1. Математический анализ (850924), страница 15
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@ 1 2 !9 ! (;3 ;2) (;2 ;1) . > . ! ! ! 5 N!!, ,, !45 ! , , ! . 10.16. . 9 4 !5 I arctg x2 ; arctg x1 x2 ; x1 ! x2 > x1:< f (x) = arctgx f (x) = 1 +1 x2 5 I !, 1 = arctg x2 ; arctg x1 2 (x x ):1 21 + 2x2 ; x1K5, ! 8 1 +1 2 1 !arctg x2 ; arctg x1 1x2 ; x1!arctg x2 ; arctg x1 x2 ; x1: >241000000 10.17. . 9 4 !5 I -j sin 3x2 ; sin 3x1j 3jx2 ; x1j:< f (x) = sin 3x f (x) = 3 cos 3x 5 I !, ; sin 3x1 3 cos 3 = sin 3xx2 ;2 x1 { , !9 x1 x2:K5, j cos 3 j 1 !1sin3x;sin3x21 1:j cos3 j = 3 x ; x2184 !, j sin 3x2 ; sin 3x1j 3jx2 ; x1j: >0 !"#!$# $ %&#'# (&1) *(, ?9 = 1 @, .. @ ( 1 ?9 1 #2) *( : a0xn + a1xn 1 + ::: + an 1x = 0 1 x = x1 91 x = x2 na0xn 1 + (n ; 1)a1xn 2 + ::: + an 1 = 09 1 91 + x1, + x2#;;;;;3) : ( f (x) = (x ; 1)x(x + 1)1, 1: f (x) = 0 ( 1, 1: 9.03) * , 9=: (;1 0) (0 1).|||||242 11 /-0 $ % / 9 4 I- 3 , 5 # .! f (x) > 0 8 x 2 (a b), f (x) ( b) (.
11.1). ! f (x) < 0 8 x 2 (a b), f (x) ( b) (. 11.2).0;0f (x) > 0f (x) < 000. 11.1. 11.2 &! f (x) 5 x0 f (x) 3 x0 ! " + " " ; "( " ; " " + "), x0 ! ( ), f (x0 ) { ( ) f (x) (. 11.3 11.4). < 3 !3 x0 5 #, f (x0 ) { f (x).0;. 11.3.
11.4243< 53 !3 #5 !4 ! 2 ! 2 4 ! x0, 9 ! ! . # 54 .(! 2 2 f (x) ! ! E 54 4 maxf (x)*x Eminf (x):x E22 ! " C! f (x) 9 4 f () !49 3:1) , 53 f (x) = 0 1 ! 9L ( I- f (x))*85 # ! 4 ! ! f (x) !5 f (x).2) ! f (x) 3 !3 *3) !5 5 f (x),! # #5*4) # = f (x). 11.1. C! 4 4, ! 4f (x) = 31 x3 ; 12 x2 ; 2x + 2 13 :< . ! 52 3 !.1) f (x) = x2 ; x ; 2*f (x) = 0 () x = ;1 ! x = 2.
f (x) ! 3 ! 53 3 , !! , ! ! !, .. 1 = ;1 2 = 2.2) M 4 ! ! f (x) ( ! # ! ) !: (;1 ;1)* (;1 2)* (2 +1). 8! f (x) 00000000244 #3 !. .! # 5 ! , , ;2 2 (;1 ;1)* 0 2 (;1 2)*3 2 (2 +1) ! f (x) #3 3.f (;2) = 4 > 0* f (0) = ;2 < 0* f (3) = 4 > 0:84 ! 5, f (x) > 0 2 (;1 ;1)* f (x) < 0 2 (;1 2)* f (x) > 0 2 (2 +1) (. 11.5).3) 1 f (x) > 0 3 2 (;1 ;1) (2 +1), 3 (;1 ;1) (2 +1) f (x) . C , f (x) < 0 2 (;1 0) !, # ! f (x)5. f (x) 3 = ;1 (! ) " + " " ; ". /!! , # !! 5 , 5f (;1) = 31 (;1)3 ; 12 (;1)2 ; 2 (;1) + 2 13 = 3 21 *0000000000;; 3 = 2 f (x) " ; " " + ", # # !! 5 , 5f (2) = ;1. N! 5 5 3 . 11.5.01&"2maxf (x) = f (;1) = 3 21minf (x) = f (2) = ;1.
11.54) ! 3, # (. 11.6). >A(;1 3 12 ) B (2 ;1). 11.6245 11.2. ! f (x) = 14 x4 ; 2x2 + 1 9 .< f (x) = x3 ; 4x*f (x) = 0 () x3 ; 4x = 0 () x1 = 0*x23 = 2 { .0 f (x), !5 f (x) (a 5), # 3 5 5 . 11.7.0001&"2;;minf (x) = f (;2) = ;3minf (x) = f (2) = ;3maxf (x) = f (0) = 1. 11.7@ ! . 11.8.. 11.8; ! 5 ! .@ , f (;x) = f (x) ! 3 , !! , . > 11.3. C! 4 4, ! 4f (x) = x + 1 ; ex:< f (x) = 1 ; ex* f (x) = 0 () 1 ; e = 0 ()x = 0 { .0 f (x), !5 f (x), # . .
11.9.2460001&"2;;maxf (x) = f (0) = 0. 11.9@ . 11.10. >. 11.10/ 3 ! 5! , 53 !4 , 53 . ! f (x0) = 0, ! = f (x) x0 !!! (! ), ! f (0) = 1, !!! (! ).( 5 5 5 f (x) ! x0 !3, f (0 ) = 0 () f (x0) = 1 ().00;00! f (x0) = 0 #, # 54 , f (x0 ) = 1 { .q 11.4.
C! = (x ; 1)2 = 1.< y (x) = 3px2 ; 1 x 6= 1: = 1 4, 3 !:q2;0(x;1)y(x);y(1)1 = 1:py (1) = xlim1 x ; 1 = xlim1=limx 1 x;1x;10, y (1) = xlim1 y (x) = xlim1 3px2 ; 1 :2470030330!!0!0!!33B , , 9, ! 9 xlimx f (x) (5 !5) f (x) 5 x0, 0!0f (x0) = xlimx f (x):00!C, 11.40y (1) = 1:/!! , = 1 { ! (1 0) ! ..! ! 3 (. 11.11).1&"20;;. 11.111 , = 1 5min y(x) = (1) = 0.@ { . 11.12. >. 11.12 11.5. C! , ! 4 4, -4q3y = (x2 ; 1)2:< y (x) = p 42x x 6= 1*3 x ;1 ;4 !4xy (;1) = xlim1 p 2 = 0 = 1*3 x ;14!4xy (1) = xlim1 p 2 = 0 = 1*3 x ;1248030!;30!3y (x) = 0 () x = 0:1 , !4 x1 = 0*x23 = 1, 3 (1 0) (;1 0) ! ! .N! 5 ! { 3 (.
11.13).01&"2;;. 11.131 , 3 = 1 5min y(x) = y(1) = 0* = 0 { !max () = (0) = 1.@ { . 11.14.. 11.148, ! ! . > !"#!$# $ %&#'# (&:;( ) (), ) A( 1) y = 2x3 ; 3x2#3) y = x2e x#2) y = px34 ; 2x2 ; 5#4) y = x2 ; 1:;;;; ;1)2)3)|||||2494) 12 /-0 $ / 9 4 3 5!, .! f (x) > 0 (< 0) 8x 2 (a b), f (x) (5) (a b), ! # ! y = f (x)! ( 2) ! ( ) (a b) (. 12.1, 12.2).000;; f (x) > 0 ) tg 1 < tg 2. 12.100f (x) < 0 ) tg 1 < tg 2.
12.200 %! f (x) 3 x0, M (x0 f (x0)) ! 5!4 4 . ! # ! 4 M , M 5 y = f (x) (. 12.3).00;. 12.3 ! " C! 9 4 II- !49 3:2501) II- f (x), 53 f (x) = 0 1 ! 9L (5 # ! 4 ! ! f (x) !5 f (x))*2) ! f (x) 3 !3 *3) !5 (f (x) > 0) 5!(f (x) < 0) f (x), (f (x) 3 x0 )*4) # y = f (x). 12.1. <5 f (x) !(0 1), ! 3 x 2 (0 1):) f (x) > 0* f (x) > 0* f (x) < 0*) f (x) > 0* f (x) < 0* f (x) < 0*) f (x) > 0* f (x) < 0* f (x) > 0*) f (x) < 0* f (x) > 0* f (x) > 0:< ) C! 4 , !494 3:00000000000000000000;;;00001 , #, f (x) < 0 ! (0 1), f (x) 5!5:00@ .
12.4.. 12.4251;;;;;;) C !49 35:@ { . 12.5.. 12.5) C 3 f (x) f (x) f (x):000@ { . 12.6.. 12.6) f (x) f (x) f (x) 3:000@ { . 12.7. >. 12.7252 12.2. ! f (x) = 13 x3 ; 21 x2 ; 2x + 2;;; # .< f (x) = x2 ; x ; 2 = 0 () x1 = ;1* x2 = 2.C !494 3:0fmax = f (;1) = 3 16 # fmin = f (2) = ;1 13 :.! ! 4 4:f (x) = 2x ; 1 =) f (x) < 0 x < 12 *f (x) > 0 x > 21 * x0 = 12 { :000000 11{ !" ! # $%f 21 = 12@ ! . 12.8.
>. 12.8 y = f (x) ! (x0 f (x0 )) 3 9 f (x0), 3 ! ! . ( . 12.9 5 5 5 : )f (x0) = 0 (! ! )* ) f (x0) = 1 (! ! )* ) f (x0 ) > 0 (! ! 5 !2530000;;; x ) ) f (x0 ) < 0 ( x).0. 12.9 12.3. f (x) = ln (1 + x2) - ! " .< f (x) = 1 +2xx2 = 0 () x = 0: $ %:0fmin= (0) = 0f2)2(1;x', f (x) = (1 + x2)2 = 000()x = 1.f( 1) = ln 2), * * + (;1 ln 2) + ", Ox, * + (+1 ln 2) { ,( .
12.9 ,).254' ! " , * x ! 0 ln(1 + x2) x2, x ! 1ln(1 + x2) 2 ln jxj.1 " . 12.10. >;;;;. 12.10 12.4. p " , ", f (x) = 2 ; x ; 1 ! " .< f (x) = ; 13 q 1 2 * * x = 1.(x ; 1)303f(1) = 2 f0(1) = 13 * x = 1 , f (x) = 29 q 1 5 .(x ; 1) %:0034* x = 1 { + * + , , ( .
12.9 +).1 " . 12.11. >. 12.11255 p:1) = ln( 2 + 2 + 2)2) = 1 + 1 + .;yxxy3x1)2)|||||256 13 !" #$"%& 6 7 * $ !.1. ) + .2. % , (*, *, *, 7 , ..).3. + *, % "% .4. * % " *% * % + , % .5. ", I- .6. ", II- .7. 6 .3 7 * ! .9 , + *, + 7 + *% 7 * ,. 6 +$* 7 . 4* * , % % 7 %!% , *, "* % * .: , + " * ! " . 9 ! ! " , + % * %. " 7 * 7 % % * , $% ! + . : * + 7 " ", I- , ", II- 7 " " ( 7 257 7 ", II- ).
13.1. 6 3+x;2xy = x3 :< 1. )+ : x 6= 0:2. 9, : y = 0 ) x3 + x ; 2 = 0: ; ,* x = 1 ! , ,* x3 + x ; 2 x ; 1: ' x ; 1* x2 + x + 2: , x3 + x ; 2 = (x ; 1)(x2 + x + 2) = 0:4 %* x2 + x +2 <, y(x) * x = 1:3. 3, $ " :3+x;2xy= 0k = lim = limx43+x;2xb = lim (y ; kx) = lim x3 = 1:x!1xx!1x!1x!1= " $ y = 1.34 y ; y = x +xx3 ; 2 ; 1 = x x;3 2 > 0 x ! 1 7 x ! 1: ;2 ! !2 = ;1 x = 0 { 4. xlim0 y = ;0 = +1 xlim+0 y = ;+0 .
1 +" + $ + "!;;!258;;;2 + 1)x3 ; 3x2 (x3 + x ; 2) 3x3 + x ; 3x3 ; 3x + 6(3x5: y ===x6x40= ;2xx4+ 6 = ; 2(xx;4 3) :{ .x = 3 { * , y(3) = 2827436. y = ;2x ; 4xx8(;2x + 6) = ;2x +x85x ; 24 = 6x x;5 24 == 6(xx;5 4) :00"x = 4 { + * +, y(4) = 3332 (4 33=32){ * +.7. , (0 1) " ,, (0 3) { " ,, (3 4) { + ,, (4 1) { + ,. > 13.2. 6 q3q3y = (x + 4)2 ; (x ; 4)2: < 1. ? + "* x.2. y = 0 x = 0.? qy(x) { *.',,qqq2y(;x) = (;x + 4) ; (;x ; 4)2 = (x ; 4)2; (x + 4)2 = ;y(x):33325933 * + * * .qq2 ; (x ; 4)2(x+4)3: k = lim=33xx!12 ; (x ; 4)2(x+4)= xlim x((x + 4)4=3 + (x + 4)2=3(x ; 4)2=3 + (x ; 4)4=3) = !16= 0= xlim (x + 4)4=3 + (x + 4)2=3(x ; 4)2=3 + (x ; 4)4=3 = 161qqb = xlim (x + 4)2 ; (x ; 4)2 =2 ; (x ; 4)2x = 0:(x+4)= xlim (x + 4)4=3 + (x + 4)2=3(x ; 4)2=3 + (x ; 4)4=3 = xlim 316x4=3@ ", y = 0(qqx ! +1y ; y = (x + 4)2 ; (x ; 4)2 >< 00 x ! ;1 7 , x ! +1 , x ! ;1:4.
3 % " ! +" ", .!1!133!1!1!133;;pp5: y = p 2 ; p 2 = 23 qx ; 4 ; x + 4 3 x+4 3 x;4(x + 4)(x ; 4)3033y6: y = ; 92 (x + 4)00min;33= (;4) = ;4y(4=3) + 2 (x ; 4)9;260max = y (4) = 4:yq3q344(4=3) = 2 (xq+ 4) ; (x ; 4) 9 ( (x + 4)(x ; 4))43x = 0 y = 0 { * +.7. 9 (;1 ;4) + ,, (;4 0) " ( ,), (0 4) " ,, (4 +1) { + ,. >;; 13.3. 6 y = ln x ;x 1 + 1:< x 1. )+ x ; 1 > 0 < , *x 2 (;1 0) (1 +1).2. 9 < y(x) = 0 ln x ;x 1 = ;1 % x = 1 ;1 e :=(x ; 1)] + 1 =3: k = xlim ln(x=(x x; 1)) + 1 = xlim lnA((x ; 1) + 1)x!1!1= xlim lnA1 + 1x=(x ; 1)] + xlim x1 = xlim x(x 1; 1) + 0 = 0!1!1!1!xb = lim ln x ; 1 + 1 = 0 + 1 = 1:@ ", y = 1(x0 x ! +1y ; y = ln x ; 1 >< 0 x ! ;1:x!1261!x4.
limln x ; 1 + 1 = (ln(+0) + 1) = ;10 0! !!1xlimln x ; 1 + 1 = ln +0 + 1 = +1:1+0C, x = 0 x = 1 { x!;x!;;;; , . D", * % 3 4 " ! +" .!x5: y = ln x ; 1 + 1 = (ln x ; ln(x ; 1)) = x1 ; x ;1 1 = ; x(x 1; 1) :0002 (x ; 1)22x ; 1 :6. y = ; x12 + (x ;1 1)2 = xx;=2 (x ; 1)2x2(x ; 1)2007. ) * * $, > 13.4. 6 y = x2e :x;< 1. ? % "*% x:2. y = 0 x = 0:2623. ' * k b ;:!x11k1 = x lim= lim = +1 = 0+ ex x + ex!2x2x22b1 = x lim(y ; k1 x) = x lim= lim = x lim= +1 = 0++ ex x + ex+ exy = 0 { , y ; y: = x2e x > 0 7 ,.E, * k2 = x lim xe x = 1 , , $!!11!!11!1!1;;!;1lim y(x) = x lim x2e x = +1:;x!;1;;;!;14. 3 % " ! +" x ! ;1:5.
y = 2xe0; x2ex;y = 2e00;miny6.= x(2 ; x)e x y = 0 x = 0 x = 2:x;x;=e= (0) = 0y y0max= (2) = 4y; 2xe ; 2xe + x2e = epp(x ; 2 ; 2)(x ; 2 + 2):;xx;x;2=e :x;(x2 ; 4x + 2) =x;p2)2p(2;y(2 ; 2) =e22p;263p2)2p(2+y(2 + 2) =:e2+2p7.;> 13.5. 6 y = ln sin x:< E 7 , * y(x) { * y(x + 2) = ln sin (x + 2) = ln sin x = y(x), ! * + 7 , 2, A; ], " 7 , " *.1.
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