1. Математический анализ (850924), страница 16
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9 " A; ] x 2 (0 ):2. y(x) = 0 ) ln sin x = 0 ) sin x = 1 ) x = 2 :3. = "% % .4. x limln sin x = (ln(+0)) = ;10+0!lim 0 ln sin x = (ln(+0)) = ;1x!;;;x = 0 x = { % .x = ctg x:5. y = cossin x0maxy= ( 2) = 0y =:6. y = ; sin12 x < 0 { ,.7. 9 " A; ] y(x) = ln sin x $ + ":00264;;;; ; ;;" * , 7 * .
> !:q322) = ( ; 1)21) = 2 +22 2 ;;39 ; 3 p4) = ln 5) = ln( 2 cos )3) = ln + 6 ; 1yyxxxxx1)3)x3yyx xxx2)4)5)|||||265yx : 14 !" #$"%&', ) ""*+,+-% $+!"" $+!"" ! +"* %+ ", , : y = y(x) " * 9x = '(t) = (14:1)y = F(t) "yx = xytt % +"*%dy = dy=dt :dx dx=dt000 14.1. 9,dy , dx9x = a cos t = y = b sin t (14:2)(a > 0 b > 0 { ).dy = b cos t =) dy = b cos t = ; b ctg t: >< dx=;asintdtdtdx ;a sin t ady , +$ , * 7, * " 6 " dx t2.
6!, + , " d y , " dy 7 dx2dx x 7* t = ' 1(x), ' 1 { + ', 7, + , ,.' d2y = d dy ! = d ; b ctg t! = d ; b ctg t! dt =dx2 dx dx dx adt adx;b 1 =; b := a sin2 t dx=dta2 sin3 t266;G , * (14.1), *!, , y = y(x).3 * !x2 + y2 = 1a2 b2;(! * " * % , * t ).3 % ! vuubt12x; a2 :vuu y = bt1; xa22 , 7 { y = ;D , ", * * (14.1) "*% , y = y(x).6 " ,, ", * (14.1)2ydyd " dx dx2 7 " *, $% + " , y = y(x), % , * ,, $% ! , y = y(x), * +. 14.2. 6 , ", * 8< x = a(t ; sin t)(a > 0): y = a(1 ; cos t) .< 3* x y % "*, t.
9,dy dy : " dxdt dt dxdx = a(1 ; cos t) dy = a sin tdtdtdy = dy=dt = a sin t = 2 sin(t=2) cos(t=2) = ctg t :dx dx=dt a(1 ; cos t)22 sin2(t=2)267G * * , :dy = 1 x = 2an(t = 2n) dxdy = 0 x = a + 2an(t = + 2n) n = 0 1 2 : dxdyC $ " ", dx;? " 2na<x<a +2an(2n < t < 2n + ), + a+2an < x < 2a+2an( + 2n < t < 2 + 2n ):? * % x = a +2an(t = + 2n)y(a + 2an) = 2a. G !% * % Ox, y (a + 2an) = 0.? * % x = 2an y(2an) = 0.
G !% * % Oy, y (2an) = 1.9, " 00d2y = d dy ! = d ctg t ! = d ctg t ! dt =dx2 dx dx dx2 dt2 dx1 =; 111=;= ; sin2(1t=2) 12 dx=dt2 sin2(t=2) a(1 ; cos t)4a sin4(t=2) :G * * x = 2na (t = 2n). 9 %, 7% 7 * * , ,.9 ( . 14.1) " ,. >;. 14.1 14.3. 6 ,, ", - * 8< x = a cos3 t: y = b sin3 t268(a > 0 b > 0):< E, * " 7 * t:88< x = a cos3 t< cos2 t = (x=a)2 3() : sin2 t = (y=b)2 3 :: y = b sin3 t==" * 722sin t + cos t = 1 * , !2=3x !2=3y+ b = 1:) " , + " * .3* x y % "*, t. 4 cos3 t sin3 t { * , 2, * " t % 0 2. 6 ! + " x + " A;a a], +" y + " A;b b]. C, 7 , * y = b x = a.dy dy 9, " dxdt dt dxadx = ;3a cos2 t sin t dy = 3b sin2 t cos t:dtdtdy = 3b sin2 t cos t = ; b tg t:dx ;3a cos2 t sin t aG * * ,, % " * :dy = 0 x = a (t = 0) x = ;a (t = ) x = a (t = 2) dxdy = 1 x = 0 (t = ) x = 0 (t = 3 ): dx22C $ " ",dy :dx;269" " , * (14.1) y = f (x), A;a +a].) y = f1(x) "*(0 t ), y = f2(x) { < ( t 2).
? y = f1(x) " ;a < x < 0 + 0 < xdy< a, * x = 0 ymax = b. 3 !, * dx = 1, , , , . ? y = f2(x) + ;a < x < 0 " 0 < x < a, * x = 0dy = 1, , ymin = ;b. 6 x = 0 dx!, * , . 3 * % (;a 0)dy = 0. 3 !% * % , (a 0) " dx ". 9, " !!2d y = d dy = d ; b tg t = ; b d (tg t) dt =dx2 dx dx dx aa dtdx1 =; b 1b== ; ab cos12 t dx=dta cos2 t ;3a cos2 t sin t 3a2 cos4 t sin t :d y > 0 0 < t < { ) : dx22d y < 0 < t < 2 { . 9 2dx" .
1 " ,.2;> , : " % % = ()270; 0 { , , { , , , 7 x * " ,* * ,9x = () cos =y = () sin :2yddy6 " dx dx2 7 " " , = () 7 + ", <: ,, * ,, $%+ " = (x) % .. 14.4. , " % % = sin 3 .< 6 " 0 =6 , * , 0 1. = * " $ *, %$ " ( 77 *, , ).
: ! $$ * * , *+ = sin 3 + . 6 $ * , ( = 0) =6 ( 7 7 *, ) ! * $ * " ( = 0) , 1. 6 ,<" =6 =3 * * $ =6 ( 7 ), * , $ = sin 3$ * " ( = 0): 3 " * " " I- .!2' "*, 2 3 3 * < 0: J ,271 7 + , ! "*, * 7* = =3 = 2=3 * , (9K).6 7 " % "" 0 2 ( " +), , * " % " ",! " % , ",. 0 ! 6 ! 3 ! 23 ! 56 ! ! 43 ! 32 ! 53 ! 2; 0 ! 1 ! 0 9K 0! 1 ! 0 9K 0 ! 1 !0 9K 0' * , %, % % * %.
' ! " * " * * ,:x = sin 3 cos = 12 (sin 4 + sin 2) y = sin 3 sin = 12 (cos 2 ; cos 4) "dy = 2 sin 4 ; sin 2dx 2 cos 4 + cos 2p3dy = 0 dy = ;p3 dy =dx dx dx =0= 6 = ' , ,= 3 ='j =0 = 0 'j = 6 = ; 3 'j = 3 = 3 : =272 =C, " " I- *, = 0 = =3 = =6 * = =6 $ " ".J* 7 , * x " " , + + ,. > 14.5.
6 ,, ", - % % = a(1 ; cos ) .E, * (;) = ():1 "*, * * , *% = C = ;C . J ! * * , ! 8C ( +, % *,), 7 * , .6! * , " 0 : 6 $ * , = 0 =2 , 7 7 *, , , * , 0 :6 ,< " =2 * *$ =2 ( 7 , * , $ = a(1 ; cos ) $ * * , , = a = 2a.' "*, 2 (0 ) " , +.
0 ! 2 ! 0 ! a ! 2a' * , %, % % * %. ' ! " 273 * " * * ,:x = a(1 ; cos ) cos = a(cos ; cos2 )y = a(1 ; cos ) sin = a(sin ; 12 sin 2):9, "dx = a(; sin + sin 2) = 2a sin cos 3 d2 2dy = a(cos ; cos 2) = 2a sin 3 sin d2 2dy = tg 3 dx2dy = 0dx =0 ' 'j =0 = 0dy dydy = 1=1=0dx = 3dx =2 3dx = =%= ,:'j = 3 = 2 'j =2 3 = 0 'j = = 2 : == C, , " = 0 = 23 = 3 = { .dy 2 A0 ]: " dx;dy" dxy (x)C, * % ,, $% " 0 =3 2=3 = y(x) " , * %, $% " =3 2=3 = (x) +.9, "d2y = d tg 3 ! = d tg 3 !, dx =dx2 dx 2d 2 d274031 := 32 cos2(31 =2) 2a sin(=2)1cos(3=2) = 4a sin(=2) cos3(3=2)d y 2 A0 ]: )*, * " dx2 " cos3 32 :C, * ,, $, 2 A0 =3] , * ,, $, 2 A=3 ] .@* , , * , " .
>2 &, (& )* )& , .1) = cos 2 a2) = cos 3 3) = (1 + cos );;1) +) (2) ,) (275a :3) -;|||||276 15 f (x) , .. maxf (x)2E minf (x) f (x)2E . f (x) ! "a b] ! $%& & & , ' maxf (x) minf (x). (22 ' , ), ) ) , )'*.1. ,% f (x) (a b):2. $ f (x) () ) ) f (x) (a b)) "a b]:3.
$ ) % f (x) & & . / ) f (x) "a b]:xxxab]xab]. 15.1. ,% & & y = x ; 5x + 5x + 1 x 2 ";1 2]:< y0 (x) = 5x ; 20x + 15x = 5x (x ; 4x + 3) = 5x (x ; 1)(x ; 3)5y0 (x) = 0 ! x = 0 x = 1 x = 3:6 x = 0 x = 1 !) . ,%) :y(0) = 1 y(1) = 2 y(;1) = ;10 y(2) = ;75y(x) = y(;1) = ;10: >maxy(x) = y(1) = 25 2min2;;5443322211x12]22x277212]3 !" #! ( )=y x:max ( ) = 132;22]xy x4x;2 2+5min ( ) = 42;22]xxy xx2 ;2 2]::||||| f (x) ! , !, f (x) ) . $ ! ! ! ) ,''* !)% ) !) ) !) !) .
15.2. 9) '(! ; 1 x < 0f (x) = ;x x+ 1 !0 x < 1:(; 1 < x < 0< f 0 (x) = ;11 !! 0 < x < 1:0:) f (x) * , ! % x = 0.f (;1) = 15 f (0) = 15 f (1) = 2:maxf (x) = 25 2minf (x) = 1: >2;;x11]x11] 15.3. 9) 'f (x) = tg x"!1) ! 0 2 5!;2) ! 2 2 :278< f 0 (x) = cos1 x : 6 ) cos x = 0 x = 2 + k k 2 Z5!1) 0 2 , ! f (0) = 0 !lim; f (x) = +1: :min f (x) = 0 2max f (x) = +152!2) ; 2 2 .
9) !) x = 2 !, lim f (x) = +1 !;lim f (x) = ;1: <),! ;min f (x) = ;1 2 ;max f (x) = +1: >2;2xx0=2)x=2x(x0=200=2)x=2+0=2=2)x(=2=2) 15.4.9) 'f (x) = 2 tg x ; tg x ! 0 x < 2 :!2tgx2(1;tgx)20< f (x) = cos x ; cos x = cos x : $ 0 2 ! ) x = 4 5 f (0) = 05 f 4 = 15 !f 2 !), !lim; f (x) = !lim; tg x"2 ; tg x] = ;1:9, 2min f (x) = ;1 2max f (x) = 1: >222xx2=20=2)0xx=200=2)% !" #!!!"# $% &!'% !"() : 111) ( ) = +2) ( ) = sin ; cos 4; 4 0 4),f xxxf x1) min( ) = ;120 4)f xx2)min2( 4x= ]f xp( ) = 22f xmax( ) = 2,20 4)xmax ( ) = +1,2(=4]xf x|||||279xx: 6 ' ) ) . 15.5.
>) ! ) !) !! . 9, ! 10 / ) ! ' 30 . , )( * ) ' 480 . . : % !) * ) 1 ! )&%? 6 ! * ) ?< : / { ) !, v / { !), ) x = v , ) { !!.9 x=30 / ! v=10 / ), = 0 03::) !) 1 v1 , ! * ) 1 ! ) f (v) = (x + 480) v1 = 0 03v + 480v :329) !' ' f (v) % v 2 (0 +1) :f 0 (v) = 0 06v ; 480v 52f 0 (v) = 0 ! v = 205f (20) = 485 f (0) f (1) !), limf (v) = lim!1 f (v ) = 1:!v9 48 . { & * ) 1 !.>) ! ) ! x = 0 038000 = 240/, ') * ) 240+480=720 /.v0> & ) % !, )% !.9) * ) ( ) )% !) ' * ).
! & ) ' !, ) % & ) & , ) % .280 15.6. C 9 %&% 5 ! , !% 15 , ! %&% ( ! ). ) !& ! 5 /, 4 /, ! ) !, !! %& .< : ! , { , { , %& , D { , ) )! .;9 )p = 9 , = 15 . : CD = ,)p AD = 81 + x , , !,81 + x . BD = (15 ; ) , ! 4! ! 15 5; x . $ ) pt(x) = 814+ x + 15 5; x :( % & t(x) "0 15] :px181 + x 55x;4pt0(x) = p;=4 81 + x 520 81 + xt0(x) = 0 ! = 12, ! t0 (x) < 0 ! x < 12 t0(x) > 0! x > 12. <), t(x) = 12.( , ) ! 3 . > 15.7. ( * &%, D 72 , ! 1:2. 6 ) , ! ! &%?2222232812< : { & , ) 2 {) . $ h ) &2x h = 725:h = 36x!3672: ! S (x) = 2 2x + x + x = 4x + 216x522228(x ; 27) 5S 0 (x) = 8x ; 216=xx322S 0 (x) = 0 ! x = 3:9, ! ! ) &%, ) * 3 , 6 4 .
> 15.8. /% ), D : 6 ) ) ), ! &%?H { . /D< : R { ) ),) R H = ) H = R : : 22S (R) = R + 2RH = R + 2R S 0 (R) = 2R ; R2 = 2(RR ; ) ss0S (R) = 0 ) R = H = : >2232233% !" #!1) - "!# .'" #" N # $'" A ('!. 15.1) !'! !! NP,!"() #" ! 1 "'$ AB, .'2") ' $'" A. 3!.'# . !! #"# , . 1 "'$. 4 . P 1.'#! !!, % ) ! .'# .'" #" N # $'" A . !! . 1 "'$ % , ! #!, AB=500 , NB=100 ?2) 6 ' .'$, #'$ .'$ ('!. 15.2). 7" .'' & $'%. 9' 2 ''2 " .'.! !# !#?px282y; ;. 15.1!p ,1) 500 ; 1003. 15.22) = 4 2+xpy1=2 ; ; 2|||||283pxx: ""! 1$ !):(x ; 2x ; 1)(x + 1) 51) lim!;x + 4x ; 5p1p+ 2x ; 3 52) lim!x;2ln(1 + sin x) 53) lim! sin 4(x ; )x ; 154) lim!ln x5) lim(1 ; ln(1 + x )):!3x41x4x022x13x3=(x2 sin x)0! 2$ !):(x ; 3x ; 2) 51) lim!;x+xp1 ; xp; 3 52) lim!;2+ x1 ; cos(10(x + )) 53) lim!e ;1px ; x + 1 ; 154) lim!ln xpx) :5) lim(cos!3x1x8x23x202x1x01=x284! 3$ !):(x + 3x + 2) 51) lim!; x + 2x ; x ; 2px;15p2) lim!x ;13x ; 5x 53) lim!sin 3x1 + cos 3x 54) lim!sin 7x!1+x25) lim:!1 + x32xx31123222x0x21=xxx2x0! 4$ !):(2x ; x ; 1) 51) lim! x + 2x ; x ; 2ppx+13;2x + 152) lim!x ;91 ; cos 2x 53) lim! cos 7x ; cos 2x4) !lim 1(;;sin4x2)x 5p5) lim(2;3):!2x1x3x0x=4322222arctgxx2= sin x0285! 5$ !):(x + 2x ; 3) 51) lim!; x + 4x + 3xpx ; 6 + 252) lim!;x +84x53) lim! tg( (2 + x))1 + cos x 54) lim!tg x!1+sinxcosx5) lim!1 + sin x cos x2x23323x32x0x12ctgx03x:! 6$ !):(x ; 2x ; 1) 51) lim!;x + 2x + 1px ; 2px ; 4 52) lim!2x53) lim! tg"2 (x + 1=2)]4) !lim tgtg3xx 5!45) lim5 ; cos x:!3x2414x16x0x=22 3x1= sinx0286! 7$ !):(1 + x) ; (1 + 3x) 51) lim!x+xp9p+ 2x ; 5 52) lim!x;21 ; cos x 53) lim!4xsin x ; tg x 54) lim!(x ; )3x0x8533x202x24px))5) lim(1;ln(1+!3xp4 3x= sin0x:! 8$ !):x ; 2x + 1 51) lim! 2x ; x ; 1p2) lim 1 ; 2x + x ; (1 + x) 52x212xarcsin 3xp 5p3) lim!2+x; 2px ; x + 1 ; 154) lim!tg x!0xx02x15) lim2;e!x02arcsinp 3=xx:287! 9$ !):x ; 3x ; 2 51) lim!; x ; x ; 2p8 + 3x + x ; 2 52) lim!x+x2 ;2 53) lim! ln(1 + 4x)cos 5x ; cos 3x 54) lim!sin x5) lim(cos x):!3x213x220x+1x0xx021=(x sin x)! 10$ !):x + 5x + 7x + 3 51) lim!; x + 4x + 5x + 2p27 + x ; p27 ; xp52) lim!x+2 xarctg 2x 53) lim! sin"2 (x + 10)]4) lim sin 7x ; sin 3x 5x1323233x0x03e ;e5) lim(1 + sin 3x)!!2x4 2x22x041= ln cos x:288 ""! 11.
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