 1)*f (1) = 0* f IV (x) = ;8 cos 2(x ; 1)*f IV (1) = ;8: ! 1!4(x;1)84f (x) 1 ; 4! (x ; 1) = 1 ; 3, !! , x0 = 1 f (x) , , !5 , #228000000000000; 4 ! 5. 1 x0 = 1 ! max f (x) = 1 (.
9.2). >. 9.2 9.9. C! f (x) = x2 ; 4x + cos2 (x ; 2) x0 = 2:< <5! f (x) 53 x0 = 2:f (2) = ;3*f (x) = 2x ; 4 ; 2 cos(x ; 2) sin (x ; 2) = 2x ; 4 ; sin 2(x ; 2)*f (2) = 0* f (x) = 2 ; 2 cos 2(x ; 2)*f (2) = 0* f (x) = 4 sin 2(x ; 2)*f (2) = 0* f IV (x) = 8 cos 2(x ; 2)*f IV (2) = 8: ! 1! f (x) ;3 + 13 (x ; 2)4 , !! ,x = 2 { f (x)* min f (x) = ;3: ; f (x) ! (1 ;3) ! !! 5 #* !5 3 (.
9.3). >000000000000;. 9.3229 !"#!$# $ %&#'# (&; : (1: :1) f (x) = 2 ln x + x2 ; 4x + 3 x0 = 1#2) f (x) = x11 + 3x6 + 1 x0 = 0:; ;1) (1 0) { f (x)2) x = 0 { f (x)||||| x0 = 0 ! 1! 54 ! B!. ! B!2nex = 1 + 1!x + x2! + : : : + xn! + Rn+1(x)*2 x42nxxncos x = 1 ; 2! + 4! ; : : : + (;1) (2n)! + R2n+2(x)*3 x52n 1xxxn1sin x = 1! ; 3! + 5! ; : : : + (;1) (2n ; 1)! + R2n+1(x)*2n2 x4xxch(x) = 1 + 2! + 4! + : : : + (2n)! + R2n+2 (x)*352n 1sh(x) = 1!x + x3! + x5! + : : : + (2xn ; 1)! + R2n+1 (x)*23nln (1 + x) = x1 ; x2 + x3 ; : : : + (;1)n 1 xn + Rn+1(x)*m(m ; 1) x2+: : :+ m(m ; 1) : : : Gm ; (n ; 1)] xn+R (x):(1+x)m = 1+ mx+n+11!2!n!;;;;230< !53 5!3 4 ! ! ! ! ! , 2.
A!1! ! 2 # . 9.10." 1) #8 2 ! ex 0 21 ! 1! n = 3:2) ( n ! 1!," 1 # 49(!49) 4 ex x 2 0 2 2 4 2 0 001:< 1) 85 !5 52 !!4 ! ! 5! ex2 x3xxe 1+x+ 2 + 6 : # 2 !R4(x) = e4! x4" 1# 2 (0 x) x 2 0 2 pe 1 pejR4(x)j 4! 24 = 384 :1 pe < 2 jR4(x)j < 1=192 < 0 01 # 2 3 0 01:2) ! 5! ex ! 1! !53 n- !4! , 2 !pe 1eRn+1 = (n + 1)! xn+1 (n + 1)! 2n+1 :.! 2 2 0 001 ! n 5!49 !4pe 121<=(n + 1)! 2n+1 (n + 1)!2n+1 (n + 1)!2n < 0 001231!2n(n + 1)! > 1000:1 23(3 + 1)! = 8 24 < 1000 24(4 + 1)! = 16 120 > 1000 ! n 4: > !"#!$# $ %&#'# (&1) 4 + 9 1 ln (1 + x) x ; x2 + x3 1x 2 0 2 # n <, )= ) ln (1 + x)2) 1x 2 0 2 +) + 0 001:21#1) 642) n 7:|||||2323 10 . + +- f (x) g() !4 !5, ! ! 2 ! x0, 2 fg((xx)) 51 , 5 00 ! 1! ! xlimx fg((xx)) 54 .! I! ! ! 5 !c 0 1 5 !0 1f (x) = lim f (x) :limx x g (x) x x g (x)!00!0!00K! # !5 ! 9f (x) :limx x g (x)00!0 10.1. (ln(1 + x) :limx 0 sin x!10(ln(1+x))(1+x)ln(1+x)< xlim0 sin x = 0 = xlim0 (sin x) = xlim0 cos x = 1:@ 2 , ! #!5 !5 : !ln(1+x)0 = lim x = 1: >lim=x 0 sin x0 x 0x!00!!!;!!8 #!53 !53 ! 5! !5 ! 4 00 : 10.2.
(x ; ln(1 + x) :limx 0x2!2330!1 ; 1=(1 + x) =x;ln(1+x)==lim< xlim0x20 x 02x= xlim0 21x+(1x+;x1) = xlim0 2(1 1+ x) = 12 :!!!!< # 5! ! 9 4 #!53 !53 . > 10.3. (x ; x2 ; 2x ; 22elimx 0x ; sin x :x ; 2x ; 2 0 !x ; x2 ; 2x ; 2 0 !2e2e= 0 = xlim0 1 ; cos x = 0 =< xlim0 x ; sin xx ; 2 0!x2e2e= xlim0 sin x = 0 = xlim0 cos x = 21 = 2:0 ! I! 2! 5, 2 53 53 53 = 0 !4 ! 00 . 8, ! ! I! 5!4.
< !, 9x2e! 2 3 53 xlim0 cos x ! x;22e9 ! 2 53 53 xlim0 sin x 9 # ! 5, 4 ,9 ! 2 53 53. 1 , ! ! I! 5!4. >C !5 5 5! !. 10.4. (ln cos 3x :limx 0 ln cos x!1 (;3 sin 3x)lncos3x0(cos3x)< xlim0 ln cos x = 0 = xlim0 (cos x) 1 (; sin x) =0!cosx3sin3x3sin3x= xlim0 cos 3x xlim0 sin x = xlim0 sin x = 0 = xlim0 3 x3x = 9:234!!!!!!!!;!;!!!!!0 ! ! ! I!, #!5 !5 . >! I! ! 5! ! . 10.5. (ln x :limx! 1 xn+(n>0)1!1=x = lim 1 = 0:lnx==lim< xlim! 1 xn1 x + nxn 1 x + nxnn>(! I! ! ! ). > 10.6.
(xn :limx + exnn 1xnxn(n ; 1) ::: 1 = 0: >< x lim=lim=:::=lim+ ex x +x +exexC #3 !, ! +1 ln x xn(n > 0)x 5 . (! 5 ! e ! 3 ln x .. ln x < xn < ex ! +1:(! (0 1) (1 ; 1) ! 00 ! 11 ! 5 53 ! ! I! (! ). 10.7. (lim(9 ; x2) ctg x:x 3(+0)!;1!!11;!1!1!1!< @ ! (0 1:) ! ! 00 :0 ! I!:2 0!9;x;2x =2lim(9;x)ctgx=(01)=lim==limx 3x 3 tg x0 x 3 cos 2 x;2 3 = ; 6 : >= cos2 3235!!;!;!1; :limx 0 sin x x< @ ! (1;1): 9!4 5 3, ! ! 00 ! ! I!: 1!0!1x;sinxlim; = (1 ; 1) = xlim0 x sin x = 0 =x 0 sin x x0!x1;cosx= xlim0 sin x + x cos x = 0 = xlim0 cos x + cossinx + x(; sin x) =0!= 2 = 0: > 1 10.8.
(!!!!! 5! ! ! {53 u(x)v(x) ! v = e xlim!x v ln u :limux x!00<49 # ! 00 10 1 ! (0 1) (1 0) v ln u: 10.9. (lim xtg x:1x 0lim tg x ln xx!0!:< xlim0 xtg x = (00) = e<5! !5 !, ! ! I! #!5 !5:1!1xlnxlim tg x ln x = (0 1) = xlim0 ctg x = 1 = xlim0 ; sin 2 x =x 02 x 0!2sinx= ; xlim0 x = 0 = ; xlim0 x = ; xlim0 x = 0:1 ,tg x = e0 = 1: >limxx 0!;!!!!!!236!; 10.10. ( 1 !sin xlim xx 0! !sin x1:! sin x ln x :< xlim0 x= (10) = e xlim/! ! xlim0 sin x ln x1 ! I! #!5 !5:1!lnx1lim sin x ln x = (0 1) = ; xlim0 sin 1 x = 1 =x 02x12xsinx= ; xlim0 ; sin 2 x cos x = xlim0 x cos x = xlim0 x cos x == xlim0 cosx x = 01 = 0:/!! , 1 !sin xlim= e0 = 1: >x 0 x10!!!;!;;!!!!! 10.11.
(lim(3x + x)1=x:x 0xxlim!0 ln(3 +x)=x!< xlim0(3x + x)1=x = (1 ) = e:<5! xlim0 x1 ln(3x + x) ! ! I!:1!!x + x) 0 !x + x) 1 (3x ln 3 + 1)ln(3(3lim x= 0 = xlim0=x 01x= xlim0 3 3lnx +3 +x 1 = ln 3 + 1:;!!!C,lim(3x + x)1=x = e ln3+1 = 3e: >x 0!0, ! 9L xlimx fg ((xx)) ! I!, !! , , # 9 , 9L xlimx f(x)g(x) :0!00!2370 10.12. (x ; sin x :limxx + sin x!1cos x = lim tg2 x< 0 ! 2 53 xlim 11 ;+ cos x x22 9, tg (x=2) ! 1 0 +1 ! !. /!! , ! I! . 8 35 ! 9, 5 5! :x ; sin x = lim 1 ; (sin x)=x = 1: >limxx + sin x x 1 + (sin x)=x!1!1!1!1/! , ! I! #5, , ! 5!4. 10.13.
(p 21+x :limx +xp!12 ) 1=22 1!x(1+xx =1+xp< x lim==lim=lim+x +x1 x +11 + x2p 2 !111+x := 1 = x lim=lim+ x (1 + x2 ) 1=2 x +x ! I! 5 9 3!, .. ! I! 5 #5,3 . .! 5! ! ! :!;1!!limx +!1p1 + x2x1!;1!vuut 1 + x2 = lim= x lim+x +x2!1!11vuut 1 + 1 = 1:2x !"#!$# $ %&#'# (&>1 1:2x #2x ; ln(1 + 2x) #1) xlim0 esin2)lim3x ; 1x 0x2xx3) xlim0 eex+;ex ;;12 #4) xlim0 lnlntgtg2xx #!!;!!2381>116) xlim0 x ; xex #5) xlim0 x ln x#!!7) xlim0(tg x)x#8) xlim0(ctg x)sin x#!!9) xlim0(2x + x)ctg x:!1) 2=3# 2) 2# 3) 2# 4) 1# 5) 0# 6) 1# 7) 1# 8) 1# 9) 2e:|||||# #" # .
# f (x)1) Ga b]2) (a b)3) " Ga b] f (a) = f (b) 2 (a b) ( , ), f ( ) = 0:0#" "'"&.". # f (x)1) Ga b]2) (a b) 2 (a b) ( , ), f (a)f ( ) = f (bb) ;;a0( )*):#" #(. # f (x) g(x)1) Ga b]2) (a b)3) g (x) 6= 0 8 x 2 (a b) 2 (a b) ( , ), 0f ( ) = f (b) ; f (a)g ( ) g(b) ; g(a)00( +,):239.
. , M2 ! 9 -5 I, .. I { 5 ! 5 M2.. g (x) = x g (a) = a g (b) = bg (x) = x = 1 g ( ) = 1: g(a) g(b) g ( ) = 1 ! M2, ! ! I. /!! , M2 ! 9 5 I. 10.14. . : ! 0000a0xn + a1xn 1 + + an 1x = 0;; ! 5 x = x0 na0xn 1 + (n ; 1)a1xn 2 + + an 1 = 0;;; ! 5 ! 2 x0:< f (x) = a0xn + a1xn 1 + + an 1xf (x) = na0xn 1 + (n ; 1)a1xn 2 + + an 1:A f (x) ! 5 N!!:1) 5 Gx0 0]2) ! (x0 0)3) 3 Gx0 0] 5 ;;0;;;f (x0 ) = f (0) = 0:# 9 2 (x0 0) f ( ) = 0 ..0na0xn 1 + (n ; 1)a1xn 2 + + an 1 = 0;;; ! 5 ! 2 x0: > 10.15.
( 5! f (x) = (x + 1)(x + 2)(x + 3)2405 , ! ! 53 f (x) = 0 !5, 53 3.< A f (x) ! 5 N!!:1) 5 3 G;3 ;2] G;2 ;1]2) !3 (;3 ;2) (;2 ;1)3) 3 G;3 ;2] G;2 ;1] 5 f (;3) = f (;2) = f (;1) = 0:# 94 1 2 (;3 ;2) ( , ) 2 2 (;2 ;1) ( , ), 53 f (1) = 0 f (2 ) = 0: 1 , 1 2 { f (x) = 0:A f (x) ! ! 3- , # f (x) { ! 2- ! 3 ! 53 .
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