1. Математический анализ (850924), страница 13
Текст из файла (страница 13)
. ; f (x0) = f (x) = x0 (. 8.1).0. 8.1 y ; y0 = f (x0 )(x ; x0) 0(8:1) 0 = f (x0 ):! f (x0) 6= 0 ! # ! = f (x) x0 ; f (1x ) 0, ! (8:2)y ; y0 = ; f (1x ) (x ; x0):0000! f (x0 ) = 0 ! !!! y = y0(8:3) ! !!! x = x0:(8:4)0 8.1. ( ! ! p = 1 ; x2 x0 = 35 :p 2< f (x) = 1 ; x = p 1 2 (;2x) = ; p x 2 * )2 1;x1;x3!3! vu 3 !2 4u33=5t= ; 4 * y0 = f 5 = 1 ; 5 = 5 :f 5 = ;q1 ; (3=5)2000210 ! (8.1), ! :!334 ; 5 = ;4 x ; 5! = ; 34 x + 45 :.! ! ! ! ! (8.2):; f (;13=5) = ; (;31=4) = 43 ) ! (.
8.2):!434y ; 5 = 3 x ; 5 ! y = 43 x: >0;. 8.2 8.2. / ! ! - = ( ; 1)3 + 2x (1 2).< 0 0 = (1) = 2. .!,f (x) = 3(x ; 1)2 + 2 ) f (1) = 2. ! ! (8.1),! ! ; 2 = 2(x ; 1) ! = 2.1 ; f 1(1) = ; 12 , ! 2 ! (8.2), 3 ! (.
8.3):y ; 2 = ; 21 (x ; 1) ! y = ; 12 x + 52 : >000211;. 8.3 8.3. ( ! ! p = 1 ; x ; 2 ; 0 = ;1.< 0, pf (x) = ( ;x ; 1)2 ) y0 = f (;1) = 0*! 1 ; p;xp1f (x) = 2( ;x ; 1) ; 2p;x = 2p;x ) f (;1) = 0: ! (8.3) ! # ! = 0 (..! 4 )./! ! (8.4) ! ! = ;1 (. 8.4). >00;.
8.4! s = s(t) 5 (t { , s { , ! ), s (t) 5 t.0212! 8! .< / t s (t) = 10t + 6: > 8.4.s(t) = 5t2 + 6t + 3.0 !"#!$# $ %&#'# (& y = f (x) x0, :1) y = x2 ; 5x + 4 x0 = ;1#2) y = x3 + 2x2 ; 4x ; 3 x0 = ;2:1) 7x + y ; 3 = 0 x ; 7y + 71 = 0#2) y ; 5 = 0 x + 2 = 0:||||{ ! = f (x) 4, 9:y = f (x + :x) ; f (x) :y = f (x):x + :x0 = (x :x) { ! :x ! 0, ! ! :x 9dy = f (x):x5 ( x).! f (x) 6= 0, :x ! ! :x ! 052 4 dy.# !53 3 j:xjj:xj jdyj ! dy ! 9 :y dy:.! dx = :x:21300/ ! , ! 5 !dy :dy = f (x)dx f (x) = dx( ! 2 ! ! .); ! . 8.5. M { ! = f (x) ( ).
1 9 N x + dx 9 ! 9 : (! AN ), ! dy 5 , 9 + dx (! AT ).00;. 8.5 8.5. .! = x2 5 : dy, 5! : dy = 1 dx = 0 1 0 01. 8 2 ! : dy.< : = y(x + dx) ; () = (x + dx)2 ; x2 = 2xdx + (dx)2 *dy = f (x)dx = (x2) dx = 2xdx:00< x = 1 dx = 0 1 : = 2 1 0 1 + (0 1)2 = 0 21*dy = 2 1 0 1 = 0 2, ..
: dy, !4 2 # !j:y ; dyj = j0 21 ; 0 2j = 0 01*! 2 ::y ; dy = 0 01 = 1:y0 21 21214(!! 5% ).< = 1, dx = 0 01 :y = 2 1 0 01+(0 01)2 == 0 0201* dy = 2 1 0 01 = 0 02, .. { : dy. <# ! ! : dy 5! 9! 2 4, !4 2 j:y ; dyj = j0 0201 ; 0 02j = 0 0001 ! 2 :y ; dy = 0 0001 = 1 :y0 0201 201.. ! 0 5%. / ! 5 5!:y dy dx = 0 1 dx = 0 01, , dy { ! :y 2 9 dx !dy ! 4 9 : >.4;1x 8.6.
y = 4 : ( dy x = 1* dx = 1.x 0+ 14 1!32x;1< dy = y (x)dx = @ x4 + 1 A dx = 1 ; x4 + 1 dx = (x48+x 1)2 dx:1 ! dy x = 1 dx = 18 13 1 = 2:(14 + 1)21 dx = 1 !, ! dy 9 ! 4;12: = (1 + 1) ; y(1) = 24 + 1 ; 0 = 1517 : >000 8.7. ( ( 4 dx) ! () x = 0, !:1) y = earcsin 3x*2) y = ln(tg2 x + 1):< 1) dy = earcsin 3x dx = earcsin 3x q 3 2 dx =1 ; (3x)= earcsin 3x p 3 2 dx )1 ; 9x2150 x = 0 earcsin 3 0 p 3 2 dx = 3dx*1;902) dy = (ln(tg2 x+1)) dx = (tg2 x2+tg1)xcos2 x dx ) x = 0 !4, 2 tg 0(tg2 0 + 1) cos2 0 dx 0: >0 !"#!$# $ %&#'# (&px ; 11) dy, y = px + 1 #332) ( dy x = 0 dx = 0 1, y = 2x2 arcsin x#3) y = 2tg x, ) '(dx) dx, x = 0#4) * x +x 1 )) ) + y(x + dx) y(x) + dy x = 0 dx = 0 1.1) dy = 3,p3 x(p32x + 1)]2 dx#2) 0 1#3) '(x) = ln 2dx#4) .) + 1=110 { 0 1 ( 10%).||||{3 2.
x ! ! x0 = x :x :x { !4 2 .! , x = x0 + dx216jdxj :x:<5! = f (x) ! 0 = f (x0 ) !y ; y0 = f (x0 + dx) ; f (x0 ) = :y:! 2 :x !, ! jdxj ! :y dy..! dy 3 !44 2 : 5! , ! !jdyj :y ! 5! ! y0 = y :y :.! = dy = x 1dx* jdyj = jx 1j jdxj jx 1j:x:;;/!! ,;:y = jx 1j:x:;:x 5 y # ! 5 2 y = :x=jy jjxj2y = jj x: 8.8. !5 x ! x = 2, !4 2 ! 0 01. <5! = 3 !44 2 : . ( ! 5 2x y .< :x = 0 01* x = jxxj = 0 005 ) y = 3 0 005 = 0 015*:y = jyj y = 23 0 015 = 0 12*y = 8 0 12: >217 !"#!$# $ %&#'# (&1) *(, ( = ax (a > 0) 1 +:3y = ax j ln aj 3x# y = 3x j ln aj#2) 4 )) ) + 1 11 x, = 3 0 01#23) *(, = ln x +y = 3xx = x#4) 4 )) + 1 1 ln x, x = 2 0 1. 1) jdyj = j(ax) dxj = ax j ln aj jdxj ax j ln aj 3x 3y = ax j ln aj3x#0y = 3jyyj = a j lnaxaj3x = 3x j ln aj#xln 2 < 1 # = ln 2 < 1 #2) 3y = 800800 y 100 1003) jdyj = j(ln x) dxj = j x1 dxj 3xx ) 3y = 3xx = x ( x)#4) 3y = 0 05:0||||{ y(x) !! 5.
(ln y(x))5 y(x). ! ! (ln y(x)) = y(1x) y (x) !y = y(ln y) { ! .@ ! ! ! 53 u(x)v(x) (u(x) > 0).21800000 8.9. ( 4y = x1=x (x > 0):< ( !4 4:ln y = x1 ln x =) (ln y) = ; x12 ln x + x12 = x12 (1 ; ln x): ! ! :y = x1=x x12 (1 ; ln x) = x1=x 2(1 ; ln x): > 8.10. ( 400;y = x5x (x > 0):< ln y = 5x ln x =) (ln y) = 5x ln 5 ln x + 5x x1 =0ln 5 x ln x + 1 ! :x ! ! :(y) = x5x 5x ln 5 x xln x + 1 = x5x 1 5x (ln 5 x ln x + 1): > 8.11.
( 4= 5x0;y = (sin x)2 sin x + (sin x)sin x ecos x (0 < x < ):< ( ! 4 y1 = (sin x)sin x:x=ln y1 = sin x ln sin x =) (ln y1 ) = cos x ln sin x + sin x cossin x= cos x(ln sin x + 1) =) y1 = (sin x)sin x cos x (ln sin x + 1):0, y2 = (sin x)2 sin x = y12, #00y2 = 2y1 y1 = 2(sin x)sin x (sin x)sin x cos x (ln sin x + 1) =00= 2(sin x)2 sin x cos x (ln sin x + 1):219 ! 3 :dy = h2(sin x)2 sin x cos x (ln sin x + 1)+dxi+(sin x)sin x cos x (ln sin x + 1) ecos x+hi+ (sin x)2 sin x + (sin x)sin x ecos x (; sin x) =h= (sin x)sin x ecos x (2(sin x)sin x + 1) cos x (ln sin x + 1);i;((sin x)sin x + 1) sin x : >A! ! 4 3!3, ! ! ! 5 ! ! , , ! 4, !! ! ! !.
8.12. ( 4p2p2x+6p 2 x + 1:y=x +3< 4 , ! . ( 9, ! ! :1 ln(x2 + 6) + 1 ln(x2 + 1) ; 1 ln(x2 + 3)*ln y = 151061 2x + 1 2x ; 1 2x =2x(ln y) = 15x2 + 6 10 x2 + 1 6 x2 + 3 (x2 + 6)(x2 + 1)(x2 + 3) :p2p2x+6p 2 x + 1 (x2 + 6)(x22+x 1)(x2 + 3) =y=x +32xqq= q 2:>(x + 6)14 (x2 + 1)9 (x2 + 3)710156010150615106! y(x) 6= 0, !53 , ! y = y(ln jyj) :220001 (ln jxj) = x1 8x 6= 0 x > 0(ln jxj) = (ln x) = x1 x < 0(ln jxj) = (ln (;x)) = ;1x (;1) = x1 : # , ! , 9 !, 9 . @! ! 5 9!5 ! . 8.13.
( 433y = (x ;(x1)+(x2)+6 5) (x 6= ;2):< 0 ! 9! ! : x 6= 1* ;5* ;2ln jyj = 3 ln jx ; 1j + 3 ln jx + 5j ; 6 ln jx + 2j =)(ln jyj) = x ;3 1 + x +3 5 ; x +6 2 = (x ; 1)(x 54+ 2)(x + 5) =)3(x + 5)32 (x + 5)2(x;1)5454(x;1)y = (x + 2)6 (x ; 1)(x + 2)(x + 5) =:(x + 2)78, ! ( ! x 6= 1* ;5* ;2) 5 ! , 5 x = 1* x = ;5.B , # ! , 5 ! 4 x = 1* x = ;5:C !:2 (x + 5)254(x;1)(x 6= ;2): >y=(x + 2)700000000 8.14.
( 4px ; 1y = px + 3 px ; 2 :4202215< 94 ! ! :1 ln jx + 3j ; 1 ln jx ; 2j =)ln jyj = 14 ln jx ; 1j ; 205(ln jyj) = 4(x 1; 1) ; 20(x1+ 3) ; 5(x 1; 2) =; (x ; 1)(x +1 3)(x ; 2) =)10px ; 1;1y = px + 3 px ; 2 @ (x ; 1)(x + 3)(x ; 2) A =1qq= ;q x > 1* x 6= 2: >(x ; 1)3 (x + 3)21 (x ; 2)60402045205 !"#!$# $ %&#'# (& (1 :1) y = xsin x#3) y = (x2 + x ; 2)3(x2 ; 1)2(x2 + 3x + 2)#1) y = xsin x0cos x ln x + sin x #x2) y = xxx+x 1 ,x ln x(ln x + 1) + 1]#3) y = 6x(2x + 3)(x + 2)3(x + 1)2(x ; 1)4#2 19x + 61) x 6= 1# ;5:4) y = ; 23 (x + 2)(x4 +q3(x ; 1) 3 (x + 5)202) y = xxx #2 3 x+54) y = (x+2)(x 1)3 :;00|||||222;p 9+ +, .
y (x) 5 y (x) y (x), ..y (x) = (y (x)) :D! !4 " y (x) = (y (x)) ::.! 53 52 5 23 ! 4 5yIV (x) yV (x) ! 5 3y(4) (x) y(5)(x) : ! n- y(n) (x) = (y(n 1) (x)) :4 y (x) ! 5 !53 54 , 4y(x) 5 y(x) = y(0) (x):E5 -! x0, ! 3 ! x, x!4 x0. 9.1. ( y (3=5)y = arcsin x:< y = (arcsin x) = p 1 2 = (1 ; x2) 1=2 =)1;x 1!21=2y = (y ) = ((1 ; x ) ) = ; 2 (1 ; x2) 3=2(;2x) = x(1 ; x2) 3=2:2230000000000000;0000000000;;0;;8401 3 !2 3=2 753y 5 = 5 @1 ; 5 A = 64 : >< 53 53 !3 ! !, 5494 4 n{ , 9 4 3 4 !4 . 9.2. ( y(n)y = 2(3x+1)=2:5 y y y : !< ( !! !233(3x+1)=2(3x+1)=2y =2ln 2 2 *y =2ln 2 2 *003!;000000000!33yln 2 2 :8, ! !! ! ln 2 32 .
#3!n(n)y = 2 ln 2 2(3x+1)=2: >000= 2(3x+1)=2 9.3. ( y(10)(1)y = (2x + 1)ex:< (3 5y = 2ex + (2x + 1)ex = (2x + 3)ex*y = 2ex + (2x + 3)ex = (2x + 5)ex*y = 2ex + (2x + 5)ex = (2x + 7)ex:< !, y(n) = (2x + (2n + 1))ex. .! ! :1) n = 1 y = ((2x +1)ex) = 2ex +(2x +1)ex = (2x +2 1+1)ex{ *2) ! n = k, ..y(k) = (2x + (2k + 1))ex =)22400000000y(k+1) = (y(k) ) = 2ex + (2x + (2k + 1))ex = (2x + 2(k + 1) + 1)ex {! n = k +1, 2 ! .1 y(10) (1) = (2 1 + (2 10 + 1)) e1 = 23e: > 9.4.
( y(n)y = sin x:< < # ! ! 94 ! ! y(n) ,! ! 5 , ! !5 :!y = cos x = sin 2 + x *! !! !y = cos 2 + x = sin 2 + 2 + x = sin 2 2 + x *! !! ! y = cos 2 2 + x = sin 2 + 2 2 + x = sin 3 2 + x : !(n)( , y = sin n 2 + x : > 9.5. ( y(n)y = cos x:< cos x = (sin x) : <! 2 ! 9.10,!:!(n)(n)(n+1)(cos x) = ((sin x) ) = (sin x)= sin (n + 1) 2 + x =!! ! = sin 2 + n 2 + x = cos n 2 + x : > 9.6. 1 !, , 4, ! S (t) = t4 ; 3t2 + 2t.
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