1. Математический анализ (850924), страница 8
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% ) +++ ) ! (+ )( < ! '$ ) )). ") # $ + )! ) ).M ! ) ) < +! + ! ) % ! )%.; '+( + < ! % $. !"# ) 8% 00 : ( '+ ! :f (x) limx x g (x) xlimx f (x) = 0 xlimx g (x) = 0:" ) % + $1( < ! ( $1) <%= f (x) g(x) % )% 0, )+( ) ! #( , ,), ..!!00!0f (x) = a(x ; x0)m + o((x ; x0)m ) a 6= 0128g(x) = b(x ; x0)l + o((x ; x0)l ) b 6= 0 f (x) = lim a(x ; x0)m + o((x ; x0)m ) =limx x g (x) x x b(x ; x0 )l + o((x ; x0 )l )8>m >< 0a m > la(x;x)0= b xlimx (x ; x )l = > b m = l>0: 1 m < l .1 0 1 1;1 '$( %; 1 00 +( + ! ).@+ %+ ! 00 0 1 *( <! , ) <..33xx4x ; 3 + o(x ) ; x;61:sinx;x=lim=;1) xlim0 x3 = xlim0x 0 x3x362323x44x ; + o(x )5 ; x2#"122x ; x = lim32) xlim0 x2 ; sin12 x = xlim0 sinx2 sin=2xx 0x2x + o(x2)]2!0!!001!!1!!!!442x2x4; + o(x )= xlim0 x2(x32 + o(x2 )) = xlim0 x34 = ; 32 :! x ln(1+sin x) = e1 = e3) xlim0 (1 + sin x)1=x = exlim% %%2 ))2 ) + o(x)ln(1+sinx)ln(1+x+o(xx+o(xlim= xlim0= xlim0=x 0xxx= xlim0 xx = 1:!!01!!!!! 1.
0 "+ 3" " "'"+" (c . 13.1).2. 0 "+ &" (c . 13.2).1293. B '? e 2 + 2!1 + 3!1 + 4!1 + 5!1 + 6!1 :4. E"? 3" 2n-' " y = cos x:5. <" sin 10 ( 0,0001.23. 3" ex, e = 1 + x + x2! + x3! + x4! + x5! + x6! + rn 2x'3456rn = e7! x 0 < < 1:x&+ x = 1. 3'"73 = 1 < 0 001:rn = e7! 0 < < 1 jrn j < 504016804. 3" " cos x = 12 + 21 cos 2x, n n !1(2x)(2x)1n (2x)=cos x = 2 + 2 1 ; 2! + 4! ; + (;1) (2n)! + o xnn= 1 ; x + x3 ; + (;1)n 2 (2n)!x + o x n :5. sin 10 0 1736: "",@ """ F' "" + " , 10 = 18 3" " "'"+" y = sin x n = 2m ; 1 = 3.22224422;122 +12 +1130 141 + + $ . & ' ( 2' 14.1.M%=+ f (x) $+ &( (() a b], + ', , )% 1 2 $%, +*, x1 < x2, ++ f (x1 ) f (x2 )(f (x1 ) f (x2 )):2' 14.2.
M%=+ f (x) $+ # &((# () (. 14.1, 14.2) a b], +', , )% 1 2 $%, +*, x1 < x2, ++ f (x1 ) > f (x2 )(f (x1 ) < f (x2 )):;; . 14.1. 14.22' 14.3.M%= '* $* $% a b] $+ a b], <%= $* '* a b], $+ # a b].2' 14.4. M%=+ f (x) $+ ( 0, )% 0 $+ f (x), * %+ %( (x0) )% x0, )8x 2 (x0 ; x0) f (x) < f (x0 )8x 2 (x0 x0 + ) f (x) > f (x0 ):M%=+ f (x) $+ &( 0, )%0 $+ & f (x), * %+131%( (x0), )% x0, )8x 2 (x0 ; x0) f (x) > f (x0 )8x 2 (x0 x0 + ) f (x) < f (x0 ):)(* a b] ! 14.1. 1) f (x) a b]2) f (x), , (a b).% #, & f (x) & &( (() a b] & , & f (x) 0 (f (x) 0) 2 (a b):@%$( + )+ '*! a b] <%=.2& .
( <%=+ f (x) { '*+ a b] <%=+, .. 8x1 x2 2 a b] $ x1 < x2 , ) f (x1 ) f (x2 ). "$($( )% x0 2 (a b), )% x0 + 2x 2 (a b), ! f (x0) = f (x0 + 0) = xlim+0 f (x0 + 22xx) ; f (x0 ) 0% %% 2x > 0 f (x0 + 2x) f (x0 ):% . ( f (x) 0 + ', x 2 (a b). "$($( x1 x2 $ a b], x1 < x2. A 1% <%= f (x) x1 x2]. A 1 * %+)% 2 (x1 x2), )f (x2 ) ; f (x1 ) = f ( ) 0:x2 ; x19 , ) f (x2 ) ; f (x1 ) 0, .. <%=+ f (x) '*+ a b]. %$.0000!001320+ ! 14.2. f (x) a b] - , , (a b).
"#, f (x) > 0 (f (x) < 0) 8x 2 (a b), f (x) # (# &) a b].%. "$( $( )% 1 2 $ a b], x1 <x2. ++ A 1 % <%= f (x) x1 x2] ),) * %+ )% 2 (x1 x2), )f (x2 ) ; f (x1 ) = f ( ) > 0x2 ; x1 (, f (x2 ) > f (x1 ). % '$, <%=+ f (x) $ a b]. %$. 14.1. : f (x) > 0 8x 2 (a b) +++ ', + $*! <%= a b].;, <%=+ y = x3 $ ;1 1]. 9 %f (x) = 3x2 '*+ 0 )% = 0.00000+ ' (() ! 14.3. f (x) 0 f (x0) > 0 (f (x0) < 0), # f (x) (&) 0.%. $ ! )% 0f > 0:f (x0) = xlim0 f (x0 + 22xx) ; f (x0) = xlim0 22x? (, * % ) > 0, ) + , j2xj < ,f > 0. ( 0 < 2x < . 2x 6= 0 ++ 22x2f = f (x0 + 2x) ; f (x0 ) > 0, ..
f (x0 + 2x) > f (x0 ), x0 < x0 + 2x.. 2x < 0, 2f = f (x0 + 2x) ; f (x0 ) < 0 .. f (x0 + 2x) < f (x0 ) x0 < x0 +2x. % '$, <%=+ f (x) $ )% x0. %$.000!!133, 2' 14.5. )% 0 ( %! <%=+ f (x) ) $+ # (# ) f (x), * %+ %( (x0) )%0, ) + , x 2 (x0) f (x) f (x0 ) (f (x) f (x0 )).;. 14.32' 14.6. )% %( % %( $+ 0. C)+ <%= ,)%, $+ 0 .2' 14.7. o)% 0 ( %! <%=+ f (x) ) $+ ## # (### ), * %+ %( (x0) )%0, ) + , x 2 (x0) x 6= x0: f (x) < f (x0) (f (x) > f (x0))..
y = sin x x 2 ; ]:)% 1 = ;=2 { )% %( , )%1 = =2 { )% %( % <%= y = sin x ; ]:)(* - ! 14.4. . 0 { 0 f (x), 0 f (x) & 0, & .%. @%$( ' .( )% 0 ( % ( )% 0 * $ + <%= f (x) f (x0) 6= 0. (, f (x0) > 0. ; , 14.3 <%=+ f (x) $ )% 0 , ( * %+ %( (x0) )%0, ) 8x 2 (x0 x0 + ) f (x0 ) < f (x) 8x 2 (x0 ; x0) f (x0 ) > f (x). , )% 0 %. ) ). %$.13400)(". ". ' '( <%=+ f (x) $% a b]. " "!# <%=+ f (x) '(# (# $)! $% a b].
. '(# $)+ (a b), $) ' $ %(, % ('(#). ;'(# $) 1 (+ %=, $%. )' !, ( 1 '! % <%= f (x) $)+ %=, $%. ;'(# $, $)! +++ '(# $) f (x) $% a b].I ) + (# $) <%= f (x).51 1( = ,1 + '(# (# $)! <%=! f (x) a b].;! )% $( %, 1* (a b),.. )% 1 2 : : : n, %, f (x) ' 0, ' *, '=:0x ax1x2 : : : xkby f (a) f (x1 ) f (x2) : : : f (xk ) f (b);'(# $ ) ff (a) f (x1 ) f (x2 ) : : : f (xk ) f (b)g { ( '(# $) <%= f (x) a b], (# $ ) ff (a)f (x1 ) f (x2 ) : : : f (xk ) f (b)g { ( (# $) <%= f (x) a b].+ -N % *( ! $ !. ! 14.5.
1) f (x) 0 2) ( ) f (x) 0, ( & - 0."#, f (x) 0 ( ) "+" "{", 0 { # f (x), "{" "+", 0 { #. . - 0 , 0 0.13500%. ( + , x 2 (x0 ; x0) f (x) > 0. <%=+ f (x), 6.2, $ $%x0 ; x0] , (, 8x 2 (x0 ; x0) f (x) < f (x0 ). I % %%+ , x 2 (x0 x0 + ) f (x) < 0, <%=+ f (x) ' $% x0 x0 + ] f (x) < f (x0 ) 8x 2 (x0 x0 + ).
T% '$,* %+ %( (x0 ; x0 +), ) 8x 2 (x0 ; x0 +)x 6= x0 : f (x) < f (x0 ), .. )% 0 { )% %( %.. + , x 2 (x0 ; x0) : f (x) < 0, <%=+ f (x) ' $% x0 ; x0] 8x 2 (x0 ; x0) f (x) > f (x0 ). I % %%+ , x 2 (x0 x0 + ) : f (x) > 0, <%=+ f (x) $ $% x0 x0 + ] f (x) > f (x0 ) : 8x 2 (x0 x0 + ). % '$,* %+ %( (x0 ; x0 + ), ) 8x 2 (x0 ; x0 + )x 6= x0 : f (x) < f (x0 ), ..
)% 0 { )% %( .( + , x 2 (x0 ; x0) : f (x) < 0 + , x 2 (x0 x0 + ) :f (x) < 0. , <%=+ f (x) ' $%,x0 ; x0] x0 x0 + ] , (, 8x 2 (x0 ; x0) :f (x) < f (x0 ) f (x) > f (x0 ) : 8x 2 (x0 x0 + ), (, )% x0 %.I ), + , x 2 (x0 ; x0) : f (x) > 0 + ,x 2 (x0 x0 + ) : f (x) > 0, , <%=+ f (x) $ $%, x0 ; x0] x0 x0 + ] , (, 8x 2 (x0 ; x0) :f (x) < f (x0 ) f (x) < f (x0 ) 8x 2 (x0 x0 + ), (, )% x0 %. %$.
! 14.6. f (x) 0 f (x0) f (x0 ), f (x0) = 0, f (x0) > 0 (f (x0) < 0). "# 0 # (# ) f (x).%. ?* ! $ ! )% 0) $ '! * $ ! f (x) % )%0 , ', ( <%= %! % U (x0).% %% f (x0) > 0, $ 14.3 , ) f (x) $ )% 0, .. * %+ %( (x0), ) + ,x 2 (x0 ; x0) : f (x) < 0, + , x 2 (x0 x0 + ) : f (x) > 0 $), ) )% 0 { %(! .. f (x0) < 0, 14.3 f (x) ' )% 0, ..0000000000000000000000000136* %+ %( (x0), ) + , x 2 (x0 ; x0) :f (x) > 0, + , x 2 (x0 x0 + ) : f (x) < 0, .. )% 0 {%(! %.
%$.00$ - !"# .* '*!14.7. f (x) 0 ((n)f (x0) ( 0. , -,f (x0) = f (x0) = ::: = f (n 1) (x0) = 0 f (n) (x0 ) 6= 0:000;"#, n { , 0 0, n { , 0 { 0, , f (n) (x0 ) >0, 0 { , f (n) (x0 ) < 0, 0 { . 1. 0 ' ""(1 " 8a b] (c .
14.2).2. >"" " x "" " f (x)? (c . 14.4).3. 0 "+ !; " 8a b](c . 14.1).4. 0 "+ " ' " 8a b] (c . 14.2).5. @"+, f (x) = x ; sin x "" " 80 2]:6. 0 ?7. 0 F " (c . 14.5 14.6).8. 0 "+ !; F " (c . 14.4).9.
0 "+ " F " ( . 14.5).10. ; ", "+, ( =x2f (x) = e x 6= 00x=00;1 x = 0 .1375. 7 f (x) = x ; sin x ( f (x) = 1 ; cos x, " "" (0 2) ' !? . 0", 14.2 x ; sin x ' "" " 80 2]:6. <, " , x ; sin x ' " " 80 2] (c . 5), " " f (x) = 1 ; cos x " F :f (0) = 0 f () = 2 f (2) = 0 (c . 14.1 14.2).10.
7( =x20f (x) = e0 xx 6==000;1 x = 0 ' , " " 8x 6= 0 f (x) > 0, " f (0) = 0(c . 14.7).138 15( + ) . (( <%=+ f (x) <<= (a b) , , %1 ! )% (x0 f (x0)) x0 2 (a b) <% <%=%(, %! :y ; f (x0 ) = f (x0)(x ; x0):02' 15.1.D+ <% <%= y = f (x) '*%( , (a b), )% <% 11 '! %(! (. 15.1) (<(f (x) < f (x0 ) + f (x0 )(x ; x0) , x x0 2 (a b) x 6= x0).0;;. 15.12' 15.2.D+ <% <%= y = f (x) '*%( $ (a b), )% %! 1# '! %(! (.
15.2) (<(f (x) > f (x0 ) + f (x0 )(x ; x0) , x x0 2 (a b) x 6= x0).0. 15.2D, '* %( , ' $( , %( $ { #.139+ () ! 15.1. . (a b) - f (x) , .. f (x) < 0, y = f (x) 0 .%. "$( $( )% x0 2 (a b) %( % %! y = f (x) )% (x0 f (x0)). ' %$, %1, ) )% %! (a b) 1 1! %(!, .. ) '! )% %! y = f (x)(# %(! 1 $) x.: %(! % %! )% (x0 f (x0 )) 00Y ; f (x0 ) = f (x0 )(x ; x0)0Y = f (x0 ) + f (x0)(x ; x0):(15:1)8$1 <%= y = f (x) % )% x0 <! ) ) < A 1:(15:2)y = f (x0 ) + f (x0)(x ; x0) + f 2!( ) (x ; x0)2 = x0 + 2x 0 < < 1 2x = x ; x0:")+ (15.1) $ (15.2), )y ; Y = f 2!( ) (x ; x0)2 < 0 8x 2 (a b) x 6= x0: , %$, ) '+ )% %! y = f (x) 11 %(! % ! %!, %% ' ' $)+ x x0 (a b).
I $), ) %+ %. %$.I ) '$ %$+ *+ . ! 15.2. . (a b) f (x) -, .. f (x) > 0 8x 2 (a b), f (x) 0 #.00000000140+ (2' 15.3.( <%=+ y = f (x) . x0 ! )% %) '%) $ . )%(x0 f (x0 )) $+ #& y = f (x), + % )( %! y = f (x) ! ) (. 15.3).;. 15.3 ! 15.3. y= f (x).. f (x0 ) = 0 f (x0) x0 f (x) , (x0 f (x0 )) #&. ( +, ) )% x0 f (x) %) '%) f (x0)).%.
( f (x) < 0 x0 ; < x < x0 f (x) > 0 x0 < x < x0 + % . , (x0 ; x0)%+ %, (x0 x0 + ) { . ? (,)% A = (x0 f (x0)) ( )% '.. 1 f (x) > 0 x0 ; < x < x0 f (x) < 0 x0 < x < x0 + , (x0 ; x0) %+ y = f (x) , (x0 x0 + ) { %. ? (, )% A = (x0 f (x0)) %1+++ )%! '. %$.?< ' '* . ! 15.4. f (x) & ( :f (x0) = f (x0 ) = : : : = f (n) (x0) = 0 f (n+1) (x) x0 f (n+1) (x0) 6= 0.
"#, n { , (x0 f (x0)) #&.. n { , x0, f (n+1) (x0) < 0, # x0, f (n+1) (x0) > 0.00000000000000000000141 15.1. 7+, ) %+ y = f (x) )% (x0 f (x0)) ', $ + <%= f (x) )% x0 ' = 1,' ;1.)(* ( ! 15.5. f (x) x0 . . 0(x0 f (x0)) #& y = f (x), f (x0) = 0.%. @%$( ' .( )% x0 * + + $ + f (x0) 6= 0. 1, , ) f (x0 ) > 0. *%+ %( )% x0, %! f (x) > 0. ; , 8.2, %+ y = f (x) ! % , (,)% (x0 f (x0 )) ( )% '. I ) +)! f (x0) < 0.
%$.. ;! ' % %!y=e x:7,. ;! y y .0000000000;000y = ;2xe x y = ;2e0;2200x2 + 4x2 e x2;;= 2e x (2x2 ; 1):;2% %% + $ + * , ! )% x, %, y = 0.00pp2e x (2x2 ; 1) = 0 2x2 ; 1 = 0 x1 = 22 x2 = ;2 2 :C, )py > 0 x < ; 22 ;200ppy < 0 ; 22 < x < 22 py > 0 x > 22 :1420000p10? (, %+ y = e x @;1 ; 22 A { 0p10 p p 1222 , @; 2 2 A { %, @ 2 1A {10p10 p22 . % '$, )% A @; e 1=2A B @ e 1=2A (;2;22)% ' <% <%= y = e 2x , ) x < 0 :C, ) $ y=;2xe2y > 0, .. <%=+ y2 = e x $ x < 0, x > 0 : y < 0,..
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