Σχολικό βιβλίο Γ΄ Λυκείου Μαθηματικά Κατεύθυνσης
TRANSCRIPT
!"#$!"%&'"
!" #$%&'
()*+,-. /.01,-.
#12*03 '+2145.)67
!"##$%&'(!
!"#$%&#'()* +,-./&"0* !!!! 1&2)3),4* 5&"%6/7,)89:- !2)";"
1&,7&$3<$)* =&79.%/:* !!!! 1&2)3),4* =/28/&* >(6&9#%-7)*
?@,)* +,@A&":* !!!! 1&2)3),4* =/28/&* >(6&9#%-7)*
?6$:-B:<,&* 1C"/":* !!!! 1&2)3),4* =/28/&* >(6&9#%-7)*
5&6&7,&-$9#)* +,&<$:* !!!! 1&2)3),4* 5&"%6/7,)89:- !2)";"
5:.<D:* E%;$3/:* !!!! 1&2)3),4* =/28/&* >(6&9#%-7)*
(!)*$(+% !,-'(.-%)%
FC8&G#)* HC'"")* !!!! 1&2)3),4* =/28/&* >(6&9#%-7)*
OMA/% %0%-*$&.!,!
!"#$%&#'()* +,-./&"0* 1&,7&$3<$)* =&79.%/:* ?@,)* +,@A&":* ?6$:-B:<,&* 1C"/":* 5:.<D:* E%;$3/:*
'1*1)'(% !)* 12%(!(* )*" 1%(/%#.#(+*" (0!)()*")*"
!#&806:-.:* I%C"9#&* !!!! >69,/8:* +<8J:-.:* ,:- 5.H.
K&(,-.:3$'A)7): E&$#@$) L0D&
+B48&,&: ?6:<,7/(&* ?/B'.)*
?% &60A&7) ,)* %..)"/(4* (-J%$"47%C* ,& #/#&(,/(' J/J.9& ,:- K)8:,/(:<,
,:- E-8"&79:- (&/ ,:- I-(%9:- ,-6;":",&/ &60 ,:" M$3&"/780 >(#07%C* K/-#&(,/(;" =/J.9C" (&/ #/&"@8:",&/ #C$%'".
!"#!$%&'# &()'*+, "-'.&'-, *-' ($+,*&!/-01)
"-'.-%1%'*# '),0'0#!0#
!"#$!"%&'" %2 03456
()*+,-. /.01,-.
#12*03 '+2145.)67
-789:;83<56 ,=>?@;7A6 !"#$%$&'( )"*+,-.&$/012 3#$*4*
*;=B;9CD956 E;BF?:@G6 !"#$%$&'( 5/#/-"( +6,"07+2.$(
/H=56 ,=HI;7G6 !"#$%$&'( 5/#/-"( +6,"07+2.$(
/J9G>KGD=;6 *L7/7G6 !"#$%$&'( 5/#/-"( +6,"07+2.$(
";J;B=;>9F856 ,=;D9G6 !"#$%$&'( )"*+,-.&$/012 3#$*4*
"G?DMG6 %:N9C@G6 !"#$%$&'( 5/#/-"( +6,"07+2.$(
89:";&<!8< ('=8<(>< =&="'%&'>; ?&?/&>;
"#$;" 1999
!"#$#%#&
To !"!#$% &%' ()*+,+- .+* /0)"* .*1 &-)"#*2!,3-" +43 5#4 +63 7*842*+"(93,
:&61 &)%!#0&-+*" *&: +% &):;)*22* .&%'<93 +41 =-+"(>1 ?*+-58'3.41 +41 @A +,B41 +%' C3"*$%' D'(-$%', +%' %&%$%' 4 -E*)2%;> *)/$F-" *&: +% ./%#"(: 0+%1 1999-2000.
G% !"!#$% *'+: &)%>#8- *&: *3*2:)E6.4 +%' !"!#$%' +63 7*842*+"(93 +41 241 (*" +41 441 <0.241 +41 @A +,B41 +%' @-3"(%5 D'(-$%' (*" *&%+-#-$+*" *&:
<5% 20)4.
! To &)9+% 20)%1, &%' E0)-" +%3 +$+#% HD@CIJH, *&%+-#-$+*" *&: <'% (-E,-
#*"*.
— To &)9+% (-E,#*"% *&%+-#-$ 2"* -".*;6;> .+4 =-6)$* +63 K"3,(63, 4 %&%$*
2-+*B5 ,##63 -$3*" 03* -);*#-$% ;"* +4 2-#0+4 +63 @-62-+)"(93 7-+*./42*+"-.293 (*" +63 @)*22"(93 L'.+42,+63, +* %&%$* 2-#-+93+*" .+% $<"% (-E,#*"%.
— G% <-5+-)% (-E,#*"% -".,;-" .+%'1 7";*<"(%51 H)"82%51, %" %&%$%" -$3*" &)%0(+*.4 +63 K)*;2*+"(93 H)"8293. M" 7";*<"(%$ H)"82%$ *3*(*#5E84(*3 +43 &-)$%<% +41 H3*;0334.41 .+43 &)%.&,8-"* -&$#'.41 -B".9.-63 +)$+%'
!*82%5. N261, .+%'1 *"93-1 &%' *(%#%584.*3 *&%<-$/+4(- 4 2-;,#4 .42*.$*
+%'1 ;"* &,)* &%##, &)%!#>2*+* +41 2*842*+"(>1 -&".+>241 (*" +63 -E*)2%-
;93 +41. G% (-E,#*"% *'+: 0/-" #4E8-$ *&: +% !"!#$% +63 7*842*+"(93 =-+"(>1 ?*+-5-
8'3.41 IA +,B41 C3"*$%' D'(-$%' +63 .';;)*E063: H<*2:&%'#%' D., I".(*-
<%'),(4 I., @*!*#, O., K%#5F%' @. (*" L!0)(%' H.
! T% <-5+-)% 20)%1, &%' E0)-" +%3 +$+#% HPHDQLR, *&%+-#-$+*" *&: +)$* (--E,#*"*.
— G% &)9+% (-E,#*"% .42*+%<%+-$ 03* 30% B-($342*. C$3*" +% &0)*.2* *&: +"1 &-&-)*.203-1 &),B-"1 .+"1 «,&-")-1 <"*<"(*.$-1». G* .&0)2*+* +41 033%"*1 +%'
%)$%' '&,)/%'3 *.E*#91 2- &%#5 .*E> (*" .';(-()"203% +):&% .+* ;)*&+, +%'
H)/"2><4. R *3,&+'B4, :261, *'+>1 +41 033%"*1 0;"3- .+* /):3"* +41 H3*;03-34.41 (*" 0(+%+- (*+0/-" (-3+)"(> 80.4 .+%3 (:.2% +63 2*842*+"(93 -33%"93. ?*+’ *)/,1 .+% (-E,#*"% *'+: &*)%'.",F%3+*" !*."(01 – (*" ><4 ;36.+01 .+%'1 2*84+01 - 033%"-1 +63 .'3*)+>.-63, (*891 (*" 2-)"(01 *(:24 !*."(01 033%"-1 +41 H3,#'.41. L+4 .'30/-"* -".,;-+*" 4 033%"* +%' %)$%' .+% "0x #, 4 033%"*
+%' %)$%' .+% $% (*" .+% &% (*" <$3%3+*" %" &"% /*)*(+4)".+"(01 "<":+4+01 +%'. G0#%1, <$3-+*" 4 033%"* +41 .'30/-"*1 2"*1 .'3,)+4.41 (*" &*)%'.",F%3+*" %" !*."(:+-)-1 "<":+4+01 +41. — L+% <-5+-)% (*" +)$+% (-E,#*"% &*)%'.",F%3+*" %" 033%"-1 +41 &*)*;9;%'
(*" +%' %#%(#4)92*+%1 *3+".+%$/61 (*" ;$3-+*" /)>.4 +63 -33%"93 *'+93 .- &%##01 -E*)2%;01. R &*),;6;%1 (*" +% %#%(#>)62* -$3*" (*+, (,&%"% +):&% %" <5% <"*E%)-+"(01 :S-"1 +%' $<"%' 3%2$.2*+%1. L- 2"* 0(E)*.> +%'1 -$3*" 4 (#$-
!" #"$ %&'(#)*+,"$ -'. #) %*/'01,, !% 233" " #'45#"#' -'. #) *6-)$ #"$ #7)-
4.2$ %,1$ -.,"#)5 -#3.
89#1 #) /./3:) ;$ ',<7=(.,) 0"*.)57>"*' 0%, %:,'. #+3%.). ? *1,)$ #71()$ >.'
,' +4)9*% !#' !4)3%:' *'$ 5!#%7' '(1 *%7.-2 471,.' +,' -'35#%7) *+!) 0.0'-
!-'3:'$ %:,'. ) ,"&23.)$ -'. %3%5<%7)$ 0.23)>)$, #), )():) " %(.!#"*),.-6 ('-
720)!" +4%. -'<.%7=!%. >.' '.=,%$ #=7'. @.’ '9#1 #) 31>) " !9>>7'&.-6 )*20'
*% .0.':#%7" .-',)():"!" <' 0+4%#'. !413.' -'. ('7'#"76!%.$ >.' #) /./3:) '(1
)().),06()#% – !9,20%3&), *'<"#6 6 233) ()3:#" – %,0.'&+7%#'. >.' #' A"#6-
*'#' #"$ ('.0%:'$. B' !413.' -'. ). ('7'#"76!%.$ *()7)5, ,' '()!#+33),#'. !#) C'.0'>;>.-1 D,!#.#)5#), E%!)>%:;, 396, 153 10 8>:' C'7'!-%96.
E27#.)$ 1999
?. F9>>7'&%:$
!"#$"%&'"()
)* '"#&+ (),-".#))
/"0),)$& 11: !234567 – -8499:5; +<=>?94>4 !"#.
1.1 ! "##$%& '$( )*#&+& 11
1.2 ,-./01/2 )%#3+4# - ,$55&)5&/%&/6.7 &-%06$8 61 )*#&+& 16
1.3 ,$55&)5&/%&/6.7 )%#3+4# 24
1.4 91461'-%+$* 61'&/:26&'%/6$* 37 1.5 ! "##$%& '$( ;-&66%+$8 /(/'<6&'$7 50
1.6 =)*5(/2 ;-&66%+$8 /(/'<6&'$7 61 '2 6"0$>$ &)&5$%?<7 '$( Gauss 52
1.7 E)*5(/2 ;-&66%+$8 /(/'<6&'$7 61 '2 6"0$>$ '4# $-%@$(/A# 63
/"0),)$& 21: ':@4A:512 )8:B912 2.1 ! B##$%& '$( C%;&>%+$8 D-%06$8 85
2.2 ,-3E1%7 /'$ F8#$5$ ! '4# C%;&>%+A# 88
2.3 C"'-$ C%;&>%+$8 D-%06$8 97
2.4 G-%;4#$61'-%+< C$-?< C%;&>%+$8 102
2.5 ,$5(4#(6%+"7 =E%/A/1%7 /'$ ! 112
.* '"#&+ ()(),C+D)
/"0),)$& 11: E8:1 - =<3FG6:4 =<3;8>H=H7 !"#.
1.1 ,-&;6&'%+$* D-%06$* 129
1.2 F(#&-'</1%7 132
1.3 C$#.'$#17 /(#&-'</1%7 - D#'*/'-$?2 /(#3-'2/2 148
1.4 H-%$ /(#3-'2/27 /'$ "0x # $157
1.5 I>%.'2'17 '4# $-*4# 165
1.6 C2 )1)1-&/6"#$ .-%$ /'$ "0x #$ 176
1.7 H-%$ /(#3-'2/27 /'$ 3)1%-$ 182
1.8 F(#":1%& /(#3-'2/27 188
!"#$%$&' 2(: )*+,(-*./0 %(1*23/0 2.1 ! "##$%& '() *&+&,-,$. 209
2.2 /&+&,0,12%34) 2.#&+'524%) - /&+6,0,$) 2.#6+'52( 222 2.3 7#4) *&+&,-,%2() 229
2.4 9.:38) 34'&;$<5) 241
2.5 =4-+(3& >"2() ?%35) @%&A$+%B$C D$,%23$C 245
2.6 E.#"*4%4) '$. =40+53&'$) >"2() ?%35) 250
2.7 ?$*%B6 &B+8'&'& 2.#6+'(2() 258
2.8 7.+'8'('& - 2(341& B&3*5) 2.#6+'(2() 272
2.9 F2C3*'0'4) - 7#4) De L’ Hospital 279
2.10 >4<"'( B&% G6+&H( '() ,+&A%B5) *&+62'&2() 3%&) 2.#6+'(2() 287
!"#$%$&' 3(: '4(.45-67*./0 %(1*23/0 3.1 F8+%2'$ $<$B<5+03& 303
3.2 >":$I$% $<$B<5+02() 309
3.3 @%&A$+%B") 4H%2-24%) 318
3.4 J+%23"#$ $<$B<5+03& 326
3.5 ! 2.#6+'(2( !"x
dttfxF#
)()( 333
3.6 =4-+(3& >"2() ?%35) J<$B<(+0'%B$C D$,%23$C 340
3.7 K3;&I8# 4*%*"I$. G0+1$. 342
89')"&:"&; – $9$<=>;"&; $;!>;"?< 365
!" #$%&'
!()$*%!
!"#$%&' - ()$**"%$ '+',-*$,$
1.1 - &##."$ ,.+ !"#$%$
(/0123
!"##$%& $%'(#)*#+& ,&%&'-'./ &0)(1+2.)-2 !BA ,, 1&+ " 342(02 '+& )( )$5$0-
)&4( 6(2)"5( )(0/ -/ ,%(/ ,"2)$ )$72+1* 7&%&1)8%+#)+1* )+/ $9./ ,58%(:(%4$/: ;%'(#)*#+( #: <#7=/ 97 DIN, 7%>2(/ '+& )8 6$)&?(5. )8/ )&7=)8)&/ &,> 0-100
km/h 10,7 sec, )$5+1. )&7=)8)& 180 km/h, 1&)&2*5-#8 #)82 ,>-58 &2* 100 km 9,5 lit, :(%(5('.#+6(+ 4,,(+ 10.
;%'(#)*#+( $: <#7=/ 100 DIN, 7%>2(/ '+& )8 6$)&?(5. )8/ )&7=)8)&/ &,> 0-100 km/h 12,9 sec, )$5+1. )&7=)8)& 191 km/h, 1&)&2*5-#8 #)82 ,>-58 &2* 100 km 11 lit, :(%(5('.#+6(+ 4,,(+ 11.
;%'(#)*#+( !: <#7=/ 45 DIN, 7%>2(/ '+& )8 6$)&?(5. )8/ )&7=)8)&/ &,> 0-100 km/h 17,9 sec, )$5+1. )&7=)8)& 140 km/h, 1&)&2*5-#8 #)82 ,>-58 &2* 100 km, 7,1 lit, :(%(5('.#+6(+ 4,,(+ 6.
;%'(#)*#+( ": <#7=/ 174 DIN, 7%>2(/ '+& )8 6$)&?(5. )8/ )&7=)8)&/ &,> 0-100 km/h 7,6 sec, )$5+1. )&7=)8)& 225 km/h, 1&)&2*5-#8 #)82 ,>58 &2* 100 km 12,5 lit, :(%(5('.#+6(+ 4,,(+ 20.
!+/ ,58%(:(%4$/ &0)"/ 6,(%(=6$ 2& )+/ ,&%(0#+*#(06$ ,+( (%'&2-6"2& -/ $-9./:
!$72+1* @&%&1)8%.
;%'(#)*#+(
<#7=/DIN
@%>2(/ '+& )86$)&?(5. )8/)&7=)8)&/&,>
0-100 km/h
!$5+1.!&7=)8)&
km/h
A&)&2*5-#8#)82 ,>58
lit &2* 100 km
B(%(5('.#+6(+ 4,,(+
#$!"
17445
10097
6,79,179,127,10
225140191180
5,121,7
115,9
206
1110
1
1 !"#$%&' – ()$**"%$ '+',-*$,$
12
!" "#$%&'($)* +,+-&./" ('0 -#%-12/$"0 "3(40 +$*("5'0, )6,$7&./" &.7" 7, "1)86,0,
!!!!
"
#
$$$$
%
&
205,122256,7174
61,71409,1745
11111919,12100
105,91807,1097
6.&, 9($ 7:'&"(;<-3/ ./"/ !"#$%$ &' 4 ()$&&*+ %$, 5 -./0'+ 4, 73/(-&9(,#",
./"/ =;/")" .1!23 4''''5 4 ")9&" ./"/ 4''''5 !"#$%$. >7(? (- 787('&"
()
(*
+
,-.,-.,.-.
337
225
123
!zy
!zx
!zyx
.
!- 787('&" "3(9 %" &=-#-87, /" ="#"7("%,; ?0 ,540:
@3/(,6,7(40
A5;7?7'(-3 x (-3 y (-3 z (-3 ! 7("%. 9#-0
1'2'3'
150
-3 0 1
2-2-7
-1 1 3
123
>(7$ -$ 73/(,6,7(.0 (?/ "1/27(?/ 7:'&"(;<-3/ (-/ 3''''4 !"#$%$
!!!
"
#
$$$
%
&
.
...
3710
1205
1231
)"$ -$ 73/(,6,7(.0 (?/ "1/27(?/ &"<; &, (-30 7("%,#-80 9#-30 (-/ 3''''5 !"#$%$
!!!
"
#
$$$
%
&
.
...
33710
21205
11231
.
B,/$)* .:-3&, (-/ ")96-3%- -#$7&9:
4567847
C$" +$*("5' "# / (- =64%-0 "#$%&2/ 7, &-#D4 -#%-1?/;-3 7:4&"(-0 &, #
1#"&&.0 )"$ " 7(46,0, 6.1,("$ !"#$%$+ .1!23 ./' 4 "=6-87(,#" ./' !"#$-
%$+.
1 !"#$%&' – ()$**"%$ '+',-*$,$
13
!"#$ %&'()*$ +"#$ ,#-."/&0"#-* ,#'123$ -* )*4(/(&( 567--(+( !BA ,, )+/.
89 (692-"& -* +"#$ "%"&"#$ ,:;-(+&0"#-* <'(' %&'()( /<5"'+(9 !"#$%&'( +"# %&'()(. !" ,+"9:*&" *'=$ "#! %&'()( $ %"# ('1)*9 ,+;' i-56(--1 )(9 j-,+1/; ,#-."/&-0*+(9 -* ij% .
>+,9 " "#! %&'()($ $ 5674*+(9: j ,+1/;
56(--1
21
21
222221
111211
i
%%%%
%%%%
%%%%%%%%
#"#j##
i"ijii
"j
"j
""""""""
#
$
%%%%%%%%
&
'
(()()())
(()()())
((((
1 ,#'+"-"56(49)7 ][ ij% , #i **1 , "j **1 .
?9( %(67@*95-(, " 22! %&'()($ ][ ij% -* ji% ij +, <:*9 ,+"9:*&( 01111 ,+,% ,
12112 +,+,% , 11221 ,+,% )(9 02222 ,+,% . A%"-<'3$, " %&'()($ (#+=$
5674*+(9 "#
$%&
' +01
10.
B 9,=+;+( -*+(CD +3' %9'7)3' "6&0*+(9 3$ *C1$:
)*+,-),
E#" %&'()*$ BA, /<-* =+9 *&'(9 '!#$, =+(' <:"#' +"' &@9" (692-= 56(--F', +"' &@9" (692-= ,+;/F' (@;/(@1 (' *&'(9 +"# &@9"# +D%"#) )(9 +( ('+&,+"9:( ,+"9-:*&( +"#$ *&'(9 &,(. ?9( '( @;/F,"#-* =+9 @D" %&'()*$ *&'(9 &,"9 5674"#-* BA ,
G%= +"' "69,-= (#+= %6")D%+*9 =+9 @D" %&'()*$ @9(4"6*+9)"D +D%"# @*' -%"6*& '( *&'(9 &,"9. G' <'($ %&'()($ <:*9 +"' &@9" (692-= 56(--F' )(9 ,+;/F', @;/(@1 *&'(9 +D%"#
""! 59( )7%"9" *!-" , +=+* " %&'()($ (#+=$ /<5*+(9 "&".(/01$234 5'1(2(4. !( ,+"9:*&( ""%%%% ...,,,, 332211 *'=$ +*+6(53'9)"D %&'()( $, /<-* =+9 ,:;-(+&-0"#' +;' 26.$( 7$(/81$# +"# $. G' +( ,+"9:*&( *'=$ +*+6(53'9)"D %&'()( $ %"# @*' .6&,)"'+(9 ,+;' )D69( @9(-5F'9" *&'(9 =/( 0, +=+* " $ /<5*+(9 7$(/81$#4 5'1(2(4. ?9( %(67@*95-(, "9 %&'()*$:
"#
$%&
'+ 30
02,
"""
#
$
%%%
&
'
100
010
001
,
""""
#
$
%%%%
&
'
+1000
0000
0020
0006
*&'(9 @9(5F'9"9 %&'()*$.
1 !"#$%&' – ()$**"%$ '+',-*$,$
14
!"#$ %&"#'#$ %() *+,- .&# ./"( 01#..2, /%3$ ( ! "3102 4*0,5#- !"#$%$&
'($))*, ,"6 *"#$ %&"#'#$ %() *+,- .&# ./"( 75248, /%3$ (
###
$
%
&&&
'
(
1
2
7
4*0,5#- !"#$-
%$& +,*-.. !"#$ %&"#'#$ %() *+,- *"# ./"( 75(-+,&(, /%3$ ( ]3[) 4*0,5#- !"-#$%$& +,/012"/. 9*4($, *"#$ 5,51#03"-'/$ %&"#'#$ 4*0,5#- ,(0'3#0%4& 5#3, /5#" /4# 5# 75(--+,&# 5() %() :1&7'("5#- ';53 #%/ 58" '<1-# =-#06"-( ,&"#- .8=,"-'; '#- ,(0-'3#0%4& %5,3, /5#" /4# 5# 75(-+,&# 5() %() :1&7'("5#- %;"3 #%/ 58" '<1-# =-#06"-( ,&"#- .8=,"-';. >-# %#1;=,-0.#, (- %&"#',$
###
$
%
&&&
'
()
200
510
063
,
####
$
%
&&&&
'
( )
3000
5100
2300
0112
,&"#- 51-03"-'(& ;"3, ,"6 (- %&"#',$
###
$
%
&&&
'
()
275
013
006
,
####
$
%
&&&&
'
(
)1602
0513
0000
0004
,&"#- 51-03"-'(& ';53.
&.$)*/(&'
1. 6/ 70!-$#4 +1*)$ !$(0+,5#20 ,/ /70%4
7"%,8/ !/8 +8#7920 ,0& !4-20& 0(1$ ,,, %$0 &. :$ !$($+,$;2" ,/ 7"%,8/ $8,4 )2 9#$# !"#$%$ ,/8 /!/"/8 %5;2 +,/012"/ #$ <$#2(=-
#20 ,/ !-*;/& ,3# 78#$,=# ,(4!3# )2,5>$-
+.& $!4 !4-. +2 !4-., (410 /!3+7*!/,2 70$-
</(2,0%* !4-.), $</? !(/.'/8)9#3& !2(5-
+/8)2 $!4 )"$ )4#/ !4-., !.1. $1$ %,-.
2+'-
?%/ 58" %/48 ! 758" ! )%;1+()" 3 51/%(-: ABA , !A" , AEA , #%/ 58" ! 758" # =," )%;1+,- 51/%($, #@(< %1*%,- "# %,1;7()., #%/ .&# ./"( %/48, #%/ 58" ! 758" $ )%;1+()" 2 51/%(- $A" , !#$ , #%/ 58" ! 758 " )%;1+,- *"#$ 51/%($ !%" , #%/ 58" ! 758" % )%;1+,- 1 51/%($ !"% '54.
1
$ % "
B A
1 !"#$%&' – ()$**"%$ '+',-*$,$
15
!"#$ "% %&$'( &)'"*% +,%-.) /0 ,0-0#"01.) +. "%/ ,)/0'0 &$,234 .$#(&%*.
3 0,25 +. "%/
5 !5 ,)/0'0
2. !" #$#%"&%#' "( )*+,-.)( %/012 %3( yx, 4/" %/2 .*.'#2 /&-5.)(:
i) "#
$%&
'(
)"#
$%&
'*(
530
12
1613
15)(32xxx
xx ii) "
#
$%&
'*
(*)"
#
$%&
' (*72
411
76
6253
yx
xyx
.+'-
i) H $#("6"0 "#
$%&
'(
)"#
$%&
'*(
530
12
1613
1)53(2xxx
xx $#78.$, 0/ '0$ +(/% 0/ #*/0261.8%*/
%$ $#("6".4
++,
++-
.
()
)*))(
5316
013
11
2)53(
2xx
x
xx
.
9 "-)"6 $#("6"0 0261.8.$ :$0 3
1)x . 9 "$+3 0*"3 "%* x .,0261.8.$ '0$ "$4 522.4
&8% $#("6".4. ;,%+</=4, %$ ,0-0,5/= $#("6".4 #*/0261.8%*/ :$0 3
1)x .
ii) 9 $#("6"0 "#
$%&
'*
(*)"
#
$%&
' (*72
411
76
6253
yx
xyx $#78.$, 0/ '0$ +(/% 0/ #*/026-
1.8%*/ %$ $#("6".4
++,
++-
.
)*)()(*)*
77
26
462
1153
yx
xy
x
.
9 &.8".-6 '0$ "-)"6 $#("6"0 :-5>%/"0$ ,-.
)*)*
62
22
yx
yx '0$ ,-%>0/?4 &./ #*/026-
1.8%*/ :$0 '0+)0 "$+3 "=/ x '0$ y. ;,%+</=4, &./ *,5-7%*/ "$+<4 "=/ yx, :$0
"$4 %,%).4 %$ ,)/0'.4 0*"%) /0 .)/0$ )#%$.
@A9BCDE
! " # $ %
! 3 0 2 1 1" 0 2 0 2 1# 2 0 2 0 1$ 1 2 0 3 1% 1 1 1 1 2
""""""
#
$
%%%%%%
&
'
21111
13021
10202
12020
11203
1 !"#$%&' – ()$**"%$ '+',-*$,$
16
$'%-'&"'
!" #$!%!&
1. ! "#$%&'() $*'&+&) ",*-',# .#& /0,#) 123",) $1"145&*016 /16) &.7',) !, /#) '*+,) ", /#) 8//,) #, /#) #41$&%*,) $, /& /902&/& % $16 $9/6-, : 123-"&, /& /902&/& & $16 "9-/:+, : 123"& +&# /16) ;&<21=) ' $16 9-,#. >'
][ ij(A ! ,*'&# 1 $*'&+&) &6/(), /(/, '& ;0,*/,: i) ?1#1) ,*'&# 1 /=$1) /16 $*'&+&.
ii) ?1#,) $%:01510*,) 2&) "*'16' /& 4/1#-,*& 241512 ,, ((( +&# 37( .
2. @*',/&# 46'/121.0&5#+3 1 44" $*'&+&) ][ ij(A ! ($16 || ji(ij #! .
A& $&0&4/84,/, /1' $*'&+& &6/(', &51= ;0,*/, /& 4/1#-,*& /16.
3. A& ;0,*/, /& yx, .#& /& 1$1*& #4-=,#:
i) $%
&'(
) *!$
%
&'(
)*#
#132
1
23
0112 yxx
yxyx
x ii) $
%
&'(
)!$
%
&'(
)
*#*
yyxx
yyx
2
0122
2
.
4. B#& $1#& /#28 /16 <,/#+1= &0#<21= x 1 $*'&+&) $%
&'(
)
##
2lnln
1ln12
2
xx
x
,*'&# "#&.7'#1).
5. A& ;0,<1=' 1# /#29) /C' )2,0[ )x+ .#& /#) 1$1*,) #4-=,#:
$%
&'(
)!$
%
&'(
)01
11
46'2,52:2:22 2
xx
xx.
1.2 !).'/&'- !"#$%0# - !.11$!1$'"$'*.' $)"/*.+ *& !"#$%$
!2345647 89:;<=:
D*& ,/�,*& $16%3,# /:%,1034,#), E6.,*&, +16F*',) +&# $%6'/80#& 4, ><8'&, G,44&%1'*+: +&# ?3/0&. !# $C%84,#) /16) 28',) H,$/92;0#1 +&# !+/7;0#1 $&-0164*&4&' /:' ,I8) +*':4::
$$$
%
&
'''
(
)
1111454312
1741032712
1561333612
B&EIHNA
@>JAKGLMNN>AOPK!D>@MH
1 !"#$%&' – ()$**"%$ '+',-*$,$
17
!"#$%&'()*+ A!"#$ %&'/() *+,-$
,-$.'()*+ A!"#$ %&'/() *+,-$
.)/&0-+'&12 18 16 13 20 18 15 345&6$ 12 10 9 15 13 12 70486#&2 9 11 8 13 10 10 */4#,"-1$ 14 12 10 17 16 8
9:0;<#=2, ,042 >40 $4,0?2 ;"#&2 01 '4#0/1(<2 :=/"'&12 ,)2 &,$1-&6$2 ",$# 01 &@"2:
)810()1612()1714(*/4#,"-1$)108()1011()139(70486#&2)129()1310()1512(345&6$)1513()1816()2018(2.)/&0-+'&1
*+,-$%&'/#6()A!"#$
!!!!!!!!!!!!
.
A# ,B-$ !&=-"'04;& ,042 :6#$(&2 ,=# :$-$:+#= :=/"'&=# <C04;&:
D1$ ,0 E&:,<;F-10:
""""
#
$
%%%%
&
'
(
101214
8119
91012
131618
A
D1$ ,0# G(,BF-10:
""""
#
$
%%%%
&
'
(
81617
101013
121315
151820
B
($1 51$ ,12 '4#0/1(<2 :=/"'&12:
""""
#
$
%%%%
&
'
(
""""
#
$
%%%%
&
'
!!!!!!!!!!!!
(
182831
182122
212327
283438
81016121714
1081011139
12913101512
151318162018
! .
O :6#$($2 ! /<5&,$1 /0(*)1&2 ,=# :1#+(=# " ($1 # ($1 '4;F0/68&,$1 ;& BA ! , >)/$>" BA! !( . D(+ <C04;& ,0# $(H/04!0 0-1';H:
,34!5,!
60(*)1&2 >40 $%) :1#+(=# ][ ij&A ( ($1 ][ ij'B ( /<5&,$1 0 $%) :6#$($2 ,04 0:0604 (+!& ',01C&60 &6#$1 ,0 +!-01';$ ,=# $#,6',01C=# ',01C&6=# ,=# " ($1 #. G :6#$($2 $4,H2 '4;F0/68&,$1 ;& BA ! . I)/$>",
][ ijij '&BA !(!
1 !"#$%&' – ()$**"%$ '+',-*$,$
18
!"# $%&'$()" *+%$,-). /,#*01# 2,.3$%"4,0$5 45/$(.
6,. /.%*2",7)., $, /&#.0"8 !!!
"
#
$$$
%
&
''(
275
548
932
A 0., !!!
"
#
$$$
%
&
'
'(
246
690
317
B /$( "&#.,
4$( &2,$( 45/$( 33) , )" 9*-: 4$# /.%./*#1 $%,-);, )/$%$5# #. /%$-4"+$5# 0., 4$ *+%$,-)* 4$(8 "&#.,
!!!
"
#
$$$
%
&
'(
!!!
"
#
$$$
%
&
'
'*
!!!
"
#
$$$
%
&
''(*
01111
11138
1229
246
690
317
275
548
932
BA ,
"#< $, /&#.0"8 !"
#$%
&'(
512
301! 0.,
!!!
"
#
$$$
%
&'(
24
32
01
" , /$( 2"# "&#., 4$( &2,$( 45-
/$( 2"# )/$%$5# #. /%$-4"+$5#. = /%*>: )" 4:# $/$&. 9%&-0$()" 4$ *+%$,-). 25$ /,#*01# ?@7"4., # $ % & ' ( &
) # * + , - . +.
"./012134 124 560783724 19: 5/:;<9:
= /%;-+"-: 41# /,#*01# @A", ,2,;4:4"8 .#*?$7"8 )" 4:# /%;-+"-: 41# /%.7).-4,0<# .%,+)<#. B(70"0%,)@#.:
+ C# !BA ,, "&#., +/) /&#.0"8, 4;4"
ABBA *(* .#4,)"4.+"4,0D !BA!BA **(** )()( /%$-"4.,%,-4,0D
+ C# ! "&#., $ +/) /&#.0.8 /$( ;?. 4. -4$,A"&. 4$( "&#., ):2@#, 4;4" 7,.
0*+" +/) /&#.0. 0 ,-A5",
AAA (*(* !!
E /&#.0.8 ! ?@7"4., !"#$%&'() *+%,',).
6,. /.%*2",7)., $, /&#.0"8 !"
#$%
&00
00, !
"
#$%
&000
000 "&#., ):2"#,0$&.
+ C# )" A' -()9$?&-$()" 4$# /&#.0. 4$( $/$&$( ;?. 4. -4$,A"&. "&#., .#4&+"-4. 41# .#4&-4$,A1# -4$,A"&1# "#;8 /&#.0. 0, 4;4" ,-A5",
!(*'('* AAAA )()(
E /&#.0.8 A' ?@7"4., ,%-+.$-/) -/0 *+%,', 0.
1 !"#$%&' – ()$**"%$ '+',-*$,$
19
!"# $#%&'(")*#, + #,-./(-+0 -+1 $.,#2# !"
#$%
&'
'207
163 (.,#" + $.,#2#0
!"
#$%
&'
''207
163.
3 $%+4(-#"%"4-"25 "'"6-7-# *#0 ($"-%8$(" ,# )%&9+1*( !BA (( )"# 2#/8,# #$6 -# .4# #/%+.4*#-# )( !BA (( , !BA (( )( . :*+.;0, #, !,"BA ,, (.,#" $.-,#2(0 -+1 .'"+1 -<$+1, -6-( 8=+1*(:
)]([)]([)()(])[( "!BA"!BA"!BA"!BA ((()((()((()(((
"!AB"!BA ((()((() ])[(])[( 2->.
2#" ($+*8,;0, *$+%+<*( ,# )%&9+1*( "!BA ((( )"# 2#/8,# #$6 -# #/%+.-4*#-# #1-&. !(,"2&, ($("'5 "4=<(" 7 #,-"*(-#/(-"25 2#" 7 $%+4(-#"%"4-"25 "'"6--7-#, *$+%(. ,# #$+'("=/(. 6-" -+ &/%+"4*# -%"?, 5 $(%"44+-8%;, $",&2;,
#AAA ...,,, 21 (.,#" -+ .'"+ 2#-& +$+"+,'5$+-( -%6$+ 2#" #, (2-(>(4-(. 7 $%64/(-47 2#" 41*@+>.A(-#" *( #AAA ((( *21 .
$./01234 56789:7
B$;0 2#" 4-7, $(%.$-;47 -;, $%#)*#-"2?, #%"/*?,, 8-4" 2#" 4-+10 $.,#2(0 7 #9#.%(47 +%.A(-#" *( -7 @+5/("# -70 $%64/(470. C1)2(2%"*8,#, #, BA, (.,#" '<+ #$+ $.,#2(0, -6-( 7 '"#9+%& BA ' +%.A(-#" ;0 (D50:
)( BABA '()'
!"# $#%&'(")*#, #, !!!
"
#
$$$
%
&
'
')
36
04
23
A 2#" !!!
"
#
$$$
%
&')
10
23
14
B , -6-(
!!!
"
#
$$$
%
&
''''
)!!!
"
#
$$$
%
&
''''
(!!!
"
#
$$$
%
&
'
')
!!!
"
#
$$$
%
&''
!!!
"
#
$$$
%
&
'
')'
26
27
31
10
23
14
36
04
23
10
23
14
36
04
23
BA .
E7>#'5, + $.,#2#0 BA ' $%+2<$-(" *( #9#.%(47 -;, 4-+"=(.;, -+1 % #$6 -# #,-.4-+"=# 4-+"=(.# -+1 &. F$6 -+10 $#%#$&,; +%"4*+<0 -70 $%64/(470 2#" -70 #9#.%(470 $%+2<$-(" 6-":
BAXABX '),)(
G%&)*#-":
— F, ABX )( , -6-( BABBX ')'( , +$6-( BAX ') , (,?
— F, BAX ') , -6-( BBABX (')( , +$6-( ABX )( .
!;<</5</36/3=>? /16@=;A =2 507/9/
: $#%#2&-; $.,#2#0 & $(%")%&9(" -"0 -"*80 $?>7470 4( '%#=*80 -%"?, 7>(2-%"-2?, ("'?, *"#0 @"+*7=#,.#0 4( '<+ 1$+2#-#4-5*#-#:
1 !"#$%&' – ()$**"%$ '+',-*$,$
20
T!"#$%&'#() *+,-#$ .-#%#$/0,(1&
2345$13-&'-!$22345$13-&'-!$1
170000100000150000
180000110000160000!"
#$%
&'A
6, 13-& -!, 5#%+$7$ -0, #15-8'#0,, $ 9($2:;3,$) 5%$-+<#-3( ,3 1&,#( =15-0-'! 20% '-3 5%$>?,-3 -$4, -?-# 5%=5#( ,3 7(32$%/8'#( -() ,=#) -(2=) '-$ 80% -0, 5%$!@$42=,0,. A( ,=#) -(2=) 58"!'!) <3 5%$1BC$4, 3, 5$""35"3'(&'$42# -() 53"(=) -(2=) 2# 0,8, ?50) /3+,#-3( '-$, 53%31&-0 5+,313:
!"
#$%
&((((((
'1700008,01000008,01500008,0
1800008,01100008,01600008,0B !
"
#$%
&'
13600080000120000
14400088000128000.
A 5+,313) ! "=@#-3( !"#$%&#' (') *+",%'- 0,8 %& ('# ./#*0* $ 13( '429$"+D#--3( 2# A(0,8 , 7!"37: #+,3( AB 8,0' . E#,(1&, =;$42# -$, 31?"$4<$ $%('2?:
O123453
6"#$%&#' &#$7 .+*!%*("0'- *+",%'- . %& 8#*# ./#*0* ]ij/A [' , "=@#-3( $
5+,313) 5$4 5%$1B5-#( 3, 5$""35"3'(&'$42# 1&<# '-$(;#+$ -$4 " 2# #. A 5+-,313) 34-?) '429$"+D#-3( 2# A. ( : .A . F!"37:,
][ ij#$A# '(
G 5%&H! 2# -!, $5$+3 9%+'1$42# -$ @(,?2#,$ 3%(<2$B 2# 5+,313 "=@#-3( % & #- # $ % # $ ' ( $ ' ) * + $ , ( - ) & . ) / % 0 1 $ 2 $.
E(3 53%&7#(@23, -$ @(,?2#,$ -$4 3%(<2$B 3)'# 2# -$, 5+,313
!"
#$%
&)))
'412
152A #+,3( $ 5+,313):
!"
#$%
&())())()())()()
'!"
#$%
&)))
)')43)1()3()2()3(
13)5()3(23
412
15233A !
"
#$%
&)))
'1236
3156.
I01234356 378 97../9./:1/:;7< /=1>;7< ;5 9?@/A/
6, BA, #+,3( 1)* 5+,31#) 13( #2, 5%3@23-(1$+ 3%(<2$+, -?-# (';B$4, $( 53%3-
1&-0 (7(?-!-#), 5$4 #+,3( &2#'! '4,=5#(3 -$4 $%('2$B:
1. #"2"A#2 +'+ )(
2. ,- ##BA# +'+ )(
3. A2##"2 )()( '
4. AA '1
I5(5"=$,, (';B#( ! ('$74,32+3:
!! ''.' A##A :0
1 !"#$%&' – ()$**"%$ '+',-*$,$
21
&.$)*/(H
N! "#$%$& ' (&)!*!+ 0 ,-! .') '('&' -/01$-:
!!!
"
#
$$$
%
&
''('
!!!
"
#
$$$
%
&'
'
14
02
13
53
10
75
21
2 X .
(M-! .2.'-! -/3.4.! $&)!- 5-! 1 2 3 4 5 4 6 71 8 3 9 : ; 1 <).
=+'-
!"#$%&
!!!
"
#
$$$
%
&
''('
!!!
"
#
$$$
%
&'
'
14
02
13
53
10
75
21
2 X
!!!
"
#
$$$
%
&'
''!!!
"
#
$$$
%
&
''(')
10
75
21
2
14
02
13
53X
!!!
"
#
$$$
%
&
''(')
!!!
"
#
$$$
%
&
''
'*
!!!
"
#
$$$
%
&
''(')
720
1420
117
3
20
1410
42
520
010
515
3 XX
!!!!!!
"
#
$$$$$$
%
&
'
'
''
()!!!
"
#
$$$
%
&
'''('')
3
7
3
203
14
3
203
1
3
17
720
1420
117
3
1)3(
3
1XX .
$'%-'&"'
67 896:6;
1. '& ()*&%+, )-. /+0 -)1)(,/2 -&1+-/34&+0, 5) 61&7/& /# ,*1#+4%) BA* ()+ /85 9+):#1, BA' , &:.4#5 :$4+(, #17;#5/)+:
i) !"
#$%
& '(
31
25A , !
"
#$%
&'
(52
36B
ii) !"
#$%
&(
8765
9876A , !
"
#$%
&(
9876
8765B
1 !"#$%&' – ()$**"%$ '+',-*$,$
22
iii) ]654[!A , ]654[ """!B
iv)
###
$
%
&&&
'
(
"
"!
126
130
215
A ,
###
$
%
&&&
'
("!
52
13
04
B
v)
###
$
%
&&&
'
(!
!"#$yx
%&'A ,
###
$
%
&&&
'
(
"""""""""
!!"#$yx
%&'B
1
1
1
.
2. !" #$"%& #$
%&'
("!
01
231A , #
$
%&'
(!
21
532A , #
$
%&'
("
"!
31
563A '%& #
$
%&'
("
!31
204A ,
"% ()#$*# *+ ,-)+&./% 4321 AAAA ))) .
3. 0% ()#$*# *% $yx ,, 1&% *% +2+$% &.34#& 5 &.6*5*%:
###
$
%
&&&
'
( "!
###
$
%
&&&
'
(
"")
###
$
%
&&&
'
(""
107
61
58
13
34
13
92
23
25 $
y
x .
4. 0% ',"#*# *&7 2),8#&7:
i) #$
%&'
( ")#
$
%&'
(" 350
6213
141
3125 ii) #
$
%&'
("
""#
$
%&'
("420
642
2
1
401
1534
iii) #$
%&'
("
"""""""
"3
2.
5. !" ###
$
%
&&&
'
( "!
01
12
13
A '%& ###
$
%
&&&
'
("!
84
20
62
B , "% ()#$*# *+97 2$"%'#7:
i) A2 ii) )3(2 A" iii) AB 25 " iv) BA2
13 " .
6. 0% :4.#*# *&7 #8&.;.#&7:
i) #$
%&'
("
"!#
$
%&'
( ")
52
414
27
153X
ii) #$
%&'
(""
"!"#
$
%&'
(" 22
1357
43
216 X .
7. 0% %2+<#$8#*# 6*& *+ ,-)+&./%
1 !"#$%&' – ()$**"%$ '+',-*$,$
23
!"
#$%
&'
(!"
#$%
& '!!!!
!!!!!
!!"#$%#$%!"
!"#$%!"!"#$%
#$%
&'%() *%(+ ,)(-.%)/+ 0'%(1(+.
8. 2% !"
#$%
&)
"#$!
X 1() !"
#$%
&''
)"#$!
Y3
2, %( 34&'5& 5)+ 5)"*+ 56% !,$,#,"
.#5& %( )#78&):
!"
#$%
& ')'
1321
81452 YX .
B! "#$%$&
1. 9( 34&'5& 5( yx, -)( 5( /0/'( )#78&):
i) !"
#$%
& ')!
"
#$%
& ((!"
#$%
&' 01
113
10
2122 yx
yyx ii) !
"
#$%
&)!
"
#$%
& '(!
"
#$%
&
'('
00
10
1
12
1
32
2
yy
yxxx.
2. N( 34&'5& 5/$+ 0'%(1&+ YX , -)( 5/$+ /0/'/$+ )#78&):
!"
#$%
&)(
30
033 YX 1() !
"
#$%
&)(
40
0425 YX .
3. 2% !"
#$%
&'
')
153
201A 1()
!!
"
#
$$
%
&
'
')1104
3
4
12
B , %( :8#&5& 5!% &;'#6#!
BAXBX 52
12)(3 '*
+,
-./ ()( .
4. <)( 3)/"!7(%'( 0/$ 1(5(#1&$=>&) 5!:&/4=#&)+, 3'%5&/ 1() 1="&4&+
*7&) ,8/ &4-/#5=#)( 0(4(-6-?+ 1% 1() 2% . @/ 1A#5/+ 0(4(-6-?+ (%= #$#1&$? ,'%&5() (#& 7):)=,&+ ,47.) #5/$+ 0(4(1=56 0'%(1&+:
@!:. B)%5. C(". @!:. B)%5. C(".
D4-(#'(E:)1=
363225
4028301 !
"
#$%
&)%
D4-(#'(E:)1=
382823
4230382 !
"
#$%
&)% .
9( 34&'5& 5/% 0'%(1( )(2
121 %% ( 1() %( &;!-?#&5& 5) &1F4=>&).
5. <)( 3)/"!7(%'( *7&) 5*##&4( &4-/#5=#)( 0(4(-6-?+ 321 ,, %%% 1()
4% , 1(G*%( (0A 5( /0/'( 0(4=-&) ,8/ 04/HA%5( 1E 1() 2E . @/ !"&-4?#)/ &0'0&,/ 0(4(-6-?+ #& "/%=,&+ 04/HA%56% ,'%&5() #5/% &0A"&%/ 0'%(1(:
1 !"#$%&' – ()$**"%$ '+',-*$,$
24
1! 2! 3! 4!
2
1
1201204080
60140180200
E
EA !
"
#$%
&' .
i) N! "#$%&$ &' ()$#*+,' $-%-$.' -!#!/0/*1, !2 !3&* !34(5$% 6!&7 10%.
ii) 8! "#$%&$ &' +92':' &(1 -!#!/0/*1 !27 -#';<2 +$ 5 )*2$1, !2 3-'&$5$% <&, &! $#/'+&7+,! .'9:$=!2 2 )*2$1 )$ &' -#'(/'9)$2'
$-%-$.' 6!, 3 )*2$1 )$ &' 2>' $-%-$.' -!#!/0/*1 (1 )*2!1 = 30
)>#$1).
1.3 !.//$!/$'"$'*.' !"#$%0#
.123456 789 :2;84<;89 =>8 ?2;@AB; ?1 3-'5>+'3)$ <&, /,! &(2 6!&!+6$3* .9' $,.@2 /:36,+)7&02 1" 6!, 2" A#$,!-
B<)!+&$ &! 3:,67 +$ kg -'3 C!%2'2&!, +&'2 -!#!67&0 32( -%2!6!:
?:$9#, D7A!#( E'9&3#'
!!!
"
#
$$$
%
&'
4,08,04,1
3,06,02,1
A/:96,+)!
/:96,+)!
2
1
"
".
F+&0 $-%+(1 <&, &' 6<+&'1 +$ .#A. &02 3:,6@2 !3&@2 !27 6,:<, /,! &! >&( 1992 6!, 1993, $%2!, <-01 .$%A2$, ' -!#!67&0 23( -%2!6!1:
1992 1993
!!!!
"
#
$$$$
%
&
'
1200900
200170
180160
B
"'9&3#'
B7A!#(
!:$9#,
G,! 2! "#'9)$ &' 6<+&'1 +$ .#!A)>1 &02 3:,6@2 &'3 /:36%+)!&'1 1" , -'::!-
-:!+,7B'3)$ &,1 -'+<&(&$1 &02 3:,6@2 )$ &,1 !2&%+&',A$1 &,)>1 6!, -#'+5>&'3)$ &! /,2<)$2! !3&7. H(:!.* &' 6<+&'1 &'3 1" &' 1992 *&!2
5649003,01706,01602,1 ')*)*) .
I -!#!-720 .,!.,6!+%! -$#,/#7C$&!, )$ &( "'*5$,! &02 -,27602 01 $4*1:
1 !"#$%&' – ()$**"%$ '+',-*$,$ 25
]564[]9003,01706,01602,1[
900
170
160
]3,06,02,1[ !"#"#"!$$$
%
&
'''
(
).
O 11* !"#$%$& [564] '()*+$, !"#$%&#' +-& !./+-& ).$001& +23 ! *!" +-# !./+-
4+1'- +23 ". 5#$'6)7&, +2 %64+2& +23 1# +2 1993 1+$#
69612003,02006,01802,1 !"#"#" .
8-'$91 !$.,4+:#*+$, 0* +2 ),#60*#2 +-& !./+-& ).$001& +23 ! *!" +-# 9*;+*.-
4+1'- +23 "
]696[
1200
200
180
]3,06,02,1[ !$$$
%
&
'''
(
).
<02"7&, +2 %64+2& +23 2# +2 1992 1+$#:
7209004,01708,01604,1 !"#"#" 1
]720[
900
170
160
]4,08,04,1[ !$$$
%
&
'''
(
),
*#/ +2 1993 1+$#:
89212004,02008,01804,1 !"#"#" 1
]892[
1200
200
180
]4,08,04,1[ !$$$
%
&
'''
(
).
< !"#$%$& $%
&'(
)!
892720
696564# 9*"=#*, +2 %64+2& +7# 9;2 )'3%,40:+7# %$+: +$ (+-
1992 %$, 1993. < !"#$%$& # !23 !.2%;!+*, 0* +2# !,2 !:#7 +.6!2 '()*+$, !"#$-
%&#' (') *+#,-, $ %& ('# *+#,-, . %$, 430>2'"?*+$, 0* BA " 1 AB , 9-'$91
$%
&'(
)!
$$$
%
&
'''
(
)
$%
&'(
)!
892720
696564
1200900
200170
180160
4,08,04,1
3,06,02,1# .
@*#,%: (=230* +2# $%6'23A2 2.,406:
O./0120
1 !"#$%&' – ()$**"%$ '+',-*$,$ 26
!" ][ ik!A ! #$"%& '"%( "#" )$"%*%( *%& ][ kj$B ! #$"%& '"%( %"" )$"%*%(, +,-
+# -.$/-01# 2( !"#$%&#' (') *+#,-, $ %& ('# *+#,-, . *%& +- 3014-5$/-01# 1# BA # 6 1# &' +-" %#" )$"%*%, +-0 -)-$-0 *78# 3+-&9#$- ij( #$"%& +- 7-
8.-&31% +2" :&"-1'"2" +2" " 3+-&9#$2" +;( i -:.%116( +-0 & 1# +% %"+$3+-&9%
" 3+-&9#$% +;( j -3+65;( +-0 '. <;5%=6,
/ji/jijiij 0101012 $$$! %2211
>9;1%+&*7
j -3+65;
&&&&&&&&
'
(
))))))))
*
+
!
&&&&&&&&
'
(
))))))))
*
+
&&&&&&&&
'
(
))))))))
*
+
,
#%#j##
i%ijii
%j
%j
"%"j""
%j
%j
#"##
i"ii
"
"
((((
((((
((((((((
$$$$
$$$$$$$$
!!!
!!!
!!!!!!
i
%%-
%%-
%%%%
----%%%%
%--
%-
%%
21
21
222221
111211
21
222221
111211
21
21
22221
11211
:.%116
?&% )%.7=#&:1%, +- :&",1#"-
&&&
'
(
)))
*
+,&
'
()*
+ ,
01
51
24
103
312 4.$3*#+%& 2( #@6(:
1311)1(043
,41)3()1(142
21
11
!#$,$#!!#,$,$#!
((
*%& 6015023
90)3(5122
22
12
!#$#$#!!#,$#$#!
((
.
A)-1'"2(,
&'
()*
+!
&&&
'
(
)))
*
+,&
'
()*
+ ,613
94
01
51
24
103
312.
B-"$/#+%& ,+& +- :&",1#"- &' -.$/#+%& ,+%" - %.&81,( +2" 3+;5C" +-0 )$"%*% &
#$"%& $3-( 1# +-" %.&81, +2" :.%11C" +-0 )$"%*% '.
>9;1%+&*7:
%#%""#AB'A
""".),(
?&% )%.7=#&:1%, %" &'
()*
+,
!214
312A ,
&&&
'
(
)))
*
+
,!
21
02
31
B *%& &&&
'
(
)))
*
+ ,,!
2120
0452
1113
) ,
+,+#, 3D1E2"% 1# +-" )%.%)7"2 -.&31,, -.$/-"+%& +% :&",1#"% A)BAAB ,, *%& #$"%&
1 !"#$%&' – ()$**"%$ '+',-*$,$ 27
!"
#$%
&'
!!!
"
#
$$$
%
&
(!"
#$%
&(
'84
07
21
02
31
214
312AB ,
!!!
"
#
$$$
%
&
((
('!
"
#$%
&(
!!!
"
#
$$$
%
&
('
136
624
9214
214
312
21
02
31
BA !"#
!"
#$%
&((
('
!!!
"
#
$$$
%
& ((
!"
#$%
&(
'86314
85134
2120
0452
1113
214
312A! ,
$%& '$% ()*+(%,"# ," -#%./$%" !"B! , !"# !# .
"./012134 156 75889789:/9:;5< 1=> 7/>?@=>
) 0% $%, $*%"# 1)"-/",#!(* ")#2/(* !"# BA, $*%"# 1*%"!$3, ,.,$ #456(7% (# 1")"!8,9 #'#.,:,$3 /$ ,:% 1)(;1.2$4: (,# ()*+(%,"# (# 1)8<$#3 1(7 4:/$#&%(-
%,"#.
!ABB!A )()( ' &'()*+,-'-)+-./
#!AB!"# *'* )( !"# !#BAA!" *'* )( *&-$*'-)+-./
AB%$$B%A )())(( '
) 0% /$ 0I 47/=(>*4(7/$ ,(% 00+ '#"-&%#( 1*%"!"
!!!!
"
#
$$$$
%
&
1000
010000100001
,----
,,,
,(7
(1(*(7 !82$ 4,(#5$*( ,:3 !7)*"3 '#"-9%*(7 $*%"# *4( /$ 1, ,.,$ -#" !82$ ,$,)"-
-9%#!. 00+ 1*%"!" # #456$#:
AAIAI 00 ''
O 1*%"!"3 "7,.3 >?-$,"# !"#$%&$'"( )'#$*$(.
@#" 1")8'$#-/", (# 1*%"!$3 !"
#$%
&'
10
012I ,
!!!
"
#
$$$
%
&'
100
010
001
3I $*%"# /(%"'#"*(#.
A(% 1*%"!" 0I 2" ,(% 47/=(>*+(7/$ "1>(64,$)" /$ 1, .,"% $*%"# 1)(B"%C3 (
,61(3 ,(7.
0% ,&)" # $*%"# ?%"3 0$+ 1*%"!"3, ,.,$ #456(7%
$"> A' !"# I; AA ' .
@#" 1")8'$#-/"
1 !"#$%&' – ()$**"%$ '+',-*$,$ 28
!"
#$%
&'
!!!
"
#
$$$
%
&
!"
#$%
&654
321
100
010
001
654
321 !"#
!"
#$%
&'!
"
#$%
&!"
#$%
&654
321
654
321
10
01.
$ %&'()*"#&#(*#!+ #,#-*.*" /"0 )%#*&1%)# 2" 3&45'6/) AB! 3#" !"712" "%- *"
8(" 3#2-/)2" )(B!A , !AB)( . 9/'8:0, "2 ",#,!,$ )82"# %82"!)0 *1*'#'#, ;(*) 2"
'&8<'2*"# *" 3#2-/)2" "# , #! , !$ *-*) 1='6/)
$B!A$#!A!$BA!$AB$!AB )]([])[()]([))((])[( ''''
!"# /%'&'>/) 2" 3&45'6/) "#!$ 3#" !"712" "%- *" 3#2-/)2" "6*4.
?)2#!4, )%)#,+ #(=>)# . %&'()*"#&#(*#!+ #,#-*.*", /%'&)8 2" "%',)#=*)8 -*# -*"2 %'@@"%@"(#4<'6/) 12"2 "/- %#24!:2 %AAA ...,,, 21 *' 3#2-/)2' 7" )82"# *'
8,#' !"*4 '%'#'2,+%'*) *&-%' !"# "2 )!*)@)(*)8 ' %'@@"%@"(#"(/-0, =:&80 --
/:0 2" "@@4A)# . ()#&4 *:2 %"&"3-2*:2 !"# (6/B'@8<)*"# /) %AAA ...21 .
C2 ' " )82"# 12"0 *)*&"3:2#!-0 %82"!"0, *-*) '&8<'2*"# *" 3#2-/)2" "", """,
"""", !*@. !"# *" (6/B'@8<'6/) /) /'&5+ ,624/):2 :0 )A+0: ...,,,, 432 AAA "-
2*#(*'8=:0. 9&8<'6/) )%8(.0 AA '1 .
C2 qp, )82"# 7)*#!'8 "!1&"#'#, !"# & %&"3/"*#!-0 "/-0, "%',)#!2>)*"# -*#:
qpqp AAA (') , pqqp AA ')( !"# ppp A&&A ')( .
!"#$%#
?2:&8<'6/) -*# 3#" *'2 %'@@"%@"(#"(/- *:2 %&"3/"*#!;2 "/;2 #(=>)#, )%#-%@1'2, !"# . "2*#/)*"7)*#!+ #,#-*.*". D.@",+, #(=>)# '((' )') 3#" '%'#'6(,+-
%'*) !*',( . $ #,#-*.*", -/:0, "6*+ &'( #(=>)# 3#" *'2 %'@@"%@"(#"(/- *:2 %#24!:2, "5'> 6%4&='62 %82"!)0 BA, /) BAAB + . ?#" %"&4,)#3/", "2
!"
#$%
&,'
35
21A !"# !
"
#$%
&,
'10
21B , *-*) BAAB + , "5'>:
!"
#$%
& ,,'!
"
#$%
&,!
"
#$%
&,'
75
41
10
21
35
21AB , )2; !
"
#$%
&,,
'!"
#$%
&,!"
#$%
&,
'35
89
35
21
10
21BA .
E%)#,+, @'#%-2, ,)2 #(=>)# . "2*#/)*"7)*#!+ #,#-*.*" '# #(-*.*)0: 222 2)( BABABA (-'- , 32233 33)( BABBAABA -(-'- ,
))((22 BABABA ,(', , ))(( 2233 BABABABA ((,', !*@.
&'( )*+,-.( /0(1-1'. F*.2 %)&8%*:(., -/:0, %'6 BAAB ' '# %"&"%42: #(-*.-
*)0 #(=>'62.
1 !"#$%&' – ()$**"%$ '+',-*$,$ 29
&.$)*/(H
!"#$#%&' $' ("#&)*+ !"
#$%
&'
01
10A )&' !
"
#$%
&((
'11
21B . ,& &($-*'.%*"
/%':
i) IA '2 , IB ('2 )&' !!!!') 22 BA
ii) IBAAB ')
iii) ABBABA 2)( 222 ))*) .
0+'-
i) !"#$% IA '!"
#$%
&'!
"
#$%
&!"
#$%
&'
10
01
01
10
01
102
IB ('!"
#$%
&(
('!
"
#$%
&((!
"
#$%
&((
'10
01
11
21
11
212
&'$ !') 22 BA .
ii) !"#$% !"
#$%
& (('!
"
#$%
&((!
"
#$%
&'
21
11
11
21
01
10AB
!"
#$%
&((
'!"
#$%
&!"
#$%
&((
'11
12
01
10
11
21BA
&'$ IBAAB ') .
iii) !"#$%
IBAABIBBAABABABABA ())')))'))') 222 ))(()(
IBAAB ')' (()*+ ,-. ii))
!"
#$%
& (('!
"
#$%
& ((''))
42
22
21
1122222 ABABBA (()*+ ,-. ii)).
&'$, ABBABA 2)( 222 ))*) .
$123425673893 :;1<=>?
+ /#+'"01234 ),% *%$ 5674 8'$*3$,%5) $'%73) ! 34 0*! 286'94% 1 $#,":,'1-
;). ,12, 812 :23<1("04,$% 34 !1
= 1(! , *%$ ,1# 181"1 %:9>4% 111 '' (( !!!! .
1 !"#$%&' – ()$**"%$ '+',-*$,$ 30
!"#$% &'(%)* +,-$ #$ -.+/0'123: “4# 5'63" 7#$8 9"#$)$8 ! 29'-':23 #$ ;-':-
23 7#$# 9"#$)$ " +7+'%'# ,0+3 #$ %0<:3% IBAAB !! ;”
=:2>.#$ 23 +'# 9'&&$9&$0%$02* 9'1 '-"0$23 2%$ +7+'%$ 3-,+?0? 7<3% #*?2$
2*#' $# ' ! 3"#$% 7#$8 +3+-$(.#%)*8 9"#$)$8. @5?(':2$0+3 7+0% 0+'# 3A/8 '-%-02*:
O!"#$%#
B0+. ! 7#$8 +3+-$(.#%)*8 9"#$)$8 +:9'1 ##" . 4# 19C-<3% +3+-$(.#%)*8 9"-#$)$8 " +:9'1 ##" , +7+'%'8 ,0+3 #$ %0<:3% IBAAB !! , +*+3 ' ! &7(3+$% &-
'()*(+,-)./0 12'&3&0 )$% ' " &'(2*(+/4/0 +'1 !.
4# 7#$8 9"#$)$8 ! 7<3% $#+"0+-'>', +*+3 $9'53%)#:3+$% *+% $1+*8 3"#$% 2'#$5%-)*8 )$% 012;'&"D3+$% 23 1#A . B+0% 7<'123:
IAAAA !! ## 11
E%$ 9$-C53%(2$, $#
$%
&'(
)!
31
21A )$% $
%
&'(
)#
#!
11
23B ,
+*+3 7<'123:
IAB !$%
&'(
)!$
%
&'(
)#
#$%
&'(
)!
10
01
11
23
31
21 )$% IBA !$
%
&'(
)!$
%
&'(
)$%
&'(
)#
#!
10
01
31
21
11
23.
F-$, ' " 3"#$% ' $#+"0+-'>'8 +'1 !.
* =:2>.#$ 23 +'# 9$-$9C#. '-%02*, ' 9"#$)$8 " 3"#$% $#+"0+-'>'8 +'1 !, *-
+$# IAB ! )$% IBA ! . 49'53%)#:3+$%, *2.8, *+%:
567!8$9
4# (%$ 51' ##" 9"#$)38 BA, %0<:3% 2%$ $9* +%8 %0*+?+38
IAB ! )$% IBA ! ,
+*+3 6$ %0<:3% )$% ? C&&?.
G3 ;C0? $1+* +' 63,-?2$, (%$ #$ $9'53"A'123 *+% 7#$8 ##" 9"#$)$8 " 3"#$% $#+"0+-'>'8 3#*8 ##" 9"#$)$ !, $-)3" #$ $9'53"A'123 2"$ 2*#' $9* +%8 %0*+?-
+38 IAB ! )$% IBA ! .
* H7&'8, $# 7#$8 9"#$)$8 ! 3"#$% $#+%0+-7I%2'8, +*+3 %0<:'1# '% %0'51#$2"38:
(i) BAXBAX 1#!+!
(ii) 1#!+! BAXBXA
J-C(2$+%, (%$ +?# (i) 7<'123:
— 4# BAX ! , +*+3 BAAXA 11 ## ! , '9*+3 BAX 1#! .
1 !"#$%&' – ()$**"%$ '+',-*$,$ 31
— !" BAX 1!" , #$#% BAAAX 1!" , &'$#% BAX " .
()&*+, -'&.%/0"1%#-/ 0-/ 2 (ii).
!"#$%#
3"+4*5&6)% $#/ 7/- #&" '&88-'8-9/-9)$ #+" '4-7)-#/0:" -4/;):" /9<1%/ %'/-'8=&" 0-/ 2 /./$#2#-: “-" 0"# !" , #$#% 0"" > 0"! ”. ? /./$#2#-, $)+,, -6#>
&'( )*+,') 7/- #&" '&88-'8-9/-9)$ #+" '/"@0+", -A&1 '.<. 7/- #&6, '*"-0%,
$%
&'(
)"
01
01A 0-/ $
%
&'(
)!
"11
00B /9<1%/ $
%
&'(
)"$
%
&'(
)!$
%
&'(
)"
00
00
11
00
01
01AB <+4*,, +9#$9&,
"- %*"-/ O"A > O"B . B28-.>:
“-./0'1 2(3 4)(56'(/ .)(789( (3 )*/,:3) 6' :/ 6;&'()85 .1(383, +901< 83-
(2(3< (3 '1(3) 6;&'()85<”.
C#2" '%4*'#+92 $)+, '&6 /9<1%/ !"AB 0-/ & ="-, -'$ #&6, '*"-0%, %*"-/ --
"#/9#4=D/)&,, #$#% & @88&, %*"-/ )2.%"/0$,. E4@7)-#/, -" & # %*"-/ -"#/9#4=D/-)&,, #$#% =<&6)% ./-.&</0@:
!"AB
!11 !! " AABA
!"IB
!"B .
A./01/23435 6.75 22* 80.9:9
F9#+ $%
&'(
)"
$%!"
A ="-, 22* '*"-0-,. G- %H%#@9&6)% '$#% -6#$, -"#/9#4=A%#-/
0-/ ;- I4&1)% #&" -"#*9#4&A$ #&6.
3/- "- -"#/9#4=A%#-/ & #, '4='%/ 0-/ -40%* "- 6'@4<%/ '*"-0-, $%
&'(
)"
&z
yxX #=-
#&/&,, :9#% "- /9<1%/ IAX " >, /9&.1"-)-,
$%
&'(
)"$
%
&'(
)$%
&'(
)10
01
&z
yx
$%!"
$%
&'(
)"$
%
&'(
)++++
10
01
$&%y$z%x!&"yz!"x
,-.
"+"+
0
1
$z%x!z"x
(C1) 0-/ ,-.
"+"+
1
0
$&%y!&"y
(C2)
!40%*, %'&)="+,, #- 969#>)-#- (C1) 0-/ (C2) "- =<&6" 8192. J- 969#>)-#- -6-
#@ =<&6"
1 !"#$%&' – ()$**"%$ '+',-*$,$ 32
!"#$$"!#
D !""
!"#
$$!
Dx ""0
1, "
"#
Dz !""0
1, !
$!
Dy !""1
0, #
"#
D% ""1
0.
$%&'()*+:
# A) 0$D , ,-,. ," /0/,1'"," (21) !"# (22) (3&0) '&)"4#!1 56/7, &%-,. & %8-)"!"+ & "),#/,9(:.,"#. ; 56/7 ,&0 (21) .8)"# ,& <.6=&+ ),( zx '.
D
$D
Dx x "" !"#
D
"D
Dz z !
"" ,
.)> 7 56/7 ,&0 (22) .8)"# ,& <.6=&+ ),( %y '.
D
!D
Dy
y !"" !"#
D
#D
D% % "" .
?9" %%%
&
'
(((
)
*
!
!
"
D
#D
"D
!D
$
X , &%-,. & "),8/,9&:&+ ,&0 & .8)"# & %8)"!"+
%&
'()
*!
!"!
#"!$
DA
11 .
# @) 0"D , ,-,. ()" ,&05A3#/,&) "%- ," /0/,1'"," (21) !"# (22) .8)"# "46)"-
,&, &%-,. & %8)"!"+ & 4.) "),#/,9(:.,"#. B9A='",#.
") @) 0$xD 1 0$yD 1 0$zD 1 0$%D , ,-,. ()" ,&05A3#/,&) "%- ,"
/0/,1'"," (21) !"# (22) C" .8)"# "46)",&.
D) @) 0"""" %zyx DDDD , ,-,. 0"""" $"!# , &%-,. !"# %A5# ," 46&
/0/,1'"," C" .8)"# "46)",".
@%&4.8E"'. 5&#%-) -,#:
# O %8)"!"+ %&
'()
*"
$"!#
A .8)"# "),#/,9(F#'&+, ") !"# '-)& ") 0$$"!#
.
# G "),8/,9&:&+ .)-+ %8)"!" %&
'()
*"
$"!#
A , ") 0%A93.#, 48).,"# "%- ,&) ,6%&
%&
'()
*!
!"!
./01
DA
11 , -%&0
$"!#
D " .
H#" %"9A4.#='":
1 !"#$%&' – ()$**"%$ '+',-*$,$ 33
!) " #$%!&!' !"
#$%
&'
21
43A !%()*(+,-.(!), /)!($ 02
21
43(' &!) 0 !%($*(+0-1'
(02 .$%!) 0
!"
#$%
&)
)')
31
42
2
11A .
3) " #$%!&!' !"
#$%
&'
12
24A !"# !%()*(+,-.(!), /)!($ 0
12
24' .
&.$)*/(H
$%#&#'() &) *%#(+", !"
#$%
& )'
43
12A +() !
"
#$%
&)
'122
311B
i) N( -."/"% & (#'%0'.&1&, '&2 *%#(+( $
ii) 3( 42/"% 5 "6%0705 BAX '
89:;
i) 4)! (0% #$%!&! ! ,5026. 01143
12('
)'D . 7+!
!"
#$%
&)
')
23
14
11
11A (1)
ii) 8#.)9: 0 #$%!&!' ! .$%!) !%()*(+,;)60', ,5026.:
!"
#$%
&)
*!"
#$%
&)
'+*'+' )
122
311
23
14
11
1)1(1 XBAXBAX
!"
#$%
&)
'+11
16X .
$'%-'&"'
<= >?<$<:
1. N! 3+.$(. (! /)%16.%! AB &!) BA *. 1#0).' !#1 ()' #!+!&<(= #.+)-#(>*.)' 0+$?0%(!):
1 !"#$%&' – ()$**"%$ '+',-*$,$ 34
i) ]123[!A ,
"""
#
$
%%%
&
'!
0
3
2
B ii) "#
$%&
'(
(!
12
24A , "
#
$%&
'!
24
12B
iii)
"""
#
$
%%%
&
'
(
(!
135
240
103
A ,
"""
#
$
%%%
&
'
((
(!
310
214
051
B
iv) "#
$%&
'((
!313
124A , .
174
210
312
"""
#
$
%%%
&
'
(!B
2. !" """
#
$
%%%
&
' (!
02
10
11
A , "#
$%&
'!
101
110B #$%
"""
#
$
%%%
&
'(!
220
101
011
! .
&$ '()*+) +,-. /*"$#).: i) AB ii) !AB ( iii) AB! .
3. T$ 0+,%1)*$ 2%$ +%. $3,%'4. #$% +," $(%536 +7" )(2$+8" 0) 9:, ,%#,-
9,3%#4. )+$%()*). " #$% # 41,-" 3) 3,(;< /%"=#7" 7. )><.:
A(%536. )(2$+8" ?'9,3$9%$*). $/,9,14. ?%9%#)-34",% !")%9*#)-+,% (0) 1%@. 9($134.)
"#
$%&
'6030
7560
B
A
,%!")%9*#)-+,%?%9%#)-34"
"#
$%&
'40
50.
&$ )#;(=0)+) 3) +A ',<5)%$ +,- /,@@$/@$0%$03,: +7" /%"=#7" +,
0:",@, +7" $3,%'8" +7" )(2$+8" 0+%. 9:, )+$%()*)..
4. B) #$5)3%= $/6 +%. /$($#=+7 /)(%/+80)%. "$ $/,9)*>)+) 6+% , /*"$-
#$. # )*"$% $"+*0+(,;,. +,- ".
i) "#
$%&
'!
21
32A , "
#
$%&
'(
(!
21
32B ii)
"""
#
$
%%%
&
'
((((
!423
352
231
A ,
"""
#
$
%%%
&
'
((
((!
11119
11017
1814
B
5. &$ '()*+) +," $"+*0+(,;,, );60," -/=(1)%, #$5)"6. $/6 +,-. /$($-
#=+7 /*"$#).:
"#
$%&
'!
21
32A , "
#
$%&
'!
12
24B #$% "
#
$%&
' (!
$$$$
!0-"A3A30-"
.
6. i) &$ '()*+) +," $"+*0+(,;, +,- /*"$#$ "#
$%&
' (%%%%
A30-"0-"A3
.
1 !"#$%&' – ()$**"%$ '+',-*$,$ 35
ii) !" #$%&'& '() &*+%,%(: !"
#$%
& '!!!!
(-%.)%.)(-
!"
#$%
&''
(!!!!
X%.)(-(-%.)
.
B! "#$%$&
1. /) !"
#$%
&(
42
21A , '0'&:
i) !" 12&+'& '34 '3-54 ',) !)yx, 63" '34 787+&4 3%9$&3 yIxAA *(2 .
ii) !" .87#76+%&'& '7.4 8+)":&4 3A :"3 4A .
2. A) !"
#$%
&'
'(
13
12A , )" 12&+'& '7) 82"6-"'3:0 "23;-0 x, <%'& )" 3%9$&3
"('' xIAA 103 .
3. !" 12&+'& '7.4 8+)":&4 !"
#$%
&(
"!
X0
1, !)!," 63" '7.4 787+7.4 3%9$&3
IX (2 .
4. /) !!!
"
#
$$$
%
&
''
'(
322
323
110
A ,
!!!
"
#
$$$
%
&
''''
(233
212
334
B , )" "87=&+*&'& 0'3:
i) IA (2 , IB (2 ii) "(' 2)( BA , IBA 4)( 2 (* .
iii) ))((22 BABABA '*+'
5. A) !"
#$%
&''
(12
23A , )" "87=&+*&'& 0'3:
i) > 8+)":"4 # ")'3%'25?&'"3 :"3 )" 12&+'& '7) 1'A .
ii) IAA $$ 2)( 1 (* ' , *#)$ .
6. A) !"
#$%
&(
xx
xxxA
%.)(-(--%.)
)( , !"
#$%
&'
(xx
xxxB
(-%.)%.)(-
)( , '0'&:
i) !" "87=&+*&'& 0'3 )2()(2 xAxA ( , )2()(2 xAxB '(
ii) !" "87=&+*&'& 0'3 "(* )()( 22 xBxA
iii) !" #$%&'& '() &*+%,%( IxBxA 2)()( 22 (' .
7. M3" 137-(9")+" &8+8#,) :7.@+)"4 59&3 =$7 &267%'A%3" 1E :"3 2E . >3 8+)":&4 % :"3 & =+)7.) '34 <2&4 &26"%+"4 87. "8"3'7$)'"3 63" '() :"'"%:&.B :A;& &8+8#7. :"3 '34 ,23"+&4 "-73154 '7. 827%,83:7$ %& =2"9-54 ")'3%'7+9,4.
1 !"#$%&' – ()$**"%$ '+',-*$,$ 36
!"#"$%&'( )*+,-. /'$%&'"$0" 1E 2E
12"345,!"24%6"7*8%.9
4,02,15,1
3,09,01
2,06,06,0
!!!
"
#
$$$
%
&'M ,
/'$%&'"$0")*+,-.!"#"$%&'(
14001350
17001600
15501500
!!!
"
#
$$$
%
&'N
i) :" ;2&0#& #.< 30<"%" !" %", <" &=>8($&#& #, &%?2*5&,.
ii) 7.,. &0<", #. %@$#.9 &28"$0"9 8," #>< 3"2"8A8( -,"9 %"24%6"9 $#. &28.$#*$,. 1E %", &<@9 3*8%.' $#. &28.$#*$,. 2E ;
8. B< !!!
"
#
$$$
%
&
((('
312
625
311
A %", !!!
"
#
$$$
%
&
((
('
433
434
110
B , <" "3.C&0=&#& @#,:
i) !'3A %", 8&<,%* !'#A , 3)#
ii) IB '2 , BB '3 %", 8&<,%* *+,
'D&#,%@93&2,##@9"<
D&#,%@9*2#,.9"<#B
#IB # .
9. E0<&#", . 30<"%"9 !"
#$%
&'
1>->-1
$'<1
)(x
x
xxA , -
./
012(3
2,
2
$$x
i) :" "3.C&0=&#& @#, )()(1 xAxA ('(
ii) :" 6F$&#& #>< &=0$A$> IxA ')( .
10. B< !!!
"
#
$$$
%
&
'100
210
1
)(
2
x
xx
xA ,
i) :" "3.C&0=&#& @#, )()()( yxAyAxA 4'
ii) :" ;2&0#& #> $G4$> -&#"=F #A< yx, H$#& . 30<"%"9 )(yA <" &0<", "<#0$#2.?.9 #.' )(xA .
iii) N" ;2&0#& #.< "<#0$#2.?. #.' 30<"%"
!!!
"
#
$$$
%
&'
100
210
111
M .
11. B< !"
#$%
&((4
'%%%%
A1
1, "3% , #@#&:
i) :" "3.C&0=&#& @#, IA '2 , AA '3 %", 8&<,%* @#,
*+,
'3&2,##@9"<,
*2#,.9"<,
#A
#IA#
1 !"#$%&' – ()$**"%$ '+',-*$,$ 37
ii) !" 2!! , "# $%&'(& ()" *'"#+# " ,-# ()" )*)') -./0&-
"#
$%&
'(
!10
111993 XA
iii) 1# 2*)3),'.&(& () 45%)-.6#
102 AAAI )))) * .
12. 7'"&(#- ) *'"#+#8 "#
$%&
'!
11
12A
i) 1# $%&'(& ()" #"('.(%)9) ()2 *'"#+# #
ii) 1# $%&'(& ()" *'"#+# " .& +45& 6-# #*: (-8 *#%#+4(; *&%-*(<-
.&-8:
#) "#
$%&
' (!
01
12AX $) "
#
$%&
'!
00
01AXA ,) AAAX 22 )! .
13. !" ""#
$
%%&
' (!13
31
2
1A , (:(&:
i) 1# #*)=&'>&(& :(- IA (!3 +#- ,&"-+4 :(-
+,-(
!*&%-((:8#",
4%(-)8#",3
$I
$%A $ .
ii) 1# $%&'(& (-8 *%#,6#(-+?8 (-6?8 ()2 x ,-# (-8 )*)'&8 -./0&-
!!)) 198919922 )2( AxAx .
1.4 (&.*&,)"%/" *&,$'0-*$,"'*/"
- 122345 637 (89:86;4<3= *865>?@:564>:3=
. @";%'A)26& #*: (B" !C D2+&')2 :(- .2"4%(B.B #*: ?"# .0")3) # .& ?"# .0-
")3) & &'"#- 6-# =-#=-+#.'# 6& (B" )*)'# +45& .()-/&') ()2 # #"(-.()-/'A&(#- .& ?"# +#- 6)"#=-+: .()-/&') ()2 &. E(B" *#%4,%#9) #2(F 5# #./)3B5)06& 6& .2-
"#%(F.&-8 ,-# (-8 )*)'&8 (# # +#- & .26*'*()2" 6& () .0")3) E (;" .B6&';" &":8 +#%(&.-#")0 &*-*?=)2 Oxy . G- .2"#%(F.&-8 #2(?8 3?,)"(#- !"#$"%&'()* $"%+,-.$+%',$)* ,%) "/*/"0) F, #*34, !"#$"%&'()* $"%+,-.$+%'-,$)*. 7B3#=F, ,&;6&(%-+:8 6&(#./B6#(-.6:8 &'-"#- )*)-#=F*)(& .2"4%(B.B
EE/:T .
M'(x',y') (
M(x,y)
O x
y 2
1 !"#$%&' – ()$**"%$ '+',-*$,$ 38
!" #$%" &' ()*+$&'(' ,)&- ' ./01*,, )(MT , &%) ('2.3%) ),( yxM 4, ()25%63-7.&,/ 2. ),( yxM !!! .
8*, #,$+9./:2, :.;2.&$/0%< 2.&,(='2,&/-(2%< .3*,/ ' ()*+$&'('
EE":T
),(),( yxMyxM #!" ,
' %#%3, ,*&/(&%/=37./ 0+4. ('2.3% ! (&%
()22.&$/01 &%) M ! ;" #$%" &%* +>%*, xx! .
$ ?&' ()*@=./, 4, ,(=%6'4%<2. 21*% 2. &%)" :.;2.&$/0%<" 2.&,(='2,&/(2%<" #%) ,#./0%*37%)* &, ('2.3, ),( yxM (&, ),( yxM !!! &;* %#%3;* %/ ()*&.&,:2@*." 93*%*&,/ ,#1 @*, (<(&'2, &'" 2%$A-"
%&'
(()!(()!
"#y$xy
%&y'xx
-, /(%9<*,2,, ,#1 2/, .>3(;(' &'" 2%$A-"
*+
,-.
/(*+
,-.
/*+
,-.
/)*
+
,-.
/!!
"%
y
x
#$&'
y
x (1)
1#%) "',&,$,#,%, #$,:2,&/0%3 ,$/42%3. B* 0)% 0,/ 0)" , &1&. ' .>3(;(' (1) #,3$*./ &' 2%$A-
*+
,-.
/*+
,-.
/)*
+
,-.
/!!
y
x
#$&'
y
x (2)
?&'* #.$3#&;(' ,)&- % :.;2.&$/01" 2.&,(='2,&/(21" 6@:.&,/ !"#$$%&'( $)*#-
+,-$#*%+$'( 0,/ % #3*,0," *+
,-.
/#$&'
6@:.&,/ ./0#&#( &%) :$,22/0%< 2.&,(='2,-
&/(2%<.
C/, #,$+9./:2,, % :.;2.&$/01" 2.&,(='2,&/(21" #%) %$37.&,/ ,#1 &% (<(&'2,
%&'
()!()!
yxy
yxx
22
37 -, /(%9<*,2,, ,#1 &'* .>3(;(' *
+
,-.
/*+
,-.
/)*
+
,-.
/!!
y
x
y
x
22
37 .3*,/ @*," :$,2-
2/01" 2.&,(='2,&/(21" 2. #3*,0, &%* *+
,-.
/22
37. D. ,)&1* &%* 2.&,(='2,&/(21 &%
('2.3% )2,1(A ,#./0%*37.&,/ (&% )6,13(A! , .*E &% ('2.3% )2,1( #B (&% )2,1( #!B ,
9'6,9- (&%* .,)&1 &%).
$ B" 4.;$-(%)2. &E$, &% :$,22/01 2.&,(='2,&/(21
M(x,y)
M((x,#y) C(
C
O x
y 3
1 !"#$%&' – ()$**"%$ '+',-*$,$ 39
!"
#$%
&!"
#$%
&'!
"
#$%
&((
y
x
!"#$
y
xT :
!"# $" %&'"(#")" (#"'*+%"$" )0,1('i)
!"#
)1,0('j)
. ,-$., / .#!-'" A( $&0 123"$&4 )0,1(A
$&0 (#"'*+%"$&4 i)
25.# +0'$.$"6%2'.4 ),( "$ ,
"7&*
!"
#$%
&'!
"
#$%
&!"
#$%
&'!
"
#$%
&((
"$
!"#$
y
x
0
1,
.'8 / .#!-'" B( $&0 123"$&4 )1,0(B $&0 (#"'*+%"$&4 j)
25.# +0'$.$"6%2-'.4 ),( !# , "7&*
!"
#$%
&'!
"
#$%
&!"
#$%
&'!
"
#$%
&((
!#
!"#$
y
x
1
0.
9"3"$/3&*%., (/:"(;, -$#:
O! "#$%&%'()*$&+ %,+ &!-.$'+ %/# 0*1'%/+, 0)(1,A , %/# 2!'$3")'%/+ (1,0)'i)
&4$'! , 015%, "%67,, &$5 /! "#$%&%'()*$&+ %,+ &!-.$'+ %/# 0*1'%/+, 1)0,(. ,
%/# 2!'$3")'%/+ j)
&4$'! , 2&3%&1, "%67, %/# 04$'-' %/# (1'))!-/3 )&%'-
"8,)'%!")/3.
<#" 1"3=(.#6%", & 63"%%#!-4 %.$"+5/%"$#+%-4, 1&0 "1.#!&')>.# $" 123"$"
)0,1(A !"# )1,0(B $?' (#"'0+%=$?' )0,1('i)
!"# )1,0('j)
+$" +/%.)" )1,3(A( !"#
)2,1(B( "'$#+$&)5?4, 25.# 1)'"!" !"
#$%
&21
13.
&/$)*0(&'
1. 94$&%'! / (1'))!-.+ )&%'"8,)'%!").+
!"
#$%
&!"
#$%
& *'!
"
#$%
&((
y
x
y
xT
01
10:
i) :' ;1&</3$ /! &!-.$&+ ),( 11 yxA ((( -'! ),( 22 yxB ((( %=$ ",)&4=$ ),( 11 yxA -'! ),( 22 yxB '$%!"%/48=+.
ii) :' '0/2&!8%&4 .%! )()( ABBA '(( .
)j
)i
%@($,") &(0,1)
%(1,0)
&'(#,!)
O x
y 4
1 !"#$%&' – ()$**"%$ '+',-*$,$ 40
.+'-
i) !"#$%&
!"
#$%
&'(!
"
#$%
&!"
#$%
& '(!
"
#$%
&))
x
y
y
x
y
x
01
10.
'(#%)*+,, #- &-./*&, 0+* ),( 11 yxA .1- ),( 22 yxB &2*1- 01 34%&21 ),( 11 xyA ') .1- ),( 22 xyB ') 1*0-30#2"+,.
ii) '2*1-
)()()()()()( 2
12
2
12
2
12
2
12 AByyxxxxyyBA ('*'('**'()) .
5161046#7%& /0- # %&013"4%10-3%/, 1$0/, 8-1046&2 0-, 1(#3093&-,. :- ;61%%--.#2 %&013"4%10-3%#2 (#$ 8-1046#7* 0-, 1(#3093&-, <);#*01- !"#$%&'(%).
2. *(+%&,! # -',$$!./) $%&,"01$,&!"$/):
!"
#$%
&!"
#$%
&(!
"
#$%
&))
y
x
y
xT
11
34:
N, 2'%3%(:
i) 4# 5'/&65#  "1$%(#6 0)(2,A) , 718,79 &# "1$%(# ),( yx$ 5#6 ,5%!.#+(-:%&,! "&# 0)(2,A) .
ii) ; %!./+, &1) %63%(,) 12: *'( xy/ .
.+'-
i) =3"7&-
!"
#$%
&!"
#$%
&(!
"
#$%
&y
x
11
34
0
2.
'(&-8> # (2*1.1, !"
#$%
&11
34 &2*1- 1*0-306)?-%#,, (#<<1(<13-9@#$%& .1- 01 87#
%)<4 %& 0#* 1*02306#A/ 0#$, (#$ &2*1- # (2*1.1, !"
#$%
&'
'41
31 .1- )"#$%& 8-18#"--
.9:
!"
#$%
&(!
"
#$%
&+!
"
#$%
&'
y
x
0
2
11
341
!"
#$%
&(!
"
#$%
&+!"
#$%
&'
'y
x
0
2
41
31
!"
#$%
&(!
"
#$%
&' y
x
2
2.
1 !"#$%&' – ()$**"%$ '+',-*$,$ 41
!"# $% &'()*% ! +,)- &./$)$#0(+/)1 )2,2( ! .
ii) 2"3)* /# 4"%5() $'/ )6*&7&' ' %8%*# )8#9':)5)$#- #8; $-1 &./$)$#0(+/)1 $7/ )-3;/7/ $7/ &'()*7/ $'1 ).:)*#1 " 3#- (;/% #8’ #.$+1. <"=0(#$-, +,%.():
"#
$%&
'("
#
$%&
'))
*"#
$%&
'+"
#
$%&
'"#
$%&
'("
#
$%&
')) !
y
x
y
x
y
x
y
x1
11
34
11
34
"#
$%&
'("
#
$%&
'))
*"#
$%&
'!
!+
y
x
y
x
41
31
"#
$%&
'("
#
$%&
'),)!
)!)+
y
x
yx
yx
4
3
-./
),)!(
)!)(+
yxy
yxx
4
3. (1)
E8%(+/71, #/ $% &'()*% ),( yxM #/>3)- &$'/ ", $;$) :# -&,5)-:
12 ,!( xy
1624 ,),)!(),)! yxyx
12 ,)!()! xy
2
1
2
1!)() xy
!"#, $% &'()*% ),( yxM ))) #/>3)- &$'/ ).:)*# 2
1
2
1: !() xy" .
299= 3#- # $ % & ' % ( ) *+ ,, #/ $% &'()*% ),( yxM ))) #/>3)- &$'/ ).:)*#
2
1
2: !
0() xy" , $;$) $% ),( yxM #/>3)- &$'/ ).:)*# 12: ,!( xy" .
?./)8@1, ' )-3;/# $'1 ).:)*#1 12: ,!( xy" )*/#- ' ).:)*# 2
1
2
1: !() xy" .
!"#$%#
28%A)-3/5)$#- ;$- 3=:) 0"#((-3;1 ()$#&,'(#$-&(;1, $%. %8%*%. % 8*/#3#1 &-
'()*(+,-.(&), #8)-3%/*B)-:
— ).:)*)1 &) ).:)*)1
— ).:50"#((# $(>(#$# &) ).:50"#((# $(>(#$# () =3"# $-1 )-3;/)1 $7/ =3"7/
— 8%9507/# &) 8%9507/# () 3%".C+1 $-1 )-3;/)1 $7/ 3%".C@/.
D-# 8#"=A)-0(#, () $% ()$#&,'(#$-&(;
1
1
O x
"E
" y 5
1 !"#$%&' – ()$**"%$ '+',-*$,$ 42
!"
#$%
&!"
#$%
&'
(!"
#$%
&))
y
x
y
xT
01
12:
!" !#$%&'" !"# () *"#+,-. )0,1(A , )3,1('B */0 )3,0(# /1)0*"'$2)!/0 3!" !#$-%&'" #BA ))) 1"+ -4)0 &. *"#+,-. !0. )0*5'). )1,2( ')A , )1,1(B) */0 )0,3(# ) !&' *"#+,6' !"+ !#0%6'"+ !"#.
7$'/0 8"90*5, 1"99-. ,"#-., -'/ 1"9:%&'" $AAA ...21 '/ !" 1/#03!;'"+() () !"' 1$'/*/
!"
#$%
&
$
$
yyy
xxx
**
21
21,
1"+ -4)0 &. 3!<9). !0. 3+'!)!/%(-'). !&' *"#+,6' !"+. ="' 1$'/*/ /+!5' >/
!"' 9-() !"#$%$ &'( !')(*+#'(. ?!30, " 1$'/*/. !"+ !"# )$'/0 " !"
#$%
& '330
011,
)'6 !"+ #BA ))) " !"
#$%
&' 011
312. E$'/0 ,/')#5 5!0
A) B) # ) ! " #
!"
#$%
& '!"
#$%
&'
(!"
#$%
&' 310
011
01
12
011
312 .
@#/ " 1$'/*/. !"+ !#0%6'"+ #BA ))) 1#"*:1!)0 /' 1"99/19/30;3"+() !"' 1$'/*/ !"+ %#/((0*": ()-!/34A(/!03(": () !"' 1$'/*/ !"+ !#0%6'"+ !"#.
B+!5 034:)0 */0 %0/ "1"0"C<1"!) 1"9:%&'".
./01234 567869:1234 869/0;<8/910834 1. ,(--.&/"$ 01 !/'1 &2# $/34 &0# $56#0#
D/9":() 3+(()!#$/ &. 1#". !A' /#4< !&' /E5'&' !" %)&()!#0*5 )*)$'" ()!/34A(/!0-3(5 () !"' "1"$" *;>) 3A()$" ),( yxM !"+
*/#!)30/'": )101-C"+ /1)0*"'$2)!/0 3!"
3+(()!#0*5 !"+ ),( yxM ))) &. 1#". !A' /#4<
!&' /E5'&'. F1&. %'&#$2"+() /15 !A' BG H+*)$"+ 034:)0
.10
01
10
01!"
#$%
&!"
#$%
&'
'(!
"
#$%
&))
+,-.
'()/'()
+,-.
'()'()
y
x
y
x
yxy
yxx
yy
xx
@#/, A 3+(()!#$/ &. 1#". !A' /#4< !&' /E5'&' )$'/0 %#/((0*5. ()!/34A(/!0-
3(5. () 1$'/*/ I'(!"
#$%
&'
'10
01.
O x
y 6
"
!%
#%
"G
!
#
O x
y 7
C%
C &(x,y)
&%(-x,-y)
1 !"#$%&' – ()$**"%$ '+',-*$,$ 43
2. !"##$%&'( )* +&,* -.,/( #0( $"1$'( ..
!"#$%&' ()&&'*+," -. /+$. 01$2" &3" ')4'," !, *$ 5'-&'*+367 '6',2$ &'*"(89&"*3(&7 &' *$2 $/$,$ 604' (9&',$ ),( yxM *$) 6"+*'(3"2$%
'/3/:;$) "/'36$2,<'*"3 (*$ ()&&'*+367 *$),
),( yxM !!! , -. /+$. *92 ')4'," !. =*92 /"+05+">$ ")*? 4" "(8$#94$%&' &' *9
()&&'*+," -. /+$. *$2 01$2" xx! , *9 ()&&'*+,"
-. /+$. *$2 01$2" yy ! 6"3 *9 ()&&'*+," -. /+$. *92 ')4'," xy " .
2(. !"##$%&'( )* +&,* %,/ -.,/( xx ! .
@A/-. 52-+,<$)&' "/7 *92 B@ C)6',$) 3(8%'3:
#$
%&'
(#$
%&'
()
"#$
%&'
(!!
*+,-
)"!."!
*+,-
)"!"!
y
x
y
x
yxy
yxx
yy
xx
10
01
10
01.
D+", 9 ()&&'*+," -. /+$. *$2 01$2" xx! ',2"3 5+"&&367. &'*"(89&"*3(&7. &'
/,2"6" #$
%&'
()10
01.
22. !"##$%&'( )* +&,* %,/ -.,/( yy! .
E/-. 52-+,<$)&' "/7 *92 B@ C)6',$) 3(8%'3:
#$
%&'
(#$
%&'
()"#
$
%&'
(!!
*+,-
."!
.)"!*
+,-
"!)"!
y
x
y
x
yxy
yxx
yy
xx
10
01
10
01.
D+" 9 ()&&'*+," -. /+$. *$2 01$2" yy! ',2"3 :2". 5+"&&367. &'*"(89&"*3(&7. &' /,2"6"
#$
%&'
()10
01.
23. !"##$%&'( )* +&,* -.,/( %4/ $"1$'( xy " .
E/-. 52-+,<$)&' "/7 *92 B@ C)6',$) 3(8%'3:
#$
%&'
(#$
%&'
("#
$
%&'
(!!
*+,-
."!
."!*
+,-
"!"!
y
x
y
x
yxy
yxx
xy
yx
01
10
01
10.
D+", 9 ()&&'*+," -. /+$. 01$2" *92 ')4',"
xy " ',2"3 :2". 5+"&&367. &'*"(89&"*3(&7. &'
/,2"6" #$
%&'
(01
10.
O3 /"+"/02- 5+"&&36$, &'*"(89&"*3(&$, ',2"3 7#$3 05,#$%&'$*.
O
y
x
!
"#(x@,y@)
"(x,y)
8
O x
y 9
C#
C
"#(x,)y)
"(x,y)
O
y
x
10
C@ C
"#()x, y) "(x,y)
O
y
x
11
C@
!
C
"#(y,x)
"(x,y)
1 !"#$%&' – ()$**"%$ '+',-*$,$ 44
3. !"#$%& '( )*+"#$ , )-. /0+1- 2.
!"#$ Oxy %&' "("#)*' "+&#,#'-*%&$& "#.
,/0/,1. 2'3 ! *3' 4,#325 5 '6&)#325 -$&0'. 7'-
8.(*, "#6.95 *, 2%. " 2'3 -$&0' ! #. -,$-
*,#632: ,2,0&. *,#'";)*'#3"*: *, #.& ./.0.
2<4, ")*,0. ),( yxM #.+ ,/3/%1.+ '"#.3;0-=,#'3 "#. /%6'> ),( yxM !!! , #.+ 13'&("*'#.>
MO ! /.+ ,0&'3 ) #,8325 4%") #.+ OM , '& '+#:
"#6'9,0 -(6$ '/: #. " 2'#< -$&0' !.
?& # ,0&'3 ) -$&0' /.+ ";)*'#0=,3 #. 13<&+"*'
OM *, #.& <@.&' xx! 2'3 $ #. *%#6. #.+ 13'&("*'#.> OM , #:#, 4' 3";(,3:
"#$
%%
#$y
#$x
)*"+&
(1) 2'3 "#$
&%!&%!
))*(
)"+&(!#$y
!#$x (2).
!#"3, 4' 3";(,3
"#$
&%!'%!
("#$
&%!'%!
!#$!#$y
!#$!#$x
!#!#$y
!#!#$x
)*)"+&("+&))*(
)*))*("+&)"+&(
))*"+&"+&)*(
))*)*"+&"+&(
"#$
&%!'%!
(!y!xy
!y!xx
"+&)*)*"+&)1(
)*
+,-
.)*
+,-
. '%)
*
+,-
.!!
(y
x
!!!!
y
x
"+&)*)*"+&
.
A6', ) "#6.95 *, 2%. " 2'3 -$&0' ! ,0&'3 -6'**32:> *,#'";)*'#3"*:> *,
/0&'2' )*
+,-
. '....
34+5'5'34+
. B3132:#,6':
') C "#6.95 *, 2%. " 2'3 -$&0' 2
%! % %;,3 /0&'2' )*
+,-
. '01
10.
D) C "#6.95 *, 2%. " 2'3 -$&0' %! % %;,3 /0&'2' I'%)*
+,-
.'
'10
01
2'3 ,0&'3 ) "+**,#60' $> /6.> #)& '6;5 #$& '@:&$&.
-) C "#6.95 *, 2%. " 2'3 -$&0' 2
3%! % %;,3 /0&'2' )*
+,-
.' 01
10.
1) E%8.>, ) "#6.95 *, 2%. " 2'3 -$&0' %! 2% %;,3 /0&'2' /%)*
+,-
.10
01
2'3 ,0&'3 ) #'+#.#325 '/,32:&3").
B(2.8' '/.1,32&(,#'3 :#3 2'3 ) "#6.95 ,0&'3 *3' 3".*,#60'.
O x
12
&'(xF,yF) $
# !
y
&(x,y)
1 !"#$%&' – ()$**"%$ '+',-*$,$ 45
4. !"#$#%&'().
!"#$%&' #"#$#%&'() &' ()*+,$ +-* ",./
+0* "12*0* ! ("3 #24$ *!!" +$ &'+"5.--
&"+35&2 &' +$* $6$7$ (89' 5-&'7$ ),( yxM
+$: '636);$: "*+35+$3.7<'+"3 5+$ 5-&'7$
),( yxM """ 6$: $,7<'+"3 "62 +-* 352+-+"
OM"MO #" .
=6'3;/ ),( yxOM # ("3 ),( yxMO ""#" , ).$:-
&': ),(),( yx"yxOM"MO #""$#"
%&'
#"#"
$"yy
"xx
%&'
(#"(#"
$"yxy
y"xx
0
0
)*
+,-
.)*
+,-
.#)
*
+,-
.""
$y
x
""
y
x
0
0.
>,",- $&$3$9'57" &' ()*+,$ +-* ",./ +0* "12*0* ("3 #24$ 0/" '7*"3 )*"?
4,"&&3(2? &'+"5.-&"+35&2? &' 67*"(" .I.
.#)
*
+,-
.0
0.
*+,*!+-.!/ .0+!+0/
@3" *" 9:&2&"5+' +$* 67*"(" +0* 6","68*0 4,"&&3(A* &'+"5.-&"+35&A*, ",-
('7 *" 9:&2&"5+' 2+3 - 6,A+- 5+/#- +$: '7*"3 $3 5:*+'+"4&)*'? +-? '3(2*"? +$:
5-&'7$: )0,1(A , '*A - ;'%+',- 5+/#- +$: '7*"3 $3 5:*+'+"4&)*'? +-? '3(2*"? +$: )1,0(B . @3" 6",8;'34&", $ 67*"("? +-? 5:&&'+,7"? 0? 6,$? +$* 81$*" xx"
'7*"3 $ )*
+,-
.010
01 6$: ).'3 43" 6,A+- ("3 ;'%+',- 5+/#- +3? 5:*+'+"4&)*'? +0*
5:&&'+,3(A* 0? 6,$? +$* 81$*" xx" +0* 5-&'70* )0,1(A ("3 )1,0(B "*+35+$7-.0?.
5. 1)2344545 "&6)7#23.
B5+0 ),( 21 ### #1
)*" ;38*:5&" +$: (",+'53"*$% '636);$: Oxy . !"#$%&' 6"-
,8##-#- &'+"C$,8 ("+8 ;38*:5&" #1
+$ 4'0&'+,3(2 '('7*$ &'+"5.-&"+35&2 &' +$* $6$7$ (89' 5-&'7$ ),( yxM +$: '636);$: "*+35+$3.7<'+"3 5+$ 5-&'7$
),( yxM """ 6$: $,7<'+"3 "62 +-* 352+-+" #MM1
#" (D.. 14).
O
y
x
13
$%(xE,yE)
">1
$(x,y)
1 !"#$%&' – ()$**"%$ '+',-*$,$ 46
!"#$%& ),( yyxxMM !"!"#" , '()*+#
),(),( 21 !!yyxx!MM #!"!"$#"%
&'(
#!"#!"
$2
1
!yy
!xx
&'(
)#")#"
$2
1
!yy
!xx
*+
,-.
/)*+
,-.
/#*
+
,-.
/""
$2
1
!!
y
x
y
x
*+
,-.
/)*+
,-.
/*+
,-.
/#*
+
,-.
/""
$2
1
10
01
!!
y
x
y
x.
,-., / ".-011/1/ +#2.3)-0 !"# #45.$ 6-.++$789 +#2.:(/+.2$:+89.
&.$)*/(&'
1. $%&' , ( )"&*%+,)*&-%)./ “%&0(12 )" 34#&0( / 3*- 5'#6* 4
01 !# ”. 7*
80"9"6 , "-3.#* C " &,/ 3*):;<,/x
yC1
: # '/ :0(/ &( )"&*%+,)*&-%). ,.
2+'-
;:2< ),( yxM '5. :/+#4) 2)* #"$"'%)* 7.$ ),( yxM """ / #$785. 2)* <9 "-)9 2)
+#2.:(/+.2$:+8 ". =82# >. $:(?#$
*+
,-.
/
**+
,
--.
/
!#*
+
,-.
/""
$*+
,-.
/
***
+
,
---
.
/
#*+
,-.
/""
y
x
y
x
y
x
##
##
y
x
2/22/2
2/22/2
4:*5
4/+-
4/+
4:*5
*+
,-.
/#*
+
,-.
/""
**+
,
--.
/
!$
!
y
x
y
x1
2/22/2
2/22/2*+
,-.
/#*
+
,-.
/""
**+
,
--.
/ !$y
x
y
x
2/22/2
2/22/2
00&
00'
(
")"#
"!"#$
)(2
2
)(2
2
yxy
yxx. (1)
%!
%!
O
y
x
14 $%(x@,y@)
$(x,y)
1 !"#$%&' – ()$**"%$ '+',-*$,$ 47
!"#$%&'(, )& *# ),( yxM )&+,-. /*0& ,)$"120 C, *3*- 4) ./51-.:
1!xy
" # 1)()(2
1 22 !$%$ yx
1)2(
)(
)2(
)(2
2
2
2
!$
%$ yx
,
#"3*- *# /0$-6# ),( yxM $$$ 4) -6&). /0$-6#
*0( ./#/,-2#1( 7"-89#2+(
1)2()2(
:2
2
2
2
!%$ yxC .
:22; ,). ! " # $ % # & ' () * , )& *# ),( yxM $$$ )&+,-. /*0& ,)$"120 C $ , *3*- *#
),( yxM 4) )&+,-. /*0& C.
<8), 0 -.,3&) *0( x
yC1
: ! , '( "8#( *0 /*8#=+ $- ,%&*8# + ,). >'&6) 4
,- %! ,
-6&). 0 7"-89#2+ 1)2()2(
:2
2
2
2
!%$ yxC . ?7&-"@(, 0 ,)$"120
xyC
1: ! -6&). $.)
7"-89#2+ "#7 "8#,1"*-. )& /*8%A#7$- *0& C $ ,)*; >'&6) 4
,- !% .
2. !"#$ 1T %&' 2T (' )*&++'%(, +-#&"./+&#'"+(, +- 0,1&%-2
&'
()*
+%
!11
211A %&' &
'
()*
+!
11
232A
&1#'"#(,.$2. 3& 4*-5-, ( 0,1&%&2 #(6 )*&++'%(7 +-#&"./+&#'"+(7 0(6 0*(-
%70#-', &1 -8&*+9"(6+- 0*:#& #(1 1T %&' ;0-'#& #(1 2T , </=&<> #(6 +-#&-
"./+&#'"+(7 12 TT , .
.+'-
B/*' ),( yxM %&) /0$-6# *#7 -"."%C#7. :& ),( yxM $$$ -6&). 0 -.,3&) *#7 . $%-/' *#7 $-*)/50$)*./$#1 1T ,). ),( yxM $$$$$$ 0 -.,3&) *#7 M $ $%/' *#7 $-*)-
/50$)*./$#1 2T , *3*- 4) ./51-.
&'
()*
+&'
()*
+%
!&'
()*
+$$
y
x
y
x
11
21 ,). &
'
()*
+$$
&'
()*
+!&
'
()*
+$$$$
y
x
y
x
11
23,
#"3*- 4) %5#7$-
C/
y=%x y=x
C
O x
y 15
1 !"#$%&' – ()$**"%$ '+',-*$,$ 48
!"
#$%
&!"
#$%
&'!
"
#$%
&(!"
#$%
&)
(!"
#$%
&'!
"
#$%
&****
y
x
y
x
y
x
30
81
11
21
11
23.
!"#, $ %&'#(#) *$+ 12 TT + ,&'#- $ !"
#$%
&30
81 %$+ ,&'#- *$ .-'/0,'$ 12 AA ( *1' %--
'2(1' *1' 0,*#3450#*-306' 2T (#- 1T . 7,'-(2:
“!" 21 , AA #$"%& '$"%(#) *+, -.%//&(0" /#1%234/%1&2/0", 1T (%& 2T %"1&-21,$35), 161# , '$"%(%) 1,7 -.%//&(,+ /#1%234/%1&2/,+ 12 TT + #$"%& ,
12 AA ( , #"0 1,7 21 TT + #$"%& , 21 AA ( ”.
$'%-'&"'
!8 9:!;!<
1. 8# ."29,*, *$+) %&'#(,) *1' ."#00-(6' 0,*#3450#*-306':
,-.
/)'*/'*
yxy
yxxT :1 ,
,-.
/'*/'*
yxy
yxxT
2:2 ,
,-.
/'*/'*
yxy
yxxT
42
2:3
(#- '# :",&*, *-) ,-(/',) *1' 350,&1' )0,1(A (#- )1,0(B .
2. 8# :",&*, *$ ."#00-(/ 0,*#3450#*-30/ %$+ #%,-($'&;,- *# %<"#*#
)0,1(A (#- )1,0(B *1' 0$'#=-#&1' =-#'+302*1' i0
(#- j0
3*# 350,&#
(i) )2,1( (#- )3,1() #'*-3*$&41), (ii) )1,1() (#- )1,2( #'*-3*$&41).
3. >&',*#- $ ."#00-(/) 0,*#3450#*-30/):
!"
#$%
&!"
#$%
&)
'!"
#$%
&**
y
x
y
xT
01
12: .
8# :",&*, *-) ,-(/',) *1' 350,&1' )0,0(O (#- )4,3(A (#- 3*5 3+'<4,-#
'# #%$=,&?,*, /*- $ ! =,' ,&'#- -3$0,*"&#.
4. >&',*#- $ ."#00-(/) 0,*#3450#*-30/):
!"
#$%
&!"
#$%
&'!
"
#$%
&**
y
x
y
xT
11
12:
i) 8# :",&*, *5' ,-(/'# "#BA **** *$+ *,*"#.6'$+ $%#" %$+ <4,- %&'#-
(# !"
#$%
&1100
0110.
ii) 8# #%$=,&?,*, /*- *$ "#%$ **** ,&'#- %@2.-$ %#"#@@5@/."#00$.
1 !"#$%&' – ()$**"%$ '+',-*$,$ 49
5. !"#$%&' ( )$%&*+,)&%'*)-.:
!"
#$%
&!"
#$%
&'!
"
#$%
&((
y
x
y
xT
23
11: .
N& /0$"%$
i) %( 10-%21( %(2 *,)$"(2 )5,0(A(
ii) %,# $'3-#& %,. $24$"&. 1: )' xy! .
6. 5' 1&0'*%6#(2# 7$8)$%0'36 (' 70&))'3(" )$%&*+,)&%'*)(" 1(2 9+(2# 1"#&3&:
i) !!"
#
$$%
&
*'
2/32/1
2/12/3A , ii) !
"
#$%
&'
20
02A
iii) !"
#$%
&+!"
#$%
&*
*'
01
10
10
01A , iv) !
"
#$%
&+!"
#$%
&*
*'
20
02
10
01A
v) !"
#$%
&+!"
#$%
&*
'01
10
10
01A , vi) !
"
#$%
&*
*+!"
#$%
&*
+!"
#$%
&'
10
01
10
01
01
10A .
B! "#$%$&
1. !"#$%&' ( 70&))'3-. )$%&*+,)&%'*)-.:
!"
#$%
&!"
#$%
&'!
"
#$%
&((
y
x
y
xT
42
21:
i) :& &1(;$"<$%$ -%' ( " &1$'3(#"=$' ->& %& *,)$"& %(2 $1'19;(2
*%,# $24$"& xy! 2: ' .
ii) N& /0$"%$ %& 10-%21& %(2 *,)$"(2 )0,0(O .
iii) :& &1(;$"<$%$ -%' %( *,)$"( )1,1(A( ;$# 9+$' 10-%21(.
2. !"#$%&' ( 70&))'3-. )$%&*+,)&%'*)-.:
!"
#$%
&!"
#$%
& *'!
"
#$%
&((
y
x
y
xT
01
11:
3&' ;?( (1('&;@1(%$ *,)$"& ),( 11 yxA , ),( 22 yxB %(2 $1'19;(2. :& &1(-
;$"<$%$ -%'
i) O T ;$# $"#&' '*()$%0"&, ;,>&;@ -%' )()( ABBA ,(( .
ii) A " &1$'3(#"=$' %( )9*( %(2 $24. %)@)&%(. #$ *%( )9*( %,. $'3--
#&. %(2 BA (( .
iii) 5( $)/&;-# %(2 %0'7B#(2 %#$ $"#&' "*( )$ %( $)/&;- %,. $'3-#&. %(2 BAO ((( .
1 !"#$%&' – ()$**"%$ '+',-*$,$ 50
3. !" #$%&'% '( )$"**+,- *%'"./0*"'+.*- 1(2 "1%+,(3&4%+ '" .0*%&"
)1,1(A ,"+ )1,1( !B .'" .0*%&":
i) )1,0(A" ,"+ )1,2(B" "3'+.'(&/56
ii) )3,6(A" ,"+ )1,3(B" "3'+.'(&/56.
7% ,"8%*+9 1%$&1'5.0 3" #$%&'% '03 %+,-3" '06 %28%&"6 xy! 2: !# .
4. !" "1(:%&;%'% -'+ ,"8<3"6 "1- '(26 1"$",9'5 )%5*%'$+,(=6 *%'"-
./0*"'+.*(=6 %&3"+ )$"**+,-6 *%'"./0*"'+.*-6.
i) > .2**%'$&" 56 1$(6 '03 %28%&" xy !# .
ii) > 1$(#(?@ 1935 .'(3 9;(3" xx" .
iii) > 1$(#(?@ 1935 .'(3 9;(3" yy" .
iv) > 1$(#(?@ 1935 .'03 %28%&" xy # .
7'0 .23</%+" 3" #$%&'% .% ,"8%*&" 1%$&1'5.0 '03 %+,-3" '(2 '%'$")A-
3(2 "#$% *% 1&3"," $%
&'(
)1100
0110 ,"+ 3" %1+#%#"+A.%'% )%5*%'$+,9
'03 "193'0.@ ."6.
5. B&3%'"+ ( *%'"./0*"'+.*-6 & *% 1&3"," $%
&'(
)#
'(
A0
0, -1(2 0** '( .
i) !" "1(:%&;%'% -'+ 0 %+,-3" '(2 ,=,?(2 1: 22 #+ yxC %&3"+ 0 <?-
?%+C0 1:2
2
2
2
#+"'y
(x
C
ii) DE(= #$%&'% '03 %+,-3" B$AO """" '(2 '%'$")A3(2 "#$%, 1(2 </%+
1&3"," $%
&'(
)1100
0110, 3" :%&;%'% -'+ )()( %OA$('B$AO ,#"""" .
1.5 - &##."$ ,.+ ()$**"%.+ '+',-*$,.'
- F98% %;&.5.0 '06 *($E@6 'x(x(x( )) #+++ .2211 , -1(2 )((( ...,,, 21 ' %&3"+ 1$")*"'+,(& "$+8*(& ,"+ )xxx ...,,, 21 9)35.'(+, ?<)%'"+ !"#$$%&' ()*+,+- $( / #!./+0123. G+ "$+8*(& )((( ...,, 21 ?<)(3'"+ +2.0(4(+053 0,. #!./+0,. ,"+ ( 0
+0#6("73 7"13.
1 !"#$%&' – ()$**"%$ '+',-*$,$ 51
!"# $#%&'(")*#, + (,-./.+ 33 !"! yx (-0#" *"# )%#**"12 (,-./.+ *( '34 #-
)05.6478. 9$-.+8, + 152 4321 "!#! xxxx (-0#" )%#**"12 (,-./.+ *( 6:..(%"8
#)05.6478, (05 4" (,".5.("8 12 2 "! yx 1#" 232 "! yx '(0 (-0#" )%#**"1:8. ;&<( '"#6(6#)*:0+ !-&'# #%"<*50 )...,,,( 21 !sss $47 ($#=+<(3(" *"# )%#**"12
(,-./.+ *( ! #)05.6478 =:)(6#" !"#$ %$& '()#*#$&. !"# $#%&'(")*#, + '"#6(6#)*:0+ 6%"&'# )2,0,1(! (-0#" =3.+ 6+8 (,-./.+8
123 !"#! "yx , #>43 123202)1(3 !"#!"#$!! .
% ?0# $=2<48 # )%#**"150 (,".5.(/0 *( ! #)05.6478 6/0 4$4-/0 @+6&*( 6"8 14"0:8 =3.("8, =:)(6#" +,-../01 #"#%$.- . '(/#2#'*3 .' / -+32#%45& 2 #-
$=43.6(%# /.& +,-../01 #"#%$.-.
!"# $#%&'(")*#, 64
'()
!"!#"#!
23
132
"yx
"yx )A( 1
(-0#" :0# 32& )%#**"1B .3.6+*#.
!(0"1&, :0# !#& )%#**"1B .3.6+*# :C(" 6+ *4%>2
**'
**(
)
"###
"###"###
#!#!##
!!
!!
$x%x%x%
$x%x%x%$x%x%x%
+
++
2211
22222121
11212111
....................................................... )A( 2
D 148 '(-16+8 i 647 .706(=(.62 ij% '(-C0(" 6+0 (,-./.+ .6+0 4$4-# #021(" 4
ij% , (05 4 248 '(-16+8 j '(-C0(" 640 &)0/.64 *( 640 4$4-4 $4==#$=#."&@(6#" 4
ij% . !"# $#%&'(")*#, 4 32% (-0#" 4 .706(=(.628 647 #)05.647 2x .6+0 6%-6+ (,--./.+ (0B8 .7.62*#648. ;&<( '"#6(6#)*:0+ !-&'# #%"<*50 + 4$4"# ($#=+<(3(" B=(8 6"8 (,".5.("8 (0B8 !#& )%#**"143 .7.62*#648, =:)(6#" !"#$ 647 .7.62*#648. !"# $#%&'(")*#, +
'"#6(6#)*:0+ 6%"&'# )1,0,1(! (-0#" =3.+ 647 .7.62*#648 )A( 1 , #>43
1130)1(2 "$#!! 1#" 21031 !"!$#! .
E '"#'"1#.-# *( 6+0 4$4-# F%-.147*( 6"8 =3.("8 (0B8 .7.62*#648 =:)(6#" '6)-!5#$ 647 .7.62*#648. ?0# .3.6+*# $47 :C(" *"# 647=&C".640 =3.+ =:)(6#" #5.7/7-#%1, (05 :0# .3-
.6+*# $47 '(0 :C(" 1#*-# =3.+ =:)(6#" -8"3-%4.
G:=48, '34 )%#**"1& .7.62*#6# $47 :C470 6"8 -'"(8 #1%"F58 =3.("8 =:)406#" /#48"3-.-.
% H( 6+ F42<("# 6/0 $"0&1/0 64 .3.6+*# )A( 1 )%&>(6#"
,-
./0
1!
",-
./0
1!##!
2
1
3
32
"yx
"yx.
1 !"#$%&' – ()$**"%$ '+',-*$,$ 52
!" 1" #$%"& '#(& )*& +,')*)-& -.)/& 012-+ )" 3+2'#02" )". 32! 412-5-
"#
$%&
'(
(131
312 )(2 ,.2)0%0,)62 )(2 -326,)(2 #0 )"2 13! 412-5- ,)/%*
"""
#
$
%%%
&
'
!y
x
)(2 -326,)(2. 74"#$2(& )" ,8,)*#- )9( 1 3:;<0)-+
"#
$%&
'(
)"""
#
$
%%%
&
'
"#
$%&
'(
(2
1
131
312
!y
x
.
=02+5')0:-, )" "#! 3:-##+5' ,8,)*#- )9( 2 3:;<0)-+
"""""
#
$
%%%%%
&
'
)
""""
#
$
%%%%
&
'
"""""
#
$
%%%%%
&
'
#"#"##
"
"
$
$$
x
x
x
%%%
%%%%%%
**+
***++
2
1
2
1
21
22221
11211
.
>2 )6:- ,.#?"%1,".#0 #0 & )"2 412-5- )(2 ,.2)0%0,)62 )(2 -326,)(2, #0 '
)"2 412-5--,)/%* )(2 -326,)(2 5-+ #0 ( )"2 412-5--,)/%* )(2 ,)-@0:62 '-
:(2, )')0 )" )9( 2 3:;<0)-+ BAX ) .
>2 "+ ,)-@0:"1 ':"+ 02'& 3:-##+5"8 ,.,)/#-)"& 012-+ '%"+ 1,"+ #0 )" #*A$2, )')0 )" ,8,)*#- %$30)-+ !"!#$%&' 5-+ ,82)"#- 3:;<0)-+ !!!!)AX .
!$%"& " 412-5-&
"""""
#
$
%%%%%
&
'
##"##
"
"
$%%%
$%%%$%%%
+****
++
21
222221
111211
4". -4")0%01)-+ -4' )"2 412-5- & )(2 ,.2)0%0,)62 )(2 -326,)(2, ,.#4%*:(-
#$2" #0 )* ,)/%* )(2 ,)-@0:62 ':(2 %$30)-+ $()*+,"&%!' (-%).)' )". ,.,)/-
#-)"&. B4(& @- A"8#0 ,)* ,.2$C0+-, " 412-5-& -.)'& 4-1D0+ ,*#-2)+5' :'%"
,)*2 041%.,* )". ,.,)/#-)"&. E 5-)-5':.<* A+-505"##$2* 3:-##/ ,)"2 04-.-
F*#$2" 412-5- 4:",)1@0)-+ -4%6& 3+- 2- F0C(:1D0+ )* ,)/%* )(2 ,)-@0:62 '-
:(2.
1.6 &!".+'- ()$**"%/+ '+',-*$,/' *& ,-
*&0/1/ $!$./"2-' ,/+ GAUSS
A4"A0+5280)-+ ')+, -2 ,0 $2- 3:-##+5' ,8,)*#- 0<-:#',".#0 #+- -4' )+& 04'-
#020& A+-A+5-,10&, )')0 4:"584)0+ +,"A82-#" ,8,)*#-:
1 !"#$%&' – ()$**"%$ '+',-*$,$ 53
— !"#$$#%& '() *+,() -./ 012,3,04"
— 5/$$#6$#,2#,78) '4" 70$3" 72#) 019,4,() 70 +"# 7( 7(-0"2:8 #;2*78.
— 5;8,*0,( '4" 70$3" 72#) 019,4,() (6/$$#6$#,2#,7+"4" 70 +"#" #;2*78)
,'# 7+$( 72#) <$$(). =',2, 8'#" +>/?70 "# $.,/?70 +"# %;#772:8 ,.,'(7# 6;/,6#*/.70, 0@#;78A/-
"'#) '2) 6;/(%/.70"0) -2#-2:#,90), "# '/ 70'#,>(7#'9,/?70 ,0 +"# <$$/ 2,/-.-
"#7/ ,.,'(7# '/? /6/9/? ( $.,( "# 09"#2 6;/@#"&). B) -/.70 '3;# 70 +"# 6#;<-02%7# 64) 0@#;78A/"'#2 :#2 64) ,?7C/$9A/"'#2 /2 ';02) #?'+) -2#-2:#,90).
=,'4 '/ %;#772:8 ,.,'(7# !"
!#
$
%&&'%&'
%&'
123
352
02
!yx
!yx
!yx
)D( 1
— 5/$$#6$#,2<A/?70 '# 7+$( '() 1() 019,4,() 1E '/? )D( 1 70 2'
:#2 '# 6;/,*+'/?70 ,'# #"'9,'/2- ># 7+$( '() 2() 019,4,() 2E '/?
)D( 1 . =',2, #6#$09@0'#2 #68 '(" 2"
/ <%"4,'/) x.
— 5/$$#6$#,2<A/?70 '# 7+$( '() 1( '/? )D( 2 70 3' :#2 '# 6;/,*+-
'/?70 ,'# 7+$( '() 3( . E!',2, #6#$09@0'#2 #68 '(" 3( / <%"4-
,'/) x.
— 5/$$#6$#,2<A/?70 '# 7+$( '()
2" '/? )D( 3 70 3
1. =',2, / ,?"'0-
,'&) '/? y %9"0'#2 1.
D?"0>9A/?70 0@#;78A/"'#) '2) 6#;#6<"4 -2#-2:#,90) 6/? 6#;2,'<"/?70 6$+/" 78"/ ,?7C/$2:<:
233 7EEE ') !"
!#
$
%''%&%&'
88
1
02
!!y
!yx
)D( 5
338
1EE ')
!"
!#
$
'%'%&%&'
1
1
02
!!y
!yx
)D( 6
)D(123
333
02
2 2122
!"
!#
$
%&&'%&
%&'')
!yx
!y
!yx
EEE
)D(17
333
02
3 3133
!"
!#
$
%''%&
%&'')
!y
!y
!yx
EEE
)D(17
1
02
3
1422
!"
!#
$
%''%&
%&')
!y
!y
!yx
EE
1 !"#$%&' – ()$**"%$ '+',-*$,$ 54
322 EEE !" #$
#%
&
!'''(!
1
0
02
!y
!yx
)!( 7
311 EEE !" #$
#%
&
!'''!
1
0
12
!y
yx
)!( 8
211 2EEE (" #$
#%
&
!'''
1
0
1
!y
x
)!( 9
"#$%&' () *+*(,-. )!( 9 $/0.% %*)&+0.-) -$ () .12%34 *+*(,-. )!( 1 , *5-#$-1./0)5-$ 4(% , 6+*, ()5 *5*('-.()7 $/0.% , (1%8&. )1,0,1( ! .
9#)1)+-$ 0. #$1%:18;)5-$ .#6)+*($1. (, &%.&%3.*/. $#/65*,7 $047 "#)
:1.--%3)+ *5*('-.()7, .0 *3$<()+-$ =7 $>'7: ?<)+ 38@$ $>/*=*, #.1%*(80$-(.% -$ -%. :1.--' ()5 $#.5>,-A0)5 #/0.3., .13$/ )% #.1.#80= -$(.(1)#A7 (=0 $>%*B*$=0 0. :/0)0(.% *(%7 :1.--A7 #$$$ ...,,, 21 ()5 $#.5>,-A0)5 #/0.3.. C% -$(.(1)#A7 .5(A7 6A:)0(.% !"#$$%&"'()*+ 3.% $/0.% )% $>'7:
,"#$$%&"'(- ./$0%1*2$3+
1. %"&''&() *+, -./+, 012 (3&##4" ji $$ *
2. 52''&6'&/7&/#8, #7&, (3&##), #9 ."&
#+ #+09"7:8 &37-#8 ii '$$ " , 0+'
3. 538/-9/+ *!" /*27;9<!" #7&, (3&##),, 62''&6'&/7&/#."!" #9 ."&" &37-#8, /*& jii '$$$ ("
&"*</*27;& /*27;9<& #7&, =''+, (3&##),.
D(.0 A2)5-$ &+) #/0.3$7 >, ? #)5 ) A0.7 #1)3+#($% .#4 ()0 866) -$ :1.--)-
#18>$%7, (4($ )% #/0.3$7 .5()/ 6A:)0(.% !"#$$%42%567#$%* ' .#6B7 *2%567#$%* 3.% :18<)5-$ B~A . "/0.% #1)<.0A7 4(%, .0 )% $#.5>,-A0)% #/0.3$7 &+) *5-
*(,-8(=0 $/0.% %*)&+0.-)%, (4($ 3.% (. *5*('-.(. $/0.% %*)&+0.-., .<)+ 3.@$--%8 :1.--)#18>, >$2=1%*(8 )&,:$/ *$ *+*(,-. %*)&+0.-) -$ () .12%34.
E(*%, , $#/65*, ()5 #1),:)+-$0)5 *5*('-.()7 -#)1$/ 0. :/0$% =7 $>'7:
,,,
-
.
///
0
1!!
!
1213
3512
0121
~122 2$$$ !"
,,,
-
.
///
0
1!
!
1213
3330
0121
~133 3$$$ !"
,,,
-
.
///
0
1
!!
!
1170
3330
0121
~22
3
1 $$ "
,,,
-
.
///
0
1
!!
!
1170
1110
0121
~233 7$$$ !"
1 !"#$%&' – ()$**"%$ '+',-*$,$ 55
!!!
"
#
$$$
%
&
''
'
8800
1110
0121
~33
8
1 !! '(
!!!
"
#
$$$
%
&
''
'
1100
1110
0121
~322 !!! '(
!!!
"
#
$$$
%
&
'
'
1100
0010
0121
~311 !!! '(
!!!
"
#
$$$
%
&
'
'
1100
0010
1021
~211 2!!! )(
!!!
"
#
$$$
%
&
'1100
0010
1001
.
! "#$#%"&'() *'+&,&) &+"-."(-/#' ."( .0."12& *+
*,
-
'...
1
0
1
"y
x
.
3*(24+5), 1 $0.1 "(% .%."62&"() #'+&- 1 "7-89& )1,0,1( ' .
:&7&"17(02# ;"- ( "#$#%"&'() *'+&,&) "5+ .%+"#$#."<+ "5+ &=+<."5+ #'+&- (
2(+&9-&'() 33/ *'+&,&). >".- 2*(7(02# +& “9-&?8.(%2#” &24.5) "1 $0.1 "(%
.%."62&"(). @-& +& &*$(*(-6.(%2# ,&- +& .%+"(2#0.(%2# &,;21 *#7-..;"#7( "1 9-&9-,&.'&
#*'$%.1) #+;) .%."62&"(), *($$4) A(74) ."( '9-( ?62& #A&72;B(%2# *#7-..;-
"#7#) &*; 2'& =7&22(*78C#-). D) $0.(%2# "<7& ,&- "( .0."12&
*+
*,
-
.))')'.'')''
.))')
636242
4232
10652
54321
54321
54321
xxxxx
xxxxx
xxxxx
.
:&'7+(%2# "(+ #*&%C124+( *'+&,& ,&- 4/(%2# 9-&9(/-,8:
!!!
"
#
$$$
%
&
''''''
'
636242
423121
1065121
~133
122
2!!!!!!
'()(
!!!
"
#
$$$
%
&
'''
'
1494000
642000
1065121
~22
2
1 !! (
!!!
"
#
$$$
%
&
'''
'
1494000
321000
1065121
~233 4!!! )(
!!!
"
#
$$$
%
&
''
''
210000
321000
205121
~33 !! '(
!!!
"
#
$$$
%
& '
210000
321000
1065121
~311
322
6
2
!!!!!!
'('(
!!!
"
#
$$$
%
&'''
210000
101000
205121
~211 5!!! '(
1 !"#$%&' – ()$**"%$ '+',-*$TA 56
!!!
"
#
$$$
%
&'
'
210000
101000
300121
.
!"#$ "% &'($)* #+#",-& ./0&$ $#%1+0&-% -. "% #+#",-&
()
(*
+
,',
,'-
2
1
32
5
4
321
x
x
xxx
.
2+0%3-. ",0 4'5", .6/#7#, 78 4'%8 90&0 :;07#"%, 4.(. 78 4'%8 1x (&3"* -&8 1$.3)%<+0.$, &=%+ % #30".<.#">8 "%3 &;05#"%3 &3"%+ ./0&$ 1) )&$ 9(%3-.
()
(*
+
,',
-',
2
1
23
5
4
321
x
x
xxx
.
?4.$1> % :;07#"%8 1x .)=':@."&$ 78 #30:'",#, "70 32 , xx , &3"* #,-&/0.$ *"$ -4%'%+-. 0& .4$<96%3-. &3A&$'9"78 "$8 "$-98 "70 32 , xx . B,<&1>, "% ;'&--$)*
#+#",-& 9(.$ :4.$'% 4<>A%8 <+#.70, "$8 1$&"."&;-90.8 4.0":1.8 )2,1,,,23( 3232 '-' xxxx , *4%3 %$ 32 , xx -4%'%+0 0& 4:'%30 %4%$.#1>4%". 4'&;-
-&"$)98 "$-98. C.(. ;$& 12 ,x , 03 ,x 9(%3-. ", <+#, )2,1,0,1,1( ' "%3 #3#">-&"%8. D ".<.3"&/%8 &4* "%38 4&'&4:07 $#%1+0&-%38 .4&36,-90%38 4/0&).8 ./0&$, *-
478 <9-., 90&8 !"#$%&"'( )*+%!),-.( /0"!)!(. E.0$):, 1/0%3-. "%0 .4*-.0%
%'$#-*:
O123453
!0&8 !". 4/0&)&8 <9;."&$ !"#$%&"'( )*+%!),-.((1) , &0 $#(+%30 #3;('*078 "& 4&'&):"7:
&) D$ -, -,1.0$)98 ;'&--98 F'/#)%0"&$ 4'$0 &4* "$8 -,1.0$)98. F) G% 4'5"% &4* &'$#".': -, -,1.0$)* #"%$(./% ):A. -, -,1.0$)>8 ;'&-->8
./0&$ "% 1 )&$ F'/#)."&$ 1.6$*".'& "%3 &0"/#"%$(%3 1 ",8 4'%,;%+-.0,8 ;'&-->8.
;) G% 4'5"% &4* &'$#".': 1 ):A. -, -,1.0$)>8 ;'&-->8 ./0&$ )&$ "% -*0% -,
-,1.0$)* #"%$(./% ",8 #"><,8 #",0 %4%/& &0>).$.
!"#$ 4.(. %$ 4/0&).8
(1) !0&8 !" . 4/0&)&8 <9;."&$, &4<58, )*+%!),-.(, &0 &) D$ -, -,1.0$)98 ;'&--98 F'/#)%0"&$ 4'$0 &4* "$8 -,1.0$)98 )&$ F) G% 4'5"% &4* "& &'$#".': -, -,1.0$)* #"%$(./% ):A. ;'&-->8 F'/#)."&$ 1.6$*".'& &4* "% &0"/#"%$(%
#"%$(./% ",8 4'%,;%+-.0,8 ;'&-->8, (7'/8 0& ./0&$ )&"&0:;), /#% -. 1.
1 !"#$%&' – ()$**"%$ '+',-*$,$ 57
!"#$% $#&'()#*% +,%($+-.*", !#/ *% 0"#$+!1
!!!
"
#
$$$
%
&'
'
2100
1110
3201
,
!!!
"
#
$$$
%
&
0100
1000
0021
2!# !"#$% $#&'()#*% +,%($+-.*".
34!.%+5 (! .*61 $#&'()#*61 +,%($+-.*71 0"#$+!1 %847!% .* 0$9$+5.- :!/9&($.
!"#$%&'
;5:! 0"#$+$1 (!.$.9)0!.$% 8! $#&'()#* +,%($+-.< 0"#$+$ (! .&# !+.),!8&
!#<1 0!0!9$8()#*6 0,=:*61 '9$((*095>!-#.
37(?-#$ (! .* :!/9&($ $6.< +5:! 0"#$+$1 !"#$% %8*27#$(*1 (! )#$# $#&'()#*
+,%($+-.< 0"#$+$. @0*9!" #$ $0*2!%4:!" <.% $6.<1 * $#&'()#*1 +,%($+-.<1 0"#$+$1 !"#$% +$% (*#$2%+<1. A 0$9$+5.- $,'<9%:(*1 ($1 2"#!% (%$ ():*2* (! .&# *0*"$ (0*9*7(! #$ B9"-8+*6(! +5:! ?*95 .* (*#$2%+< $6.<# $#&'()#* +,%($+-.< 0"#$+$.
'()*$+!&*,
-%&' 1.: C9"8+*6(! .&# 09/.& 8.=,& .*6 0"#$+$ 0*6 0!9%)4!% (& (&2!#%+<
8.*%4!"*.
-%&' 2.: @!.$?)9*6(! 8.*# 0"#$+$ 09/.& .& '9$((= 0*6 0!9%)4!% (& (&2!-#%+< 8.*%4!"* .&1 8.=,&1 ('9$((*095>& 1).
-%&' 3.: ;5#*6(! .* (& (&2!#%+< 8.*%4!"* .&1 8.=,&1 (*#52$ ('9$((*095-
>& 2).
-%&' 4.: ;5#*6(! <,$ .$ 8.*%4!"$ .&1 8.=,&1 0*6 !"#$% +5.- $0< .& (*#52$
(&2!#%+5 ('9$((*095>& 3).
-%&' 5.: D'#**7(! .&# 09/.& '9$((= .*6 0"#$+$ +$% !0$#$,$(B5#*6(! .$
B=($.$ 1 )-1 4 '%$ .%1 !0<(!#!1 '9$(()1 .*6 0"#$+$. D# <(-1 *% '9$(()1 0*6 $0)(!%#$# !"#$% (&2!#%+)1, 0&'$"#*6(! 8.* 6* B=($.
-%&' 6.: D0< '9$((= 8! '9$((= 49&8%(*0*%/#.$1 .* 09/.* $0< $9%8.!95 1
+5:! '9$((=1 +$% .& '9$((*095>& 3 +5#*6(! (&2)# <,$ .$ 8.*%-4!"$ .&1 8.=,&1 8.&# *0*"$ B9"8+!.$% & (*#52$ $6.=.
!!!!
"
#
$$$$
%
&'
0000
3100
1010
2001
,
!!!
"
#
$$$
%
&'
'
21000
10510
30301
,
!"
#$%
&000
000
1 !"#$%&' – ()$**"%$ '+',-*$TA 58
! "#$#"%&' #()*$+,-./, ".0 .&.-%123#+ 4#+ !"#$%&'()* +), Gauss, .(.4(5-
$6&23#+ *3#& 72 4%,2 -5 -582&+49 )$#--9 3.0 ":# 3. "$63. #"* #$+732$% 1
2:&#+ 4#+ 3. -*&. -5 -582&+4* 73.+;2:. 35/ 739(5/ 735& .".:# #&942+. <37+ )+# &# (=7.0-2 >&# )$#--+4* 7=735-# -2 3.& #()*$+,-. 3.0 Gauss, -23#-
3$>".0-2 3.& 2"#0?5->&. ":% 3.0 72 >&#& +7.8=&#-. #&5)->&. 4(+-#4'3*
":#.
@+# "#$%82+)-#, #/ (=7.0-2 3. 7=735-#
!"
!#
$
%&'&'%&&''
'%&'&&'
2322
533
43322
54321
54321
54321
xxxxx
xxxxx
xxxxx
.
A;5-#3:1.0-2 3.& 2"#0?5->&. ":# 3.0 70739-#3./
((((
)
*
++++
,
-
''
''
'''
232211
511133
413322
4#+ >;.0-2 8+#8.;+4%:
31 !! . (B9-# 2.)
((((
)
*
++++
,
-
'''
''
''
413322
511133
232211
133
122
2
3
!!!!!!
&/'/
(B9-# 4.)
(((
)
*
+++
,
-
''''
''
077700
187700
232211
).2B9-#(
)5.B9-#()$#--95135&C)&..=-2
32 !! .
(((
)
*
+++
,
-
''''''
187700
077700
232211
( 227
1 !! / B9-# 3.)
(((
)
*
+++
,
-
''''''
187700
011100
232211
233 7!!! &/ (B9-# 4.)
(((
)
*
+++
,
-
'''''
110000
011100
232211
1 !"#$%&' – ()$**"%$ '+',-*$,$ 59
)!3"#$%(
)5!"#$%(&'%$$#(2)(*&+!!,$-
33 !! !"
###
$
%
&&&
'
(!!!
110000
011100
232211
311
322
3!!!!!!
!"!"
("#$% 6!)
###
$
%
&&&
'
(!!!!!
110000
101100
102211
211 2!!! !" ("#$% 6!)
###
$
%
&&&
'
(!!
!
110000
101100
100011
.
. )-/-0)%1!2 31+%4%2 -1+%5 %+(&$6+!2 4/5$%47)82 4%5 %+)59)!5:-1 9)! 9,9)($%
)*
)+
,
-!-!
-!
1
1
1
5
43
21
x
xx
xx
3!0 -1+%5 59!;,+%$! $- )! 9,9)($%
)*
)+
,
-.!-
.-
1
1
1
5
43
21
x
xx
xx
.
<3!$6+72, )! 9,9)($% 6:-5 =3-5'! 3/#>!2 /,9-7+ )(2 $!'?#2
)1,,1,,1( 4422 xxxx .!. , !/42 , xx .
0 *2 /,9!0$- )@'% 4%5 )! 9,9)($%: )*
)+
,
-.!.-!.!-.!.
2434
122
132
"zyx
"zyx
"zyx
.
A:($%)1B!0$- )!+ -3%0C($6+! 31+%4% )!0 909)#$%)!2 4%5 6:!0$- ;5%;!:54=:
###
$
%
&&&
'
(
!!!
!
21434
11212
11321
~133
122
4
2
!!!!!!
!"!"
####
$
%
&&&&
'
(
!!!
!!!
!
23850
13850
11321
~22
5
1 !! !"
####
$
%
&&&&
'
(
!!!
!
!
23850
5
1
5
3
5
810
11321
~233 5!!! ."
####
$
%
&&&&
'
(
!
!
!
10000
5
1
5
3
5
810
11321
.
1 !"#$%&' – ()$**"%$ '+',-*$TA 60
!"# $%& 3% '()**+ $,- $./.-$)0,- "0&)1) 23,-*. #$4 10000 !"### !zyx +
10 !" , ",- 5%*)0&.4 #$4 $, 565$%*) .0&)4 )76&)$,. 8.&419,
!" #$%& %'" ()*+,-' ("./ -,-%01$%2/ 1( %' 3204(5$ %2, ()$,6'17"2, )*"$-
#$ )$82,-5$-%(* 15$ 98$110 %'/ 128:0/ .|0...00 , 1( 0$. , %.%( %2 -;-%'-
1$ (*"$5 $<;"$%2.
=>?@A?
!"# $) 5-5$+*)$) ",- /65)*. *23(4 $:(), ")()$%(,6*. #$4, #5) )"# )-$9 .0-&)4 5-*;4;)5$9, + 23,-& *0) *,&)741+ /65% + 23,-& 9".4(, "/+<,= /65.>&. ?%-
/)7+, 7.& .*@)&05$%1. 565$%*) ",- &) 23.4 ".(455#$.(.= )"# *0) )//9 ".".()-
5*2&,- "/+<,-= /65.4=. !",7.41&6.$)4 #$4 )-$# 4536.4 '.&419 '4) $) '()**419
5-5$+*)$).
&/$)*0(H
B$ +,4(* %2 -;-%'1$
%&
%'
(
"#"#"#
1
1
1
yx
1yx
y1x
, !!!!)1 .
2+'-
A3%*)$0B,-*. $,& .")-C%*2&, "0&)1) $,- 5-5$+*)$,= 1)4 23,-*. 74)7,3419:
***
+
,
---
.
/
111
11
11
""
~31 ## 0
***
+
,
---
.
/
11
11
111
"" ~
122 ### !!0
***
+
,
---
.
/
!!!
"""
110
010
111
~233 ### #0
***
+
,
---
.
/
!!
""
100
010
111
.
1 !& 1$1 , $#$. $, 565$%*) .0&)4 )76&)$,.
1 !& 1"1 , $#$. , $./.-$)0,= "0&)1)= '(9@.$)4 ***
+
,
---
.
/
000
000
111
.
D",*2&>=, $, 565$%*) 45,7-&)*.0 *. $%& .C05>5%
xyyx !"2"# 11
1)4 2$54 23.4 9".4(, "/+<,= /65.>& $%= *,(@+= )1,( xx ! , !)x .
1 !"#$%&' – ()$**"%$ '+',-*$,$ 61
$'%-'&"'
!" #$!%!&
1. !" #$%&'(' (" )*)(+,"("
i) !"#
$%&$%&$
22
1532
!yx
!yx -". ii)
'!
'"
#
%&%&$%&
63
12623
2
!y
!yx
yx
)(/ ,0$1+ BAX % , 230* " 0 345"-"6 (75 )*5('8')(95 (75 "#59-
)(75, # 0 345"-"6 (75 "#59)(75 -". $ 0 345"-"6 (75 )(":'$95 2-
$75.
2. !" #$%&'(' (" #$",,.-% )*)(+,"(" 30* 3'$.#$%10*5 0. .)2(/('6:
i)
(((
)
*
+++
,
-$%
((((
)
*
++++
,
-
(((
)
*
+++
,
-
$
$
3
2
1
3403
4312
1221
%!y
x
ii)
((((
)
*
++++
,
-$
%(((
)
*
+++
,
-
((((
)
*
++++
,
-
$$
$
1
3
1
2
324
213
132
311
!y
x
;(/ )*5<='." 5" #$%&'(' (0*6 '3"*>/,<50*6 345"-'6 (75 )*)(/,%(75
"*(95.
3. !" 8?)'(' (" )*)(+,"(" 30* "5(.)(0.=0?5 )(0*6 "5/#,<50*6 -8.,"-
-7(0?6 345"-'6:
4. M' (05 "8#2$.:,0 (0* Gauss 5" 8?)'(' (" )*)(+,"(":
i) '!
'"
#
%$&%&$%&&
02
3
1
zyx
zyx
zyx
ii) '!
'"
#
%&$%&&%&$
134
12
0
zyx
zyx
zyx
iii) '!
'"
#
%&$$%&$
%$&
023
32
22
zyx
zyx
zyx
.
(((
)
*
+++
,
-$
2100
1010
3001
i)
()
*+,
-$ 4210
3301ii)
(((
)
*
+++
,
- $
41000
30100
20011
iii)
1 !"#$%&' – ()$**"%$ '+',-*$TA 62
5. O!"#$% &' ()(&*!'&':
i) !"
!#
$
%&''%'&&%'''
322
12
22
!zyx
!zyx
!zyx
ii)!"
!#
$
%&'&&%''&%&'&
0332
12
12
zyx
!zyx
!zyx
iii) !"
!#
$
%'%'&'%&'&
03
22
032
zx
!zyx
!zyx
6. +!"#$% &' ()(&*!'&':
i)
!!"
!!#
$
%'&%&
%&%'
13143
16
443
32
xx
yx
yx
yx
ii)
!!"
!!#
$
%'&%&%&'%'&
1410
123
02
13
zyx
yx
zyx
zyx
B! "#$%$&
1. ,' -./#&/ &' "#,$, 0(&/ &" (1(&2!' !"
!#
$
%'&%&'%''
33
12
0
"!#yx
"!y$x
"!#y$x
3' 45/6 $% 71-
(2 &23 )1,1,1(),,( &%!yx .
2. ,' -./#&/ &23 /8#($(2 &2% 9'.'-"7*% "#x$xy ''% 2 , !("#,$, ,
9") :64.5/&'6 '9; &' (2!/#' )0,1( , )0,2( <'6 )6,1(& .
3. =3 2
0%$,#," )) <'6
!!"
!!#
$
%&'
&%'&%&'
2
1/>()32!2
133/>()322!4
2/>()342!2
"#$
"#$"#$
, 3' '9":/#8/&/ ;&6
3
%"#$ %%% .
4. =3 ***
+
,
---
.
/
&%
122
121
322
A <'6 ***
+
,
---
.
/%!y
x
X , 3' 71(/&/ &" ?.'!!6<; (1(&2!'
XAX 4% .
5. =3 *+
,-.
/&&
%22
11A , 3' -./#&/ "7")% &")% 9#3'</% & ?6' &")% "9"#")% 6-
(51/6:
XAAX % .
1 !"#$%&' – ()$**"%$ '+',-*$,$ 63
6. !" #$%&'& '" %(%')*"'":
i) !"
!#
$
%&&%&&%&&
2104
42
1
!"yx
!"yx
"yx
ii) !"
!#
$
%&&%&&%&&
#$"yx
"yx
"yx
2
1032
6
iii) !"
!#
$
%&&%&%&
3)1(2
1
y%x
%%yx
yx
, !'% .
1.7 &!".+'- ()$**"%/+ '+',-*$,/' *& ,-
*&0/1/ /)"2/+'3#
&4567879
+',- ./0,10$*&-, ."/21/"30 .&/41/25"*& *4" *67080 *& ',- 0.09" *.0/0$*& -" :/9%;0(*& ', #$%, &-<= 1/"**4;o$ %(%')*"'0=. >*?=, <.?= 6@0(*& 8&4 %'" 1/"**4;2 %(%')*"'" *& 8$0 "1-A%'0(=, &9-"4 @/)-
%4*0 -" 6@0(*& ;"4 6-"- '$.0, 0 0.090= -" &;3/2B&4 '4= #$%&4= &-<= 1/"**4;0$
%(%')*"'0= ?= %(-2/',%, '?- %(-'&#&%'A- '0(.
C4 '$.04 .0( 7" :/0$*& 1&-4;&$0(- '0(= '$.0(= .0( )8, D6/0(*& 14" ',- .&/9-.'?%, &-<= 1/"**4;0$ %(%')*"'0= 22( .
E" ./0@?/)%0(*& %',- "-"B)',%, &-<= '6'040( '$.0( .0( '" &/1"#&9" 14" ',- "-&$/&%) '0( &9-"4 04 !"#$!%&'(.
/:;<=>56 ?@AB 22( C;@6D6
F%'? 0 22( .9-";"= )*
+,-
.%
2221
1211
!!!!
A .
C "/47*<= 122122 6666 /11 #61&'"4 !"#$!%&) *!% +#,)-) $ ;"4 %(*:0#9B&'"4 *&
|| A ) *& 2221
1211
!!!!
. G,#"8),
12212211
2221
1211|| 6666
6666
A /%% .
H.&48) , 0/9B0(%" "(') "-'4%'04@&9 %& 6-"- 22( .9-";", #61&'"4 !"#$!%&) 2.( */0.(. I4" ."/28&41*", , 0/9B0(%"
— '0( .9-";" )*
+,-
.//
%22
13A &9-"4 4262)1()2(3|| /%&/%///%A
— '0( .9-";" )*
+,-
.%
10
01I &9-"4 101|| %/%I
— '0( .9-";" )*
+,-
.%
00
00" &9-"4 0|| %" .
1 !"#$%&' – ()$**"%$ '+',-*$TA 64
./012345 6789 33! :075;5
! "#$%"&'( )*+, 33! -$*(.( "#$%)/(0 1) /2* 3"45)0( /2, "#$%"&'(, 22, /672, 8, )74,:
9'/8 o 33! -$*(.(, """
#
$
%%%
&
'(
333231
232221
131211
!!!!!!!!!
A .
: (#051+, 3231
2221
13
3331
2321
12
3332
2322
11 5555
55555
55555
5 )* ;<=)/(0 !"#$!%&' (!% )#*'-
+' $ .(0 '&13";$%)/(0 1) || A 4 1)
333231
232221
131211
555555555
.
>2;(?4
3231
2221
13
3331
2321
12
3332
2322
11
333231
232221
131211
||!!!!
!!!!!
!!!!!
!!!!!!!!!!
A )*(( . (1)
@-)0?4 2 "#$%"&'( (&/4 (*/0'/"0A)$ ') <*(* 33! -$*(.(, ;<=)/(0 !"#$!%&' 3,- (./,-.
B0( -(#6?)0=1(, (* """
#
$
%%%
&
' *(
021
310
512
A , /+/)
20)1(5)3(1)6(221
105
01
30)1(
02
312|| *(*)*)*()**(A
! -(#6'/('2 (1) 1) /2* "-"$( "#$%)/(0 2 || A ;<=)/(0 '*.)(%01' (,- || A 2- )"!- (' &(!345#' (,- )"6(,- 0"'117-.
C) )./<;)'2 /8* -#67)8* '/" (*6-/&=1( (&/+ <A"&1):
)()()(|| 223132211323313321122332332211 !!!!!!!!!!!!!!!A *)***(
)()()( 311232112331133311221332331221 !!!!!!!!!!!!!!! ***)**(
3231
1211
23
3331
1311
22
3332
1312
21 !!!!
!!!!!
!!!!!
! *)*( (2)
H -(#6'/('2 (2) ;<=)/(0 '*.)(%01' (,- || A 2- )"!- (' &(!345#' (,- 859(5-",- 0"'117-.
:1"$8,, 1-"#"D1) *( ?0(-0'/E'"&1) +/0 0'AD)0 .(0
1 !"#$%&' – ()$**"%$ '+',-*$,$ 65
2221
1211
33
2321
1311
32
2322
1312
31||!!!!
!!!!!
!!!!!
!A !"# , (3)
!"# $%&'()* !"#$%&'(! || A )* $+,* %! -%,./01! %2* %+1%2* '+!((3*.
+),)(-,"./' 0(* 1' 2)3%4) )!0 () )4)!(.&/)() (-5 || A , 263' 1("*7'8" ij! (-5 )4(81("*7-5 &,)//95 !"$$)!$)1*6:'()* /' (-4 ",8:"#1) 2-5 (6;-5 ("# !84)2)
!"# !,"2.!('* )!0 ("4 ", )4 !),)$'8<"#/' (- &,)//9 2)* (- 1(9$- ("# 1("*-7'8"# ij! . = ",8:"#1) )#(9 $%&'()* 044#-)" ,+15,&-! %,& -%,./01,& ij. 2)* 1#/>"$8:'()* /' ijM .
+),)(-,"./' '!81-5 0(* 263' 0,"5 '405 )4)!(.&/)("5 (-5 || A %7'* !,01-/" !
9 ", 8?*" /' (" !,01-/" ("# ji!" )1( . @" &*40/'4" ijji M!"1)( $%&'()* !4'06+.78
-&($43+)(! %,& -%,./01,& ij. 2)* 1#/>"$8:'()* /' ijA .
A-$)?9
ijji
ij MA !"# )1( .
B' ("#5 1#/>"$*1/".5 )#(".5 () )4)!(.&/)() (1), (2) 2)* (3) &,6C"4()*:
131312121111|| A!A!A!A !!#
232322222121|| A!A!A!A !!#
333332323131|| A!A!A!A !!# .
M' '2(%$'1- (D4 !,6;'D4 !,"2.!("#4
313121211111|| A!A!A!A !!#
323222221212|| A!A!A!A !!#
333323231313|| A!A!A!A !!#
!"# '84)* )46!(#&/) (-5 || A D5 !,"5 () 1("*7'8) (-5 1-5, (-5 2-5 2)* (-5 3-5 1(9$-5 )4(*1("87D5.
9:;:<=;=>=
E(-4 !,6;- (" )46!(#&/) !"# 7,-1*/"!"*"./' &*) ("4 #!"$"&*1/0 /*)5 ",8:"#-
1)5 '84)* D5 !,"5 (- &,)//9 9 1(9$- !"# %7'* () !',*110(',) /-?'4*26.
F*) !),6?'*&/), )4 $$$
%
&
'''
(
) "#
010
302
211
A , (0('
1)43(02
110
32
211
30
210|| #""#
"!"
"#A .
1 !"#$%&' – ()$**"%$ '+',-*$TA 66
./012345 6789 77 ! :075;5
!"#$%&' &()"* +,"% +-. /"#0/1$% '.23 22! 4#.%5% 5%* &' +- 6/78'*9 +-3 +-. /"#0/1$% '.23 33! 4#.%5%.
!"#0/1&' '4#$-3 :3 /"#0/1$% '.23 4#.%5% &' (.% $+/*)'#/ ][ 11! , .% '#.%* +/ #;*/
+/ $+/*)'#/.
<'.*59, &4/"/=&' .% /"#$/1&' +-. /"#0/1$% " +9>-3 3"# &' +- 6/78'*% +/1
/"*$&/= +-3 /"#0/1$%3 1$# +9>-3. ?.%3 +(+/*/3 /"*$&23 @(A'+%* 6 : 5 < = < > ;
8 9. B1A5'5"*&(.%, ($+: / ""! 4#.%5%3
%%%%
&
'
((((
)
*
+
""""
"
"
!!!
!!!!!!
A
,---
,,
21
22221
11211
.
!./&90/1&' /"#0/1$% +/1 4#.%5% # 5%* +- $1&6/@#0/1&' &' || A 7
""""
"
"
!!!
!!!!!!
,---
,,
21
22221
11211
+/. %"*8&2
"" A!A!A!A 1112121111|| ...+ ,
24/1 ijji
ij MA .$+ )1( 5%* ijM , - )1( $" +9>-3 /"#0/1$% +/1 4#.%5% 4/1 4"/5=-
4+'* %42 +/. #, %. 4%"%@'#C/1&' +- A"%&&7 5%* +- $+7@- +/1 $+/*)'#/1 ij! .
D4:3 5%* $+*3 /"#0/1$'3 3-3 +9>-3, - /"#0/1$% ijM @(A'+%* !"#$$%& '()*'+$,
-'+ $-'./!)'+ ij5 5%* +/ A*.2&'./ ijji M.$1)( @(A'+%* ,"0!1(.23 $+45"6(%4,
-'+ $-'./!)'+ ij5 .
E 4%"9$+%$- ""A!A!A! 1112121111 ... , &' +-. /4/#% /"#$%&' +-. || A @(A'+%*, 24:3 5%* $+*3 /"#0/1$'3 3-3 +9>-3, ,-+04, -78 '()*'+$,8 2,-# -, $-'.-/!), -78 178 0(,4468.
F4/;'*5.='+%* +/ '42&'./ 8',"-&%:
9:;<=>?
<*% /4/*%;74/+' A"%&&7 i 7 $+7@- j '.23 ""! 4#.%5% # *$)='*: (%) i"i"iiii A!A!A!A ...+ ,2211|| 5%* (6) "j"jjjjj A!A!A!A ...+ ,2211||
1 !"#$%&' – ()$**"%$ '+',-*$,$ 67
! "#$%&'#&( (#) )*+,'#- !"#$%&'(! %)* +,-.+&/!* ./ "$0/ '# &'0-1,2# '(/ i
+$#334/, ,56 ( (7) !"#$%&'(! %)* +,-.+&/!* ./ "$0/ '# &'0-1,2# '(/ j &'4-
)(/.
012134245675
1) 825#- 9#5,$: :'- #5 '# &'0-1,2# 3-#/ +$#334/ 4 &'4)(/ ,5:/ "25#;# ! ,25#- :)# 3(<*5, ':', 0|| !A .
2) =5 *5#/ "25#;#/ ,25#- '$-+.5-;:/ %5. 4 ;%'., ':', #"0<,-;5>,'#- :'- ( 0$2-?0@&% '0@ ,25#- 2&( 3, '0 +-5:3,50 '.5 &'0-1,2.5 '(/ ;@$2#/ <-#+.520@. A-#
"#$%<,-+3#
13
1)3(1
3
100
30
3ln21
5 "!"!" " .
3) ="0<,-;5>,'#- ,"2&(/ :'-, #5 <>0 +$#33*/ (<>0 &'4),/) ,5:/ "25#;# ,25#- 2&,/ 4 #5%)0+,/, ':', ( 0$2?0@&# '0@ "25#;# ,25#- 2&( 3, 3(<*5. A-# "#$%<,-+3#,
0
151210
963
321
!"
"
,
#90> 12 3## ! .
&.$)*/(H
8! 9,:;+<" %! !=':9,>?# /&($=),@(!%! %A" /%+>B:-A" %)* 2)* /%C=)* %)* +,-.+&/!* %A" $>"#?A"
i)
###
$
%
&&&
'
("
"!
113
202
121
A ii)
###
$
%
&&&
'
( "!
00
00B
2
2
/&"0/&"2010
/&"20(4)
?!> /%) /&"DB:>! "! &$+=+'>/;+<" +> || A , || B .
1+'-
i) B10@3,
1 !"#$%&' – ()$**"%$ '+',-*$TA 68
8)62(13
22)1( 21
12 !"""!"
"! #A
43113
11)1( 22
22 !#!"
"! #A
0)22(22
11)1( 23
32 !""!"
""! #A .
!"#$%&'(, 16014082102|| 322212 !$#$#$!##! AAAA .
ii) )*#+$,
0-+&-+&2
00)1(
2
21
12 !"! #
!!B
!!!!!!!
B 2-+&-+&.$4-+&-+&2
2-+&.$4)1( 222
2
222
22 #!"
"! #
12-+&2.$2-+&)-+&.$2( 2222 !#!#! !!!!!
000
2-+&.$4)1(
223
32 !"
"! # !!B
!"#$%&'(
1010|| 322212 !$##$! BBBB .
&./0123 44 % 567889:;< 212=>87=;? 8@ =3 8AB;C; =;1 Cramer
/"'( ,012$, -32 "4#.5#6$,&2, %&2 542$$789 "#% -6-3.$2 $"#4,0 &2 %*,7 $#-
&2178: ;6-. : <",74# ";:=#( ;6-,'& : &2 ,0&27 216&23#. >3.& ,7178: ",40"3'-.
"#+ 3# -6-3.$2 ,0&27 ""% , 3# ,"9$,&# =,?4.$2, 3#+ #"#0#+ . 2"91,7@. "242-
;,0",327, $2( ";.4#A#4,0 "93, 3# -6-3.$2 2+39 %*,7 $#&2178: ;6-. 827 "93, %*,7 <",74# ";:=#( ;6-,'& : ,0&27 216&23#.
!"#$%&'
)-3' 3# ""% 542$$789 -6-3.$2 BAX ! .
& B& 0|| 'A , 393, 3# -6-3.$2 %*,7 $#&2178: ;6-. 3.& )...,,,( 21 "xxx $,
D
Dxx
D
Dxx
D
Dxx "
" !!! ...,,, 22
11
9"#+ D ,0&27 . #40C#+-2 || A 3'& -+&3,;,-3?& 3'& 25&?-3'& 827 iDx ,
"i ...,,3,2,1! ,0&27 . #40C#+-2 "#+ "4#86"3,7 2"9 3.& D 2& 2&378232-3:-#+-
$, 3.& i -3:;. 3'& -+&3,;,-3?& 3#+ 25&?-3#+ ix $, 3. -3:;. 3'& -32=,-4?& 94'&.
& B& 0|| !A , 393, 3# -6-3.$2 : ,0&27 216&23# : %*,7 <",74# ";:=#( ;6-,'&.
1 !"#$%&' – ()$**"%$ '+',-*$,$ 69
!"# $% &'()*+, ,-$# ")%./"$'0 #$0:
!"#$%&'
1% %+%2'345 6/6$*+, !!AX ,
" 47'0 +#3% $* +*8'30.9 :/6*, ,3 .,0 +#3% ,3 0|| #A .
" 47'0 .,0 +* +*8'30.45 :/6'05 (;"'0)% ":9&%5), ,3 .,0 +#3% ,3 0|| !A .
%(")$'
1) <3, !!$ 2),++0.# 6/6$*+, BAX ! +' 0|| #A , :42'$,0 .,0 *+*,-./
Cramer, * 8' '"=:-6* $%- 6-6$9+,$%5 ,-$%/ ,3,>4)'$,0 .,0 ?5 0/121/3 ,45 Cramer. @ .,3#3,5 $%- Cramer 8'3 '=3,0 ,"%8%$0.9 +4&%8%5 20, 3,
7)*60+%"%0*&'= 6$* :/6* 6-6$*+;$?3 +' 43, +'2;:% ,)0&+# 'A06(6'?3, 20,-
$= ")4"'0 3, -"%:%206$%/3 "%::45 %)=B%-6'5 +'2;:*5 $;A*5. C0,-$# 6$* 6--
347'0, +' $%3 .,3#3, ,-$#3 &, '"0:/%-+' +#3% 22$ .,0 33$ 2),++0.; 6--
6$9+,$,.
D5 ")%5 $%-5 ,)0&+*$0.%/5 -"%:%206+%/5 * +4&%8%5 '"=:-6*5 6-6$9+,$%5 +' $%3 ,:2#)0&+% $%- Gauss -"')$')'= $%- .,3#3, $%- Cramer. E+?5, % .,3#-
3,5 $%- Cramer '=3,0 080,=$'), 7)960+%5 6' &'?)*$0.; B*$9+,$,.
2) C0, $*3 '"=:-6* '3#5 !!$ 2),++0.%/ 6-6$9+,$%5 BAX ! +' 0|| !A ')2,-
B#+,6$' 6-39&?5 +' $* +4&%8% ,",:%0>95 $%- Gauss.
3) !3 43, !!$ 2),++0.# 6/6$*+, '=3,0 %+%2'345, $#$' !!! ...21 DxDx
0!! !Dx , ,>%/ #:'5 %0 %)=B%-6'5 47%-3 +0, +*8'30.9 6$9:*.
&.$)*/(&'
1. N/ 65789 ,4 *+*,-./ %&
%'
(
!))!*)!*
252
12
1
0yx
0yx
0x
1+'-
<7%-+' 04
521
112
101
#!**
!D , 7
522
111
101
!**
!Dx
7
521
112
111
*!**
!Dy .,0 3
221
112
101
!!D" .
1 !"#$%&' – ()$**"%$ '+',-*$TA 70
!"#$%&'(, )*$+'&, $- .#& /,&0&, .#1 Cramer, -2&,3
4
7!!
D
Dxx ,
4
7"!!
D
Dxy ,
4
3!!
D
D!!
456,47 .# )*).5$, %8-3 .5 $#&,43/7 6*)5 #$%
&'( "
4
3,
4
7,
4
7.
2. !" #$%&' () *+*(,-" )*
)+
,
!--!"-!"-
1
1
1
./yx-
/.yx
/y.x.
0+'-
98#1$-
)1)(1()1(1)1(1)1(
11
11
112 -"!-"""-!
"""
! """"""""
""
D
)1)(1()1(1)1(1)1(
11
11
1112 -"!""-"-!"
"! """""
""Dx
)1)(1()11(1)1()1(
11
111
11
-"!-"""-!"
""
! """"""
"Dy
)1)(1()1(1)11()1(
111
11
11
-"!---""!"
! """""""
D! .
:3 .3$%( .5( ",;,$%.;#1 " "#1 $54-&2<#1& .5& #;2<#1), )1)(1( -"! """D -2&,3 #3 1,1,0 " .
— =3, 0." /,3 1"." /,3 1." -2&,3 0.D /,3 -"#$%&'(
""""""
D
Dxx
1
)1)(1(
)1)(1(!
-"-"
!! , "D
Dyy
1!! /,3
"D
D!! 1!!
456,47 .# )*).5$, %8-3 .5 $#&,43/7 6*)5 #$%
&'(
"""1
,1
,1
.
— =3, 0!" , .# )*).5$, >2&-.,3 )*
)+
,
!-"!"!"
1
1
1
yx
!x
!y
. ?& ";#)@%)#1$- /,.A $%65
1 !"#$%&' – ()$**"%$ '+',-*$,$ 71
!"# $%& !'(')!*+'# ',"-.-'"# /0+-1&)2' !& -%-!32* !"
!#
$
%&%&%&
2
1
1
!y
!x
!y
, !& &4&+&
40&5*6.# '+6*" *$%6*!&.
— 7"* 1%" , !& -%-!32* 8+6'!*"
"#$
%''&%&'
(!"
!#
$
%''&%&'%&'
1
1
1
1
1
!yx
!yx
!yx
!yx
!yx
"#$
%%&'
(22
1
y
!yx ))
*
+,,-
.',"-.-'"#!"#29(3
1*!:40&-;9-*2'
"#$
%%
(1y
!x.
<4&296=#, !& -%-!32* 9>'" :4'"0'# (%-'"# !3# 2&05?# ),1,( !! , !/! .
— 7"* 1&%" , !& -%-!32* 8+6'!*"
"#$
%&&%&'&
(!"
!#
$
%&'&%&&%&'&
1
1
1
1
1
!yx
!yx
!yx
!yx
!yx
"#$
%&%&'&
(22
1
!!yx
))*
+,,-
.',"-.-'"#!"#29(3
1*!:40&-;9-*2'
"#$
&%%'&
(1
0
!yx
"#$
&%%
(1!
yx.
<4&296=#, !& -%-!32* 9>'" :4'"0'# (%-'"# !3# 2&05?# )1,,( &yy , !/y .
3. !" #$%&' () )*)+&,-. /0/(1*" !"
!#
$
%''%''%''
0
0
0
./yx
/.yx
/y.x.
0+'-
@>&)2'
)2()1()1()1()1(
11
11
1122 '&%&'&&&%% """"""
""
"D .
1 !"#$%&' – ()$**"%$ '+',-*$TA 72
!" #"$%& #'& ()*)$%#*+, ! (+, $'-./01+,/ #'/ +*01+,2) D .0/)" +" 1 3)" 2! .
— 4") 1"! 3)" 2!"! .0/)" 0"D 3)" .(+$%/5& #+ 262#'$) %7." $+/)-"38
962' #' $'-./"38 )0,0,0( .
— 4") 1#! #+ 262#'$) :0/.#)" $%
$&
'#(()
#((#((#((
0
0
0
0
"yx
"yx
"yx
"yx
3)" %7." ;(."*.& 962."& #'& $+*<8& !*!! "y"y"y ,),,,( .
— 4") 2!#! #+ 262#'$) :0/.#)"
$%
$&
'
#!(#(!#((!
02
02
02
"yx
"yx
"yx
$%
$&
'
#!#!#((!
)033
033
02
"x
yx
"yx
++,
-../
0-6+;99.&#"&)(=
.>0252'(*?#'#'/)<)"*%2)$.
%&'
##
)x"xy
.
@A*), #+ 262#'$) %7." ;(."*.& 962."& #'& $+*<8& ),,( xxx , !*x
$'%-'&"'
!" #$!%!&
1. B) ,(+9+:02.#. #"& +*01+,2.&
i)
5000
320
0530
! , ii)
8
100
51
3
2
ee
e
, iii)
##
##
'$22,/010
2,/-2'$
iv)
0
0
111
22 $%$% , v)
2log01
5log01
111
!, vi)
ee
e
0
111
10
2
.
2. B) 962.#. #"& .>"2?2."&:
i) 0
10
12
111
#!!
x
x
x
ii) 0
33
1
11
#!
x
xx
x
1 !"#$%&' – ()$**"%$ '+',-*$,$ 73
iii) 0
1
3
21
!xx
xx
x
iv) 0
11!"11!"111
2
!"x
x .
3. #$ %&'()( )$ '*')+"$)$
i) #$%
"!&""!"
73
425
yx
yx ii)
'#
'$
%
!"!&"!&&
366
11644
11443
yx
!yx
!yx
iii) '#
'$
%
!&&!&"!&&
07
0
0432
zyx
zyx
zyx
iv) #$%
!&!&
2'*,2!"3
1'*,!"2
xx
xx, )2,0[ "x( .
4. N$ -.(/)( )01 )0"21 )3* !(# 40$ )01 353/(1 )$ 5$.$67)8 '*')+"$)$
293*, 6$0 "! "!:(,0621 %&'(01.
i) '#
'$
%
!"&&!&""!&"
0)1(3
0
0)2(
!#yx
!#yx
yx# ii)
'#
'$
%
!&&!&&!&&
0
0
0
#!yx
!#yx
!y#x.
B! "#$%$&
1. #$ %&'()( 40$ )01 :07;3.(1 )0"21 )8, !($#, )$ '*')+"$)$:
i) '#
'$
%
!&&&!&&
!&&
222
1
1
!y#x#!#y#x
!yx
ii) '#
'$
%
!&&"!&&
!"&
133
1
1
$!yx
$$!y$x!y$x
.
2. #$ %&'()( )3 '&')!"$:
'#
'$
%
!&&!&!&&
0
2
0)(
!y$x!y$x
!y$x
.
3. #$ -.(/)( )! '9()06+ <2'! )8, (*<(0=,
i) '#
'$
%
!"!&"!&
523:
12:
12:
3
2
1
yx%yx%yx%
ii) '#
'$
%
!&"!&!&
53:
152:
52:
3
2
1
yx%yx%yx%
iii) '#
'$
%
!&!&!&
1:
324:
02:
3
2
1
yx%yx%yx%
iv) '#
'$
%
!&!&!&
562:
03:
193:
3
2
1
yx%yx%yx%
.
1 !"#$%&' – ()$**"%$ '+',-*$TA 74
4. !"#$%&'" ()* "+,-#-) 02 !"" !"t#t , 0## ./0 (% -&-()'/
$%
$&
'
!"!"!""
02
02
0
"y!x#y"x
zyx
.
i) 1/ /2%3",+"(" 4(0 /* ) "+,-#-) 56"0 3&% $,7"8 9*0-"8, (4(" (% -&--()'/ 56"0 '%*/30.: ;&-). 1/ "+"(9-"(" /* 0-6&"0 (% /*(,-($%<%.
ii) =* ) "+,-#-) 56"0 '0/ 302;: $,7/, */ "+"(9-"(" 24-"8 ;&-"08 56"0 (% -&-()'/.
5. =* >0/ (%?8 !(#,",! ?29$6%?* ($yx ,, ), 2%? 3"* ",*/0 4;%0 ')35*, (5(%0%0 @-(" */ 0-6&"0
"$!yx "! , !x#$y "! ./0 #y"x$ "!
*/ /2%3",+"(" 4(0 12222 !""" #"!!"# .
6. 1/ ;&-"(" (/ -?-(:'/(/+
i)$%
$&
'
"!"!""
"!""
12
1)1(
1)1(
%yx
y%x
%yx%, !(% , ii)
$%
$&
'
!**!""!"*
0
05
02
%yx
y%x%yx
.
7. A,*"(/0 % 2,*/./8 +,
-./
0**
!21
32A .
i) 1/ B$",(" (08 (0'58 (%? !(% >0/ (08 %2%,"8 ?29$6"0 ') ')3"*0.48
2,*/./8 +,
-./
0!
y
xX (5(%0%8, @-(" &%'& ! (1).
ii) C0/ (08 (0'58 (%? % 2%? D/ B$",(" */ ;&-"(" ()* "+,-#-) (1).
(&#"%&' $'%-'&"'
!" #$%&%'
1. A,*"(/0 % 2,*/./8
+,
-./
0 *!
xx
xxxA
-?*)')'-?*
)( .
1/ /2%3",+"(" 4(0
1 !"#$%&' – ()$**"%$ '+',-*$,$ 75
i) )()()( yxAyAxA !"#
ii) )()]([ 1 xAxA $"$
iii) )()]([ !xAxA ! " !"# $%&' *!%! .
2. ()*'+#" , -)*#$#. &&&
'
(
)))
*
+"
000
100
010
M .
i) /# #-,0')1'+' 2+" ""3M $#" !'*"$% ""!M !"# $%&' 3,! .
ii) 3* "#M#MA !!" 2 $#" "#MM##B !$$" 2)1( , #%# , +2+' *#
45')+' +, !"*26'*, $% $#" *# #-,0')1'+' 2+" AB "$1 .
3. 3* &'
()*
+"
10
01I $#" &
'
()*
+$
"01
10J +2+' *# #-,0')1'+' 2+":
i) IJ $"2
ii) 7, %&5,"86# $#" +, !"*26'*, -"*%$9* +:. 6,5;<. &J#I ! , 2-,=
#,& -5#!6#+"$,) #5"&6,) ')*#" '-)8:. -)*#$'. +:. )0"#. 6,5;<..
iii) > -)*#$#. &J#I ! ?@'" #*+)8+5,;,, #* $#" 62*, #* 022 -! &# .
4. /# #-,0')1'+' 2+" : 8=66'+5)# 9. -5,. %1,*# +:* '=&')# ' +,= 0"-A#*,B 8@<6#+,. ')*#" !5#66"$2. 6'+#8@:6#+"862. 6' -)*#$#
&'
()*
+$
"((((
A28=*2:62:628=*
.
C-,5')+' #-2 +,* -)*#$# $ *# 45')+' +,=. -)*#$'. +9* 8=66'+5"D*
9. -5,. +,* %1,*# xx. , +,* %1,*# yy. , +:* '=&')# xy " $#" +:* '=-
&')# xy $" ;
5. E8+9 , !5#66"$2. 6'+#8@:6#+"862.
&'
()*
+&'
()*
+"&
'
()*
+..
y
x
)*&#
y
xT : , 6' 0-
)*&#
.
i) /# #-,0')1'+' 2+" , + ')*#" 8=*%5+:8: "11" $ .
ii) /# 45')+' +,* #*+)8+5,;, +,= 6'+#8@:6#+"86,B +.
( O
y
x
'
,-(xF,yF)
,(x,y)
1 !"#$%&' – ()$**"%$ '+',-*$TA 76
iii) !" #$%&'% '()* "+'&,'$(-()* .%'",/0."'1,.(2* '3+ ,)..%'$14+ '0* ,'$(-5* 6"1 '0* (.(1(7%,&"*.
6. !" 82,%'% '( ,2,'0.":
!"
!#
$
%&%&%&&
0
0
0
22 y!x"!y"x
zyx
, * , !'!" .
7. !" 82,%'% 91" '1* :1;-($%* '1.<* '() ),0[ #"' '( ,2,'0.":
!"
!#
$
%&&%&&%&&
0),)+()0.(
0),)+()0.(
0
22 $y"x"$y"x"$yx
.
8. !" #$%&'% 91" '1* :1;-($%* '1.<* '() !'% '1* ,/%'16<* 7<,%1* '3+ %)7%14+
1:1 %& yx& , %yx& %&:2 1:3 %& y%x& .
9. !" 82,%'% '( ,2,'0.":
!"
!#
$
%&%(%&&(
0
0
0)1(
'$y
$'y$yx'
, !'' .
10. =,'3 ( >&+"6"* )*
+,-
.((
%1
3
""
A . ?+ )>;$/()+ .0 .0:%+16@* >&+"6"*
)*
+,-
.%
y
xX 6"1 >$"9."'16@* "$17.@* ' '<'(1(1, 4,'% +" 1,/2%1 'XAX % ,
+" ">(:%&A%'% @'1 22 //( " .
11. i) !" 82,%'% '( ,2,'0.":
"#$
&%&&%&
232
4532
'yx
'yx, !'' .
ii) !" #$%&'% '1* '1.<* '() ' 91" '1* (>(&%* (1 %A1,4,%1*:
0)45(32 2 %&(& 'tt
0)23(22 %&(& 'tt
</()+ 6(1+5 $&B". C(1" %&+"1 0 6(1+5 $&B" ,% 6;7% .1" >%$&>'3,0;
1 !"#$%&' – ()$**"%$ '+',-*$,$ 77
&).,-'&"' %$,$#/-'-'
!. "# $%&#'() %*+ ,(- *%.%$),/ *#.(*,01#(- 2% $3$401#,# ,5 6.)''% 7,
%2 5 (183.(1'+- #92%( %4:&;- $%( ,5 6.)''% <, %2 5 (183.(1'+- #92%( =#3>;-, %(,(545602,%- 1368.+2/- ,:2 %*)2,:1; 1%-.
1. !" !!" !" )1( 2 , "#" , #$#% !!! . ! &
2. !" !!$ !" )4( 2 , "#" , #$#% '(#’ (")*'+ %,"(- !!! . ! &
3. A" !!$"" ))(1( 2 #!"" , "#" , #$#% #! ! . ! &
4. !" $!# 2! , $./0 #!, #%#1(*2"-'/, .,"('%3, #$#% /- #!, %,"(- ("#-4#156-7/- .,"('%3.
5. !" $!# 2! , $./0 #!, #%#1(*2"-'/, .,"('%3, #$#% $#! 2! .
6. !" !!$ 2)( IA #$#% '(#’ (")*'+ 8( %,"(- $A ! .
7. !" IA !2 #$#% '(#’ (")*'+ 8( %,"(-
IA ! 9 IA $! .
8. !" !!2A #$#% / ! :%" %,"(- ("#-4#156-7/3.
9. ;4<=%- .)"#/#% 222)( BAAB ! .
10. !" BXAX ! , #$#% '(#’ (")*'+ 8( %,"(- BA ! .
11. > ("#,4#1/?/3 #/0 .,"('( %&
'()
*32
21 %,"(- / .,"('(3
%&
'()
*$
$12
23.
12. @( 404#97(#(
+,
+-
.
$!""!""!""
153
032
02
zyx
zyx
zyx
'(- +,
+-
.
$!"!"!""
12
0
0
zy
zy
zyx
%,"(- -4/:="(7(.
! &
! &
! &
! &
! &
! &
! &
! &
! &
1 !"#$%&' – ()$**"%$ '+',-*$TA 78
13. !" #"$ %&$''()* +,+-.'$ #/0( 123 1($43&0-()#5 6,-+0(5, -*-0 7$ #/0( 890(&05 6,+0(5.
14. :"$ 3'3%0"#5 +,+-.'$ '93&0; "$ 0;"$( $1,"$-3.
15. <3 +,+-.'$ !"#
$%$&%
0
0)1(
y!x
y!x! #/0( )$( '. '.10"()#5
6,+0(5.
16. !" . 0=;+>+. 02 $%% "t#t$ , '0 0'$ ,0;"$( $1,"$-.
+-3 (, -*-0 -3 +,+-.'$ !"#
$%$%#y#x$$y"x#
2
2 #/0( '3"$1()?
6,+..
17. <3 +,+-.'$ )!
)"
#
$%$%$%
%yx%y%x
yx
2
0
, 2&'% , 0;"$( $1,"$-3. ! @
18. :+-> (A) #"$ 33* %&$''()* +,+-.'$.
i) A" 0|||||| +%% zyx DDD , -*-0 -3 +,+-.'$ 0;"$( 3'3%0"#5.
ii) !" 0)1( 222 $%&% zyx DDD , -*-0 -3 +,+-.'$ 0;"$( 3'3%0"#5.
!!. "# $%$&'()*) *+ (,(*- #./0*+(+ () $#1)23/ #.4 *35 .#6#$/*, .)-63.*'()35
1. B 9;"$)$5 ,,-
.
//0
1
%&%&
330
0442
2
!!!!
0;"$( 3 '3"$1($;35, $"
!) 2$! , C) 0$! , D) 1$! , E) 3$! .
2. :+-> & #"$5 '*2 9;"$)$5. !" 298&/0( 9;"$)$5 ( -#-3(35, F+-0 XAAX $ , -*-0 3 ( 0;"$( -,932
!) 22* , C) 32* , D) 23* , E) 33* .
3. :+-> & #"$5 $"-(+-&#G('35 9;"$)$5 )$( 0'% . B $"-;+-&3435 -32
9;"$)$ %& 0;"$( 3 9;"$)$5
!) A%1
, C) 1&A% , D) 11 &A%
, E) 1&& A% .
! @
! @
! @
! @
! @
! @
1 !"#$%&' – ()$**"%$ '+',-*$,$ 79
4. !" #$#%&'( !"#
$%&%&
1
1
y!x
yx! )*+(, (-$+(%", (+
.) 0%! , /) 1%! , 0) 1$%! , 1) 1,1,0 $'! .
5. 2, )34)*)5
12 %& yx , 222 %& yx , 22 %& yx .
.) 6)7+"$+ (89 %" *-," #&')*".
/) :;&'(%*<"3+ %7*=>+".
0) ?*+(, 8(7@AA&A)5 (+@ -$".
1) 2, -$" )*+(, 8(7@AA&A)5 B(, & %7*%& %,5 %C'+),.
?) 2, -$" #3'8*8%"3+ B(, & %7*%& %,5 %C'+),.
!!!. "# #$%&'%(&)*'+%+ ,-.+ /+%#')0/#%&'/1 %02 345%02 '%6702 '%($ 3*-$#,- %(8 %02 9+:%+402 '%6702
;<=>?@A;>=!?;O? B!">C>?
1. :%7"DE B(%@ =>+*( o90 B(, #%& #3-
+C;),( #3'')%7*( >5 87"5 %&+ )34)*( xy % .
2. :3'')%7*( >5 87"5 %"+ @F"+( xx( B(, #%& #3+C;),( #3'')%7*( >5 87"5 %&+ )34)*( xy % .
3. :3'')%7*( >5 87"5 %&+ (7;E %>+
(F9+>+ B(, #%& #3+C;),( #3'')%7*( >5 87"5 %&+ )34)*( xy % .
(. )*
+,-
. $01
10
G. )*
+,-
.$10
01
=. )*
+,-
. $10
01
-. )*
+,-
.$
$01
10
). )*
+,-
.$ 01
10
1 !"#$%&' – ()$**"%$ '+',-*$TA 80
"',.)"%. '-*&"/*$
.0 “123445627890:” ;’ <=3 >0=<?0>5 62@ABC43 D5E 35E 30F=3 6.G.
!" #$%&#'" $(%)*+&, $("-(.#/,0 ,$% &0, ,(.,1, 20'#3024 56**"74 $("-)*+&8/9' &# /1/*" “:''-, 2#;8*,0, 5/+ &,<+&,/024 /-.'+”. = *>5+ $"6 ?1'#/,0 #2#1 56&$1$/#0 "650,5/028 &# /+ 5>7.("'+ &-<"?" /"6 “#$,6@+&--'"6 $1',2,” 2,0 /9' “7(,&&"$(8@#9'”.
3 !"#$%&' #&'( )'*+( ,-.)/#&!+(, 2 !"#$%&' #&'( #0%1&'( ,-.)/#&!+( )'& 1 !"#$%& #&'( )')+( ,-.)/#&!+( !23/-3 39 dou ,&%$1&.
2 !"#$%&' %4( )'*+(, 3 !"#$%&' %4( #0%1&'( )'& 1 !"#$%& %4( )')+( ,-.)/-#&!+( !23/-3 34 dou ,&%$1&.
1 !"#$%& %4( )'*+(, 2 !"#$%&' %4( #0%1&'( )'& 3 !"#$%&' %4( )')+( ,-.)/-#&!+( !23/-3 26 dou ,&%$1&.
5' 61"7"2 89,/ ,&%$1& !23"& 03' !"#$%& '89 )$7" "2!/( ,-.)/#&!+(.
!" $(%)*+&, ,6/% ,'87#/,0 54&#(, 5/+' #$1*65+ #'%A 7(,&&02"> 565/4-&,/"A /(0B' #@05B5#9' &# /(#0A ,7'B5/"6A x, y, z:
2632
3432
3923
!""!""!""
zyx
zyx
zyx
.
C/" ,(.,1" 2#1&#'", 5/" "$"1" ?#' 6$8(."6' 2,<%*"6 5>&)"*,, ?1'"'/,0 "?+71#A 70, /+' /"$"<-/+5+ /9' ,(0<&B' 5/0A 2,/,2%(6;#A 5/4*#A #'%A 8),2, 5>&;9', &# /"' #@4A /(%$":
393426
113
232
321
= $,(,$8'9 ?08/,@+ &#/,5.+&,/13#/,0 5/+ 56'-.#0, 9A #@4A:
= 2+ 5/4*+ $"**,$*,5083#/,0 #$1 3 2,0 2,/%$0' ,;,0(#1/,0 ,$’ ,6/4' 2 ;"(-A + 3+ 5/4*+, &# ,$"/-*#5&,:
392426
113
252
31
1 !"#$%&' – ()$**"%$ '+',-*$,$ 81
!"#$%&' ( 1( )#*+( %,++"%+")&-./#"& /%0 3 1"& "%’ "2#*' "3"&4/0#"& ( 3( )#*+(, 5/ "%,#6+/)5":
392439
118
254
3
76+,8, ( 1( )#*+( %,++"%+")&-./#"& /%0 5 1"& "%’ "2#*' "3"&4/0#"& 4 3,-468 ( 2( )#*+(, 5/ "%,#6+/)5"
392499
1136
25
3
7, "49&1$ ):)#(5" 69/& +,&%$' 5/#")9(5"#&)#/0 )#,
9936
245
3923
!!"!""
z
zy
zyx
"%$ #, ,%,0, 2%,+,;0./#"& "56)<8 , z 1"& 5/ =&"=,9&168 "'#&1"#")#-)/&8, ,& x, y. >#, "49"0, 1/05/',, 5/ 5&" "'-+,;( =&"=&1")0" %,2 /1#/+/0#"& %--'< )#,' -?"1", %4,)=&,40./#"& ( +:)( #,2 %4,?+*5"#,8:
1 =/5-#& #(8 1"1*8 )2;1,5&=*8 =0'/& 4
32 dou )&#-4&
1 =/5-#& #(8 56#4&"8 )2;1,5&=*8 =0'/& 4
14 dou )&#-4&
1 =/5-#& #(8 1"+*8 )2;1,5&=*8 =0'/& 4
19 dou )&#-4&.
>#, 64;, “@4&A5(#&1-”, #,2 B++('" 5"A(5"#&1,: #(8 @+/C"'=4&'*8 %/-4&$=,2 D&$3"'#,2, 2%-49,2' %,++- %4,?+*5"#" %,2 "'-;,'#"& )#(' /%0-+2)( ;4"55&1E' )2)#(5-#<'. >#, /%$5/',, %,2 /0'"& #, %4$?+(5" 19 #,2 %4E#,2 ?&?+0,2, , #4$%,8 /%0+2)(8 ?40)1/#"& %,+: 1,'#- %4,8 #, ):;-94,', "+;/?4&1$ #4$%, )16F(8:
!"#$%& '())*#*+ *#,-./0+ 123+ /, '#$,+ 4*.5*&1.$&/, '/" 4/,2/0 "2$#(-63),& $2,'*6-(&', *#,-.7. (8* 5#$-/0& 4 *#,-./% ('), 7)'$ 4*.5*&1.$&/, *&9 '#$,+ &* :$2$#&/0& '/& 944/ ;*'9 </-(&'* *#,-.1).
G D&$3"'#,8 %"4,2)&-./& #( +:)( #,2 %4,?+*5"#,8 56)" "%$ 5&" /&=&1* %/40%#<)( (%,2 ;/'&1/:/#"& -5/)"). B)#<, ;4-3/&, $#& ,& *, 5, = C/%/4-',:' #,' < 1"#- 20, ,& 5, =, < C/%/4',:' #,' * 1"#- 30, ,& =, <, * C/%/4-',:' #,' 5 1"#- 40 1"& ,& <, *, 5 C/%/4',:' #,' = 1"#- 50. 7, %4$?+(5", $%<8 /0'"& 3"'/4$, "'-;/#"& )#(' /%0+2)( #,2 ;4"55&1,: )2)#*5"#,8:
1 !"#$%&' – ()$**"%$ '+',-*$TA 82
50
40
30
20
!"!!!"!!!"!!!"!!
#$%&$%&#%&#$&#$%
! "#$%&'()*, ) )+),)* -. /012#3)+)#., .#-#45 2637)8& 9#& (1' +0$2:.21 4&# (1' #2$(1(&, 86'.# () +0$7813& 3. (1' .#2&9;9< .'$* 7)1:1(#4)6 &-9'=2()>, +)> .4%05?.# () 5:0)#23& (;' 4 ?1()63.';' &0#:3='. @ 3A:)--$* ()>, 3. 269/0)') 2>37)8#23$, 2>')B,?.(&# ;* .C<*: D' x2"!!! &#$% , ($(. &+$ (1' 20!"!! &#$% A/)>3.
202 !"!!! &&#$% < 2022 !" &x < 10'" x& . E3)#&, &+$ (#* 588.* .-C#2=2.#* +&,0')>3. 15'" x% , 20'" x$ 4&# 25'" x# . D+$ (#* 4 (.8.>-
(&,.* #2$(1(.*, 3. +0$2:.21 +&,0')>3. 704 '"!!! x&#$% <
7042 '" xx < 35"x 4&# 50& )# ?1()63.')# &0#:3), .,'&#
201535 "'"% , 152035 "'"$ , 102535 "'"# , 251035 "'"&
- ./012 34567892 48:; “</=><?9:; ” 93 ./@AB24: :.:B<85C; ! G.W. Leibniz, 2. 3#& .+#2()8< ()> +0)* ()' l’ Hospital 2(#* 28-4-1693, +0$(.#'. 3#& 3A:)-) /012#3)+),121* (;' &0#:3=' 9#& (1' A4%0&21 9.'#-4=' 2/A2.;', $+;* &40#7=* 9,'.(&# 3. (1 /0<21 (;' 90&335(;'. F* +&-05-.#93& A-;2. A'& 90&33#4$ 262(13& 3 .C#2=2.;' 3. 2 &9'=2()>*, 90&33A') 2(1 3)0%<:
0323130
0222120
0121110
"!!"!!"!!
yx
yx
yx
(1)
!# 2>'(.8.2(A* ()> 2>2(<3&()* -.' .4%05?)>' .-= &0#:3)6* &885 8.#-()>09)6' $+;* 4&# )# -#+8), -.,4(.* +)> /012#3)+)#)6'(&# 2<3.0& 9#& (1' +&052(&21 (;' 2()#/.,;' .'$* +,'&4&. D>(), )# “B.>-)&0#:3),” ($-+;* ()>* &+)4&8., ) Leibniz) -.,/')>' 3. () +0=() B1%,) ()>* (1' .C,-2;21 2(1' )+),& 70,24)'(&# 4&# 3. () -.6(.0), () 90533& 2() )+),) &-'<4)>'. G012#3)+)#='(&* (1 3A:)-) (;' &'(,:.(;' 2>'(.8.2(=', ) Leibniz &+&8.,%.# ()' 59';2() y, &0/#45 &+$ (1' 11 4&# 21 .C,2;21 4&# 4&($+#' &+$ (1' 11 4&# 31. H(2# +0)46+()>' )# .C#2=2.#*:
031.1230.1232.1132.10
021.1220.1222.1122.10
"''!"''!
xx
xx
I(1 2>'A/.#& &+&8.,%.# () x &+$ (#* ->) (.8.>(&,.* .C#2=2.#* (86')'(&* 4&:.3#5 ;* +0)* x 4&# .C#2=')'(&* (& &+)(.8A23&(&) 4&# %(5'.# 2(1' #-2$(1(&:
30.21.1232.20.1131.22.1031.20.1230.22.1132.21.10 !!"!!
<
030.21.1232.20.1131.22.1031.20.1230.22.1132.21.10 "'''!! .
1 !"#$%&' – ()$**"%$ '+',-*$,$ 83
!"#$%& '’ $()* +,( '&-./$ $+,).#.0 )"1 $1$23$0$ '(14&3" 25$ 1$ 67.5 #8'" ), '8')"-$ (1) (, -"%.15'-*9 )"9 ,/0:,('$9 ),( .+$(;"-61,( +01$-3$ ),( '(')&-$),9). <, +/,"2,8-.1, $+,)6#.'-$ ),( Leibniz 67.5 3(/0=9 )" '"-$'0$ )"9 !-
"!#$%&'()!* .1*9 2/$--53,8 '(')&-$),9, %"#$%& )"9 '76'"9 -.)$;8 )=1 '(1).#.')>1 " ,+,0$ +/,38+).5 *)$1 201.)$5 $+$#,5?& *#=1 )=1 $21>-')=1. @)"1 0%5$ .+5'),#& , Leibniz %01.5 3$5 61$ 2.153* 3$1*1$ 25$ ),1 (+,#,25'-* )"9 $+$#.0?,('$9 .1*9 ,+,5,(%&+,). 2/$--53,8 '(')&-$),9, +,( 67.5 .+$/3& $/54-* .;5'>'.=1 25$ )"1 $+$#,5?& *#=1 )=1 $21>')=1.
./ 012304567 89/ 0/ :2;9867 <7 9;6=>1?@?67 A;;0/67 A 7/"'5-,+,0"'" )=1 ,/5:,('>1 25$ )"1 .+0#('" 2/$--53>1 '(')"-B)=1 6251. +/>)" ?,/B $+* ),1 C. MacLaurin ), 1729, $##B " -64,%,9 $()& 6-.51. 21=')& -. ), *1,-$ ),( G. Cramer, , ,+,0,9 )"1 +$/,('0$'. '), C5C#0, ),( “D5'$2=2& ')"1 $1B#('" )=1 $#2.C/53>1 3$-+8#=1 2/$-->1” (1750). E6#,1)$9 1$ +/,'%5,/0'.5 -5$ 3$-+8#" +,( %56/7.)$5 $+* 5 21=-')B '"-.0$ 3$5 67.5 .;0'='" )"9 -,/?&9
022 !""""" xExyDyCxByA
, Cramer 3$)$#&2.5 '’ 61$ 2/$--53* '8')"-$ 5 .;5'>'.=1 -. $21>'),(9 )$ EDCBA ,,,, . F5$ 1$ #8'.5 $()* ), '8')"-$, +./52/B?.5 -5$ -64,%,
(+,#,25'-,8 )=1 $21>')=1 -. 3$)$'3.(& 3#$'-B)=1, )=1 ,+,0=1 , 3,5-1*9 +$/,1,-$')&9 3$5 ,5 $/54-")69 +/,'%5,/0:,1)$5 $+* ),(9 '(1).#.-')69 ),( '(')&-$),9 '8-?=1$ -. 2.153,89 3$1*1.9. G()& .01$5 ,('5$')5-3B " '"-./51& -64,%,9 )=1 ,/5:,('>1 $##B , Cramer %. 7/"'5-,+,5.0 3B+,5, .5%53* *1,-$ & '8-C,#, 25$ )"1 611,5$ )"9 ,/0:,('$9. H #$)5153*9 */,9 determinantem (,/0:,('$) 7/"'5-,+,5&4"3. 25$ +/>)" ?,/B $+* ),1 C.F. Gauss ), 1801 $##B *75 -. )" '"-./51& '"-$'0$. A +/>)" '(')"-$)53& %5$+/$2-B).('" )"9 4.=/0$9 )=1 ,/5:,('>1 6251. $+* ),1 A.L. Cauchy '. -5$ ./2$'0$ ),( +,( %"-,'5.8)"3. ), 1815. A #6;" “,/0:,('$” 7/"'5-,+,5.0)$5 .3.0 -. )" '"-./51& '"-$'0$ .1> .5'B-2.)$5 3$5 " ).)/$2=153& %5B)$;" )=1 '),57.0=1 )"9 -. )" C,&4.5$ )=1 %5-+#>1 %.53)>1:
nnn
n
n
n
a
##
###
#
##
$%%
$$
%
2
222
112
1
21
11
(,5 2 3B4.).9 2/$--69 25$ ), '(-C,#5'-* -5$9 ,/0:,('$9, 7/"'5-,+,5&4"-3$1 25$ +/>)" ?,/B $+* ),1 A. Cayley ), 1841).
@. $1)04.'" $+* )" '"-./51& #,253& '.5/B +$/,('0$'"9, " 611,5$ ),( +0-1$3$ (+&/;., 5'),/53B, -.)$2.16')./" $+* )"1 611,5$ )"9 ,/0:,('$9. <, 2.2,1*9 *)5 " ,/0:,('$ %.1 .01$5 -*1, 61$9 $/54-*9 $##B '('7.)0:.5 $()*1 ),1 $/54-* -. -5$ ).)/$2=153& %5B)$;" '),57.0=1, ,%&2"'. C$4-
1 !"#$%&' – ()$**"%$ '+',-*$TA 84
!"#$# %&' !()*&' #+&,- &'- $."#- &'- ."/�'-, #1(0/2&'&# #34 &'1 &"!, &'- 52$65+%#-. 7 425- matrix (!,&2#, 8#)593"), 35+ %,!(2# :2'%"!535"-($&#" ."(;1<- ="# &'1 *115"# &5+ 3$1#8#, 32>&5(!?#1$%&'8( %( !"# (2=#-%$# &5+ J.J. Sylvester &5 1850, ="# 1# ."#:>2"%&($ ' *115"# &'- 52$65+%#- #34 &'1 &(&2#=>1"8, ."/�' &>1 %&5":($>1 35+ &'1 3#2/=(". @5 1855 5 A. Cayley ("%,=#=( ="# 32<&' ?52/ &5+- 3$1#8(- %&' !()*&' &>1 =2#!!"-8<1 !(&#%:'!#&"%!<1, (1< &5 1858 %&'1 (2=#%$# &5+ “B"# 32#=!#&($# %&' ;(>2$# &>1 !'&2<1”, #1*3&+0( %+%&'!#&"8/ 4)' &' C#%"8, ;(>2$#. D3>- =2/?(" 5 Cayley, %&'1 ".(/ &5+ 3$1#8# *?&#%( &4%5 #34 &'1 *115"# &'- 52$65+%#- 4%5 8#" #34 &'1 #1/=8' (14- C5)"859 &2435+ *8?2#%'- &>1 (0"%<%(>1 byxx !"# $ , dycxy !"# (14- !(&#%:'!#&"%!59, 5 535$5- #3("851$6(" &5 %'!($5 ),( yx %&5 %'!($5 ),( yx ## . E- *1# &*&5"5 &2435 *8-
?2#%'-, 5 Cayley ("%/=(" &51 3$1#8#
%%&
'(()
*dc
b$
8#" !( C/%' &"- "."4&'&(- &>1 !(&#%:'!#&"%!<1 52$6(" &"- 32/0("- &>1 3"-1/8>1 8#" 325%."52$6(" &"- "."4&'&*- &5+-. F.:., #1 &51 325'=59!(15 !(-&#%:'!#&"%!4 #34 &5 ),( yx %&5 ),( yx ## #85)5+;,%(" *1#- 1*5- !(&#%:'-
!#&"%!4- #34 &5 ),( yx ## %&5 ),( yx #### 8#" !( (0"%<%("- yhxgyyfxex #!#"###!#"## , &4&(, 43>- (985)# !352($ 1# #35.(":;($, "%:9-
(":
yfdebxfcex )()( !!!"## $
yhdgbxhcgy )()( !!!"## $ .
G&%" 5 Cayley 52$6(" &51 35))#3)#%"#%!4 3"1/8>1, !( C/%' &5 324&+35 &'- ."#.5:"8,- (8&*)(%'- &>1 .+5 !(&#%:'!#&"%!<1, >- (0,-:
%%&
'(()
*!!!!
"%%&
'(()
*%%&
'(()
*hdgbhcg
fdebfce
dc
b
hg
fe
$$$
.
H&'1 $."# (2=#%$# (3"%'!#$1(" (3$%'- 4&" #+&4- 5 35))#3)#%"#%!4- ($1#" !"# 32/0' !" #$%&!'%#('%&)* 8#;<- 8#" &5 =(=514- 4&" +3/2:5+1 !' !'-.(1"85$ 3$1#8(- 35+ *:5+1 >- ="14!(15 &5 !'.(1"84 3$1#8#. 7 )5="%!4- &>1 3"1/8>1 #1#3&9:;'8( &# (34!(1# :241"# %( !"# #+&5&(), !#;'!#&"8, ;(>2$#, 35+ #35&()($ !*25- (14- (+29&(25+ 8)/.5+ &>1 B#-;'!#&"8<1, &'- I2#!!"8,- J)=(C2#-. @5 1925, 5 W. Heisenberg (C2#C($5 K4!3() L+%"8,- 1932) :2'%"!535$'%( &' ;(>2$# &>1 3"1/8>1 ="# 1# (8-?2/%(" &# !' #1&"!(&#;(&"8/ B#;'!#&"8/ 35+ 3(2"=2/?5+1 &# ?#"14!(1# &'- 8C#1&"8,- !':#1"8,-, (1< #2=4&(2# ' :2,%' &>1 3"1/8>1 (3(8&/;'8( 8#" %( /))(- (3"%&,!(-.
2 !"#$%"&'" $(")!'" (1)
2.1 * +,,'"$ -'. !"#$%"&'. $(")!'.
+/012324 ! "#$%&'()*+ ,-. $%)+"%/0. +(%1$0. &23*43,+% 5,#. 6(&56713%+ 36*4'5#8 ,-.
39%5053-. 3&' :+1$&;. <. 5,#. 023 !""" !x"x#ax 1=5&'$3 a
#yx
3#! /+%
3/,34=5&'$3 ,%8 6(793%8, ,>,3 6(&/;6,3% $%+ 39*5-5# ,#8 $&(2?8 qpxx "!3 .
@,%8 +(A=8 ,&' 16&' +%0.+ &% B,+4&* +4)3:(%5,=8 S. del Ferro /+% N. Tartaglia +.+/74'C+. $%+ $=1&"& 36*4'5#8 ,=,&%-. 39%5053-., 6&' $3 5#$3(%.> 5'$:&-4%5$> %5&"'.+$3* $3 ,&. ,;6&
33
22D
qD
qx #""! , >6&'
22
32$%
&'(
)#$%
&'(
)!pq
D .
@,#. 63(*6,-5# 6&' # “"%+/(*.&'5+” D 3*.+% 13,%/?, & ,;6&8 +',>8 "*.3% +$=-5-8 $%+ (*D+ ,#8 39*5-5#8. E%+ 6+(7"3%)$+, 5,#. 2893 "! xx 3*.+% 169!D /+% & ,;6&8 "*.3% 4!x , 6&' 3*.+% # $&.+"%/? 6(+)$+,%/? (*D+. F%+6%5,01#/3, >-
$-8, ,>,3 =.+ 2+%.>$3.& ,343*-8 "%+2&(3,%/> +6> ,#. 63(*6,-5# ,-. 39%5053--. 2&' :+1$&;: G67(A&'. 39%5053%8 $3 6(+)$+,%/=8 (*D38, >6-8, )%+ 6+(7"3%)-$+, # 4153 "! xx , 6&' =A3% $%+ 6(&2+.? (*D+ ,& 4 (&% 74438 ";& 3*.+% &%
32 "# , 32 ## ), +447 # "%+/(*.&'5+ D 3*.+% +(.#,%/?! H ,;6&8 5,# 5')/3-/(%$=.# 63(*6,-5# "*.3%
33 12121212 ##"#"!x . (1)
I6-8 3*.+% 2+.3(>, &% $+1#$+,%/&* :(=1#/+., 3"0, $6(&5,7 53 =.+ "*4#$$+: ? 1+ =6(363 .+ 3)/+,+43*C&'. ,# $=1&"& ,-. Ferro-Tartaglia -8 )3.%/? $=1&"& 36*4'5#8 39%5053-. 3&' :+1$&; ? .+ "3A,&;. >,% =.+8 5')/3/(%$=.&8 +(%1$>8, >6-8 ,& 4, $6&(3* .+ 3/2(+5,3* $3 6+(+5,753%8 6&' 63(%=A&'. ,3,(+)-.%/=8 (*D38 +(.#,%/0. +(%1$0.. ! "3;,3(# 76&C# 2+%.>,+. +"%+.>#,# +447 +',> "3.
(1) To /3274+%& +',> =A3% 4#213* +6> ,& :%:4*&: «J+1#$+,%/7 KL M79#8 N.%+*&' O'/3*&' P3,%/?8 Q+,3;1'.-5#8» ,-.: <"+$>6&'4&' O., K%5/+"&'(7/# K., E+:+47 F., R&4;D&' E. /+% @:=(/&' <.
2 !"#$%"&'" $(")M'"
86
!"#$%&'! ()* !+,-"./0 (1* ,2/!3-&45* #-67!1* '! (8(.&!9 #,-,'(6'!&9. :(, "8', (.; 16.; ,&5*, . R. Bombelli, 46*.*(,9 (.2")-89 ;#.<8'!&9, 3-04! $(&
&'=>!& ! " 1212123
#$%#$ 4,& ! " 1212123
##%## . ?*(&4,<&'(5*(,9 ,;(89 (&9 &'$()(!9 '()* (1) #-.4>#(!& ,"8'19 $(& 4%x , %)2,%0 (. ,%&,*$)(. /@*!(,& #-,/",(&4$()(,! A& ,-&<".@ ()9 ".-+09 i!a $ "! 1#%i , #.; .*."6'()4,* ,-=&46 +,*(,'(&4.@ 4,& ,-/$(!-, "&/,%&4.@, 8/&*,* ,#$ ($(! ,*,#$'#,'(. !--/,2!@. (1* B,<)",(&45* 4,& (1* !+,-"./5* (.;9 '(&9 622!9 !#&'(0"!9. A J. Hadamard, . .#[email protected] (. 1896 ,#8%!&7! "! =-0') ()9 "&/,%&409 ,*62;')9 (. “<!5-)", (1* #-5(1* ,-&<"5*”, 8/-,C! $(&:
“" #$%&'()&*+', -+)(', .%/(*#. #* -0' .123*4*, #&' 5*-6' &7% 5+.8(.&49:% 5*+%/ (;#. .5) &' 5*-6' &7% (48.-49:%”.
*+ ,-.+/+ &&&& 01. !2345267. $829:7.
D*1-@E.;"! $(& ) %!;(!-.36<"&, !7@'1') "! ,-*)(&40 %&,4-@*.;', %!* 8=!& 2>-
') '(. '>*.2. ! (1* #-,/",(&45* ,-&<"5*. F&%&4$(!-, ) !7@'1') 12 #%x %!* 8=!& 2>') '(. '>*.2. ! (1* #-,/",(&45* ,-&<"5*, ,+.> (. (!(-6/1*. 46<! #-,/",(&4.> ,-&<".> !@*,& ") ,-*)(&4$9 ,-&<"$9. D&, *, 7!#!-6'.;"! ()* “,-%;*,"@,” ,;(0, %&!;->*.;"! (. '>*.2. ! '! 8*, '>*.2. ", (. .#.@. *, 8=!& (&9 @%&!9 #-67!&9 "! (. !, (&9 @%&!9 &%&$()(!9 (1* #-67!1* ,;(5* 4,& '(. .#.@.
*, ;#6-=!& "@, (.;26=&'(.* -@E, ()9 !7@'1')9 12 #%x , %)2,%0 8*, '(.&=!@. i ,
(8(.&., 5'(! 12 #%i . :>"+1*, "! (&9 #,-,%.=89 ,;(89 (. %&!;-;"8*. '>*.2. "
<, 8=!& 19 '(.&=!@,:
' G2.;9 (.;9 #-,/",(&4.>9 ,-&<".>9 ' G2, (, '(.&=!@, ()9 ".-+09 i! , #.; !@*,& /&*$"!*, (1* '(.&=!@1* (.; ! "! (. i 4,&
' G2, (, ,<-.@'",(, ()9 ".-+09 i!. $ , "! . 4,& ! #-,/",(&4.>9 ,-&<".>9. H, '(.&=!@, (.; " 28/.*(,& !"#$%"&'( $)"*!'( 4,& (. " +,-'.' /0- !"#$%"-&1- $)"*!1-. F#."8*19:
H. '>*.2. " (1* "&/,%&45* ,-&<"5* !@*,& 8*, ;#!-'>*.2. (.; ';*$2.; !
(1* #-,/",(&45* ,-&<"5*, '(. .#.@.:
' F#!4(!@*.*(,& .& #-67!&9 ()9 #-$'<!')9 4,& (.; #.22,#2,'&,'".> 8('&, 5'(! *, 8=.;* (&9 @%&!9 &%&$()(!9 $#19 4,& '(. !, "! (. ")%8* (0) *, !@*,& (. .;%8(!-. '(.&=!@. ()9 #-$'<!')9 4,& (. 8*, (1) (. .;%8(!-. '(.&=!@. (.; #.22,#2,'&,'".>,
' I#6-=!& 8*, '(.&=!@. i (8(.&., 5'(! 12 #%i ,
' J6<! '(.&=!@. z (.; " /-6+!(,& 4,(6 ".*,%&4$ (-$#. "! () ".-+0
i!.z $% , $#.; (!., !.
2 !"#$%"&'" $(")!'"
87
! "#$%&'( !!" !"i!" ,, )*+&, &#%,-./ 0,1, 2"3) !"#$%"&' $(")!'. 4*+&, (
'5+6)'( 758 &%,63.+, 189 :%&;3&1,#85 " #&, 189 i! , 18+ 8:8*8 8+83<=893) *$+,$-,"&' $(")!'. > " 2";)1&, .($#!$,"&' !/(01 189 z #&, '(3),.+)1&,
)Re(z , )+. 8 ! 2";)1&, *$+,$-,"&' !/(01 189 z #&, '(3),.+)1&, )Im( z . 4:,-:2"8+, '18 " #<6) :%&;3&1,#0/ &%,630/ " )#$%<=)1&, ?/ i" 0" , )+. #<6) $&-
+1&'1,#0/ &%,630/ i! )#$%<=)1&, ?/ i!"0 .
@1( '9+"A),&, 01&+ 2"3) 8 3,;&7,#0/ i!"z "# , )++8853) 01, 8, " #&, ! )*+&, :%&;3&1,#8* &%,638* #&, 18 ;);8+0/ &910 7) 6& 18+*=)1&, ,7,&*1)%&.
"*+,-,. !/0.1/234 $5/6734 4:),7B #<6) 3,;&7,#0/ &%,630/ z ;%<$)1&, 3) 38+&7,#0 1%0:8 '1( 38%$B
i!" " , 758 3,;&7,#8* &%,638* i!" " #&, i#$ " )*+&, *'8,, &+ #&, 30+8 &+ $" #
#&, #! # . C(2&7B ,'A5),:
$"#" i#$i!" $" # #&, #! # .
4:83"+?/, ):),7B i000 "# , "A893)
$#" 0i!" 0#" #&, 0#! .
D)1< 18+ 8%,'30 1(/ ,'01(1&/ 3,;&7,#.+ &%,63.+ 7(3,89%;)*1&, 18 )%.1(3& &+ 7,&1<''8+1&, 8, 3,;&7,#8* &%,638*. E+?%*=893) 01,, #&1< 1(+ ):"#1&'( &:0 18
# '18 %, 8, :%<F),/, ( 7,<1&F( #&, 8, ,7,01(1)/ &91.+ 8, 8:8*)/ ,'A589+ '18 #,
3)1&$"%8+1&, #&, '18 %. G& *7,& '93-&*+89+ #&, ;,& 1,/ ):)#1<'),/ &:0 18 % '18
& #&, &:0 18 & '18 '. @1(+ ):"#1&'(, 03?/, &:0 18 ' '18 (, )+. 8, :%<F),/ #&, 8, ,7,01(1)/ &91.+ :89 ,'A589+ '18 ' )F�+ +& ,'A589+ #&, '18 (, )+ 18518,/ ( 7,<1&F( #&, 8, ,7,01(1"/ 1(/ 7) 3)1&$"%8+1&,.
#8978,5/2: ;.5<*,.*- !/0.1/234
K<6) 3,;&7,#0 &%,630 i!" " 3:8%853) +& 18+ &+1,'18,A*'893) '18 '(3)*8 ),( !"M )+0/ #&%-
1)',&+85 ):,:"789. H22< #&, "%&'(&)*+,-, #<-
6) '(3)*8 ),( !"M 189 #&%1)',&+85 &9185 ):,-:"789 3:8%853) +& 18 &+1,'18,A*'893) '18
3,;&7,#0 i!" " . G8 '(3)*8 M 2";)1&, 2"&'+$
189 3,;&7,#85 i!" " . A+ 6"'893) i!"z "# ,
101) 18 '(3)*8 ),( !"M 3:8%853) +& 18 '93-
-82*=893) #&, 3) )(zM .
I+& #&%1)',&+0 ):*:)78 189 8:8*89 1& '(3)*& )*+&, ),#0+)/ 3,;&7,#.+ &%,63.+ 6& &+&$"%)1&, ?/ !"#$%"&' 2.3.2%0. > <F8+&/ xx) 2";)1&, .($#!$,"&'1 450-
+$1, &$85 &+B#89+ ') &910+ 1& '(3)*& )0,("M :89 )*+&, ),#0+)/ 1?+ :%&;3&1,-#.+ &%,63.+ i"" 0"# , )+. 8 <F8+&/ yy) 2";)1&, *$+,$-,"&'1 <F8+&/, &$85
!
. a x
y
M(",!) / 0(z)
1
2 !"#$%"&'" $(")M'"
88
!"#$%&" '( !&)*" )! '+,(-! ),0( !M .%& (-"!/ (/$*"(0 )1" 2!")!')/$3"
i!i! !" 0
4"!0 ,/5!6/$*0 i!"z !" .!7/')8"()!/ (.-'+0 $!/ ,( )+ 6/!"&',!)/$# !$)-"!,
OM , )%& '+,(-%& ),( !"M .
2.2 *($+,"- -.' -/0'1' #### .20 !"#$%"&20
9:,21"! ,( )%" %7/',* )%& #, + .7*';('+ $!/ % .%<<!.<!'/!',*0 6:% ,/5!6/-$3" !7/;,3" 5-"%")!/ *.10 !$7/=30 $!/ %/ !")-')%/>(0 .78?(/0 ,( 6/3"&,!
x!" ! ')% $, *.%& =@=!/! !")- 5/! x @>%&,( )% i . 4)'/:
% A/! )+" !"#$%&$' 6:% ,/5!6/$3" !7/;,3" i!" ! $!/ i#$ ! @>%&,(:
i34563i54i6 )()()()( !!!"!!! .
A/! .!786(/5,!, iiii 28)64()53()65()43( &"&!!"&!! .
% A/! )+" ()(*"&$' )%& ,/5!6/$%: !7/;,%: i#$ ! !.* )%" i!" ! , (.(/6# % !-
")-;()%0 )%& ,/5!6/$%: i#$ ! (-"!/ % ,/5!6/$*0 i#$ && , @>%&,(:
i#!$"i#$i!"i#$i!" )()()()()()( &!&"&&!!"!&! .
B+<!6#
i34563i54i6 )()()()( &!&"!&! .
A/! .!786(/5,!
iiii 102)64()53()65()43( !&"!!&"&&! .
C" ),(1 !"M $!/ ),(2 #$M (-"!/ %/ (/$*"(0 )1" i!" ! $!/ i#$ ! !")/')%->10 ')% ,/5!6/$* (-
.-.(6%, )*)( )% 8;7%/',!
i#!$"i#$i!" )()()()( !!!"!!!
.!7/')8"()!/ ,( )% '+,(-% ),( #!$"M !! .
D.%,@"10, 21 OMOMOM !" , 6+<!6#:
% x
y
M("+$,!+#)
M1(",!)
M2($,#)
2
2 !"#$%"&'" $(")!'"
89
“! "#$%&'($)#*+ $*),%$ )-& $./-,'($)-0 )1% (#2$"#*3% *i+ ! *$# i,- !
4,%$# )- 5./-#'($ )1% "#$%&'($)#*3% $*),%1% )-&0”. !"#$%&, % '()*+,-
i!"#$i!#i"$ )()()()( "!"#!"!
"),($.-/0.)( 10 .+ $%10#+ ),( !"#$N "" .
!"+12/3&, 21 OMOMON "# , '%4)'5:
“! "#$%&'($)#*+ $*),%$ )60 "#$7-/50 )1% (#2$"#*3% *i+ ! *$# i,- ! 4,%$# 6 "#$7-/5 )1% "#$%&'($)#*3% $*),%1% )-&0”.
$ 6() .+/ 8-99$89$'#$'(: '7+ 1(8)'(9:/ i"$ ! 9)( i!# ! 2;+<10:
#!!!#!!!#!! ))(()()())(( i!i"i"#i$!$#i!#i"i!#$i!#i"$
i"#$!"!$#"!i"#i$!$#i"!i"#i$!$# )()(2 !!"#"!!#!!!# .
=%4)'5,
i*-+,*,+-,i-*i+ )()())(( !!"#!! .
6() "),-'0(81),
iiiiiii 239)1820()2415(24201815)65()43( 2 !#"!!#"!"#"%! .
!('(9>.0,), 2;+<10: 22))(( "$i"$i"$ !#"! . ? ),(@1>& i"$ " 4280.)( '&;&2+0
.+< i"$ ! 9)( $<1A+4#B0.)( 10 i"$ ! . =%4)'5,
*i+*i+ "#! .
!"0('5 0#/)( 9)( i"$i"$ !#" , +( i"$ ! , i"$ " 428+/.)( '&;&24,0 (#2$"#*-,.
$ C24+&, 8() /) 09*,-$+<10 .+ 869,*- i!#i"$
!!
, >"+< 0&! i!# , $.% 1+,*5
i%& ! , "+44)"4)$(-B+<10 .+<& >,+<& .+< 94-$1).+& 10 .+ $<B<85 .+< "),+-
/+1)$.5 9)( 2;+<10:
i!#$!"#
!#"!$#
!#i$!"#"!$#
i!#i!#i!#i"$
i!#i"$
222222
)()(
))((
))((
!"
!!!
#!
"!!#
"!"!
#!!
.
'
(3("#,"!)
)($"#,""!)
(2(#,!)
(1($,")
x
y 3
2 !"#$%"&'" $(")M'"
90
!"#$%&,
i*+,*-+
*+-*,+
*i+-i,
2222 !"
!!!
#!!
.
'($ )$*+%,(-.$: iiiii
ii
ii
i
i
10
7
10
1
10
71
91
362
)31)(31(
)31)(2(
31
2 2
!"#!"
#!
!!!#
!"!!
#"!
.
%./,01 !2+,*234.
/( %01+.,(2 ,132 .(-$%(456 $*(7.56 z ., ,4789" $48*$(5 5*:;519$( $4*(<=2 3)>2 4$( ?9502 )*$-.$9(4562, %"#$%& 5*:;50.,:
zz #1 , ,...,2 zzz $# 4$( -,1(4+ zzz !! $# "1 ,
-($ 4+7, 7,9(45 $48*$(5 ! , ., 1%! . @):?"2, $1 0&z , 5*:;50.,
!!
zzz
1,10 ## " -($ 4+7, 7,9(43 $48*$(5 !.
'($ 9(2 %01+.,(2 9>1 .(-$%(4=1 $*(7.=1 (?A6501 5( :%(,2 (%(39"9,2 )50 (?A6501 4$( -($ 9(2 %01+.,(2 9>1 )*$-.$9(4=1 $*(7.=1. B%($:9,*$ -($ 9(2 %01+.,(2 950 i
8A50.,: iiiiiiii "##"### 23210 1,,1, .
C9" ?018A,($, )$*$9"*56., 39( ,:1$(:
iiiiiiiiiiiiiiiii "#$##"#$###$#### 3347224645224 1,11,1,1 ,
%"#$%&, .,9+ 95 4i 5( 9(.82 950 !i ,)$1$#$.<+1519$(. D*$, -($ 1$ 0)5#5-:?50-
., ?0-4,4*(.81" %61$." 950 i , -*+E50., 951 ,4789" ! ?9" .5*E& "#! !# 4 ,
3)50 # 95 )"#:45 4$( " 95 0)3#5()5 9"2 ,04#,:%,($2 %($:*,?"2 950 ! ., 95 4,
5)39, 8A50.,:
''(
'')
*
#"###
###### !
3$1,
2$1,1-
1$1,
0$1,1
1)( 444
"i
""i
"
iiiiiiii ""#"#"#"#!
'($ )$*+%,(-.$:
1224314 "### !$ iii
iiii "### !$ 334419
1004416 ### !$ iii
iiii ### !$ 115421 .
"*2561678 9:;:+</ @),(%& 5( ?0;0-,:2 .(-$%(45:, 3)>2 7$ %56., ?9(2 ,)3.,1,2 )$*$-*+E502, .$2 %(,045#61501 ?9" .,#89" 9>1 .(-$%(4=1 $*(7.=1, 7$ $1$E,*756., (%($(98*>2 ?, $09562.
2 !"#$%"&'" $(")!'"
91
! !"# $%&'(%)* +,-,+(# #% +%)*.+/ ),( !"M )'% ),( !"M "# (0# 1232&4. $%&'(%)4. i!"z $% )'% i!"z "% +-.'% 15$+-' 12$$+"6%)7 8/ ,6#/ "#.
,6'&$'"%)* 79#.'.
! :%' (0# 1232&+-/ $%&'(%)#0/ '6%;$#0/ i!"z $% )'% i!"z "% $,#6#0$+ +0)#<', $+ +)"=<+15 "8. ,679+8., .' (%',%-
1"41#2$+ *"%:
*zz 2%$
+izz 2%" .
! >. i!"z $%1 )'% i#$z $%2 +-.'% (2# $%&'(%)#- '6%;$#-, "*"+:
1. 2121 zzzz $%$
2. 2121 zzzz "%"
3. 2121 zzzz &%&
4. 2
1
2
1
z
z
z
z%''(
)**+
,.
?% %(%*"5"+/ '2"=/ $,#6#0. .' ',#(+%@"#0. $+ +)"=<+15 "8. ,679+8.. :%' ,'-67(+%&$' =@#2$+:
i#!$"i#$i!"zz )()()()(21 $$$%$$$%$
21)()()()( zzi#$i!"i#!$" $%"$"%$"$% .
?% ,'6',7.8 %(%*"5"+/ 1 )'% 3 %1@0#2. )'% &%' ,+6%11*"+6#2/ ',* (2# $%&'(%-)#0/ '6%;$#0/. A-.'% (5<'(B:
,, zzzzzz $$$%$$$ -- 2121
,, z...zzzzz &&&%&&& 2121 ... .
C(%'-"+6', '. +-.'% zzzz % %%%% ...21 , "*"+ 5 "+<+2"'-' %1*"5"' &-.+"'%:
,, zz )()( %
& x
)(zM #
M(z)
y 4
2 !"#$%"&'" $(")M'"
92
!"# $#%&'(")*#,
33
3
54
32
54
32
54
32!"#
$%&
'(
)**+
,
--.
/!"#
$%&
('
)**+
,
--.
/!"#
$%&
('
i
i
i
i
i
i.
*+,-./0 102 *3,/4/02 02 )(( 56z7z !" !!!!056,7, 879 017
+$("', 12 ')i -#" 1)( 22 '))' ii , . (/0121. 12 ')x 34(" 156 178696 2 528 *")#'"-:8 #%";*:8 '76 971("<, 5"< ix )1 -#" ix ')2 . =*602<, . (/0121.
42 ')x 34(" 156 178696 528 *")#'"-:8 #%";*:8 '76 971("<, 5"< ix 21 ) -#" ix 22 ') , #>67
izizizz 2)2(44 22222 )3)34)3') , iz 2') .
+7-69#, ?*2<, *$6%67*( 8# '"#$"15:16@*( ?5" -#" -&;( (/0121. '(75(%6@ A#;*67 *( $%#)*#5"-67< 1@85(9(153< 34(" $&85# 971. 156 178696 C. B%&)*#5", 3152 . (/0121.
02 )(( !z"z# , *( !0!"# ,, -#" 01# .
+%)#C?*#15( ?$2< 15.8 #850156"4. $(%0$521. 156 ! -#" 5. *(5#14.*#-50C6@*(, *( 5. *3;6'6 1@*$9,%21.< 5(5%#):828, 15. *6%>,:
2
2
42 #$
#"
z )!"#
$%& ( ,
?$6@ #!"$ 42 ') . '"#-%086@1# 5.< (/0121.<. D51", 346@*( 5"< (/,< $(%"$5:-
1("<:
5 06$ . T?5( . (/0121. 34(" '76 $%#)*#5"-3< 971("<: #$"
z2
2,1
7')
5 0)$ . T?5( 34(" *"# '"$9, $%#)*#5"-, 971.: #"
z2
')
5 08$ . T?5(, ($("',
2
2
22
22 2)2(
)(
4
))(1(
4 !!"
#$$%
& ')
')
'')
#$%
#$i
#$
#$
, . (/0121. )%&-
>(5#": 22
22 !!"
#$$%
& ')!
"
#$%
& (#$i
#"
z .
E%# 6" 971("< 5.< (08#":
#
$i"z
22,1
'7') , (1)
6" 6$60(< (08#" 1@C@)(0< *")#'"-60 #%";*60. !"# $#%&'(")*#, . (/0121. 0652 )(' zz 34(" 012425 6)')$ -#" 6" 971("<
5.< (08#": 32
151 )
()z , 2
2
152 )
')z . F*2<, . (/0121. 0222 )(' zz 34("
2 !"#$%"&'" $(")!'"
93
0484 !"#"#! !"# $# %&'(#) *+) (,-"# $# './.0(,) 1#0"2#!$, "3#41$,:
ii
z $#$
# 12
421 , i
iz "#
"# 1
2
422 .
!"#"$%#%&% 5"3"*+3$&1( 6*# !"# (27 #'8&$.- $# '89'(#):
*+
zz"
#$ 21 !"# *,
zz #21 .
-.$(!'#-/
1. '() *(+ ,(-./01+ *(23+ */4 51*(6/7 )630)(/4 0 8) 49/:/;(<*1= */ -50/(-<2)
0iiiiS $$$$# %32 .
12/3
:# ;3$'4(*9$# *$. "43$,'1"*$) 98$.- ;%<4$) & !"# (,-"# 2#"2$8#!$, 63$# 0(=-
1(*3#!<) ;3$62$. 1( ;37*$ 63$ i !"# %60$ (;,'+) i . >;$19-=), 1
1
""
#i
iiS
"
,
$;6*(, %60= *+) #'6*+*") #$" $# 4 *+) (.!%(,2(#") 2#",3('+) *$. & 1( *$ 4,
98$.1( *#) (?<) ;(3#;*7'(#):
' 0## . @6*( & (#4 , $;6*( S ii
#""
#1 1
10
' 1## . @6*( & (# $4 1 , $;6*( S ii
ii#
""
#1
1
' 2## . @6*( & (# $4 2 , $;6*( S ii
i
i
i
i
i ii#
" ""
#""
#"
#$
# " $1 1
1
2
1
2
1
2 1
21
( )
' 3## . @6*( & (# $4 3 , $;6*( S ii
i
i
i#
" ""
#""
# "1
1
1
11 .
2. >) ?0151= */ <78/:/ *@8 1(6A8@8 *@8 2(;),(6B8 z <*(+ 910(9*B<1(+ 6)-
*- *(+ /9/=1+ / )0(52A+ iz
z
2
1
""
1=8)( )) .)8*)<*(6A+ ?) 90);2)*(6A+.
12/3
A- z x yi# $ , *6*(:
2 !"#$%"&'" $(")!'" 94
z
z i
x yi
x y i
!!
"! ## !
1
2
1
2
( )
( )
22
22
)2(
)22()2(
!#!##!#!
"yx
yxiyyxx
iyx
yx
yx
yyxx2222
22
)2(
22
)2(
2
!#!#
#!#!#!
" .
!"#$%&'(:
)) * )+,-$.( z
z i
!!
1
2 /0&), 1)&2)32,4.(, )& 4), $.&# )& 0
)2(
222
22
"!#!#!
yx
yyxx, 56-
7)58, )& 4), $.&# )& 0222 "!#! yyxx 4), 0)2( 22 $!# yx 8, ,3#59&)$),
4
5)1(
2
1 2
2
"!#%&'
()* ! yx 4), ( , ) ( , )x y $ 0 2 .
:+), 2# 39# 2'& /,4.&'& 2#; z /0&), 2) 36$/0) 2#; 4947#; $/ 4%&2+#
%&'
()*" 1,
2
1K 4), )420&)
5
2, $/ /<)0+/36 2# 36$/0# A( , )0 2 .
=) * )+,-$.( z
z i
!!
1
2 /0&), "+)>$)2,4.(, )& 4), $.&# )& 0
)2(
2222"
!#!#
yx
yx, 567)58,
)& 4), $.&# )& 2 2 0x y# ! " 4), 0)2( 22 $!# yx .
:+), 2# 39# 2'& /,4.&'& 2#; z /0&), 2) 36$/0) 26( /;-/0)( $/ /<03'36 2 2 0x y# ! " , $/ /<)0+/36 2# 36$/0# A( , )0 2 .
$*&+*,"*
!" #$!%!&
1. ?) =+/02/ 2,( 2,$%( 2#; +! ,, @32/ # )2)(3( ii!z !#" &) /0&),: )) "+)>$)2,4.( )+,-$.( =) 1)&2)32,4.( )+,-$.(.
2. ?) =+/02/ 2#;( "+)>$)2,4#9( )+,-$#9( yx, >,) 2#;( #"#0#;( ,3A9/,:
)) iiyxyx !"!## 3)()(
=) iixxx #"!#!# 2)3(63 22
>) yiyxi !#"! )23(279 .
2 !"#$%"&'" $(")!'"
95
3. !"# $%&'(%)+* +,-,+(# .' /0$+%1/+"+ "%2 +%)*.+2 )'% "%2 (%'.3/$'"%-)42 ')"-.+2 "5. $%&'(%)1. '6%7$1.: i!1 , 1, i, i2" , i43! , i43" , 5, 0.
4. 8' ,+6%&69:+"+ &+5$+"6%)9 "# /;.#<# "5. +%)*.5. "5. $%&'(%)1.
'6%7$1. z ,#3 %)'.#,#%#;. "%2 /=4/+%2: ') ># ,6'&$'"%)* $46#2 "#3 z +-.'% -/# $+ $0(4.. ?) ># @'."'/"%)* $46#2 "#3 z +-.'% -/# $+ $0(4.. &) ># ,6'&$'"%)* $46#2 "#3 z +-.'% -/# $+ "# @'."'/"%)* "#3 $46#2.
5. !+ )'7+$%9 ',* "%2 ,'6')9"5 ,+6%,"1/+%2 .' +)"+<4/+"+ "%2 ,69A+%2
,#3 /0$+%1.#."'% )'% .' &69:+"+ "# ',#"4<+/$' /"0 $#6@B i!" !
') )27()64( ii "!!" ?) )46()23( ii !""
&) )35()78()43( iii !!""!! () )54)(23( ii !!
+) )6(3 ii ! /") )34)(34( ii "!
C) )2)(3( iii "! .
6. 8' &69:+"+ "#32 ,'6')9"5 $%&'(%)#;2 /"0 $#6@B i!" ! :
') i"1
1 ?) 6i
&) 122 !! ii () 2)31( i!
+) i
i
"!
2
3 /")
21
26
i
i
!
".
7. 8' ?6+-"+ "#32 #yx, $, &%' "#32 #,#-#32 %/=;+%:
') yixiyxi "%!"" )()23( 2
?) iiyxi
i!%
!!&
'(
)*+"!
11
1
12
&) 12)2(2)2)(23( "!"%"" iiyxiyxi .
8. 8' 3,#<#&-/+"+ "%2 ,'6'/"9/+%2:
') 56463626166 iiiiii !!!!!
?) 1023754111
1111
iiii"!" .
9. D#%#2 +-.'% # z , *"'.:
') iz 75!"% ?) iz 94""% &) iz 4%
() 11%z +) iz "% /") 0%z .
2 !"#$%"&'" $(")!'" 96
10. !" #$%"& '())"*+,"& )#$+$-. ./ #+$0-1$(. /#2 *3. "%02./ *$( )%-4/5%0$- yixz !" $% "%02."& *6. )%4/5%07. zz #, 0/% z# ;
11. 8. i
iz
47
951 !
#" 0/%
i
iz
47
952 #
!" , ./ 5",9"*" 2*% $ 21 zz ! ",./% #+/4)/*%-
02& /+%:)2&, ".7 $ 21 zz # ;/.*/'*%02& /+%:)2&. 12. </ #"+%4+=1"*" 4"6)"*+%0= *$ '-.$>$ *6. "%02.6. *6. )%4/5%07.
/+%:)7. z #$( %0/.$#$%$-. *%& #/+/0=*6 '?@'"%&: /) izz 6"# A) 22 zz " 4) 22 zz #" 5) zz #"2
13. </ >-'"*" '*$ '-.$>$ *6. )%4/5%07. /+%:)7. *%& "9%'7'"%&:
/) 0232 "!# xx A) 0322 "!# xx 4) 11"!
xx .
14. 8. )%/ +,B/ *3& "9,'6'3& 02 2 "!! !x"x , 2#$( !$!", , ",./% i23! ,
./ A+",*" *%& *%)@& *6. " 0/% !.
!" #$%&%'
1. 8. !"# ,, 0/% $ ",./% #+/4)/*%0$, /+%:)$,, ./ "9"*='"*" #2*" *$ #3-
>,0$ i$!i"#
!!
",./% #+/4)/*%02& /+%:)2&.
2. 8. 2
31 iz
#" , ./ A+",*" *3. *%)C *3& #/+='*/'3&
zz #2
1.
3. </ A+",*" *3. *%)C *3& #/+='*/'3& 2020 )1()1( ii ##! .
4. D2'"& 5%/;$+"*%0@& *%)@& )#$+", ./ #=+"% 3 #/+='*/'3 %% ii #! ;
5. </ >-'"*" *%& "9%'7'"%& /) 2zz " A) 3zz " .
6. E'*6 $ )%4/5%02& z )" 0%z . </ 5",9"*" 2*% $ z
z
z
z! ",./% #+/4)/*%-
02& 0/% 2*% 22 &!&#z
z
z
z.
7. </ /#$5",9"*" 2*% 0)()( 1010 "#!! i#"i"# , 2#$( !$"#, .
8. /) F%/ @./ )%4/5%02 /+%:)2 z ./ /#$5",9"*" 2*%:
2 !"#$%"&'" $(")!'"
97
! ! z "#$%& '(%)*%+&,-., %$ ,%& *-$/ %$ zz "
! ! z "#$%& 0%$+%1+&,-., %$ ,%& *-$/ %$ zz #" .
2) 3$ 1
1
1
zz " ,%&
2
2
1
zz " ,%& 121 #$% zz , $% %'/4"#5"+" -+& / %(&6-
*-. 21
21
1 zz
zzu
&&
" "#$%& '(%)*%+&,-., "$7 / %(&6*-. 21
21
1 zz
zzv
&#
" "#-
$%& 0%$+%1+&,-..
9. 8% 2("#+" +/ )"9*"+(&,- +-'/ +9$ "&,-$9$ +9$ *&)%4&,7$ z )&% +/:. /'/#/:. &1;<"&:
%) )Re(51
Re zz
z "'()
*+, & 2) )Im(3
1Im z
zz #"'
()
*+, & .
2.3 !*+(' !"#$%"&', $(")!', =1+9 ),( yxM > "&,-$% +/: *&)%4&,/< yixz &"
1+/ *&)%4&,- "'#'"4/. !(#?/:*" 9. !"#$% +/: z +>$ %'-1+%1> +/: M %'- +>$ %(;@ O , 4>A%4@
22|||| yxOMz &""
B&% '%(C4"&)*%, 5)4(3|43| 22 "#&"# i .
D+%$ / *&)%4&,-. z "#$%& +>. */(0@. !-"&" xixz 0 , +-+" ||0|| 22 xxz "&" , '/: "#$%& > )$91+@ *%. %'-A:+> +&*@
+/: '(%)*%+&,/< %(&6*/< x.
3$ yixz &" , +-+" yixz #" ,%& yixz ##"# . E'/*E$9.,
! |||||| zzz #""
! zzz %"2||
!& "'-*"$". &4&-+>+". %$%0E(/$+%& 1+&. 1;E1"&. '/: 1:$4E/:$ +/ )&$-*"$/ ,%& +/ '>A#,/ *&)%4&,7$ *" +% *E+(% +/:. ,%& "#$%& #4&". *" +&. %$+#1+/&;". &4&-+>+". +9$ %'-A:+9$ +&*7$ '(%)*%+&,7$ %(&6*7$. 3$ 21 , zz "#$%& *&)%4&,/# %(&6*/#, +-+"
x
M(x,y)
|z |
!
"
a
y 5
2 !"#$%"&'" $(")!'" 98
! |||||| 2121 zzzz "#"
! 2
1
2
1
z
z
z
z#
!"#$%&'(, )*+,%-:
22
21
2212121 |||||||||||| zzzzzzzz "#"$"#"
22112121 ))(( zzzzzzzz """#""$
22112121 zzzzzzzz """#"""$
.&(, -/-(01 2 '-3-,'&4& (56'2'& (5*7-(, 8& (5*7-( .&( 2 (5+079&%2 &"*(.1.
:9#3+$& &/+0-(.97-'&( .&( 2 0-7'-"2 (0(6'2'&. ;-9(.#, &/+0-(.97-'&( 6'(
|z|...|z||z|zzz| ** """# 2121 |...
.&( -(0(.6'-"& *|z||z| #! .
<)3+=, &/6 '2 $9>5'1 %&= '"($>9(.1 &9(-56'2'& .&( &/6 '2 $->%-'"(.1 -"%29-4&
'+, &8"+45%&'+= 21 zz % .&( '2= 0(&?+"#=
21 zz & 07+ %($&0(.@9 /"+.7/'-( 6'(:
|||||||||| 212121 || zzzzzz %'%'&
A/452=, -49&( ?&9-"6 6'( '+ %)'"+ '+, 0(-
&975%&'+= ON -49&( 45+ %- '+ %)'"+ '+, 0(&975%&'+= 12 MM . A/+%)9>=:
“"# $%&'# &() *+,-#'.) */# $+0,*+12! 34!,+ 45# $3 &(! ,675&,5( &8! 3+17-
!8! 	)”.
B23&01:
||)( 2121 zzMM &#
C'5(, $(& /&"#0-($%&, 2 -D45>52 3|)2(| #%& iz
-/&328-7-'&( %69+ &/6 '+,= %($&0(.+7= z /+, )*+,9 '29 (0(6'2'& +( -(.69-= '+,= 9& &/)*+,9 &/6 '29 -(.69& '+, %($&0(.+7 i%2 , 023&01
&/6 '+ 52%-4+ )1,2(K , &/65'&52 3 %+9#0-=. A/+%)9>=, 2 -D45>52 &,'1 -49&( -D45>52 .7-.3+, %- .)9'"+ '+ 52%-4+ )1,2(K .&( &.'49&
3#! .
"
M3(&z2)
N(z1&z2)
M(z1+z2) M2(z2)
M1(z1)
x
y 6
K(2,1)
" x
y 7
2 !"#$%"&'" $(")!'"
99
!"#$%&, ' "()*+*'
0,0 !"# **|zz|
,-.$*/&#"$ /0# !"!#$ 1" %2#/.0 /0 *'1")0 K z( )0
%-$ -%/)#- !.
3,)*'4, ' "()*+*' |)31(||)21(| iziz $##"$# ",--
5'6"7"/-$ 18#0 -,8 /094 1$:-;$%074 z ,09 2<09# /'# $;$8/'/- 0$ "$%8#"4 /094 #- $*-,2<09# -,8 /$4 "$%8#"4 /+# 1$:-;$%=# i21$ %-$ i31$# , ;'5-;>
-,8 /- *'1")- )2,1(A %-$ )3,1(#B . 3,012#+4, '
"()*+*' -9/> ")#-$ "()*+*' /'4 1"*0%-62/09 /09 "967:.-1109 /1>1-/04 "K . !"#$%&, ' "()*+*'
||| 21 zzzz| #"#
,-.$*/&#"$ /' %&'$!()&*$ /09 /1>1-/04 1" &%.- /- *'1")- A z( )1 %-$ B z( )2 .
+,$(!'#+-
1. +, -./ *$01 %.-/2.!$"1 .zzz ,...,, 21 .'3"&.
12
2
1
1 %$#
$$$#
$$#
iz
iz
iz
iz
iz
iz
.
.& ,
,/ /4$2&.3*&5 6*. !/,7,/1 /46 /0*$"1 2&, &5,/. 48/-%/*.!61 /8.)%61. $/'%+"01
?# 2#-4 -,8 /094 #zzz ,...,, 21 , :$- ,-.&;"$:1- 0 $z , >/-# ,.-:1-/$%84, /8/" 0$
1$:-;$%0) iz$ # %-$ iz$ $ 6- >/-# *9@9:")4 %-$ ",012#+4 1"$#
"$#
iz
iz
iz
iz
$
$
$
$ ,
-A07 /- 12/.- ;70 *9@9:=# 1$:-;$%=# ")#-$ )*-. B8/" 81+4 6- ")<-1"
12
2
1
1 '$#
$$$#
$$$#
$$#
iz
iz
iz
iz
iz
iz
iz
iz
(
(
)
) && ,
,09 ")#-$ &/0,0.
2. +, -./ *$ %.-/2.!6 z .'3"&. 22(2 "$# |i)z| , ,/ 98&)&5:
/) : -&;%&*8.!61 *64$1 *<1 &.!6,/1 *$0 z '*$ %.-/2.!6 &454&2$.
B(#1,3)
A(1,2)
% x
y 8
2 !"#$%"&'" $(")!'" 100
!) " #$%&'() *+& ) ,-./&'() (� (12 |z| .
*+,-
!) " #$%&'&! | ( )|z i! " #2 2 2 ()!*'+(,(&!# !)% %*-./ &-./ 0#1!2#3-,/ z )-.
45-.6 &'6 #2#%&'&! -# (#3%6(/ &-./ 6! !)45-.6 !)% &- $'0(7- K( , )2 2 $&!+(89
!)%$&!$' 7$' 0( 2 3!# 0%6- !)% !.&-,/. :)-046;/, - <'&-,0(6-/ 1(;0(&8#-3%/ &%)-/ (76!# - 3,3*-/ 0( 346&8- K( , )2 2 3!# !3&76! $ # 2 , 2'*!29 - 3,3*-/
2)2()2( 22 #!"! yx .
=) >- | |z (76!# ' !)%$&!$' &'/ (#3%6!/ M z( )
!)% &'6 !859 O( , )0 0 , 2'*!29 &- 093-/ ( )OM .
?)% &' @(;0(&87!, %0;/, 16;87<-.0( %&# !6 ' (.+(7! O ! &406(# &-6 3,3*- $&! A 3!# B , &%&( ( ) ( ) ( )OA OM OB% % , )-. $'0!76(# %&# ' 041#$&'
	 &-. | |z (76!# &- 093-/ ( )OB 3!# ' (*A5#$&'
&- 093-/ ( )OA .
" (B7$;$', %0;/, &'/ (.+(7!/ O! (76!# ' y x# .
:)-046;/, -# $.6&(&!1046(/ &;6 $'0(7;6 A 3!# B +! (76!# -# *,$(#/ &-. $.-$&90!&-/
&'
&()
##!"!
,
2)2()2( 22
xy
yx
)-. (76!# &! <(,1' ( , )11 3!# ( , )3 3 . C8!, ' 041#$&' 	 &-. | |z (76!# 7$' 0(
( )OB # " #3 3 3 22 2 3!# ' (*A5#$&' 7$' 0( ( )OA # " #1 1 22 2 .
$,&-,.",
34 563738 1. D! =8(7&( &! 04&8! &;6 0#1!2#3E6 !8#+0E6:
i"1 , i!1 , i43 " , i43 ! , i5! , 4! , i
i
!"
1
1,
42 )1()1( ii "*! , )21()2( ii "*! 3!# i
i
34
3
!"
.
2. D! =8(7&( &! 04&8! &;6 0#1!2#3E6 !8#+0E6:
2)1( i" ,
2
1
1+,-
./0!"
i
i,
2
1
1+,-
./0"!
i
i,
2
++,
-../
0!"
i"#i"#
, %)-. !1"#, 0( 0|||| 2" "# .
y=x
K(2,2)
B
A $(z)
% x
y 9
2 !"#$%"&'" $(")!'"
101
3. !" #$%&'% '()* +,-".,/(0* yixz !" -," '()* (1(&()* ,230%,:
") 22 || zz " #) zz "# |1| -) ziz 2|| "! .
4. !" #$%&'% 1(0 "45/()4 (, +,-".,/(& z -," '()* (1(&()* ,230%,:
") 1|| "z #) 1|| "# iz
-) 3|21| "!! iz .) 2||1 $$ z
%) 2|| %z .
5. !" #$%&'% 1(0 "45/()4 (, %,/64%* '74 +,-".,/84 z -," '()* (1(&()*
,230%,:
") |2||1| izz #"! #) |1||| !&# ziz
6. 94 !'x , 4" #$%&'% '( -%7+%'$,/6 '61( ':* %,/64"* '() +,-".,/(0 z,
61() ix
xiz
!!
"1
.
7. 916 '()* +,-".,/(0* z, -," '()* (1(&()* ,230%, 2|4| "# iz , 1(,(* ;3%, '( %<=3,2'( /", 1(,(* '( +;-,2'( .)4"'6 +;'$(;
8. 94 -," '()* +,-".,/(0* z ,230%, 1|| "z , 4" #$%&'% 1() "45/()4 (, %,/64%* '74 +,-".,/84 w +% 12 !" zw .
9. >," .0( +,-".,/(0* "$,?+(0* 1z /", 2z 4" "1(.%&@%'% 6',
22
21
221
221 ||2||2|||| zzzzzz !"#!! .
!" #$%&%'
1. !" .%&@%'% 6', -," /=?% +,-".,/6 z ,230%,:
|)Im(||)Re(|||2 zzz !%( .
2. A2'7 ( +,-".,/6* z , -," '(4 (1(&( ,230%, 1#)z . !" "1(.%&@%'% 6',:
94 1|| "z , '6'% ( 1
1
!#
"z
zw %&4", B"4'"2',/6* "$,?+6* /", "4',2'$6-
B7*.
3. A2'7 ( +,-".,/6* z +% 0)z . !" "1(.%&@%'% 6',: C z
zw1
!" %&4",
1$"-+"',/6*, "4 /", +64( "4 ( z %&4", 1$"-+"',/6* 5 1|| "z .
4. A2'7 ( +,-".,/6* z +% i!z ) , 61() *!'! . !" "1(.%&@%'% 6',: (
!iz
i!zw
!!
" %&4", B"4'"2',/6*, "4 /", +64( "4 ( z %&4", B"4'"2',/6*.
2 !"#$%"&'" $(")!'" 102
5. !" # $%&'"( )*+ ,%-(.%&*/ z ("0&$% 1)*" &/&2* &3")4*+ )0,0(O &(% (&)5"(6 1!! , "( .$57$)$ ')% )* 5.%* %18/$% &(% -%( )#" $%&'"( )*+ ,%-
-(.%&*/ 2
2
"#
!iz
izw .
6. !" -%( )* ,%-(.%&' z %18/$% |2||12| #!# zz , "( .$57$)$ ')% # $%&'"(
)*+ z ("0&$% 1)*" &/&2* ,$ &3")4* )0,0(O &(% (&)5"( 1!! .
7. !" -%( )* ,%-(.%&' z %18/$% 1|| !z , "( 94$5)$ )#" )%,0 )#6 :(4;1)(-
1#6 22 |1||1| zzA #""! . <( $4,#"$/1$)$ -$=,$)4%&; )* 1+,:34(1,(.
8. <( 94$5)$ )* -$=,$)4%&' )':* )=" $%&'"=" " )=" ,%-(.%&>" z , -%(
)*+6 *:*5*+6 %18/$%: |4||1| izz "!" . ?*%* (:' )( 1#,$5( " (:38$% )#" $2;8%1)# (:'1)(1# (:' )#" (480 )0,0(O .
9. !" 1M &(% 2M $5"(% *% $%&'"$6 )=" ,%-(.%&>" 1z &(% 2z (")%1)*58=6
&(% 1
12
4
zzz "! , "( (:*.$57$)$ ')%: @)(" )* 1M &%"$5)(% 1$ &/&2* &3-
")4*+ )0,0(O &(% (&)5"(6 4, )')$ )* 2M &%"$5)(% 1$ ,%( 322$%A#.
10. () !" 1|| !z , "( .$57$)$ ')% z
z1
! .
9) !" -%( )*+6 ,%-(.%&*/6 #zzz ,...,, 21 %18/$% 1||...|||| 21 !!!! #zzz ,
"( (:*.$57$)$ ')%: #
#zzz
zzz1
...11
|...|21
21 """!""" .
2.4 *("#+,'!-*("&. !'(/. !"#$%"&'0
-1234546 B (:*.*80 )=" ,%-(.%&>" (4%C,>", $&)'6 (:' )%6 .+"()')#)$6 :*+ ;"*%7$ 1)#" $:52+1# )=" $7%1>1$=", 3.=1$ ,$-;2# $+$2%75( 1)*" (2-$94%&' 2*-%1,'. D%(
:(4;.$%-,(, # :(4;1)(1# 22 yx " ,:*4$5 )>4( "( :(4(-*")*:*%#C$5 1)# ,*4E0
))(( yixyix #" . F% ,(C#,()%&*5 $&,$)(22$/)#&(" (+)' )* -$-*"'6 1$ :*22; G#-
)0,()(, ':=6 $5"(%, -%( :(4;.$%-,(, * :*22(:2(1%(1,'6 &(% # .%(54$1# )=" )'-7=" $"'6 &/&2*+. H* 1739 * A. de Moivre, 1+".+;G*")(6 )*" +:*2*-%1,' )="
2 !"#$%"&'" $(")!'"
103
!"#$!%& '$(%& )*'*+,-+./& ,01 23'451 !ia ! ()3" .24*&6(3&,*$ +,3& ,7)3
.)68"+01 ,01 qpxx !"3 ) 2. ,0& ,'$9/&32.,'$!5 ,*",:,0,*
### 302024023 3 "$ , ;</+. ,$1 )'%,.1 $<;.1 9$* ,0& ,'$9/&32.,'$!5 )*'-+,*+0
,/& 2$9*<$!%& *'$=2%&. >3 1748 3 L. Euler, ?.!$&%&,*1 *): ,0& *&-8"+0 ,01 $+:,0,*1 102+"& 22 "! "" +,0 23'45 1)02+"&)(02+"&( "$! "i""i" , ,:&$+. ,0
+02*+6* ,/& )*'*+,-+./& ,01 23'451 "i" 02+"& ! !*$ ;<.$?. :,$ "!! )02+"&)(02+"&( yiyxix !! )+"&( yx )02( yxi ! . @.&$!.73&,*1 ;4,*+. +,0 +A;-
+0 nzinzziz n 02+"&)02+"&( %"% ()3" +52.'* 4;'.$ ,3 :&32* ,3" de Moivre),
*): ,0& 3)36*, 2. A'5+0 ,3" <$/&"2$!37 *&*),792*,31, #'5!. ,7)3"1 9$* ,* nz02 !*$ nz+"& .
B. :8.1 ,$1 )'309372.&.1 ).'$),%+.$1 3$ 2$9*<$!36 *&,$2.,/)6(3&,*& /1 !*=*'- +"2#38$!;1 )*'*+,-+.$1, )3" <.& *).$!:&$(*& !-)3$* +"9!.!'$2;&0 )'*92*,$-!:,0,*. C ,'$9/&32.,'$!5 )*'-+,*+0 ;</+. :2/1 ,0 <"&*,:,0,* &* A'0+$23-)3$0=37& (*): ,3& C. Wessel ,3 1799 !*$ ,3& R. Argand ,3 1806) 9$* ,0& *&*-8",$!5 ;!4'*+0 ,01 <$.7="&+01 +,3 .)6).<3, *!'$#%1 :)/1 3$ =.,$!36 !*$ *'&0-,$!36 A'0+$23)3$37&,*$ 9$* ,0 <$-!'$+0 ,01 43'-1 +,0& ."=.6*. D",;1 3$ .?.86-?.$1 <$.7'"&*& ,$1 .4*'239;1 ,/& 2$9*<$!%& !*$ -&3$?*& ,3 <':23 9$* ,0 9./2.-,'$!5 .'20&.6* ,3"1, ,0& 3)36* !*=$;'/+. 3 C.F. Gauss ,3 1831.
*+,-./ !,0/1,234 E+,/ ;&*1 20 20<.&$!:1 2$9*<$!:1 *'$=2:1
yixz !" !*$ OM 0 *&,6+,3$A0 <$*&"+2*,$!5 *!,6-&* ,3".
F&32-(3"2. !"#$%& ,3" 2$9*<$!37 z !*=.2$- *): ,$1 9/&6.1 )3" ;A3"& *'A$!5 )8."'- ,0& 02$."=.6* Ox !*$ ,.8$!5 )8."'- ,0& 02$."=.6* OM .
D): :8* ,* 3'6+2*,* ,3" z ;&* *!'$#%1 #'6+!.,*$ +,3 <$-+,02* )2,0[ # . D",: 8;9.,*$ '"()*+,- !"#-$%& ,3" 2$9*<$!37 z !*$ +"2#386(.,*$ 2. )(zArg .
G6&*$ 4*&.': :,$:
& T3 )(zArg .6&*$ 0 9/&6* )3" +A02*,6(.$ 0 <$*&"+2*,$!5 *!,6&* ,3" 2$9*<$!37
z 2. ,3& -?3&* xx' .
& H"3 3'6+2*,* ,3" z <$*4;'3"& !*,- 9/&6* $#2 , !($ .
@$* ,3 2$9*<$!: 0"z <.& 3'6(.,*$ :'$+2*. @$’*",:, +,0 +"&;A.$*, :,*& *&*4.-':2*+,. +. :'$+2* 2$9*<$!37, =* .&&3372. :,$ 0)z .
5+,0673.89+,2: !3+;: !,0/1,234
% "
x
&(z)
y10
2 !"#$%"&'" $(")!'" 104
!"#$ % &'()*'+,- 0!"# yixz , .%/ 012' &0#3% 22 yx* "# . 45 ! 265)' 05)
,3'"&) #%/ z, #,#2, )., #%5 %3'"&, #$5 #3'($5%&2#3'+75 )3'8&75 "2 %38%+)5%-5'+, "9"#:&), 01%/&2
*x+ #!"# +)'
*y+ #$%
%.,#2 !"x "/5# +)' !"y :&#
;.%&05$-, % &'()*'+,- z (3<=2#)'
i!"!"yixz $"#"# :&"/5 ,
*:>)*? .)6352' #: &%3=?
):&"/5( !i!"z "#
@ #3,.%- )/#,- (3)=?- #%/ &'()*'+%9 z >0(2#)' &'()*#+%,&'(-. ? /+0(-.
%+'1. &+" z .
A') .)3<*2'(&), )5 iz "%# 3 , #,#2 #% &0#3% #%/ z 265)' 21)3( 22 #"%#"
+)' (') +<82 ,3'"&< #%/ ! '"19%/5:
2
3"/5 %#! +)' 2
1:& #! .
;.%&05$-, &') #'&? #%/ %36"&)#%- 265)' : 6
5#! # . B3), 01%/&2
&'(
)*+ "#
6
5:&6
5"/52#
i#
z ? (25'+,#23):
,-
./0
1&'(
)*+ ""&
'(
)*+ "#
6
52:&
6
52"/52
#$#i#$#z , !2$ .
4.%*2'+592#)' ,#', )5 (') 05)5 &'()*'+, )3'8&, z '"192' ):&"/5( !i!rz "# , ,.%/
03r +)' "2! , #,#2 : .)3<"#)": ):&"/5( !i!r " 265)' #3'($5%&2#3'+? &%3=?
#%/ &'()*'+%9 )3'8&%9 z .
;.2'*? 6"%' &'()*'+%6 )3'8&%6 01%/5 #:5 6*') 2'+,5) "#% &'()*'+, 2.6.2*% +)' )5#'"#3,=$-, 01%/&2 #% )+,>%/8% +3'#?3'% '",#:#)- &'()*'+75:
“2"+ %$ %$3,#(-+4 %()53(-+4 5'(6%+4 ,4#5( 4!+(, 5# -5( %7#+ 5# 89+"# 4!5
%8&'5 -5( $ 3(51+': &*# +'(!%:&*# &+"; ,4#5( 5-8'5(+ /+005/0:!(+ &+"
2,”.
y
"
%
!
M(x,y)
x x
y11
2 !"#$%"&'" $(")!'" 105
!"#$%&:
'( )")*+(( 1111 !i!"z !" ,$- )")*+(( 2222 !i!"z !" ./($- 01-23(4).01-,56 )41756 03( )-2$%-,8( 1z ,$- 2z , 090.:
#" 21 zz ( 21 "" " ,$- #$!! 221 $"% , !&$ ).
*+,-./0123+,45 !0+65 #,/017/08 !,-9:,4;/
'( )")*+(( 1111 !i!"z !" ,$- )")*+(( 2222 !i!"z !" ./($- 4- 01-23(4).01-,56 )41756 %:4 )-2$%-,8( $1-;)8( 21 ,$- zz , 090. 2-$ 04 2-(9).(9 04+6 5<4+).:
)")*+(()")*+(( 22211121 !i!"!i!"zz !$!"$
)]")*+(()")*+([( 221121 !i!!i!"" !$!"
)]")*+(*+(")()")")*+(*+([( 2121212121 !!!!i!!!!"" !!%"
)]")()*+(([ 212121 !!i!!"" !!!" .
=)4/36, 2-$ 04 >"#/,4 04+6 2
1
z
z, 5<4+).:
)]")*+()(")*+((
)")*+()(")*+((
)]")*+((
)")*+((
2222
2211
2
1
222
111
2
1
!i!!i!!i!!i!
""
!i!"!i!"
z
z
%!%!
"!!
"
2
22
2
2211
2
1
")*+()]")(-)*+((-)[")*+((
!!!i!!i!
""
!
!!"
)]")()*+(([ 2121
2
1 !!i!!""
%!%" .
'>4%./?$). #4->9( 90-:
'( )")+*+(( 1111 !i!"z " ,$- )")+*+(( 2222 !i!"z " ./($- %+4 )-2$%-,4/ *. 01-23(4).01-,& )417&, 090.
)]")()*+(([ 21212121 !!i!!""zz !!!"$
)]")()*+(([ 2121
2
1
2
1 !!i!!""
z
z%!%" .
@-$ >$1A%.-2)$, $( '()
*+, !"
3
2")3
2*+(21
#i
#z ,$- '
()
*+, !"
6
11")6
11*+(32
#i
#z ,
090. 5<4+).:
2 !"#$%"&'" $(")!'" 106
i!
i!!!
i!!
zz 62
5!"2
5#$%66
11
3
2!"6
11
3
2#$%3221 !"#$
%&' (!)
*
+,-
."#$
%&' (("
#$
%&' (/!/
&'(
)*
+,-
."#$
%&' 0("
#$
%&' 0!)
*
+,-
."#$
%&' 0("
#$
%&' 0!
6
7!"6
7#$%3
2
6
11
3
2!"6
11
3
2#$%3
2
2
1 !i
!!!i
!!z
z
3
1
3
3
2
1
2
3
3
2ii (
0!"
"#
$%%&
'(! .
)*+ ,(- ,.(/0%1"2,.(&3- "1.43- ,1$ /(%1"3%1$ &'( ,1$ *!56&1$ "(/'7(&8% *.1-&9*,1$% 1( (7(+,!,2-
|||||| 2121 zzzz /! &'( ||
||
2
1
2
1
z
z
z
z! ,
,(- 1*162- 3:1$"2 #$%'%,;#2( &'( #,!% § 2.3.
< /20"2,.(&; 2."!%26' ,1$ /(%1"3%1$ &'( ,1$ *!56&1$ 791 "(/'7(&8% 4'6%2,'( #,' *'.'&=,0 #:;"',':
M(z1z2)
M2(z2)
M1(z1)
y
x "(1) O
#2
#2
#1+#2
#1
12
>' ,.6/0%' $"%2 &'( $%1% 26%'( +"1('
M(z1/z2)
M2(z2)
M1(z1)
y
x "(1) O
0#2
#2
#10#2 #1
13
>' ,.6/0%' $"%2 &'( $%%1 26%'( +"1('
?9"40%' "2 ,' *'.'*=%0:
1 @ *155'*5'#('#"+- ,1$ "(/'7(&19 )!"#$%( 1111 #i#&z (! "2 ,1 "(/'7(&+
)!"#$%( 2222 #i#&z (! #!"'6%2( #,.14; ,!- 7('%$#"',(&;- '&,6%'- ,1$ 1z &',=
/0%6' 2# &'( "2,= *155'*5'#('#"+ ,!- "2 2& (?:. 12). A*1"3%0-, 1 *155'*5'-
#('#"+- 2%+- "(/'7(&19 z "2 ,1 "(/'7(&+ #i# !"#$% ( #,.342( "+%1 ,! 7('%$-
#"',(&; '&,6%' ,1$ z &',= /0%6' # , '419 1|!"#$%| !( #i# . A(7(&+,2.', 1 *15-
5'*5'#('#"+- ,1$ z "2 i #,.342( ,! 7('%$#"',(&; '&,6%' ,1$ z &',= /0%6'
2
!, '419
2!"
2#$% !
i!
i (! .
2 !"#$%"&'" $(")!'" 107
! ! "#$%&'() *+, -#.$"#/+0 ))-(,1( 111 !i!z "# -' *+ -#.$"#/2
))-(,1( 2222 !i!"z "# ()-$%1'# (*&+34 *)5 "#$1,(-$*#/45 $/*%1$5 *+, 1z /$*6
.71%$ 2!$ /$# -'*6 8+99$89$(#$(-2 *)5 -' 2
1
" (:;. 13).
)*+,-./ 012 De Moivre
<1 ))-(,1( !i!"z "# '%1$# =1$5 -#.$"#/25 $&#>-25 (' *&#.71+-'*&#/4 -+&34,
(0-371$ -' *$ 8&+).+0-'1$ =;+,-':
%"#""#%# ))-2(,12())-(,1())-(,1( 22 !i!"!i!"!i!"zzz
%"#""#%# ))-3(,13())-(,1())-2(,12( 3223 !i!"!i!"!i!"zzz
?-+%75, @&%(/+,-' 2*#
%"# ))-4(,14(44 !i!"z
%"# )5)-(,15(55 !i!"z
A'1#/6, #(;0'# *+ '82-'1+ >'B&)-$:
!"#$%&' 1
<1 ))-(,1( !i!"z "# '%1$# =1$5 -#.$"#/25 $&#>-25 (' *&#.71+-'*&#/4 -+&-
34 /$# # '%1$# =1$5 >'*#/25 $/=&$#+5, *2*'
)])-()(,1([ #!i#!"z ## "# .
$3'%4"56
C(*7 )(#P ) #(2*)*$ 8+, >=9+,-' 1$ $8+"'%D+,-'.
! A#$ 1## ) #(2*)*$ .%1'*$# )])-(1)(,1(1[11 !i!"z %"%# 4, #(+"01$-$,
))-(,1( !i!"z "# , 8+, #(;0'#. E&$ ) )1(P '%1$# $9)>45.
! F$ $8+"'%D+,-' 2*#, $1 )(#P $9)>45, *2*' )1( "#P $9)>45 ")9$"4, $1
)])-()(,1([ #!i#!"z ## "# , *2*' ]1)+)-(1)+(,1([11 !#i!#"z ## "# "" .
G&6.-$*#, =;+,-'
))-(,1()])-()(,1([1 !i!"#!i#!"zzz ### "%"#%#"
]1)+)-(1)+(,1([1 !#i!#"# "# " .
E&$, ) )(#P $9)>'0'# .#$ 29+,5 *+,5 >'*#/+05 $/'&$%+,5 # . !
2 !"#$%"&'" $(")!'" 108
!"# $#%&'(")*#, #+ iz !" 3 , ($("', #$%
&'( !"
6-*
6./+2
!i
!z , 012/*(:
1998
1998
6-*
6./+2 )
*
+,-
.#$%
&'( !"
!i
!z #
$%
&'( !"
6
1998-*6
1998./+21998 !i
!
!i! -*333333./+(21998 !" ) !i! -*./+(21998 !" ) 19982/" .
32 $%2-)24*(+2 5(6%-*# #$2'7'(8#" .82 *#5-*#8"9: De Moivre 9#" )"’ #/8: ;0%(" 82 :+2*& 82/.
32 <(6%-*# 82/ De Moivre ".14(" 9#" :8#+ 2 (9508-= (7+#" #%+-8"9:= #90%#"2=.
>%&)*#8", 012/*(
"""
#i#$#i#$
)-*./+(
1)]-*./+([
!"! /
))-*()./+((
)0-*0./+(1
"#i"#$i
" !0!0
"
)]--*(0)-./+(0[ "#i"#$ " !" /
)]-*(-)./+(-[ "#i"#$ " !" / .
*+$(!'#*,
1. !" #$%&%' () *+,)-) (., %/01,., (., 2/3"4/05, z, 3/" ()67 )8)')67 /-
*9+%/ 61
1 -z
zArg "#
$%
&'(
!/
.
./,0
?+ yixz !" , 8:8( iyx
y
yx
yx
yix
yix
z
z2222
22
)1(
2
)1(
1
)1(
)1(
1
1
!!!
!!/!
"!!!/
"!/
.
@%#,
BiAz
z!"
!/
1
1, :$2/
22
22
)1(
1
yx
yxA
!!/!
" 9#" 22)1(
2
yx
yB
!!" .
2 !"#$%"&'" $(")!'" 109
!"#$%&'(, ) *+&,-.) 61
1 !z
zArg !"
#$
%&'
()
/0&12
2*#34&1$) $/ 52( *6%*/2(:
*+
*,
-
.
!)(/
*+
*,-
.
!
0
3
1
1
2
0
6/7 22
y
yx
y
"
!A
B
*+
*,-
.!)(/
0
2)3( 222
y
yx.
891, 5# *4&#:# 5'& /2.;&'& 5#+ z /0&12 5# 5;<# 5#+ .4.:#+ .%&59#+ )3,0(K
.12 1.50&1( 2!# "#+ /0&12 "=&' 1"; 5#& =<#&1 xx0 .
2. !" 0#$#$#$ !(( *+, %&' 0()"()"()" !(( *+, , "& &*+,-'./-0 1/'
&) !(( *+, ()"3()"3()"3 *)+, (((()"3
2) )#$(3#$3#$3#$3 *+,*+, ((!(( .
-./0
>*5' #2 $2?132.#0 $i$ )$*+&a (! , %i%b )$*+& (! , &i&c )$*+& (! . >6#+$/
000))$)$)$()*+&*+&*+&( !(!(((((!(( i&%$i&%$cba
.12 /"#$%&'(, bccb a3333 !((a .
@/ 1&52.15=*51*) 5'& b,a .12 c %6#+$/ 3213#62.=:
!((((( 333 ))$*+&())$*+&())$*+&( &i&%i%$i$
))$*+&)()$*+&)()$*+&(3 &i&%i%$i$ (((!
!((((( ))$3*+&3())$3*+&3())$3*+&3( &i&%i%$i$
)])$()*+&([3 &%$i&%$ (((((!
!((((( ))$3)$3)$3()*+&3*+&3*+&3( &%$i&%$
))$(3)*+&(3 &%$i&%$ (((((! .
!<2*A&#&51( 51 "91?$152.= .12 51 71&51*52.= $%9) 5'& 34# $/:A& %6#+$/:
)*+&(3*+&3*+&3*+&3 &%$&%$ ((!(( .12
))$(3)$3)$3)$3 &%$&%$ ((!(( .
K( , )0 3
B()1,0) A(1,0)
' x
y14
2 !"#$%"&'" $(")!'" 110
$*&+*,"*
!" #$!%!&
1. !" #$%&'(' )' ($*#+,-.'($*/0 .-$10 (-23 .*#"4*/-53:
") i31! 6) i31"
#) i31"" 4) i31!"
') 4 )() 4" .
2. !" /%,'(' (*3 7$%8'*3:
") )9.3030)2,(6)9.1515)2,(4 0000 ii !#!
6) $%&
'() !#$
%&
'() !
8
39.8
3)2,28
9.8
)2,5!
i!!
i!
#) $%&
'() !#$
%&
'() !
10
39.10
3)2,10
29.10
2)2, !i
!!i
!.
3. !" /%,'(' (*3 7$%8'*3
") )1009.100)2,(5
)1609.160)2,(2500
00
i
i
!!
6)
39.
3)2,
6
59.6
5)2,6
!i
!
!i
!
!
$%&
'() !
#) ))20(9.)20()2,(14
)09.13130)2,(700
00
"!"!
i
i.
4. !" 6$':(' (*3 42,%.'*3
") 300 )]9.20)2,20(2[ i! 6)
8
4
59.4
5)2,3 *+
,-.
/$%&
'() !
!i
!
#)
16
49.
4)2, $$
%
&''(
)$%&
'()"!$
%&
'()"
!i
!.
5. !" 27-;-#:)'(' (9, 7"$%)(")9
6
2
1"
$$%
&''(
) ! i.
6. <, 2
31 iz
!0 , ," 27-;-#:)'(' (-, 2000z .
2 !"#$%"&'" $(")!'" 111
7. !" iz !" 31 #$% iz #" 32 , "$ &'()(*+,-.- ./" '$01,.$,/
!! zz 21 ! , 2'(& ! 3-.%#24 $#50$%(4. 8. 6$ -07/"-8,-.- *-97-.0%#1 ./ :%$+0-,/ -"24 7%*$:%#(8 z 7- i .
9. !" 31 iz !" #$% iw !" 1 , "$ :-+;-.- 2.% 12
"w
zArg "$
%&
'()
#$% "$ <0-+.-
.( 12
/7 " #$% .( 12
,&" " .
10. 6$ <0-+.- .( 75.0( #$% .( <$,%#2 20%,7$ .(& 7%*$:%#(8 0*z $" zz "2 .
!" #$%&%'
1. $) 6$ <0-+.- .( 75.0( #$% .( 20%,7$ .(& 7%*$:%#(8 w, 2'(&
!
#i##i#
w $$%
&''(
)#!!!
"/7,&"1
/7,&"1, *!+! #$% "+!* $"$# ,)12( .
<) 6$ <0-+.- ./" .%7= ./4 '$01,.$,/4 100
222
222$$%
&''(
)
#!!!
i
i.
2. $) 6$ :-+;-.- 2.% $" !! ii )1()1( #"! , 2'(& *!+! .2.- !+" $$! ,4
<) !" !!
ii!f $$%
&''(
) #!$$
%
&''(
) !"
2
1
2
1)( , "$ :-+;-.- 2.% 0)()4( "!! !f!f .
3. 6$ $'(:-+;-.- 2.%
|||||| 2121 zzzz !"! , $" #$% 72"( $" )()( 21 zArgzArg " .
4. 6$ <0-+.- .( *-97-.0%#2 .2'( .9" -%#2"9" % .9" 7%*$:%#>" z , *%$
.(&4 ('(+(&4 %,?8-%:
$) 6
)("
izArg "# <) 4
)1("
zArg "! *) 2
"iz
zArg "$
%&
'()
#.
5. @-.$;8 2)9" .9" 7%*$:%#>" z '(& %#$"('(%(8" ./ ,&"3=#/
2|52| ,#! iz , "$ <0-+.- -#-+"(" '(& 5?-%:
$) A( 7%#02.-0( '09.-8(" 20%,7$
<) A( 7-*$)8.-0( '09.-8(" 20%,7$.
2 !"#$%"&'" $(")!'" 112
6. !" !i!z #$%&" !" , "' '()*+,-+.+ /.0:
') )%&"(21 "!z
z "" "! 1) )#$(2
1 "!iz
z "" "# .
7. !" 20' .)&3 $02'*04)53 z 4'0 w 0%65)&" 1|| "z 4'0 ziw )3( #" , ./-
.+: ') 7' 18+,.+ .) 2+9$+.804/ ./() .9" +04/"9" .)& $02'*04)5 w.
1) 7' 18+,.+ .#" +04/"' +4+,")& .)& $02'*04)5 '(/ .)&3 w, 20' .)"
)(),) 0%65+0 4
)(#
wArg " .
8. !" 22
2
1
2
1
1
$$
i$$
z!
!!#
" 4'0 3
)(#
zArg " , "' 18+,.+ .)" (8'2$'.04/ '-
80:$/ $
9. ;,"+.'0 .) .80<"&$) )||1)(||1(||2)( 22
2121
2 zzxzzxxf !!!#!" , /()&
1z 4'0 2z +,"'0 *5) $02'*04), '80:$),. 7' '()*+,-+.+ /.0 0)( $xf ,
20' 4=:+ !%x . >/.+ 0%65+0 # 0%/.#.';
2.5 *'+,-.,!"&/0 /1"0-0/"0 02' &&&&
/3456768 ? +(,@&%# .9" +-0%<%+9" 3)& 4'0 4)& 1':$)5, # “'"'24'%.04A” +('BA $+ .)&3 $02'*04)53 '80:$)53 20' .#" C4B8'%# .9" (8'2$'.04<" 80D<" 4'0 # +-C@0-# .)&
'@2+1804)5 @)20%$)5 *#$0)582#%'" %.03 '86C3 .)& 17)& '0<"' .03 (8)E():C%+03 20' .#" '"=(.&-# $0'3 2+"04A3 :+98,'3 .9" ()@&9"&$04<" +-0%<%+9" %.#" F@-
2+18'. G'%04= %.)06+,' '&.A3 .#3 :+98,'3 *+" A.'" $/") )0 $C:)*)0 +(,@&%#3, '@@= 4'0 *)$04= D#.A$'.', /(93 )0 %6C%+03 80D<" 4'0 %&".+@+%.<" $0'3 +-,%9-
%#3, 4':<3 4'0 # %6C%# '"=$+%' %.) 1':$/ 4'0 %.) (@A:)3 .9" 80D<". H) .+-@+&.',), ()& 4':0+8<:#4+ '82/.+8' 93 !"#"$%&'"( !"&)*#+ ,*( -$."/)+(
“$%!& #'()*")+,$- &./0*01 " 23!+'4 56&, 07' 04"'(' 7*" +,839,$:" " 3$;,2:< ;/=&<”,
*0'.&(<"+.'0 %.#" '86A *0%.'4.04=, 4':<3 )0 $02'*04), *+ :+98)5".'0 '4/$#
0%/.0$)0 (8)3 .)&3 &(/@)0()&3 '80:$)53. I R. Descartes, %.) 101@,) JJJ .#3 “La
Geometrie” (1637) 28=B+0 /.0: “4=:+ +-,%9%# $()8+, "' C6+0 ./%+3 *0'B)8+.04C3 8,D+3 /%+3 4'0 )0 *0'%.=%+03 [*#@. ) 1':$/3] .#3 =2"9%.#3 ()%/.#.'3 %.#" +-,-%9%#”, '@@= )")$=D+0 .03 :+.04C3 8,D+3 “'@#:0"C3”, .03 '8"#.04C3 “K+5.04+3” 4'0 +0%=2+0 20' (8<.# B)8= .)" /8) “B'".'%.04C3” 20' .03 &(/@)0(+3:
2 !"#$%"&'" $(")!'" 113
“…!"# $%&'&($! ") *!+',-&.$! /01 2 !34-+-2 010136 23 !"#" xxx 5-6!1 0'!17 '48!7, !" 0&(0&17 .%9'6!1 $4) $/"& %'):$)01;, '48), 0& 2, !"# &1 9<<!7 =(& %)')$5"&." >)"0)-01;57”.
!" #$%$&'()$* #$(+,%- .,* /&0$1+-* 2+3'4$ 5- -6"7.2 $8-'+$.'79 4,%-4:- %$ .,5 -526.;8, .,* <52&;4,*, 7-#(* , 6-+-0"5."6":,4, .=5 6"&;=5>%=5 ?6-'@$ 6+=.-+3'7A +A&" 4."5 ;6"&"0'4%A "&"7&,+=%2.=5 ()'246-4, +,.(5 7&-4%2-
.=5 4$ -6&2 7&24%-.-). B G.W. Leibniz ?#$4$ ." 1702 -;.A ." @9.,%- '43;+'-@A%$5"* (&-#$%?5-) A.' ." 6"&;(5;%" 44 )x # )$5 -5-&>$.-' 4$ 0'5A%$5" 6-+--
0A5.=5 1"; 9 2"; 1-#%"> %$ 6+-0%-.'7">* 4;5.$&$4.?*. !" 0$0"5A* -;.A ")9-
0,4$ 4.'* 6+(.$* 4;4.,%-.'7?* 6+"462#$'$* 5- -6")$'3.$: A.' 72#$ 6"&;(5;%"
%$ 6+-0%-.'7">* 4;5.$&$4.?* -5-&>$.-' 4$ 0'5A%$5" 6-+-0A5.=5 1"; 9 2";
1-#%">, 6"; -6".$&$: %'- 2&&, '4")>5-%, %"+C9 ."; #$%$&'()";* #$=+9%-."*. D4.$+- -6A "+'4%?5$* ,%'.$&$:* 6+"462#$'$* .=5 d’Alembert (1746), L. Euler
(1749) 7-' J.L. Lagrange (1772), " C.F. Gauss ?)=4$ .,5 6+(., -;4.,+9 -6A-
)$'8, ." 1799 (4$ ,&'7:- 22 3+"5(5), 4., )')-7."+'79 ."; )'-.+'19 6"; $:3$ .:.&": “E?- -6A)$'8, ."; #$=+9%-."* A.' 72#$ -7?+-'- +,.9 4;52+.,4, %'-* %$.-1&,.9* %6"+$: 5- -5-&;#$: 4$ 6+-0%-.'7">* 6-+20"5.$* 6+(."; 7-' )$>.$-+"; 1-#%">”.
* E+,-.-/ 1!0z
F5=+:@";%$ A.' 4." 4>5"&" .=5 6+-0%-.'7(5 -+'#%(5 , $8:4=4, 1!"z ?3$' %'-
&>4,, .,5 1!z , -5 " " $:5-' 6$+'..A* 7-' )>" &>4$'*, .'* 1!z 7-' 1"!z , -5 "
" $:5-' 2+.'"*. <* &>4";%$ .(+- 4." 4>5"&" $ .=5 %'0-)'7(5 -+'#%(5 %$+'7?* $8'4(4$'* .,* %"+C9* 1!"z , A6"; " #$.'7A* -7?+-'"*. G3";%$:
% 0)1)(1(011 233 !##"&!"&! zzzzz
01!"& z 9 012 !## zz
1!& z 9 2
31 iz
#"! 9
2
31 iz
""! ,
),&-)9 , $8:4=4, ?3$' 4." $ .+$'* +:@$*.
% 0)1)(1(011 2244 !#"&!"&! zzzz
012 !"& z 9 012 !#z
1!& z 9 iz ! 9 iz ! 9 iz "! ,
),&-)9 , $8:4=4, ?3$' 4." 4>5"&" $ .?44$+'* &>4$'*.
2 !"#$%"&'" $(")!'" 114
!"#$%& $'()"$ *+ ",-."#+ /"012.3: !"#$%&' 2
4*+ ')#+5+ *6# .$738$%0# 31$/.0# 2 "9:'6'2 1!!z , -,+; ! /"*$%-< 3%=-13$+<, =("$ ! 3%1$>0< 8$3?+1"*$%=< 5)'"$<, +$ +,+:"< 8:#+#*3$ 3,- *+# *),+:
1,...,3,2,1,0,22.2';# "!#! !"!"#
i!"#
z " .
$*'%+",-
@'*6 )2.';#( $i$r # .$3 5)'2, '" *1$76#+."*1$%A .+1?A, *2< "9:'6'2< 1!!z .
B-*", 1)]2.';#([ !# !$i$r ,
+,-*" 02.0';#))(2.)(';#( i!$i!$r ! #!#
C13, 1!!r %3$ "#!$ 20 !" , 7$3 %&,+$+ !$" , +,-*" 1!r %3$ !"#$ 2
! . D,+.=-
#6<, +$ 5)'"$< *2< "9:'6'2< 1!!z , /3 ":#3$ *2< .+1?A<
!"#
i!"#
z"22.2';# #! , !$" . (1)
E55& %3$ %!&'(&)*+,-, %&/" .$738$%-< *2< .+1?A< !"#
i!"#
z "22.2';# #! ,
!$" ":#3$ 5)'2 *2< "9:'6'2< 1!!z , 3?+)
1)2(2.)2(';#22.2';# !#!%&'
()* #! "#i"#
!"#
i!"#
z
!
" .
C13, +$ 5)'"$< *2< "9:'6'2< 1!!z ":#3$ -5+$ +$ 31$/.+: *2< .+1?A<
!"#
i!"#
z "22.2';# #! , !$" . (1)
!$3 0!" =(+;." *2# ,1+?3#A 5)'2 10 !z .
E# /='+;." ,!#
i!#
z !%&'
()*#%
&'
()*!
22.2';#1 , *-*" 7$3 *$< 1:F"< *2< 1!!z , /3 $'()"$:
""
" ,!#
i!#
!"#
i!"#
z !%&'
()* #!#!
22.2';#22.2';# , !$" .
2 !"#$%"&'" $(")!'" 115
!"#$% &'%()#:
111212
2222
11
11
22
1
0 11
!!!!
""
""
!! #$##
#$###$####
#
###
!!!!!
!!!
!!!
!!
!! """"z
""""z
""""z
"z
"z
"z
"z
z
%% *.+.&.
,$-$+.-'/01, &'%()#, )+% '% &/21%3 +.3 1#!z ('4 5"#'#+$% $() +.# (1) 51# 1"-#$% )&13 5%$6'-1+%*73 01+$8/ +'43. 9$ 181+:2'401 ;%$ ('%13 +%073 +'4 # 7<'401 5%$6'-1+%*73 &/21%3. !(1%5= ;%$ *:>1 !&# 4(:-<'4# $*7-$%'% $ *$% % +7+'%-'%, ?2+1 #$ %2</1% %$!# "# 01 !% '(0 , 7<'401:
%%%$!%$!# """""" #$#$## " 1)( .
@.&$5=, ;%$ *:>1 !&# . &/2. #z +$4+"A1+$% 01 0%$ $() +%3 132 ...,,,,,1 !!"""" .
9$ 51"8'401 +?-$ )+% '% &/21%3 13210 ...,,,,,1 !# !""""" 1"#$% 5%$6'-1+%*73 01+$8/ +'43. B2+C )+% 51 240D$"#1% $4+). E)+1 >$ 4(:-<'4# 642%*'" 21 , && 01
!&& ''( 210 , +7+'%'%, ?2+1 21 && "" # , '()+1 >$ 7<'401 5%$5'<%*::
!'&
i!'&
!'&
i!'& 2211 2
.02
24#2
.02
24# "#"
'(!'&
!'&
222 11 $#! , ;%$ *:('%' !&(
!(&& #! 21 , ;%$ *:('%' !&( .
F() +.# +1&14+$"$ %2)+.+$ (-'*/(+1% )+% ' $*7-$%'3 ! 5%$%-1" +. 5%$6'-:
21 && ! . F4+) )0C3 1"#$% :+'(', $6'/ !&& '!' 210 . !('07#C3, '% &/21%3 +.3
18"2C2.3 z) #1 1"#$% '% ! 5%$6'-1+%*'" $-%>0'"
132 ,...,,,1, !*++++ , )('4 *+,
-./"*
+,
-./#
*,
i*,+ 2!"2#$% .
G% &/21%3 $4+73 &7;'#+$% *$% %&'#()* +,-.* (!* "'%/01*. !
234564
G% 1%*)#13 1210 ,...,,, !!AAAA +C# &/21C# 132 ,...,,,,1 !!"""" +.3 18"2C2.3 1#!z 1"#$%
*'-4673 *$#'#%*'/ ('&4;?#'4 01 ! (&14-
-73, 1;;1;-$007#'4 21 */*&' 01 *7#+-'
)0,0(O *$% $*+"#$ 1#r . ,%' 24;*1*-%07#$:
A0
A7
A6
A5
A4
A3
A2
A1
) x
y15
2 !"#$%"&'" $(")!'" 116
! ! "#$%&' A0 ()$*+,-./* ,0 12+0 1.
! ! /(34/.0 "#$%&' A1 ()$*+,-./* ,0 12+0 "#$
%&'("
#$
%&')
!"
i!"# 2042+%. .
! ! "#$%&' 2A ()$*+,-./* ,0. 2# ")* ($#"2(,/* )(3 ,0. # 4/ +,$#&' ,#%
5*).2+4),#6 1OA "),- 78.9) !"2
, 501)5' "),- 78.9) 9+0 4/ ,0. "/.,$*"'
78.9) ,#% ").#.*"#2 !-78.#%.
! H "#$%&' 3A ()$*+,-./* ,0. 3# ")* ($#"2(,/* )(3 ,0. # 4/ +,$#&' ,#%
5*).2+4),#6 1OA "),- 78.9) !"22 *
",1.
* E+,-.-/ a)0z , 0a +
:+,8 )!"#$%(a 1i12 () 4*) ,$*78.#4/,$*"' 4#$&' ,#% 4*7)5*"#2 a. ;3,/ )(3
,#. ,2(# ,#% de Moivre <=#%4/: 0
0
01
i012 ""
#
$%%&
'"#$
%&' () !"#$%a .
>. ?<+#%4/ "#$
%&' ()
!$
i!$%z ! 04+%.0 , ,3,/ 0 /@9+8+0 a)!z 7$-&/,)* !! zz 0)
', *+#52.)4),
10
)""#
$%%&
'!
z
z
A(#4<.86, ,# 0z
z 4(#$/9 .) (-$/* ,*6 ! 5*)&#$/,*"<6 ,*4<6
132 ,...,,,,1 ,!#### , 3(#% "#$
%&'("
#$
%&')
!"
i!"# 2042+%. ,
#(3,/ #* 12+/*6 ,06 /@9+8+06 a)!z /9.)* #* )$*?4#9
-.
/01
2"#$
%&'("
#$
%&'*"
#$
%&' ()*)
!&"
i!&"
!$
i!$%#zz !&
&2042+%.04+%.0
1,...,2,1,0,2042+%. ,)"
#$
%&' (
((
) !&!$&"
i!$&"%! .
>(#5/9@)4/ 1#*(3. 3,*:
2 !"#$%"&'" $(")!'" 117
!"# $%&#'# "(& )*+,-*./& ,0*1)/& 2 345$($2 a!!z , 67#8 ! 13"*.69 ,.:-0,*#9 .,* )2)$8&( "i"# "!a , $# ! , :;3* ! -*,<#03"*.:9 '%$3*9 #* #7#539 -5-&#&",* ,76 "#& "%7#:
1,...,2,1,0,22)2$8& #!$
%&
'() "
""
! !%!"%&
i!"%&#z !
% .
=* 3*.6&39 "(& '%$3(& "29 345$($29 a!!z $"# )*+,-*.6 37573-# 35&,* .#08<:9 .,&#&*.#% 7#'8+/ )3 ! 7'380:9, 3++3+0,)): $3 .%.'# )3 .:&"0# )0,0(O
.,* ,."5&, ! # , 67#8 || a!# .
>$"( +*, 7,0?-3*+), 2 345$($2
)3(165 iz "! . (1)
@73*-A $%&
'() "!"
62)
6$8&32)3(16
&i
&i , #* '%$3*9 %z "29 345$($29 (1) -5-#&",*
,76 "#& "%7#
.4,3,2,1,0,30
122)30
+12$8&25
62
2)5
62
$8&325 !$%&
'() "
"!$$$$
%
&
''''
(
) ""
"! %&%&
i&%&
&%&i
&%&z %
B*# $8+.3.0*):&, #* '%$3*9 35&,*:
$%&
'() "!
302)
30$8&20
&i
&z ,
$%&
'() "!
30
132)30
13$8&21
&i
&z ,
$%&
'() "!
30
252)30
25$8&22
&i
&z ,
$%&
'() "!
30
372)30
37$8&23
&i
&z ,
$%&
'() "!
30
492)30
49$8&24
&i
&z .
=* '%$3*9 ,8":9 35&,* .#08<:9 .,&#&*.#% 73&",+/ 3++3+0,)): $3 .%.'#
,."5&,9 2!# .
*+,-./-01234 561787914 09 *:;<0;=12+>4 ?-/=9,97=34
C7(9 ,&,<:012.3 $"2& 3*$,+(+A, .?13 7#'8(&8)*.A 345$($2 0)( !zP , &*#$"#%
D,1)#%, -2',-A .?13 345$($2 "29 )#0<A9
0,0011
1 *!"""" ## !
!!
!! $$z$z$z$ + ,
A4
A3
A2
A0
A1
'
y
x
16
2 !"#$%"&'" $(")!'" 118
!"#$ %&' %()'*' &+) ,$-./$01) .2$3,1) ! .02$415 267#5. 8) !zzz ,...,, 21 #6).$ '$ 267#5 &'9 :'*9+)(,'9 )(zP ('$ ':'6#5 /#) #6).$ 0.&.);-
-0< /$.='2#&$0!5), &>&# .:'/#$0)(#&.$ >&$ &' :'*91)9,' .).*(#&.$ %# -$)>,#)' :.2.->)&+) +5 #?@5:
)(...))(()( 21 ** zzzzzz+zP !""!!#
A:',!)+5, < #:6*9%< :'*9+)9,$01) #?$%1%#+) %&' $ -6)#&.$ ,# &$5 6/$#5 ,#3>-
/'95 :'9 "2<%$,':'$'()&.$ 0.$ %&' %()'*' % &+) :2.-,.&$01) .2$3,1).
B&< %9)!"#$. 3. :#2$'2$%&'(,# %# :'*9+)9,$0!5 #?$%1%#$5 ,# :2.-,.&$0'(5 ,>)' %9)&#*#%&!5. C"'9,# @/< *(%#$ &< /#9'4;3,$. #?6%+%<, < ':'6., >:+5 #6/.,#, !"#$ /('
267#5, '$ ':'6#5, .) /#) #6).$ :2.-,.&$0!5, #6).$ ,$-./$0!5 %979-#65. 85 *(%'9,# &12. ,6. .)+&!2'9 4.3,'(, -$. :.2;/#$-,. &<) 0353 33 #!&! zzz , :'9 #6).$ :'*9+)9,$0@ &26&'9 4.3,'(. D# %"@,. Horner !"'9,#:
0320)1)(32(0353 2223 #&!'#!&!'#!&! zzzzzzzz @ 01 #!z .
E,+5, izzz 210322 &#'#&! @ iz 21!# .
F2., '$ 267#5 &<5 #?6%+%<5 #6).$ i21& , i21! 0.$ 1. G.$ %&<) :#26:&+%< .9&@
:.2.&<2'(,# >&$ '$ ,$-./$0!5 267#5 &<5 #?6%+%<5 #6).$ %979-#65. H' %9,:!2.%,.
.9&> -#)$0#(#&.$ -$. ':'$./@:'&# :'*9+)9,$0@ #?6%+%< ,# :2.-,.&$0'(5 %9-
)&#*#%&!5.
!"#$%&' 3
8) ' ,$-./$0>5 .2$3,>5 i"#z � #6).$ 267. ,$.5 :'*9+)9,$0@5 #?6%+%<5 ,# ()*+,*-./012 345-6763-82, &>&# 0.$ ' %979-@5 &'9 i"#z !#0 #6).$ 267.
&<5 #?6%+%<5 .9&@5.
$,'%-"./
D$. :'*9+)9,$0@ #?6%+%<, >:+5 -)+267'9,#, !"#$ &< ,'2=@:
0011
1 #&&&& !! #z#z#z# !!
!! ( , >:'9 !### ,...,, 10 )% ,# 0*!# .
8='( ' .2$3,>5 0z #6).$ 267. &<5 #?6%+%<5, !"'9,# 0.&; %#$2;:
00011
010 #&&&& !! #z#z#z# !!
!! (
00011
010 #&&&& !! #z#z#z# !!
!! (
00011
010 #&&&& !! az#z#z# !!
!! (
00011
010 #&&&& !! #z#z#z# !!
!! (
2 !"#$%"&'" $(")!'" 119
00011
010 !"""" ## !z!z!z! ""
"" $ , !"#$ !%"!!! ...,,, 10 .
%&!, # 0z '()!* +!* !,-./ &(0! -1/ '2(3431/. !
*+$(!'#*,
1. !" &'(
)*+"&
'(
)*+!
-.
i-
2./ 2#$%&" , "' '()*+,-.+/ 0.,:
') 0132 !""""" #-////1 $ 1) 1132 1)(1 ## #!,,,,, --/.../// .
$0'%*"12
!) 56#,7' 01
11
1
1...1 132 !
##
!##
!""""" #
#######"
"
8) )1(...321132 ...1 #""""# !,,,,, "" #####
2
)1( #
!""
#
2
)1(
21723,)
#
&'(
)*+ "!
""
"$
i"$
""$"
i""$"
2
)1(2172
)1(23,) #"
#!
$"i$" )1(17)1(3,) #"#!
1)173,)( #"! "$i$
1)1( ##! "
2. 2' 3&4+/ # +5/%6%# 01%&"22 !",# x3x . !" 21 , xx +/"', ), 7/8+9 .#9 +5/-%6%#9 '&.:9, "' ;'.'%;+&'%.+/ +5/%6%# 2)& 1'4$)< ()& "' =-+, 7/8+9 .,9
-- xx 21 , .
45,2
56#,7' 0174)13,)(443,)4 222 -#!#!#! %%%& . 9:#7;)4/,
%i%i%%x 173,)
2
1723,)22,1 .!
,.! .
< 01-#$7')1 '2(3431 =! '()!* 1
0)( 2122 !,""# /
"""" xxxxxx .
2 !"#$%"&'" $(")!'" 120
!"#$%&
)'%()($)()'%($)(1 !"i!""i"x !! !"!"
*+,
)'%(-)($)(-)]'%(-)($)([)'%($)(2 !"i!""i""i"x !!! !"!#"#"
)'%()($)( !"i!" #" .
-.#%/)01:
)($)(2)'%()($)()'%()($)(21 !"!"i!"!"i!"xx !! "#!!"!
*+,
10'%+($)0)'%(-)($)(-())'%()($)((21 ""!!"$ i!"i!"!"i!"xx !!
23+, ' 4'5#6%&)' &78(0(' &8)+, ':
01)($)(22 "!# x!"x .
3. !" "#"$%&'( )' *+#,-'#. /.$%0#1-0# 2. /.$%3#%-.
6543)( 23 !!!" xxxxP ,
"# *#04(5.%-' ,2+ 67'+ 4(5" 2. -+*"8+9, "4+&-, i21! .
*+,-
9:#6 5# )(xP /"&, 384+ 5#) +3,;%< ix 210 !" , ;+ /"&, 384+ *+, 5# ($4$=> 5#$,
ix 210 #" . -.#%/)01, 5# .#?$@)$%# )(xP A,+,3&85+, %& 5# =,)<%&)#
))(()( 00 xxxxxQ ##" , =,+ 5# #.#8# /"#$%&
]2)1][(2)1[()]21()][21([)( ixixixixxQ !###"##!#"
22 )2()1( ix ##"
2212 !#!" xx
322 !#" xx .
9) *B)#$%& 5' A,+83&(' 5#$ .#?$0)6%#$ )(xP %& 5# .#?$@)$%# )(xQ , C38-(*#$%& .'?8*# 23 !x . -.#%/)01 &8)+,
)23)(32()( 2 !!#" xxxxP .
:;<=><
2 !"#$%"&'" $(")!'" 121
!"#$%&, '()* +#+,-./0%" %+$ 120# "$1+3)34, %&/" (5678#795 9" (.+39+2$%5:* 17#2"6"12-* 9(5."; #+ 3.+,"; )* 3$#'9"#5 (.)25<&/9$)# %+$ ="72".5<&/9$)#
(+.+3'#2)# 9" (.+39+2$%5:* 17#2"6"12-*, '(57 5$ ="72".5<&/9$5$ (+.&35#2"* (+# 7(&.>57#) ->57# +.#02$%4 =$+%.;#571+.
$*&+*,"*
!" #$!%!&
1. ?+ 6:1"2" 2$* "@$181"$* %+$ #+ (+.+1241"2" 2$* 6:1"$* 125 9$3+=$%'
"(;("=5:
+) 13 !z <) 14 !z 3) 16 !z .
2. ?+ 6:1"2" 2$* "@$181"$*:
+) iz "!3 <) #$%
&'( )!
3
4093
417#164 **iz
3) #$%
&'( )!
6
5096
517#2435 **iz .
3. ?+ 6:1"2" 2$* "@$181"$*:
+) 2
)1(23 iz
)! <)
2
314 iz
"! 3) 646 "!z .
4. ?+ 6:1"2" 2$* "@$181"$*:
+) 843 23 !)) zzz <) 045 24 !)) zz .
5. A# 5 9$3+=$%'* i)2 ";#+$ .;B+ 20* "@;1)10*
0107103 23 !))" xxx ,
#+ <.";2" %+$ 2$* &66"* .;B"* 20*.
6. A# w ";#+$ 9$+ %7<$%4 .;B+ 20* 95#&=5*, 9" 1+w , #+ <.";2" 20# 2$94
20* (+.&12+10* )1)(1( 22 wwww "))"
7. ?+ 6:1"2" 20# "@;1)10
01 5432 !))))) xxxxx .
8. ?+ 6:1"2" 20# "@;1)10 0933 23 !))) zzz %+$ #+ =";@"2" '2$ 5$ "$%'-
#"* 2)# .$B8# ";#+$ %5.7,-* $1'(6"7.57 2.$38#57.
2 !"#$%"&'" $(")!'" 122
!" #$%&%'
1. !" #$%&'& '() &*(%+%&(): ") iz !"13 ,) 0)1)(1()1( 33 "#!!! ziz .
2. !" #$%&'& '-. &*/%0%- 0)1(222 223456 "###### zzzzzz .
3. !" #$%&'& '-. &*/%0%- 017 "#z 1"( %'- %2.34&(" ." ,5&/'& '" '5(+-
.26" 6& 75"86"'(19$) %2.'&#&%'3) 792 &/."( 7"5:89.'&) '92 79#20-
.$692 123456 #!#!#! zzzzzz .
4. ;'9 %$.9#9 '0. 6(8"<(1+. "5(=6+. ." ,5&/'& '() 19(.3) #$%&() '0. &*(%+%&0. 0)1( 322 "### zzz 1"( 012 1416 "## zz .
5. !" ,5&/'& '92) 6(8"<(19$) "5(=69$) z , 8(" '92) 979/92) (%4$&( 137 "zz .
6. >. - &*/%0%- *,)1()1( !$!"# !izpiz !! 34&( 75"86"'(1? 5/@", ." "-
79<&/*&'& A'( 1|| "p .
7. B/.&'"( - &*/%0%- 0422 "#! xx 6& 5/@&) '() 1x 1"( 2x .
") !" 279#98/%&'& '() '(63) '0. 7"5"%':%&0. 21 xx # , 21 xx 1"( 22
21 xx # .
,) >. - &*/%0%- 02 "## qpxx 34&( 0) 5/@&) '() 21x 1"( 2
2x , ." ,5&/'& '() '(63) '0. p 1"( q .
8. ") !" #$%&'& '-. &*/%0%-
0)%2.45(%2.2%2. 22 "!#%!% "z"z" , 22
&'
&((! .
,) !" "79<&/*&'& A'( 1"=+) '9 ' 6&'",:##&'"( %'9 <(:%'-6"
)*+
,-. !
2,
2
&&, 9( &(1A.&) '0. #$%&0. '-) &*/%0%-) 1(.9$.'"( %& 6("
27&5,9#?.
8. !" #$%&'& '-. &*/%0%-
01459 "!#! xxx .
2 !"#$%"&'" $(")!'" 123
#*+"&*, $,&-,*",
1. !"#$%&' ( )*#+,%()( f -$ zz
zzzf
!!"
#)1)(1(
)( -$ !$z .&' 0)Re( %z .
&) /& &012$"3$%$ 4%' )(1
zfz
f #&'(
)*+" .
5) 6)%7 !", 2*1 ()%&8$,1") 0,&9-&%'.1" &,'8-1" 2'&:1,$%'.1" &04 %1
0. /& 5,$"%$ %1 $"21; %(; .&-0<=(; )%(# 101"& &#>.1*# %& )(-$"&
),( yxM , -$ 0%x , 9'& %& 101"& 1' -'9&2'.1" &,'8-1" yi!x"z !# '-.e'1<# %( )?@)( 0))(Re( #zf .
2. A$7,1<-$ %1*; -'9&2'.1<; ,z w .&' ,1w 9'& %1*; 101"1*; ')?<1*#:
zizw "# .&' iw ,,!#
11 , 401* "$" . /& 2$"3$%$ 4%' &# %1 , -$%&-
5+==$%&' )%1 *" .&' ')?<$' 1ww # , %4%$ ( $'.4#& P %1* z )%1 -'9&-
2'.4 $0"0$21 .'#$"%&' )$ -'& *0$,51=>.
3. A$7,1<-$ %1*; -'9&2'.1<; "$"!!# #i##z ,)13(2 .
&) /& 5,$"%$ %1 9$7-$%,'.4 %401 %7# $'.4#7# %1* -'9&2'.1< z
5) /& 5,$"%$ %1 9$7-$%,'.4 %401 %7# $'.4#7# %1* -'9&2'.1< w 9'& %1# 101"1 ')?<$' )1( izw !!#
9) /& 5,$"%$ %1 -'9&2'.4 z 01* @?$' %(# 0=()'@)%$,( $'.4#& )%(# &,-
?> )0,0(O .
4. /& 9,&--1).'+)$%$ %1 %->-& %1* -'9&2'.1< $0'0@21* 01* 1,"B1*# 1' $'.4#$; %7# -'9&2'.C# z , 9'& %1*; 101"1*; ')?<$': &) |||12| izz !-! 5) )Re(1|1| zz !#" .
5. /& &012$"3$%$ 4%' &# 1' -'9&2'.1" .zzz ,...,, 21 @?1*# %'; $'.4#$; %1*; )%1 "2'1 (-'$0"0$21 -'&; $*8$"&; 01* 2'@,?$%&' &04 %(# &,?> )0,0(O ,
%4%$ ')?<$' 0...21 %!!! .zzz .
6. /& &012$"3$%$ 4%' 1' $'.4#$; %7# =<)$7# %(; $3")7)(; // zz #" )1(
$"#&' )(-$"& %(; $*8$"&; 2
1#x .
7. D# %1 %,'C#*-1 01, !! xx 2 -$ 0,&9-&%'.1<; )*#%$=$)%@; .&' 0%,
2$# @?$' 0,&9-&%'.@; ,"B$;, #& &012$"3$%$ 4%': &) E'& 101'1*)2>01%$ 0,&9-&%'.1<; &,'8-1<; $ .&' # ')?<$'
2 !"#$%"&'" $(")!'" 124
0))(( 22 !"""" #$%&%#$'&' .
!) "#$ %&%#%'()*&%+, ('-'.,/0 1#.$)#2%30 1z 2$# 2z )#$4%5,+#2%30 $&6 +#0 5/-,0 +%' +5#7831%' #(93,# ,&/(:0
0))(( 2221
21 !"""" #$&#$& zzzz .
8. "875/-%8+$0 6+# .#$ +#0 8#%(+;0 5/-,0 +:0 1%8<)$0 121 ,...,,,1 (!zzz #(93-
,# 0..1 121 )"""" (!zzz , 8$ $&%),/=,+, +#0 +$'+6+:+,0:
$) 0)12(
:1...6:14:12:1 )
(""""
!"!
!"
!"
!"
,
!) 1)12(
('8...6('84('82('8 ()
(""""
!"!
!"
!"
!"
.
*(+,-.*". &$,$/'-.-.
1. >$ !<?,+, (, 232?% +: (7(+* $&<8+:(::
(i) @8 (+% (38%?% +78 1#.$)#2A8 $5#B1A8 #(93,# 022 )" vu , +6+, : @. 0)u C. 0)v
". 0)) vu D. E/&%+$ $&6 +$ &5%:.%31,8$.
(ii) F $5#B160 * + * +10105353 iiz ("") ,/8$#:
@. G$8+$(+#260 C. H:);8
". I5$.1$+#260 D. E/&%+$ $&6 +$ &5%:.%31,8$.
2. I%#,0 $&6 +#0 ,&61,8,0 #(6+:+,0 $?:B,3%'8 .#$ 2<B, 1#.$)#26 z :
@. 22 |||| zz ) C. 2
zzz ), ". 2zzz ),
D. zzz ), J. 22 zz ) .
3. K31478$ 1, +: ('8B*2: &%' #2$8%&%#%38 %# 1#.$)#2%/ z 2$# $8$4;-5,+$# (+:8 &5A+: (+*?:, 8$ +%'0 $8+#(+%#9/(,+, (+:8 ,'B,/$ +:0 ),3-
+,5: (+*?:0 &%' $8*2,# : ,#268$ +%'0:
!"#$%&' ("$)*+
A. iziz ")( $. 1)x
B. 11 ")( zz !. yy -
2 !"#$%"&'" $(")!'" 125
!. izz !"!1 ". xy "
#. xy !"
$. izz #"#1 %. xx$
4. &' '()*+),*-.+%)% /01% 2*"'#*/3 z )45 678)45 +)9:45 +), 37*+20
),; )45 #%<)%745 +)9:45:
!"#$%"&'( ( k >0 ) )*"+,$
=. kik # '. -450
>. kik ! ?. 2250
!. kik !! ". 450
$. kik #! #. 1800
%. 600
@. 1350
5. &' ?0:%)% +% /</:, )*5 +A+)B5 '6'()9+%*5. C '7*1235 )A( 67'"2')*/8( 7*@8( 2*'5 6,:;A(;2*/95 %D.+A+45 5,;
?'12,< 2% 67'"2')*/,<5 +;()%:%+)B5 26,7%. (' %.('*:
=. 1 >. 2 !. 3 $. 4 E. 5
6. &' "70F%)% ),;5 2*"'#*/,<5 6,; B-,;( A5 %*/3(%5 )' +4-
2%.' =, >, ! /'* $ ),; #*-6:'(,< +-92'),5:
=:
>:
!:
$:
7. =( z %.('* , 2*"'#*/35 6,; B-%* A5 %*/3(' ), =, (' '()*+),*-.+%)% /0-
1% 2*"'#*/3 )45 678)45 +)9:45 +)4( %*/3(' ),; 6,; '('GB7%)'* +)4
#%<)%74 +)9:4 /'* +42%*8(%)'* +), 6'7'/0)A +-92':
!
B
450
A
"
O x
y
1
2 !"#$%"&'" $(")!'" 126
!"#$%"&'( )"&'*$
z !
z2
1 "
z
1 #
z! $
z! %
&
'
"
B A
-(
( $
&
% '
#
O x
y
2 1 0.5
!" #$%&'
()(*+',
O!"# - $%&'('") $%&)!*+$+$
1.1 ,!)-.)*"/#" )!"0.#"
*1 234151 674 89:;<:6=>?4 :9=@<?4 !"#$%&'()*&'# +,- ,* ./$*0* ! ,1$ "234'3,-56$ 32-%'6$ 3"*,#0#(,3- 3"+ ,*&7 28,*/7 53- ,*&7 9228,*&7 32-%'*/7 53- "32-.,9$#,3- '# ,3 .8'#(3 #$+7 9:*$3, ! " # $ % " & ' ! ( & ) * ' + , ' ! - . / & ' * - 0 , / &. (;<. 1)
x= x
" e 3
5 4 3 2 1 0 "1 "2 "3 "5 "4
1
>8,*( 32-%'*( 0?4*$,3- *- 32-%'*( "*& ?<*&$ @ '"*2*/$ $3 "92*&$ ,8 '*2A@ 1'
,
+"*& ', 1 35?23-*- '# 0#1 . B* ./$*0* ,1$ 28,6$ 32-%'6$ .&'C*0()#,3- '# $.
D($3-, E803E@,
%&'
()*
#+ 0'#35?23-*- 1',11'
! .
!"#$%&'()*&'# +,- ,* ./$*0* ,1$ 35?23-1$ 32-%'6$ #($3- ,*
...},3,2,1,0,1,2,3,{ """+ ," ,
#$6 ,* ./$*0* ,1$ A&.-56$ 32-%'6$ #($3- ,*
1
1 !"#! - $%&'('#) $%&)"*+$+$ 130
...},3,2,1,0{!! .
!"# $# %&'()# "#! ,, *#" " "%+&,":
$"#! ### .
-# %&'()# }0{},0{},0{ $$$ "#! *#" }0{$$
$# %./0()12(./, %.'$(/3$,4# /, *! , *# , *"
*#" *$ #'$"%$(1+56.
,-./012 341 51.64/7 869 $$$$
7$( %&'()( $ $5' 84#9/#$"*:' #4";/:' (41-%$<*#' (" 84=>,"6 $<6 843%;,%<6 *#" $(. 8())#8)#%"#%/(& *#" /, $< 0(?;,"=
$(.6 < #@#14,%< *#" < A"#14,%<. B" "A"3$<$,6 $5' 84=>,5' #.$:' ,1'#" 9'5%$C6 #83 84(<9(&/,',6 $=>,"6. 7$< %.'C+,"# (41%$<*, < C''("# $<6 A"=$#><6, (" %8(.A#"3$,4,6 "A"3$<$,6 $<6 (8(1#6 ,1'#" (":
1) A' !" % *#" #! % , $3$, #" %
2) #!#"!" &%&'%
3)
03$#',
03$#',
,':(
)
*+
*,
-
.'%
%'%
#
#
!#"#!"
!#"#!".
4) .$3$,,
$3$,,
0
*#"*#"
A'
*#"D'
!$"#
$!#"
",!,#,$
$#!"$#!"
%
&%&
**+
**,
-
///
0
1
222
3
4
)
%%
%%
5) D' 0%",! *#" *!5% , $3$, "%+&," < "%(A.'#/1#:
%% !"!" %'%
6) )0*#"0(0 6%'% !"!!"
7) A' 0)"! , $3$, "%+&," < "%(A.'#/1#
!"!" 11
.'% .
:1486;<464 =-4><4613?@ 4-1A<?@
2
$
" # %
1 !"#! – $%&'('#) $%&)"*+$+$ 131
! !" !"!," #$ "! # , %&%$ '"'#()'*#$ !"#$%&'#%# '( )*+# %# ,,- +,-.",
,/& %, /,0,+(%1 23"'4,:
:}|{)( "x!x!," ##"$ ! #,-"*%. !")$%/'#
:}|{][ "x!x!," %%"$ ! *0("$%. !")$%/'#
:}|{),[ "x!x"! #%"$ ! *0("$%.-#,-"*%. !")$%/'#
:}|{]( "x!x!," %#"$ ! #,-"*%.-*0("$%. !")$%/'#.
! !" !"! , %&%$ '"'#()'*#$ '/ 1+#2-
'3,# !"#$%&'#%# '( )*+- %- , +,-.",
,/& %, /,0,+(%1 23"'4,:
}|{),( !xx! &"$'( !
}|{),[ !xx! )"$'( !
}|{),( !xx! #"$*( !
}|{],( !xx! %"$*( !
5/& #'067 89,2%7#,%': %' 23"'4' ! %' 2*#;'4<)'*#$ #$ ),( '(*( .
=, 2>#$<, $"&: 89,2%7#,%': #, /'* $<",9 89,6'0$%9+( ,/& %, (+0, %'*, 4.?'"%,9 ($4%(+"*) $/'(5# %'* #.
)./0123 2456 .7,85,249:; ,74<5:;
@ #6.07%/ %"'& $"&: /0,?#,%9+'3 ,09-#'3 !, /'* 2*#;'4<)$%,9 #$ || ! , '0<)$-%,9 1: $A7::
+,-
#*)
$0,",
0,",||
!!!!
!
B$1#$%09+(, > ,/&4*%> %9#7 %'* ! /,092%("$9 %>" ,/&2%,2> %'* ,09-#'3 ! ,/&
%' #>8.",
. /00 10 0 xC x 4 3 , 2
|a |
1 0 *1 *2 *3 *4
5
$"D > ,/&4*%> %9#7 %'* "! * /,092%("$9 %>" #6.$%#$/ %4, #+"8'9, , *#" -.
a
a
a
a
"
"
"
"
3
a
a
a
a
4
1 !"#! - $%&'('#) $%&)"*+$+$ 132
! "# ## $# # # x! x 4 3 , - 2
|a%! |
1 0 %1 %2 %3 %4
6
"#$%&'( )*+ ,%( -).%&'( %/%+,0,#( ,0( )*+12,0( ,%34( #56)% 7% #84(:
1) 22|| "" & 2) ||2 "" &
3) |||||| !""! '& 4) ||
||
!"
!"&
5) |||||||||| !"!"!" ()()% 6) #xx#x#xx (**%+*% 000 || , 0,#
'.)"/!0+ !" #$"%&'( )*+ ,&$%- ./"01-,"1&2 - 3(4052 ./"015,614( 1" 0'(&7":
i) -./
012 )& 1
1
x|xA ii)
-./
012
*%& 12x
xA | 1.
1%$+
i) E56)%
xx
1101
1%)+)
x
x 10
%)+
0)1( 3%+ xx &)% 04x
0x *+ 4 13x .
9$) ),1[)0,( (56%5&A .
ii) :56)%
121
1121
*%*%+*%xx
31
1 **+x
3
11 ,,+ x .
1 !"#! – $%&'('#) $%&)"*+$+$ 133
!"# !"#
$%&' 1,
3
1A .
1.2 $%&)"*+$'#$
+ ,--./0 123 4506701/893 :;-<512:23
$ %&&'(# )*+ ,-&.")*,*+ /0&#( 1&2,)3 #45 4"'*1'67/&/+ ).8/(+. 9)*& 4#".-
1"#:' #-)3 -4/&;-70<'-7/ )'& '"(,75 )*+ 4"#17#)(=3+ ,-&.")*,*+ 7/ 4/>0'
'"(,7'6 %&# -4',6&'?' )'- !, /4#&#?#7@.&'-7/ 1&2,)%+ %&&'(/+ =#( )%?'+ '"0<'-7/ 4".8/(+ 7/)#86 )2& 4"#17#)(=A& ,-&#")3,/2&.
!"#$%!$
B,)2 ! %&# -4',6&'?' )'- !. C&'7.<'-7/ &'()*(+,-. /012'+3/3 7/ &4567
7',/*78 +7 ) 7(# >(#>(=#,0# (=#&5&#) f , 7/ )*& '4'0# =.;/ ,)'(D/0' Ax( #-
&)(,)'(D0</)#( ,/ %&# 75&' 4"#17#)(=5 #"(;75 y. E' y '&'7.</)#( +,*. +39 f
/+7 x =#( ,-7@'?0</)#( 7/ )(xf .
F(# &# /=:".,'-7/ )* >(#>(=#,0# #-)3, 1".:'-7/:
!)Af :
)(xfx ) .
— E' 1".77# x, 4'- 4#"(,).&/( '4'('>34')/ ,)'(D/0' )'- ! ?%1/)#( (14:2'-
+3+3 *4+(;<3+., /&A )' 1".77# y, 4'- 4#"(,).&/( )*& )(73 )*+ f ,)' x, ?%-1/)#( 4:('+3*=13 *4+(;<3+..
— E' 4/>0' '"(,7'6 ! )*+ ,-&.")*,*+ f ,-&3;2+ ,-7@'?0</)#( 7/ fD .
— E' ,6&'?' 4'- %D/( 1(# ,)'(D/0# )'- )(+ )(7%+ )*+ f ,/ 5?# )# Ax( , ?%1/)#( /817<7 +,*>1 )*+ f =#( ,-7@'?0</)#( 7/ )( Af . G0&#( >*?#>3:
)(|{)( xfyyAf '' 1(# =.4'(' }Ax( .
?"!$!@A
9)# /457/&# =#( ,/ 5?* )*& %=)#,* )'- @(@?0'- :
— H# #,D'?*;'67/ 75&' 7/ ,-&#")3,/(+ 4'- %D'-& 4/>0' '"(,7'6 5,2/+3*(
3 =1B/3 5,(/+3*2+B1.
— IC)#& ;# ?%7/ 5)( “A /012'+3/3 f 461(, 7',/*=13 /’ =1( /817<7 C”, ;#
/&&''67/ 5)( )' " /0&#( -4',6&'?' )'- 4/>0'- '"(,7'6 )*+. 9)*& 4/"04)2,*
#-)3 7/ )(Bf ;# ,-7@'?0<'-7/ )' ,6&'?' )2& )(7A& )*+ f ,/ =.;/ Bx( .
G0&#( >*?#>3:
1 !"#! - $%&'('#) $%&)"*+$+$ 134
)(|{)( xfyyBf !! !"# $%&'"' }Bx" .
$,-./0/123453 6,-72.8689
()*#+, &#-#&%./ 01" !"# .# '-"21,) +"# 23.%-1424, f #-$,) .# *'5'6. *6' 21'"-7,)#:
# 1' &,*)' '-"2+'6 148 $#"
# 4 1"+9 148, )(xf , !"# $%5, x 1'3 &,*)'3 '-"2+'6 148.
:3.95/8, 0+/8, #.#;,-0+#21, 2, +"# 23.%-1424 f *).'.1#8 +0.' 1'. 16&' +, 1'. '&')' ,$;-%<,1#" 1' )(xf . :, +"# 1=1'"# &,-)&1/24 5# ! " # $ % & ' " ( )'
* + , - . / 01" 1' &,*)' '-"2+'6 148 f ,).#" 1' 26.'>' 0>/. 1/. &-#!+#1"$?. #-"5+?. x, !"# 1'38 '&')'38 1' )(xf =7," .04+#. @12", !"# &#-%*,"!+#, #.1) .#
>=+, “*).,1#" 4 23.%-1424 !$%& ]2,(:f , +, xxf 24)( %! ” 5# >=+, “*).,1#" 4
23.%-1424 f +, 16&' xxf 24)( %! ” 9, &"' #&>%, “*).,1#" 4 23.%-1424
xxf 24)( %! ”, 9 “*).,1#" 4 23.%-1424 xy 24 %! ”.
':)";!<+ !"#o $%&'# (" )$*%" "+#,-". (/0 ,1&2+(/,/0 f -$ (.)":
i) x
xxxf
23)(
2 '%! ii) xxf ln1)( %!
=%$+
i) H 23.%-1424 f '-)<,1#", 01#. $#" +0.' 01#.
0232 ('% xx $#" 0)x .
A' 1-"?.3+' 0+/8 232 '% xx =7," -)<,8 1'38 #-"5+'68 1 $#" 2. @12", 4 #.)2/24
0232 ('% xx #>45,6,", 01#. $#" +0.' 01#.
1*x 9 2(x .
(&'+=./8, 1' &,*)' '-"2+'6 148 f ,).#" 1' 26.'>' ),2[]1,0()0,( '&++%&!A .
ii) B 23.%-1424 f '-)<,1#", 01#. $#" +0.' 01#.
0ln1 (% x .
().#" 0+/8
1ln0ln1 *,(% xx
1 !"#! – $%&'('#) $%&)"*+$+$ 135
ex lnln !"
ex !#" 0 .
!"#$%&'(, )# "*+,# #-./$#0 )1( f *,&2. )# /0# ],0( eA $ .
,-./012 3.-456.57 5894-6757:
4/)' f $.2 /5&6-)1/1 $* "*+,# #-./$#0 ! 72. Oxy %&2 /0/)1$2 /5&)*)28$%&'&
/)# *","*+#. 9# /0# )'& /1$*,'& ),( yxM 8.2 )2 #"#,2 ./:0*. )(xfy $ , +1-
32+; )# /0# )'& /1$*,'& ))(,( xfxM , Ax% , 3%8*)2. !"#$%&' (#")*+#*,
)1( f 72. /5$<#3,=*)2. /5&;>'( $* fC . ? *@,/'/1, 3#."A&, )(xfy $ *"231>*0-
*)2. $A&# 2"A )2 /1$*,2 )1( fC . !"#$%&'(, 1 )(xfy $ *,&2. 1 *@,/'/1 )1( 8-2B.7;( "2-6/)2/1( )1( f.
!"*.+; 76>* Ax% 2&)./)#.:,=*)2. /* %&2 $A&# !%y , +*& 5"6-:#5& /1$*,2 )1( 8-2B.7;( "2-6/)2/1( )1( f $* )1& ,+.2 )*)$1$%&1. C5)A /1$2,&*. A). 76>* 72-
)27A-5B1 *5>*,2 %:*. $* )1 8-2B.7; "2-6/)2/1 )1( f )# "#30 %&2 7#.&A /1$*,#
(D:. 72).
4)/., # 7073#( +*& 2"#)*3*, 8-2B.7; "2-6/)2/1 /5&6-)1/1( (D:. 7<).
EF)2& +,&*)2. 1 8-2B.7; "2-6/)2/1 fC $.2( /5&6-)1/1( f, )A)*:
2) 9# "*+,# #-./$#0 )1( f *,&2. )# /0# ! )'& )*)$1$%&'& )'& /1$*,'& )1( fC .
<) 9# /0# ).$G& )1( f *,&2. )# /0# )(Af )'& )*)28$%&'& )'& /1$*,'& )1( fC .
8) ? ).$; )1( f /)# Ax %0 *,&2. 1 )*)28$%&1 )#5 /1$*,#5 )#$;( )1( *5>*,2(
0xx $ 72. )1( fC (D:. 8).
& '( ( )( ( O x
y
(a)
Cf
!
O x
y 7
C
(")
1 !"#! - $%&'('#) $%&)"*+$+$ 136
Cf
O
y
x
(!)
"
Cf
O
y
x
(#)
f (")
Cf
O
x=x0
A(x0, f (x0))
x0
y
x
($)
f (x0)
8
!"#$ %&$'"#( ) *+#,(-. /#+01"#1) fC , 2(#3 14$0+")1)3 f 2/5+562', '/&1)3, $#
17'%(01542' -#( "(3 *+#,(-83 /#+#1"01'(3 "9$ 14$#+".1'9$ f! -#( || f .
#) : *+#,(-. /#+01"#1)3 ")3 14$0+")-
1)3 f! '&$#( 1422'"+(-., 93 /+53 "5$ 0;5$# xx" , ")3 *+#,(-.3 /#+0-
1"#1)3 ")3 f, *(#"& #/5"'<'&"#( #/= "#
1)2'&# ))(,( xfxM !" /54 '&$#( 1422'-"+(-0 "9$ ))(,( xfxM , 93 /+53 "5$ 0-
;5$# xx" . (>7. 9).
?) : *+#,(-. /#+01"#1) ")3 || f #/5"'-<'&"#( #/= "# "2.2#"# ")3 fC /54
?+&1-5$"#( /0$9 #/= "5$ 0;5$# xx"
-#( #/= "# 1422'"+(-0, 93 /+53 "5$ 0;5$# xx" , "9$ "2)20"9$ ")3 fC /54
?+&1-5$"#( -0"9 #/= "5$ 0;5$# #4"=$. (>7. 10).
,-./012 345/012 5674.895-/2
>")$ /#+0*+#,5 #4". %&$542' "(3 *+#,(-83 /#+#1"01'(3 2'+(-@$ ?#1(-@$ 14-
$#+".1'9$, "(3 5/5&'3 *$9+&1#2' 1' /+5)*562'$'3 "0;'(3.
! "#$%&'%()*+ ,%'-./0,0 34xxf #$)(
O
y
x
9
%&(x,!f (x))
y=f (x)
y=!f (x)
%(x,f (x))
O
y
x
10
y=f (x) y=| f (x)|
1 !"#! – $%&'('#) $%&)"*+$+$ 137
11
a>0
O x
y
a<0
O x
y
a=0
O x
y
! "#$%&'%()*+ ,%'-./0,0 2)( ,xxf ! , 0", .
O x
y
!>0
x O
y
!<0
12
! "#$%&'%()*+ ,%'-./0,0 3)( ,xxf ! , 0", .
O x
y
!>0
O x
y
!<0
13
! .0/+ ,%'-./0,0 x
,xf !)( , 0", .
O x
y
!>0
O
x
y
!<0
14
1 !"#! - $%&'('#) $%&)"*+$+$ 138
!" #$%&'()#*"+ xxf !)( , |x|xg !)( .
y x!
O x
y
y x! | |
O x
y 15
!"#$%&
0
0
,
,
)(
"
#
$%
$&' (
!x
x
x
xxg , ' ()*+$,& "*)-./*.' /'0 || xy ! *"1/#2#3/*$ *"1
%41 ,2-%150. 6 78*0 #38*$ ' ()*+$,& "*)-./*.' /'0 xy ! ,*$ 1 -2210 ' .59-
9#/)$,& /'0 :0 ")10 /18 -;18* yy) .
1 !"#! - $%&'('#) $%&)"*+$+$ 138
!" #$"%&'"()* +,'-$#.+/"* : xxf 01)( ! , xxf +,')( ! , xxf /2)( !
O
y=!"x
2# #
1
"1
y
x
O
y=$%&x
2# #
1
"1
y
x
3#/2 #/2 "#/2 O
y='(x
y
x
( ! )
(" )
(# )
16
)#'&*%"+,-%"' ./0, -0 $%&12/3$'04 xxf !")( ! 510 $%&x)( !xf '+&10 #'20-60574 "' #'2+-6- $T 2! , '&8 ! $%&92/!$! xxf '()( ! '+&10 #'20-6053 "' #'2+-6-
$T ! .
3 /(4/#"(. +,'5$#0+0 x,xf !)( , 10 #$ , .
#
1
1 O x
y
(#) #>1
O x
y
(") 0<#<1
# 1
1
17
1 !"#! – $%&'('#) $%&)"*+$+$ 139
!"#$%&'()*&'# +,-:
!" 1!, , #$#%: 2121 xx,, xx "#"
#$.
!" 10 "" , , #$#%: 2121 xx,, xx !#" .
& '()!*+,-+./ 01"2*#303 xxf ,log)( $ , 10 %" ,
!
1
1 O x
y
(!) !>1
!
1
1
O x
y
(") 0<!<1
18
!"#$%&'()*&'# +,-:
1) x,yx y, $#$log 4) 2121 loglog)(log xxxx ,,, &$
2) x, x, $log .!+ x, x, $log 5) 21
2
1 logloglog xxx
x,,, '$((
)
*++,
-
3) 1log $,, .!+ 01log $, 6) 11 loglog x-x ,k
, $
7) !" 1!, , #$#%: 2121 loglog xxxx ,, "#"
#$.
!" 10 "" , , #$#% : 2121 loglog xxxx ,, !#" .
8) ,xx e, ln$ , !4(5 ,e, ln$ .
/- "010"2$3 ,4"*- -564*&$ '# ,7$ "1*8"+%#57 +,- ,0 6175-'*"*-*4'#$0 54'-
9*:0 ;6*&$ $+7'0.
<# ,7 9*=%#-0 ,3$ "010"2$3 >10?-@.$ "0105,25#3$ '"*1*4'# $0 56#A-25*&-
'# ,-B >10?-@;B "0105,25#-B #$+B '#>2:*& 01-%'*4 5&$01,=5#3$, +"3B 5,7$ "010@2,3 #?01'*>=.
1 !"#! - $%&'('#) $%&)"*+$+$ 140
',)"-!.+
!" #"$"%&'%(&( )$"*+,- ,-.( /+" "#0 &+1 #"$",-&2 %34"$&'%(+1:
i) |)( x|xf ! , ii) |
)(x|
xg1
! , iii) |1
)(-x|
xh1
! .
/%$+
i) !"#$%& '("$)*&+,-./ 0"(1$%& *2 )-+&"*2)2
xx! !)( %($ 3'/$*( *2+ |)(|)( x!xf ! .
ii) !"#$%& '("$)*&+,-./ 0"(1$%& *2 )-+&"*2)2
xx!
1)( ! %($ 3'/$*( *2+ |)(|)( x!xg ! .
iii)4'/$56 )1()( "! xgxh ,2 0"(1$%6 '("&)*()2
*27 h '",%8'*/$, (+ ./*(*,'9),-./ *2 0"(-
1$%6 '("&)*()2 *27 g %(*& .9( .,+&5( '",7 *( 5/:$&.
#012324 056472809:6 ;)*< ,$ )-+("*6)/$7:
1)(
2
3
##
!x
xxxf %($ xxg !)( .
=("(*2",8./ >*$:
y= |x| y=x
O x
y 19
y
x!
1
y
x!
1
| |
O x
y20
21
y
x!
"1
1| |
1
1
y
x!
1
| |
O x
y
1 !"#! – $%&'('#) $%&)"*+$+$ 141
— !" #$%&'()#*"+ f ,&" g -.!$% ,!"%/ 0*12! !'"#3!4 (! #4%!5! !!A ,&"
— 6"& ,78* Ax" "#.4*" )()( xgxf ! , &9!4
)(1
)1(
1)(
2
2
2
3
xgxx
xx
x
xxxf !!
##
!##
! .
:(;% 0*'20(<#; &$() 5-3* /(" !" #$%&'()#*"+ f ,&" g *2%&" 2#*+. =*%",7:
O!"#$%#
>4! #$%&'()#*"+ f ,&" g 5-6!%(&" &'() /(&%:
$ -.!$% (! 21"! 0*12! !'"#3!4 ! ,&"
$ 6"& ,78* Ax" "#.4*" )()( xgxf ! .
="& %& 1;5?#!$3* /(" 14! #$%&'()#*"+ f ,&" g *2%&" 2#*+ 6'79!$3* gf ! .
@#(< (?'& f, g 14! #$%&'()#*"+ 3* 0*12&
!'"#3!4 !, " &%("#(!2.<+ ,&" # -%& $0!-
#4%!5! (<% ! ,&" ". A% 6"& ,78* #x"
"#.4*" )()( xgxf ! , (/(* 5-3* /(" !" #$-
%&'()#*"+ f *+, g (&-+, &'() './ '0-/-1/ ,. (:.. 22)
="& 0&'71*"63&, !" #$%&'()#*"+
1
1)(
2
%%
!x
xxf ,&"
x
xxxg
#!
2
)( ,
0!$ -.!$% 0*12& !'"#3!4 (& #4%!5& }1{%!!A ,&" }0{%!!B &%("#(!2.<+, *2-%&" 2#*+ #(! #4%!5! }1,0{%!!# , &9!4 6"& ,78* #x" "#.4*"
1)()( #!! xxgxf .
-./0123 41 5678.9:5123 @#(< !" #$%&'()#*"+
xxf %! 1)( , xxg !)(
,&" !"
xxx$ #%! 1)(1 , xxx$ %%! 1)(2 , xxx$ &%! 1)(3 , x
xx$ %!
1)(4 .
B&'&(;'!43* /(":
&) C! 0*12! !'"#3!4 (<% 21 ,$$ ,&" 3$ *2%&" (! #4%!5! ]1,0[ , 0!$ *2%&" ; (!3)
(<% 0*12<% !'"#3!4 ]1,(%'!A ,&" ),0[ #'!B (<% f, g, *%? (! 0*12! !'"#3!4
(;+ 4$ *2%&" (! #4%!5! ]1,0( , 0!$ *2%&" ; (!3) (<% !, " &% *D&"'-#!$3* (& x
6"& (& !0!2& "#.4*" 0)( !xg , ,&"
()*
x
y
%
#
A B
22
1 !"#! - $%&'('#) $%&)"*+$+$ 142
!) )()()(1 xgxfx! !" , )()()(2 xgxfx! #" , )()()(3 xgxfx! $" , )(
)()(4
xg
xfx! " .
"#$ %&'()*+%,#$ 321 ,, !!! -(# 4! *#$ ./0, 12)3#%0(, 4#(53)1, 6#'70,'3 -(# 89-
.:-3 ('*#%*3:;<$ *<' gf , . =,'#-1:
>):?3&0, <$ !"#$%&'( gf ! , )%(*$#! g-f , +%,-'.,$ fg -(# /0123$ g
f 4@3
%&'()*+%,<' f, g *#$ %&'()*+%,#$ 0, *@83&$
)()())(( xgxfxgf !"!
)()())(( xgxfxgf #"#
)()())(( xgxfxfg "
)(
)()(
xg
xfx
g
f"%%
&
'(()
*.
"3 8,4:3 3)#%03@ *<' gf ! , gf # -(# fg ,:'(# 9 *30+ BA+ *<' 8,4:<' 3)#-%03@ " -(# # *<' %&'()*+%,<' f -(# g ('*#%*3:;<$, ,'A *3 8,4:3 3)#%03@ *9$
g
f ,:'(# *3 BA+ , ,B(#)3&0/'<' *<' *#0A' *3& x 83& 094,':?3&' *3' 8()3'3-
0(%*+ )(xg , 49.(4+ *3 %@'3.3
Axx ,|{ -(# Bx, , 0, }0)( -xg .
$,-./01 02-34560/7-
C%*< 9 %&'1)*9%9 1)( #" xx! . D *#0+ *9$ ! %*3 x 083),: '( 3)#%*,: %, 4@3
51%,#$ <$ ,B+$:
() E*3 !,x ('*#%*3#;:?3&0, *3' ()#207 1#" xy -(# %*9 %&'/;,#(
!) %*3 1#" xy ('*#%*3#;:?3&0, *3' ()#207 1#" xy , ,57%3' 01.#" xy .
y x" #1
g f/
y=x#1
23
g f
x
E*9 4#(4#-(%:( (&*+ ,05(':?3'*(# 4@3 %&'()*+%,#$:
() 9 1)( #" xxf , 83& /;,# 8,4:3 3)#%03@ *3 %@'3.3 !"A ($% !&'() -(#
!) 9 yyg ")( , 83& /;,# 8,4:3 3)#%03@ *3 %@'3.3 ),0[ !0"B ()% !&'().
C*%#, 9 *#0+ *9$ ! %*3 x 6)15,*(# *,.#-1
1 !"#! – $%&'('#) $%&)"*+$+$ 143
))(()( xfgx! ! .
! "#$%&'("( ! )*+,'-. "#$%&"' (') f *& ('$ g /-. "#012)34,'-. 0, gof .
52 6,732 2&."028 '(9 ! 7,$ ,3$-. 2):/)(&2 '2 6,732 2&."028 + '(9 f, -))% 6,&.-2&34,'-. "'- Ax" +.- '- 2623- ( '.0; )(xf -$;/,. "'2 6,732 2&."028 , '(9 g,
7()-7; ,3$-. '2 "8$2)2 ),1[1 #$!A . <,$./%:
!"#$%!$
=$ f, g ,3$-. 782 "#$-&';",.9 0, 6,732 2&."028 +, , -$'."'23>?9, ':', 2$2-
0%42#0, &'()*&+ ,+- f .* ,+( g, /-. '( "#012)342#0, 0, gof , '( "#$%&'("(
0, '862
))(())(( xfgxgof ! .
g f%
g(B) A
g
B f (A)
f
A1
g( f (x))
f (x)
x
24
52 6,732 2&."028 '(9 gof -62',),3'-. -6: :)- '- "'2.>,3- x '2# 6,732# 2&.-"028 '(9 f +.- '- 2623- '2 )(xf -$;/,. "'2 6,732 2&."028 '(9 g. @()-7; ,3$-. '2 "8$2)2
})(|{1 BxfAxA ""! .
A3$-. B-$,&: :'. ( gof 2&34,'-. -$ &'1A , 7()-7; -$ &'( BAf )( .
/"!$!01
C'( "#$*>,.- /-. ", :)( '($ */'-"( '2# 1.1)32#, D- -">2)(D280, 0:$2 0, "#-
$-&';",.9 62# 2. "#$D*",.9 '2#9 *>2#$ 6,732 2&."028 234&,+.5 ; 6(7&+ 235-&,+.4,7(.
',)"-!.+
8&,7 93 &:(5;,<&*3- xxf ln)( ! =53 xxg !)( . >5 ?;*@,* ,3- &:(5;,<&*3-:
i) gof ii) fog .
1 !"#! - $%&'('#) $%&)"*+$+$ 144
,%$+
! "#$%&'("( f )*+, -+./0 0&,"102 '0 ),0( !"#fD , +$3 ( g '0 ).0[ "!#gD .
i) 4,5 $5 0&/6+'5, ( -5&%"'5"( ))(( xfg -&)-+,:
fDx$ 75, gDxf $)( (1)
8, ,"0.2$515,
11
0
0ln
0
0)(
0%&
'()
%*
&'()
%*
&'()
%*
xx
x
x
x
xf
x,
.(95.8 -&)-+, 1%x . :-01)$;<, 0&/6+'5, ( gof 75, +/$5,
xxgxfgxgof ln)(ln))(())(( ### , =,5 7%>+ )1[ "!$x .
ii) 4,5 $5 0&/6+'5, ( -5&%"'5"( ))(( xgf -&)-+,:
gDx$ 75, fDxg $)(
8, ,"0.2$515,
00
0
0
0
0)(
0*&
'()
*%
&'()
*%
&'()
*%
xx
x
x
x
xg
x,
.(95.8 -&)-+, 0*x . :-01)$;<,0&/6+'5, ( fog 75, +/$5,
xxfxgfxfog ln)())(())(( ### , =,5 7%>+ )0( "!$x .
!"#$%&
++++ ?'($ -5&5-%$; +@5&10=8 -5&5'(&021+ A', foggof , . 4+$,7%, 5$ f, g +/$5, .20 "#$5&'8"+,< 75, 0&/60$'5, 0, gof 75, fog , 'A'+ 5#')< ! " # " $ # % & ' ( ) * + " , - & . / /"+<.
+ B$ hgf ,, +/$5, '&+,< "#$5&'8"+,< 75, 0&/6+'5, ( )(gofho , 'A'+ 0&/6+'5, 75, ( ofhog)( 75, ,"*2+,
ofhoggofho )()( # .
C( "#$%&'("( 5#'8 '( 9)1+ "2$>+"( ';$ f, g 75, h 75, '( "#1D09/60#1+ 1+ hogof . ! "2$>+"( "#$5&'8"+;$ =+$,7+2+'5, 75, =,5 -+&,""A'+&+< 5-A '&+,< "#$5&'8"+,<.
1 !"#! – $%&'('#) $%&)"*+$+$ 145
)$,+$'#$
!" #$!%!&
1. !"#" $%&'# (" )$*%" "+#,-". (/& )'+'01(/ ,2&'+(3,$/&;
i) 23
2)(
2 !"!
#xx
xxf , ii) xxxf "!"# 21)( 3
iii) x
xxf
21)(
"# iv) )1ln()( xexf "#
2. 4#' )"#56 (#-56 ("2 !$x 7 8+'9#03 )'+1,(',7 (76 ,2&1+(7,76 f
:+%,0$('# )1&/ '); ("& 1<"&' xx% , ;('&:
i) 34)( 2 !"# xxxf , ii) x
xxf
"!
#1
1)( , iii) 1)( "# xexf .
3. 4#' )"#56 (#-56 ("2 !$x 7 8+'9#03 )'+1,(',7 (76 ,2&1+(7,76 f
:+%,0$('# )1&/ '); (7 8+'9#03 )'+1,(',7 (76 ,2&1+(7,76 g, ;('&:
i) 12)( 3 !!# xxxf 0'# 1)( !# xxg
ii) 2)( 3 "!# xxxf 0'# 2)( 2 "!# xxxg .
4. =# '&>+/)"?;8"# $0(#-".& "(# (" .@"6 ("2 '&>+A)"2 *%&$('# '); (#6 ,2&'+(3,$#6:
64,7089,2)( !# xxA (8#' ("26 1&*+$6) 0'#
48,7175,2)( !# xx! (8#' (#6 82&'%0$6)
;)"2 x ,$ $0'(",(1, (" -30"6 ("2 :+'B%"&'. C$ -%' '&',0'93 :+5>7-
0$ 5&' ",(; '); :+'B%"&' -30"26 0,45 m.
') D& )+"5+B$('# '); 1&*+' )"#" 3('& (" .@"6 ("2;
:) A& )+"5+B$('# '); 82&'%0' )"#" 3('& (" .@"6 (76;
5. C.+-' -30"26 20#& cm 0;:$('# ,$ *." 0"--1(#' -$ -307 x cm 0'# )20( x" cm. E$ (" )+A(" 0"--1(# ,B7-'(%F"2-$ ($(+18/&" 0'# -$ ("
*$.($+" #,;)?$2+" (+%8/&". G' :+$%($ (" 1>+"#,-' (/& $-:'*A& (/& *." ,B7-1(/& /6 ,2&1+(7,7 ("2 x.
6. N' )'+',(3,$($ 8+'9#01 (7 ,2&1+(7,7:
1 !"#! - $%&'('#) $%&)"*+$+$ 146
i) 1||
)( !"x
xxf , ii) ||)( xxxf " ,
iii) 1
1
,
,
1
3)(
#$
%&'
!!(
"x
x
x
xxf iv) |ln|)( xxf " .
!"# "$% &' ()"*#+, $")-.&".' /" $)0.1#0)2.3&3 &0 .4/050 &6/ &#-78/ &'9 f .3 +":37#- $3)2$&6.'.
7. ;" 3<3&-.3&3 .3 $0#39 "$% 	 $")"+-&6 $3)#$&8.3#9 32/"# gf " .
=	 $3)#$&8.3#9 $0> 32/"# gf ) /" $)0.1#0)2.3&3 &0 3>)4&3)0 1>-
/"&% >$0.4/050 &0> * .&0 0$020 #.?43# )()( xgxf " .
i) 2)( xxf " +"# 2)()( xxg "
ii) ||
1)(
2
2
xx
xxf
!(
" +"# ||
11)(
xxg ("
iii) 1
1)(
(
("
x
xxf +"# 1)( !" xxg .
8. @2/0/&"# 0# .>/")&,.3#9
xxf
11)( !" +"#
x
xxg
("
1)( .
;" A)32&3 	 .>/")&,.3#9 fggfgf ,, (! +"# g
f.
9. B70269 (#" 	 .>/")&,.3#9
xxxf
1)( !" +"#
xxxg
1)( (" .
10. ;" $)0.1#0)2.3&3 &' .>/-)&'.' gof , "/
i) 2)( xxf " +"# xxg ")( , ii) xxf '7)( " +"# 21)( xxg ("
iii) 4
)(!
xf " +"# xxg 3*)( " .
11. @2/0/&"# 0# .>/")&,.3#9 1)( 2 !" xxf +"# 2)( (" xxg . ;" $)0.1#0-
)2.3&3 	 .>/")&,.3#9 gof +"# fog .
12. ;" 3+*)-.3&3 &' .>/-)&'.' f 69 .4/:3.' 140 , $3)#..0&C)6/ .>-
/")&,.36/, "/
i) )1('7)( 2 !" xxf , ii) 13'72)( 2 !" xxf
1 !"#! – $%&'('#) $%&)"*+$+$ 147
iii) )1ln()( 2 !" xexf , iv) )3(!")( 2 xxf " .
B! "#$%$&
1. #$ %&'()*'&+(,-, -! (./0&-!(! f -!1 '%'+$1 ! 2&$3*45 %$&0(-$(!
,+/$*:
1
2 1 O
i) y
x
ii) 2
2 1 O
x
y
4 3
1
2 1 O
iii) y
x
2. 6/$ 4'.-+ 4.7*/)&*4'8 (95"$-'1 :9,* $4-+/$ ;0(!1 x cm 4$* <24' 628
cm3. =' .7*4< ->/ ;0(,>/ 4'(-+?,* 4 )&9. $/0 cm2, ,/@ -' .7*4< -!1 4.7*/)&*451 ,%*30/,*$1 1,25 )&9. $/0 cm2. #$ ,43&0(,-, -' (./'7*4<
4<(-'1 >1 (./0&-!(! -'. x. A<(' 4'(-+?,* :/$ 4'.-+ ", $4-+/$ ;0(!1 5 cm, 4$* 8B'1 8 cm;
3. C-' )*%7$/< (95"$ ,+/$* 1"AB , 3"!" 4$* 2""# .
#$ ,43&0(,-, -' ,";$)</ -'. 2&$""'(4*$(":/'. 9>-
&+'. >1 (./0&-!(! -'.
AMx " , <-$/ -' $ )*$-
2&03,* -' ,.D82&$""' -"5-
"$ !".
4. 6/$ '&D'2@/*' %&$' 8-
B'.1 x cm ,+/$* ,22,2&$"-
":/' (, :/$ -&+2>/' !("
;0(!1 10"B" cm 4$* 8-
B'.1 5"!# cm. #$ ,43&0-
(,-, -' ,";$)< ) 4$* -!/ %,&+",-&' * -'. '&D'2>/+-'. >1 (./0&-!(! -'. x.
5. #$ %$&$(-5(,-, 2&$3*40
-! (./0&-!(!:
i) 2
|1||1|)(
!##"
xxxf , ii)
2
|!"|!")(
xxxf
#" , ]2,0[ +x$ .
A%< -! 2&$3*45 %$&0(-$(! -!1 f /$ %&'()*'&+(,-, -' (8/'7' -*"@/ -!1 (, 4$D,"*0 %,&+%->(!.
6. #$ ;&,+-, (./0&-!(! f -:-'*$, @(-, /$ *(98,*:
% & '
A
'
) #
" B M x
) $
" & #
x x
K B
N
A
1 !"#! - $%&'('#) $%&)"*+$+$ 148
i) 22))(( 2 !!" xxxfog , !" 1)( !" xxg
ii) 21))(( xxfog !" , !" 2)( xxg #"
iii) |#$"|))(( xxgof " , !" 21)( xxg #" .
7. %&"'"(!) ') #$"!*(+#,)- 1)( !" xxf .!) 2)( !" !xxg . /)! 0')! ()1+
('$ !$! )#23,) goffog " .
8. %&"'"(!) ') #$"!*(+#,)-:
!x
"!xxf
#!
")( , 1, 2!" #% .!) 12)( !#" xxxg .
4! !0'5,&6,(, 7()
!) xxff "))(( , 8)! .9:, }{!x #$! .!)
;) xxgg "))(( , 8)! .9:, ]1,0[$x .
9. <) 0'=,'571') 1)!- 07=>- ,.()1'3" 7(), 7(!" ' 0=>:$#17- # (>- 07-
=>- ,&"!) x ,.!('"(95,- 2)=)95,- 9('1!, :! $09*2'$" #(>" 07=>
)(210 2 xxN !" 2)=)95,- !$('.&">(!. ?*,$",- 5,&2"'$" 7() #, t @(>
!07 #+1,*! ' 0=>:$#17- (>- 07=>- :! ,&"!) 4!t ,.!('"(95,- 2)-=)95,- 9('1!.
i) 4! ,.A*9#,(, ('" !*):17 $ (B" !$('.)"+(B" (>- 07=>- B- #$-
"9*(>#> ('$ t.
ii) C7(, :! $09*2'$" #(>" 07=> 120 2)=)95,- !$('.&">(!.;
1.3 ,!&!*!&'$ $%&)"*+$'#$ -
)&*#$*"!-+ $%&)"*+$+
,./.0./12 34/5607378
&&&& <) @""'),- “8">#&B- !36'$#! #$"9*(>#>”, “8">#&B- A:&"'$#! #$"9*(>#>”
,&"!) 8"B#(@- !07 0*'>8'31,"> (96>. D$8.,.*)1@"!, 19:!1, 7():
!"#$%!$
E)! #$"9*(>#> f =@8,(!)(1) :
(1) E)! #$"9*(>#> f =@8,(!), !0=F-,:
1 !"#! – $%&'('#) $%&)"*+$+$ 149
!!!! !"#$%&' ()*+,$( !’ "#$ ! " # $ % & ' ( ) %&' ()*+&' &,-!.&/ %01, 2%$# 3-$
&(&-$*4(&%) )xx "21 , .) 21 xx # -!5/)-:
)()( 21 xfxf # (65. $)
!!!! !"#$%&' -.%"+,$( !’ "#$ ! " # $ % & ' ( ) %&' ()*+&' &,-!.&/ %01, 2%$# 3-$
&(&-$*4(&%) )xx "21 , .) 21 xx # -!5/)-:
)()( 21 xfxf $ (65. 7)
% &'' (' ' )
* (a)
x2 x1 x
y
f (x2)
f (x1)
% &'' (' ' )
* x2 x1
f (x1)
f (x2)
x
25 y
(+)
8-$ #$ *09:!&'.) 2%- 0 f )+#$- 3#0!+;1 $/<&'!$ ($#%-!%&+5;1 3#0!+;1 =>+-#&'!$) !) "#$ *-?!%0.$ ), 3,?=&'.) f ) ($#%-!%&+5;1 f )).
8-$ ($,?*)-3.$, 0 !'#?,%0!0 2)( xxf ) :
— )+#$- 3#0!+;1 $/<&'!$ !%& ),0[ *+ , $=&/ 3-$
210 xx #, "5&'.) 22
21 xx # , *09$*4
)()( 21 xfxf #
— )+#$- 3#0!+;1 =>+#&'!$ !%& ]0,(-+ , $=&/ 3-$
021 ,# xx "5&'.) 120 xx -#-, , &(2%) 21
220 xx #, , *09$*4
)()( 21 xfxf $ .
@# .-$ !'#?,%0!0 f )+#$- 3#0!+;1 $/<&'!$ 4 3#0!+;1 =>+#&'!$ !’ "#$ *-?!%0-
.$ ) %&' ()*+&' &,-!.&/ %01, %2%) 9".) 2%- 0 f )+#$- !"#$%&' /+"01+"# $1+ ,. 6%0# (),+(%;!0 (&' %& ()*+& &,-!.&/ %01 f )+#$- "#$ *-?!%0.$ ) A$- 0 f )+#$- 3#0!+;1 .%� !’ $'%2, %2%) >$ 9".), $(9:1, 2%- 0 f )+#$- 3#0!+;1 .%&-
#0.
)-./0101 2345.06267
! ()*+,$( !’ "#$ *-?!%0.$ ), 2%$# 3-$ &(&-$*4(&%) )xx "21, .) 21 xx # -!5/)-
)()( 21 xfxf , .
! -.%"+,$( !’ "#$ *-?!%0.$ ), 2%$# 3-$ &(&-$*4(&%) )xx "21, .) 21 xx # -!5/)-
)()( 21 xfxf . .
O x
y=x2
y 26
1 !"#! - $%&'('#) $%&)"*+$+$ 150
!" #$$%"&' “(#)"*+%”, “&,-."*+%”, */$-0+1*1' &2$3" 43" 3/+#' )$5*+#' 367 60%-
1)%8(&$&' +-9&"'. :/)4&40"(#$3 (-;3(& 7+":
!"#$%!$
<"3 */$-0+1*1 f (& 6&=2% %0"*(%8 ! ;3 ,#(& 7+":
! >30%/*"-?&" *+% Ax "0 (%,"47) &'()*+,, +% )( 0xf , 7+3$
)()( 0xfxf # )"3 4-;& Ax" (:.. 273)
! >30%/*"-?&" *+% Ax "0 (%,"47) -./0)*+,, +% )( 0xf , 7+3$
)()( 0xfxf $ )"3 4-;& Ax" (:.. 27@).
(a) Cf
f (x0)
f (x)
O
x
y
x0 x
(")
C f
f (x0)
f (x)
O
x
y
x0 x
27
A"3 630-=&")(3:
— B */$-0+1*1 1)( 2 %&' xxf (:.. 283) 630%/*"-?&" (#)"*+% *+% 00 'x , +%
1)0( 'f , 3C%8 )0()( fxf # )"3 4-;& !"x .
— B */$-0+1*1 |1|)( &' xxf (:.. 28@) 630%/*"-?&" &,-."*+% *+% 10 'x , +%
0)1( 'f , 3C%8 )1()( fxf $ )"3 4-;& !"x .
())
y=&x2+1
1
O x
y
y=|x&1|
1 O x
y
(=)
28
1 !"#! – $%&'('#) $%&)"*+$+$ 151
— ! "#$%&'("( xxf ())( ! (*+. 29,) -,&.#"/%01/ )23/"'., '. 1!y , "1 4,52$,
,-6 ', "()17, 2
2!
k! " , !#k 4,/ 18%+/"'., '. 1$!y , "1 4,52$, ,-6 ', "()17,
22
!k! $ , !#k , ,9.: 1()1 %%$ x 3/, 4%51 "#x .
— ! "#$%&'("( 3)( xxf ! (*+. 29;) <1$ -,&.#"/%01/ .:'1 )23/"'., .:'1 18%+/-"'., ,9.: 17$,/ 3$("7=> ,:?.#",.
O
(1)
y=()x
2- 5-/2 3-/2 $-/2
-/2 -
1
$1
y
x
O x
y
("')
y=x3
29
@-=> 17<,)1 4,/ "', -&.(3.:)1$, -,&,<173),',, %881> "#$,&'A"1/> -,&.#"/%-
0.#$ )6$. )23/"'., %881> )6$. 18%+/"'., %881> 4,/ )23/"'. 4,/ 18%+/"'. 4,/ %881> .:'1 )23/"'. .:'1 18%+/"'..
B. (.8/46) )23/"'. 4,/ '. (.8/46) 18%+/"'. )/,> "#$%&'("(> f 823.$',/ (.8/4%)
!"#$%!%! '(> f.
$,-./0121 11$
C"'= ( "#$%&'("( x
xf1
)( ! D,&,'(&.:)1 6'/ 3/,
.-./,<A-.'1 0, 21 &xx /"+:1/ ( "#$1-,3=3A:
“A$ 21 xx & , '6'1 )()( 21 xfxf & ”,
-.# "(),7$1/ 6'/:
“B, </,9.&1'/4% "'./+17, fDxx #21 , 2+.#$ -%$'.-
'1 </,9.&1'/42> 1/46$1>”.
E63= '(> '181#',7,> /</6'(',> ( "#$%&'("(
xxf
1)( ! 8231',/ "#$%&'("( 11$ (2$, -&.> 2$,). F1$/4%:
&'()*&)
G/, "#$%&'("( "'Af : 8231',/ +,-.#%/+/ 11$ , 6',$ 3/, .-./,<A-.'1 Axx #21 , /"+:1/ ( "#$1-,3=3A:
,$ 21 xx & , '6'1 )()( 21 xfxf & .
f (x2)
f (x1)
x2 x1 O x
y
x!
1 y 30
1 !"#! - $%&'('#) $%&)"*+$+$ 152
!" #$#%&%' (" )*+$+ #$+,"-./0"*#- 1*-:
!-# (2/)3*4(4 !!Af : "5/#- !"#$%&'!' 11" , #/ .#- 61/+ #/ %-# +$+-#,'-
$+*" Axx #21 , -(70"- 4 (2/"$#%&%':
#/ )()( 21 xfxf $ , *1*" 21 xx $ .
8*(- %-# $#3),"-%6#:
— 9 (2/)3*4(4 !x"xf %$)( , 6" 0&" "5/#- (2/)3*4(4 11" . (:7. 31#, ;)
O x x2 x1
f (x1)
f (x2)
(a)
y
O x
y
x2 x1
f (x2)
f (x1)
(!)
O x
y
x2
!
x1
(#)
31
#<+0, #/ 2$+=>(+26" 1*- )()( 21 xfxf $ , *1*" >7+26" ,-#,+7-.):
!"x!"x %$% 21
21 "x"x $
21 xx $ .
— 9 (2/)3*4(4 !xf $)( ,"/ "5/#- (2/)3*4(4 1-
1 (:7. 31%), #<+0 !xfxf $$ )()( 21 %-# +$+-#-
,'$+*" !#21 , xx ,
— 9 (2/)3*4(4 2)( xxf $ (:7. 32) ,"/ "5/#- (2-
/)3*4(4 11" , #<+0 1)1()1( $$" ff #/ .#- "5/#- 11&" .
()*+,-
' ?$1 *+/ $#3#$)/& +3-(61 $3+.0$*"- 1*- 6-# (2/)3*4(4 f "5/#- 11" , #/ .#- 61/+ #/:
— @-# .)=" (*+-7"5+ y *+2 (2/1A+2 *-6B/ *4C 4 "D5(&(4 yxf $)( >7"- #.3-;BC 6-# A0(4 &C $3+C x.
— E"/ 2$)37+2/ (46"5# *4C %3#<-.'C *4C $#3)(*#(4C 6" *4/ 5,-# *"*#%6>/4. ?2*1 (46#5/"- 1*- .)=" +3-F1/*-# "2="5# *>6/"- *4 %3#<-.' $#3)(*#(4 *4C f *+ $+A0 (" >/# (46"5+ (:7. 33#).
"1 1
1
y=x2
O x
y 32
1 !"#! – $%&'('#) $%&)"*+$+$ 153
x
y
!"#$%&'!' 1-1
O
O x2 x1
B A
x
y
!"#$%&'!' ()* 1-1
33
! +# ,*- !"#$%&'!' ./#-* 0#'!/12 ,3#(&3#',
&(&. 4%35-#62, ./#-* !"#$%&'!' "11" " .
7&!*, 3* !"#-%&8!.*2 !"xxf #$)(1 , 0%" ,
32 )( "xxf $ , 0%" , x"xf $)(3 , 10 %& " 9-*
xxf "log)(4 $ , 10 %& " , ./#-* !"#-%&8!.*2 11" . :4$%)3"#, (,12, !"#-%&8!.*2 43" ./-
#-* 11" -;;$ <.# ./#-* 0#'!/12 ,3#(&3#.2, (412 0*- 4-%$<.*0,- ' !"#$%&'!'
'(
')*
+
,$
0,1
0,)(
xx
xxxg (=). 34).
),-./-0123 /4,50-3/3
!!!! 7!&1 ,*- !"#$%&'!' !-Af : . +# "43>?-!3",. (&* -"&8 ./#-* 11" , &(&. 0*- 9$>. !&3*)./3
y &3" !"#(;3" &*,6#, )(Af , &'2 f "4$%).* ,3-
#-<*9( !&3*)./3 x &3" 4.</3" 3%*!,3@ &'2 # 0*-
&3 343/3 *!)@.* yxf $)( . A43,?#12 3%/B.&-* ,*-
!"#$%&'!'
!-)(: Afg
,. &'# 343/- 9$>. )(Afy. -#&*!&3*)/B.&-* !&3
,3#-<*9( Ax. 0*- &3 343/3 *!)@.* yxf $)( .
+4( &3# &%(43 43" 3%/!&'9. ' g 4%39@4&.* (&*:
— ?).* 4.</3 3%*!,3@ &3 !@#3;3 &*,6# )(Af &'2 f,
— ?).* !@#3;3 &*,6# &3 4.</3 3%*!,3@ # &'2 f 9-*
— *!)@.* ' *!3<"#-,/-:
xygyxf $/$ )()( .
O x
y
y=g(x)
34
O x
x
y=f (x)
y 35
1 !"#! - $%&'('#) $%&)"*+$+$ 154
!"#$ %&'()*+, $#,, (* & f (*#,%#-,.)/+, #- x %#- y, #$#+ & g (*#,%#-,.)/+, #- y %#- x 0(, (-*#,%#1$234. 5&6(78 & g +)*(, & (*#)%#1-2& 7,(7,0(%)( #&4 f. 9,( #- 6$:- ("#$ & g 6;:+#(, !"#$%#&'() %*"+&#)%) #&4 f 0(, %"'<-6)/+-#(, '+ 1!f . =>-';*34 ;.-"'+
xyfyxf "#" ! )()( 1
->$#+ Axxxff $"! ,))((1 0(, )(,))(( 1 Afyyyff $"! .
9,( >(1?7+,:'(, ;%#3 & +0@+#,08 %"*?1#&%&
x!xf ")( . A>34 +)*(, :*3%#$ & %"*?1#&%&
("#8 +)*(, 11! '+ >+7)- -1,%'-B #- ! 0(, %B-
*-6- #,'C* #- ),0( %& . =>-';*34 -1)/+#(, &
(*#)%#1-2& %"*?1#&%& 1!f #&4 f. D %"*?1#&-
%& ("#8, %B'23*( '+ $%( +)>('+ >1-&:-"';-*34,
— ;.+, >+7)- -1,%'-B #- ),0( %&
— ;.+, %B*-6- #,'C* #- ! 0(,
— (*#,%#-,.)/+, 0?@+ ),0( %&$y %#- '-*?7,0$ !$x :,( #- ->-)- ,%.B+,
y! x " . =>+,78 $'34
yxy! !x log"#"
@( +)*(, yyf !log)(1 "! . =>-';*34, & (*#)%#1-2& #&4 +0@+#,084 %"*?1#&%&4 x!xf ")( , 10 '( ! , +)*(, & 6-:(1,@',08 %"*?1#&%& xxg !log)( " . E"*+>C4
!!!!$" xx, x, ,log ,!- )(0,,log %&$" xx, x,
) !4 >?1-"'+ #C1( ',( 11! %"*?1#&%& f
0(, (4 @+318%-"'+ #,4 :1(2,0;4 >(1(%#?-
%+,4 C 0(, C * #3* f 0(, #&4 1!f %#- )7,-
%B%#&'( (F$*3* (E.. 37). =>+,78
xyfyxf "#" ! )()( 1 ,
(* ;*( %&'+)- ),( "!M (*80+, %#& :1(2,08
>(1?%#(%& C #&4 f , #$#+ #- %&'+)-
),( !"# * @( (*80+, %#& :1(2,08 >(1?%#(%&
g
f
f(A) A
y=f(x) g(y)=x
36a
(0,+%)
f -1
f
!
y=ax logay=x
36"
y=x
C$
C
O x
M$(",!)
M(!,") y
37
1 !"#! – $%&'('#) $%&)"*+$+$ 155
C ! !"# 1"f $%& %'!&(!)*+,#. -% ("./0%, *.,#, %1!2 /0'%& (1../!)&$2 ,# 3)4# !"' /15/0% 341 6&74!4./0 !&# 8,'0/# xOy $%& yOx !! . 934.:',#:
;& 8)%+&$:# 3%)%(!2(/&# C $%& C ! !,' (1'%)!<(/,' f $%& 1"f /0'%& (1../-!)&$:# ,# 3)4# !"' /15/0% xy # 341 6&74!4./0 !&# 8,'0/# xOy $%& yOx !! .
=!(&, 4& 8)%+&$:# 3%)%(!2(/&# !,' (1'%)!<(/,' x!xf #)( $%& xxg !log)( # ,
10 $% ! , /0'%& (1../!)&$:# ,# 3)4# !"' /15/0% xy # .
',)"-!.+
!" "#$%&'()&* +)' , -./01),-, 12)( 23 &# "xexf &*/"' 11" 2"' /" 31&4&* ,
"/)*-)1$56 ),7.
/%$+
— =(!, !'21 , xx ./ )()( 21 xfxf # . >% 6/0?41./ *!& 21 xx # . @)28.%!& :741./ 6&%647&$2:
)()( 21 xfxf #
1212 223213 &#& "" xx ee
223213 22 "" # xx ee
223213 "" # xx ee
2323 21 "#" xx
21 33 xx #
21 xx # .
— A&% '% B)4C./ !"' %'!0(!)4+" !"# f 5:!41./ )(xfy # $%& DC'41./ ,# 3)4# x. =741./ D4&3*':
yeyxf x #&(# " 12)( 23
12 23 "#( " ye x
2
123 "#( " y
e x
2
1ln23
"#"(
yx , 1)y
22
1ln3 &
"#(
yx , 1)y
1 !"#! - $%&'('#) $%&)"*+$+$ 156
3
2
2
1ln
3
1!
"#$
yx , 1%y .
!"#$%&'(, 3
2
2
1ln
3
1)(1 !
"#" y
yf , 1%y , #")*+ , -&*./*0#1, *,( f +.&-2 , /3-
&40*,/,
3
2
2
1ln
3
1)(1 !
"#" x
xf , 1%x .
)$,+$'#$
!" #$!%!&
1. N- 50+.*+ "#2+( -") *2( "-0-64*' /3&-0*7/+2( +.&-2 8&,/.'( -9:#3-/+( 6-2 "#2+( 8&,/.'( 1;./+( i) xxf "# 1)( ii) 1)2ln(2)( ""# xxf
iii) 13)( 1 !# "xexf iv) 1)1()( 2 ""# xxf , 1&x .
2. <- 50+.*+ "#2+( -") *2( "-0-64*' /3&-0*7/+2( +.&-2 "11" " 6-2 82-
6-;+$.- -"’ -3*%( &- 50+.*+ *,& -&*./*0#17 *,( i) 23)( "# xxf v) )1ln()( xxf "#
ii) 1)( 2 !# xxf vi) 1)( !# "xexf
iii) 1)2)(1()( !""# xxxf vii) 1
1)(
!"
#x
x
e
exf
iv) 3 1)( xxf "# viii) |1|)( "# xxf .
3. =.&#&*-2 #2 80-126%( "-0-/*4/+2( *'& /3&-0*7/+'& !gf ,, 6-2 " .
O x
y
y=f (x)
O x
y
y=g(x)
1 !"#! – $%&'('#) $%&)"*+$+$ 157
O x
y
y=!(x)
O x
y
y="(x)
!" #$%&'% ()*%+ "(, '*+ -./"$'0-%*+ "!,gf ,, 12)./ "/'&-'$)34 5"* 6*" 5"7%8&" "(’ ".'1+ /" 2"$9:%'% '4 6$"3*50 ("$9-'"-4 '4+ "/'&--'$)30+ '4+.
4. !" ;%&:%'% ,'*:
i) </ 8*" -./9$'4-4 f %&/"* 6/4-&=+ ">:).-" -% 1/" ;*9-'48" #,
','% 4 -./9$'4-4 f! %&/"* 6/4-&=+ 37&/).-" -') #.
ii) </ ;>) -./"$'0-%*+ gf , %&/"* 6/4-&=+ ">:).-%+ -% 1/" ;*9-'4-
8" #, ','% 4 -./9$'4-4 gf " %&/"* 6/4-&=+ ">:).-" -') #.
iii) </ ;>) -./"$'0-%*+ gf , %&/"* 6/4-&=+ ">:).-%+ -% 1/" ;*9-'4-
8" # 5"* *-2>%* 0)( #xf 5"* 0)( #xg 6*" 597% #x$ , ','% 4 -.-
/9$'4-4 fg %&/"* 6/4-&=+ ">:).-" -') #.
</9?)6" -.8(%$9-8"'" ;*"'.(@/)/'"*, "/ )* gf , %&/"* 6/4-&=+ 37&-/).-%+ -% 1/" ;*9-'48" #.
1.4 !"#! $%&)"*+$+$ $*! !!!!$0x
',-./0/1
A 1//)*" '). )$&). 6%//0745% -'4/ ($)-(97%*" '=/ 8"748"'*5@/ /" "("/'0-
-)./ -% %$='08"'" ,(=+:
— B* )/)89C).8% -'*68*"&" '"2>'4'" %/,+ 5*/4')>;
— T* )/)89C).8% %3"(')81/4 8*"+ 5"8(>?4+ -% 1/" -48%&) '4+;
— B* )/)89C).8% %8#";, %/,+ 8*5',6$"88). 2=$&).;
D'*+ ("$"6$93).+ (). "5)?).7)>/, "$2*59 ($)-%66&C).8% '4/ 1//)*" '). )$&).
“;*"*-74'*59”, -'4 -./12%*" ;*"'.(@/).8% ')/ ".-'4$, 8"748"'*5, )$*-8, ').
)$&). 5"* 8%$*51+ #9-*51+ *;*,'4'1+ '). 5"* '1?)+, %*-96).8% '4/ 1//)*" '4+ -./12%*"+ 8*"+ -./9$'4-4+.
1 !"#! - $%&'('#) $%&)"*+$+$ 158
+ ,--./0 1.2 .34.2
! !"#$ % "&'()#%"%
1
1)(
2
""
#x
xxf .
* "&'()#%"% +&#, -./0 1/234 4)0"546 #4 "6'474
}1{"#!fD 8+0 9)(:/#+0
11
)1)(1()( $#
"$"
# xx
xxxf , 1%x .
;145-'$<, % 9)+:08, #%< 1+)("#+"% /3'+0 % /&=/3+ 1$# xy 5/ />+3)/"% #4 "%-
5/34 A(1,2) (?.. 38). ?#4 ".,5+ +&#@, 1+)+#%)465/ @#0:
“A+=B< #4 x, 80'465/'4 5/ 41404'2,14#/ #)@14 1('$ "#4' (>4'+ xx& , 1)4"/9-93C/0 #4' 1)+95+#08@ +)0=5@ 1, #4 )(xf , 80'465/'4 1('$ "#4' (>4'+ yy & , 1)4-
"/993C/0 #4' 1)+95+#08@ +)0=5@ 2. A+0 5(70"#+, 40 #05-< )(xf /3'+0 #@"4 84'#(
"#4 2 @"4 =-74&5/, 90+ @7+ #+ 1%x 14& /3'+0 +)846'#$< 84'#( "#4 1”.
?#%' 1/)31#$"% +&#, 9)(:4&5/
2)(lim1
#'
xfx
8+0 20+D(C4&5/
“#4 @)04 #%< )(xf , @#+' #4 x #/3'/0 "#4 1, /3'+0 2”.
E/'08(:
F#+' 40 #05-< 50+< "&'()#%"%< f 1)4"/993C4&' @"4 =-74&5/ -'+' 1)+95+#08@
+)0=5@ ( , 8+=B< #4 x 1)4"/993C/0 5/ 41404'2,14#/ #)@14 #4' +)0=5@ 0x , #@#/ 9)(:4&5/
(#'
)(lim0
xfxx
8+0 20+D(C4&5/
“#4 @)04 #%< )(xf , @#+' #4 x #/3'/0 "#4 0x , /3'+0 ( ” ,
“#4 @)04 #%< )(xf "#4 0x /3'+0 ( ”.
f (x)
f (x)
f x( )0 # (
(a)
O x0 x x x
y
f(x0)
(!)
f(x)
f(x)
O x0
(
x x x
y
(")
f(x)
f(x)
O x0
(
x x x
y39
!"#$%#
f(x)
f(x)
2
O 1 x x x
y38
1 !"#! – $%&'('#) $%&)"*+$+$ 159
!"# $% "%&%"'() *+,-%$% "%&%$.&/0-1 #$2:
— 32% (% %(%4.$,*/5-1 $/ #&2/ $.6 f *$/ 0x , "&7"12 . f (% /&841$%2 #*/ 97-:/5-1 “;/($' *$/ 0x ”, <.:%<, . f (% 18(%2 /&2*-7(. *’ 7(% *0(/:/ $.6 -/&=,6
),(),( 00 !xx" ! , ),( 0x" , ),( 0 !x .
— >/ 0x -"/&18 (% %(,;12 *$/ "1<8/ /&2*-/0 $.6 *5('&$.*.6 (?+. 39%, 39@) ,
(% -.( %(,;12 *’ %5$# (?+. 39A).
— B $2-, $.6 f *$/ 0x , #$%( 5"'&+12, -"/&18 (% 18(%2 8*. -1 $/ #&2# $.6 *$/
0x (?+. 39%) , <2%=/&1$2;, %"# %5$#. (?+. 39@).
" C*$), $D&%, . *5('&$.*.
#$%
&'()'
*1,5
1,1)(
xx
xxxf ,
$.6 /"/8%6 . A&%=2;, "%&'*$%*. %"/$1:18$%2 %"#
$26 .-21591816 $/5 <2":%(/0 *+,-%$/6. E%&%$.&/0-1 #$2: — F$%( $/ x "&/*1AA8412 $/ 1 %"# %&2*$1&'
)1( )x , $#$1 /2 $2-76 $.6 f "&/*1AA84/5( #*/ 97-:/5-1 $/( "&%A-%$2;# %&29-# 2. ?$.( "1&8"$)*.
%5$, A&'=/5-1:
2)(lim1
*(+
xfx
.
— GH$%( $/ x "&/*1AA8412 $/ 1 %"# <1I2' )1( &x , $#$1 /2 $2-76 $.6 f "&/*1AA8-4/5( #*/ 91:/5-1 $/( "&%A-%$2;# %&29-# 4. ?$.( "1&8"$)*. %5$, A&'=/5-1:
4)(lim1
*'+
xfx
.
31(2;':
— GH$%( /2 $2-76 -2%6 *5('&$.*.6 f "&/*1AA84/5( #*/ 97:/5-1 $/( "&%A-%$2;#
%&29-# 1, , ;%9D6 $/ x "&/*1AA8412 $/ 0x %"# -2;&#$1&16 $2-76 )( 0xx ) , $#$1 A&'=/5-1:
1)(lim0
,*(+
xfxx
;%2 <2%@'4/5-1: “$/ #&2/ $.6 )(xf , #$%( $/ x $18(12 *$/ 0x %"# $% %&2*$1&', 18(%2 1, ”.
— GH$%( /2 $2-76 -2%6 *5('&$.*.6 f "&/*1AA84/5( #*/ 97:/5-1 $/( "&%A-%$2;#
%&29-# 2, , ;%9D6 $/ x "&/*1AA8412 $/ 0x %"# -1A%:0$1&16 $2-76 )( 0xx & , $#$1 A&'=/5-1:
2
0
)(lim ,*'+
xfxx
;%2 <2%@'4/5-1: “$/ #&2/ $.6 )(xf , #$%( $/ x $18(12 *$/ 0x %"# $% <1I2', 18(%2 2, ”.
4
f(x)
f(x)
2
O 1 x x x
y40
1 !"#! - $%&'('#) $%&)"*+$+$ 160
!1
!2
(a)
f(x)
f(x)
O x0 x x x
y
(!)
f(x)
f(x)
O x0
!1
!2
x x x
y
f(x0)
(")
f(x)
f(x)
O x0
!1
!2
x x x
y41
!"#$ %&'()"*$ )(lim0
1 xfxx "#
$! +%' )(lim0
2 xfxx %#
$! ,"#$ -.)/ !"#$%&'( )%&* ,0$ f
1," 0x +%' 1#2+/+&').3% ," 1! *%&+,#%) )%&- ,0$ f 1," 0x , /34 ," 2! .#/&)
)%&- ,0$ f 1," 0x .
567 ,% 6%&%6839 1:;)%,% <%=3/,%' 7,':
!$#
)(lim0
xfxx
, %3 +%' )73" %3 !$$%#"#
)(lim)(lim00
xfxfxxxx
>'% 6%&8?/'2)%, 0 1#38&,010 x
xxf
||)( $ (@:. 42) ?/3
.:/' 7&'" 1," 00 $x , %<"*:
— 2'% 0&x /=3%' 1)( "$"
$x
xxf , "67,/ 1)(lim
0
"$"#
xfx
,
/34
— 2'% 0'x /=3%' 1)( $$x
xxf , "67,/ 1)(lim
0
$%#
xfx
,
+%' .,1'
)(lim)(lim00
xfxfxx %#"#
(
!,-./01 234 3,534 .23 !!!!)0x
* @,% 6&"02"*)/3% 239&=1%)/ ,03 .33"'% ,"# "&="# ?'%'1(0,'+8. A=?%)/ 7,', 7,%3 2&8<"#)/ !$
#)(lim
0
xfxx
, /33""*)/ 7,' "' ,').$ )(xf B&=1+"3,%' 71" (.-"#-
)/ +"3,8 1," ! , 2'% 7-% ,% 0xx ( ,% "6"=% B&=1+"3,%' “%&+"*3,9$ +"3,8 1,"
0x ”. >'% 3% ?'%,#641"#)/, ,4&%, ,% 6%&%6839 1/ )%(0)%,'+; 2-411% /&2%C7-
)%1,/ 9$ /D;$:
"1= f(x)
O
f(x)=1
x x x
y42
1 !"#! – $%&'('#) $%&)"*+$+$ 161
— !"# $%&# "#' ()*&#' “+, ",-%' )(xf .)/-&0+1"2, +&+345+"6 $%7+8-6 0+1"* &"+ ! ”
9)#&,-+5+,+:-6 "#1 21,&;"#"2
!xf "# |)(| ! , (1)
;5+8 ! +5+,+&345+"6 $6",0;' 2),$-;'.
— !"# $%&# "#' ()*&#' “<,2 ;72 "2 0xx $
5+8 .)/&0+1"2, 2)0+:1"=' 0+1"* &"+ 0x ”
9)#&,-+5+,+:-6 "#1 21,&;"#"2
"xx "#" ||0 0 , (2)
;5+8 " 6/12, %12' 2)0+:1"=' -,0);' $6",0;' 2),$-;'. (> 21,&;"#"2 ||0 0xx #"
3#7?16, ;", )0xx $ .
— @,2 12 &813%&+8-6 ",' 38+ 28"%' ()*&6,' &:-(=12 -6 "+1 3,2,&$#",0; +),-&-; 7%-6 ;", <,2 +5+,+1345+"6 $6",0; 2),$-; ! -5+)+:-6 12 .)+:-6 %121 $6",0;
2),$-; " "%"+,+1 ?&"6, 21 "+ x ,021+5+,6/ "# (2), ";"6 "+ )(xf $2 ,021+5+,6/ "#1 (1). A9+8-6 3#7234 "+1 20;7+8$+ +),&-;:
!"#$%!$%%%%
A&"= -,2 &81*)"#&# f +),&-%1# &6 %12 &:1+7+ "#' -+)(4' ),(),( 00 #xx$ & .
B2 7%-6 ;", # f %96, &"+ 0x ;),+ !'! , ;"21 <,2 0*$6 0(! 85*)96, 0(" "%-"+,+', ?&"6 <,2 0*$6 ),(),( 00 #xx$x &' , -6 "xx "#" ||0 0 , 12 ,&9:6,:
!xf "# |)(| !
C5+36,01:6"2, ;",, 21 -,2 &81*)"#&# f %96, ;),+ &"+ 0x , ";"6 28"; 6/12, -+12-
3,0; 02, &8-.+7/D6"2,, ;5=' 6/32-6, -6 )(lim0
xfxx)
.
!"# &81%96,2, ;"21 <)*(+8-6 !*)
)(lim0
xfxx
, $2 611++:-6 ;", 85*)96, "+ ;),+ "#'
f &"+ 0x 02, 6/12, /&+ -6 ! .
!81%56,2 "+8 52)25*1= +),&-+: 6/12, +, 20;7+8$6' ,&+3812-/6':
(2) !*)
)(lim0
xfxx
+ 0))((lim0
*#)
!xfxx
(.) !*)
)(lim0
xfxx
+ !*,)
)(lim 00
hxfh
- C1 -,2 &81*)"#&# f 6/12, +),&-%1# &6 %12 3,*&"#-2 "#' -+)(4' ),( 0 #x 02, "#1 21,&;"#"2 "xx "#" ||0 0 "#1 21",02"2&"4&+8-6 -6 "#1 "xxx ,"" 00 , ";"6 %9+8-6 "+1 +),&-; "+8 )(lim
0
xfxx ,)
, 61? 21 # f 6/12, +),&-%1# &6 %12 3,*&"#-2
!#!
!,!
!
y=f(x)
f(x)
O x
x0+" x0#" x0 x
y43
1 !"#! - $%&'('#) $%&)"*+$+$ 162
!"# $%&'(# ),( 0x! )*+ !", *,+-.!"!* "xx !"! ||0 0 !", *,!+)*!*-!(-%/$0 $0 !", 00 xx"x !!" , !.!0 12%/$0 !%, %&+-$. !%/ )(lim
0
xfxx "#
.
34%50+),60!*+ .!+:
3, $+* -/,7&!"-" f 08,*+ %&+-$1," -0 1,* -6,%9% !"# $%&'(# ),(),( 00 #xx! $ ,
!.!0 +-260+ " +-%5/,*$8*:
%&#
)(lim0
xfxx
' %&&(#"#
)(lim)(lim00
xfxfxxxx
) 3, $+* -/,7&!"-" f 08,*+ %&+-$1," -0 1,* 5+7-
-!"$* !"# $%&'(# ),( 0 #x , *997 50, %&8:0!*+ -0 5+7-
-!"$* !"# $%&'(# ),( 0x! , !.!0 %&8:%/$0:
)(lim)(lim0
0
xfxfxxxx (##
& .
;+* 4*&750+<$*,
0limlim00
&&(##
xxxx
(=2. 44)
) 3, $+* -/,7&!"-" f 08,*+ %&+-$1," -0 1,* 5+7-!"-
$* !"# $%&'(# ),( 0x! , *997 50, %&8:0!*+ -0 5+7-!"$*
!"# $%&'(# ),( 0 #x , !.!0 %&8:%/$0:
)(lim)(lim00
xfxfxxxx "##
& .
;+* 4*&750+<$*,
0limlim00
&"&""##
xxxx
(=2. 45)
!"#$%#
34%50+),60!*+ .!+ !% )(lim0
xfxx#
08,*+ *,0>7&!"!% !?, 7)&?, #!, !?, 5+*-!"$7-
!?, ),( 0x! )*+ ),( 0 #x -!* %4%8* @0?&%6$0 .!+ 08,*+ %&+-$1," " f.
A!-+ <+* 4*&750+<$*, *, @19%/$0 ,* B&%6$0 !% .&+% !"# -/,7&!"-"#
1
|1|)(
""
&x
xxf -!% 00 &x , 40&+%&+:.$*-!0 -!%
/4%-6,%9% )1,0()0,1( $" !%/ 4058%/ %&+-$%6 !"#, -!% %4%8% */!( 4*8&,0+ !" $%&'(
11
)1()( "&
"""
&x
xxf .
C4%$1,?#, .4?# '*8,0!*+ )*+ *4. !% 5+49*,.
-2($*, !% :"!%6$0,% .&+% 08,*+ 1)(lim0
"&#
xfx
.
O x
y44
y x&
O x
y45
y x& "
y=1
y="1
O 1 "1 x
y46
1 !"#! – $%&'('#) $%&)"*+$+$ 163
!"#$%!&
!"# $%&'()*+, ,"+& -'.) ,"* .*+ $%&/0"#$# f '()* '()*+ ,*( 0x .*+ *1*,"#"+ !
2+ )&&334.) ,"* *$(4)* .*+ +5, "*6 5+0+7/"8 "0)*6 $%&297)6:
+) : f );&+* 30*$.'&# $) '&+ $4&3-3 "#6 .30<96 ),(),( 00 "xx# ! 7+* $"3 $4&3-
-3 +%", '()* "#& *1*,"#"+ !.
=) : f );&+* 30*$.'&# $) '&+ $4&3-3 "#6 .30<96 ),( 0x# , '()* $’ +%", "#& *1*,-
"#"+ !, +--/ 1)& 30;>)"+* $) $4&3-3 "#6 .30<96 ),( 0 "x .
?) : f );&+* 30*$.'&# $) '&+ $4&3-3 "#6 .30<96 ),( 0 "x , '()* $’ +%", "#& *1*,-
"#"+ !, +--/ 1)& 30;>)"+* $) $4&3-3 "#6 .30<96 ),( 0x# .
@*+ 5+0/1)*?.+, # $%&/0"#$# x
xxf
#.)( " );&+* 2)"*79 73&"/ $"3 00 "x , +<34
30;>)"+* $"3 $4&3-3 #$%
&'(!#
$%
&'()
2,00,
2
$$ 7+* );&+* 2)"*79 $) +%",.
,-./ 0120/0.345 - 60178-45 629:-0;6;5
A) "# =392)*+ "3% 30*$.34 "3% 30;3% +531)*7&4)"+* ,"*:
f(x0)=x0
(a)
f(x)
f(x)
x0 O x x x y=x
y
y=c
(")
x0 O x x x
y47
: 50B"# *$,"#"+ 1#-B&)* ,"* "3 ,0*3 "#6 "+%"3"*796 $%&/0"#$#6 xxf ")( (!(. 47+) $"3 0x );&+* ;$3 .) "#& "*.9 "#6 $"3 0x , )&B # 1)4")0# *$,"#"+ 1#-B&)* ,"* "3 ,0*3 "#6 $"+2)096 $%&/0"#$#6 cxg ")( (!(. 47=) $"3 0x );&+* ;$3 .) c.
ccxx
"* 0
lim 00
lim xxxx
"*
1 !"#! - $%&'('#) $%&)"*+$+$ 164
)$,+$'#$
!" #$!%!&
1. N! "#$%&$ &' )(lim0
xfxx!
(!) &' )( 0xf , $*+,'- ./0#1'.-, +&!- 2 3#!-
*)(4 /!#0,&!,2 &25 ,.-0#&2,25 f $%-!):
O
x0=3
y=f (x)
3
2
x
y
O
y=f (x)
x0=2
2
4
2
x
y
O
x0=1,2
y=f (x)
1
1
2
2 x
y
O
x0=1,2,3
y=f (x)
1
1
2
3 2 x
y
2. 6! 1!#07$&$ &2 3#!*)(4 /!#0,&!,2 &25 ,.-0#&2,25 f (!) 8$ &2
"'49$)! !.&45 -! "#$%&$, $*+,'- ./0#1$), &' )(lim0
xfxx!
, +&!-:
i) 2
65)(
2
"#"
$x
xxxf , 20 $x ii)
%&
%'
(
)
*$
1,1
1,
)(x
x
xx
xf , 10 $x
iii)%&
%'(
)#"
*$
1,1
1,)(
2
xx
xxxf , 10 $x iv)
x
xxxf
2
)( #$ , 00 $x .
3. :8'%;5 +&!-:
i) 1
33)(
2
23
"""#
$x
xxxxf , 10 $x 4 10 "$x
ii) 13
169)1()(
2
"#"#
$x
xxxxf ,
3
10 $x .
1 !"#! – $%&'('#) $%&)"*+$+$ 165
4. !"#$%&' ( )*#+,%()( f -.* $"#&' .,')/0#( )%. ),2[ !"# 1&' 02$' 3,&-
4'15 -&,+)%&)( -.* 4&"#$%&' )%. -&,&1+%6 )25/&. 7& $8$%+)$%$ -.'.' &-9 %.*: $-9/$#.*: ')2*,')/.;: $"#&' &<(=$":.
i) 2)(lim2
$#%
xfx
ii) 1)(lim1
$!%
xfx
iii) 2)(lim1
$%
xfx
iv) 3)(lim2
$%
xfx
v) 4)(lim3
$%
xfx
vi) 3)(lim4
$%
xfx
5. !"#$%&' /'& )*#+,%()( f .,')/0#( )%. ),(),( 00 !xx" & , /$
6)(lim 2
0
#$#%
#xfxx
1&' #xfxx
$!%
)(lim0
. 7& >,$"%$ %': %'/0: %.* !'# , 3'&
%': .-."$: *-+,2$' %. )(lim0
xfxx%
.
1.5 #,#!*+*'$ *-& !"#-&
./01 230 4056378
?'& %. 9,'. 1&' %( @'+%&8( &-.@$'1#;$%&' 9%' ')2;.*# %& -&,&1+%6 =$6,5/&%&.
!"#$%&' 1(
( A# 0)(lim0
)%
xfxx
, %9%$ 0)( )xf 1.#%+ )%. 0x (B2. 48&)
( A# 0)(lim0
*%
xfxx
, %9%$ 0)( *xf 1.#%+ )%. 0x (B2. 48>)
O
+
(a)
Cf
! " x0
x
y
O
+
(!)
Cf
x0 x
y
! "
48
O
y=f (x)
1
1
4
3
2
4 3 -2 2 x
y
1 !"#! - $%&'('#) $%&)"*+$+$ 166
!"#$%&' 2(
!" #$ %&"'()*%+$, gf , -.#&" /($# %)# 0x 0'$ $%.1+$ )()( xgxf ! 0#")2 %)#
0x , )/)+ )(lim)(lim
00
xgxfxxxx ""
!
O
(a)
Cg
Cf
! " x0 x
y
O
(!)
Cg
Cf
! " x0 x
y49
,-./ 0/. 1-234.5
3' 41# 5'%$02 /($' 00
lim xxxx
#"
, ccxx
#" 0
lim 0'$ )# 6+7(89' :#& '0#;#&6+< 4$+&-
0#;1"#&" )#" &:#;#=$%9/ )>" #(<>".
!"#$%&'
!" &:2(.#&" )' /($' )>" %&"'()*%+>" f 0'$ g %)# 0x , )/)+:
1. )(lim)(lim))()((lim000
xgxfxgxfxxxxxx """
$#$
2. )(lim))((lim00
xf#x#fxxxx ""
# , =$' 026+ %)'6+(2 !%#
3. )(lim)(lim))()((lim000
xgxfxgxfxxxxxx """
&#&
4. )(lim
)(lim
)(
)(lim
0
0
0 xg
xf
xg
xf
xx
xx
xx
"
"
"# , +?/%#" 0)(lim
0
'"
xgxx
5. )(lim|)(|lim00
xfxfxxxx ""
#
6. kxx
k
xxxfxf )(lim)(lim
00 ""# , +?/%#" 0)( (xf 0#")2 %)# 0x .
@$ $4$/)8)+, 1 0'$ 3 )#& 6+>(*9')#, $%.1#&" 0'$ =$' :+($%%/)+(+, ':/ 4&# %&-
"'()*%+$,. A9+%8 %&"-:+$' '&)#1 +<"'$:
$
xx
$
xxxfxf )*+
,-.#
"")(lim)]([lim
00
, *"%$
1 !"#! – $%&'('#) $%&)"*+$+$ 167
!"# $#%&'(")*#,
!!
xxxx 0
0
lim !"
.
— +,-. -/%# -0 $012/32*0
011
1)( "x"x"x"xP !!
!! ####! $
$ % 4#" !&0x .
56*7.3# *( -"8 $#%#$&3. "'"9-:-(8 ;<02*(:
)(lim)(lim 0
1
100
"x"x"xP !!
!!
xxxx###! $
$""%
00
1
100
lim)(lim)(lim "x"x"xx
!!
xx
!!
xx "
$$""
###! %
00
1
01
0
limlimlim "x"x"xx
!
xx!
!
xx! "
$
"$"###! %
)( 001
010 xP"x"x" !!
!! !###! $
$ % .
=$0*;3.8,
)()(lim 00
xPxPxx
!"
.
!"# $#%&'(")*#,
4227262)276(lim 2323
2$!$'#'$!$#$
"xxx
x.
— +,-. : %:-> ,23&%-:,: )(
)()(
xQ
xPxf ! , 9$02 )(xP , )(xQ $012/32*# -02 x 4#"
!&0x *( 0)( 0 (xQ . ?9-(,
)(
)(
)(lim
)(lim
)(
)(lim)(lim
0
0
0
0
00 xQ
xP
xQ
xP
xQ
xPxf
xx
xx
xxxx!!!
"
"
"".
=$0*;3.8,
)(
)(
)(
)(lim
0
0
0 xQ
xP
xQ
xP
xx!
", (79,03 0)( 0 (xQ
!"# $#%&'(")*#,
9
8
1222
42
12
4lim
2
2
2
2
2!
#'##
!##
#" xx
x
x.
!"#$%#
@-#3 0)( 0 !xQ , -9-( '(3 (7#%*9A(-#" : "'"9-:-# 4 -02 $#%#$&3. B(.%>*#-08. 5-:3 $(%C$-.,: #2-> (%)#A9*#,-( 9$.8 ,-:3 (7#%*0)> 1 ii), $02 #40102B(C.
1 !"#! - $%&'('#) $%&)"*+$+$ 168
',)"-!.'$
1. N! "#$%&'( )! *!#!+,)- .#/!:
i) ]|11)[(lim 392
0!"#
$x|x
x ii)
4
65lim
2
23
2 !#!
$ x
xxx
x iii)
1
23lim
2
1 !!#
$ x
xx
x.
/%$+
i) !"#$%&
|1|lim)1(lim|1|)1(lim 3
0
92
0
392
0][ !"#%!#
$$$xxxx
xxx
)1(lim)1(lim 3
0
92
0][ !"#%
$$xx
xx
1|1|19 %!"% .
ii) '(&)*+ 0)4(lim 2
2%!
$x
x, *&, %(#-#.%& ,/ &0/-%12#$%& 3#, 4/,1,/ 3#$ (567-
4#$ ()*)1353/ 4). 8/-/35-#.%& 1%9: 13) ;)/ 2%x %5*&,7<#,3/) 4/) #) *.# 1-#)
3#$ 46=2%/3#:. >(13& 5 2$,=-3525 4
65)(
2
23
!#!
%x
xxxxf , ;)/ 2&x , ;-=0&3/):
2
3
2
)3(
)2)(2(
)3)(2(
)2)(2(
)65()(
22
#!
%#!
%#!!!
%#!#!
%x
xx
x
xx
xx
xxx
xx
xxxxf .
'(#%?,9:,
2
1
22
234
2
3lim)(lim
2
22!%
#"!
%#!
%$$ x
xxxf
xx.
iii) @)/ 1%x %5*&,7<#,3/) #) 1-#) 3#$ 46=2%/3#:. A35, (&-7(3925 /$3+ &-;/-
<1%/23& 9: &B+::
8#66/(6/2)=<#$%& 4/) 3#$: *.# 1-#$: 3#$ 46=2%/3#: %& xx 232 ## 4/) ?32) ?"#$%&:
'()*
+, ##!
!'()*
+, #
%'()*
+, ##!
'()*
+, ##'()*
+, !#
%!!#
%xxx
xx
xxx
xxxx
x
xxxf
23)1(
)2(3
23)1(
2323
1
23)(
2
22
2
2
222
xx
x
xxx
xx
xxx
x
23
)1(3
23)1(
)1)(1(3
23)1(
33
222
2
##
#!%
'()*
+, ##!
#!!%
'()*
+, ##!
#!% .
1 !"#! – $%&'('#) $%&)"*+$+$ 169
!"#$%&'(,
2
3
24
6
23lim
)1(3lim
23
)1(3lim)(lim
2
1
1
211
)(!"
#
!"
$%&'
() ##
#!"
##
#!"
*
*
**xx
x
xx
xxf
x
x
xx.
2. N! "#$%$&, !' ()*#+$,, -. /#,. 0-. 10 "x -12 0('*#-1012
+,
+-
.
/!
0!"
1,1
14,3
)(
2
xx
xx
xf .
,%$+
)& 10x , *+*, 43)( 2 !" xxf , #"+*,
1413)43(lim)(lim 22
11
!"!1"!"*!*
xxfxx
.
)& 12x , *+*, x
xf1
)( !" , #"+*,
11
lim)(lim11
!"$%&
'()!"
*#* xxf
xx
.
!"#$%&'( 1)(lim1
!"*
xfx
.
-./01./2 34.5672819
-"#.%*#/$, +*0 “1#&*2 3*# 0x ” $04 3/&25-
*636 f “,718'9:;,*40” (<=. 50) 4&2$,34
3, >?# 3/&45*@3,0( h 140 g. )&, 14.A( *# x
*,:&,0 3*# 0x , #0 g 140 h %=#/& 1#0&+ +50#
3 , *+*,, +"'( B4:&,*40 140 3*# 3=@$4, 6 f
.4 %=,0 *# :>0# +50# 3 . )/*+ >:&,0 *6& 0>%4
*#/ "45412*' .,'5@$4*#( "#/ ,:&40 7&'-
3*+ '( 3#,-4#,. )!#$5".642.
789:;<=
C3*' #0 3/&45*@3,0( hgf ,, . )&
4 )()()( xgxfxh 55 1#&*2 3*# 0x 140
4 3""**
)(lim)(lim00
xgxhxxxx
,
*+*, 3"
*)(lim
0
xfxx
.
O
3
Ch
Cf
Cg
! " x0 x
y50
1 !"#! - $%&'('#) $%&)"*+$+$ 170
!"# $#%&'(")*#, 01+*lim
0!"
#$
%&'
( xx
x.,%&)*#-", )"# 0)x ./01*(
||1+*||
1+* xx
xx
x *+! ,
0$2-(
||1+*|| xx
xx **, .
3$("'4 0||lim)||(lim00
!!,((
xxxx
, 56*789# *( -0 $#%#$&98 :%"-4%"0, ./01*(:
01+*lim
0!"
#$
%&'
( xx
x.
T,-./01234,-56 7,-8
;0 :%"-4%"0 $#%(*<0=4> (?9#" $0=6 /%45"*0 )"# -09 1$0=0)"5*2 0%"5*.989 -%"-)890*(-%":@9 0%?89. A%/":& #$0'(":9601*( 2-":
|||+*| xx * , )"# :&B( !-x
(+ "52-+-# "5/6(" *290 2-#9 0!x )
)9!:'#;+....
— !"# 0!x $%07#9@> "5/6(" + "52-+-#.
— !"# "#$
%&'-
2,0!
x #$2 -0 '"$=#92 5/4*# ./01*(
xMAMAMMx !//! )-0C()()(+* 1 .
D%#
|||+*| xx / , )"# :&B( "#$
%&'-
2,0!
x (1)
— !"# "#$
%&',- 0,
2
!x (?9#" "
#$
%&'-,
2,0!
x , 0$2-( =2)8 -+> (1) ./01*(, |||)(+*| xx ,/,
4, "50'69#*#, |||+*| xx / .
— !"# "#$
%&',0
2,
2
!!x (?9#" |+*|1
2|| x!
x 121 , 0$2-( |||+*| xx / .
E( 2=(>, =0"$29, -"> $(%"$-@5("> "5/6(" |||+*| xx * , *( -+9 "52-+-# 9# "5/6(" *2-
90 )"# 0!x . !
F( -+ <04B("# -+> $#%#$&98 #9"52-+-#> :#" -01 :%"-+%?01 $#%(*<0=4> B# #$0-
'(?C01*( 2-":
M
A
x
x M1 O x
y51
1 !"#! – $%&'('#) $%&)"*+$+$ 171
),!-'#.+
! !"#$%& '( ()*+,-.*/0, 12$
030lim0
"#
xx
%($ 14/5lim0
"#
xx
(1)
6"&70(2$:
— 890:;5( 0, 235 )"*37*90,53 (5$41232( <#*/0, |||30| xx $ , *)12,
||30|| xxx $$% .
=),$+> ||lim0)||(lim00
xxxx ##
""% , 490:;5( 0, 2* %"$2>"$* )(",0?*@>A, '( ,-5($
030lim0
"#
xx
.
— B5;"-C*/0, 12$ xx 22 3014/5 %" , *)12,
xx 23014/5 %" , 7$( %&', &'(
)*+,&
'(
)*+%-
2,00,
2
!!x .
=)*0<5;A
10130lim1301lim4/5lim 2
0
2
00"%"%"%"
###xxx
xxx.
! D( ()*+,-.*/0,, 2E"(, 12$ 00
3030lim xxxx
"#
. 6"&70(2$. <#*/0,
)304/54/530(lim)(30lim30lim 000
000
hxhxhxxhhxx
."."###
hxhxhh30lim4/54/5lim30
00
00 ##
."
000
)1(
3004/5130 xxx "/./" .
! !5&@*7( ()*+,$%59,2($ %($ 12$ 04/54/5lim0
xxxx
"#
. !
! 00
3030lim xxxx
"#
! 00
4/54/5lim xxxx
"#
1.
! () 130
lim0
"# x
x
x
! ?) 014/5
lim0
"%
# x
x
x
2.
1 !"#! - $%&'('#) $%&)"*+$+$ 172
),!-'#.+ !!!!
— !" 2
0!
x "" , #$#% &'$ #( )*'+&"$ ,-./&
'0(12'#%* $#*
)%/3(#0*4)%/3(#(/)%/3(#0*4 "#$"#%"#% "" ,
o'$#% 5-(6/% )*&)(-*17:
xxx %812
1
2
19/12
1##""##
"x9/ x x%8" .
xx
x
,6"1
9/1 ""
19/,6" ""
x
xx .
— !" 02
""$ x!
, #$#% 2
0!
x "$" , ('$#% 5-(6/% 1)(9/
)(,6" "$$
"$x
xx
1&* 70&
19/,6" ""
x
xx .
:'(/5";<, 4*& 17=% %&'
()*+%
&'
()*$,
2,00,
2
!!x *,-2%* 1
9/,6" ""x
xx .
:'%*). 1,6"lim0
-.
xx
, &'$ #( 10*#.0*( '&0%/3(+.< '0(12'#%* $#* 19/
lim0
-. x
x
x.
x
3' 2' ' O
1
yx
x-9/
-'
y53
-3' -2'
3) >-(6/%
)1,6"(
)1,6")(1,6"(lim
1,6"lim
00 //$
-$
.. xx
xx
x
x
xx
)1,6"(
1,6"lim
2
0 /$
-. xx
x
x
),6"1(
9/lim
2
0 xx
x
x /$-
.
M
N
A x
M1 O x
y52
01
2 %8x
9/x
3
45
65
1 !"#! – $%&'('#) $%&)"*+$+$ 173
!"
#$%
&'
()*+ x
x
x
x
x !"#1
$%$%lim
0
x
x
x
x
xx !"#1
$%lim
$%lim
00 '()*
++
001 *()* . !
,-./ 01234567 0829-56067
&' ()* )+),($('* -." /#/012."%' %132) (42/ %-.2.5%' #/ -2.!+).26!."%' (/
,2)/ /-74# !"#/2(8!'9#. :#, ,%9*, ;17."%' #/ "-.7.<6!."%' (. ))((lim0
xgfxx+
,
($* !5#;'($* !"#=2($!$* gf , !(. !$%'6. 0x , (,(' '2</>,%/!(' 9* '?8*: 1. @1(."%' )(xgu * .
2. A-.7.<6>."%' (/# "-=23')) (. )(lim0
0 xguxx+
* B/)
3. A-.7.<6>."%' (/# "-=23')) (. )(lim 0
ufuu+
*- .
:-.+')B#5'(/) ,(), /# 0)( uxg . B.#(= !(. 0x , (,(' (. >$(.5%'#. ,2). '6#/) 6!.
%' - , +$7/+8 )!35'): )(lim))((lim
00
ufxgfuuxx ++++++++
**** .
!"#$#%&
C($ !"#13')/ B/) !' ,7$ ($# 1B(/!$ (." D)D76." (/ ,2)/ ($* %.208* ))((lim
0
xgfxx+
%' (/ .-.6/ ;/ /!3.7$;.5%' ;/ '6#/) (1(.)/, 4!(' #/ )B/#.-.)'6(/)
$ !"#;8B$: “ 0)( uxg . B.#(= !(. 0x ” B/) <)/"(, +'# ;/ '71<3'(/).
E)/ -/2=+')<%/:
/) F!(9 ,() ;17."%' #/ "-.7.<6!."%' (. ,2).
!"#
$%& '
+ 4$%lim 2
0
!x
x.
:# ;1!."%' 4
2 !xu '* , (,('
44limlim 2
00
!!xu
xx*!
"#
$%& '*
++, .-,('
2
2
4$%$%lim
4$%lim
4
2
0***!
"#
$%& '
++
!u
!x
!u
x.
D) F!(9 ,() ;17."%' #/ "-.7.<6!."%' (. ,2).
x
x
x
3$%lim
0+.
1 !"#! – $%&'('#) $%&)"*+$+$ 174
!"#$%
x
x
x
x
3
3&'3
3&'! .
()*%, $# +,*-.'/ xu 3! , )0)/ 03limlim00
!!""
xuxx
, -10)/
313&'
lim33
3&'lim3
3&'lim
000!#!!!
""" u
u
x
x
x
x
uxx.
)$,+$'#$
!" #$!%!& 1. N$ 23/")/ )$ 03%$:
i) )524(lim 35
0$%%
"xxx
x ii) )12(lim 310
1%$%
"xxx
x
iii) 208
1)32(lim $$
%"xx
x iv) ][ |32|)5(lim 23
3%%%
"xxx
x
v) 3
52lim
4
1 $%$
" x
xx
x vi)
1
|2||3|lim
2
2
0 $$%$%
" xx
xxx
x
vii) 3 2
1)2(lim $
"x
x viii)
34
22lim
2
2
1 $$%$$
" xx
xx
x.
2. (*)4 '%$ *.#53)&*& f '/ 4)(lim2
!"
xfx
. 6$ 23/")/ )- )(lim2
xgx"
$#:
i) 5))((3)( 2 %! xfxg ii)1))((
|11)(2|)(
2 $%
!xf
xfxg
iii) )3)()(2)(()( %$! xfxfxg .
3. 6$ 23/")/ )$ 03%$
i) 8
16lim
3
4
2 %%
" x
x
xii) ii)
1
132lim
2
2
1 %$%
" x
xx
x
iii)
2
1 11
11
lim
x
xx
%
%
" iv)
x
x
x
27)3(lim
3
0
%$"
.
1 !"#! – $%&'('#) $%&)"*+$+$ 175
4. !" #$%&'% '" ($)"
i) x
x
x !!
" 9
3lim
9 ii)
2
2
0
11lim
x
x
x
!!"
iii) 35
22lim
22 !#
!#" x
x
x iv)
45
2lim
24 #!!
" xx
x
x.
5. !" #$%&'% ("* +,-$.%)), '/ ($)/ '01 f 2'/ 0x "*:
i) $%&
'(
)1,5
1,)(
2
xx
xxxf 3") 10 )x
ii) $%&
!*#!+!
)1,1
1,2)(
2 xx
xxxf 3") 10 !)x .
6. !" #$%&'% '" ($)":
i) x
x
x
304lim
0" ii)
x
x
x
%5lim
0" iii)
x
x
x 204%54
lim0""""
iv) ,-.
/01 !
" x
xx
x
04lim
0 v) ,
-.
/01
#" xx
x
x 30
04lim vi)
245
504lim
0 !#" x
x
x.
7. !" #$%&'% '" ($)":
i) x
x
!x 2+*1
04lim
2
#" ii)
x
x
x 2042+*1
lim2
0
!"
iii) x
x
x 20404
lim0"
.
8. N" #$%&'% '/ )(lim0
xfx"
, "*:
i) 22 1)(1 xxfx #((! 6)" 3-7% !2x
ii) x
xfx2
4
2+*1
)(1 ((! 6)" 3-7% ,-.
/01!2
2,
2
!!x .
9. 8&*%'") 0 2+*-$'020 $%&
'#(#
)3,3
3,2)(
x"x#x"x#
xf . !" #$%&'% ')1 ')491 ':*
!2"#, , 6)" ')1 /,/&%1 )2.;%) 10)(lim3
)"
xfx
.
!" #$%&%'
1. !" #$%&'% '" ($)":
1 !"#! – $%&'('#) $%&)"*+$+$ 176
i) 8
2lim
3
23
2 !!!!
" x
xxx
x ii)
1
)1(lim
1
1 !##!#
" x
!x!x!
x
iii) 2
1lim
1 !#
!" xxx
x
x
2. N! "#$%&$ '(! !)' &! )!#!*+&, '#-! .)+#/0.1
i) 5
2510lim
2
5 ###
!" x
xx
x ii)
5
54|5|lim
2
5 !!!#!
!" x
xxx
x
iii) 5
54|5|lim
2
5 !!!#!
#" x
xxx
x
iv) 1
lim2
1 !!
" x
xx
x.
3. 2&0 3-)4!1' (/56! &0 &#%7,10 "#$ $%1!-
0#80791-0 6$ 1$% . :! .)0407%($&$ &!
'#-!:
i) )(lim
2
&'()
!"
, ii) )(lim 22
2
&'()
!"
iii) '&
()2
lim"
.
4. :! "#$%&$ &0 )(lim1
xfx""""
, !1:
i) 10)42)(4(lim1
!!!!$$$$!!!!####""""
xxfx
ii) 11
)(lim
1$
!" x
xf
x.
1.6 ,+ -'-'")$,'&! !"#! $*! !!!!%0x
— 2&0 (/56! 54 ;/0.6$ &< 7#!=-*5 )!#+(&!-
(< 6-!> (.1+#&<(<> f *01&+ (&0 0x . ?!#!&<-
#0@6$ '&-, *!89> &0 x *-10@6$10 6$ 0)0-0135-
)0&$ &#')0 )+1, (&01 +A01! xx& )4<(-+B$- &01 )#!76!&-*' !#-86' 0x , 0- &-6;> )(xf !.A+10-
1&!- !)$#-'#-(&! *!- 7%101&!- 6$7!4@&$#$> !)' 0)0-0135)0&$ 8$&-*' !#-86' *. 2&<1 )$#%-)&,(< !.&5 4;6$ '&- < (.1+#&<(< f ;/$- (&0
0x '#-0 #' *!- 7#+=0.6$
#'$"
)(lim0
xfxx
.
$
# " %=1
' &
)
O x0 x x
M
f (x)
x
y 54
1 !"#! – $%&'('#) $%&)"*+$+$ 177
— !"# $%&'( 55 )%#*'+ ", -.(/01& 2(.3$"(-
$, '0(4 $*53.",$,4 f 1#5"3 $"# 0x . 6(.(",-
.#7'+ 8"0, 1(9:4 "# x 105#7'+5# '+ #2#0#5;&-
2#"+ ".82# 235< $"#5 3=#5( xx! 2>,$03?+0 "#5 2.(-'("018 (.09'8 0x , #0 "0')4 )(xf +>("":-
5#5"(0 (2+.08.0$"( 1(0 -@5#5"(0 '01.8"+.+4 (28 #2#0#5;&2#"+ (.5,"018 (.09'8 M"
)0( #M . !",5 2+.@2"<$, (*"& >)'+ 8"0 , $*-
53.",$, f )%+0 $"# 0x 8.0# "$ 1(0 -.3/#*'+ "$%
&)(lim
0
xfxx
.
!"#$%!$''''
A$"< '0( $*53.",$, f 2#* +@5(0 #.0$')5, $+ )5( $75#># ",4 '#./&4 ),(),( 00 !xx" ( . B.@?#*'+
) *$%&
)(lim0
xfxx
, 8"(5 -0( 139+ 0#M *23.%+0 0## ")"#0#, :$"+ -0( 139+
),(),( 00 !xx"x (+ , '+ #xx ,", ||0 0 5( 0$%7+0
Mxf #)(
) "$%&
)(lim0
xfxx
, 8"(5 -0( 139+ 0#M *23.%+0 0## ")"#0#, :$"+ -0( 139+
),(),( 00 !xx"x (+ , '+ #xx ,", ||0 0 5( 0$%7+0
Mxf ",)(
C53>#-#0 #.0$'#@ '2#.#75 5( ;0("*2<9#75 8"(5 "& 0xx 1(0 *& 0xx .
(a)
O
Cg
Cf
x0 x
y
(!)
O x0 x
y 56
Cg
Cf
D2<4 $",5 2+.@2"<$, "<5 2+2+.($')5<5 #.@<5 )"$0 1(0 -0( "( 32+0.( 8.0( $*-
5(."&$+<5, 2#* #.@?#5"(0 $+ )5( $75#># ",4 '#./&4 ),(),( 00 !xx" ( , 0$%7#*5 #0 2(.(13"< 0$#;*5('@+4:
Ox0 x x
-M
f (x)
x
y55
1 !"#! – $%&'('#) $%&)"*+$+$ 178
!"##$!"#!%&%%
)(lim)(lim)(lim000
xfxfxfxxxxxx
&"##$&"#!%&%%
)(lim)(lim)(lim000
xfxfxfxxxxxx
.
!" #$ %&'(")* #&+ &,)-.&/ *0&1")23/&3#*) &) 0*,*24#5 )1)6#$#"7:
' 83 !"#%
)(lim0
xfxx
, #6#" 0)( (xf 2&3#4 -#& 0x , "39
*3 &"#%
)(lim0
xfxx
, #6#" 0)( )xf 2&3#4 -#& 0x .
' 83 !"#%
)(lim0
xfxx
, #6#" &"#&%
))((lim0
xfxx
, "39
*3 &"#%
)(lim0
xfxx
, #6#" !"#&%
))((lim0
xfxx
.
' 83 !"#%
)(lim0
xfxx
' &" , #6#" 0)(
1lim
0
#% xfxx
.
' 83 0)(lim0
#%
xfxx
2*) 0)( (xf 2&3#4 -#& 0x , #6#" !"#% )(
1lim
0 xfxx, "39 *3
0)(lim0
#%
xfxx
2*) 0)( )xf 2&3#4 -#& 0x , #6#" &"#% )(
1lim
0 xfxx.
' 83 !"#%
)(lim0
xfxx
' &" , #6#" !"#%
|)(|lim0
xfxx
.
' 83 !"#%
)(lim0
xfxx
, #6#" !"#%
k
xxxf )(lim
0
.
:/.;53* ." #)7 )1)6#$#"7 *+#<7 <=&+.":
!"!"!"!"####%%%% 20
1lim
xx 2*) >"3)24 !"!"!"!"####
%%%% !20
1lim
xx, *!*! (:=. 57*)
y
x#
12
(")
O x
y
y
x#
1
(#)
O x
y57
1 !"#! – $%&'('#) $%&)"*+$+$ 179
!"!"!"!"####!!!!$$$$ xx
1lim
0
!"# $%&#!' !"!"!"!"####!!!!!!!!$$$$
120
1lim
,x x
, !%!
%&(
&"&"&"&"####&&&&$$$$ xx
1lim
0
!"# $%&#!' &"&"&"&"####!!!!&&&&$$$$
120
1lim
!xx
, !%! ()*. 57+).
,-./0&12, 3%& 4-'5*%# 67. /830& 7. 95#. 782 12
1)(
!#
!xxf , !%! .
:#" 7" 95#" ";5.<6/"7.2 !"# $#&./0&.4 3=. 64&"57>6%1& "-.3%#!&=.&7"# 7" -"5"!'71 ;%15>/"7":
!"#$%&' 1( ()*+( ,-*(./0,1(2)
?& 67. x0%'
7. 95#. 782 f %<&"#: "%' "%' !" -" !" -"
!"# 7. 95#. 782 g %<&"#: !" -" !" -" -" !"
797% 7. 95#. 782 gf ! %<&"#: !" -" !" -" ; ;
!"#$%&' 2( ()*+( 3+4(054(6)
?& 67. x0%',
7. 95#. 782 f %<&"#: ">0 "<0 ">0 "<0 0 0 +" +" -" -"
!"# 7. 95#. 782 g %<&"#: +" +" -" -" +" -" +" -" +" -"
797% 7. 95#. 782 f·g %<&"#: +" -" -" +" ; ; +" -" -" +"
)7.42 -<&"!%2 71& -"5"-'&1 ;%158/'71&, 9-.4 4-'5*%# %5178/"7#!9, 68/"<&%# 97# 7. 95#. ("& 4-'5*%#) %@"57'7"# !';% A.5' "-9 7#2 64&"57>6%#2 -.4 -"<5&.4-/%. )7#2 -%5#-7(6%#2 "4702 B0/% 97# 0*.4/% ,7*(/8+)*+/19 0(*:;. C8B"3>, "-5.63#95#67%2 /.5A02 $#" 7" 95#" ";5.<6/"7.2 !"# $#&./0&.4 64&"57>6%1& %<&"# .#:
)()( &"&"&"&"!!!!!"!"!"!" !"# )("("("(")))) (0 .
,-%#3> )( gfgf &!#& !"# g
fg
f 1)# , "-5.63#95#67%2 /.5A02 $#" 7" 95#" 782
3#"A.5'2 !"# 7.4 -8B<!.4 64&"57>6%1& %<&"# .#:
)()( !"!"!"!"&&&&!"!"!"!" , )()( &"&"&"&"&&&&&"&"&"&" !"# 0
0,
""""((((("("("("
.
:#" -"5'3%#$/":
1 !"#! – $%&'('#) $%&)"*+$+$ 180
— !" #$%&'() *+, -'"!%*.-)+, 2
1)(
xxf !" /!+
2
1)(
xxg " , *0*) 12&'():
!#"$%&
'()!"
** 200
1lim)(lim
xxf
xx, +#""
** 200
1lim)(lim
xxg
xx
/!+
011
lim))()((lim2200
"$%&
'() +!"+
** xxxgxf
xx
)"3,
— !" #$%&'() *+, -'"!%*.-)+, 11
)(2+!"
xxf /!+
2
1)(
xxg " , *0*) 12&'():
!#"$%&
'() +!"
**1
1lim)(lim
200 xxf
xx, +#""
** 200
1lim)(lim
xxg
xx
/!+
11lim1
11
lim))()((lim02200""$
%&
'() ++!"+
*** xxx xxxgxf .
4"$5&6! #!%!7)86(!*! (#&%&9() "! 73-&'() /!+ 6+! *+, $55), (&%:1,.
',)"-!.'$
1. N! "#$%&'( )! *#+!:
i) |x|
xx
x 1
65lim
2
1 !+!
* ii)
22 2)(
23lim
!+!
* x
x
x.
/%$+
i) ;#)+7. 0|1|lim1
"!*
xx
/!+ 0|1| ,!x /&"*$ -*& 1, )8"!+ +#"!* |1|
1lim
1 xx. E#)+7.
)#+#51&" )8"!+ 2)65(lim 2
1"+!
*xx
x, 12&'():
#+"-.
/01
2+!3
!"
!+!
**)65(
|1|
1lim
|1|
65lim 2
1
2
1xx
xx
xx
xx.
ii) ;#)+7. 0)2(lim 2
2"!
*x
x /!+ 0)2( 2 ,!x /&"*$ -*& 2, )8"!+ +#"
!* 22 )2(
1lim
xx.
;#)+7. )#+#51&" )8"!+ 4)23(lim2
!"+!*
xx
, 12&'()
!#"-.
/01
2+!3
!"
!+!
**)23(
)2(
1lim
)2(
23lim
2222x
xx
x
xx.
1 !"#! – $%&'('#) $%&)"*+$+$ 181
2. !" #$%&'() *" +,%-$./0 1$." *23 4-)0$*2423 2
1)(
2
!"!
#x
xxxf 4*'
20 #x /". 4*2 4-)56%." )" %7%*"4&%8, ") -+0$6%. *' 1$.' *23 )(xf 4*' 2.
,%$+
— !"#$%& 0)2(lim2
#!$
xx
'($ 02 %!x )$( 2%x , #*+($ "&#!"$ 2
1lim
2 xx
. !"#$%& #-
"$",-.+ 3)1(lim 2
2
#"!"$
xxx
, -/.01#
"&#'()
*+, "!!
#!"!
"$"$)1(
2
1lim
2
1lim 2
2
2
2
xxxx
xx
xx
.
— !"#$%& 0)2(lim2
#!$
xx
'($ 02 -!x )$( 2-x , #*+($ !&#!!$ 2
1lim
2 xx
. !"#$%& #-
"$",-.+ 3)1(lim 2
2
#"!!$
xxx
, -/.01#
!&#'()
*+, "!.!
#!"!
!$!$)1(
2
1lim
2
1lim 2
2
2
2
xxxx
xx
xx
.
2(3(453.61# 74$ 4( %6. ",#03$'8 73$( %#+ #*+($ *9(. !".1-+:; %#+ 0"83/#$ 73$. 45; f 94. 2.
)$-+$'#$
9: ;<9=9>
1. N( <3#*4# ((+ 0"83/#$) 4. 73$. 45; f 94. 0x 74(+:
i) 24 3
5)(
xx
xxf
""
# , 00 #x ii) 4)1(4
32)(
!!
#x
xxf , 10 #x
iii) ||
11)(
xxxf !# , 00 #x .
2. =( <3#*4# ((+ 0"83/#$) 4. 73$. 45; f 94. 0x , 74(+:
i) 21
4
1
3)(
xxxf
!!
!# , 10 #x ii)
||
23)(
2
xx
xxxf
!"# , 00 #x
iii) /01
234 "#
3
2 11)(
xxxf , 00 #x .
1 !"#! – $%&'('#) $%&)"*+$+$ 182
!" #$%&%'
1. !" #$%&'% (%()*+, -./$0%1) '+ 842
9lim
4 !""
"# xxxxx
.
2. !" ".+2%&3%'% )'1:
i) 4 *-,/$'5*5 xxf %()( $ 2%, 60%1 )$1+ *'+ 2
!.
ii) 4 *-,/$'5*5 xxf *()( $ 2%, 60%1 )$1+ *'+ 0.
3. 7&,+,'"1 +1 *-,"$'8*%19
1
2)1()(
2
2
""!"
$x
xx"xf :"1
x
#xxxg
!!$
2)(
2
.
N" #$%&'% '19 '1;69 '<, !%#", =1" '19 +.+&%9 -./$0+-, *'+ ! '" )-
$1"
)(lim1
xfx#
:"1 )(lim0
xgx#
.
>'5 *-,60%1" ," -.+?+=&*%'% '" ."$"./,< )$1".
4. !" #$%&'% '+ )(lim1
xfx#
, )'",:
i) !&$"
# )(
4lim
1 xf
x
x ii) "&$
!# 2
)(lim
1 x
xf
x iii) !&$"
#)]23)(([lim 2
1xxf
x.
1.7 O"#) $%&)"*+$+$ $*! ),'#"!
>'" ."$":/'< *08;"'" 60+-;% '19 =$"(1:69 ."$"*'/*%19 '$1@, *-,"$'8*%<,
hgf ,, *% 6," 21/*'5;" '59 ;+$(89 ),( !&$ .
Cf
f (x)
(a)
O
'
+& x x
y
Cg
g(x)
(%)
O +&
+&
x x
y
Ch
h(x)
O
(&)
+&
"&
x
x
y58
A"$"'5$+B;% )'1 :"C@9 '+ x "-3/,%'"1 ".%$1)$1*'" ;% +.+1+,28.+'% '$).+,
1 !"#! – $%&'('#) $%&)"*+$+$ 183
— !" )(xf #$"%&''()&* +%" ,-."/0& !"1 #$2'02!*3+ 2$*,0+ ! . 4!51 #&$(#!6%5
2/!7 .-0& +!* 5 f -8&* %!" "# +$*" !" ! 32* '$9:"/0&
!$"#%
)(lim xfx
— !" )(xg 2/;91&!2* 2#&$*+$*%!2. 4!51 #&$(#!6%5 2/!7 .-0& +!* 5 g -8&* %!"
"# +$*" !" "# 32* '$9:"/0&
"#$"#%
)(lim xgx
— !" )(xh 0&*<1&!2* 2#&$*+$*%!2. 4!51 #&$(#!6%5 2/!7 .-0& +!* 5 h -8&* %!"
"# +$*" !" &# 32* '$9:"/0&
&#$"#%
)(lim xhx
.
!"#"$%#%&%
=#+ !2 #2$2#916 #$"3>#!&* +!* '*2 12 212)5!7%"/0& !" +$*" 0*2? %/19$!5%5? f %!" "# , #$-#&* 5 f 12 &(12* "$*%0-15 %& @*9%!502 !5? 0"$:7? ),( "#! .
=19."'"* "$*%0"( 0#"$">1 12 @*2!/#6,">1, +!21 &#%x '*2 0*2 %/19$!5%5
#"/ &(12* "$*%0-15 %& @*9%!502 !5? 0"$:7? ),( "&# . AB!%*, '*2 !*? %/12$!7%&*? hgf ,, !61 #2$239!6 %8509!61 -8"/0&:
!
f (x)
(!)
O &# x x
y
Cf
Cg
O
g(x)
(")
&#
+#
x x
y
Ch
O
h(x)
(#)
&#
&#
x x
y59
!$&#%
)(lim xfx
"#$&#%
)(lim xgx
32* &#$&#%
)(lim xhx
.
C*2 !"1 /#"."'*%0+ !"/ "$("/ %!" "# 7 &# &1+? 0&'9."/ 2$*,0"> %/12$!7%&-61 8$&*2)+02%!& !2 #2$239!6 D2%*39 +$*2:
"#"#"#"#$$$$"#"#"#"#%%%%
,
xxlim 32* 0
1lim $
"#% ,x x, *!!!!',
(((())))****########""""
$$$$&#&#&#&#%%%% '()*++,-./,-
0)+*1-./,lim
,,
x ,x
32* 01
lim $&#% ,x x
, *!!!!', .
C*2 #2$9@&*'02,
&#$&#%
3lim xx
, "#$"#%
2lim xx
32* 01
lim2$
"#% xx.
1 !"#! – $%&'('#) $%&)"*+$+$ 184
!"# $# %&"# '$( !" , #" "')*(+, (" -,.'$/0 "1"%$2$30 $., (&4., '$( 0x 53 $2, 6&(76%83'2 %$": — (" '+,#&$9'3"0 34,#" (&"'5/,30 '3 :#$;<<2<# '*,(<# :#" — 13, :#$#<9-(+53 '3 #6&('1"%&"'$2 5(&=9.
,-./ 0/123425.678 69. -:;78 <24=-;:<:8
$ >'$. 2 '+,;&$2'2 1252)( 23 #!#% xxxxf . ?, 3=#&5%'(+53 $"0 "1"%$2$30 $., (&4., -"# $(, +6(<(-"'5% $(+ )(lim xf
x !"&, :#$#<9-(+53 '3 #6&('1"%&"'$2
5(&=9. @$2, 63&46$.'2 #+$9 3&-#A%5#'$3 .0 3B90: !"# 0'x /)(+53
()*
+,- #!#%
32
3
2
11
2
512)(
xxxxxf .
C63"19
100012
11
2
51lim
32%#!#%(
)*
+,- #!#
!"& xxxx :#" !"%
!"&)2(lim 3x
x
/)(+53 )2(lim)(lim 3xxf
xx !"&!"&%!"% .
!3,":;
!"# $2, 6(<+.,+5":9 '+,;&$2'2 01
1)( !x!x!xP ""
"" !!!% #
# . , 53 0'"! "-')*3":
)(lim)(lim ""
xxx!xP
!"&!"&% :#" )(lim)(lim "
"xx
x!xP#"&#"&
%
!"# 6#&;13"-5#,
#"%%!#!##"&#"&
)4(lim)7634(lim 5235 xxxxxxx
.
$ >'$. $D&# 2 '+,;&$2'2 75
13)(
3
2
#!!#
%xx
xxxf .
!"# 0'x /)(+53:
32
2
3
2
32
3
2
2
5
7
5
11
3
1
3
11
5
3
5
7
5
115
3
1
3
113
)(
xx
xx
x
x
xxx
xxx
xf
#!
!#/%
()*
+,- #!
()*
+,- !#
% .
C63"19
1
5
7
51lim
3
1
3
11lim
5
7
5
11
3
1
3
11
lim
32
2
32
2
%()*
+,- #
0!
()*
+,- !#
%#!
!#
!"&
!"&
!"&
xx
xx
xx
xx
x
x
x
:#"
1 !"#! – $%&'('#) $%&)"*+$+$ 185
05
3lim
5
3lim
3
2
!"#$
%&'!""
#
$%%&
'()*()* xx
x
xx
!"#$%&
""#
$%%&
'!!
()*()* 3
2
5
3lim0)(lim
x
xxf
xx.
'&()*+,
'), -. /.-0 1$(+/-.1. 01
11
011
1)(!x!x!x!"x"x"x"
xf ##
##
$$
$$
((((
((((!
++
++
,
,, 0-$" , 0-#!
)1"2&):
""#
$%%&
'!
()*()* ##
$$
xx x!x"
xf lim)(lim *,) ""#
$%%&
'!
+)*+)* ##
$$
xx x!x"
xf lim)(lim
'), 3,/+4&)5%,,
3
5
3
5lim
23
165lim
2
2
2
2
!""#
$%%&
'!
+((+
+)*+)* x
x
xx
xx
xx.
,-./ 01203.145 - 678/-.29.145 :;<=-3>:>5
63#4&)*(2&-,)(1) 7-):
. 6( 1/" (8". 60), -7-&
0lim !+)*
x
x" , ()!
()*
x
x"lim
+)!*
x"x
loglim0
, ()!()*
x"x
loglim
. 6( 10 00 " (8". 61), -7-&
()!+)*
x
x"lim , 0lim !
()*
x
x"
()!*
x"x
loglim0
, +)!()*
x"x
loglim
?0@0-/:9A<7 B-.7 /1767;2C/5
(1) 9 ,374&):. 3,/,;&<3&-,).
y=ax
y
1
1 y=logax
O x
60
y=ax
y=logax
1
1
O x
y61
1 !"#! – $%&'('#) $%&)"*+$+$ 186
! "##$%& '() &*$+$,-.&) /.#&% 0#12'3 &45 46$(0$78/#/) '9:/%). ;,0*/*6%8"#&:
!"#$%!$
&'()(*+,- $#$89</'&% *9-/ 46&08&'%*3 2,#96'(2( !" !*:! .
! /%*5#& )("! '() &*$+$,-.&) ! 2,8=$+.</'&% 2,#3-1) 8/ "! , /#> ( &*$+$,-.&
! 2,8=$+.</'&% 8/ )( "! . ?%& 4&69@/%08&, ( 2,#96'(2( "
!"1
" , *"#" /.#&% 8%&
&*$+$,-.&.
A4/%@3 '$ 4/@.$ $6%28$7 *9-/ &*$+$,-.&), /.#&% '$ ...},4,3,2,1{* "" , "B/% #5(8&
#& 8/+/'32$,8/ '( 2,84/6%C$69 '() 0%& 4$+7 8/09+/) '%8") '$, ", @(+&@3 5'&# $%!" .
D $6%285) '$, $6.$, &*$+$,-.&) /.#&% 	+$0$) '$, $6%28$7 '$, $6.$, 2,#96-'(2() 2'$ $% *&% @%&',4>#/'&% 1) /:3):
!"#$%!$
E& +"8/ 5'% ( &*$+$,-.& )( "! "B/% 56%$ '$ #& ' *&% -& 069C$,8/ &"%! ""!lim ,
5'&# 0%& *9-/ 0(# , ,496B/% *0 "#" '"'$%$, >2'/ 0%& *9-/ 0"" ( #& %2B7/%
#!" )* || &
D% 0#12'") %@%5'('/) '1# $6.1# 2,#&6'32/1# 5'&# $%!x , 4$, 8/+/'32&8/ 2'& 46$(0$78/#&, %2B7$,# *&% 0%& '%) &*$+$,-./). F/ '( =$3-/%& '1# %@%$'3'1# &,'># 84$6$78/ #& ,4$+$0.<$,8/ 56%& &*$+$,-%>#. ?%& 4&69@/%08&,
2
1
4
2lim
14
532lim
2
2
2
2
""*$$*
%!%! ""
""""
"".
)$,+$'#$
&. !%&/&$ 1. N& =6/.'/ '& 56%&:
i) )5210(lim 3 *$*$%!
xxx
ii) )125(lim 3 $**%!
xxx
iii) 8
5lim
3 $*%! xx
ii) 23
125lim
3
34
$**$*
$%! xx
xxx
x v)
24
12lim
23
3
$**$
$%! xx
xx
x vi)
3
2lim
10 $$$
$%! xx
x
x
vii) +,-
./0
$*
$$%! 2
5
1lim
2 xx
x
x viii) ++
,
-../
0$$
*$
*%! 2
35lim
22
x
x
x
x
x.
1 !"#! – $%&'('#) $%&)"*+$+$ 187
2. !" #$%&'% '" ($)":
i) 324lim 2 !"!#$
xxx
ii) 910lim 2 !!"#$
xxx
iii) )( 231lim 22 !"!!!#$
xxxx
iv) !"x!x"xx
%"!!"#$
),))(((lim
v) )( 34412lim 2 !"""!#$
xxxx
.
3. !" #$%&'% '" ($)":
i) x
x
x
1lim
2 !!#$
ii) )1(lim 2 xxx
"!!#$
iii)x
x
x
1lim
2 !"#$
iv) )1(lim 2 xxx
!!"#$
v) 1
1lim
2
2
""
!"!#$ xx
xx
x vi) )22(lim 22 xxxx
x"!!
!#$.
B! "#$%$&
1. *)" ')+ ,)-./$%+ 0$"12"')34+ ')24+ '/5 #, 6" 50/7/1&8%'% '" 0"$"3--'9 ($)":
i) )1(lim 2 x#xx
!!"#$
ii) 65
32)1(lim
2
23
!"!!"
!#$ xx#xx#
x.
2. N" 0$/8,)/$&8%'% '/ !&$ , :8'% '/ )105(lim 2 x$xxx
"!!!#$
6" 50-$-
;%) 8'/ !.
3. <6 !x"x
xxf !"
!!
'1
1)(
2
, 6" #$%&'% ')+ ')24+ '96 !&!", , 1)" ')+ /-
0/&%+ )8;=%) 0)(lim '!#$
xfx
.
4. !" #$%&'% '" ($)":
i) 23
|5|lim
2
2
!"!"
"#$ xx
xxx
x ii)
2
2
34
51lim
xx
xx
x !!
"!!"#$
iii) 1
||lim
2
""
!#$ x
xx
x.
1.8 $%&'('#) $%&)"*+$+$
1 !"#! – $%&'('#) $%&)"*+$+$ 188
O,-./01 231 .45678-91
!"#$ %& "'()*#+",&- hgf ,, #$( %.%/$( %& 0*)1&23- .)*)"#4",&- 5/(%(#)& "#)
.)*)24#$ "6+7)#).
(a)
Cf
O
!" f x( )0
x0 x
y
Cg g(x0)
O
(!)
!
x0 x
y
Ch
O
(")
!1
!2
x0 x
y62
8)*)#9*%:7, ;#&:
— < "'(4*#9"9 f ,/()& %*&"73(9 "#% 0x 2)& &"6:,&:
)()(lim 00
xfxfxx
"#
— < "'(4*#9"9 g ,/()& %*&"73(9 "#% 0x )==4
)()(lim 00
xgxgxx
$#
.
— < "'(4*#9"9 h ,/()& %*&"73(9 "#% 0x )==4 5,( '.4*6,& #% ;*&; #9-.
>.; #&- #*,&- 0*)1&23- .)*)"#4",&- #%' "6+7)#%- 7;(% 9 0*)1&2+ .)*4"#)"9
#9- f 5, 5&)2;.#,#)& "#% 0x . ?/()&, ,.%73($-, 1'"&2; () %(%74"%'7, !"#$%&
!'( 0x 7;(% #9 "'(4*#9"9 f. @,(&24, 36%'7, #%( )2;=%'A% %*&"7;.
)*+,-),
B?"#$ 7&) "'(4*#9"9 f 2)& 0x 3() "97,/% 0x #%' .,5/%' %*&"7%: #9-. C) =3-7, ;#& 9 f ,/()& !"#$%&. !'( 0x , ;#)(
)()(lim 00
xfxfxx
"#
@&) .)*45,&07), 9 "'(4*#9"9 ||)( xxf " ,/()& "'(,6+- "#% 0, )1%:
)0(0||lim)(lim00
fxxfxx
"""##
.
D:71$() 7, #%( .)*).4($ %*&"7;, 7&) "'(4*#9"9 f /$# $0#12 "'(,6+- ", 3()
"97,/% 0x #%' .,5/%' %*&"7%: #9- ;#)(: )) E,( '.4*6,& #% ;*&; #9- "#% 0x +
F) G.4*6,& #% ;*&; #9- "#% 0x , )==4 ,/()& 5&)1%*,#&2; ).; #9( #&7+ #9-, )( 0xf ,
"#% "97,/% 0x .
1 !"#! – $%&'('#) $%&)"*+$+$ 189
!"# $#%&'(")*#:
— + ,-.&%/0,0 !"#
$%&'
(0#.,2
0#.,1)(
2
xx
xxxf '(. (1.#" ,-.(234 ,/5 0, #657
1)1(lim)(lim 2
00
('()%)
xxfxx
, (.8 2)2(lim)(lim00
(%()')
xxfxx
,
5$9/( '(. -$&%2(" /5 9%"5 /04 f ,/5 0.
— + ,-.&%/0,0 *!
*"
#
(
+%%
(
1#.,3
1#.,1
1
)(
2
x
xx
x
xf '(. (1.#" ,-.(234 ,/5 1, #657
2)1(lim1
)1)(1(lim)(lim
111('(
%'%
()))
xx
xxxf
xxx, (.8 3)1( (f .
:1# ,-.&%/0,0 f $5- (1.#" ,-.(234 ,( 9;# /# ,0*(1# /5- $('15- 5%",*57 /04, <# ;=)(/#", #$;&, !"#$%&' !"#()*+!+.
!"# $#%&'(")*#:
— ,(-$ ./0"1#"234& !"#()*+!+ " $5#63 !"#$%&', #657 )"# >&<( !,0x ",27("
)()(lim 00
xPxPxx
()
.
— ,(-$ )+*& !"#()*+!+ Q
P $5#63 !"#$%&', #657 )"# >&<( 0x /5- $('15- 5%"-
,*57 /04 ",27("
)(
)(
)(
)(lim
0
0
0 xQ
xP
xQ
xP
xx(
).
— 73 !"#6)*&!$3' xxf +2)( ( 463 xxg !"#)( ( $5#63 !"#$%$5', #657 )"# >&<( !,0x ",27("
00
0*0*lim xxxx
()
>#" 00
,-.,-.lim xxxx
()
.
?=;54, #$5'(">.7(/#" 9/":
— 73 !"#6)*&!$3' x,xf ()( 463 xxg ,log)( ( , 10 +- , $5#63 !"#$%$5'. -./0123 41 56718193 567,.:;5123
@$9 /5. 5%",*9 /04 ,-.=2("#4 ,/5 0x >#" /"4 "'"9/0/(4 /A. 5%1A. $%5>7$/(" /5
$#%#>&/A <(8%0*#:
89:;<=>
1 !"#! – $%&'('#) $%&)"*+$+$ 190
!" #$ %&"'()*%+$, f -'$ g +."'$ %&"+/+., %)# 0x , )0)+ +."'$ %&"+/+., %)# 0x
-'$ #$ %&"'()*%+$,:
gf ! , fc " , 01#& !#c , gf " , g
f, || f -'$ ! f
2+ )3" 1(#4105+%3 0)$ #(.6#")'$ %+ 7"' 8$9%)32' 1#& 1+($7/+$ )# 0x .
:$' 1'(98+$;2': — <$ %&"'()*%+$, xxf !")( $ -'$ xxg #")( $ +."'$ #$%!&!'( =, 13>.-' %&"+-/?" %&"'()*%+=".
— @ %&"9()3%3 23)( %$ xxf +."'$ %&"+/*, %)# 1+8.# #($%2#A )3, &'(
)*+ !,,
3
2,
'B#A 3 %&"9()3%3 23)( %$ xxg +."'$ %&"+/*,.
— @ %&"9()3%3 |32|)( xxxf $ +."'$ %&"+/*,, 'B#A +."'$ )3, 2#(B*, |)(|)( xgxf $ , 01#& xxxg 32)( $ 3 #1#.' +."'$ %&"+/*, %&"9()3%3 =, ;$"02+"#
)=" %&"+/?" %&"'()*%+=" xxf $)(1 -'$ xxf 32)(2 $ .
C7>#,, '1#8+$-"A+)'$ 0)$ ;$' )3 %A"5+%3 %&"+/?" %&"'()*%+=" $%/A+$ )# '-0->#&5# 5+?(32':
)*+,-./
!" 3 %&"9()3%3 f +."'$ %&"+/*, %)# 0x -'$ 3 %&"9()3%3 g +."'$ %&"+/*, %)#
)( 0xf , )0)+ 3 %A"5+%* )#&, gof +."'$ %&"+/*, %)# 0x .
:$' 1'(98+$;2', 3 %&"9()3%3 )1(32)( 2 %$ xx"
+."'$ %&"+/*, %+ -95+ %32+.# )#& 1+8.#& #($%2#A )3, =, %A"5+%3 )=" %&"+/?" %&"'()*%+="
1)( 2 %$ xxf -'$ xxg 32)( $ .
',)"-!.+
012 3412 5167 54$ / 8 #$%9:58#8 -.
-/
0
1
2!$
02%,86
02%,22
xx
x
x/x
)(xf !'%21 #$%!&7(;
#=32y=32(x2%1)
y=x2%1
g f g3
f x
1 !"#! – $%&'('#) $%&)"*+$+$
191
,%$+
— !"# $%&'"()* )0,(!" ( f +,-% "./# !xxf 2)( 2 #$ 0*% -/#)+123 -41*% '51--,63 23 /#75215)%06.
!"# $%&'"()* ),0( #" ( f +,-% "./# x
xxf
())( $ 0*% -/#)+123 -41*% '51-,63 23
/(740# '51-,81 '51*9"6'-21.
:%* 1* -41*% ( f '51-,63, *90-4 1* -41*% '51-,63 0*% '"# 00 $x , $(7*$6 *90-4 )0()(lim
0fxf
x$
%. ;,#5)- <)23:
!!xxfxx
2)2(lim)(lim 2
00
$#$%!%
,
1()
lim)(lim00
$$%#% x
xxf
xx
0*%
!f 2)0( $ .
=/#)+123, *90-4 12 $! 6, %'#$.1*)*, 2
1$! .
$-./0123 4-.5678489 41 :25478;3 <32 =342<5 >1?6@;373
>#77& */< "* ?-296)*"* "(3 @1&75'(3 *1*A+9#1"*% '- '51*9"6'-%3 #% #/#4-3 -41*% '51-,-43 '- $%*'"6)*"* "#5 /-$4#5 #9%')#. "#53. =41*%, -/#)+123, */*-9*4"("# 1* B1294C#5)- "% -11##.)- <"*1 7+)- <"% )%* '51&9"('( f -41*% '51--,63 '’ +1* $%&'"()*.
!"#$%!$
& D%* '51&9"('( f ?* 7+)- <"% -41*% &'()*+, &) -(. .(/0123 405&267.
),( =3 , <"*1 -41*% '51-,63 '- 0&?- '()-4# "#5 ),( "! . (!,. 63*)
& D%* '51&9"('( f ?* 7+)- <"% -41*% &'()*+, &) -(. 18)0&23 405&267.
],[ =3 , <"*1 -41*% '51-,63 '- 0&?- '()-4# "#5 ),( "! 0*% -/%/7+#1
)()(lim !fxf!x
$#%
0*% )()(lim "fxf"x
$!%
(!,. 63E)
y
( ) O
(!) " a x
y
[ ] O " a x
63
(")
@1&7#B#% #9%')#4 $%*"5/81#1"*% B%* $%*'"6)*"* "(3 )#9A63 ],( "! , ),[ "! .
1 !"#! – $%&'('#) $%&)"*+$+$
192
!"# $%&'()* '+',-.-/* -01 &"1/231 &"1%4-5&/01 &/ +'%&-56%-% /(7489#1-%' %:, -% :%4%(8-0 ;/0456%-%:
!"#$%&' ()* Bolzano
<-# +':=%1, &256% )2#"6/ -. >4%7'(5 :%-48&-%&. 6'%* &"1/2#?* &"184-.&.* f &-#
],[ !" . @:/'+5 -% &.6/A% ))(,( "f"A (%' ))(,( !f!B $4A&(#1-%' /(%-)40;/1 -#" 8B#-
1% xx! , . >4%7'(5 :%48&-%&. -.* f -)61/' -#1 8B#1% &/ )1% -#"=82'&-#1 &.6/A#. <">(/(4'6)1% '&2?/' -# :%4%(8-0 ;/34.6% -#" #:#A#" . %:,+/'B. :%4%=/A:/-%'.
!+,-./0
C&-0 6'% &"184-.&. f , #4'&6)1. &/ )1% (=/'&-, +'8&-.6% ],[ !" . D1:
" . f /A1%' &"1/25* &-# ],[ !" (%', /:':=)#1, '&2?/'
" 0)()( #$ !f"f ,
-,-/ ":842/' )1%, -#"=82'&-#1, ),(0 !"x % -)-#'#, 3&-/
0)( 0 &xf .
!.=%+5, ":842/' 6'%, -#"=82'&-#1, 4A9% -.* /BA&0&.* 0)( &xf &-# %1#'(-,
+'8&-.6% ),( !" .
123453
D:, -# ;/34.6% -#" Bolzano :4#(?:-/' ,-':
— D1 6'% &"184-.&. f /A1%' &"1/25* &/ )1% +'8&-.6% # (%' +/ 6.+/1A9/-%' &’
%"-,, -,-/ %"-5 5 /A1%' ;/-'(5 >'% (8;/ #x% 5 /A1%' %41.-'(5 >'% (8;/ #x% ,
+.=%+5 +'%-.4/A :4,&.6# &-# +'8&-.6% #. (<2. 65)
y
f (x)>0
O ! a x
(")
y
f (x)<0
O
! a x
65
(!)
— E'% &"1/25* &"184-.&. f +'%-.4/A :4,&.6# &/ (%;)1% %:, -# +'%&-56%-%
&-% #:#A% #' +'%+#2'()* 4A9/* -.* f 204A9#"1 -# :/+A# #4'&6#? -.*.
!!x0 !x0
x0
y
B(!, f (!))
$(", f (")) f (a)
f (!)
O ! a
x
64
1 !"#! – $%&'('#) $%&)"*+$+$
193
x
y
!5
!4 !3
!2
!1
+
! ! +
! +
66
!"#$ %&' ()*"+,-./*) 0#,/ 12,0(),2)0%$ #," 12,03%," #4' f 5)& #)' ()67,2*' #)%8' #," x. 9"5+*+2)%8/&, , 12,0(),2)0%$' &"#$' 5:/*#&) ;' *<3':
&) =2:0+,"%* #)' 2:>*' #4' f.
?) 9* +&@8/& &1$ #& "1,()&0#3%&#& 1," ,2:>,"/ ,) ()&(,A)+8' 2:>*', *1)-85,"-%* 8/&/ &2)@%$ +&) ?2:0+,"%* #, 12$04%, #4' f 0#,/ &2)@%$ &"#$. B, 12$-04%, &"#$ *:/&) +&) #, 12$04%, #4' f 0#, &/#:0#,)A, ()60#4%&.
C)& 1&26(*)5%&, 80#; $#) @8-,"%* /& ?2,.%* #, 12$04%, #4' 0"/62#404'
xxxf 0"/4%)( !" , ]2,0[ "x# .
!2A)+6 "1,-,5:>,"%* #)' 2:>*' #4' 0)( "xf 0#, ]2,0[ " . DA,"%*
41*70"/4%00"/4% "
xxxxxx "$"$"$"! 3 4
5"x " .
D#0) ,) 2:>*' #4' f A;2:>,"/ #, 1*(:, ,2)0%,. #4' 0#& ()&0#3%&#&
%&'
()*
4,0"
, %&'
+,-
4
5,
4
"" +&) ./
0+,- ""
2,4
5.
E 1&2&+6#; 1:/&+&' (*:A/*) #& &1,#*-80%&#& #," *-85A," #," 12,03%," #4' f 0* +6@* ()60#4%&.
F)60#4%& %&'
()*
4,0"
%&'
+,-
4
5,
4
"" ./
0+,- ""
2,4
5
G1)-*5%8/,' &2)@%$' 0x 0
2
" "2
)( 0xf 1! 1 1!
H2$04%,
! 1 !
G1,%8/;', 0#& ()&0#3%&#& %&'
()*
4,0"
, ./0
+,- ""
2,4
5 *:/&) 0)( 2xf , */I 0#, ()60#4%&
%&'
+,-
4
5,
4
"" *:/&) 0)( 3xf .
1 !"#! – $%&'('#) $%&)"*+$+$
194
!"#$%&' "()*+&",-( .*&#(
!" #$%&#'" (#)*+&, ,$"-#.#/ 0#'/1#23+ -"2 (#4*5&,-"6 -"2 Bolzano 1,7 #/',7 0'43-% 46 (#)*+&, #'879"' -7&)'.
!/01234
:3-4 &7, 32'9*-+3+ f, + "$"/, #/',7 "*73&;'+ 3# ;', 1.#73-% 8793-+&, ],[ !" .
<': ! + f #/',7 32'#=56 3-" ],[ !" 1,7
! )()( !f"f "
-%-#, 07, 19(# ,*7(&% + &#-,>? -4' )("f 1,7 )(!f 2$9*=#7 ;',6, -"2.9=73-"' ),(0 !"x # -;-"7"6, )3-#
#xf $)( 0
),!-'#.+
<6 2$"(;3"2&# %-7 )()( !f"f % . !%-# (, 73=?#7 )()( !f#"f %% (@=. 67). <' (#4*53"2&# -+ 32'9*-+3+ #xfxg &$ )()( , ],[ !"x# , $,*,-+*"?&# %-7:
! + g #/',7 32'#=56 3-" ],[ !" 1,7
! 0)()( %!g"g ,
,A"?
0)()( %&$ #"f"g 1,7 0)()( '&$ #!f!g .
B$"&;'46, 3?&A4', &# -" (#)*+&, -"2
Bolzano, 2$9*=#7 ),(0 !"x # -;-"7", )3-# 0)()( 00 $&$ #xfxg , "$%-# #xf $)( 0 . !
567897
<' &7, 32'9*-+3+ f 8#' #/',7 32'#=56 3-"
8793-+&, ],[ !" , -%-#, %$46 A,/'#-,7 1,7 3-" 87$.,'% 3=5&,, 8#' $,/*'#7 2$"=*#4-7-19 %.#6 -76 #'879#6 -7&;6.
! C# -+ D"5(#7, -"2 (#4*5&,-"6 #'87,&;34' -7&)' ,$"8#71'?#-,7 %-7:
E #71%', )($f #'%6 87,3-5&,-"6 $ &;34 &7,6 32'#="?6 1,7 &+ 3-,(#*56 32-
'9*-+3+6 f #/',7 8793-+&,.
(x0 x0 ((x0
y
B(!, f (!))
f (a)
f (!)
O !
y=# #
a x
67
%(" , f ("))
y
f (a)
f (!)
O
y=# #
x
68
! a
1 !"#! – $%&'('#) $%&)"*+$+$
195
y
( ) O
(!) " a x
y
( ) O
(") " a x
69
y
[ ) O
(#) " a x
y
[ ] O
($) " a x x1 x2
%
m
m
%
!"#$ %&'&() *%+,*"-.# */0 "/ & %,$1& 2$1 (3%&."4 '&5."#61 ],[ "! , &.78%& "/
*1+1(5"- 9%:+#61.
!"#$%&' (&()*+,-. /0* 1234*+,-. ,*56.)
;$ f %,$1& .0$%7)< .0$5+"#.# ."/ ],[ "! , "4"% # f *1,+$%& ."/ ],[ "! 6&1 62-=&."# "&6) % (1& 6&1 %357&."# "&6) m. (!7. 69')
>#31'), 0*5+7/0$ ],[, 21 "!xx ! "2"/&1, :."%, 1$ )( 1xfm " (1& )( 2xfM " , $1
&.78%&
Mxfm ## )( , )*0 /371 ],[ ,-x! .
89:;<:
;*4 "/ *1+1*5$- 9%:+#61 (1& "/ 9%:+#61 %$'&56%.-$ "&6:$ *+/(8*"%& 4"& "/
+=>?2? ,*5@> 6&1< .0$%7/8< .0$5+"#.#< f 6% *%',/ /+&.6/8 "/ ],[ "! %,$1& "/
(3%&."4 '&5."#61 ],[ Mm , 4*/0 m # %357&."# "&6) (1& % # 62=&."# "&6) "#<.
?&1 *1+5'%&=61, # .0$5+"#.# xxf #6)( " ,
]2,0[ 'x! 27%& .8$/3/ "&6:$ "/ ]1,1[$ , 1@/8 %,-$1& .0$%7)< ."/ ]2,0[ ' 6% 1$"m (1& 1"M .
O 2* 3*/2
*/2 *
1
$1
y
x
70
1 !"#! – $%&'('#) $%&)"*+$+$
196
! !"#$%, &'$()*+,-).&* /.*: A, 0*& 12,34.515 f )6,&* !"#$%&' ()*+,$( +&* $,"-./' 1) ",& &,$*+./ (*3-
1.50& ),( !" , ./.) .$ 1-,$#$ .*07, .5% 1.$ (*31.50& &2./ )6,&* .$ (*31.50&
),( #$ (89. 71&), /'$2
)(lim xf$"x "#
$ +&* )(lim xfB!x %#
$ .
:,, /0;%, 5 f )6,&* !"#$%&' 01%"+,$( +&* $,"-./' 1.$ ),( !" , ./.) .$ 1-,$-
#$ .*07, .5% 1.$ (*31.50& &2./ )6,&* .$ (*31.50& ),( AB (89. 71<).
y
( ) O
(") !
B
A
a x
y
( ) O
(!) !
$
#
a x
71
=*& '&43()*>0&,
— !$ 1-,$#$ .*07, .5% 1ln)( "$ xxf , ),0( ex& , 5 $'$6& )6,&* >,516;% &-?$21&
+&* 12,)9@% 12,34.515 (89. 72), )6,&* .$ (*31.50& )2,(%' , &A$-
%'$"#
)(lim0
xfx
+&* 2)(lim $%#
xfex
.
— !$ 1-,$#$ .*07, .5% x
xf1
)( $ , )1,0(&x , 5 $'$6& )6,&* >,516;% AB6,$21& +&*
12,)9@% 12,34.515, (89. 73) )6,&* .$ (*31.50& ),1( "' , &A$-
"'$"#
)(lim0
xfx
+&* 1)(lim1
$#
xfx
.
:,3#$>& 120')4310&.& "9$20) +&* /.&, 0*& 12,34.515 f )6,&* 12,)9@% +&* >,516;% 0$,/.$,5 1) (*&1.@0&.& .5% 0$4A@% ],[ !" , ),[ !" +&* ],( !" .
e
2
O x
y72
1
1
O x
y73
1 !"#! – $%&'('#) $%&)"*+$+$
197
',)"-!.+ !" #$%&'$( )'% * $+(,-,* 4,./ !" xx 0&$% 1%", '2.34&%,'2/, 5(6" ,'2 #%4-
,'*1" )2,( // .
)0!1'#2+
!"#$%&'" () *+,-$()*) 4*+,)( #"! xxxf , ]2,[ !!x$ . ./(":
%%%% 0 f "1,23 *+,"456 *(% ]2,[ !! #6 -7$%3*'2 *+,"48, *+,2$(5*"#,.
% 91,23 0)2()( &' !f!f , 2:%&
054*+,)( &#!#"! !!!!f ;23 03242*+,2)2( (#!#"! !!!!f .
9<%'=,#6, *&':#,2 '" (% 7"8$)'2 (%+ Bolzano +<-$4"3 =,2, (%+>-43*(%,, )2,(0 !!x $ (=(%3%, 8*(" 0)( 0 !xf , %</(" 04*+, 00 !#" xx ;23 *+,"<86
4*+, 00 !" xx . ?$2, ) "@1*#*) 4*+, !" xx =4"3 '32 (%+>-43*(%, $1A2 *(% B3--
*()'2 )2,( !! .
)$3+$'#$
78 9:7;7< 1. C(2 <2$2;-(# *45'2(2 B1,%,(23 %3 D$2:3;=6 <2$2*(-*"36 B+% *+,2$-
(5*"#,. E2 F$"1(" (2 *)'"12 *(2 %<%12 2+(=6 B", "1,23 *+,"4"16.
2. E2 '">"(5*"(" #6 <$%6 () *+,=4"32 *(% 0x (36 <2$2;-(# *+,2$(5-
*"36:
i) )*+
,&"
!2,
2,4)(
3
2
xx
xxxf , 2, 20 !x
O
1
1
3
2
3 2 x
y
O
#1
1
1
2
3,5 3 2 x
y
1 !"#! – $%&'('#) $%&)"*+$+$
198
ii) !"
!#$
%&
'&(
1,3
1,1)(
2
xx
xxxf , !" 10 (x
iii) !"
!#
$
)()
)*&)&
(
2,3
2,2
2
)(
2
x
xx
xx
xf , !" 20 )(x .
3. #! $%&%'()%'% *+ ,-.+ '/ )0"12%3! '3+ ,!-!45'* )0"-'()%3+ 4!3 $%-
'5 "! 2!-56%'% '/ 7-!834( '.0+ ,!-5)'!)/, !"
i) !"
!#$
+
,(
1||,2
1||,2)(
2
xx
xxxf ii)
!"
!#
$
(
*)
&)(
2,5
2,2
65
)(
2
x
xx
xx
xf
iii) "#$
%'
(1,ln
1,)(
xx
xxxf iv)
"#$
+&),
(0,1
0,)(
2 xx
xexf
x
.
4. #! $%&%'()%'% *+ ,-.+ '/ )0"12%3! '3+ )0"!-'()%3+
i) !"
!#
$
+)
)
,)
(1,
1
1
1,32
)(
2
xx
x
xx
xf ii) !"
!#
$
%
'(
0,)0"
0,/$
)(
xx
xx
x
xf .
5. #! !,.9%:6%'% ;'3 .3 ,!-!45'* )0"!-'()%3+ %:"!3 )0"%2%:+:
i) )/$()0")( xxf ( ii) )1ln()( 2 &&( xxxf
iii) -./
012
&(
1
1/$)(2x
xf iv) xexf /$)( (
v) )ln(ln)( xxf (
6. #! !,.9%:6%'% ;'3 / %6:)*)/ 01/$ (&) xx 12%3 $:! '.0&523)'." &<)/
)'. 935)'/$! ),0( ! .
7. =3! 45>% $:! !,; '3+ ,!-!45'* ,.&0*"0$341+ )0"!-'()%3+ f, "! ?-%:-
'% 1"!" !41-!3. " '1'.3.", @)'% )'. 935)'/$! )1,( &"" / %6:)*)/
0)( (xf "! 12%3 $:! '.0&523)'." -:A!
i) 1)( 3 )&( xxxf ii) 12)( 5 &&( xxxf
iii) 42)( 4 )&( xxxf iv) 2)( 3 &&)( xxxf .
1 !"#! – $%&'('#) $%&)"*+$+$
199
8. !" "#$%&'(&)& *)+ , &('-.-,
0))(())(())(( !""#""#"" !x"x#$x"x%$x!x& ,
*#$/ 0,, $#%& 0"+ $!" %% , 12&+ %/$ 3'4&5 67+-&5, 8+" -)$ %+6-),8"
),( !" 0"+ 8+" -)$ ),( $! .
9. !" 93&')& )$ #3*-,8" ),5 -/763),-,5 f :+" *;&5 )+5 #3":8")+015 )+-
815 )$/ x, *)"7:
i) 22)( 23 ""#! xxxxf ii) 24 9)( xxxf "!
iii) 3&<)( "! xxf , ),( ''x "& iv) xxxf -/7,8)( #! , ]2,0[ 'x& .
10. !" 93&')& )$ -=7$;$ )+8>7 ).7 -/7"3)?-&.7
i) 1ln)( "! xxf , ],1[ ex& ii) 2)( #"! xxf , )2,0(&x
iii) 1,82)( #! xxf , '()
*+,&
6,0'
x iv) 1)( #! xexf , ]0,("-&x .
B! "#$%$&
1. @7 ./0
$#1#"
!2,5
2,))(()(
xx(x(x(x
xf , 7" #3$-%+$3'-&)& )$ (, >-)& , f
7" &'7"+ -/7&2?5 -)$ 20 !x .
2. @7 2.
2/
0
$#!%"#
!1,
1,5
1,12
)(
22
x%x&x
xx%x&xf , 7" 93&')& )+5 )+815 ).7 !&%&, :+"
)+5 $#$'&5 , f 7" &'7"+ -/7&2?5 -)$ 10 !x .
3. i) A-). 8'" -/763),-, f , $#$'" &'7"+ -/7&2?5 -)$ 00 !x . !" 93&'-
)& )$ )0(f , "7 :+" 06B& *!&x +-2=&+
1-/7)( "! xxfx .
ii) C8$'.5, 7" 93&')& )$ )0(g :+" ), -/763),-, g #$/ &'7"+ -/7&2?5 -)$ 00 !x 0"+ :+" 06B& !&x +-2=&+
2|,8)(| xxxxg 1" .
4. @7 $+ -/7"3)?-&+5 gf , &'7"+ $3+-817&5 0"+ -/7&2&'5 -)$ ]1,0[ 0"+ #;,3$=7 )+5 -21-&+5 )0()0( gf % 0"+ )1()1( gf $ , 7" "#$%&'(&)& *)+ /#632&+ 17" )$/;62+-)$7 )1,0(&) )1)$+$ >-)& )()( )g)f ! .
1 !"#! – $%&'('#) $%&)"*+$+$
200
5. !" "#$%&'(&)& *)+ $+ &(+,-,&+.:
") 02
1
1
1 64
!"#
#"#
x
x
x
x /) 0
2
ln
1!
"#
" x
x
x
ex
01$23 4+", )$2561+,)$3, 7'8" ,)$ )2,1( .
6. 9& :";&4+6 "#* )+. #"7":6)< #&7+#)-,&+. 3" "#$%&'(&)& *)+ $+ =7">+-:0. #"7",)6,&+. )<3 ,23"7)?,&<3 f :"+ g 01$23 03" ":7+/-. :$+3* ,@4&'$
i) xexf !)( :"+ x
xg1
)( ! ii) xxf ln)( ! :"+ x
xg1
)( !
7. i) A,)< f 4+" ,23&1?. ,2367)@,@ ,)$ %+6,)@4" ]1,1[" , =+" )@3 $#$'"
+,1B&+ 1)(22 !# xfx =+" :6;& ]1,1["$x .
") !" /7&')& )+. 7'8&. )@. &(',<,@. 0)( !xf .
/) !" "#$%&'(&)& *)+ @ f %+")@7&' )$ #7*,@4* )@. ,)$ %+6,)@4" )1,1(" .
=) C$+$. 4#$7&' 3" &'3"+ $ )B#$. )@. f :"+ #$+" @ =7">+:? )@. #"76-,)",@;
ii) D& "365$=$ )7*#$ 3" /7&')& )$3 )B#$ )@. ,23&1$B. ,2367)@,@. f ,)$ ,B3$5$ !, =+" )@3 $#$'" +,1B&+
22 )( xxf ! =+" :6;& !$x .
8. E'3&)"+ )$ )&)76=<3$ !"#$ )$2 %+#5"3$B ,1?4")$. :"+ 4'" ,2-3&1?. ,)$ ]1,0[ ,2367)@,@ f
)@. $#$'". @ =7">+:? #"76,)"-,@ /7',:&)"+ $5*:5@7@ 40," ,)$ )&)76=<3$ "2)*.
i) !" /7&')& )+. &(+,-,&+. )<3 %+"=<3'<3 )$2 )&)7"=-3$2 :"+
ii) !" "#$%&'(&)& 4& )$ ;&-7@4" )$2 Bolzano *)+ @ fC )043&+ :"+ )+. %B$ %+"=-3+&..
9. 9)$ %+#5"3* ,1?4" @ :"-
4#B5@ C &'3"+ @ =7">+:? #"76,)",@ 4+". ,2367-)@,@. f #$2 &'3"+ ,23&1?. ,)$ ],[ %& :"+ )$
),( 000 yx' &'3"+ 03" ,@-
4&'$ )$2 &#+#0%$2,
y
!(0,0)
#(1,1)
"(1,0)
$(0,1)
x
y
'(x, f (x))
"(&, f (&))
B(%, f (%))
O % a x
'0(x0,y0)
1 !"#! – $%&'('#) $%&)"*+$+$
201
i) !" #$%&'% '() '*+( ',- "+./'"/,- )()( 0MMxd ! '(0 /,1%&(0
),( 000 yxM "+. '( /,1%&( ))(,( xfxM ',- fC 23" 456% ],[ !"x" .
ii) !" "+(7%&8%'% .'3 , /0)5$',/, d %&)"3 /0)%9:- /'( ],[ !" 4"3 /', /0-
);9%3" .'3 0+5$9%3 ;)", '(0<593/'(), /,1%&( ',- fC +(0 "+;9%3 "+.
'( 0M <32.'%$( "+. .'3 "+;9(0) '" 0+.<(3+" /,1%&" ',- 4"3 ;)",
'(0<593/'(), /,1%&( ',- fC +(0 "+;9%3 "+. '( 0M +%$3//.'%$( "+.
.'3 "+;9(0) '" 0+.<(3+" /,1%&" ',-.
'",*+$'#$ -)*)&!+$+$
!. "# $%&#'() %*+ ,(- *%.%$),/ *#.(*,01#(- 2% $3$401#,# ,5 6.)''% 7,
%2 5 (183.(1'+- #92%( %4:&;- $%( ,5 6.)''% <, %2 5 (183.(1'+- #92%( =#3>;-, %(,(545602,%- 1368.+2/- ,:2 %*)2,:1; 1%-.
1. =) xxf ln)( ! 4"3 xexg #!)( , '.'%
") x
xfg1
))(( !$ , *!"x = >
#) xxgf #!))(( $ , !"x = >
2. =) !"!#%
lx
xf
x 1
)(lim
1, '.'% 0)(lim
1!
%xf
x. A >
3. ?&)"3 01
lim01
limlim1
lim2020020
!&
'!&
'!()
*+,
-./0
123
& %%%% xxxxx
xxx
xxxx. A >
4. =) 1)( 4xf 23" 456% !"x 4"3 0+5$9%3 '( )(lim0
xfx%
, '.'%
4"'’ ")524, 1)(lim0
4%
xfx
.
5. @/9*%3: ") 11,1lim !./0
123
&5% xx
x = >
#) 1,1
lim !&5% x
x
x. = >
= >
1 !"#! – $%&'('#) $%&)"*+$+$
202
6. !" 1)(0 !! xf #$"%& '%$ 0, %(%) 0))((lim 2
0"
#xfx
x. ! *
7. !" 2
1)(
xxf ! , ),( $%& !x , %(%) #+%’ +"&,#- .+ )/"+0
0)(lim "$%#
xfx
.
8. !" 12&34)0 %$ ))()((lim6
xgxfx#
, %(%) )/"+0 /'$ 5) )6()6( gf ' . ! *
9. !" 1|)(|lim0
"#
xfxx
, %(%) #+%’ +"&,#- .+ )/"+0
1)(lim0
"#
xfxx
6 1)(lim0
("#
xfxx
.
10.!" 0|)(|lim0
"#
xfxx
, %(%) 0)(lim0
"#
xfxx
. ! *
11. !" - f )/"+0 '1")467 '%$ ! #+0 ,0+ 4)x 0'48)0
4
127)(
2
($(
"x
xxxf , %(%) %$ )4(f )/"+0 /'$ 5) 1.
12. !" - f )/"+0 '1")467 '%$ ]1,1[( #+0 4)1( "(f , 3)1( "f ,
%(%) 12&34)0 23+,5+%0#(7 +30.5(7 )1,1(0 (&x %9%$0$7, :'%) "xf ")( 0 .
!!. "# $%$&'()*) *+ (,(*- #./0*+(+ () $#1)23/ #.4 *35 .#6#$/*, .)-63.*'()35
1. !" lxfxx
"#
)(lim0
, mxgxx
"#
)(lim0
, !&ml, #+0 )()( xgxf * #$"%& '%$ 0x ,
%(%) #+%’ +"&,#- .+ )/"+0:
!) ml * ;) ml ! <) ml +
=) ml " >) lm * .
2. ?$ (30$ 32
32
)1(
)21(lim
$(
$%# x
x
x )/"+0 /'$ 5):
!) 8 ;) 1 <) 0 =) $% >) 8( .
3. ?$ 2
2323 |1|lim
x
xxxx
x
$((($%#
)/"+0 /'$ 5):
!) $% ;) (% <) 1 =) 1( >) 0.
! *
! *
! *
! *
1 !"#! – $%&'('#) $%&)"*+$+$
203
4. !" #$ xx
xxx
xx !!!
" 3
23 2lim
0
%&" '()*+&,, #-#&:
!) 00 #x .) 20 #x /) 10 !#x 0) 10 #x .
!!!.
1. 01"$"#2, $, 3'"2*#43&,5
1)2(
1)(
2$
!#
xxf 62,
1
1)(
2 !#
xxg .
!(- #$'5 72*26)#8 ,3+'*,39$:5 ;)<$5 &1"2, $:
!) = g &1"2, 3'"&+45 3#$ 2
.) = f &1"2, 3'"&+45 3#$ 1
/) = g >+&, %'$ 3=9&12 3#2 $($12 %&" &1"2, 3'"&+45
0) 1)(lim #$%"
xfx
.
2. 7$,2 2(- #2 (2*26)#8 -*,2 &1"2, 62;?5 $*,39>"2;
!) 1lim 20
0$!
"xx
x .) 1lim 20
0!!
"xx
x
/) 13lim 9 !$$%"
xxx
0) 13lim 9 !$!%"
xxx
@) )]1[ln(lim 3
0$$
"xx
x AB) )]1[ln(lim 3
0!$
"xx
x.
3. 01", = 3'")*#=3= f = $($12 &1"2, 3'"&+45 3#$ %,)3#=92 ]3,0[#! ,
9& 2)0( #f , 1)1( #f 62, 1)3( !#f .
7$,$5 2(- #$'5 (2*26)#8 ,3+'*,39$:5 %&" (*$6:(#&, 62#’ 2")C6= 2(- #,5 '($<>3&,5;
!) D()*+&, )3,0(0 &x #>#$,$5, ?3#& 0)( 0 #xf .
.) 1)(lim3
!#!"
xfx
.
/) )2()(lim2
fxfx
#"
.
0) )(]2,1[ !f'! .
@) E 9>C,3#= #,94 #=5 f 3#$ ]3,0[ &1"2, #$ 2 62, = &;)+,3#= #,94 #=5 #$ 1! .
1 !"#! – $%&'('#) $%&)"*+$+$
204
#$*!"#,! $+-'#.-)
+ /00123 456 7809:45756 ! "##$%& '() *+#,-'(*(), .) "/0-&*( 1%&) 23,-'(*() &#,12*& *2 45$ *+6/2/-%1"#2) 7$*8'('2), 210	:2'&% 1’ "#&# +7$#$$512#$ '-87$ ;4( &78 '(# &-<&%8'('&. =#& <&-&/'(-%*'%/8 7&-,42%61& &7$'2>$5# $% 79#&-/2) <$-4?# '() “@>1&6"*'()”, '$+ =>>(#& 1&A(1&'%/$5 /&% &*'-$#81$+ '() &>23-%#;) 72-%84$+ B>&54%$+ C'$>21&9$+. D'( 1%& *';>( &+'?# '.# 7%#,/.# +7,-<$+# '& 1;/( '.# '83.# 2#8) /5/>$+ /&% *'(# ,>>( '& 1;/( '.# &#'9*'$%<.# <$-4?#. E-(*%-1$7$%?#'&) '(# "##$%& '$+ (1%'8#$+ *'$# 1$#&4%&9$ /5/>$ 17$-$512 #& 2/0-,*$+12 &#&>+'%/, '( “*+#,-'(-*(” '.# 7%#,/.# '$+ C'$>21&9$+ .) 23;):
<$-4; '83$+ 2
(122)(x
AMABx !!! .
F2 '$# 94%$ +7$#$$512#$ '-87$ ( "##$%& '() *+#,-'(*() 210	:2'&% *'$+) >$6&-%A1%/$5) 79#&/2) 7$+ /&'&*/2+,*'(/&# *'%) &-<") '$+ 170+ &%?#&. G& 626$#8'& 7$+ "4.*&# &7$0&*%*'%/; ?A(*( *'(# &#,7'+3( '() "##$%&) '() *+#,-'(*() ;'&# ( 4(1%$+-69& '() H>62I-&) (<-;*( 6-&11,'.# /&% 2%4%/?# *+1I8>.# 6%& '(# &#&7&-,*'&*( 1&A(1&'%/?# 7-,32.#, *<"*2-.#, &6#?*'.# /.>7.) /&% '() &#&>+'%/;) 62.12'-9&) (<-;*( '$+ &>62I-%-/$5 *+1I$>%*1$5 *2 62.12'-%/, 7-$I>;1&'&). J Descartes, *'$ "-6$ '$+ “La Geometrie” (1637), 7&-$+*%,:$#'&) '( 1"A$4$ 7-$*4%$-%*1$5 1%&) /&175>() &78 1%& 239*.*( .) 7-$) x /&% y ('& $7$9& 2/0-,:$+# '& 2+A5-6-&11& '1;1&'&-*+#'2'&61"#2) '.# *(129.# '() /&175>()), 72-%"6-&K2 6%& 7-?'( 0$-, '( 4+#&'8'('& &#&>+'%/;) &#&7&-,*'&*() 1%&) *<"*() 23,-'(*() &#,12*& *2 12'&I>('") 7$*8'('2):
“!" #$%&'" &()$*+, -%.-$/%0( 1". (&,%)$ $4 -%.5$),6%07" 6%+7" 8%. 6$ 6+2+. y 6'6, 3. &)$09:,% 1". (&,%)$ $4 6%+7" 8%. 6$ 6+2+. x 0.% ,&$+1";4 +%. .&,%)<. -%.5$),6%07" =>+,<;", +, 6> ?$23,%. 6;" $&$<;" +&$),< ". =/,-%.=6,< > @>6$9+,"> 0.+&9#>”.
J 8-$) “*+#,-'(*(” (&78 '$ >&'%#%/8 -;-1& fungor, 7$+ *(1&9#2% 2/'2>?, >2%'$+-6?) 210	*'(/2 6%& 7-?'( 0$-, '$ 1673 *’ "#& <2%-86-&0$ '$+ Leibniz 12 '9'>$ “! &#'9*'-$0( 1"A$4$) '.# 20&7'$1"#.# ; 72-9 *+#&-';*2.#” (Methodus tangentium inversa, seu de functionibus), *'$ $7$9$ 232',:2'&% $ +7$>$6%*18) '.# '2'&61"#.# y '.# *(129.# 1%&) /&175>() 8'&# 29#&% 6#.*'; /,7$%& %4%8'('& '.# &-#'9*'$%<.# 20&7'$1"#.#. J 8-$) &+'8) ,-<%*2 #& &7$/', &78 2/29#( '(#
A
M
B
A x
y
x
1 !"#! – $%&'('#) $%&)"*+$+$
205
!"#$% &'( ')'(*+!,- .-&(.*( /'( +-0 (0("(,1.+(.- "#.#+%+20 "#3 !4(,-+50+(' ("6 177!8 &!+(97-+:8 "#.6+-+!8, ')'(*+!,( 6+(0 - !41,+-.- (3+% &"#,!* 0( "1,!' +- &#,;% &'(8 (0(73+'<%8 :<;,(.-8. = J. Bernoulli :)2-.! +# 1718 +#0 !"6&!0# /!0'<6 #,'.&6:
“!"#$%&' ()"%*+,(, -"./ $-+012,+#3 $-456#)/ $70 8#(.+,+0 8#) (9,$0-+:&-+07 $- #8#7#;<8#+- +*.8# 08. 0)+. +# $-+012,+. $54-6#/ =07 08. (+0-6-*5/”.
> (0+*7-?- +-8 .301,+-.-8 28 “(0(73+'<%8 :<;,(.-8” <3,'1,$-.! /'( :0( &!/17# $,#0'<6 )'1.+-&(, .+- )'1,<!'( +#3 #"#*#3 - &(@-&(+'<% (-0173.- #,*A#0+(0 28 - /!0'<% !"'.+%&- +20 &!+(97-+50 <(' +20 .30(,-+%.!50 +#38. = !"6&!0#8 #,'.&68, "#3 +(3+*A!' +-0 :00#'( +-8 .301,+--.-8 &! (3+%0 +-8 “(0(73+'<%8 :<;,(.-8”, )6@-<! ("6 +#0 L. Euler +# 1748, .+# :,/# +#3 “B'.(/2/% .+-0 ("!',#.+'<% (0173.-”.
“>)"%*+,(, $70/ $-+012,+</ 8#(.+,+0/ #"#$%&-+07 $70 0"02)+7=< 5=?*0-(, 8#) (9,$0+:&-+07 $- #8#7#;<8#+- +*.8# 08. 0)+< +, $-+012,+< 8#(.+,-+0 =07 0*76$#3/ < (+06-*5/ 8#(.+,+-/”.
> "(,(":,( !4:7'4- +-8 :00#'(8 +-8 .301,+-.-8 ",#%7@! <3,*28 ("6 +-0 ",#."1@!'( &(@-&(+'<%8 !,&-0!*(8 ;3.'<50 ",#97-&1+20, 6"28 ".$. +# ",697-&( &'(8 "(776&!0-8 $#,)%8, .+!,!2&:0-8 .+( )3# 1<,( +-8. C’ (3+6 +# ",697-&(, "#3 ("(.$67-.! ')'(*+!,( +#38 !"'.+%�!8 .+- )'1,-<!'( +#3 18#3 ('50(, A-+!*+(' 0( ",#.)'#,'.+!* &'( .301,+-.- +-8 &#,-
;%8 ),( txfy ! "#3 "!,'/,1;!' +# .$%&( +-8 $#,)%8 .! &'( )!)#&:0- $,#-
0'<% .+'/&% t. D# !*)#8 6&28 +20 .30(,+%.!20 "#3 3"!'.:,$#0+(' .’ (3+6 +# A%+-&( !*0(' +6.# /!0'<6, "#3 (01/<(.! +#38 &(@-&(+'<#E8 0( (0(@!-2,%.#30 +-0 <(@'!,2&:0- (0+*7-?- 6+' <1@! .301,+-.- +(3+*A!+(' &! &'( (0(73+'<% :<;,(.- <(' 0( (0(A-+%.#30 /!0'<6+!,#38 #,'.&#E8. = L. Euler, %)- ("6 +# 1755 )'(+E"2.! :0( +:+#'# #,'.&6, ("(77(/&:0# ("6 +-0 1&!.- (0(;#,1 .+-0 :00#'( +-8 “(0(73+'<%8 :<;,(.-8”.
“@" =%8#7-/ 8#(.+,+-/ -A0*+B"+07 08. %22-/ 8#(.+,+-/ $- +5+#7# +*.8# B(+-, .+0" #7 +-2-)+0:-/ 022%&#)" ()$10:"-7 +# :;7# =07 $- +7/ 8*B+-/, +.+- #7 8*B+-/ #"#$%&#"+07 ()"0*+<(-7/ +'" +-2-)+0:'". @)+./ # #*7($./ -:"07 8#23 -)*3/ =07 8-*720$1%"-7 =%6- $56#;# $- +," #8#:0 $70 8#(.+,+0 60 $8#*#3(- "0 8*#(;7#*7(+-: 08. %22-/. @" 2#78." +# x )8#;,2B"-7 $70 $--+012,+< 8#(.+,+0, +.+- .2-/ #7 8#(.+,+-/ 8#) -A0*+B"+07 08. +# x $- #-8#7#;<8#+- +*.8# < 8*#(;7#*:&#"+07 08. 0)+., #"#$%&#"+07 ()"0*+<(-7/ +#) x”.
=' 0:!8 (3+:8 (0+'7%?!'8 #)%/-.(0 9(@&'(*( .+-0 :00#'( +-8 .301,+-.-8 28 (3@(*,!+-8 (0+'.+#'$*(8 (01&!.( .+( .+#'$!*( )3# .306720, "#3 )!0 (<#7#3@!* 3"#$,!2+'<1 <1"#'# “06&#”. = J. Fourier, +# 1822, !"'.%&(0! ,-+1 (3+6 +# .-&!*# &! +-0 !4%8 "(,(+%,-.-: “C-"7=%, , ()"%*+,(, )(xf
80*7(+%"-7 $70 ;70;#9< +7$B" < +-+04$5"'", =06-$7% 08. +7/ #8#:-/ -:"07 0)60:*-+,. @" ;#6-: $70 08-7*:0 +7$B" (+," +-+$,$5", x, 60 )8%*9#)" :(#)
82<6#)/ +-+04$5"-/ )(xf . D2-/ 59#)" 8*04$0+7=5/ 0*76$,+7=5/ +7$5/, 6-+7-=5/ < 0*",+7=5/ < $,;5". E-" 8*#F8#65+#)$- .+7 0)+5/ #7 +-+04$5"-/ )8.-
1 !"#! – $%&'('#) $%&)"*+$+$
206
!"#$%&# '’ ($& !)#$* $*+) . ,#&,(-)$%&# . +#& %.$ /00. +" )1)#),21)%" %3*-1) !&# !&4"+#/ &1* &5%(6 ,7$"%&# '&$ $& 2%&$ +#& +)$&,#!2 1)'*%.%&”.
+ ,--./0 123 45-,67/03 !"# $%&'()( $(* " +##(,- ."/ 0*#1&-."0"/ .-*.'2(#.-# 3% -*.4# ."/ “-#--5*.,64/ +67&-0"/”, *$4&8-# )*( ),--7(&%.,6+/ -#.,549%,/ :,- ."# +##(,- ."/ 0*#+8%,-/. ; 3'- -$< -*.+/, 3% 6--=-&1 :%>3%.&,64 $&(+5%*0", %?+7&-2% ."# ,),<.".- 3,-/ 6-3$@5"/ #- 3" $--&(*0,12%, “),-6($+/”. " 155", 3% $&(+-5%*0" 6*&'>/ -$< ." 7*0,64, %?+7&-2% ."# ,),<.".- %#</ 7-,#(3+#(* #- -6(-5(*=%' .(# '),( “#<3(”, ."# ,),<.".- 3,-/ 0*#1&."0"/ #- ),-."&%' ."# '),- -#-5*.,64 +67&-0" 0’ (5<65"&( .( $%)'( (&,03(@ ."/. A’ -*.4# ."# .%5%*-.-'- -#.'5"9" $%&' 0*#+8%,-/ 106"0% +#.(#" 6&,.,64 ( A. L. Cauchy .( 1844, 0"3%,B#(#.-/ .- %?4/: “8%& (39& %:$ Euler !&# Lagrange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uler !&# Langrange "<&3+*'#+) '" *0& %& "7,. %:$ '5$&3%2'":$, %*%" +#& &102 &00&92 %)5 '5+@)0#'+)= "7$&# '5-$/ &3!"%2 9#& $& +"%&'-.+&%7-'"# +#& '5$"-2 '5$/3%.'. '" &'5$"-2 !&# &$%7'%3)<&. C%'# 1.-., &$ %) x '5+@)07;"# +#& 13&9+&%#!2 +"%&@0.%2, %*%" . '5$/3%.'. 1)5 #')=%&# +" x!
2 x" , &$/0)9& +" %) &$ . +"%&@0.%2 x "7$&# 4"%#!2 2 &3$.%#!2, 13(1"# 9#&
%) 0*9) &5%* $& %)1)4"%.4"7 '%.$ !0/'. %:$ &'5$"-A$ '5$&3%2'":$ . *+:6 . 7,#& '5$/3%.'. 4& +1)3)='" $& 4":3.4"7 :6 '5$"-26 *%&$ 93&<"7 '%.
+)3<2 2x (1) .
C%'#, ) -&3&!%23&6 %.6 '5$(-"#&6 %:$ '5$&3%2'":$, 4":3)=+"$)6 &1* %) '.+"7) *1)5 )# 9":+(%3"6 '%&+/%.'&$ 9#& 13A%. <)3/, "7$&# &'&<26 !&# &@(@&#)6. D &@"@&#*%.%& *+:6 4& ">&<&$#'%"7, &$ '%. 4('. %)5 )3#'+)=
(1) C'#-, 7-#%&< <., ( Cauchy 8&"0,3($(,%' %)B, 8>&'/ #- ."# (#(312%,, ." 0*#1&."0" -$<5*." .,34.
O
y
x0 x
D0*#+8%,- 0.( x0 5<:> ),--6($4/ ."/ 6-3$@5"/ 0’-*-.< .( 0"3%'(
O
y
x0 x
D0*#+8%,- 0.( x0 5<:> 3%-.-E(54/ ."/ -#-5*.,64/ +6-7&-0"/ 0’-*.< .( 0"3%'(.
1 !"#! – $%&'('#) $%&)"*+$+$
207
!"# Euler $%!&'$!$(!)("#*+ $#!,% -"# ./0 12(+& (!" '+345$&" 66 !"# .7-8"# *"# “958+:7&') $%45#(;” …”.
! "#$%&'(, %)"* "+"," -*-./#0)-$ 012 " Cauchy, -+")030, "4%$-%)$56 )7* +#2)7 -+'+0$#- &03/)7( )7( /**"$-( )7( %4*/80$-( &0 3"9$5: -4%)7-#')7)-. ;+"%4*1/"*)-( -4):* )7* /**"$- -+' 56<0 90=&0)#$5: 0+"+)0,- 5-$ 0>6#)7%7 -+' )7* /**"$- )7( “-*-34)$5:( /5.#-%7(”, )7 &0)-%87&6-)$%0 %0 &$- 5-<-#6 -#$<&7)$5: $1$')7)- )=* %4*-#):%0=*, +"4 &+"#0, *- 9,*0$ -*)$50,&0*" 3"9$%&"?. ! "#$%&'( -4)'( )"4 Cauchy, +"4 1'<750 )" 1821, /80$ =( 0>:(: (/*-* +-#'&"$" "#$%&' 0,80 12%0$ 5-$ " B. Bolzano )" 1817).
“<(!0 )(xf *&$ (#%47!;(; !;= *+!$:5;!)= x '$& $= #-">.("#*+ ,!& 8&$
'4>+ !&*) !"# x (’ .%$ 1"(*.%" 1&4(!;*$ ; (#%47!;(; $#!) ./+& -4%!"!+ *&$ *"%$1&') '$& -+-+7$(*.%; !&*). 9% 12("#*+ (!;% *+!$:5;!) x *&$ $--+&7"+54/&(!; $?@;(; $, ; (#%47!;(; >$ $#@;>+A '$!4 !; 1&$3"74
)()( xfxf !"# , ; "-"A$ +@$7!4!$& $-, !; %.$ *+!$:5;!) $ '$& !;% !&*)
-"# +A/+ !" x. B’ $#!)% !;% -+7A-!0(;, ; (#%47!;(; )(xf >$ "%"*4C+!$& ,-./012 (!" 1&4(!;*$ !;= *+!$:5;!)= x, $% 8&$ '4>+ !&*) !"# x (’ $#!, !"
1&4(!;*$, ; $-,5#!; !&*) !;= 1&$3"74= )()( xfxf !"# *&'7$A%+& +-’ 4-+&-7"% *$CA *’ $#!)% !"# $. D+ 455$ 5,8&$, ; )(xf >$ -$7$*.%+& (#%+/)= 0= -7"= x, $% *&$ $-+&7"+54/&(!; $?@;(; !;= *+!$:5;!)= -$748+& -4%!"!+ *&$ $-+&7"+54/&(!; $?@;(; !;= A1&$= !;= (#%47!;(;=”.
+ 3..456 74- 4894-
@ /**"$- )7( %4*/80$-( 5-<2( 5-$ "#$%&/*0( 6330( A-%$5/( /**"$0( )7( -*634%7( +"4 <- 9*=#,%"4&0 %)- 0+'&0*- 50.63-$- ('+=( +.8. 7 +-#6-9=9"( 5-$ )" "3"53:#=&-) +0#$0,8-*, %)- +#2)- %)61$- )7( 0>/3$>:( )"4(, "#$%&/*0( -%6.0$0(, +"4 ".0,3"*)-* 54#,=( %)7* -14*-&,- )=* &--<7&-)$52* *- 1$-+#-9&-)04<"?* &0 3"9$5: -4%)7#')7)- )7* /**"$- )"4 -+0,#=( &$5#"? 5-$ )"4 -+0,#=( &0963"4. ;4): 7 -14*-&,- "179"?%0 +"33"?( *- -&.$%A7)"?* )- <0&/3$- +6*= %)- "+",- %)7#,B"*)-* )" "$-5"1'&7&- )7( &-<7&-)$5:( -*634%7( 5-$ *- %4*1/"4* )- 0*)4+=%$-56 -+")03/%&-)6 )7( &0 "#$%&/*0( &0)-.4%$5/( 0#&7*0,0(. !$ &-<7&-)$5", +#"%+6<7%-* *- >0+0#6%"4* -4)/( )$( 14%5"3,0( 0$%69"-*)-( )7* $1/- )"4 "#,"4, &0 )7* "+",-, -#8$56, 05.#6B"*)-* 7 14*-)')7)- &$-( &0)-A-33'&0*7( +"%')7)-( *- +#"%099,B0$ 0+’ 6+0$#"* &$- %)-<0#: +"%')7)- 8=#,( %)7* +#-9&-)$5')7)- *- )7 .)6*0$ +")/. ! d’ Alembert '#$%0 )" 1765 -4):* )7* /**"$- %)7* “C95453"+-,10$-” )"4 Diderot =( 0>:(: “EF%$ *.8+>"= "%"*4C+!$& :854 +%,= 455"# ,!$% !" 1+?!+7" *-"7+A %$ -7"(+88AC+& !" -72!" (+ *&$ $-,(!$(; "("1)-"!+ *&'7), $% '$& .%$ *.8+->"= 1+% *-"7+A %$ @+-+7%4 -"!. !" *.8+>"= -"# -7"(+88AC+&. .!(& 2(!+ ; 1&$3"74 *&$= !.!"&$= -"(,!;!$= $-, !" ,7&, !;= %$ +A%$& +%!+52= $*+5;!.-$”.
1 !"#! – $%&'('#) $%&)"*+$+$
208
!"#$%&' ()*+,& #’ '-.,& .)& )/*0#,, +)- +1/*2(131* .4& 5&&)*' .46 23-&4046 %6 #*' 7*'7*2'03' +/)0588*046, ) '/*9#,6 2 13&'* .) ,/*) .46 '2)-()-93'6 1,9 1,99 1,999 1,9999 …, '((: ,;* ,/*) '2)()-93'6 1,9 1,99 2 2 … (8*'.3 '-.< “$.:&1*” .) 2), )".1 ,/*) .46 '2)-()-93'6 1,9 2,01 1,9999 2,0001 … (8*'.3 '-.< =1+1/&: .) 2). > ./,+)6 #1 .)& )+)3) )* #'94#'.*2)3 ;/40*#)+)*)"0'& .4& 5&&)*' '-.< .)- )/3)- $'3&1.'* ;'/'2.4/*0.*2: 0.) 1+,#1&) +'/:71*8#', 0.) )+)3) )
S.F. Lacroix '+)71*2&"1* .) 1810 ,.* !!
!"
##$% x
x
xlim :
“!"#$ %#& '()*#+& , "-)./#,", !
!#x
x, "#,) 010(+ -1023#0-4* %#& #0 x +--
5.)*#+& 2*#&6. 7$/(8 #3908. :&+&/;)#+8 +/&24,#< 6+& 1+/0)04+"#< 4* #0 x,
=/("60-4*
x
!!
#1
, 3)+ +10#39*"4+ 10- '*(7)*& 6+2+/. %#& , "-)./#,", 2+
1+/+43)*& 1.)#0#* 4&6/%#*/, +1% #0 + +99. 2+ 1/0"*>>(?*& "-)37*&+ +-#<)
#,) #&4<, +@0A #0 43/08 x
! #0- 1+/0)04+"#< 4*&;)*#+& %90 6+& 1*/&""%#*-
/0 6+& 410/*( )+ 4*&$2*( %"0 2390-4*. B '&+@0/. +).4*"+ "#0 '0"43)0 69."4+ 6+& #,) #&4< + *6@/.?*#+& $8
!!
!!!
#"
#&
xx
x 2
6+& *1043)$8 >()*#+& 0903)+ 6+& 1&0 4&6/<, %"0 #0 x >()*#+& 4*>+9A#*/0, 6+& 410/*( )+ >()*& 4&6/%#*/, +1% 010&+'<10#* 10"%#,#+, 0"0'<10#* 4&-6/<. C-)*1;8, #0 '0"43)0 69."4+ 410/*( )+ 1/0"*>>(?*& #0 + %"0 60)#.
2390-4*: ./+ #0 + *()+& #0 %/&0 #,8 "-)./#,",8 !
!#x
x $8 1/08 #,) +%/&"#,
+A5,", #0- x”.
?*' &' .-+)+)*<0)-#1 '-.<& .4& #'2/)021(< 7*'7*2'03', )* #'94#'.*2)3 +/)0+:940'& &' '+)0-&750)-& .4& 5&&)*' .)- )/3)- '+, .4& 5&&)*' .46 23&4046 2'* &' .4& )/30)-& #1 2'9'/: '/*9#4.*2)"6 ,/)-6, 5.0* @0.1 &' 83&1* 5&' '&.*213#1&) #'94#'.*2)" ()8*0#)". A) '+).5(10#' '-.<6 .46 +/)0+:91*'6 -+</=1 ) 04#1/*&,6 “0.'.*2,6” )/*0#,6 #1 .4 B)<91*' .%& '&*0).<.%& 2'* .46 '+,(-046 .*#<6, +)- 7*'.-+@9421 '+, .)& Weierstrass 0.' #50' .)- 19)- '*@&'. C1 '-.,& .)& )/*0#,, 4 5&&)*' .)- )/3)- '+)8-#&@9421 '+, 2:91 0.)*;13) 1+)+.13'6 '((: 58*&1 5.0* 7-&'., &' '+)71*;9)"& #1 ()8*2< '-0.4/,.4.' )* *7*,.4.16 .%& )/3%& 2'* &' .--+)+)*4913 4 7*'7*2'03' -+)()8*0#)" .)-6.
2 !"#$%&"'%( )%*"(+%(
2.1 , -..%"# /,( 0#&#*1*%2
(34564787 379:3;37
!" #$%&'()*+$ ,-. (/+. 0)* 12-$34.2 1.45 +'1)" $-6" 57)-. 1.2 ." *0)#,()*-
+$ 642 )(tSS ! $3-.2 8 4$4+8+,-8 4)* (/+.4)" .*4)9 48 :&)-21' (42+' t.
x=S(t)
M
O
A
S(t0)
S(t)
M0
1
H (*-5&48(8 S 1.#)&3;$2 48 #,(8 4)* (/+.4)" 48 :&)-21' (42<+' t 1.2 )-)+5-;$4.2 !"#$%&'!' ()!'* 4)* 12-84)9.
!" *0)#,()*+$, 4/&., 642 150)2. :&)-21' (42<+' 0t 4) 12-846 =&3(1$4.2 (48
#,(8 0M 1.2 642 +$45 .06 0.&,>$*(8 :&6-)* h, ?8>.?' 48 :&)-21' (42<+'
htt """"!!!! 0 , =&3(1$4.2 (48 #,(8 !. (@:. 1). @4) :&)-216 ?25(48+. .06 0t ,%" t 8
+$4.4602(8 4)* 12-84)9 $3-.2 3(8 +$ )()( 0tStS #### . A&., ' +)!' &,-.&'&, 4)*
12-84)9 (’ .*46 4) :&)-216 ?25(48+. $3-.2
:&6-)"+$4.4602(8)()(
0
0 !!!!########
tt
tStS.
B() 4) t $3-.2 0>8(2,(4$&. (4) 0t , 46() 8 +,(8 4.:9484. 4)* 12-84)9 ?3-$2 +$ 1.>94$&8 0&)(,<<2(8 4) " # $ % & ' ( ( ' ) * + 48" #,(8" 4)* 12-84)9 1)-45
(4) 0t . C2. 4) >6<) .*46 4) 6&2) 48" +,(8" 4.:9484.", 1.#/" 4) t 4$3-$2 (4) 0t ,
4) )-)+5;)*+$ !&/0+/,1, &,-.&'&, 4)* 12-84)9 48 :&)-21' (42<+' 0t 1.2 48
(*+=)>3;)*+$ +$ )( 0t# . D8>.?':
0
0
00
)()(lim)(
tt
tStSt#
tt ##
!$
.
2 !"#$%&"'%( )%*"(+%( 210
!"# $#%&'(")*#,#+ tttS 4)( 2 !"# (,+#" - ./+&%0-.- 12.-3 (+43 5"+-067 (89.2:),
x=S(t)
t=4 t=2
4 3 2 1 O
t=0
(!)
x=S(t)
2 4
4
O
x
t
2
(")
040( - .0")*"#,# 0#970-0# 06/ 5"+-067 5#0& 0"3 9%6+"523 .0")*23 11 #t , 22 #t
5#" 33 #t (,+#" #+0".06,9;3:
$ 21
)3)(1(lim
1
34lim
1
)1()(lim)1(
1
2
11#
""""
#"
"!"#
""
#%%% t
tt
t
tt
t
StS#ttt
$ 02
)2)(2(lim
2
44lim
2
)2()(lim)2(
2
2
22#
""""
#"
"!"#
""
#%%% t
tt
t
tt
t
StS#ttt
$ 23
)3)(1(lim
3
34lim
3
)3()(lim)3(
3
2
33"#
""""
#"
"!"#
""
#%%% t
tt
t
tt
t
StS#ttt
.
!"#$%#
<=0#+ 2+# 5"+-04 5"+(,0#" $%63 0# '(>"&, 040( 56+0& .06 0t ".97("
0)()(
0
0 &""
tt
tStS, 6$40( (,+#" 0)( 0 't# , (+?, 40#+ 06 5"+-04 5"+(,0#" $%63 0# #%"-
.0(%& 56+0& .06 0t ".97(" 0)()(
0
0 (""
tt
tStS, 6$40( (,+#" 0)( 0 )t# .
,-./0123 45367829:1;
@,+#" )+;.04 #$4 0-+ @/5A(,'("# !(;*(0%,# 40" (B#$06*2+- (+43 575A6/ .( 2+# .-*(,6 06/ $ 6-+6*&C6/*( 0-+ (/1(,# - 6$6,# 29(" *( 06+ 575A6 2+# *4+6 56"+4 .-*(,6, 06 $. = 6%".*43 #/043 '(+ *$6%(, +# )(+"5(/0(, )"# 6$6"#'D$60( 5#*$7A-,
)"#0,, *( 2+#+ 0206"6 6%".*4 - $#%#:6AD 2xy #
1# (,9( .06 .-*(,6 )1,1(A '76 (B#$04*(+(3 % 5#" &
(89. 4#), (+? - 3xy # '(+ 1# (,9( .06 .-*(,6
)1,1(A 5#*,# (B#$06*2+- (89. 4:).
O
$
%
3
2 !"#$%&"'%( )%*"(+%( 211
!
" #
x
A(1,1)
($)
y=x2
y
(%)
! x
y=x3
y 4
A(1,1)
!"#$%&'(, ")%"*+ &, ,&,-./01#2$* %&,& 344#& #)+1$5 /.( *6,"/#$%&.( /#2 7874#2, # #"#9#( &, $"#)*9 &, :*&+7*2/*9 :+, 54*( /+( 7,$"84*(.
;*')#8$*, 4#+"5&, %&, 344# 1.$*9# & /#2 7874#2 (<=. 5). >, 1.$*9, MA, #)9-#2& $+,
/%$, /#2 7874#2, /.& *2?*9, AM . @,-?A( /# 1.$*9# &, 7+$*&# "3&' 1/#& 78-74# "4.1+3-*+ 1/# ', . /%$, '& 6,9&*-/,+ &, %=*+ '( “#)+,70 ?%1.” /.& *6,"/#$%&. /#2 7874#2 1/# '. >. B+,"91/'1. ,2/0 ?, B#8$*, /A),, "'( $"#)#8$* &, /.& ,C+#"#+01#2$* :+, &, #)9-1#2$* /.& *6,"/#$%&. /.( :),6+70( ",)3-1/,1.( $+,( 12&3)/.1.( 1* %&, 1.$*9# /.(.
! D1/' f $9, 12&3)/.1. 7,+ ))(,( 00 xfxA %&, 1.$*9# /.( :),6+70( /.( ",)3-
1/,1.(.
E& "3)#2$* %&, ,75$. 1.$*9# ))(,( xfxM , 0xx " , /.( :),6+70( ",)31/,1.( /.( f 7,+ /.& *2?*9, '& "#2 #)9-#2& /, 1.$*9, ' 7,+ M, ",),/.)#8$* 5/+: @,?A( /# x /*9&*+ 1/# 0x $* 0xx # , . /%$, '& 6,9&*/,+ &, ",9)&*+ $+, #-
)+,70 ?%1. " (<=. 6,). >.& 9B+, #)+,70 ?%1. 6,9&*/,+ &, ",9)&*+ 7,+ 5/,& /# x
*6,"/#$%&. 1/# '
'
&
&
O
&
5
"
x O
Cf
x
(%) x0
M(x, f (x))
y
M
A(x0, f (x0))
6
"
x O
Cf
x x0
M(x, f (x))
A(x0, f (x0))
y
M
($)
2 !"#$%&"'%( )%*"(+%( 212
!"#$"% &!' 0x (" 0xx ! ()*. 6+). ,-$ '.%/01 23&- !-4 !" 2/ (5'.'6&/(" $/
!-$ '$'(7&'8(" "9/5!'(3$- !-4 :./9%014 5/.7&!/&-4 !-4 f &!' !. ;5"%<1 -
0=#&- !-4 !3($'8&/4 !" "#$/% #&- (" 0
0 )()(
xx
xfxf
""
, "#$/% =':%0> $/ /$/(3$'8("
>!% - "9/5!'(3$- !-4 fC &!' &-("#' ))(,( 00 xfxA 2/ 3*"% 0=#&- !'
0
0 )()(lim
0 xx
xfxf
xx ""
#.
?!&% <#$'8(" !'$ 5/./07!@ '.%&(>.
!"#$%!$
?&!@ f (%/ &8$7.!-&- 0/% ))(,( 00 xfxA 3$/ &-("#' !-4 fC . A$ 857.*"% !'
0
0
0
)()(lim
xx
xfxf
xx ""
# 0/% "#$/% 3$/4 5./:(/!%0>4 /.%2(>4 #, !>!" '.#B'8(" @4 "9/-
5!'(3$- !-4 fC &!' &-("#' !-4 !, !-$ "82"#/ $ 5'8 <%3.*"!/% /5> !' ! 0/% 3*"% &8$!"="&!1 <%"628$&-4 #.
;5'(3$@4, - "C#&@&- !-4 "9/5!'(3$-4 &!' &-("#' ))(,( 00 xfxA "#$/%
)()( 00 xx,xfy "$" ,
>5'8
0
0
0
)()(lim
xx
xfxf#
xx ""
$#
.
D%/ 5/.7<"%:(/, 3&!@ - &8$7.!-&- 2)( xxf $ 0/% !' &-("#' !-4 )1,1(A . ;5"%<1
1
1lim
1
)1()(lim
2
11 ""
$""
## x
x
x
fxf
xx
2)1(lim1
$%$#
xx
,
'.#B"!/% "9/5!'(3$- !-4 fC &!' &-("#' !-4 )1,1(A . E "9/5!'(3$- /8!1 3*"% &8$!"="&!1 <%"6-
28$&-4 2$# 0/% "C#&@&- )1(21 "$" xy .
%-./012 34-456578 /89:-;</<2 /= /<0=>7
)!/ 5.'-:'6("$/, '% '.%&('# !-4 &!%:(%/#/4 !/*6!-!/4 "$>4 0%$-!'6 0/% !-4 "-9/5!'(3$-4 &" &-("#' (%/4 0/(56=-4 (/4 '<1:-&/$ &" 3$/ >.%' !-4 ('.914
0
0
0
)()(lim
xx
xfxf
xx ""
#.
D%/ !-$ %<%/#!".- 5".#5!@&- 5'8 !' 5/./57$@ >.%' 857.*"% 0/% "#$/% 5./:(/!%-0>4 /.%2(>4, <#$'8(" !'$ /0>='82' '.%&(>:
A(1,1) y=x2
x O
y 7
2 !"#$%&"'%( )%*"(+%( 213
!"#$%!$
!"# $%&'()*$* f +,-. /)" .0&#" &'(')*)+,-./ ,’ 01' ,/.2+3 0x )1% 2.301%
1("$-14 )*5, #& %2'(6." )1
0
0 )()(lim
0 xx
xfxf
xx !!
"
7#" .0&#" 2(#8-#)"7/5 #("9-/5. :1 /("1 #%)/ 1&1-';.)#" &'(4)*)35 6/5 f ,63 0x 7#" $%-<1+0;.)#" -.
)( 0xf # . =*+#3>:
0
00
)()(lim)(
0 xx
xfxfxf
xx !!
$#"
.
?"# 2#('3."8-#, #& 1)( 2 %$ xxf , )/). $)1 10 $x ,61%-.
2)1(lim1
)1)(1(lim
1
1lim
1
)1()(lim
11
2
11$%$
!%!
$!!
$!!
""""x
x
xx
x
x
x
fxf
xxxx.
@21-,&A5, 2)1( $#f .
B&, )C(#, $)*& "$/)*)# 0
0
00
)()(lim)(
xx
xfxfxf
xx !!
$#"
9,$1%-. hxx %$ 0 , )/). ,-
61%-.
h
xfhxfxf
h
)()(lim)( 00
00
!%$#
".
D1++,5 E1(,5 )1 0xxh !$ $%-<1+0;.)#" -. x! , .&C )1 $!% )()( 00 xfhxf
)()( 00 xfx!xf !% $%-<1+0;.)#" -. )( 0xf! , 12/). 1 2#(#2'&A )4215 8('E.)#":
!x
x!fxf
!x
)(lim)( 0
00 "$# .
F ).+.%)#0# "$/)*)# 13>8*$. )1 Leibniz &# $%-<1+0$." )*& 2#('8A81 $)1 0x -.
dx
xdf )( 0 > 0
)(xx
dx
xdf$ . G $%-<1+"$-/5 )( 0xf # .0&#" -.)#8.&,$).(15 7#" 1E.0+.)#"
$)1& Lagrange.
@0&#" E#&.(/ /)", #& )1 0x .0&#" .$A).("7/ $*-.01 .&/5 3"#$)>-#)15 )1% 2.301%
1("$-14 )*5 f, )/).:
F f .0&#" 2#(#8A80$"-* $)1 0x , #& 7#" -/&1 #& %2'(61%& $)1 & )# /("#
0
0
0
)()(lim
xx
xfxf
xx !!
!",
0
0
0
)()(lim
xx
xfxf
xx !!
%"
7#" .0&#" 0$#.
2 !"#$%&"'%( )%*"(+%( 214
!"# $#%&'(")*#,
— + ,-.&%/+,+ !"#
$%&
'0,
0,)(
2
2
xx
xxxf (0.#" $#%#)1-
)0,"*+ ,/2 0 *( 0)0( '(f , #324
00
lim0
)0()(lim
2
00
'&&
'&&
)&) x
x
x
fxf
xx
5#"
00
lim0
)0()(lim
2
00
'&
'&&
)*) x
x
x
fxf
xx
,
(.6
— + ,-.&%/+,+ !"#
$%
'0,5
0,)(
3
xx
xxxf '(. (0.#" $#-
%#)1)0,"*+ ,/2 0, #324
00
lim0
)0()(lim
3
00
'&
'&&
)&) x
x
x
fxf
xx
5#"
505
lim0
)0()(lim
00
'&
'&&
)*) x
x
x
fxf
xx
.
!"#$%&
74*31.# *( /2. $#%#$&.1 2%",*8:
+ 9 ,/")*"#0# /#:4/+/# (.8; 5".+/24, /+ :%2."5< ,/")*< 0t , (0.#" + $#%&)1)2; /+; ,-.&%/+,+; =>,+; )(tSx ' /+ :%2."5< ,/")*< 0t . ?+@#'<, (0.#"
)()( 00 tSt, (' .
+ A ,-./(@(,/<; '"(4=-.,+; /+; (3#$/2*>.+; ! /+; fC *"#; $#%#)1)0,"*+; ,-.&%/+,+; f, ,/2 ,+*(02 ))(,( 00 xfxA (0.#" + $#%&)1)2; /+; f ,/2 0x . ?+@#-
'<, (0.#"
)( 0xf- (' ,
2$8/( + (B0,1,+ /+; ! " # $ % & ' ( ) * + ! (0.#":
))(()( 000 xxxfxfy &('&
C+. 5@0,+ )( 0xf ( /+; (3#$/2*>.+; ! ,/2 ))(,( 00 xfxA =# /+ @>*( 5#" '()*+ ,+-
fC *,. # < '()*+ ,+- f *,. 0x .
'./.012,34 53.6/789:4
O x
y
y=&x2
y=x2
8
O x
y
y=x3
y=5x
9
2 !"#$%&"'%( )%*"(+%( 215
! !" #$%&', ()*+, +, &-$*$%&' ,+ $*./$0&' '1+-($&2,3 (3" 4*+1567" -+*8/(+/3" &5+" /0,'9$%" /0,8*(3/3" f /’ 2,+ /3&'.$ (3"
))(,( 00 xfxA , :(+, 3 f #', '.,+5 -+*+4;4./5&3
/($ 0x .
— </(; 45+ -+*8#'54&+ 3 /0,8*(3/3
xxf ")( (=9. 10).
> /0,8*(3/3 +0(7 '.,+5 /0,'97" /($ 0, +??8 #', '.,+5 -+*+4;4./5&3 /’ +0(:, +1$%
#$"""%%
&&& xx
x
x
fxf
xxx
1limlim
0
)0()(lim
000.
@+*+(3*$%&' :&;" :(5, +, ))(,( xfxM , 0'x , '.,+5 2,+ /3&'.$ (3" fC , (:(', 6+A)" ($ x ('.,'5 /($ 0, 3 (2&,$0/+ !" 1+.,'(+5 ,+ -+.*,'5 ;" $*5+67 A2/3 (3, 6+(+6:*013 '0A'.+ -$0 -'*,8'5 +-: ($ !, #3?+#7 ('.,'5 ,+ /0&-2/'5 &' ($, 8B$,+ yy ( . =(3, -'*.-(;/3 +0(7 ;" '1+-($&2,3 (3" 4*+1567" -+*8/(+/3" (3" f /($ )0,0(O $*.C$0&' (3, 6+(+6:*013 '0A'.+ 0"x .
— DE/(; ()*+ 6+5 3 /0,8*(3/3
||)( xxf " . (=9. 11)
> /0,8*(3/3 +0(7 '.,+5 /0,'97" /($ 0, +??8 #', '.,+5 -+*+4;4./5&3 /’ +0(:, +-1$%
%$"%
%"
%"
%%
%&%&%& xx
x
x
fxf
xxx
1limlim
0
)0()(lim
000
6+5
#$"""%%
#&#&#& xx
x
x
fxf
xxx
1limlim
0
)0()(lim
000
.
@+*+(3*$%&' :&;" 6+5 '#) :(5, +, ))(,( xfxM , 0'x , '.,+5 2,+ /3&'.$ (3" fC ,
(:(', 6+A)" ($ x ('.,'5 /($ 0, 3 (2&,$0/+ !" ('.,'5 ,+ /0&-2/'5 &' ($, 8B$,+
yy ( . =(3, -'*.-(;/3 +0(7 ;" '1+-($&2,3 (3" fC /($ )0,0(O $*.C$0&' (3, 6+-
(+6:*013 '0A'.+ 0"x .
F',568:
!"#$%!$
y x"
O x
M(x, f (x))
y10 +
O x
M(x,f (x)) M(x,f (x))
y11
+ %
2 !"#$%&"'%( )%*"(+%( 216
!" #$% &'"()*+&+ f ,-"%$ !"#$%&' &*. 0x /%$ $&01,$ #$% %23 *$4 2%)%/(*5
&'"67/,4:
%) !"#$$
%0
0 )()(lim
0 xx
xfxf
xx (7 )$"
8) !"#$$
$%0
0 )()(lim
0 xx
xfxf
xx
/%$ $"#$$
!%0
0 )()(lim
0 xx
xfxf
xx
,
9) $"#$$
$%0
0 )()(lim
0 xx
xfxf
xx
/%$ !"#$$
!%0
0 )()(lim
0 xx
xfxf
xx
,
*3*, .)-:.'#, 54 $()*+,-.#/ *+4 fC &*. &+#,-. ))(,( 00 xfxA *+" /%*%/3)'-
;+ ,'6,-% 0xx # .
<$% 2%)(=,$9#%, + 9)%;$/7 2%)(&*%&+ *+4 &'"()*+&+4
&'
&()
*
+$$#
0,
0,)(
xx
xxxf (>0. 12)
=?0,*%$ &*. &+#,-. *+4 )0,0(O /%*%/3)';+
,;%2*.#?"+, *+" 0#x , %;.1 ,-"%$ &'",074 &*. 0 /%$ $&01,$
!"#$
#$$
#$$
$%$%$% xx
x
x
fxf
xxx
1limlim
0
)0()(lim
000
.1
limlim0
)0()(lim
000!"###
$$
!!! %%% xx
x
x
fxf
xxx
, !" #$% &'"()*+&+ f 0$# ,-"%$ 2%)%95-
9-&$#+ &*. 0x /%$ 0$# $&01.'" .$ 2)[email protected]?&,$4 *.'
2%)%2("5 .)$&#.1, *3*, 0$# ,123,"-$ ,;%2*.#?"+
*+4 fC &*. &+#,-. ))(,( 00 xfxA .
<$% 2%)(=,$9#%, + 9)%;$/7 2%)(&*%&+ *+4 &'"()-*+&+4
'()
*+
#0,
0,)(
2 xx
xxxf ,
=," ?0,$ ,;%2*.#?"+ &*. )0,0(O , %;.1
1lim0
)0()(lim
00
##$$
%$% x
x
x
fxf
xx
,
O x
y12
+
+
O x
y
y=x
y=x2
13
2 !"#$%&"'%( )%*"(+%( 217
!"#
0limlim0
)0()(lim
0
2
00
!!!""
##$#x
x
x
x
fxf
xxx
.
,-./01023 4-5 6789:;5-
$%&' ( %)"*+&(%( ||)( xxf ! . , f !-"./ %)"!012 %&3 00 !x , .44* 5!" !-"./ 6.+.7'7-%/8( %’ .)&9,
.:3;
1lim0
)0()(lim
00
!!""
#$# x
x
x
fxf
xx
, !"#
1lim0
)0()(lim
00
"!"
!""
#"# x
x
x
fxf
xx
.
<.+.&(+3;8!, 5(4.51, 9&/ 8/. %)"*+&(%( f 863+!- ". !-"./ %)"!012 %’ =".
%(8!-3 0x 0'+-2 ". !-"./ 6.+.7'7-%/8( %’ .)&9. >", 98'2, ( f !-"./ 6.+.7'7--%/8( %&3 0x , &9&! ?. !-"./ @./ %)"!012 %&3 0x , 5(4.51 /%0;!/ &3 6.+.@*&'
?!#+(8.:
!"#$%&'
>" 8/. %)"*+&(%( f !-"./ 6.+.7'7-%/8( %’ =". %(8!-3 0x , &9&! !-"./ @./ %)-
"!012 %&3 %(8!-3 .)&9.
#,%!<"=>
A/. 0xx % =03)8!
)()()(
)()( 0
0
00 xx
xx
xfxfxfxf "&
""
!" ,
369&!
'(
)*+
,"&
""
!"##
)()()(
lim)]()([lim 0
0
0
00
0
xxxx
xfxfxfxf
xxxx
)(lim)()(
lim 000
0
0
xxxx
xfxf
xxxx"&
""
!##
00)( 0 !&-! xf ,
.:3; ( f !-"./ 6.+.7'7-%/8( %&3 0x . B638="'2, )()(lim 00
xfxfxx
!#
, 5(4.51 ( f
!-"./ %)"!012 %&3 0x . !
()*+,*
O x
y 14
2 !"#$%&"'%( )%*"(+%( 218
!" #$% &'"()*+&+ f ,-" -."%$ &'"-/01 &’ 2"% &+#-.3 0x , *4*-, &5#67"% #- *3
8)3+935#-"3 :-;)+#%, ,-" #83)-. "% -."%$ 8%)%979.&$#+ &*3 0x .
,$#&+%*-
!"# $%"&' (")*' (%+ !!. , , -+./0(,-, "#$
%&&'&&
(0,1
0,)(
3
22
x.xx
x.xxxf &1.#":
i) -+.&23' -(% 00 (x ; ii) $#0#4541-"), -(% 00 (x ;
)/(-
i) < f -."%$ &'"-/01 &*3 00 (x , %" =%$ #4"3 %"
)0()(lim)(lim00
fxfxfxx
((&)*)
0, $&3,5"%#%,
112 (+( !! 0 1*(! .
ii) >$%=)."3'#- *$1 -?01 8-)$8*;&-$1:
, !" 11,*-. , + &'"()*+&+ f ,-" -."%$ &'"-/01 =%$ -83#2"71 ,-" -."%$ 8%)%-
979.&$#+.
, !" 1(. , + &'"()*+&+ 9)(6-*%$
"#$
%&&'&&
(0,1
0,1)(
3
2
xxx
xxxxf .
— @$% 0'x , 2/3'#-
1)1(11
0
)0()( 2
&(&
(*&&
(**
xx
xx
x
xx
x
fxf,
384*-
1)1(lim0
)0()(lim
00
(&(**
)*)x
x
fxf
xx
.
— @$% 0.x 2/3'#-
1)1(11
0
)0()( 223
&(&
(*&&
(**
xx
xx
x
xx
x
fxf,
384*-
1)1(lim0
)0()(lim 2
00
(&(**
)&)x
x
fxf
xx
.
2 !"#$%&"'%( )%*"(+%( 219
!"#
0
)0()(lim
0
)0()(lim
00 !!
"!!
#$!$ x
fxf
x
fxf
xx
$#% &'()*+,-, .%# 1"! / f &0+#% '#"#.,.01%)/ 12( 00 "x .
% 3+ 1!", , / 14+5"2/1/ ."56&2#%
&'(
)#!*##
"0,1
0,1)(
3
2
xxx
xxxxf
— 7%# 0*x , *8(4)&
1)1(11
0
)0()( 2
#"#
"!##
"!!
xx
xx
x
xx
x
fxf,
('92&
1)1(lim0
)0()(lim
00
"#"!!
$!$x
x
fxf
xx
.
— 7%# 0+x *8(4)&
1)1(11
0
)0()( 223
!"!
"!#!
"!!
xx
xx
x
xx
x
fxf,
('92&
1)1(lim0
)0()(lim 2
00
!"!"!!
$#$x
x
fxf
xx
.
!"#
0
)0()(lim
0
)0()(lim
00 !!
,!!
#$!$ x
fxf
x
fxf
xx
$#% &'()*+,-, .%# 1!"! / f :&+ &0+#% '#"#.,.01%)/ 12( 00 "x .
#('-(."(
!" #$!%!&
1. ;# <"&02& 2/+ '#"5.,.( 2/- 14+5"2/1/- f 12( 1/)&0( 0x , 92#+
i) 1)( 2 #" xxf , 00 "x ii) 2
1)(
xxf " , 10 "x
iii) xxf 2/))( " , 00 "x .
2. ;# <"&02& (#+ 4'5"8&%) 2/+ '#"5.,.( 2/- 14+5"2/1/- f 12( 1/)&0(
0x , 92#+
2 !"#$%&"'%( )%*"(+%( 220
i) ||)( xxxf ! , 00 !x ii) |1|)( "! xxf , 10 !x
iii) |3|)( 2 xxxf "! , 10 !x iv)#$
#%&
'(
)((!
0,1
0,1)(
2
xx
xxxxf , 00 !x .
3. !" # $%"&'(#$# f )*"+, $%")-./ $(0 0, "+ +102)*3)() 4(, # $%"&'(#-$# )()( xxfxg ! )*"+, 1+'+565*$,7# $(0 0.
4. !809 7):)(.$)() 6/ 1'0/ (# $%";-),+ $(0 0x (,/ 1+'+<&(6 $%"+'-
(.$),/, "+ )3)(&$)() +" )*"+, 1+'+565*$,7)/ $(0 $#7)*0 +%(4.
i) #$
#%&
'
)(!
0,
0,1)(
3
2
xx
xxxf , +" 00 !x ii) 1|1|)( ("! xxf , +" 10 !x .
5. =+ >')*() (#" )3*$6$# (#/ )8+1(07;"#/ (#/ fC (+" 0'*?)(+,) $(0
))(,( 00 xfxA 5,+ <&@) 7*+ +14 (,/ $%"+'(.$),/ (6" +$<.$)6" 1 <+, 2.
!" #$%&%'
1. =+ >')*() (#" 1+'&5650 (#/ $%"&'(#$#/ ||#72)( xxxxf ("! $(0
$#7)*0 00 !x .
2. !" 5,+ 7*+ $%"&'(#$# f ,$-9), 32332)1( hhhhf (((!( , 5,+ <&@) !*h , "+ +102)*3)() 4(,:
i) 2)1( !f ii) 3)1( !+f .
3. !" #$
#%
&
'(
)"!
0,1#7
0,1
1
)(
xx
xxxf , "+ +102)*3)() 4(, 0'*?)(+, )8+1(07;"# (#/
5'+8,<./ 1+'&$(+$#/ $(0 $#7)*0 )1,0(A <+, $-#7+(*?), 7) (0" &30"+
(6" x 56"*+ 4
!.
4. =+ >')*() (#" 1+'&5650 (#/ $%"&'(#$#/ #$
#%
&
!
,"
!
0,0
0,$%"1
)(
x
xx
x
xf $(0
00 !x .
5. !" 1)(1 2 ((--( xxxfx , 5,+ <&@) !*x , "+ +102)*3)() 4(,:
i) 1)0( !f
2 !"#$%&"'%( )%*"(+%( 221
ii) 1)0()(
1 !"#
" xx
fxf, !"# 0$x $#"
1)0()(
1 !%#
% xx
fxf, !"# 0&x
iii) 1)0( '(f .
6. %& '"# ()&*+,-(- f ./&#" ()&.012 (,3 (-'./3 00 'x $#" !"# $*4. !)x "(05.":
4242 -')(-' xxxxfxx !%%#
&# #637./8.,. 9,"
i) 0)0( 'f ii) 1)0( '(f .
7. A& - ()&*+,-(- f ./&#" ()&.012 (,3 0 $#" 4)(
lim0
'* x
xf
x, &# #637./8.,.
9,":
i) 0)0( 'f ii) 4)0( '(f .
8. N# #637./8.,. 9,", #& '"# ()&*+,-(- f ./&#" 6#+#!:!/("'- (,3 0x ,
,9,.
i) )()()(
lim 000
0xf
h
xfhxf
h(#'
##*
ii) )(2)()(
lim 000
0xf
h
hxfhxf
h('
##!*
.
9. ;,3 6#+#$*,: (01'# 7/&3&,#" 3" !+#<"$=2 6#+#(,*(."2 ,:& ()&#+,1-
(.:& 4=(.:2 ,+">& $"&-,>& 63) $"&14-$#& 6*&: (,3& *83&# xx( (,3
0+3&"$9 7"*(,-'# #69 0sec =:2 8sec. ?# @+./,.:
8 7 6 5
4 2
O
t (sec)
x=S(t) $"&-,9 A
$"&-,9 %
$"&-,9 B
i) C3"3 $"&-,9 8.$/&-(. #69 ,-& #+01 ,3) *83&# $/&-(-2;
ii) C3"3 $"&-,9 $"&14-$. '9&3 6+32 ,# 7.8"*;
2 !"#$%&"'%( )%*"(+%( 222
iii) !"#" $#%&'( )**+,- ."/) $0%&1&2 '& 3/"%#$4 1'#564 2!t sec,
7"#" '& 3/"%#$4 1'#564 4!t sec $+# 7"#" '& 3/"%#$4 1'#564
5!t sec;
iv) !"#" $#%&'( $#%48&$- 7/"2 '+ +/#1'-/) 1- (*" '" 3/"%#$( 9#)-1'&6+ +7( 0sec :;2 4sec;
v) !"#" $#%&'( '-/6)'#1- 7#" $"%') 1'&% +/34 '"< ),"%+ $0%&1&2;
vi) !"#" $#%&'( 9#)%<1- '" 6-5+*='-/" 9#)1'&6+;
2.2 ,#&#*-*"("+.( (/0#&12(."( -
,#&#*-*%( (/0#&12(2
" >1'; f 6#+ 1<%)/'&1& 6- 7-90" "/#16"= :%+ 1=%"*" !. ?+ *:6- ('#:
— H f -0%+# 7+/+5;501#6& 1'" ! 4, +7*), !"#"$%$&'()*, ('+% -0%+# 7+/+5;50-1#6& 1- $)8- 1&6-0" Ax #0 .
— @ f -0%+# !"#"$%$&'()* '+ ,-" "-.(/01 2(3'0*)" ),( 34 '"< 7-90"< "/#16"=
'&2, ('+% -0%+# 7+/+5;501#6& 1- $)8- 1&6-0" ),(0 "#x # .
— @ f -0%+# !"#"$%$&'()* '+ ,-" /4+('01 2(3'0*)" ],[ 34 '"< 7-90"< "/#16"=
'&2, ('+% -0%+# 7+/+5;501#6& 1'" ),( "# $+# -7#7*:"% #13=-#
!#$$
%& #x
#fxf
#x
)()(lim $+# !#
$$
$& "x
"fxf
"x
)()(lim .
" >1'; f 6#+ 1<%)/'&1& 6- 7-90" "/#16"= ! $+# 1A 'o 1=%"*" ';% 1&6-0;% '"<
! 1'+ "7"0+ +<'4 -0%+# 7+/+5;501#6&. A%'#1'"#30B"%'+2 $)8- 1Ax# 1'" )(xf ' ,
"/0B"<6- '& 1<%)/'&1&
),(
: 1
xfx
RAf
'&&'
& "7"0+ "%"6)B-'+# !#50* !"#3$%$.6 0*6 f 4 +7*) !"#3$%$.6 0*6 f. H
7/C'& 7+/)5;5"2 '&2 f 1<6D"*0B-'+# $+# 6- dx
df 7"< 9#+D)B-'+# “%'- -. 7/"2
%'- 3#”. E#+ 7/+$'#$"=2 *(5"<2 '&% 7+/)5;5" 1<%)/'&1& )(xfy '! 8+ '& 1<6-
D"*0B"<6- $+# 6- ))(( '! xfy .
A% <7"8:1"<6- ('# '" 1! -0%+# 9#)1'&6+ 4 :%;1& 9#+1'&6)';%, '('- & 7+/)5;-
5"2 '&2 f ' , +% <7)/3-#, *:5-'+# 2+70+#* !"#3$%$.6 0*6 f $+# 1<6D"*0B-'+# 6- f '' .
2 !"#$%&"'%( )%*"(+%( 223
!"#$%$&'( )*+,-.#& / !"#$%& '()*+,+#- %.- f, 0- 3!! , '#& 1203)4+,-.#& 0- )(,f . 5/4#67
][ 1)()( "# $,, ff , 3!, .
8 -9*-1/ ./: "#*#$;$)2 12<(*./1/:, 0- 3(1/ .)< )*&10= ")2 6;1#0-, 6-< -+-<#& "(<.# -9')4/. >./ 12<?@-&# A# 6)90- 0-*&'?: 3#1&'?: "-*&".;1-&: "#*#$;-$&1/: 12<#*.71-%<, ")2 A# .&: @*/1&0)")&)90- 1./< -9*-1/ "#*#$;$)2 12<#*-.71-%< (#<.+ <# @*/1&0)")&)90- .)< )*&10= '(A- B)*().
-./012134 56/789, :.;789, ;<,./=>;62,
% C!1.% / 1.#A-*7 12<(*./1/ cxf #)( , !&c . 8 12<(*./1/ f -+<#& "#*#$%-
$+1&0/ 1.) ' '#& &1@9-& 0)( #" xf , 6/4#67
0)( #"c
D*($0#.&, #< 0x -+<#& ?<# 1/0-+) .)2 ', .=.- $&# 0xx ( &1@9-&:
0)()(
00
0 #$$
#$$
xx
cc
xx
xfxf.
!")0?<%:,
0)()(
lim0
0
0
#$$
) xx
xfxf
xx,
6/4#67 0)( #"c . !
% E1.% / 12<(*./1/ xxf #)( . 8 12<(*./1/ f -+<#& "#*#$%$+1&0/ 1.) ' '#& &1@9-& 1)( #" xf , 6/4#67
1)( #"x
D*($0#.&, #< 0x -+<#& ?<# 1/0-+) .)2 ', .=.- $&# 0xx ( &1@9-&:
1)()(
0
0
0
0 #$$
#$$
xx
xx
xx
xfxf.
!")0?<%:,
11lim)()(
lim00
0
0
##$$
)) xxxx xx
xfxf,
6/4#67 1)( #"x . !
% E1.% / 12<(*./1/ !xxf #)( , }1,0{$&"! . 8 12<(*./1/ f -+<#& "#*#$%$+1&-
0/ 1.) ' '#& &1@9-& 1)( $#" !x!xf , 6/4#67
1)( $#" !! x!x
2 !"#$%&"'%( )%*"(+%( 224
!"#$%&'(, &) 0x *+)&( ,)& -.%*+/ '/0 !, '1'* $(& 0xx " (-23*(:
100
21
0
100
210
0
0
0
0 ))(()()( ######
$$$%#
$$$#%
##
%## !!!
!!!!!
xxxxxx
xxxxxx
xx
xx
xx
xfxf&
&,
/41'*
10
10
10
10
100
21
00
0
0
)(lim)()(
lim #######
''%$$$%$$$%
## !!!!!!!
xxxxx!xxxxxxx
xx
xfxf&& ,
5.6&57 1)( #%( !! x!x . !
) 8-'9 . -0)#"'.-. xxf %)( . : -0)#"'.-. f *+)&( 4&"&$9$+-(%. -'/
),0( $* ;&( (-23*( x
xf2
1)( %( , 5.6&57
+ ,x
x2
1%
(
!"#$%&'(, &) 0x *+)&( ,)& -.%*+/ '/0 ),0( $* , '1'* $(& 0xx " (-23*(:
+ ,+ ,+ , + , 000
0
00
00
0
0
0
0 1
)()(
)()(
xxxxxx
xx
xxxx
xxxx
xx
xx
xx
xfxf
$%
$#
#%
$#
$#%
#
#%
##
,
/41'*
0000
0
0 2
11lim
)()(lim
xxxxx
xfxf
xxxx%
$%
##
'',
5.6&57 + ,x
x2
1%
(.
<49= *+5&%* -'.) 4&"#$"&>/ 3.1 . xxf %)( !"# "$#%& '%(%)*)$+&,- +./
0. !
) 8-'9 -0)#"'.-. xxf .%)( % . : -0)#"'.-. f *+)&( 4&"&$9$+-(%. -'/ ! ;&( (-23*( xxf -0))( %( , 5.6&57
xx -0)).%( %(
!"#$%&'(, $(& ;#?* -x ! ;&( 0"h (-23*(
h
xhxhx
h
xhx
h
xfhxf .%.%-0)-0).%.%)(.%)()( #.$.%
#$%
#$
2 !"#$%&"'%( )%*"(+%( 225
h
hx
h
hx
!"#$%)1#$%(!" !"#
!$ .
&'()*+
1!"
lim0
$% h
h
h ,-) 0
1#$%lim
0$
#% h
h
h,
./0$"(
xxxh
xfhxf
h#$%1#$%0!")()(
lim0
$!"!$#"
%.
1!2-*+, xx #$%)!"( $& . !
' 3#45 ! #$%674!#! xxf #$%)( $ . 8 #$%674!#! f (9%-) '-7-:5:9#)"! #40 (
,-) )#/;() xxf !")( #$& , *!2-*+
xx !")#$%( #$&
<76:"-4), :)- ,6=( )x ( ,-) 0*h )#/;():
h
xhxhx
h
xhx
h
xfhxf #$%!"!"#$%#$%#$%)(#$%)()( #!#!$
#"$
#"
h
hx
h
hx
!"!"1#$%#$% !##
!$ ,
0'>4(
+,
-./
0 !#+,-
./0 #
!$#"
%%% h
hx
h
hx
h
xfhxf
hhh
!"!"lim
1#$%#$%lim)()(
lim000
xxx !"1!"0#$% #$!#!$ .
1!2-*+, xx !")#$%( #$& . !
!"#$%#
?- >7)-
1!"
lim0
$% x
x
x, 0
1#$%lim
0$
#% x
x
x,
4- 0'09- /7!#)"0'0)+#-"( :)- %- $'020:9#0$"( 4!% '-76:5:0 45% #$%-74+#(-5% xxf !")( $ , xxg #$%)( $ (9%-) ! '-76:5:0@ #40 00 $x 45% #$%-74+#(5%
gf , -%4)#409/5@, -A0;
)0(0
0!"!"lim
!"lim
00f
x
x
x
x
xx&$
##
$%%
)0(0
0#$%#$%lim
1#$%lim
00g
x
x
x
x
xx&$
##
$#
%%.
2 !"#$%&"'%( )%*"(+%( 226
! !"#$ % "&'()#%"% xexf ")( . *+,-./0'1.#2/ 3#/ % f .4'2/ +2)25$54"/6% "#,
# 02/ /"71./ xexf "$ )( , -%82-9
xx ee "$)(
! !"#$ % "&'()#%"% xxf ln)( " . *+,-./0'1.#2/ 3#/ % f .4'2/ +2)25$54"/6%
"#, ),0( %& 02/ /"71./ x
xf1
)( "$ , -%82-9
xx
1)(ln "$
,$#&+%*,(
1. !" #$%&%' () *+,%') (+- .$"/012- 3"$4*("*+- (+- *564$(+*+- xxf ln)( " , *() )3)') + %/"3(),76+ 807$9%("0 "3: (+6 "$92 (;6 "<:6;6.
)-(.
:+./-9 x
xxf1
)(ln)( "$"$ , % .;4"$"% #%< .=2-
+#,6>'%< ! #%< fC ". >'2 "%6.4, ))(,( 00 xfxM
.4'2/
)(1
ln 0
0
0 xxx
xy '"' .
? .&@.42 ! -/>)7.#2/ 2+3 #%' 2)79 #$' 2;3'$'
)0,0(O , 2' 02/ 63', 2'
exxxx
x "("('"' 000
0
0 1ln)0(1
ln0 .
A)2, #, B%#,16.', "%6.4, .4'2/ #, )1,(eM .
1
y
O e
"
15
2 !"#$%&"'%( )%*"(+%( 227
2. !"# $%&'()* +,-.( #% /01/2/3 1, 4(% 2,
/2)(% #% /5(&"*./)/3 "63 78(5%4-3 &(89-
+"(+63 "63 +0)98"6+63 xxf 6.)( ! +"( +6-
./2( 0)(0,O 4(% ,0)(-A ()"%+"#2,:3. ;(
<8/1#=):
i) >% /?%+@+/%3 ":) 1, 4(% 2,
ii) A# /.<($*) "#0 "8%7@)#0 � +,6.("2-B#0) #% 1, , 2, 4(% # 9?#)(3 ":) x.
).(/
i) !"#$%& xxxf '())*+()( !"!" , #,)-$ 1)0( !"f .-$ 1)( #!" !f /"01# /$ 1" , 2"
23/() #4$'5'#$6
xy ! .-$ )( !xy ##!
-)1$'1/,376.
ii) 8) 9:'/(+# 1/ ':'1*+- 17) "-;-"<)7 %:/ #4$'5'#7) =;,'./(+# 01$ /$ #(-
>#,#6 1" , 2" 12+)/)1-$ '1/ '*+#,/ $%&
'()
2,
2
!!# . ?;-, 1/ 1;,@7)/ $%# 23#$ #+=-%0)
422
1 2!!!& !*! .
#('/(0"(
CD >ECFC!
1. A- =;#,1# 1*) "-;<@7@/ 1*6 '()<;1*'*6 f '1/ '*+#,/ 0x 01-):
i) 4)( xxf ! , 10 #!x ii) xxf !)( , 90 !x
iii) xxf '())( ! , 6
0
!x ! iv) xxf ln)( ! , ex !0
v) xexf !)( , 2ln0 !x .
2. N- =;#,1#, 0"/( /;,B#1-$, 1*) "-;<@7@/ 17) '()-;1&'#7):
i) +,
+-.
/
0!
1,
1,)(
2
xx
xxxf ii)
+,
+-.
/
0!
0,
0,*+)(
xx
xxxf
A(",0)
y
x
"2 "1
#
O(0,0)
16
2 !"#$%&"'%( )%*"(+%( 228
iii) !"
!#$
%
&'
2,
2,)(
4
3
xx
xxxf iv)
!"
!#$
(
)'
3/2,
3/2,)(
3
2
xx
xxxf .
3. N! !"#$%&'%(% )(* $%+ ,"-./#,+ 012%&! (13 "!.!4#563 2xy ' 0(!
#"#&! #* %7!"()2%+%3 (13 8.!7*963 "!.-0(!013 +! %&+!* 2%(!': (#,3 "!.-5515%3. ;0/:%* (# &$*# 8*! (1 8.!7*96 "!.-0(!01 (13 0,+-.(1013
3)( xxf ' ;
4. <! "!.!0(60%(% 8.!7*9-
(1+ "!.-8=8# (13 0,+-.(1-
013 f (#, $*"5!+#: 0/62!-
(#3.
5. <! "!.!0(60%(% 8.!7*9- (1
0,+-.(101 *]8,0[:f +, 1
#"#&! %&+!* 0,+%/63, 2% 0)0( 'f , 9!* (13 #"#&!3 1 "!-
.-8=8#3 "!.*0(-+%(!* 8.!-
7*9- 0(# $*"5!+) 0/62!.
!" #$%&%'
1. <! 4.%&(% (*3 (*2>3 (=+ !, " 8*! (*3 #"#&%3 1 0,+-.(101
!"
!#$
%,
&'
#x"x!
#xxxf
,
,12)( , %&+!* "!.!8=8&0*21 0(# #x '0 .
2. ?0(= 1 0,+-.(101 xxf ')( 9!* (# 012%&# ))(,( $f$A , 0-$ (13 8.!7*963 "!.-0(!013 (13 f. <! !"#$%&'%(% )(* 1 %,@%&! "#, $*>./%(!* !") (! 012%&! ))(,( $f$% 9!* )0,( $& . %7-"(%(!* (13 fC 0(# %.
3. <! !"#$%&'%(% )(* 1 %7!"(#2>+1 (13 8.!7*963 "!.-0(!013 (13 3)( xxf ' 0% #"#*#$6"#(% 012%&# (13 ),( 3!!M , 0-! >/%* 2% !,(6+
9!* -55# 9#*+) 012%&# ' %9()3 (#, (. A(# 012%&# ' 1 95&01 (13 fC
%&+!* (%(.!"5-0*! (13 95&013 (13 0(# (.
4. ?0(= ) 1 %7!"(#2>+1 (13 8.!7*963 "!.-0(!013 (13 0,+-.(1013
xxf
1)( ' 0% >+! 012%&# (13 //
0
1223
4$
$M1
, . B+ &%, %&+!* (! 012%&! 0(!
9 6 4 2
.2
.2
4
2
O
y
x
y=f (x)
8 4 2 .1
1 2
O
y
x
y=f * (x)
2 !"#$%&"'%( )%*"(+%( 229
!"!#$ % ! &'()*+ &!,- ./!)*- xx! 0$+ yy! $)&+1&!#23-, )$ $"!4*#/*&* 5&+
i) 6! " *#)$+ ('1! &!, #$.
ii) 6! *(7$45) &!, &8+9:)!, %#$ *#)$+ 1&$;*85, 4%<$4= $)*/.8&%&!
&!, *!"& .
2.3 '#,%,-( .#&#*/*"(0(
.12345467 18269:;1<67
!"#$%&' 1
>) !+ 1,)$8&=1*+- gf , *#)$+ "$8$939#1+(*- 1&! 0x , &5&* % 1,).8&%1% gf #
*#)$+ "$8$939#1+(% 1&! 0x 0$+ +12?*+:
)()()()( 000 xgxfxgf !#!$!#
#.%!-"=0
@+$ 0xx % , +12?*+:
0
0
0
0
0
00
0
0 )()()()()()()()())(())((
xx
xgxg
xx
xfxf
xx
xgxfxgxf
xx
xgfxgf
&&
#&&
$&
&&#$
&#&#
.
A"*+4= !+ 1,)$8&=1*+- gf , *#)$+ "$8$939#1+(*- 1&! 0x , '2!,(*:
),()()()(
lim)()(
lim))(())((
lim 00
0
0
00
0
00
0
0
xgxfxx
xgxg
xx
xfxf
xx
xgfxgf
xxxxxx!#!$
&&
#&&
$&
#&#'''
4%<$4=
)()()()( 000 xgxfxgf !#!$!# . !
>) !+ 1,)$8&=1*+- gf , *#)$+ "$8$939#1+(*- 1’ ')$ 4+.1&%($ ', &5&* 9+$ 0.;* 'x" +12?*+:
)()()()( xgxfxgf !#!$!# .
6! "$8$".)3 ;*:8%($ +12?*+ 0$+ 9+$ "*8+115&*8*- $"5 4?! 1,)$8&=1*+-. B%<$-
4=, $) kfff ...,,, 21 , *#)$+ "$8$939#1+(*- 1&! ', &5&*
)()()()()( 2121 xfxfxfxfff kk!##!#!$!### (( .
@+$ "$8.4*+9($,
xxx exxexxexx ##$!#!#!#!$!### 21,))3()()()%(()3%(( 22 .
.12345467 4>?6;@?6A
2 !"#$%&"'%( )%*"(+%( 230
!"#$%&' 2
!" #$ %&"'()*%+$, gf , +-"'$ .'('/0/-%$1+, %)# 0x , )2)+ 3'$ 4 %&"5()4%4
gf ! +-"'$ .'('/0/-%$14 %)# 0x 3'$ $%67+$:
)()()()()()( 00000 xgxfxgxfxgf "#"$"!
#,%!-"./
8$' 0xx % $%67+$:
0
00
0
0 )()()()())(())((
xx
xgxfxgxf
xx
xgfxgf
&&
$&
!&!
0
0000 )()()()()()()()(
xx
xgxfxgxfxgxfxgxf
&&#&
$
0
0
0
0
0 )()()()(
)()(
xx
xgxgxfxg
xx
xfxf
&&
#&&
$ .
9.+$:* #$ gf , +-"'$ .'('/0/-%$1+,, 5(' 3'$ %&"+6+-, %)# 0x , ;6#&1+:
0
0
00
00
0
00
0
0
)()(lim)()(lim
)()(lim
))(())((lim
xx
xgxgxfxg
xx
xfxf
xx
xgfxgf
xxxxxxxx &&
#&&
$&
!&!''''
)()()()( 0000 xgxfxgxf "#"$ ,
:4<':*
).()()()()()( 00000 xgxfxgxfxgf "#"$"! !
!" #$ %&"'()*%+$, gf , +-"'$ .'('/0/-%$1+, %’ ;"' :$5%)41' !, )2)+ /$' 35=+ !x( $%67+$:
)()()()()()( xgxfxgxfxgf "#"$"! .
8$' .'(5:+$/1',
xexexexexe xxxxx 1
ln)(lnln)()ln( #$"#"$" , 0)x .
># .'('.5"0 =+?(41' +.+3)+-"+)'$ 3'$ /$' .+($%%2)+(+, '.2 :7# %&"'()*%+$,. @)%$, /$' )(+$, .'('/0/-%$1+, %&"'()*%+$, $%67+$:
)()()()()()()()()())()()(( )()(])[( xhxgxfxhxgxfxhxgxfxhxgxf "!#!$!$" ""
)()()()()]()()()([ xhxgxfxhxgxfxgxf "#"#"$
)()()()()()()()()( xhxgxfxhxgxfxhxgxf "#"#"$ .
8$' .'(5:+$/1',
2 !"#$%&"'%( )%*"(+%( 231
)(ln!"ln)!"(ln!")()ln!"( !""#"!"#""!$!"" xxxxxxxxxxxx
x
xxxxxxxx
1!"ln#$%ln!"2
1""#""#"$ , 0%x .
&% f '(%)* +),)-.-(#*"! #$%/,0!#! #’ 1%) 2*/#0!") ! 3)* &c ', '+'*24
0)( $!c , #5"6.%) "' 07 8'9,!") (2) 1:7$"':
)())(( xfcxcf !$!
;*) +),/2'*-"),
2233 1836)(6)6( xxxx $"$!$! .
,-./01023 4567829
!"#$%&'
&% 7* #$%),04#'*< gf , '(%)* +),)-.-(#*"'< #07 0x 3)* 0)( 0 (xg , 0=0' 3)* !
#$%/,0!#! g
f '(%)* +),)-.-(#*"! #07 0x 3)* *#:5'*:
20
00000
)]([
)()()()()(
xg
xgxfxgxfx
g
f !)!$
!
**+
,--.
/
> )+=2'*?! +),)@'(+'0)*.
&% 7* #$%),04#'*< gf , '(%)* +),)-.-(#*"'< #’ 1%) 2*/#0!") ! 3)* -*) 3/8' !x& *#:5'* 0)( (xg , 0=0' -*) 3/8' !x& 1:7$"':
2)]([
)(()()()(
xg
xgx)fxgxfx
g
f !)!$
!
**+
,--.
/.
;*) +),/2'*-"),
2
2
2
222
)15(
5)15(2
)15(
)15()15()(
15 )"))
$)
!)))!$
!
**+
,--.
/) x
xxx
x
xxxx
x
x
2
2
2
22
)15(
25
)15(
5210
))
$)))
$x
xx
x
xxx,
5
1(x .
A,!#*"7+7*9%0)< 0*< +,7!-75"'%'< +,70/#'*< "+7,75"' 09,) %) B,75"' 0*< +),)-9-7$< "',*39% )3="! B)#*39% #$%),04#'.%.
0 C#0. ! #$%/,0!#! "xxf )$)( , *!&" . > #$%/,0!#! f '(%)* +),)-.-(#*"!
#07 *" 3)* *#:5'* 1)( )))$! "x"xf , 2!@)24
1)( ))) )$! "" x"x
D,/-")0*, -*) 3/8' *"&x 1:7$"':
2 !"#$%&"'%( )%*"(+%( 232
1
2
1
2)(
)(1)1(1)( !!
!! !"
!"#!#
"#$%&
'()"# !
!
!
!
!!
!! x!
x
x!x
xx
xx . !
!"# $#%&'(")*#,
5
5144 444)(
xxxx !"!"!"# !!!! , 0*x .
+,'#*(, -*./, $"0 $%"1 -2" 1)( !"# !! x!x , )"# 3&4( 567"3- 1+! . +$0*81./, #1 }1,0{!,!" , 2-2(
1)( !"# ,, ,xx .
- 972. : 761&%2;7: xxf (5)( " . < 761&%2:7: f (,1#" $#%#).),7"*: 720
}0761|{1 "!" xx"#" 3#" "7=>(" x
xf2761
1)( "# , ':?#';
xx
27611
)(5( "#
@%&)*#2", )"# 3&4( 1",x 8=06*(:
x
xxxx
x
xxxx
x
xx
22 761:*:*761761
761)761(:*761):*(
761:*
)(5(.
"#!#
"#$%
&'(
)"#
xx
xx22
22
7611
761:*761
".
" . !
- 972. : 761&%2:7: xxf 75)( " . < 761&%2:7: f (,1#" $#%#).),7"*: 720
}0:*|{2 "!" xx"" 3#" "7=>(" x
xf2:*
1)( !"# , ':?#';
xx
2:*1
)75( !"#
-$#&+%*-(
1. !" #$%&%' ( )"$*+,+-. /(. 012*$/(0(. 1
ln)(
!"
x
xxxf .
2 !"#$%&"'%( )%*"(+%( 233
),(-
!"#$%&:
22 )1(
ln)1)(1(ln
)1(
)1(ln)1()ln(
1
ln)(
!!!"
#!
$!!!$#
$%&'
()*
!#$
x
xxxx
x
xxxxxx
x
xxxf
22 )1(
ln1
)1(
ln1lnln
!!!
#!
!!"!#
x
xx
x
xxxxxx
2. !" "#$%&'()&* +,' $' -."/'012 #"."3,43&'2 ,56 376".,83&56
1
1)(
"#
xxf 0"' 1)( 2 "!# xxxg 1($76 0$'68 &/"#,$916: 3,$ 0$'6+ ,$72 3:-
9&*$ 1)(0,A 0"' 6" ;.&)&* : &<*353: ,:2 &/"#,$916:2 "7,82.
),(-
'()&* +, -&*.#$%& /01 )0()0( gf $#$ . !"#$%&:
22 )1(
1
)1(
)1(1)1()1(
1
1)(
"!#
"
$"!"$#
$%&'
()*
"#$
xx
xx
xxf
),1 12)1()( 2 !#$"!#$ xxxxg ,
#2/0& 1)0( !#$f ),1 1)0( !#$g .
3(,
)0(1)0( gf $#!#$ .
4 &.*5657 078 &9,20#%:+78 50# 57%&*# )1,0(A &*+,1:
1)0(11 "!#+!!#! xyxy .
./0123245 6789:;<5 6=810;<6<5 !506 /01 ;70<%& 07+ 2,(<=6=# 078 5$+<(07578 xy 27%# , 7 #2#*, &*+,1 5>+?&-57 078 xxg 2)( # ),1 078 xxf 7%)( # . @2&1-A xxx 5$+7%227% ,# , :"#$%&
)5$+(7%25$+)7%(2)5$+7%2()27%( $"$#$#$ xxxxxxx
xxxxx 5$+22)7%5$+(27%25$+2 2222 #!#!# .
B,(,07(#>%& /01 7 2,(<=6=#8 078 xy 7%2# -&+ &*+,1 7 5$+<(0757 xy 25$+# ,
/268 *568 ?, 2&(*%&+& ),+&*8 ,2/ 0#+ 0>2# xx 5$+)7%( #$ . '$0/ &.7=&*0,1 %& 0#
2,(,)<06 ?&C(7%,:
2 !"#$%&"'%( )%*"(+%( 234
!"#$%&'
!" # $%"&'(#$# g )*"+, -+'+./.*$,0# $(1 0x 2+, # f )*"+, -+'+./.*$,0# $(1
)( 0xg , (3() # $%"&'(#$# gf ! )*"+, -+'+./.*$,0# $(1 0x 2+, ,$45),
)())(()()( 000 xgxgfxgf "#"$"!
6)",2&, +" 0,+ $%"&'(#$# g )*"+, -+'+./.*$,0# $) 7"+ 8,&$(#0+ ! 2+, # f )*-"+, -+'+./.*$,0# $(1 )(!g , (3() # $%"&'(#$# gf ! )*"+, -+'+./.*$,0# $(1 !
2+, ,$45),
)())(()))((( xgxgfxgf "#"$" .
9#:+8;, +" )( xgu $ , (3()
uufuf "#"$" )())(( .
<) (1 $%0=1:,$03 (1% Leibniz, +" )(ufy $ 2+, )(xgu $ , 741%0) (1" (5-1
dx
du
du
dy
dx
dy#$
-1% )*"+, ."/$(3> /> ()*+*), -., )/0123),.
4'$'5%$%6%
?1 $50=1:1 dx
dy 8)" )*"+, -#:*21. @(1" 2+"3"+ (#> +:%$*8+> +-:& $%0-)',A7')-
(+, /> -#:*21, -'&.0+ -1% )%21:5"), (#" +-10"#03")%$# (1% 2+"3"+.
B0)$# $%"7-),+ (1% -+'+-&"/ C)/';0+(1> )*"+, (+ )D;>:
% E $%"&'(#$# "xxf $)( , !"&'" )*"+, -+'+./.*$,0# $(1 ),0( () 2+, ,$45), 1)( &$" "x"xf , 8#:+8;
1)( &$" "" x"x (1)
F'&.0+(,, +" x"" exy ln$$ 2+, C7$1%0) x"u ln$ , (3() 741%0) uey $ . G-107-"/>,
1ln 1)( &$#$##$"#$"$" ""x"uu x"
x
"x
x"eueey .
% E $%"&'(#$# x"xf $)( , 0*" )*"+, -+'+./.*$,0# $(1 + 2+, ,$45),
""xf x ln)( $" , 8#:+8;
""" xx ln)( $"
(1) !-18),2"5)(+, 3(,, .,+ 1*" # f )*"+, -+'+./.*$,0# 2+, $(1 $#0)*1 00 $x 2+, # -+'&./.3>
(#> )*"+, *$# 0) 0, )-107"/> 8*")(+, +-3 (1" *8,1 (5-1.
2 !"#$%&"'%( )%*"(+%( 235
!"#$%&'(, &) !xx e!y ln!! *&( +,-./%0 !xu ln! , '1'0 ,2./%0 uey ! . 34.%,-)56,
!!!eueey x!xuu lnln)( ln !"!#"!#!# .
$ 7 -/)#"'8-8 ||ln)( xxf ! , *!%x 09)&( 4&"&$5$9-(%8 -'. *! *&( (-2:0(
xx
1)||(ln !#
!"#$%&'(.
— &) 0&x , '1'0 x
xx1
)(ln)||(ln !#!# , 0);
— &) 0'x , '1'0 )ln(||ln xx (! , .41'0, &) +,-./%0 )ln( xy (! *&( xu (! , ,-2./%0 uy ln! . 34.%,)56,
xxu
uuy
1)1(
1
1)(ln !(
(!#"!#!#
*&( #"& x
x1
)||(ln !# .
<)&*0=&>&(;).)'&6, &) 8 -/)#"'8-8 )( xfu ! 09)&( 4&"&$5$9-(%8, '1'0 ,2./-
%0:
uu!u !! #"!# (1)( uu
u #"!#2-/)
1)0=(
uu
u #"!#2
1)( u
uu #"(!#
28%1
)-=(
uuu #"!# -/))8%(
uee uu #"!#)(
uuu #"(!# 8%)-/)(
u!!! uu #"!# ln)(
uu
u #"!#1
)||(ln
,$#&+%*E(
1. !" #$%&'() '* +"$,-.-'* /.) 01)"$/20%.)
i) 92 5)()( )! 3xxf ii) 12
)( )(! xexg iii) 1ln)( 2 )! xxh .
2 !"#$%&"'%( )%*"(+%( 236
),(-
i) !" #$%&'() 53 2 !" xu , *+*) , %'"-.*,%, )(xfy " /.-0)*12
9uy " ,
&3+*) $4&'()
uuuy #$"#"# 89 9)(
)53()53(9 282 #!$!" xx
xx 6)53(9 82 $!"
82 )53(54 !" xx .
5(&678, $4&'()
ii) )1()()( 21212#!%"#"# !%!% xeexg xx (#$%1() )12 !%" xu
)2(12
xe x %" !%
12
2 !%%" xxe
iii) )( 11
1))1(ln()( 2
2
2 #!$!
"#!"# xx
xxh (#$%1() 12 !" xu )
)1(12
1
1
1 2
22
#!$!
$!
" xxx
1
2)1(2
122 !
"$!
"x
xx
x.
2. !" #$%&%' ( %)'*+*( ,(- %."/,0123(- . ,04 565704 222: /yxC """"!!!! *,0 *(1%'0 ,04
),( 111 yxM .
),(-
!" 9:%&'() *," );6%7%, *&' <:<9&' 78 3.&8 y, =.6%<&'() +*2
22 x!y %" , 1" 0&y <12 22 x!y %%" , 1" 0'y .
>3&($"78, & <:<9&8 C 13&*)9)6*12 13+ *1 %,()61 *7" /.102<?" 31.1%*-%)7"
*7" %'"1.*@%)7"
22
1 )( x!xf %" <12 22
2 )( x!xf %%"
"1(x1,y1)
A(!,0) A#(-!,0)
C
y
x
$
O
17
2 !"#$%&"'%( )%*"(+%( 237
!" !#!$%& %$'(" !)"*+,'%& *-! ./%"*-0 1"2*-3+( ],[ !!! .(" #()(454$*"+%& *-!
('!".-0 1"2*-3+( ),( !!! .
6', -7)(, +% )(xfy " *8+9!/$*!8+% %.%$'3 (#0 -"& #()(#2'5 *8'()-:*%"& *-3' !#!$( (':.%" -! ),( 111 yxM , -0-% ;( "*<=%"
)( 1xf"# #" (1) .(" 222 )( !xfx "$ (2)
>-*", +% #()(474"*3 .(" -5' 1=! +%/7' -3& (2), ,<!8+%
0)()(22 "#$ xfxfx
!#0-%, 4"( 1xx " , ;( "*<=%"
0)()( 111 "#$ xfxfx .
>-*", /045 -3& (1) ;( ,<!8+%
011 "%$ #"yx
!#0-%, 4"( 01 &y , ;( %$'("
1
1
y
x,
!!!!""""# .
?)(, 3 %@(#-!+,'3 # ,<%" %A$*5*3:
)( 1
1
11 xx
y
xyy !!!!!!!!""""!!!! ,
3 !#!$( 4)2@%-(" 1"(1!<".2:
211
211 xxxyyy $!"!
21
2111 yxyyxx $"$
211 -yyxx """"$$$$ , (3)
(@!= 221
21 !yx "$ .
6' 01 "y , #!8 *8+9($'%" 0-(' -! *3+%$! ),( 111 yxM %$'(" -! )0,(!A : -!
)0,( !$ !# , -0-% %=.!/( (#!1%".'=%-(" 0-" !" %@(#-0+%'%& -3& fC *-( *3+%$(
(8-2 %$'(" !" .(-(.0)8@%& %8;%$%&
!x " .(" !x !"
('-"*-!$<5&. B(" !" 18! (8-,& %A"*7*%"& 1$'!'-(" (#0 -!' #()(#2'5 -=#! (3)
4"( )0,(),( 11 !yx " .(" )0,(),( 11 !yx !" ('-"*-!$<5&.
C% ('2/!4! -)0#! 9)$*.!8+% -3' %A$*5*3 -3& %@(#-!+,'3& !#!"(*1:#!-% 2/-
/3& .5'".:& -!+:&.
2 !"#$%&"'%( )%*"(+%( 238
#(',(-"(
!" #$!%!&
1. !" #$%&'% '() *"$+,-,. '-) /0)"$'1/%-)
i) 16)( 47 !"!# xxxxf ii) 3ln2)( 3 !"# xxxf
iii) xxxx
xf !"!#234
)(234
iv) 3ln(23/0))( "!# xxxf .
2. 32.&-4 '-) /0)"$'1/%-):
i) )3)(1()( 2 !!# xxxf ii) xexf x(2)( #
iii) 2
2
1
1)(
x
xxf
"!
# iv) x
xxxf
/0)1
/0)(2)(
""
#
v) xxxxf /0)(2)( 2# .
3. O2.&-4 '-) /0)"$'1/%-):
i) x
exf
x
ln)( # ii) xxxf /5%5)( "#
iii) xe
xxf
(2)( # iv)
1
1
1
1)(
!"
!"!
#x
x
x
xxf .
4. N" #$%&'%, 6*.0 .$&7%'"8, '() *"$+,-,. '-) /0)"$'1/%-):
i) $%
$&'
("
)"#
0,612
0,32)(
2
xxx
xxxxf ii)
$%
$&'
*
+"#
0,
0,(2)(
2
xx
xxxxf .
5. N" #$%&'% '" /(2%&" '(4 ,$"58914 *"$+/'"/(4 '(4 f, /'" .*.&" .8 %5"*'62%)%4 %&)"8 *"$+::(:%4 /'.) +;.)" '-) x, 6'")
i) x
xxf4
)( "# ii) xe
xxf #)( iii)
x
xxf
1)(
2 "# .
6. A) 1
)1(2)(
!"
#x
xxf 9"8
1
1
1
1)(
"!
"!"
#x
x
x
xxg , )" #$%&'% '84 /0)"$'1-
/%84 gf ,,, . </=>%8 gf ,#, ;
2 !"#$%&"'%( )%*"(+%( 239
7. !" "#$%&'(&)& *)+ $+ &,"#)*-&.&/ )0. 12",+34. #"2"5)65&0. )0.
57."2)85&0. 2)( xxf ! 3"+ 2
1
2
1)( "!
xxg 5)$ 3$+.* 59-&'$ )$7/
)1,1(A , &'."+ 36:&)&/.
8. ;'.&)"+ 9 57.62)959 !x
!x!xf
""
!)( , *!#! . !" <2&')& )+/ )+-=/ )$7 !,
1+" )+/ $#$'&/ 9 3>'59 )9/ fC 5)$ 59-&'$ )9/ )1,0(A &'."+ '59 -& 2
1.
9. !" <2&')& )" 59-&'" )9/ 12",+38/ #"265)"59/ )9/ 57.62)959/ 53)( 3 "$! xxxf , 5)" $#$'" 9 &,"#)$-=.9 &'."+:
i) #"26>>9>9 #2$/ )9. &7:&'" 19 "! xy
ii) 36:&)9 #2$/ )9. &7:&'" xy $! .
10. !" <2&')& )9. &('5059 )9/ &,"#)$-=.9/ )9/ 12",+38/ #"265)"59/ )9/ 2)( xxf ! 9 $#$'" 61&)"+ "#* )$ 59-&'$ )1,0( $A .
11. ;'.&)"+ 9 57.62)959 "x#x!xf ""! 2)( , !#"#! ,, . !" <2&')& )+/ )+--=/ )0. !#"#! ,, 1+" )+/ $#$'&/ 9 fC , %+=2?&)"+ "#* )$ 59-&'$
)2,1(A 3"+ &,6#)&)"+ )9/ &7:&'"/ xy ! 5)9. "2?8 )0. "(*.0..
12. !" <2&')& )9. #"26101$ )0. 57."2)85&0.:
i) 234 )43()( $"! xxxf ii) 3/2)1()( $! xxf
iii) %&'
()*"
!21
19-)(x
xf iv) %&'
()* $! x
xxf
1ln)(
v) 2
)( xexf $! .
13. N" <2&')& )9. #"26101$ )9/ 57.62)959/ f 5)$ 59-&'$ 0x *)".:
i) 32 1)( xxxf "! , 20 !x ii) 3/23/1 )2()2()( xxxf "! , 40 !x
iii) )(9-)( 33 x$xxf ! , 6
10 !x iv)
x
xxf
$"
!2
2)(
2
, 30 !x .
14. N" <2&')& )9. #"26101$ )0. 57."2)85&0.:
i) xxxf ln)( ! ii) 352)( $! xxf
iii) xxxf )(ln)( ! , 1+x iv) xexxf 57.9-)( ,!
15. A. xxf 29-)( ! , ." "#$%&'(&)& *)+ 2)(4)( !"-- xfxf .
2 !"#$%&"'%( )%*"(+%( 240
B! "#$%$&
1. N! !"#$%&'%(% )(* #* +,!-*./0 "!,!1(21%*0 (34 154!,(61%34
xxf
1)( ! .!* 1)( 2 "#! xxxg /7#54 /4! 8)4# .#*4) 198%&#, 1(# #"#&#
#* %-!"(#8/4%0 (#50 %&4!* .2:%(%0.
2. ;! !"#$%&'%(% )(* 9 %5:%&! 23 #! xy /7%* 8% (9 +,!-*.6 "!,21(!19
(90 1542,(9190 3)( xxf ! $<# .#*42 198%&! .!* %-2"(%(!* !5(60 1% /4! !") (! 198%&! !5(2.
3. =&4#4(!* #* 154!,(61%*0 2)( 2 ""! x!x"xf .!* x
xg1
)( ! . ;! >,%&(%
(! $!", % +*! (! #"#&! #* +,!-*./0 "!,!1(21%*0 (#50 /7#54 .#*46
%-!"(#8/49 1(# 198%&# 8% (%(898/49 10 !x .
4. =&4#4(!* #* 154!,(61%*0 xexf !)( .!* xxxg ##! 2)( . ;! !"#$%&'%(% )(* 9 %-!"(#8/49 (90 fC 1(# 198%&# )1,0(A %-2"(%(!* .!* 1(94 gC .
5. ;! >,%&(% "#?5@458# (,&(#5 >!:8#< (/(#*#, @1(% 4)0( !f ,
2)1( !#&f , 4)2( !&&f .!* 6)1()3( !f .
6. ;! !"#$%&'%(% )(* $%4 5"2,7%* "#?5@458# f $%<(%,#5 >!:8#< (#5
#"# 9 +,!-*.6 "!,21(!19 4! %-2"(%(!* (34 %5:%*@4 1"! xy .!* 13 #! xy 1(! 198%&! )1,0(A .!* )2,1(B !4(*1(#&730.
7. A4 8&! 1542,(919 !!':f %&4!* "!,!+3+&1*89 1(# 198%&# "x !0 ,
4! !"#$%&'%(% )(*
i) )()()()(
lim "fa"f"x
"f"xxf
"x&"!
##
'
ii) ))()(()()(
lim "f"fe"x
"fexfe ""x
"x&"!
##
'.
8. ;! >,%&(% (! 198%&! (90 +,!-*.60 "!,21(!190 (90 1542,(9190
xxxf 2982298)( #! , ]2,0[ #x$ ,
1(! #"#&! 9 %-!"(#8/49 (90 %&4!* "!,2??9?9 1(#4 2'#4! (34 x.
9. ;! >,%&(% (94 "!,2+3+# (34 154!,(61%34
i) 3 2)( xxf ! , ii)
3 4)( xxf !
2 !"#$%&"'%( )%*"(+%( 241
!"# $%& $'()*+#" %&( +,-$.$& %&/ +0"1%23)(&/ %&/ fC $%2 )0,0(O $+ !"4+3#" 1+5-1%.$& *.5#$%6.
10. 7$%. f 3#" 1"5"8.8-$#3& $%2 ! $'(65%&$& 8#" %&( 212-" #$*9+# 1)1( "#f !"# g & $'(65%&$& 12' 25-:+%"# "1; %&( #$;%&%"
1)1()( 2 $%%" xxfxg , &x !. <" "12=+-,+%+ 2%# & +0"1%23)(& %&/ fC
$%2 ))1(,1( fA +061%+%"# %&/ gC $%2 ))0(,0( gB .
11. 7$%. 3#" 1"5"8.8-$#3& $'(65%&$& f 8#" %&( 212-" #$*9+#
xexf x$'()&3( " , 8#" !64+ &x !
i) <" >5+-%+ %&( )0(f #
ii) <" "12=+-,+%+ ;%# & +0"1%23)(& %&/ fC $%2 $&3+-2 ))0(,0( fA
$*&3"%-:+# 3+ %2'/ 6,2(+/ #$2$!+?)/ %5-8.(2.
2.4 &,-+%( +./#0%)1(
@%&( "5*A %2' !+0"?"-2' "'%29, 25-$"3+ %& $%#83#"-" %"*9%&%" +(;/ !#(&%29 %&
*52(#!A $%#83A 0t ./ %2 ;5#2
)()()(
lim 00
0
0
tStt
tStS
tt#"
$$
'.
B2 ;5#2 "'%; %2 ?)3+ !"# !"#$% $&'()*+,- %&/ %+%3&3)(&/ S %2' !#(&%29 ./ 152/ %2 *5;(2 t %& *52(#!A $%#83A 0t . C+(#!6,
./012.1
D( =92 3+%">?&%6 3+8)4& yx, $'(=)2(%"# 3+ %& $*)$& )(xfy " , ;%"( f +-("# 3#" $'(65%&$& 1"5"8.8-$#3& $%2 0x , %;%+ 2(236:2'3+ !"#$% $&'()*+,- '*"
y 3- 4!*- '* x 5'* 56$&7* 0x %&( 1"568.82 )( 0xf # .
C#" 1"56=+#83", 2 5'43;/ 3+%">2?A/ %&/ %"*9%&%"/ ! ./ 152/ %2 *5;(2 t %&
*52(#!A $%#83A 0t +-("# & 1"568.82/ )( 0t!# , %&/ %"*9%&%"/ ! ./ 152/ %2 *5;(2 t
%& *52(#!A $%#83A 0t . E 1"568.82/ )( 0t!# ?)8+%"# &48'9:";56 %2' !#(&%29 %&
*52(#!A $%#83A 0t !"# $'3>2?-:+%"# 3+ )( 0t2 . F-("# =&?"=A
)()()( 000 tSt3t2 ##"#" .
@%&( 2#!2(23-", %2 !;$%2/ 1"5"8.8A/ ", & +-$15",& # !"# %2 !)5=2/ $ +!056-
:2(%"# $'("5%A$+# %&/ 12$;%&%"/ x %2' 1"5"8;3+(2' 152G;(%2/. 7%$#, & 1"56-
8.82/ )( 0x" # 1"5#$%6(+# %2 5'43; 3+%">2?A/ %2' !;$%2'/ " ./ 152/ %&( 12$;-
2 !"#$%&"'%( )%*"(+%( 242
!"!# x, $!#% 0xx ! &#' ()*+!#' !"#$%& %&'(!) '(! 0x . ,%-(.*#, ./01.%!#' &#' .' )%%.'+2 !"#$%* +,'-"$./ '(! 0x &#' !"#$%& %0"1!) '(! 0x .
,$#&+%*,(
1. 23$ 4&('$5! -!6 ",73+($# '+ 8,$ 5,83/ -"!%$5+, %6%5#%& %68$(#'8&.
9,$ '6'%+6* 80("/'/) 1+,73+# &(# (/ 7"!3#%* '(#:8* 0t -!6 / $%(,3$ r (!6
%68$(#'8!; +,3$# 50 cm, ! "6<8&) 8+($4!5*) (/) r +,3$# 20 cm/sec. =$ 4"+-<+, ! "6<8&) 8+($4!5*) (!6 +84$1!; , -!6 -+"#%5+,+($# $-& (! %6%5#%& %;-
8$, (/ 7"!3#%* '(#:8* 0t .
)-(.
34+'56 2r!E "! &#' " #&!0%# r +0%#' 78%-/!"7" !.8 9/$%.8 t, )9.8:+
)()( 2 tr!tE ! ,
.4$!+ )()(2)( trtr!tE #"!# .
34.:)%;2,
!!trtr!tE 200020502)()(2)( 000 !""!#"!# (cm2/sec).
2. A3 (! '63!5#%& %&'(!) -$"$:>:*) x 8!3?1>3 +3&) 4#!8/7$3#%!; -"!@&-
3(!) +,3$# )(x' %$# / '63!5#%* +,'-"$./ $-& (/3 -A5/'* (!6) +,3$# )( xE ,
(&(+ )()()( xKxExP $! +,3$# (! '63!5#%& %0"1!) %$# x
xKxK /
)()( ! +,3$# (!
80'! %&'(!).
i) =$ $-!1+,.+(+ &(# ! "6<8&) 8+($4!5*) (!6 %0"1!6) 8/1+3,B+($# &($3 !
"6<8&) 8+($4!5*) (!6 %&'(!6) %$# ! "6<8&) 8+($4!5*) (/) +,'-"$./) +,3$# ,'!#.
ii) =$ $-!1+,.+(+ &(# ! "6<8&) 8+($4!5*) (!6 80'!6 %&'(!6) 8/1+3,B+($# &($3 (! 80'! %&'(!) +,3$# ,'! 8+ (! !"#$%& %&'(!).
)-(.
i) < /8=:$2 :+!#>.(62 !.8 &)/5.82 +0%#'
)()()( xKxExP #$#!# .
34.:)%;2,
)()(0)()(0)( xKxExKxExP #!#%!#$#%!# .
ii) < /8=:$2 :+!#>.(62 !.8 :)7.8 &$7!.82 +0%#'
2 !"#$%&"'%( )%*"(+%( 243
2
)()()(
x
xKxxKxK !
!"#$# .
!"#$%&'(
0)()(0)( $!"#%$# xKxxKxK !
x
xKxK
)()( $#%
)()( xKxK !$#% .
#(',(-"(
!" #$!%!&
1. M)* +,*)-)./ $"01* 2)#&)#3 *-2456) &* 178&6). 9 *.:4&* :;(, "#7
61*::8&6:*), <4&6:*) +6 cm *"= :#& :3"# 24 tr !$ , ="#7 t # 2-=&#( +6 sec. >* ?-64:6 :# -7@$= $6:*?#1/( :;( 6"),0&6)*( " .*) :#7 =A.#7 V
:;( $"01*(, =:*& 1$t sec. (B7$;@64:6 =:) 24 r#E $ .*) 3
3
4r#V $ ).
2. C =A.#( V 6&=( +,*)-).#3 $"*1#&)#3 "#7 ,#7+.8&6) *7D0&6:*) $6 -7@$= 100cm3/sec. E6 "#)# -7@$= *7D0&6:*) ; *.:4&* :#7 r :; 2-#&)-./ +:)A$/ 0t , "#7 *7:/ 64&*) 4+; $6 9cm;
3. To .=+:#( "*-*A'A/(, )(xK , .*) ; :)$/ "81;+;(, )(x$ , x $#&0<'&
6&=( ?)#$;2*&).#3 "-#F=&:#( <4&#&:*) *"= :)( +7&*-:/+6)(
1000600203
1)( 23 &&!$ xxxxK .*) xx$ 420)( $ *&:)+:#42'(. >*
?-64:6 "=:6 # -7@$=( $6:*?#1/( :#7 .%-<#7(, )()()( xKx$xP !$ , 64-&*) @6:).=(.
4. G3# "1#4* 1$ .*) 2$ *&*2'-#3& +7A2-=&'( *"= %&* 1)$0&) %. H#
"1#4# 1$ .)&64:*) *&*:#1).0 $6 :*23:;:* 15km/h .*) :# 2$ ?=-6)*
$6 :*23:;:* 20km/h.
i) >* ?-64:6 :)( +7&*-:/+6)( @%-+6'( :'& 1$ .*) 2$
&'()*+,
-*../0
d=d(t)
% $1
$2
2 !"#$%&"'%( )%*"(+%( 244
ii) !" "#$%&'(&)& *)+ , "#*-)"-, )( 21!!d ! )./ %0$ #1$'./ "0(2-
/&)"+ 3& -)"4&5* 5043* )$/ $#$'$ 6"+ /" #5$-%+$5'-&)&.
5. 7/" 6+/,)* " (&6+/2 "#* ),/ "589 )./ "(*/./ 6"+ 6+/&')"+ 6")2 39-
6$: ),: 6"3#;1,: 2
4
1xy ! , 0"x . <& #$+$ -,3&'$ ),: 6"3#;1,: $
5043*: 3&)"=$19: ),: )&)3,3>/,: x )$0 " &'/"+ '-$: 3& )$ 5043* 3&-)"=$19: ),: )&)"?3>/,: )$0 y, "/ 0#$)&4&' *)+ 0)( #$ tx ?+" 624&
0"t .
!" #$%&%'
1. @/ , &#+A2/&+" 3+": -A"'5": "0(2/&)"+ 3& 5043* 10cm2/sec, /" =5&'-)& )$ 5043* 3& )$/ $#$'$ "0(2/&)"+ $ *?6$: "0)9: *)"/ 85!r cm.
2. 7-). # )$ &3="%*/ )$0 )5+?B/$0 $%& #$0 $5'C$0/ )" -,3&'" )0,0(O ,
)0,(xA 6"+ )ln,0( xB , 3& 1#x . @/ )$ x 3&)"=211&)"+ 3& 5043*
4cm/sec, /" =5&')& )$ 5043* 3&)"=$19: )$0 &3="%$; #, *)"/ 5!x cm.
3. 7/": 2/45.#$: -#5B8/&+ >/"
6$0)' -), 523#" )$0 %+#1"/$;
-893")$: 6"+ )$ 6$0)' 6+/&')"+ 3& )"8;),)" 3m/s. !" =5&')& #*-$ ?59?$5" "/0DB/&)"+ )$
6$0)', %,1"%9 )$ 5043* 3&)"-
=$19: )$0 y.
4. 7/" "&5*-)")$ % "A9/&+ )$ >-%"A$: -& "#*-)"-, 100m "#*
>/"/ #"5"),5,)9 ! 3& )"8;),)"
50m/min. E& #$+$ 5043* "0(2-
/&)"+ , ?./'" ' #$0 -8,3")'C&+ , %! 3& )$ >%"A$: ), 85$/+69
-)+?39 6")2 ),/ $#$'" )$ 3#"1-
1*/+ =5'-6&)"+ -& ;D$: 100m.
5. M'" ?0/"'6" ;D$0: 1,60m "#$3"65;/&)"+ "#* ), =2-, &/*: A"/$-
-)2), ;D$0: 8m 3& )"8;-
),)" 0,8m/s. E& #$+" )"-
8;),)" "0(2/&)"+ $ '--6+$: ),:;
5m y s
100m
h
' !
A
(
x
) $ !
s x
1,6
*
8
2 !"#$%&"'%( )%*"(+%( 245
6. !"# $%&'$()'*+ ! *'"%,-#' *#-. /0*(1
-21 *#/$3)21 3
3
1xy !" , 0#x $)2-
4'.5("-#1 -2" #*-0 *#' ( $&(6()7#1 -(8 9:-,5%' *#-%8;%,#" %/$&+1 (<=0-
/#). >" ( &8;/+1 /%-#6()01 -21 -%--/2/7"21 -(8 $%&'$()'*(3 ?,"%-#' #$+
-(" -3$(
)()( t"t" !"$
"# 6&%,-% -( &8;/+ /%-#6()01 -21 -%--/2/7"21 -(8 42/%,(8 # -21 #*-01 4-( ($(,( $79-(8" -# 9@-# -(8 $&(-
6()7# -2 =&("'*0 4-'A/0 *#-. -2" ($(,# -( $%&'$()'*+ 7=%' -%-/2/7-"2 3! .
7. B,# 4*.)# /0*(81 3m %,"#' -($(-
;%-2/7"2 4’ 7"#" -(,=(. C( *.-:
/7&(1 -21 4*.)#1 A)84-&.%' 4-(
?.$%?( /% &8;/+ 0,1m/sec. C2
=&("'*0 4-'A/0 0t , $(8 2 *(&890
-21 4*.)#1 #$7=7' #$+ -( ?.$%?(
2,5m, "# 6&%,-%:
i) C( &8;/+ /%-#6()01 -21 A:",-#1 $ (<=0/#).
ii) C2" -#=3-2-# /% -2" ($(,# $79-%' 2 *(&890 ! -21 4*.)#1.
8. !"# *'"2-+ *'"%,-#' 4% *8*)'*0 -&(='. /% %D,4:42 122 "% yx . E#;@1
$%&".%' #$+ -( 42/%,( &&'
())*
+
2
3,
2
1A , 2 -%-#A/7"2 y %)#--@"%-#' /% &8;-
/+ 3 /(".?%1 -( ?%8-%&+)%$-(. F# 6&%,-% -( &8;/+ /%-#6()01 -21 -%--/2/7"21 x -2 =&("'*0 4-'A/0 $(8 -( *'"2-+ $%&".%' #$+ -( !.
2.5 TO ,-.&/+# +-(/( 0"+/(
<-2" $#&.A	( #8-0 ;# A":&,4(8/% 7"# #$+ -# $)7(" 6#4'*. ;%:&0/#-# -(8
G'#9(&'*(3 H(A'4/(3 $(8 %,"#' A":4-+ :1 !"#$%&' ()*%+ ,-&.+. >&='*.
?'#-8$@"(8/% -( I%@&2/# -(8 Rolle, -( ($(,( %,"#' %'?'*0 $%&,$-:42 -(8 I%:-
&0/#-(1 B7421 C'/01 *#' 4-2 48"7=%'# ?'#-8$@"(8/% -( I%@&2/# B7421 C'-/01, -( ($(,( #$(?%'*"3%-#' /% -2 6(0;%'# -(8 I%:&0/#-(1 -(8 Rolle.
!/012(3 (Rolle)
y x" !
1
33
A a
a( , )!
3
3
x
y
!%&' B M (
y
$ x
3m
A
) (
2 !"#$%&"'%( )%*"(+%( 246
!" #$% &'"()*+&+ f ,-"%$:
! &'",./0 &*1 23,$&*4 5$(&*+#% ],[ !"
! 6%)%787-&$#+ &*1 %"1$2*4 5$(&*+#% ),( !" 2%$
! )()( !f"f "
*4*, '6().,$ 9"%, *1'3(.$&*1", ),( !"# # *9*1$1, :&*,:
0)( "$ #f
;,8#,*)$2(, %'*4 &+#%-",$ 4*$ '6().,$ 9"%,
*1'3(.$&*1", ),( !"# # *9*1$1, :&*, + ,<%-
6*1#9"+ *+0 fC &*1 ))(,( #f#M "% ,-"%$ 6%)(33+3+ &*1" (=1"% *8" x.
;$% 6%)(5,$7#%, 9&*8 + &'"()*+&+
54)( 2 %&" xxxf , ]3,1[#x . (>.. 19)
?6,$5/ + f ,-"%$ &'",./0 &*1 ]3,1[ , 6%)%78-
7-&$#+ &*1 )3,1( , #, 42)( &"$ xxf 2%$ )3(2)1( ff "" , &@#<8"% #, *1 A,:)+#%
Rolle, A% '6().,$ 9"%0 %)$A#40 )3,1(##
*9*1$10, :&*, 0)( "$ #f .
;$% *+" ,@),&+ *1' %)$A#1@ #, 9.1'#,:
20420)( "'"&'"$ ###f .
!"#$%&' (&()*+ ,-./+ 0-1234-536 738-).36 !.&.,.)
!" #$% &'"()*+&+ f ,-"%$:
! &'",./0 &*1 23,$&*4 5$(&*+#% ],[ !" 2%$ ! 6%)%787-&$#+ &*1 %"1$2*4 5$(&*+#% ),( !"
*4*, '6().,$ 9"%, *1'3(.$&*1", ),( !"# # *9*1$1, :&*,:
"!"f!f#f
&&
"$ )()()(
y
O x ! #$ # "
%(#,f (#))
&(!,f (!)) '(",f ("))
18
M
B A 2
3 #=2 1 O x
y19
2 !"#$%&"'%( )%*"(+%( 247
!"#$"%&'(), *+%, -.$*/0"' ,%' +1)&2"' 30*,
%4+5)2'-%40, ),( !"# ! %3%4'4, 6-%" . "7*-
1%4$30. %.8 9&*7'(:8 1*&)-%*-.8 %.8 f
-%4 -.$"/4 ))(,( #f#M 0* "/0*' 1*&)55.5.
%.8 "+;"/*8 $%.
!'* 1*&)<"'9$*, 3-%# . -+0)&%.-.
xxf ")( , ]4,0[!x .
=1"'<: . f "/0*' -+0"2:8 -%4 ]4,0[ (*' 1*-
&*9#9/-'$. -%4 )4,0( , $" x
xf2
1)( "# , ->$-
7#0* $" %4 ;"6&.$* $3-.8 %'$:8, ;* +1)&2"' 30*8 *&';$,8 )4,0(!# %3%4'48, 6-%"
2
1
04
)0()4()( "
$$
"# ff#f .
!'* %.0 ">&"-. %4+ *&';$4> #, 324+$":
112
1
2
1
2
1)( "%"%"%"# ##
##f .
,$#&+%*,(
1. N! !"#$%&'(%) *(&:
i) + ,-./0(1,1 x-x-xxf 1)()( 23 &&&&$$$$&&&&"""" , *!!- , &2!.#"#&%) (&3 -"#45,%&3 (#- 4%6078!(#3 (#- Rolle ,(# $&/,(18! 1][0, .
ii) + %9),6,1 01)(23 2 """"&&&&$$$$&&&& -x-x , *!!- 5'%& 8&!, (#-:/'&,(#., 0);! ,(#
$&/,(18! 1)(0, .
#.%!,"/0
i) ? -+0)&%.-. f '(*0414'"/ %'8 +14;3-"'8 %4+ ;"#&:$*%48 Rolle -%4 ]1,0[
*74>
' "/0*' -+0"2:8 -%4 ]1,0[ #8 145+#0+$'(:
' "/0*' 1*&*9#9/-'$. -%4 )1,0( $" )1(23)( 2 &$&"# &xx&xf (*'
' '-2>"' 0)1()0( "" ff .
%(!,f (!))
! #' # a x
y
(
M(#,f (#))
A(a,f (a))
20
y x"
x
y
((0,0) 4 1
$(4,2) )(1,1)
21
2 !"#$%&"'%( )%*"(+%( 248
ii) !"#$, %#&'(), *&+ ,- ./)01,-.- x!xx!xf )1()( 23 !"!# , *!$! &.2$#/) #& /'#34.5&6 ,#/ 357189+,#6 Rolle, 3+ /'0125& )1,0($" ,4,#&#, :.,5 0)( #% "f 8,
&.#;$)+9+, 0)1(23 2 #!"! !"!" . <'#94)76, ,# )1,0($" 3+ 5=)+& 1=>+ ,-6 5?=-
.7.-6 0)1(23 2 #!"! !xx! .
2. !" "#$%&'()&* +)' ,'" )- ./012)-.- ,-x.xxf !!!!!!!!#### 2)( , 0&. 3"' ,'" $-
#$'$%4#$)& %'1.)-5" ],[ 21 xx , $ "2'65+7 ),( 210 xxx $$$$ , #$/ '3"0$#$'&* )$ ./-
5#82".5" )$/ 9&:245")$7 ;8.-7 <'547, &*0"' )$ 380)2$ )$/ %'".)45")$7
],[ 21 xx , %-="%4 &*0"' 2
21
0
xxx
!# .
#/%!0"12
@ ./)01,-.- #x$x%xf !!# 2)( 5=)+& ./)5286 .,# ],[ 21 xx 76 '#%/7)/9&A8 A+& '+1+*7*=.&9- .,# ),( 21 xx , 95 $x%xf !#% 2)( . <'#94)76, .$9"7)+ 95 ,# 35:-
1-9+ 94.-6 ,&986 /'0125& ),( 210 xxx $ , ,4,#&#, :.,5
12
12
0
)()()(
xx
xfxfxf
""
#% . (1)
<=)+& (976:
12
121212
12
1
2
12
2
2
12
12 )())(()()(
xx
xx$xxxx%xx
#x$x%#x$x%xx
xfxf
""!!"
#"
"""!!#
""
$xx%xx
$xx%xx!!#
"!!"
# )(])()[(
21
12
2112 .
<'#94)76, - .24.- (1) *10"5,+&:
2)(2 21
0210
xxx$xx%$x%
!#'!!#! .
3. >0" "/)$3*0-)$ %'40/.& 5*" %'"%2$54 200 ('='$58)2:0 .& 2,5 ?2&7. !"
"#$%&'(6&* +)' 31#$'" (2$0'34 .)',54, 3")1 )- %'123&'" )-7 %'"%2$547, -
)"(@)-)" )$/ "/)$3'04)$/ 4)"0 80 ('='+5&)2" )-0 ?2".
#/%!0"12
B.,7 )(tSx # , ]5,2,0[$t - ./)01,-.- 34.-6 ,#/ A&)-,#$. !1A5= )+ ;5=?#/95 (,& /'0125& ]5,2,0[0 $t , ,4,#&+ :.,5 80)()( 00 #%# tSt& .
@ ./)01,-.- S 5=)+& ./)5286 .,# ]5,2,0[ A+& '+1+*7*=.&9- .,# )5,2,0( . <'#-
94)76, .$9"7)+ 95 ,# C5:1-9+ D4.-6 E&986 /'0125& )5,2,0(0 $t ,4,#&#, :.,5
805,2
0200
5,2
)0()5,2()()( 00 #
"#
"#%#
SStSt& 2&%&(95,1+ ,-) :1+.
2 !"#$%&"'%( )%*"(+%( 249
#(',(-"(
!" #$!%!&
1. N! "#"$%&"$" '()"* !'+ $)* '!,!-%$. &/0!,$1&")* )-!0('()(20 $)* /'(34&")* $(/ 3".,15!$(* Rolle &$( 6)%&$75! '(/ !0!84,"$!), -!) &$7 &/049")!, :)! "-";0"* '(/ )&92"), 0! <,";$" +=! $! ),( !"# ! :)!
$! ('(;! )&92") 0)( "# #f .
i) 12)( 2 $%" xxxf , ]2,0[ ii) xxf 375)( " , &'(
)*+
3
2,0$
iii) xxf 2&/01)( $" , ],0[ $ iv) ||)( xxf " , ]1,1[% .
2. >! "#"$%&"$", '()4* !'+ $)* '!,!-%$. &/0!,$1&")* )-!0('()(20 $)* /'(34&")* $(/ ?".,15!$(* @4&7* A)51* &$( 6)%&$75! '(/ !0!84,"-$!) -!) &$7 &/049")!, :)! "-";0"* '(/ )&92") $( 3"B,75!, 0! <,";$"
+=! $! ),( !"# ! :)! $! ('(;! )&92") "!"f!f#f
%%
"# )()()( .
i) xxxf 2)( 2 $" , ]4,0[ ii) xxf 2753)( " , &'(
)*+
2,0$
iii) ,-
,./
%0%
%1$"
1,
1,22)(
3 xxx
xxxf , ]2,3[%
3. C0 !" 2 , 0! !'(6";#"$" +$) () &/0!,$1&")* xexf ")( -!) xxg ln)( "
)-!0('()(20 $)* /'(34&")* $(/ [email protected]. &$( 6)%&$75! ],[ !" -!) &$7
&/049")! +$):
!"!
" e"!ee
e 2%%
2 -!) ""!
"!!
1lnln12
%%
2 .
D)! $7 &/0%,$7&7 xxg ln)( " /'(34$(/5" "')'=4(0 +$) !" 220 .
'" #$!%!&
1. E;0"$!) 7 &/0%,$7&7 12520)( 234 $%%%" xxxxxf
i) >! !'(6";#"$" +$) 7 "#;&.&7 0)( "xf 49") 5)!, $(/=%9)&$(0, ,;F!
&$( 6)%&$75! )0,1(% -!) 5)!, $(/=%9)&$(0, &$( 6)%&$75! )1,0( .
ii) >! !'(6";#"$" +$) 7 "#;&.&7 0150604 23 "%%% xxx 49") 5)!, $(/-
=%9)&$(0, ,;F! &$( 6)%&$75! )1,1(% .
2 !"#$%&"'%( )%*"(+%( 250
2. !"#$%&' ( )*#+,%()( xxxf (-)1()( !" . .& &/01$"2$%$ 3%': i) 4 $2")5)( 0)( "# xf 67$' -'&, %0*8+7')%0#, ,"9& )%0 �':%3 1'+-
)%(-& )1,0( .
ii) 4 $2")5)( xx !" 1$; 67$' -'&, %0*8+7')%0#, ,"9& )%0 �':%3 1'+-
)%(-& )1,0(
3. i) !"#$%&' -'& )*#+,%()( f -$ 1)( $# xf <'& :+=$ !%x . .& &/01$"-2$%$ 3%' ( $2")5)( xxf ")( 67$' %0 /08> -'& /,&<-&%':? ,"9&.
ii) .& &/01$"2$%$ 3%' ( $2")5)( xx"
2(- &8(=$>$' -3#0 <'& 0"x .
4. i) .& &/01$"2$%$ 3%' 2
1
1 2&
' x
x, <'& :+=$ %x (.
ii) A# f $"#&' -"& )*#+,%()( /&,&<5<")'-( )%0 (, -$ 21
)(x
xxf
'"# ,
#& &/01$"2$%$ 3%' <'& 38& %& !%!", ')7>$':
||2
1|)()(| "!"f!f !&! .
5. @)%5 -'& )*#+,%()( f ( 0/0"& $"#&' )*#$7?A )%0 ]4,0[ :&' ')7>$' 5)(2 &#& xf <'& :+=$ )4,0(%x . B# 1)0( "f , #& &/01$"2$%$ 3%' 21)4(9 && f .
6. @)%5 -'& )*#+,%()( f ( 0/0"& $"#&' )*#$7?A )%0 ]1,1[! :&' ')7>$' 1)( &# xf <'& :+=$ )1,1(!%x . B# 1)1( !"!f :&' 1)1( "f , #& &/01$"2$-
%$ 3%' 0)0( "f , $;&,-390#%&A %0 C.D.E. <'& %(# f )$ :&=6#& &/3 %&
1'&)%?-&%& ]0,1[! :&' ]1,0[ .
7. .& &/01$"2$%$ -$ %0 =$F,(-& %0* Rolle 3%' 0' <,&;':6A /&,&)%+)$'A %5# )*#&,%?)$5#
xxf 2)( " :&' 12)( 2 ''!" xxxg
670*# &:,'GFA 1*0 :0'#+ )(-$"& %& )1,0(A , )2,1(B .
2.6 (,-./.".( 0%, 1.2&3+#0%(
0H( +.(3( 0"+3(
E0 C$F,(-& D6)(A E'-?A %0* 1'&;0,':0> 80<')-0> =$5,$"%&' -"& &/3 %'A )/0*1&'3%$,$A /,0%+)$'A %(A &#+8*)(A, &;0> -$ %( G0?=$'+ %0* &/01$':#>0#%&' /088+ +88& =$5,?-&%&. C& 7,()'-0/0'?)0*-$ %F,& %0 C.D.E. <'& #& &/01$"-20*-$ %& $/3-$#& 1>0 G&)':+ =$5,?-&%&.
2 !"#$%&"'%( )%*"(+%( 251
!"#$%&'
!"#$ %&' "()*+#,", f -+&"%.), "/ .)' 0&*"#,%' !. 1)
! , f /2)'& "()/345 "#- ! 6'& ! 0)( "# xf 7&' 6*8/ " # $ % " & ' ( ) ",%/2- x #-( !,
#9#/ , f /2)'& "#'8/+4 "/ 9:- #- 0&*"#,%' !.
#,%!-"./
1+6/2 )' ';-0/2<-(%/ 9#& 7&' -;-&'04;-#/ !xx $21 , &"3=/& )()( 21 xfxf " .
>+*7%'#&
! 1) 21 xx " , #9#/ ;+-?')@5 )()( 21 xfxf " .
! 1) 21 xx % , #9#/ "#- 0&*"#,%' ],[ 21 xx , f &6')-;-&/2 #&5 (;-8."/&5 #-( 8/$-
+4%'#-5 %.",5 #&%45. A;-%.)$5, (;*+3/& ),( 21 xx* $ #.#-&-, @"#/
12
12 )()()(
xx
xfxf*f
&&
"# . (1)
A;/&04 #- * /2)'& /"$#/+&69 ",%/2- #-( !, &"3=/& 0)( "# *f ,-;9#/, :97$ #,5 (1),
/2)'& )()( 21 xfxf " . 1) 12 xx % , #9#/ -%-2$5 ';-0/&6)=/#'& 9#& )()( 21 xfxf " .
B/ 9:/5, :-&;9), #&5 ;/+&;#@"/&5 /2)'& )()( 21 xfxf " . !
()$*+&'
!"#$ 0(- "()'+#4"/&5 gf , -+&"%.)/5 "/ .)' 0&*"#,%' !. 1)
! -& gf , /2)'& "()/3/25 "#- ! 6'&
! )()( xgxf #"# 7&' 6*8/ " # $ % " & ' ( ) ",%/2- x #-( !,
#9#/ (;*+3/& "#'8/+* c #.#-&', @"#/ 7&' 6*8/ !x$ )' &"3=/&:
cxgxf '" )()(
#,%!-"./
C "()*+#,", gf & /2)'& "()/345 "#- ! 6'& 7&' 6*8/ /"$#/+&69 ",%/2- !x$ &"3=/&
0)()()()( "#&#"#& xgxfxgf .
A;-%.)$5, "=%?$)' %/ #- ;'+';*)$ 8/@+,-
%', , "()*+#,", gf & /2)'& "#'8/+4 "#- !.
D+', (;*+3/& "#'8/+* C #.#-&', @"#/ 7&' 6*-
8/ !x$ )' &"3=/& cxgxf "& )()( , -;9#/ cxgxf '" )()( . !
y
O x
y=g(x)+c
y=g(x)
22
2 !"#$%&"'%( )%*"(+%( 252
!"#$%#
!" #$%$#&'( )*+%,-$ .$)+/ .$0 1" #2%03-& 1"4 0356"4' 3* 70&31,-$ .$0 250 3* 8'(3, 70$31,-&1('. 90$ #$%&7*0:-$, 831( , 34'&%1,3,
!"
!#$
%
&'(
0,1
0,1)(
x
xxf .
;$%$1,%"6-* 210, $' .$0 0)( () xf :0$ .&)* ),0()0,( *+,'+-x , *'1"61"0/ , f
7*' *<'$0 31$)*%= 31" ),0()0,( *+,'+ .
,$#&+%*H
&'()*+, -'+ ./(01*2.2 f 3,+ *2( 454'+ ,.67),
)()( xfxf () 3,+ 809) -x .... (1)
i) :+ +54;),6*)' <*, 2 ./(01*2.2 xe
xfx- )(
)( ( )'(+, .*+9)1= 8+,
ii) :+ >1)9)' 4 *754? *2? f, +( ;'()*+, )5,5@A4( <*, 1(0) (f .
).(/
i) 90$ .&)* -x . 85"4-*:
0)()(
)(
)()()()(
)1(
2(
')(
')(
)/0
123
4()xx
xx
x e
xfxf
e
exfexf
e
xfx! ,
>#"-8'(/, , ! *<'$0 31$)*%= 31" ..
ii) >#*07= , ! *<'$0 31$)*%=, 4#&%5*0 -c . 181"0", +31* cx! ()( :0$ .&)* -x .
=, 03"76'$-$, ce
xfx
()(
:0$ .&)* -x .. >#"-8'(/
xcexf ()( :0$ .&)* -x ..
>#*07= 1)0( (f , 85"4-* c(1 , "#21*
xexf ()( :0$ .&)* -x ..
2 !"#$%&"'%( )%*"(+%( 253
+,-,.,-/0 12-34.5156
!"#$ % "&'()#%"% 2)( xxf ! . *+)+#%),-./ 0#1 "#,
21("#%.+ )0,("# , "#, ,3,4, % f /4'+1 5'%"4$6 784-',&"+, 1"9-/1 02)( $!% xxf , /': "#, 21("#%.+
),0( &# , "#, ,3,4, % f /4'+1 5'%"4$6 +-;,&"+, 1-"9-/1 02)( '!% xxf . <=>3,&./, 2%=+2?, 0#1 &3()9/1 .1+ "9>"% +'(./"+ "#% .,',#,'4+ @+1 "#, 3)0"%.,
#%6 3+)+5:5,& #%6 "&'()#%"%6. A&5@/@)1.>'+ 1-"9-/1:
!"#$%&'
!"#$ .1+ "&'()#%"% f, % ,3,4+ /4'+1 ! " # $ % & ' "/ >'+ 21("#%.+ (.
( B' 0)( '% xf "/ @(8/ $ ! ) * $ + , - . "%./4, x #,& (, #0#/ % f /4'+1 5'%-
"4$6 +-;,&"+ "/ 0=, #, (.
( B' 0)( $% xf "/ @(8/ $ ! ) * $ + , - . "%./4, x #,& (, #0#/ % f /4'+1 5'%-
"4$6 784',&"+ "/ 0=, #, (.
#7%!8"9:
(((( B3,2/1@'-,&./ #, 8/:)%.+ "#%' 3/)43#$"% 3,& /4'+1 0)( '% xf .
!"#$ (xx )21 , ./ 21 xx $ . C+ 2/4;,&./ 0#1 )()( 21 xfxf $ . *)(5.+#1, "#, 21(-
"#%.+ ],[ 21 xx % f 1@+',3,1/4 #16 3),D3,8>"/16 #,& C.E.F. G3,.>'$6, &3()9/1
),( 21 xx/ ) #>#,1,, :"#/ 12
12 )()()(
xx
xfxf/f
""
!% , ,30#/ >9,&./
))(()()( 1212 xx/fxfxf "%!"
G3/12? 0)( '% /f @+1 012 '" xx , >9,&./ 0)()( 12 '" xfxf , ,30#/ )()( 21 xfxf $ .
( A#%' 3/)43#$"% 3,& /4'+1 0)( $% xf /)5+H0.+"#/ +'+=05$6. !
I1+ 3+)(2/15.+:
— % "&'()#%"% xxf !)( , /4'+1 5'%"4$6 +--
;,&"+ "#, ),0[ #& , +7,- /4'+1 "&'/9?6 "#,
),0[ #& @+1 1"9-/1 02
1)( '!%
xxf 51+ @(8/
),0( #&)x .
y
O x
y=x2
f 0(x)>0 f 0(x)<0
23
y x!
x
y
1
24
2 !"#$%&"'%( )%*"(+%( 254
— ! "#$%&'!"! xxxf 2)( 2 !" ()$*+ ,$!")-. */01#"* "'1 ),1[ #$ , *21/ ()$*+ "#$(34. "'1
),1[ #$ 5*+ 0)1(2)( %!"& xxf ,+* 5%6( ),1( #$'x , ($7 ()$*+ ,$!")-. 26)$1#"* "'1
]1,(!# , *21/ ()$*+ "#$(34. "'1 ]1,(!# 5*+ 0)1(2)( (!"& xxf ,+* 5%6( )1,(!#'x .
— ! "#$%&'!"! x
xf1
)( " ()$*+ ,$!")-. 26)$1#-
"* "( 5*68$* *9: '* ;+*"'4<*'* )0,(!# , 5*+
),0( #$ , *21/ 01
)(2(!"&
xxf ,+* 5%6(
)0,(!#'x 5*+ ,+* 5%6( ),0( #$'x .
!"#$%#
=1 *$')"'&121 '1# 9*&*9%$- 6(-&4<*'1. &'( )-*+,'). >!?*;4, *$ ! f ()$*+ ,$!")-. */01#"* (*$'+-"'1)3-. ,$!")-. 26)$1#"*) "'1 !, ! 9*&%,-,:. '!. &'( '-(.) /01+2'34)56 6('+54 (*$'+"'1)3-. *&$!-
'+54) "'1 ("-'(&+5: '1# !.
@+* 9*&%;(+,<*, ! "#$%&'!"! 3)( xxf " , *$ 5*+ ()-$*+ ,$!")-. */01#"* "'1 ), ($'1/'1+. 83(+ 9*&%-
,-,1 23)( xxf " ! 191)* ;($ ()$*+ 6('+54 "( :?1 '1
), *21/ 0)0( "&f . A"3/(+ :<-. 0)( *& xf ,+* 5%6( 'x ).
,$#&+%*,(
1. N. 72'81,( 4. &).*49:.4. *4. 101-. ; */(624;*; 132)( 23 $!" xxxf
'-(.) <(;*-3= .,>1/*., <(;*-3= ?8-(1/*..
)-(.
B "#$%&'!"! f ()$*+ 9*&*,-,)"+<! <( )1(666)( 2 !"!"& xxxxxf . =1 9&:"!<1
'!. f & ;)$('*+ "'1$ 9*&*5%'- 9)$*5*
x !# 0 1 $#
)(xf & + 0 ! 0 +
x 2
1
y
"
25
y=x2!2x
yx
"1
x
y
"
26
x
y=x3
y
"
27
2 !"#$%&"'%( )%*"(+%( 255
E!"#$%&', ( )*%+,-()( f:
— ./%01 2%()/&' 034"*)0 )-" ]0,(!" , 05"3 ./%01 )*%.67' )-" ]0,(!" 801 1)63.1 0)( #$ xf )-" )0,(!" .
— ./%01 2%()/&' 59/%"*)0 )-" ]1,0[ , 05"3 ./%01 )*%.67' )-" ]1,0[ 801 1)63.1 0)( %$ xf )-" )1,0( .
— ./%01 2%()/&' 034"*)0 )-" ),1[ "& , 05"3 ./%01 )*%.67' )-" ),1[ "& 801 1)63.1 0)( #$ xf )-" ),1( "& .
:" !,;)(#" -(' f $ 801 -" ./<"' #"%"-"%/0' -(' f )-0 <10)-7#0-0 ]0,(!" ,
]1,0[ 801 ),1[ "& )*28.%-,=%"%-01 )*%"!-18+ )-"% !0,08+-& !/%080:
x !" 0 1 +"f >(x) + 0 ! 0 +
f (x)
f(1)
f(0)
2. i) !" "#$%&'()&* +)' , -./01),-, 2-./)( !!' xxxf , ][0,,x( &*/"' 2/,-
-*34 "56$.-" 7"' /" 81&*)& )$ -5/$9$ )':;/ ),4.
ii) !" "#$%&'()&* +)' , &6*-3-, 2-./ !' xx <(&' "71'8;4 :'" 95-, -)$
][0,, .
#-%!."/0
i) ?/%01 0(#1)2)*%()( #&'$!!'$ xxxxf , 210 8+9. ],0[ ! .
?!"#$%&', ( f ./%01 2%()/&' 034"*)0 )-" ],0[ ! . ?!.1<7 ( f ./%01 )*%.67' 801 2%()/&' 034"*)0, )3#5&%0 #. -(% !0,+2,05" 1.8, -" )3%"@" -1#=% -(' ./%01 -"
<1+)-(#0 ]1,3[)](),0([ !!' !!ff .
ii) A6"*#.:
02)*%2)*% '!!)!' xxxx ,
0)( ') xf , ],0[ !x( .
?!.1<7 -" )3%"@" -1#=% -(' f ./%01 -" <1+-
)-(#0 ]1,3[ !! ! , !"* !.,1$6.1 -" 0, 90 *-
!+,6.1 $%0 -"*@+61)-"% ),0(0 !x ( , -$-"1"
=)-. 0)( 0 'xf . ?!.1<7 .!1!@$"% ( f ./%01 2%()/&' 034"*)0 )-" ],0[ ! , ( 0x ./%01 #"-
%0<187 ,/B0 -(' 0)( 'xf )-" <1+)-(#0 0*-
y
O
!2
!
y=x!2 y=)*%x 1
2 !/2 x0
x
28
2 !"#$%&"'%( )%*"(+%( 256
!". # $%&' '(!), "*+, -'%./!'0 1'0 2!3 24)5' 28, /%.'0 6 !/!5657.6 !3( 265/%-3( !35), !6, 2!" xy 1'0 !6, xy 2(." .
#(',(-"(
!" #$!%!&
1. 8. 90' !0, 2(.'$!)2/0, gf , 024:3(.:
)()( xgxf "# 1'0 )()( xfxg !"# 90' 1;</ $x %,
.' '*3=/%>/!/ "!0 6 2(.;$!626 22 )]([)]([)( xgxfx! &" /%.'0 2!'</$).
2. ?' @$/%!/ !' =0'2!)5'!' 53.3!3.%', !+. 2(.'$!)2/+.:
i) 43)( 3 !&" xxxf ii) xxxxf 1232)( 23 !!"
iii) 1
)(2 &
"x
xxf
3. O53%+, !+. 2(.'$!)2/+.:
i) '(
')*
+&
,!"
1,2
1,4)(
2
xx
xxxf ii) |1|)( 2 !" xxf
4. O53%+, !+. 2(.'$!)2/+.:
i) xe
xxf ")( ii) xxxf !" ln)( iii) |65|65)( xxxf &" , ]2,0[ "x$ .
5. A%.3.!'0 30 2(.'$!)2/0, 65)( 5 !&" xxxf 1'0 32)( !&" xxxg .
i) ?' '*3=/%>/!/ "!0 o0 gf , /%.'0 9.62%+, ':>3(2/,. ii) ?' @$/%!/ !3 2:.3B3 !05C. !3(,.
iii) ?' '*3=/%>/!/ "!0 30 />02C2/0,: 0655 "!& xx 1'0 032 "!& xx
743(. '1$0@C, 5%' $%&' !6. 1"x .
6. ?' '*3=/%>/!/ "!0: i) H 2(.;$!626 )1ln(1)( &&!" xexf x /%.'0 9.62%+, ':>3(2'.
ii) # />%2+26 )1ln(1 &!" xe x 74/0 '1$0@C, 5%' B:26 !6. 0"x .
'" #$!%!&
1. 8. 90' 5%' 2(.;$!626 f *3( /%.'0 3$0257.6 2’ "B3 !3 % 024:/0
2 !"#$%&"'%( )%*"(+%( 257
2)(|)()(| yxyfxf !"! !"# $%# &# #yx, $,
'# #()*+,-+&+ $&" . f +,'#" /�+12.
2. i) 3# #()*+,-+&+ $&" . /4'51&./. !xxxf %!& 3)( 3 +,'#" !'./,67 80,')4/# /&) *"5/&.9# ]1,1[! .
ii) 3# :1+,&+ &) /;')%) &"9<' &.7 f /&) *"5/&.9# ]1,1[! .
iii) =' 22 ''! ! , '# #()*+,-+&+ $&" . +-,/6/. 033 &%! !xx >-?+" #@1":<7 9,# %;/. /&) *"5/&.9# )1,1(! .
3. A 0>/. +'$7 @"'.&); (5'6 /+ >'#' 5-)'# &. ?1)'"@2 /&"!92 t *,'+&#" #($ &. /4'51&./.:
16016188)( 234 %!%!&& tttttSx , 50 "" t .
3# :1+,&+ &.' &#?;&.&# @#" &.' +("&5?4'/. &)4 @"'.&); @#" /&. /4'>-?+"# '# #(#'&2/+&+ /&# #@$%)40# +16&29#&#:
i) B$&+ &) @"'.&$ >?+" &#?;&.&# 9.*>';
ii) B$&+ &) @"'.&$ @"'+,&#" (1)7 &# *+-"5 @#" ($&+ (1)7 &# #1"/&+15;
iii) B$&+ . @,'./. &)4 @"'.&); +,'#" +("&#?4'$9+'. @#" ($&+ +(":1#-
*4'$9+'.;
4. A &"92 V (/+ ?"%"5*+7 *1#?9>7) +'$7 (1)C$'&)7, t 92'+7 9+&5 &.' (#-
1#!6!2 &)4, *,'+&#" #($ &)' &;()
2
2
)2(
2550
%!&
t
tV .
3# #()*+,-+&+ $&" &) (1)C$' /4'+?<7 4()&"95&#" ?61,7, $967, . &"92
&)4 '# 9()1+, '# !,'+" 9"@1$&+1. #($ &) 9"/$ &.7 #1?"@27 &"927 &)4.
5. 3# #()*+,-+&+ $&":
i) A /4'51&./. 1
9)(
2
3
!!
&x
xxxf +,'#" !'./,67 #;-)4/# /+ @#0>'# #($
&# *"#/&29#&# &)4 (+*,)4 )1"/9); &.7 @#" '# :1+,&+ &) /;')%) &6' &"9<' &.7 f /+ @#0>'# #($ &# *"#/&29#&# #4&5.
ii) H +-,/6/. 0923 &%!! !xx!x +,'#" "/)*;'#9. 9+ &.' !xf &)(
@#" /&. /4'>?+"# $&" >?+" &1+"7 (1#!9#&"@>7 1,D+7 !"# @50+ #! $.
6. 3# :1+,&+ &"7 &"9>7 &)4 *!#! !"# &"7 )(),+7 . /4'51&./.
13)( 23 %%%& xxx!xf +,'#" !'./,67 #;-)4/# /&) $.
2 !"#$%&"'%( )%*"(+%( 258
7. !" "#$%&'(&)& $)*:
i) H +,-./)0+0 xxxxf +,-01)( !" &'-"* 2-0+'34 "5($,+" +)$ 67&*-
+)8 %*.+)01" #$%
&'(
2,0!
.
ii) 0+,-01 )! xxx , 2*" 6.9& *+,
-./0
2,0!
x .
iii) H +,-./)0+0 x
xxf
01)( " &'-"* 2-0+'34 :9'-$,+" +)$ "-$*6)8
%*.+)01" *+,
-./
2,0!
.
8. !" "#$%&'(&)& 8)*:
i) H +,-./)0+0 xxxxf 3&:012)( !1" , *+,
&'(0
2,0!
x &'-"* 2-0+'34 "5-
($,+".
ii) xxx 3&:012 21 , 2*" 6.9& *+,
&'(0
2,0!
x .
2.7 TO,"'# #'&%-#-# (./#&-0(0(
0 122345 637 638493: 59;36<637
;)$ %*#7"-8 +<=1" ><$,1& )0 2/"-:*6= #"/.+)"+0 1*"4 +,-./)0+04 f +’ >-" %*.+)01" ],( "# . ?"/")0-
/$51& 8)* +)$ +01&'$ 0xx " 0 )*1=
)04 +,-./)0+04 &'-"* 1&2"75)&/0 "#8 )0- )*1= )04 +& 6.9& “2&*)$-*-68” +01&'$ )$, 0x . ;)0- #&/'#)3+0
",)= 7>1& 8)* 0 f #"/$,+*.@&* +)$
0x )$#*68 1>2*+)$. A$ '%*$ +,1B"'-&* 6"* +)" +01&'" 1x 6"* 2x . C&-*6. ><$,1& )$- "687$,9$ $/*+18:
!"#$%!$
D*" +,-./)0+0 f, 1& #&%'$ $/*+1$5 $, 9" 7>1& 8)* #"/$,+*.@&* +)$ Ax 00
&'()*+ ,-.)/&', 8)"- ,#./<&* 0)% , )>)$*$ E+)& )()( 0xfxf 3 2*" 6.9& ),( 00 %x%xAx 1!40 .
A$ 0x 7>2&)"* 0-/1 = /1,23' &'()*'4 ,2.3/&'5, &-E )$ )( 0xf &'()*+ ,-.)/&'
&16 f.
y
O
A(x0,f (x0))
Cf
a x0 x2 x1 " x
29
2 !"#$%&"'%( )%*"(+%( 259
A! " #!$%&'"'# )()( 0xfxf ! $%()*$ +$# ,-.* Ax" , '&'*, &/01 *23#4* %'"! /#-
5-+5#67 1.3, " f /#578%$-9*$ %'7 Ax "0 !"#$% &'(#)*! : #/;- &'(#)*!, '7
)( 0xf .
<'7 3$/;#!& %(:4# /#5#'"57)4* &'$ %'7 %"4*27 0xx # " '$4: '"1 %8!-5'"%"1 *2!#$ 4$,5&'*5" #/& '"! '$4: '"1 %* ,-.* “+*$'7!$,&”
%"4*27 '78 0x . <'"! /*52/'0%" #8-
': ;=4* &'$ " f /#578%$-9*$ %'7 0x
'7/$,& *;-($%'7. >7 23$7 %84?#2!*$ ,#$ %'# %"4*2# 1x ,#$ !. @*!$,-,
=(784* '7! #,&;78.7 75$%4&:
+,-./+.
A2# %8!-5'"%" f, 4* /*327 75$%47) ", .# ;=4* &'$ /#578%$-9*$ %'7 Ax "0
*!0#$% 1"23#)*!, &'#! 8/-5(*$ 0$# , '='7$7 B%'*
)()( 0xfxf % , +$# ,-.* ),( 00 #x#xAx &'(" .
>7 0x ;=+*'#$ 4')5 : )5&16! *!0#$!7 1"836)*!9, *!B '7 )( 0xf *!0#$% 1"23#-)*! *5: f.
C! " #!$%&'"'# )()( 0xfxf % $%()*$ +$# ,-.* Ax" , '&'*, &/01 *23#4* %'"! /#-
5-+5#67 1.3, " f /#578%$-9*$ %'7 Ax "0 !"#$% 1"23#)*! : #/;- 1"23#)*!, '7
)( 0xf .
># '7/$,- 4=+$%'# ,#$ '7/$,- *;-($%'# '"1 f ;=+7!'#$ *!0#$2 8$;%*8*8 #8':1, *!B '# %"4*2# %'# 7/72# " f /#578%$-9*$ '7/$,- #,5&'#'# ;=+7!'#$ 4')1#: *!-
0#$<= 8$;!*2*>=. >7 4=+$%'7 ,#$ '7 *;-($%'7 '"1 f ;=+7!'#$ !"#$2 8$;%*8*8 : #/;- 8$;%*8*8 #8':1.
@$# /#5-3*$+4#, " %8!-5'"%"
)*
)+,
$
!#
1#!,1
1#!,)(
2
xx
xxxf
/#578%$-9*$:
i) %'7 0#x '7/$,& *;-($%'7, '7 0)0( #f ,
'7 7/727 *2!#$ ,#$ 7;$,& *;-($%'7 ,#$
ii) %'7 1#x '7/$,& 4=+$%'7, '7 1)1( #f .
D %8!-5'"%" f #! ,#$ /#578%$-9*$ '7/$,& 4=+$%'7, *!'7)'7$1 3*! /#578%$-9*$ (7;$,&) 4=+$%'7.
y
O
Cf
a x0 x1 ! x
30
y
1
1
O
Cf
x
31
2 !"#$%&"'%( )%*"(+%( 260
!"#$%&
i) !"# $%&'() *+,'-$% *&%./0 "# /0"#' *'(.)$/.% #&) +"# $%&'() /123'-$% (43.32#).
y
x4 x3 x2 x1
(a) O x
(!)
y
O
min
max
a ! x
32
x4 x3 x2 x1
ii) 5" *'# -6"2.$7-7 f &#.%6-'28/' *+,'-$%, $)$/ #6$) 9# /0"#' $% */,#1:$/.% #&) $# $%&'(2 *+,'-$#, /"; #" &#.%6-'28/', /123'-$%, $)$/ #6$) 9# /0"#' $% *'-(.)$/.% #&) $# $%&'(2 /123'-$#. (43. 32<). =% */,#1:$/.% )*>? #&) $# $%&'(2 *+,'-$# *0#? -6"2.$7-7? @/" /0"#' &2"$%$/ *+,'-$% #6$A?. B&0-7? $% *'(.)$/.% #&) $# $%&'(2 /123'-$# *0#? -6"2.$7-7? @/" /0"#' &2"$%$/ /123'-$% $7? -6"2.-$7-7? (43. 32#).
,-./01.-1/234 567 5.819:7 ;9-.5<567 C/ *'# &.%-/($'(A &#.#$A.7-7 $%6 -3A*#$%? 32< <1+&%6*/ )$' #" -’ +"# /->-
$/.'() -7*/0% 0x /")? @'#-$A*#$%? $%6 &/@0%6 %.'-*%: $7? 7 f &#.%6-'28/' $%&'() #(.)$#$% (#' /&'&1+%" /0"#' &#.#,>,0-'*7 -$% -7*/0% #6$), $)$/ -$%
-7*/0% ))(,( 00 xfxA 7 /D#&$%*+"7 $7? ,.#D'(A? &#.2-$#-7? $7? f /0"#' %.'8)-
"$'#, @71#@A '-3:/' 0)( 0 !" xf . 56$) /&'</<#';"/$#' #&) $% &#.#(2$> 9/;.7*#,
&%6 /0"#' ,">-$) >? '()*+,- ./0 Fermat.
'12345& (Fermat)
!-$> *'# -6"2.$7-7 f %.'-*+"7 -’ +"# @'2-$7*# " (#' 0x +"# (67.(*89:
-7*/0% $%6 ". 5" 7 f &#.%6-'28/' ./;89: -9*:.-./ -$% 0x (#' /0"#' ;-*--
<7<=68,+ -$% -7*/0% #6$), $)$/:
0)( 0 !" xf
#,%!=">?
5? 6&%9+-%6*/ )$' 7 f &#.%6-'28/' -$% 0x
$%&'() *+,'-$%. B&/'@A $% 0x /0"#' /->$/.'()
-7*/0% $%6 " (#' 7 f &#.%6-'28/' -’ #6$) $%-
&'() *+,'-$%, 6&2.3/' 0## $+$%'%, ;-$/
"#x#x $%& ),( 00 (#'
)()( 0xfxf ' , ,'# (29/ ),( 00 #x#xx %&( . (1)
y
O
f (x0)
x0&# x0+# x0 x
33
2 !"#$%&"'%( )%*"(+%( 261
!"#$%&, #"$"'()*, + f #,*-$ "-.-/0/,1$2+ 13) 0x , $145#$
0
0
00
0
00
)()(lim
)()(lim)(
xx
xfxf
xx
xfxfxf
xxxx !!
"!!
"#$%!%
.
!")2(*06,
— -* ),( 00 x!xx !& , 373#, '7/0 3+6 (1), 8- #,*-$ 0)()(
0
0 '!!
xx
xfxf, )"73# 8-
(4)92#
0)()(
lim)(0
0
00 '
!!
"#!% xx
xfxfxf
xx
(2)
— -* ),( 00 !xxx $& , 373#, '7/0 3+6 (1), 8- #,*-$ 0)()(
0
0 (!!
xx
xfxf, )"73# 8-
(4)92#
0)()(
lim)(0
0
0
0 (!!
"#$% xx
xfxfxf
xx
. (3)
:31$, -"7 3$6 (2) ;-$ (3) (4)92# 0)( 0 "# xf .
< -"7%#$=+ /$- 3)"$;7 #'>4$13) #,*-$ -*>')/+. !
!"#$%#
?52@0*- 2# 3) ".)+/)52#*) 8#A.+2-, 3- #103#.$;> 1+2#,- 3)9 ", 13- )"),- +
f # #,*-$ %$-@).#3$;& -"7 3) 2+%(*, %#* #,*-$ 8(1#$6 3)"$;A* -;.)3>30*. !")-
2(*06, 7"06 @-,*#3-$ ;-$ 13- 14&2-3- 29 ;-$ 30, )$ # $ % & ' ( ) % ( * + $ ) , -' , . # $ / 0 ' & / 1 . , 2 , - ' 2$-6 19*>.3+1+6 f 1’ (*- %$>13+2- " #,*-$:
1. &' ()*+(,-./ )01(2' +34 ! )+' 3532' 0 5',/6*637 +07 f 108(92:(+'-.
2. &' ()*+(,-./ )01(2' +34 ! )+' 3532' 0 f 8(9 5','6*62:(+'-.
3. &' /.,' +34 ! (-* -*&;)9* 13) "#%,) ).$12)5 3+6).
B- + * - , + 1 $ / 2 1+2#,- 3)9 " 13- )"),- + f %#* "-.-/0/,C#3-$ & + "-.>-/0/76 3+6 #,*-$ ,1+ 2# 3) 2+%(*, '(/)*3-$ .,2)-1' )01(2' 3+6 f 13) %$>13+2- ". D$- "-.>%#$/2-, (130 + 19*>.3+1+
)*+
'!,
"1,)2(
1,)(
2
3
xx
xxxf .
< f #,*-$ 19*#4&6 13) - ;-$ "-.-/0/,1$2+
13) - }1{! 2#
)*+
.!,
"#1,)2(2
1,3)(
2
xx
xxxf .
E$ .,C#6 3+6 0)( "# xf #,*-$ )$ 0 ;-$ 2.
y
O
Cf
2
1
1 x
34
2 !"#$%&"'%( )%*"(+%( 262
E!"#$% & f ! '&$"()*"+,# -+, -&'"), 0 .,# 2, "(/ $"( 0!123"# -+4 1, +, .2)-#',
-&'"), +&5 f ")(,# 4# ,2#6'4) 0, 1 .,# 2. 7'85, 9!85 :,)("+,# -+4 -3%',, +, -&'"), 1 .,# 2 ")(,# 6;-"#5 +4!#./( ,.24+1+8(, "(/ +4 -&'")4 0 $"( ")(,# 6;-& +4!#.4< ,.24+1+40. =2, $"( ")(,# 9>, +, .2)-#', -&'"), 6;-"#5 +4!#./( ,.24-+1+8( +&5 f. ?!4';(85, 32"#,*9',-+" ;(, .2#+%2#4 +4 4!4)4 (, ',5 !>&24:42") !4#, ,!9 +, .2)-#', -&'"), +&5 f ")(,# 6;-"#5 +4!#./( ,.24+1+8( ,0+%5. @3"-+#.1 #-3<"# +4 !,2,.1+8 6"/2&',:
!"#$%&'
A-+8 '#, -0(12+&-& f !,2,B8B)-#'& -’ ;(, $#1-+&', ),( !" , '" "C,)2"-& )--85 ;(, -&'")4 +40 0x , -+4 4!4)4 9'85 & f ")(,# ()*+,-..
i) D( 0)( "! xf -+4 ),( 0x" .,# 0)( #! xf -+4 ),( 0 !x , +9+" +4 )( 0xf ")(,# +4-
!#.9 ';B#-+4 +&5 f. (@3. 35,)
ii) D( 0)( #! xf -+4 ),( 0x" .,# 0)( "! xf -+4 ),( 0 !x , +9+" +4 )( 0xf ")(,# +4-
!#.9 ">13#-+4 +&5 f. (@3. 35E)
iii) A( & )(xf ! $#,+&2") !29-&'4 -+4 ),(),( 00 !xx" $ , +9+" +4 )( 0xf $"( ")(,# +4!#.9 ,.29+,+4 .,# & f ")(,# B(&-)85 '4(9+4(& -+4 ),( !" . (@3. 35B).
#,%!-"./
i) E!"#$% 0)( "! xf B#, .16" ),( 0x"x% .,# & f ")(,# -0("3%5 -+4 0x , & f ")-(,# B(&-)85 ,<C40-, -+4 ],( 0x" . A+-# ;340'"
)()( 0xfxf & , B#, .16" ],( 0x"x% . (1)
?!"#$% 0)( #! xf B#, .16" ),( 0 !xx% .,# & f ")(,# -0("3%5 -+4 0x , & f ")(,# B(&-)85 :6)(40-, -+4 ),[ 0 !x . A+-# ;340'":
)()( 0xfxf & , B#, .16" ),[ 0 !xx% . (2)
y
O
f (x0)
f #<0 f #>0
! a x0 x
y
O
f #<0 f #>0
! a x0 x
35a
f (x0)
?!4';(85, >9B8 +8( (1) .,# (2), #-3<"#:
)()( 0xfxf & , B#, .16" ),( !"x% ,
!40 -&',)("# 9+# +4 )( 0xf ")(,# ';B#-+4 +&5 f -+4 ),( !" .,# 12, +4!#.9 ';B#--+4 ,0+%5. ii) ?2B,*9',-+" ,(,>9B85.
2 !"#$%&"'%( )%*"(+%( 263
y
O
f !<0 f !>0
" a x0 x
y
O
f !<0 f !>0
" a x0 x
35"
iii) !"#$ %#&
0)( !" xf , '&( )*+, ),(),( 00 "xx#x #$ .
y
O
f !>0
f !>0
" a x0 x
y
O
f !>0
f !>0
" a x0 x
35$
-.,&/0 1 f ,23(& "43,506 "#7 0x +( ,23(& '31"2$6 (8974"( ", )*+, :3( (.% #(
/&("#0;(#( ],( 0x# )(& ),[ 0 "x . -.7;:3$6, '&( 201 xxx %% &"58,& )()()( 201 xfxfxf %% . <=( #7 )( 0xf /,3 ,23(& #7.&)% ()=%#(#7 #16 f. >( /,2-
974;,, #?=(, %#& 1 f ,23(& '31"2$6 (8974"( "#7 ),( "# . @=*';(#&, :"#$
),(, 21 "#xx $ ;, 21 xx % .
— A3 ],(, 021 x#xx $ , ,.,&/0 1 f ,23(& '31"2$6 (8974"( "#7 ],( 0x# , +( &"58,& )()( 21 xfxf % .
— A3 ),[, 021 "xxx $ , ,.,&/0 1 f ,23(& '31"2$6 (8974"( "#7 ),[ 0 "x , +( &"58,& )()( 21 xfxf % .
— B:C76, (3 201 xxx %% , #%#, %.$6 ,2/(;, )()()( 201 xfxfxf %% .
-.7;:3$6, ", %C,6 #&6 .,=&.#?",&6 &"58,& )()( 21 xfxf % , 7.%#, 1 f ,23(& '31-
"2$6 (8974"( "#7 ),( "# .
D;72$6, (3 0)( %" xf '&( )*+, ),(),( 00 "xx#x #$ . !
E&( .(=*/,&';(, :"#$ 1 "43*=#1"1 34 4)( xxxf &' .74 ,23(& 7=&";:31 "#7 (. F
f ,23(& .(=('$'2"&;1 "#7 (, ;, 23 124)( xxxf &'" . D& =2G,6 #16 0)( '" xf ,23(& 0'x (/&.C0) 0 3'x , #7 /, .=%"1;7 #16 f " H(23,#(& "#73 .(=()*#$ .23()(:
x &) 0 3 *)
)(xf " & 0 & 0 +
2 !"#$%&"'%( )%*"(+%( 264
!"#$%&' #( )* +','+-&% .,/)0,/*, 1 23&-,)121 f (4&'/ 5&124%6 $74&*32' 2)*
8/-2)1#' ]3,(!" , 5&124%6 '"9*32' 2)* 8/-2)1#' ),3[ #" .'/ +',*32/-:(/ ;&'
#<&* )*+/.< '.,<)')*, 235.(.,/#;&' *=/.< (=->/2)* 5/' 3$x , )* 27)3( !$f .
!"#$%&
% ?@+%6 (48'#( 2)1& '+<8(/91 )*3 +','+-&% 7(%,0#')*6 2)1& +,A)1 +(,4+)%-
21 )* )( 0xf (4&'/ 1 #;5/2)1 )/#0 )16 f 2)* ),( !" , (&A 2)1 8(")(,1 +(,4+)%21
)* )( 0xf (4&'/ 1 (=->/2)1 )/#0 )16 f 2)* ),( !" .
% B& #/' 23&-,)121 f (4&'/ 23&(>06 2’ ;&' .=(/2)< 8/-2)1#' ],[ !" , <+%6 5&%,4:*3#( (C(A,1#' § 1.8),1 f +',*32/-:(/ #;5/2)* .'/ (=->/2)*. D/' )1& (",(21 )*3 #;5/2)*3 .'/ (=->/2)*3 (,5':<#'2)( %6 (906:
1. '()*+,-./ 01 +()*2.1 *3./)1 034 f.
2. 56,7,8)9,-./ 024 02.:4 034 f *01 *3./)1 1-0; +12 *01 ;+(1 0<= >21*03-
.;0<=.
3. &6? 1-0:4 024 02.:4 3 ./817@0/(3 /)=12 0, .:82*0, +12 3 .2+(?0/(3 0,
/7;A2*0, 034 f.
D/' +',-8(/5#', ;2)% 1 23&-,)121 1924152)( 23 ##!$ xxxxf , ]5,0[&x . E>*3-
#( 24306)( 2 #!$' xxxf , ]5,0[&x . @/ ,4:(6 )16 0)( $' xf (4&'/ */ 1$x , 4$x .
F+*#;&%6, )' .,42/#' 21#(4' )16 f (4&'/ )' 1$x , 4$x . @/ )/#;6 )16 f 2)'
.,42/#' 21#(4' .'/ 2)' -.,' )*3 8/'2)0#')*6 ]5,0[ (4&'/ 30)1( $f , 3)4( $f , 19)0( $f .'/ 14)5( $f .
G,', 1 #;5/2)1 )/#0 )16 f 2)* ]5,0[ (4&'/ 421 #( 30 .'/ +',*32/-:()'/ 5/'
1$x , (&A 1 (=->/2)1 )/#0 )16 f (4&'/ 421 #( 3 .'/ +',*32/-:()'/ 5/' 4$x .
% D/' &' ($',#<2*3#( )* +,*15*"#(&* 7(A,1#' '+'/)(4)'/ &' +,*28/*,42*3#( )* +,<21#* )16 f ' (.');,%7(& )*3 0x . H)'& * +,*28/*,/2#<6 '3)<6 8(& (4&'/ (".*=*6 0 (4&'/ '8"&')*6, )<)( )* +','.-)% 7(A,1#', )*3 *+*4*3 1 '+<8(/91
+','=(4+()'/, #+*,(4 &' #'6 +=1,*$*,02(/ '& )* 0x (4&'/ 7;21 )*+/.*" '.,<)'-
)*3.
BCDEFG&
E2)% #/' 23&-,)121 f +','5%542/#1 2( ;&' 8/-2)1#' ),( () .'/ 0x ;&' 21-
#(4* )*3 ),( () 2)* *+*4* 1 f (4&'/ 83* $*,;6 +','5%542/#1.
% B& 0)( 0 $' xf .'/ 0)( 0 *'' xf , )<)( )* )( 0xf (4&'/ )*+/.< #;5/2)*.
% B& 0)( 0 $' xf .'/ 0)( 0 +'' xf , )<)( )* )( 0xf (4&'/ )*+/.< (=->/2)*.
D/' +',-8(/5#', ;2)% <)/ 7;=*3#( &' I,*"#( )' )*+/.- '.,<)')' )16 23&-,)1-216
xxxf 23&2)( #$ , )2,0[ #x& .
2 !"#$%&"'%( )%*"(+%( 265
!"#$%&
xxf '%21)( !"# ()* xxf +$,2)( !"## ,
#-./& #* 012&3 /'3 f # &1,)* #* 6
$ ()*
6
5$.
4*) 6
$"x , &1,)* 03
6%!"&
'(
)*+## $
f , &,5 6*) 6
5$"x , &1,)* 03
6
5,"&
'(
)*+## $
f .
!/+* 7"#$%&
)) 06
"&'(
)*+# $f ()* 0
6%&
'(
)*+## $
f , #-./& /# &'(
)*+
6
$f &1,)* /#-*(. %76*+/# /'3 f.
8) 06
5"&
'(
)*+# $
f ()* 06
5,&
'(
)*+## $
f , #-./& /# &'(
)*+
6
5$f &1,)* /#-*(. &9:"*+/# /'3 f.
,$#&+%*,(
1. N! "#$%$& '( ]3[0,-x )'*+, ,*'$ '( (#%(-
-,.+( #-*! '(/ 0+12!.(3 *456!'(7 .! )4$+ 6)--+*'( $6"!08.
).(/
;# &%8)<. /#$ #0=#6>,1#$ &1,)*
xxxx!AABxE 62)3(2))(()( 32 .!"!"" .
!"#$%& )1)(1(666)( 2 !.!".!"# xxxxE . ?* 012&3 /'3 0)( "# xE &1,)* #* 1!"x , 1"x . @ %#,#/#,1)
()* /) )(0./)/) /'3 " A)1,#,/)* +/#, -)0)(:/> -1,)()
x 0 1 3
EB(x) + 0 !
E(x) max
min 0
4
min 0
C0), ' %76*+/' /*%D /#$ &%8)<#E &1,)* 1+' %& 4 ()* -)0#$+*:2&/)* ./), 1"x .
y
! #
O
y=3!x2
A(x,0) B(!x,0) x
36
2 !"#$%&"'%( )%*"(+%( 266
2. !"#$ % "&'()#%"% xxxf ln1)( !!" .
i) *+ ,-.-#%/-0 $1 2)31 #% ,3'3#3'0+ 4+5 #+ +4)6#+#+.
ii) *+ +237-58#-0 6#5 1ln !# xx , 95+ 4(/- 0$x .
),(-
i) !"#$%& x
x
xxf
111)(
!"!"% , ),0( &'(x . '
&()*+*, 0)( "% xf -"&. %)/ %01# 2)3/, 4,1 1"x . ' %#1#4#1)/ 5/. 4/ /5204/4/ 4,6 f 7/)1#14/. *4#1 8/2/594+ 8)1/5/:
x 0 1 +'f :(x) ! 0 +
f (x)
min 0
ii) ;8&.<= , f >./ 1"x 8/2#$*.93&. #?.50 &?9".*4#, >./ 59@& ),0( &'(x .*"A&.:
0ln1)0()( )!!*) xxfxf
1ln !#* xx .
' .*04,4/ .*"A&. %01# 04/1 1"x .
3. :0+ ;53,%8+'0+ 4+/3)0<-5 #%' #5,= 2>.%"%1 )(x. 4(/- ,3'(7+1 -'61 2)3?6'#31, "&'+)#="-5 #3& 2.=/3&1 x #$' ,3'(7$' 2+)+9$9=1, "@,A$'+ ,- #3' #@23 xx. 640000)( !" . B3 46"#31 2+)+9$9=1 ,5+1 ,3'(7+1 -0'+5 4000
7)8. C' % ;53,%8+'0+ 2.%)>'-5 A6)3 1200 7)8.95+ 4(/- ,3'(7+ 2)3?6'#31, '+ ;)-/-0 26"-1 ,3'(7-1 2)3?6'#31 2)D2-5 '+ 2+)(9-5 % ;53,%8+'0+, >"#- '+
D8-5 #3 ,D95"#3 7&'+#6 4D)731.
),(-
' &)*82/(, /80 4,1 8B?,*, x %#19<+1 8/2/>+>=6 &)1/.
xxxxx!xxE 400006)640000()()( 2 &!"!"" .
C# 50*4#6 /80 4,1 8/2/>+>= x %#19<+1 &)1/. xxK 4000)( " .
C# #?.50 50*4#6 %&49 4,1 8?,2+%= 4#$ 702#$ &)1/.:
xxxxK "# 520012004000)( "&" .
;8#%-1+6, 4# 5-2<#6 4,6 D.#%,"/1)/6 &)1/.
)()()( xKxExP "#!"
y
y=x!1
y=lnx
1 O x
37
2 !"#$%&"'%( )%*"(+%( 267
xxx 5200400006 2 !"!#
xx 348006 2 "!# .
!"#$%& 3480012)( "!#$ xxP , #'()& * 0)( #$ xP +"&, -./0 )*1 2900#x .
2 %#1#)#1.0 30, )0 03-()0)0 )*4 ! 5)# ),0( "% 60.1#1)0, 5)#1 '0-037)8 '.10-
30:
x 0 2900 +%P9(x) + 0 !
P(x) max 50460
:'#%+184, )# %+;,5)# 3+-<#4 '0-#$5,7/&)0, ()01 * =,#%*"01.0 '0-7;&, 2900 %#17<&4 0'( )# '-#>(1 0$)( 30, &.10, .5# %& 50460 ",?,7<&4 <-".
#(',(-"(
!" #$!%!&
1. H '0-7;8;#4 %,04 5$17-)*5*4 f &.10,
)3()2()1(3)( 23 !!!#$ xxxxf .
@,0 '#,+4 ),%+4 )#$ x * f '0-#$5,7/&, )#',3( %+;,5)# 30, ;,0 '#,&4 '0-#$5,7/&, )#',3( &?7",5)#;
2. 0) A0 %&?&)B5&)& 84 '-#4 )* %#1#)#1.0 30, )0 03-()0)0 ),4 5$10--)B5&,4:
i) 133)( 23 ""!# xxxxf ii) 23)( 3 "!# xxxg
iii) 132)( 23 !!# xxxh .
=) A0 =-&.)& )# '?BC#4 )81 '-0;%0),3D1 -,/D1 )81 &E,5D5&81:
0133 23 #""! xxx , 0233 #"! xx , 0132 23 #!! xx .
3. A0 %&?&)B5&)& 84 '-#4 )* %#1#)#1.0 30, )0 03-()0)0 ),4 5$10-)B-5&,4:
2 !"#$%&"'%( )%*"(+%( 268
i) !"#
$%
&' 1,
1,)(
1
2
xe
xxxf
x ii)
!"#
()'*)'
&1,34
1,12)(
2
2
xxx
xxxxg .
4. N! "#$#%&'#%# () *+,) %- ",.,%,./! 0!1 %! !0+2%!%! %1) '3.!+%&-'#1):
i) xexf x '&)( ii) xxxf &)( , 0$x .
5. N! 4+#/%# %1) %1"5) %(. +!", , 61! %1) ,*,/#) - '3.7+%-'-
13)( 23 )')& xx!x"xf *!+,3'178#1 %,*107 !0+2%!%! '%! '-"#/!
11 '&x 0!1 12 &x . 9! 0!:,+/'#%# %, #/;,) %(. !0+,%7%(..
6. 9! !*,;#/<#%# 2%1, !*2 2$! %! ,102*#;! '=&"!%,) ,+:,6(./,3 "# #"-4!;2 400m2, %, %#%+76(., =+#178#%!1 %- "10+2%#+- *#+/>+!<-.
7. ?# '3+"!%2*$#6"! "&0,3) 80m :5$,3"# .! *#+1>+7<,3"# ,102*#;, '=&"!%,) ,+:,6(./,3. 9! 4+#/%# %1) ;1!'%7'#1) %,3 ,10,*5;,3 *,3 5=#1 %, "#6!$@%#+, #"4!;2..
8. ?/! A+! "#%7 %- $&B- x mgr #.2) !.%1*3+#%10,@, - "#/('- %-) :#+-
",0+!'/!) #.2) !':#.,@) ;/.#%!1 !*2 %- '3.7+%-'- 4
)(3
2 xxxT '& ,
30 ** x . 9! 4+#/%# *,1! *+5*#1 .! #/.!1 - ;2'- %,3 !.%1*3+#%10,@,
A'%# , +3:"2) "#%!4,$&) %-) "#/('-) %-) :#+",0+!'/!) () *+,) x, .! 6/.#1 "561'%,).
9. C/.#%!1 %#%+76(., #$%& %,3 ;1*$!.,@ '=&"!%,) "# *$#3+7 2cm. D. %, %#-%+76(., '()* 5=#1 %1) 0,+3>5) %,3 '%1) *$#3+5) %,3 #$%&,
i) .! #0>+7'#%# %-. *$#3+7 '( '3-.!+%&'#1 %,3 x.
ii) .! 4+#/%# %, x 5%'1, A'%# %, #"4!-
;2. )(xE %,3 '()* .! 6/.#1 #$7=1'%,.
10. E, 02'%,) %-) -"#+&'1!) *!+!6(6&) x ",.7;(. #.2) 41,"-=!.10,@
*+,F2.%,) #/.!1 1000600203
1)( 23 ))'& xxxxK =1$17;#) ;+!="5),
1050 %% x . G #/'*+!<- !*2 %-. *A$-'- %(. x ",.7;(. #/.!1 22420)( xxxE '& =1$17;#) ;+!="5). 9! 4+#:#/ - -"#+&'1! *!+!6(6&
%,3 #+6,'%!'/,3, 61! %-. ,*,/! %, 05+;,) 6/.#%!1 "561'%,.
!" #$%&%'
x x
x x
&
A '
(
B
*
H %
'(x)
2 !"#$%&"'%( )%*"(+%( 269
1. !"#$%&' ( )*#+,%()( 3(-2)( !"# xxxf , ],0[ !x$
i) .& -$/$%0)$%$ %(# f 12 3,42 %( -4#4%4#"& 5&' &5,6%&%&.
ii) .& &347$"8$%$ 6%' ( $8")1)( 2
3
2
1(- "# xx 9:$' &5,';<2 -"& ,"=&
)%4 ),0( ! .
2. i) .& -$/$%0)$%$ 12 3,42 %( -4#4%4#"& 5&' %& &5,6%&%& %( )*#+,-%()(
1ln)( "!# xxxf
5&' #& ;,$"%$ %'2 ,"=$2 5&' %4 3,6)(-6 %(2.
ii) .& -$/$%0)$%$ 12 3,42 %( -4#4%4#"& 5&' %& &5,6%&%& %( )*#+,-%()(
34ln2)( 2 !"!# xxxxx"
iii) .& &347$"8$%$ 6%' 4' >,&?'592 3&,&)%+)$'2 %1# )*#&,%0)$1#
xxxg ln)( # 5&' 2
32
2
1)( 2 "!"# xxxh
9:4*# 9#& -6#4 54'#6 )(-$"4 )%4 434"4 9:4*# 5&' 54'#0 $?&3%4-9#(.
3. .& &347$"8$%$ 6%' >'& 5+@$ 0%x '):A$'
i) &) xe x !%1 ii) &) 2
2
11)*# xx "%
;) 2
2
11 xxe x !!% ;) 3
6
1(- xxx "%
iii) &) x#x # !%! 1)1( , !$# -$ 2&#
;) 2
2
)1(1)1( x
##x#x # "!!%! , !$# -$ 3&# .
4. .& &347$"8$%$ 6%', &# >'& -'& )*#+,%()( f, 34* $"#&' 3&,&>1>")'-(
)%4 ', '):A$'
162)(6)(2 33 !!#! xxxfxf ,
%6%$ ( f 7$# 9:$' &5,6%&%&.
2 !"#$%&"'%( )%*"(+%( 270
5. !"# $%&'()* +,-.( /,#0.1 "%2 34(5%6/2 &(4(+"7+1%2 $8# &(-
4(393:+%.9) +0)(4"-+19) gf ,
+’ /)( $%7+";.( ],[ !" . <# +;-
.1:# ),( !"# ! 1:)(% "# +;.1:#
+"# #&#:# ; 6(4(6*405; (&*-
+"(+; )($% .1"(=8 "9) fC 6(%
gC &(:4)1% "; .13('8"14; "%.-.
>( (&#$1:=1"1 *"% #% 15(&"*.1)12 "9) fC 6(% gC +"( +;.1:(
))(,( #f#A 6(% ))(,( #g#% 1:)(% &(47'';'12.
6. >( (&#$1:=1"1 *"% ; +0)74";+;
222 )()()()( &x!x"xxf """# , .1 &!" $$
/,1% "4:( "#&%67 1'7,%+"( 6(% $8# "#&%67 ./3%+"(.
7. ?1 /)( +84.( .-6#02 4m 6("(+6107@#0.1 /)( %+*&'104# "4:39)# &'10472 x m 6(% /)( "1"4739)# &'10472 y m.
i) >( A41:"1 "# 7B4#%+.( "9) 1.A($C) "9) $8# +,;.7"9) +0)(4"--+1% ";2 &'10472 x "#0 %+#&'184#0 "4%3C)#0.
ii) D%( &#%( "%.- "#0 x "# 1.A($*) 3:)1"(% 1'7,%+"#.
8. E:)1"(% ; +0)74";+; xxf #)( 6(% "# +;.1:# %&'
()*
0,2
9A .
i) >( A41:"1 "# +;.1:# ' ";2 fC � (&/,1% (&* "# +;.1:# $ "; .%-64*"14; (&*+"(+;.
ii) >( (&#$1:=1"1 *"% ; 15(&"#./); ";2 fC +"# ' 1:)(% 67B1"; +";)
$'.
9. F&92 3)94:@#0.1, # +":A#2 "#0 6'(+%6#8 (B';"%+.#8 (&#-"1'1:"(% (&* /)( #4B#3C)%# 6(% $8# ;.%686'%(. G) ; &14:-.1"4#2 "#0 +":A#0 1:)(% 400m, )( A41:"1 "%2 $%(+"7+1%2 "#0, C+"1 "# 1.A($*) "#0 #4B#39):#0 ./4#02 )( 3:)1% ./3%+"#.
10. H )(8'9+; .%(2 64#0(@%/4(2 (&(%"1: +0..1"#,- "#0'7,%+"#) 100 (-"*.9). G) $;'C)#0) (64%AC2 100 7"#.(, "# ()":"%.# ()/4,1"(% +1 100 ,%'%7$12 $4(,./2 "# 7"#.#. D%( 67B1 1&%&'/#) 7"#.# "# ()":"%.# ()7 7"#.# .1%C)1"(% 6("7 500 $4,. I*+( 7"#.( &4/&1% )( $;'C+#0) +0..1"#,-, C+"1 )( /,#0.1 "( &14%++*"14( /+#$(.
y
O
Cf
Cg
g(#)
f (#) $
%
! " # x
x
x E(x)
2 !"#$%&"'%( )%*"(+%( 271
11. !"#$ ! #% &'()*+, #%- .-./0.%1 *)-.#-/2%- #%- *03/),%1 "45')#%6. 73%-89#%-'& %#0 #: 4;%,0.5 "#0<'5 0!t
&2,)0 31 !r cm .)0 52 !r cm .)0 +#0 <0)
0"t : ).#2,) 1r )-=>,&#)0 '& "#)8&-;+ ;-8'+ 0,05cm/s, &,? : ).#2,) 2r
)-=>,&#)0 '& "#)8&;+ ;-8'+ 0,04 cm/s. @) (;&2#&:
i) 3+#& 8) ':*&,0"#&2 #% &'()*+, #%- .-./0.%1 *).#-/2%- .)0
ii) 3+#& 8) '&<0"#%3%0:8&2 #% &'()*+, #%- .-./0.%1 *).#-/2%-.
12. A9/%-'& ,) .)#)".&->"%-'& 9,) .)-,>/0 #%- %3%2%- : .>8&#: *0)#%'5 "#$% B)2,&#)0 "#% *03/),+ "45').
i) @) )3%*&2=&#& +#0 #% &'()*+, #:6 *0)#%'56 "#$% &2,)0 2"% '&
)"-,1(:'4 &&E #!
ii) C0) 3%0) #0'5 #%- & #% &'()*+, #:6 .>8&#:6 *0)#%'56 '&<0"#%3%0&2-#)0;
13. !,)6 .%/-'(:#56 ' (;2".&-#)0 "#: 8>/)"") 100ft(1) ')-.;0> )3+ #% 3/:"09"#&;% ":-'&2% " '0)6 &-81<;)'':6 )-.#56, &,? #% "32#0 #%- ( (;2-".&#)0 300ft ').;-> )3+ #% ":'&2% ". 73%89#%-'& +#0 % .%/-'(:#56 '3%;&2 ,) .%/-'(5"&0 '& #)-41#:#) 3ft/s .)0 ,) #;9=&0 "#:, ).#5 '& #)41#:#) 5ft/s.
i) @) )3%*&2=&#& %#0 <0) ,) *0),1"&0 #: *0)*;%'5 ')( #%- *03/),%1 "45')#%6 4;&0>D&#)0 4;+,%
5
300
3
100)(
22 xxx* $
##
! .
ii) C0) 3%0) #0'5 #%- x o .%/-'(:#56 8) 4;&0)"#&2 #% /0<+#&;% *-,)#+ 4;+,% <0) ,) B8>"&0 "#% "32#0 #%-;
14. !,)6 &;<%/>(%6 &308-'&2 ,) 4#2"&0 9,) "32#0 "#% *;+'% 3%- "-,*9&0 *1%
&;<%"#>"0) 1E .)0 2E #)
%3%2) (;2".%,#)0 "& )-
(1) 1ft = 30,48 cm
r2
r1
2m 2m
2m #
$ %
" & &
M x
'
( A
100 ft
300 ft
x !2 !1
12km
(
2 !"#$%&"'%( )%*"(+%( 272
!"#$%#& 12km '%( )'!*+!,-. '%!." +) !%/,0*1 ! '%( P8 %.$(-#$,2031. 4. & !-'."$&$% $,- '%!.,5 #) +(% %!"#$%#& d %!" *.% $*-$,(, )/6,#$7#(, )2.%( %.78,6& $&1 !%/,091 '%!.,5 $,- )/6,#$%#2,- '%( %.$(#$/":31 %.78,6& $,- $)$/%6;.,- $&1 %!"#$%#&1 d, .% </)2$) #) !,(% %!"#$%#& x %!" $, )/6,#$7#(, 1E !/*!)( , )/6,87<,1 .%
0$2#)( $, #!2$( 6(% .% *0)( $& 8(6"$)/& =-.%$9 /5!%.#&. (>%/,09 '%-!.,5 +(%1 '%!.,="0,- )."1 )/6,#$%#2,- 8*6)$%( & !,#"$&$% $,- '%-!.,5 !,- )'!*+!)$%( %!" $&. '%!.,="0, #$& +,.7=% $,- 0/".,-).
2.8 K,&-%-.-# – (.+/"# '#+0.( (,1#&-.(.(
'2345 - 6789: ;7<:89=;=>
! ?#$3 ,( #-.%/$9#)(1 2)( xxf " '%( ||)( xxg " (@0. 38).
(a)
x
y
"
y=f (x)
y=g(x)
x
y
"
38
(#)
A( !8&/,:,/2)1 $(1 ,!,2)1 +%1 =2.)( & !/;$& !%/7636,1 6(% $& #-+!)/(:,/7 '7B) +(%1 %!" $(1 =5, #-.%/$9#)(1, "!31 :%2.)$%( '%( #$, #09+% 38 )2.%( 2=()1. C&8%=9 ,( #-.%/$9#)(1,
— )2.%( 6.ç :B2.,-#)1 #$, ]0,(#$
— )2.%( 6.ç %5D,-#)1 #$, ),0[ $%
— !%/,-#(7E,-. $,!('" )870(#$, 6(% 0"x , $, ,!,2, )2.%( 2#, +) 0.
F+31, ,( #-.%/$9#)(1 %-$*1 *0,-. =(%:,/)$('*1 6/%:('*1 !%/%#$7#)(1. C&8%=9, “%.*/0,.$%(” '%( “'%$*/0,.$%(” +) =(%:,/)$('" $/"!, #) '7B) *.% %!" $% =(%-
#$9+%$% ]0,(#$ '%( ),0[ $% . G!,+*.31, ,( !8&/,:,/2)1 !,- +%1 =2.)( $, !/"-
#&+, $&1 !/;$&1 !%/%6;6,- =). )2.%( ('%.*1 6(% $& 07/%D& $&1 6/%:('91 !%/7-#$%#&1 +(%1 #-.7/$&#&1. 41 B)3/9#,-+) $;/% $(1 6/%:('*1 !%/%#$7#)(1 $3. !%/%'7$3 #-.%/$9#)3. #$,
=(7#$&+% ),0[ $% .
2 !"#$%&"'%( )%*"(+%( 273
(a)
x
y
+
!
y=x2
!"#$% &' x "()*+,&"- . ,/"0&'12+. &.% Cf
3&42/,&"- 5"&* &. #,- &-56 /'4*
y x!
x
y
!
39
""""
(")
!"#$% &' x "()*+,&"- . ,/"0&'12+. &.% Cg
3&42/,&"- 5"&* &.+ "4+.&-56 /'4*
7"4"&.4'81, 9&- 5"#$% &' x "()*+,&"-:
— . 5:;3. )(xf # &.% fC "()*+,&"-, <.:"<6 . f # ,;+"- =+.3;>% "8)'(3" 3&'
),0[ $% , ,+$
— . 5:;3. &.% )(xg # &.% gC ,:"&&$+,&"-, <.:"<6 . g # ,;+"- =+.3;>% /#;+'(3"
3&' ),0[ $% .
?&.+ 04$&. 0,4;0&>3. :21, 9&- . 3(+*4&.3. f 3&42/,- &" 5';:" 04'% &" *+>
3&' ),0[ $% , ,+$ 3&. <,8&,4. 0,4;0&>3. :21, 9&- . g 3&42/,- &" 5';:" 04'% &"
5*&> 3&' ),0[ $% . @,+-5* <;+'(1, &'+ 0"4"5*&> '4-319:
!"#$%!$
A3&> 1;" 3(+*4&.3. f # $ % & ' ( ) 3’ 2+" <-*3&.1" * 5"- + , - , . / . 0 # 1 2 3 3&' & # / 4 & - 1 5 6 &'( *. B" :21, 9&-:
& C 3(+*4&.3. f 3&42/,- &" &'()* +,'- .* /01 6 ,;+"- &2,.3 3&' *, "+ .
f # ,;+"- =+.3;>% "8)'(3" 3&' & # / 4 & - 1 5 6 &'( *.
& C 3(+*4&.3. f 3&42/,- &" &'()* +,'- .* &/.1 6 ,;+"- &'()4 3&' *, "+ .
f # ,;+"- =+.3;>% /#;+'(3" 3&' & # / 4 & - 1 5 6 &'( *.
D0'0&-5*, 1;" 3(+*4&.3. f ,;+"- 5(4&6 ("+&-3&';E>% 5';:.) 3, 2+" <-*3&.1" *,
9&"+ 2+" 5-+.&9, 0'( 5-+,;&"- 0*+> 3&. fC , =-" +" <-"=4*F,- &' &9)' 0'( "+&--3&'-E,; 3&' <-*3&.1" * 0420,- +" 3&4"/,; 5"&* &. #,&-56 ("+&-3&';E>% "4+.&--56) /'4*. (?E. 40)
" "
+
y
O x
Cf
+
40
2 !"#$%&"'%( )%*"(+%( 274
!"# $# %&'()*+,- ).*$ /0$#1# ,-.#2*'($ 3." ,"# )+$45.&)& f -0$#" 1+5.6 (#$.").*0789 1*0'&) )- :$# %"4).&,# !, 75&)",*/*"*;,- .* )+,2*'"),3 (#-$.").*0789 ).
!"#$%#
</*%-"1$;-.#" 3.", #$ ,"# )+$45.&)& f -0$#" 1+5.6 (#$.").*0789 1*0'&) )’ :$# %"4).&,# !, .3.- & -=#/.*,:$& .&9 >5#="169 /#54).#)&9 .&9 f )- 14?- )&,-0* .*+ ! 250)1-.#" “14.8” (#$.").*0789 “/4$8”) #/3 .& >5#="16 .&9 /#54).#)& (@7. 39), ,- -A#05-)& .* )&,-0* -/#=69 .*+9.
! B ,-':.& ,"#9 )+$45.&)&9 89 /5*9 .# 1*0'# 1#" 1+5.4 %"-+1*';$-.#" ,- .&
2*6?-"# .*+ -/3,-$*+ ?-856,#.*9, /*+ -0$#" 4,-)& )+$:/-"# .*+ /5*&>*;,-$*+ *5"),*; 1#" .*+ ?-856,#.*9 ,*$*.*$0#9.
&'()*+,
CD).8 ,"# )+$45.&)& f " # $ % & ' ( )’ :$# %"4).&,# ! 1#" %+* =*5:9 /#5#->8>0)",& ).* % " ) * % + , - . .*+ !.
! <$ 0)( "## xf >"# 14?- % " ) * % + , - . )&,-0* x .*+ !, .3.- & f -0$#" 1+5.6 ).* !.
! <$ 0)( $## xf >"# 14?- % " ) * % + , - . )&,-0* x .*+ !, .3.- & f -0$#" 1*0-'& ).* !.
!"# /#54%-">,#, & )+$45.&)& 3)( xxf % (@7. 41),
— -0$#" 1*0'& ).* ]0,(&' , #=*; 06)( $%## xxf , >"#
)0,(&'(x 1#" & f -0$#" )+$-769 ).* ]0,(&' -$(,
— -0$#" 1+5.6 ).* ),0[ ') , #=*; 06)( "%## xxf , >"#
),0( ')(x 1#" & f -0$#" )+$-769 ).* ),0[ ') .
!"#$%#
E* #$.0).5*=* .*+ ?-856,#.*9 %-$ ")7;-". !"# /#54-
%-">,#, :).8 & )+$45.&)& 4)( xxf % (@7. 42). D/-"%6
& 34)( xxf %# -0$#" >$&)089 #;A*+)# ).* *, &
4)( xxf % -0$#" 1+5.6 ).* *. D$.*;.*"9, & )(xf ## %-$
-0$#" ?-."16 ).* *, #=*; 0)0( %##f .
(,-./0 10-234 @.& >5#="16 /#54).#)& .&9 )+$45.&)&9 3)( xxf % (@7. 41) /#5#.&5*;,- 3.", (#) ).* )&,-0* )0,0(O & >5#="16 /#54).#)& .&9 f :7-" -=#/.*,:$& 1#" (2) -1#.:58?-$ .*+ 00 %x , & 1+5.3.&.# .&9 1#,/;'&9 #''4F-".
y=x3
O x
y41
y=x4
O x
y42
2 !"#$%&"'%( )%*"(+%( 275
!"#$ %&'(%")*# +,"- ./0& 1"2 # 3'+425- %+'6*"+*# "#7 f “560%"&"+2” *"8
*#0&(8 )0,0(O . 98 *#0&(8 ! ./3&"+2 !"#$%& '(#)*+ ,"+ fC . :&$256 ;($8,0& "8$ %+'+56") 8'2*01.
-./01-0
<*") 02+ *,$6'"#*# f %+'+3)3(*20# *’ /$+ ;26*"#0+ ),( "# , 0& &=+('&*# (-*)7 /$+ *#0&(8 "8, 0x . >$
! # f &($+2 5,'"- *"8 ),( 0x# 5+2 58(.# *"8 ),( 0 "x , - +$"2*"'14)7, 5+2
! # fC /?&2 &4+%"80/$# *"8 *#0&(8 ))(,( 00 xfxA ,
"1"& "8 *#0&(8 ))(,( 00 xfxA 8$806@&"+2 !"#$%& '(#)*+ "#7 3'+425-7 %+'6-
*"+*#7 "#7 f.
A"+$ "8 ))(,( 00 xfxA &($+2 *#0&(8 5+0%-7 "#7 fC , "1"& ./0& 1"2 # f )(2&3!45-
6$4 !,& 0x '(#)* 5+2 "8 0x ./3&"+2 78!" !"#$%&3 '(#)*+. !"+ *#0&(+ 5+0%-7 # &4+%"80/$# "#7 fC “;2+%&'$6” "#$ 5+0%B.#. >%8;&25$B&"+2, &%2%./8$, 1"2:
9:;.<1=
>$ "8 ))(,( 00 xfxA &($+2 *#0&(8 5+0%-7 "#7 3'+425-7 %+'6*"+*#7 "#7 f
5+2 # f &($+2 ;,8 48'/7 %+'+3)3(*20#, "1"& 0)( 0 "## xf .
!B04)$+ 0& "8 %+'+%6$) C&D'#0+, "+ &*)"&'256
*#0&(+ &$17 ;2+*"-0+"87 $ *"+ 8%8(+ # f ## &($+2 ;2+48'&"25- +%1 "8 0#;/$ ;&$ &($+2 C/*&27 *#0&()$ 5+0%-7. E%80/$)7, % & ' & ( # ) * + ( * , - & + , . / - 0 1 ) 2 # / ' 3 + 02+7 *,$6'"#*#7 f *’ /$+ ;26-*"#0+ $ &($+2: i) ,( $!>,$24'5 !"#$%( ,&3 ! !,( &)&%( " f ## #"-
?$@%6$,(4, '(4 ii) ,( $!>,$24'5 !"#$%( ,&3 ! !,( &)&%( ?$@ 3)52-
A$4 " f ## (!?. 43).
:2+ %+'6;&230+, /*") # *,$6'"#*#
$%&
'()
"1,)2(
1,)(
4
3
xx
xxxf (!?. 44)
F f &($+2 ;B8 48'/7 %+'+3)3(*20# *"8 * }1{( 0&
$%&
+()
"##1,)2(12
1,6)(
2 xx
xxxf .
<"*2 /?8,0& "8$ %+'+56") %($+5+:
y=f (x)
x x1 x2 O
y43
y
O
Cf
2
1
1 x
44
2 !"#$%&"'%( )%*"(+%( 276
x !" 0 1 2 #"
)(xf $$ ! 0 + + 0 +
)(xf !"#$% !&'() !&'() !&'()
*+,-.) % f $$ /%.,0#1,(2- 3(2 3%/,#2 0 !2- 2, ,04 .,0 &+5'6,- 3(" 1, "- +-72089 783,-9 (:0 3%/,#:0 !2/+)9 ,#02- (2 3%/,#2 0, 1 !2- 2. ;/:9, <+:9 =2#0,(2- 3("0 +2'2+50: +#02!2 !2- 3(" 36)/2, (2 3%/,#2 1 !2- 2 .,0 ,#02- 783,-9 3%-/,#:0 !2/+)9, 2="> 3’ 2&(5 % f .,0 2$$51,- !&'(<(%(2, ,04 (" 3%/,#" 0 ,#02- 783% 3%/,#"& !2/+)9, 2="> 3(" ))0(,0( fO &+5'6,- ,=2+("/80% (%9 fC !2- % f
3(" 0 2$$51,- !&'(<(%(2. ?2'2(%'">/, $"-+<0 <(- 2+< (-9 +-72089 783,-9 3%-/,#:0 !2/+)9, 783% 3%/,#"& !2/+)9 ,#02- /<0" (" 0, ,!2(8':7,0 ("& "+"#"& %
f $$ 2$$51,- +'<3%/". @,0-!5:
A3(: /-2 3&05'(%3% f o'-3/80% 3’ 802 .-53(%/2 ),( !" !2- ),(0 !"x % . B0
& % f $$ 2$$51,- +'<3%/" ,!2(8':7,0 ("& 0x !2-
& "'#1,(2- ,=2+("/80% (%9 fC 3(" ))(,( 00 xfxA ,
(<(, (" ))(,( 00 xfxA ,#02- 3%/,#" !2/+)9.
,$#&+%*H
N! "#$%&'$#'%($)* +! &'!%+,-!+! %+! $"$.! / %0*1#+/%/
56)( 24 #!' xxxf ,
2.*!' 30#+, , 3$.4/ 3!' *! 5#2($)* +! %/-2.! 3!-",6 +/6 7#!8'3,6 +/6 "!-#1%+!%/6.
)-(.
i) C f ,#02- .>" ="'89 +2'2D:D#3-/% 3(" (, /, )1)(1(12)( #!'$$ xxxf . E" +'<-
3%/" (%9 f $$ =2#0,(2- 3("0 2!<$"&7" +#02!2:
x !" -1 1 +"f FF(x) + 0 ! 0 +
G.H. G.H.
0 0 f (x)
*+"/80:9, % f ,#02- !&'() 3, !27802 2+< (2 .-23()/2(2 ]1,( !!" !2- ),1[ #" !2- !"#$% 3(" .-53(%/2 ]1,1[! .
-
2 !"#$%&"'%( )%*"(+%( 277
!"#$%& ' f !! ('%#)*+#,-$ .,- .'(#*- 1,1" /-$ #/-,0123#) -445+#$ "16.'(7, ,-
.'(#*- )0,1("A /-$ )0,1(B #*)-$ .'(#*- /-("&8 ,'8 fC . 9- .:("#15.(-,- -:,5
/-,-;217<),-$ .,') ,#4#:,-*- =1-((& ,7: "-1-"5)2 "*)-/-.
#(',(-"(
!" #$!%!&
1. >- ?1#*,# ,- %$-.,&(-,- .,- 7"7*- 7$ "-1-/5,2 .:)-1,&.#$8 #*)-$ /:1,08 & /7*4#8 /-$ )- "17.%$71*.#,# (-) :"51;7:)) ,- .'(#*- /--("&8 ,2) =1-@$/A) ,7:8 "-1-.,5.#2)
i) 253)( 45 #"$ xxxf ii) 3
2 23)(
x
xxg
"$ .
2. B(7*28 =$- ,$8 .:)-1,&.#$8:
i) xxexf "$ 1)( ii) )5ln2()( 2 "$ xxxg
iii) %&'
(##")#"
$0,13
0,13)(
23
2
xxx
xxxh .
3. O(7*28 =$- ,$8 .:)-1,&.#$8:
i) 2
)( xexf "$ ii) *+,
-./"0$
2,
2,#@)(
!!xxxg
iii) ||)( xxxh $ iv) ||)( xx" $
v) 1%
1&'
()""$
0,
0,)(
xx
xxx# .
4. C,7 "-1-/5,2 .;&(- %*)#,-$ ' =1-@$/& "-15.,-.' ,'8 "-1-=A=7: (*-8 .:)51,'.'8 f .,7 %$5.,'(- ]10,1[" .
10
8 7 6 5 4
2
1 x
y=f$(x)
y
O -1
2 !"#$%&"'%( )%*"(+%( 278
!" #$%&'(%$)&*+* +" '("&+,-"+" &+" %#%)" . f *)/"( 0/.&)12 "34%5-&", 0/.&)12 67)/%5&", 85$+,, 8%)9. 8"( +(2 7:&*(2 +%#(8;/ "8$%+<-+1/ 8"( &.-*)1/ 8"-#,2.
5. =+% '(#9"/> &?,-" ')/*-+"( . 0$"6(8, #"$<&+"&. C +.2 &5/<$+.&.2 7:&*12
)(tSx ! */>2 8(/.+%3 #%5
8(/*)+"( #</1 &* :/"/ <-4%/". @/ . C #"$%5&(<A*( 8"-#, +(2 ?$%/(8:2 &+(0--:2 1t 8"( 3t , /" B$*)+*:
i) C>+* +% 8(/.+> 8(/*)+"( 8"+< +. 7*+(8, 6%$< 8"( #>+* 8"+< +./ "$/.+(8, 6%$<.
ii) C>+* . 8)/.&. +%5 8(/.+%3 *)/"( *#(+"?5/>-*/. 8"( #>+* *#(B$"-'5/>-*/..
!" #$%&%'
1. !" B$*)+* +" &.-*)" 8"-#,2 +.2 0$"6(8,2 #"$<&+"&.2 +.2 &5/<$+.-&.2:
1)(
2 "!
x
xxf
8"( /" "#%'*)4*+* >+( '3% "#> "5+< *)/"( &5--*+$(8< 12 #$%2 +% +$)-+%.
2. !" "#%'*)4*+* >+( . 0$"6(8, #"$<&+"&. +.2 &5/<$+.&.2: 22)( xexf !x #! #
:?*( 0(" 8<7* +(-, +%5 $! %, "8$(B;2 :/" &.-*)% 8"-#,2 #%5 B$)&8*-+"( &+./ #"$"B%9, 22 "#! xy .
3. !" "#%'*)4*+* >+( 0(" 8<7* )2,2(#$! . &5/<$+.&.
1262)( 234 """#! xxx!xxf *)/"( 85$+, &* >9% +% %.
4. D)/*+"( . &5/<$+.&. 23)( 23 "#! xxxf .
i) !" "#%'*)4*+* >+( . f #"$%5&(<A*( :/" +%#(8> -:0(&+%, :/" +%#(-8> *9<?(&+% 8"( :/" &.-*)% 8"-#,2.
ii) A/ 21 , xx *)/"( %( 7:&*(2 +1/ +%#(8;/ "8$%+<+1/ 8"( 3x . 7:&.
+%5 &.-*)%5 8"-#,2, /" "#%'*)4*+* >+( +" &.-*)" ))(,( 11 xfxA ,
))(,( 22 xfxB 8"( ))(,( 33 xfx" *)/"( &5/*57*("8<.
x=S(t)
x=S(t)
O
t2
t3 t1 t
2 !"#$%&"'%( )%*"(+%( 279
5. !"#$ f %&' "()*+#,",, -(. /.+01 2'+'3$34"&%, "#. ]2,2[! , 3&' #,) .2.4' &"567&
03)(2)( 22 "!#! xxfxf .
8' '2.-7497#7 :#& , f -7) 057& ",%74' ;'%2<1.
2.9 #(,+-./.0( - '#1%10( DE L’ HOSPITAL
A234567689
$$$$ !"#$ , "()*+#,", x
xf1
)( " (=5. 45).
>2$1 74-'%7:
#%""#&#& x
xfxx
1lim)(lim
00
.
?(#: ",%'4)7& :#&, ;'@A1 #. x #74)7& "#. 0 '2: @7#&;01 #&%01, , 3+'/&;< 2'+*"#'", #,1 f #74)7& )' "(%20"7& %7 #,) 7(@74' 0"x . =#,) 27+42#$-
", '(#< B0%7 :#& , 7(@74' 0"x 74)'& !"#"!$-
%&'( ")*+,#-#( #,1 fC . C7)&;*:
./012.1
?) 0)' #.(B*5&"#.) '2: #' :+&' )(lim0
xfxx #&
, )(lim0
xfxx !&
74)'& #% < !% , #:#7 ,
7(@74' 0xx " B037#'& !"#"!$%&'( ")*+,#-#( #,1 3+'/&;<1 2'+*"#'",1 #,1 f.
$ C&' #,) 4-&' "()*+#,", 2'+'#,+.6%7 :#&:
01
lim)(lim ""#%&#%& x
xfxx
.
?(#: ",%'4)7& :#&, ;'@A1 #. x #74)7& "#. #% , , 3+'/&;< 2'+*"#'", #,1 f #74)7& )' "(%20"7& %7 #,) 7(@74' 0"y . =#,) 27+42#$", '(#< B0%7 :#& , 7(-
@74' 0"y 74)'& 3%45$6#4" ")*+,#-#( #,1 fC "#. #% .
D24",1 2'+'#,+.6%7 :#&
01
lim)(lim ""!%&!%& x
xfxx
.
?(#: ",%'4)7& :#&, ;'@A1 #. x #74)7& "#. !% , , 3+'/&;< 2'+*"#'", #,1 f #74)7& )' "(%20"7& %7 #,) 7(@74' 0"y . =#,) 27+42#$", '(#< B0%7 :#& , 7(@74' 0"y
74)'& 3%45$6#4" ")*+,#-#( #,1 fC "#. !% . C7)&;*:
yx
"1
x
y
!
45
2 !"#$%&"'%( )%*"(+%( 280
!"#$%!$
!" !"#$%
)(lim xfx
(#"$%&$'()*+ ))(lim !"&$%
xfx
, $,$- . -/0-(# !"y 123-$#% &'(-
)*+,(- -./01,2,3 $.+ 34#5%67+ 8#49&$#&.+ $.+ f &$' #$ (#"$%&$'()*+ &$' &$ ).
' :&$* . &/"94$.&.
x
xxf1
1)( #&"
6#% . -/0-(#
1)( &" xxg (;). 46).
<8-%=7 01
lim)]()([lim ""&#$%#$% x
xgxfxx
, 6#0>+
$' x $-("-% &$' #$ , '% $%?2+ $.+ f 84'&-33(-@'/" $%+ $%?2+ $.+ g. A.1#=7, . 34#5%67 8#-49&$#&. $.+ f 84'&-33(@-% $." -/0-(#
1&" xy . ;$." 8-4(8$*&. #/$7 12?- ,$% . -/0-(# 1&" xy -("#% -./01,2,3
(1456(-) $.+ fC &$' #$ . B-"%69:
!"#$%!$
C -/0-(# !x"y #" 123-$#% -./01,2,3 $.+ 34#5%67+ 8#49&$#&.+ $.+ f &$'
#$ , #"$%&$'()*+ &$' &$ , #"
0)]()([lim "#&#$%
!x"xfx
,
#"$%&$'()*+ 0)]()([lim "#&
&$%!x"xf
x.
C #&D?8$*$. !x"y #" -("#% &'()*+,(- #" 0"" , -"> #" 0(" 123-$#% 145-
6(- #&D?8$*$..
B%# $'" 84'&=%'4%&?, $*" #&/?8$>$*" ?%#+ &/"94$.&.+ %&)D-% $' 8#4#69$* 0->4.?#, $'/ '8'('/ . #8,=-%E. 8#4#1-(8-$#%.
789":%;
C -/0-(# !x"y #" -("#% #&D?8$*$. $.+ 34#5%67+ 8#49&$#&.+ $.+ f &$' #$ ,
#"$%&$'()*+ &$' &$ , #" 6#% ?,"' #"
)"#$%
"x
xf
x
)(lim * 6#% )"&
#$%!x"xf
x])([lim *,
#"$%&$'()*+
)"&$%
"x
xf
x
)(lim * 6#% )"&
&$%!x"xf
x])([lim *.
$<!=#;
Cf
&1
f (x)&(x&1)
1 x
y
#
46
2 !"#$%&"'%( )%*"(+%( 281
1. !"#$%&'()%*+& ,*&:
— -& "#./0(/1&'23 4/(+5*64%&3 7+81#) 1%9+.)*%5#/ 6 :4#/ *#/ 2 $%( 2;#/( +4)1"*0*%3.
— -& 5<*23 4/(+5*64%&3 )(
)(
xQ
xP, 1% 7+81, *#/ +5&81<*6 )(xP 1%9+.)*%5#
*#/.=;&4*#( '+*= $)# *#/ 7+81#) *#/ "+5#(#1+4*6, $%( 2;#/( ".=9&%3 +4)1"*0*%3.
2. >)1?0(+ 1% *#/3 "+5+"=(0 #5&41#)3, +4)1"*0*%3 *<3 95+?&'63 "+5=4*+4<3 1&+3 4/(=5*<4<3 f +(+@<*#)1%:
— >*+ ='5+ *0( $&+4*<1=*0( *#/ "%$:#/ #5&41#) *<3 4*+ #"#:+ < f $%( #-5:@%*+&.
— >*+ 4<1%:+ *#/ "%$:#/ #5&41#) *<3, 4*+ #"#:+ < f $%( %:(+& 4/(%;63.
— >*# !" , #" , %?,4#( < 4/(=5*<4< %:(+& #5&412(< 4% $&=4*<1+ *<3 1#5-
?63 ),( !"! , +(*&4*#:;03 ),( !#" .
,$#&+%*H
N! "#$%&'( &) !*'+,-.-$/ -0/ 1#!2)34/ ,!#5*-!*0/ -0/ *6(5#-0*0/
xxxf
1)( !$ .
)-(.
A"%&$6 < f 2;%& "%$:# #5&41#) *# *! '+& %&(+& 4/(%;63 4’ +/*,, 8+ +(+@<*64#/-1% '+*+',5/?< +4)1"*0*< 4*# 0 '+& ".=9&%3 4*# #" '+& !" . A:(+&
!"$!%
)(lim0
xfx
'+& #"$#%
)(lim0
xfx
,
B5+, < %/8%:+ 0$x %:(+& '+*+',5/?< +4)1"*0*< *<3 95+?&'63 "+5=4*+4<3 *<3 f.
EC%*=@#/1%, *D5+, +( /"=5;%& 4*# !" +4)1"*0*< *<3 1#5?63 "x#y !$ . E-
;#/1%:
11
lim)(
lim2
2
$!
$!"%!"% x
x
x
xf
xx, #",*% 1$# '+&
01
lim1
lim))((lim2
$$&&'
())*
+#
!$#
!"%!"%!"% xx
x
xx#xf
xxx, #",*% 0$" .
2 !"#$%&"'%( )%*"(+%( 282
!"#$%&'(, ) *+,*-. xy ! *-&./ "012/. .34$"5'5) 5)( 26.7/89( ".6135.3)( 5)( f 35# "# . :&10#2. ;6-38#+$* <5/ ) *+,*-. xy ! *-&./ "012/. .34$"5'5) 5)( 26.7/89( ".6135.3)( 5)( f 8./ 35# $# .
',-.-/0 de L’ Hospital
=35' ) 3+&165)3) 3
1)(
x
exf
x $! . >/. &. *?*513#+$* .& ) *+,*-. 0!x *-&./ 8.-
5.8<6+7) .34$"5'5) 5)( fC , @6*/1A*5./ &. +"#0#2-3#+$* 5#
30
1lim
x
e x
x
$%
. (1)
B.6.5)6#4$* <5/, .& *7.6$<3#+$* 5#& 8.&<&. 5#+ #6-#+ ")0-8#+, ".6#+3/1A*-
5./ ."6#3C/#6/35-. 5)( $#679( 0
0. D/ $%,#C#/ "#+ *7.6$<3.$* 35# 8*710./#
5#+ #6-#+ 2/. 5)& 163) 5)( ."6#3C/#6/35-.( (."0#"#-)3) 850.) C*& *7.6$<A#-&5./ 35# "/# "1&' <6/#.
>/. 5. <6/. ")0-8#+ "#+ #C)2#4& 3* ."6#3C/<6/35*( $#67%( 0
0,
#&&#
, /3@4#+&
5. *"<$*&. ,*'69$.5. () ."<C*/?9 5#+( ".6.0*-"*5./), "#+ *-&./ 2&'351 '( !"#$#%& de l’ Hospital.
'()*+,- 1. (/.012 0
0 )
:& 0)(lim0
!%
xfxx
, 0)(lim0
!%
xgxx
, },{0 #"$#'(!x 8./ +"16@*/ 5# )(
)(lim
0 xg
xf
xx ))
%
("*"*6.3$%&# 9 1"*/6#), 5<5*:
)(
)(lim
)(
)(lim
00 xg
xf
xg
xf
xxxx ))
!%%
.
=53/ 5# ".6."1&' <6/# (1) +"#0#2-A*5./ '( *?9(: =@#+$*:
0)1(lim0
!$%
x
xe , 0lim 3
0!
%x
x
8./
"#!!))$
%% 2030 3lim
)(
)1(lim
x
e
x
e x
x
x
x
!"#$%&'(:
"#!$
% 30
1lim
x
e x
x,
"#+ 3)$.-&*/ <5/ ) *+,*-. 0!x *-&./ 8.5.8<6+7) .34$"5'5) 5)( 26.7/89( ".-
6135.3)( 5)( f.
2 !"#$%&"'%( )%*"(+%( 283
!"#$%&' 2( ()(*+, !!!!""""
"!"!"!"! )
!" "!#$
)(lim0
xfxx
, "!#$
)(lim0
xgxx
, },{0 !"%!&'!x #$% &'()*+% ,- )(
)(lim
0 xg
xf
xx ((
$
('+'+)$./0"- 1 ('+%)-), ,2,+:
)(
)(lim
)(
)(lim
00 xg
xf
xg
xf
xxxx ((
#$$
.
3%$ '$)(4+%5/$, - &'-6-5%./27 ,-& x
e x
x "!$lim 58"+,$% 97 +:17:
;*-&/+: "!#
"!$
x
xelim , "!#
"!$x
xlim
#$%
"!##((
"!$"!$ 1lim
)(
)(lim
x
x
x
x
e
x
e.
<'-/0"97:
"!#"!$ x
e x
xlim .
-./01'
1. =- >+?)@/$ 2 %.*A+% #$% 5%$ ,%7 /-)B07 !%
"!,
!"%!
, !%
%!.
2. =$ '$)$'("9 >+9)1/$,$ %.*A-&" #$% 5%$ '6+&)%#( 2)%$ #$% /'-)-A/+, $" *)+%(C+,$%, "$ ,$ +B$)/2.-&/+ '+)%..2,+)+7 B-)07, $)#+8 "$ '6@)-A",$% -% ')-D'->0.+%7 ,-&7.
,$#&+%*E(
1. 2345678 9 :;4<*69:9 1
42)(
"%"#
x
x
e
exxf . =7 7>(?58@653 A68:
i) % 5;B537 2"# xy 53478 7:C)>6D69 69E fC :6( %!
ii) % 5;B537 2-xy # 53478 7:C)>6D69 69E fC :6( "! .
#-%!,"./
i) !)#+8 "$ 4+8:-&/+ 2,%
0)]2()([lim #"%%!$
xxfx
.
2 !"#$%&"'%( )%*"(+%( 284
!"#$%&'(, )*+,%-
010
04
1
4lim)]2()([lim !
"#$
!"
$!"$
$%&$%& x
x
xx e
exxf .
ii) ."/-0 1& 2-03+,%- 4'(
0)]2()([lim !$$"%&
xxfx
.
!"#$%&'(, )*+,%-
01
4lim
1
44lim)]2()([lim !
"!'
(
)*+
,
"$!$$
"%&"%&"%& xxx
x
xx ee
exxf .
2. !" #$%&'() '* "+(,-./.%0 .10 2$"3*450 -"$6+."+10 .10 +7)6$.1+10
xe
xxf !)( .
),(-
56-(27 8 9,1#"'898 f -01&( 9,1-*7: 9'+ - 8 $"&;(/7 '8: 6&"#9'&98 2-1 )*-( /&'&/4",;-: &9<%6'='-:. >& &1&?8'79+,%-, -6+%)1=:, &9<%6'='-: 9'+ $% /&( 9'+ "% .
. @(& 1& -01&( 8 !x"y "! &9<%6'='8 '8: fC 9'+ $% , &"/-0 '& 4"(&
x
xf"x
)(lim
$%&! /&( ])([lim x"xf!
x$!
$%& 1& -01&( 6"&$%&'(/+0 &"(A%+0. 56-(27
"%!! $
$%&$%&
x
xxe
x
xflim
)(lim , 8 fC 2-1 )*-( &9<%6'='8 9'+ $% .
. @(& 1& -01&( 8 !x"y "! &9<%6'='8 '8: fC 9'+ "% , &"/-0 '& 4"(&
x
xf"x
)(lim
"%&! /&( ])([lim x"xf!
x$!
$%& 1& -01&( 6"&$%&'(/+0 &"(A%+0. B*+,%-:
01
lim)(
lim !!!"%&"%& xxx ex
xf" /&(
xxx e
xx"xf!
"%&"%&!$! lim])([lim
)(
)(lim
//
!"%&
01
234
5%"
"%
xx e
x (C&141&: De L’ Hospital)
01
lim !!"%& xx e
.
D"&, 8 -,A-0& 0!y , 28E&27 + #3+1&: xx/ , -01&( &9<%6'='8 '8: fC 9'+ "% .
2 !"#$%&"'%( )%*"(+%( 285
#(',(-"(
!" #$!%!&
1. N! "#$%&$ (!' ()*#+,(') &-. /!&!/0#(1$. !234)&5&$. &5' 6#!1-/7' )!#!2&*2$5' &5' 2('!#&82$5':
i) 2
1)(
!"
xxf ii) xxf $1)( " , #
$%
&'(!)
2,
2
!!x
iii) 1
23)(
2
!*!
"x
xxxf iv)
+,
+-
.
/
0"
0,1
0,
)(x
x
xx
xf .
2. N! "#$%&$ &-. ,#-90'&-$. !234)&5&$. &5' 6#!1-/7' )!#!2&*2$5' &5' 2('!#&82$5':
i) 1
1)(
2
2
***
"x
xxxf ii) xxxf !*" 1)( 2 .
3. N! "#$%&$ &-. !234)&5&$. &5' 6#!1-/7' )!#!2&*2$5' &5' 2('!#&8-2$5':
i) 1
2)(
2
!!!
"x
xxxf ii)
2
3)(
2
!!
"x
xxf iii) xxxf *" 2)( .
4. N! (),:,6%2$&$ &! )!#!/*&5 0#-!:
i) )1ln(
;4lim
0 *1 x
x
x ii)
4
2
0
2('1lim
x
x
x
!1
iii) x
xx
x 2('1
;4lim
0 !!
1.
B" #$!%!&
1. <%'$&!- ; 2('*#&;2; 22)( 2 **" xxxf /!- ,- $(=$%$. 1:1 !!" xy"
/!- 1:2 *" xy" . >! !),?$%@$&$ 0&-
i) H 1" $%'!- !234)&5&; &;. fC 2&, !2 , $'7 ; 2" $%'!- !234)&5&;
&;. fC 2&, *2 .
ii) A-! /*=$ )x 3 -2+3$- 0)1(22 22 4*/** xxx /!- 2&; 2('B+$-! '!
!),?$%@$&$ 0&- ; fC "#%2/$&!- )*'5 !)0 &;' 1" /,'&* 2&, !2 /!- )*'5 !)0 &;' 2" /,'&* 2&, *2 .
2 !"#$%&"'%( )%*"(+%( 286
2. !" #$%&'% '() "*+,-'.'%) '/) 0$"1(23) -"$4*'"*/) '/) f 5'"6:
i) x
xxf
2)(
2
! ii) x
xxf
ln)( ! .
3. N" #$%&'% '() '(,7) '.6 "!", #, 8*'% / *964$'/*/
$%&
'()
!0,
0,/,)(
xe
x"xxf
x!
6" %&6"( -"$"0.0&*(,/ *': 00 !x .
4. ;&6%'"( / *964$'/*/ *$
*%
&
!+
,-+!
1,1
10,1
ln
)(
x
xx
xx
xf .
!" "-:<%&=%'% 5'(:
i) / f %&6"( *96%>3) ii) 2
1)1( +!.f .
5. ;&6:6'"( :( *96"$'3*%()
*$
*%
&
!
,+
)+!
1"6,0
1"6,1
)22ln(
)(
2
x
xx
xx
xf 2"( *$
*%
&
')
(!
1"6,ln
1
1"6,
)(
2
xx
x
xx
xg .
!" "-:<%&=%'% 5'(:
i) ? f %&6"( *96%>3) 2"( -"$"0.0&*(,/ *': 10 !x , %68
ii) ? g %&6"( *96%>3) "@@4 ,/ -"$"0.0&*(,/ *': 10 !x .
6. ;&6%'"( / *964$'/*/
$%&
"+!
! + ]1,0(,ln)1(
0,0)(
xxe
xxf
x.
i) !" 9-:@:0&*%'% '" 5$("
x
e x
x
+
/
+1lim
0 2"( xx
xlnlim
0/
ii) N" "-:<%&=%'% 5'( / f %&6"( *96%>3) *': 0.
iii) !" #$%&'% '/6 %=&*.*/ '/) %1"-':,76/) '/) fC *': */,%&:
)0,0(O .
2 !"#$%&"'%( )%*"(+%( 287
2.10 +,),-. '#" /#�. -.( *&#$"'.(
1#&#(-#(.( +"#( (23#&-.(.(
!"#$ %&'()'&*+ &,"- .& /+012 %34, 12 "# 5+-.26& "7$ %8#'+*+'63$ %+, &%+-9"-:&12 1;<'6 "3'&, 1%+'+012 $& <&'(=+,12 "# )'&*69- %&'(:"&:# 16&4 :,-$('"#:#4 12 69&$+%+6#"69- &9'>526&. ? %+'2>& "#$ +%+>& &9+8+,.+012 8;)2"&6 !"#$%& %&' ()*+,%&(&' 9&6 %2'68&15($26 "& %&'&9("7 5-1&"&:
1- @'>:9+,12 "+ %2/>+ +'6:1+0 "#4 f.
2o E=2"(A+,12 "# :,$;<26& "#4 f :"+ %2/>+ +'6:1+0 "#4.
3- @'>:9+,12 "64 %&'&)3)+,4 f ! 9&6 f !! 9&6 9&"&:92,(A+,12 "+,4 %>$&924 "7$ %'+:-17$ "+,4. B2 "# 5+-.26& "+, %'+:-1+, "#4 f ! %'+:/6+'>A+,12 "& /6&-
:"-1&"& 1+$+"+$>&4 9&6 "& "+%69( &9'C"&"& "#4 f, 2$3 12 "# 5+-.26& "+, %'+-
:-1+, "#4 f !! 9&.+'>A+,12 "& /6&:"-1&"& :"& +%+>& # f 2>$&6 9,'"- - 9+>8#
9&6 5'>:9+,12 "& :#12>& 9&1%-4.
4- B282"+012 "# “:,1%2'6*+'(” "#4 :,$('"#:#4 :"& (9'& "7$ /6&:"#1("7$ "+, %2/>+, +'6:1+0 "#4 (+'6&9;4 "61;4, &:01%"7"24, 9"8.)
5- !,)92$"'3$+,12 "& %&'&%($7 :,1%2'(:1&"& :’ ;$& :,$+%"69C %>$&9& %+, 8;)2"&6 9&6 ./*010' !"%02-#3* %&' f 9&6 12 "# 5+-.26( "+, <&'(::+,12 "#
)'&*69- %&'(:"&:# "#4 f. D6& 9&80"2'# :<2/>&:# "#4 fC 9&"&:92,(A+,12 ;$&$ %>$&9& "613$ "#4 f.
456786
1) E%74 2>$&6 )$7:"C, &$ 16& :,$('"#:# f 12 %2/>+ +'6:1+0 "+ ! 2>$&6 "
# $ % &, "C"2 # fC ;<26 (=+$& :,112"'>&4 "+$ (=+$& yy ! , 2$3 &$ 2>$&6 ' ( # %$ $ ), # fC ;<26 9;$"'+ :,112"'>&4 "#$ &'<- "7$ &=C$7$ *. F%+1;$74, )6& "# 12-8;"# 16&4 ";"+6&4 :,$('"#:#4 1%+'+012 $& %2'6+'6:"+012 :"& Ax" , 12 0#x .
2) G$ 16& :,$('"#:# f 2>$&6 ' ( # % + , % - ) 12 %2'>+/+ ., "C"2 %2'6+'>A+,12 "# 128;"# "#4 fC :’ ;$& /6(:"#1& %8("+,4 ..
,$#&+%*E(
1. 90 !"#"%&:"/ 10; *0 .0,0(%0:"/ <,0=;1+ & ()*+,%&(&
114)( 34 $%& xxxf .
2 !"#$%&"'%( )%*"(+%( 288
),(-
1. H f !"#$ %#&'( ()$*+(, -( !.
2. . f #'/0$ *1/#"23 *-( ! 43 %(514/1+$62.
3. 7"(1+#
)3(4124)( 223 "#"#$ xxxxxf .
8$ )'9#3 -:3 f $ #'/0$ ($ 3#x , 0#x (&$%52)
60$ -( %);*:+; -:3 &'/(/-0$ *-( &$%50/; %'/0-60, 0%; -(/ (%('( %)(*&$()'9(1+# -0 &$0*-2-+0-0 +(/(-(/'03 60$ -0 -(%$6< 06);-0-0.
7"(1+# #%'*:3
)2(122412)( 2 "#"#$$ xxxxxf .
8$ )'9#3 -:3 f $$ #'/0$ ($ 0#x , 2#x 60$ -(
%);*:+; -:3 &'/(/-0$ *-( &$%50/; %'/060, 0%; -(/ (%('( %)(*&$()'9(1+# -0 &$0*-2+0-0 *-0 (%('0 : f #'/0$ 61)-2 2 6('5: 60$ =)'-*6(1+# -0 *:+#'0 60+%23.
4) . *1/<)-:*: f &#/ !"#$ 0*,+%-4-#3 *-( %& 60$ "& , 0>(, #'/0$ %(514/1-+$62 -!-0)-(1 =0?+(,. @'/0$ ;+43:
%&##%"%&'%&'
434 lim)114(lim xxxxx
60$ %&##%"
"&'"&'
434 lim)114(lim xxxxx
.
5) A":+0-'9(1+# -(/ %'/060 +#-0=(5B/ -:3 f 60$ "0)<**(1+# -: C)0>$62 %0-)<*-0*: -:3 f.
x "& 0 3 +&f D(x) " 0 " 0 +
f (x)
E.@.
"16
x "& 0 2 +&f DD(x) + 0 " 0 +
f (x)A.F. A.F.
11 "5
y
-5
-16
11
3 2 O x
47
x "& +&f D(x) " " " +
f DD(x) + " + +
f (x)
A.F.
A.F.
E.@.
0 0
0 0
3 0 2
11
"16
+& +&
"5
2 !"#$%&"'%( )%*"(+%( 289
2. !" #$%$&'($) *"+ ," -"."/&"($) 0."1+*2 ' /3,2.&'/'
1
4)(
2
!"!
#x
xxxf .
),(-
1. H f !"#$ %#&'( ()$*+(, -( $ }1{! .
2. . f #'/0$ *1/#"23 43 )5-2.
3. 6"(1+#
2
2
2
22
)1(
32
)1(
4)1)(12(
1
4)(
!!!
#!
!"!!!#
%
&&'
())*
+!"!
#%x
xx
x
xxxx
x
xxxf .
7$ )'8#3 -53 f % #'/0$ 3,1! 90$ -( %):-
*5+: -53 &'/(/-0$ *-( &$%;0/: %'/090, 0%: -(/ (%('( %)(*&$()'8(1+# -0 &$0-*-2+0-0 +(/(-(/'03 90$ -0 09):-0-0.
6"(1+# #%'*53
34
22
)1(
8
)1(
)32)(1(2)1)(22()(
!#
!!!!!!!
#%%xx
xxxxxxf .
. f %% &#/ !"#$ )'8#3 90$ -( %):*5+: -53 &'/#-0$ *-( &$%;0/: %'/090, 0%: -(/ (%('( %)(*&$()'-8(1+# -0 &$0*-2+0-0 *-0 (%('0 5 f #'/0$ 91)-2 2 9(';5.
4) <%#$&2 ",#"-
)(lim1
xfx
, !,#!-
)(lim1
xfx
, 5 #1=#'0 1#x #'/0$ 90-09:)1>5 0*,-
+%-4-5 -53 fC .
<?#-@8(1+# -A)0 0/ 1%@)"#$ *-( ", 0*,+%-4-5 -53 +()>23 !x"y "# . 6"(1-
+#:
14
lim)(
lim2
2
#!"!
#",-",- xx
xx
x
xf
xx, (%:-# 1#"
90$
01
4lim
1
4lim))((lim
2
#!
#&&'
())*
+!
!"!
#!",-",-",- x
xx
xxx"xf
xxx, (%:-# 0#! .
<%(+!/43, 5 #1=#'0 xy # #'/0$ 0*,+%-4-5 -53 fC *-( ", .
B/@;(C0 D)'*9(1+# :-$ 5 #1=#'0 xy # #'/0$ 0*,+%-4-5 -53 fC 90$ *-( !, .
<%'*53 !"(1+#:
x !, !1 3 +,f E(x) + 0 ! ! 0 +
f (x) F.G.F.<.
1
!3
5
x !, +,f EE(x) ! +
f (x)
1
2 !"#$%&"'%( )%*"(+%( 290
!"#!$!
#!"%!"% 1
4lim)(lim
2
x
xxxf
xx !"# $"#
!$!
#$"%$"% 1
4lim)(lim
2
x
xxxf
xx.
5) $%&'"()*+,'- (+. /)."!" '-("0+12. (&3 f !"# %"4566+,'- (& 74"8#!9 (&3 /"456("6&.
#(',(-"(
!" #$!%!&
1. N" '-1-(96-(- !"# ." /"4"6(96-(- 74"8#!5 (#3 6,."4(96-#3:
i) 1193)( 23 $!!# xxxxf ii) 1
1)(
!$
#x
xxf iii) 24 2)( xxxf !# .
2. O'+):3 (#3 6,."4(96-#3:
i) x
xxf1
)( $# ii) 1
2)(
2
!!!
#x
xxxf .
3. ;" '-1-(96-(- !"# ." /"4"6(96-(- 74"8#!5 (& 6,.54(&6&
xxxf &')( $# 6(+ <#56(&'" ],[ !!! .
O
5
-3
-4
3 1
-1
y=x
x=1
x
y48
x !" +"f =(x) + ! ! +
f ==(x) ! ! + +
f (x)>.?.
>.@.
0 0
3 !1 1
!3 5
!"
+" +"
!"
2 !"#$%&"'%( )%*"(+%( 291
*,-"',( #('.(,"(
!" #$%&%'
1. i) !" "#$%&'(&)& *)+ $+ ,-".+/01 #"-"2)32&+1 )45 265"-)72&45 x
xf1
)( !
/"+ 33)( 2 "#! xxxg , ),0( $"%x 08$65 /$+57 &."#)$905: 2)$ 2:9&'$ )1,1(A .
ii) !" ;-&')& ): 28&)+/7 <02: )45 fC /"+ gC 2)$ %+32):9" ),0( $" .
2. =5 gf , &'5"+ #"-",4,'2+9&1 265"-)72&+1 2)$ &, 9&
)0()0( gf ! /"+ )()( xgxf '(' ,+" /3<& %x &,
5" "#$%&'(&)& *)+
)()( xgxf ) 2)$ )0,(#$ /"+ )()( xgxf ( 2)$ ),0( $" .
3. >2$2/&?01 )-',45$ &'5"+ &,,&,-"9905$ 2& /@/?$ 9& "/)'5" 1. =5 ! &'5"+ : ,45'" 9&)"(@ )45 '245 #?&6-A5 )$6 )-+,A5$6, 5" "#$%&'(&)& *)+ )$ &9;"%*5 )$6 &'5"+ !!E :9)2651( "! . !" ;-&')& ):5 )+97 ):1 ,45'"1 ),0( "!% ,+" ):5 $#$'" &9;"%*5 )$6 )-+,A5$6 9&,+2)$#$+&')"+.
4. B5" 2@-9" 97/$61 20m %+")'<&)"+ ,+" ):5 #&-'.-"(: &5*1 "5<*/:#$6 287-9")$1 /6/?+/$@ )$90". !" ;-&')& ):5 "/)'5" r )$6 /@/?$6, "5 &#+<69$@9& 5" 08$69& ): 9&,"?@)&-: %65")7 &#+-
.35&+" )$6 /7#$6.
5. C@$ %+3%-$9$+ #?3)$61 1m )095$5)"+ /3<&)" (D879" ). !" ;-&')& )$ 9&,"?@)&-$ %6-5")* 97/$1 9+"1 2/3?"1 #$6 9#$-&', "5 9&)".&-<&' $-+E*-5)+", 5" 2)-'F&+ 2): ,45'".
()*+,-./:
i) !" &/.-32&)& )" #$, #% 265"-)72&+ ):1 ,45'"1 !, 2
0"! )) .
r
#
!
1m
1m
%
$
& #
' !
!
2 !"#$%&"'%( )%*"(+%( 292
ii) !" "#$%&'(&)& *)+ )(,-.
1
/01
)( !f!!
"# !"! .
iii) !" 12&')& )/. )+03 )/4 56.'"4 !, 5+" )/. $#$'" )$ "# 5'.&)"+ &78-9+,)$.
6. i) !" 0&7&)3,&)& :"+ ." #"2",)3,&)& 52";+:8 )/ ,-.82)/,/
x
xxf
ln)( ! .
ii) !" "#$%&'(&)& *)+ $$ $$ )1(1 "#" 5+" :8<& e$ # .
iii) !" "#$%&'(&)& *)+ 5+" 0#x +,9=&+ )2()(2 2 fxfxx !$! :"+ ,)/
,-.>9&+" ." "#$%&'(&)& *)+ / &(',6,/ 22 xx ! >9&+ %=$ ":2+1?4 7=,&+4, )+4 21 !x , 42 !x .
7. i) @. 0, #%$ :"+ 5+" :8<& %x & +,9=&+ 2'" xx %$ , ." "#$%&'(&)& *)+ 1!$% .
ii) @. 0#$ :"+ 5+" :8<& %x & +,9=&+ 1"' x$ x , ." "#$%&'(&)& *)+ e$ ! .
8. i) !" "#$%&'(&)& *)+ / ,-.82)/,/ xexf !)( &'."+ :-2)3, &.? /
xxg ln)( ! &'."+ :$'7/.
ii) !" 12&')& )/. &;"#)$0>./ )/4 fC ,)$ ,/0&'$ )1,0(A :"+ )/4 gC
,)$ )0,1(B .
iii) !" "#$%&'(&)& *)+:
") %"' xxe x ,1 & 1) ),0(,1ln ("%)* xxx .
A*)& +,9=$-. $+ +,*)/)&4; iv) B fC 12',:&)"+ #8.6 "#* )/. gC .
9. i) !" 12&')& )/. &789+,)/ )+03 )/4 ,-.82)/,/4
x&exf x )!)( , 0#& .
ii) !" 12&')& )/ 0&5"7=)&2/ )+03 )$- 0#& 5+" )/. $#$'" +,9=&+
x&ex ' , 5+" :8<& !%x .
iii) C+" )/. )+03 )$- & #$- <" 12&')& #"2"#8.6 ." "#$%&'(&)& *)+ /
&-<&'" x&y ! &;8#)&)"+ )/4 52";+:34 #"28,)",/4 )/4 ,-.82)/-
,/4 xexg !)( .
10. D'.&)"+ / ,-.82)/,/ +,
+-.
!
/!
0,0
0,1/0
)(2
x
xx
xxf .
2 !"#$%&"'%( )%*"(+%( 293
y=!x2
y=x2
O x
y
!" "#$%&'(&)& *)+
i) , f &'-"+ #"."/0/'1+23 1)$ 00 "x 4"+ 1)3 15-67&+" *)+ 3 &58&'"
0"y ($ 9($-": xx# ) &'-"+ 3 &;"#)$26-3 )3: fC 1)$ )0,0(O .
ii) < 9($-": xx# 67&+ 2& )3- fC 9#&+." 4$+-9 132&'", #".*=$ #$5
&;9#)&)"+ )3: fC .
iii) , &58&'" xy " &'-"+ "1>2#)0)3 )3: fC 1)$ $% 4"+ 1)$ !% .
11. A. ?1)0 2+" 15-9.)313 ! )6)$+", @1)&
0)0( "! , 0)0( "#! 4"+ 0)()( "$## x!x! /+" 498& &x ' (1)
!" "#$%&'(&)& *)+:
i) , 15-9.)313 22 )]([)]([)( x!x!x" $#" &'-"+ 1)"8&.A 1)$ ' 4"+ -"
B.&')& )$- )>#$ )3:.
ii) 0)( "x! /+" 498& &x '.
!. ?1)0 %>$ 15-".)A1&+: f 4"+ g )6)$+&: @1)&:
0)0( "f , 1)0( "#f 4"+ 0)()( "$## xfxf /+" 498& &x '
1)0( "g , 0)0( "#g 4"+ 0)()( "$## xgxg /+" 498& &x '.
!" "#$%&'(&)& *)+:
i) <+ 15-".)A1&+: xxfx! 32)()( !" 4"+ xxgx" 15-)()( !" +4"-$#$+-$>- )+: 5#$861&+: (1) )$5 &.0)A2")$: C.
ii) xxf 32)( " 4"+ xxg 15-)( " /+" 498& &x '.
2 !"#$%&"'%( )%*"(+%( 294
12. !"# $%&'()* +,-.( # /0/'#1 2,3% (/"4)( 1cm /(% 5 ! 367&"3"(% +3 (8-"*) +"# +5.34# ". 9# "*:# "# 34)(% $ rad /(% "# 38;. ".-.( "% 34)(% $ cm. < 38;34( #% "2-.)3% "#) 7:#)( xx! +"# +5.34#
)0,(xP . =( $34:3"3 *"%:
i) )(5.5.+8) $x$$$$$
x "##
"
ii) 2)(lim0
#"$
$x$
.
13. >)(1 &3?#&*@#1 & :3/%)73% (&* 2)( +5.34# " /(% A($4?3% B0@C (&* .%( /8/'%/- '4.)5 (/"4)(1 2"' km .3 "(,0"5"( 4"( km/h. A) S 34)(% "#
.-/#1 "#8 "*:#8 "& /(% % "# .--
/#1 "51 (&*+"(+51 "& "#8 &3?#&*-@#8 (&* "# +5.34# 3//4)5+51 "5 ,@#)%/- +"%.- t:
A) =( (&#$34:3"3 *"%
i) 2
S$ " /(% 2
5.4$
"% , ii) tS 4" , t$ 2" /(% t5.4"% .
D) =( A@34"3 "# @8;.* .3"(A#'-1 "51 (&*+"(+51 % . E#%#1 34)(% #
@8;.*1 .3"(A#'-1 "51 (&*+"(+51 % , *"()
() 3
2)$ " , A) )$ " /(% B) 3
4)$ " ;
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
,&-./(,"( '#.#0%/(/(
y
*
#
$ x
"(1,0)
N(1,$)
P(x,0)
s l
& 4 km/h
2 km * $
"
2 !"#$%&"'%( )%*"(+%( 295
!. "# $%&#'() %*+ ,(- *%.%$),/ *#.(*,01#(- 2% $3$401#,# ,5 6.)''% 7,
%2 5 (183.(1'+- #92%( %4:&;- $%( ,5 6.)''% <, %2 5 (183.(1'+- #92%( =#3>;- >($%(545602,%- 1368.+2/- ,:2 %*)2,:1: 1%-.
1. !" # $%"&'(#$# f )*"+, $%")-./ $(0 ]1,0[ , 1+'+232*$,4#
$(0 )1,0( 5+, 0)( !" xf 2,+ 67+ (+ )1,0(#x , (6() )1()0( ff ! .
2. !" # $%"&'(#$# f 1+'+232*8)(+, $(0 ],[ !" 4) )()( "f!f $ , (6() %1&'-), ),(0 !"x # (9(0,0, :$()
0)( 0 $" xf .
3. !" 0, gf , )*"+, $%"+'(.$),/ 1+'+232*$,4)/ $(0 ],[ !" , 4) )()( "g"f % 5+, )()( !g!f % , (6() %1&'-), ),(0 !"x # (9-
(0,0, :$() $(+ $#4)*+ ))(,( 00 xfxA 5+, ))(,( 00 xgxB 0, );+1(64)")/ "+ )*"+, 1+'&77#7)/.
4. !" )2()1()( 2 &&%" xxxf 2,+ 5&<) #x ', (6():
+) (0 )1(f )*"+, (01,56 492,$(0 (#/ f A =
>) (0 )2(f )*"+, (01,56 )7&-,$(0 (#/ f ! =
5. +) ? 2'+;,5. 1+'&$(+$# 4,+/ 107%3"%4,5./ $%"&'(#$#/ &'(,0% >+<40@ 9-), 1&"(0() 0',86"(,+ );+1(049"#.
>) ? 2'+;,5. 1+'&$(+$# 4,+/ 107%3"%4,5./ $%"&'(#$#/ 1)',((0@ >+<40@ 9-), 1&"(0() 0',86"(,+ );+1(049"#.
6. ? $%"&'(#$# #x$x!x"xf (((% 23)( 4) ##$!" ,,, ' 5+, 0!" 9-), 1&"(+ 9"+ $#4)*0 5+41./.
7. !" 0, $%"+'(.$),/ gf , 9-0%" $(0 0x $#4)*0 5+41./, (6() 5+, # gfh )% 9-), $(0 0x $#4)*0 5+41./.
8. A*")(+, 6(, # $%"&'(#$# f 1+'+232*8)(+, $(0 ! 5+, 6(, # 2'+;,5. (#/ 1+'&$(+$# )*"+, 1&"3 +16 (0" &B0"+ xx" .
!" %1&'-), 5&10,0 $#4)*0 ))(,( 00 xfxA (#/ fC (0% 010*0%
# +16$(+$# +16 (0" &B0"+ xx" )*"+, 492,$(# (. )7&-,$(#),
(6() $) +%(6 (0 $#4)*0 # );+1(049"# (#/ fC )*"+, 0',86-
"(,+.
! =
! =
! =
! =
! =
! =
! =
! =
2 !"#$%&"'%( )%*"(+%( 296
9. ! "#$"%& 1!x "%'&( )&*&)+,#-. &/012*3*. *.4 5,&-()64 2&,7/*&/.4 *.4 /#'7,*./.4:
&) 1
23)(
2
"#"
!x
xxxf 8 9
:) 2
2
)1(
23)(
"#"
!x
xxxg 8 9
10.8' 5,&-()6 2&,7/*&/. *.4 /#'7,*./.4 f ;%'"*&( &2+ *< 2&,&)7*3 /=61&, *+*":
4 1 O x
y
i) *< 2";%< <,(/1<0 *.4 f $1
"%'&( *< )4,1( 8 9
ii) *< 2";%< <,(/1<0 *.4 f $1
"%'&( *< ]4,1[ 8 9
iii) 0)( %$ xf 5(& )7$" )4,1(&x 8 9
iv) #27,="( 0)(:)4,1( 00 !$& xfx . 8 9
11. ! /#'7,*./. 1)( 3 ##! xxxf >="(: &) 1(&, *<#?7=(/*<', ,%@& /*< )1,0( 8 9
:) 1(&, &),(:A4, ,%@& /*< )0,1(" 8 9
5) *,"(4 2,&51&*()>4 ,%@"4 8 9
12. 8' 5(& *(4 2&,&535%/(1"4 /*< ' /#'&,*6/"(4 gf , (/=0-
<#' 4)0( !f , 3)0( !$f , 6)5( !$f , 5)0( !g , 1)0( !$g ,
2)4( !$g , *+*" )0()()0() $!$ fggf (( 8 9
!I.
"# $%&#'() %*+ ,(- *%.%$),/ *#.(*,01#(- 2% $3$401#,# ,5 1/1,6
%*)2,515
1. B< h
!h
!
h
6"-
6"-
lim0
")*+
,-. #
/ (/<0*&( 1":
2 !"#$%&"'%( )%*"(+%( 297
!) 3
3 ")
3
4 #) 3 $) 0 %)
4
3.
2. &' h
xhxh
11
lim0
!"
# ()'*+,( -.:
!) 2
1
x ")
2
2
x! #)
2
1
x! $)
x
2! %) 0
3. !/ xxf 35)( $ +0+. 1 )(xf % ()'*+,( -.:
!) 1353 !xx ") 5ln3
53x
#) x253 &
$) x353 & %). 125ln53x
4. !/ )1()2/)( 3 "$ xxf +0+. 1 )(!f % ()'*+,( -.:
!) )1(1-)1()2/3 3 "" !! ") )1()2/3 2 "!
#) )1(1-)1()2/3 2 ""! !! $) )1()2/3 2 "!!
5. !/ 32 )1()( !$ xxf +0+. 1 345'-1 6,789:9'; ,2+<; )+' 0 ()'*+,( -.:
!) 1 ") 1! #) 0
$) 27 %) 5./ 2687=.(.
6. !/ '( .>,6+0-./.; +:/ )2/,7+<).:/ xxf ln)( $ ?,( 22)( xxg $
)+, )1-.@, -. +.+-1-3/1 0x .@/,( 6,78AA1A.;, +0+. +' 0x .@/,(:
!) 0 ") 4
1 #)
2
1 $) 1 %) 2.
7. !/ x"x# exgexf $$ )(,)( ?,( )(
)(
)(
)(
xg
xf
xg
xf
%%
$%
''(
)**+
,, +0+. +' # :; )2/87-
+1)1 +'2 " ()'*+,( -.:
!) 2
1
"" !
") 1
2
"""
#) 2
1
"" "
$) 12
2
!""
%) 1
2
!""
.
8. !/ 0)( -% xf 9(, ?8B. ]1,1[!.x ?,( 0)0( $f , +0+.:
!) 1)1( !$f ") 0)1( -!f
#) 0)1( -f $) 0)1( $!f .
!!!.
2 !"#$%&"'%( )%*"(+%( 298
1. !" "#$%&$'%()&*$* +",*-%. "/0 $%1 &2#"3$4&*%1 ", 5, 6, 7 &* *+*)#8 "/0 $%1 &2#"3$4&*%1 9, :, ;, <, =, Z /'2 #'-)>*$* 0$% *)#"% 8 /"3.6?-601 $81.
____________________________________________________________
(a)
x
y
1
O x
y (!)
O x
y (")
O x
y (#)
_______________________________________________________
O x
y ($)
O x
y (%)
O x
y
-1
(& )
O x
y (')
O x
y (()
O x
y ())
2. @",*-%. "/0 $%1 /"3"+.$? &2#"3$4&*%1 #" "#$%&$'%()&*$* &$8# *2,*)" /'2 *)#"% "&A-/$?$8 $81 63"B%+41 $81 /"3.&$"&81 &$' !" .
2 !"#$%&"'%( )%*"(+%( 299
!"#$%&'!' $!"()&*&'
1. 2
1)(
xxxf !" !. 2"y
2. xe
xxf1
1)( !!#" ". 1#" xy
3. 2
32)(
#!"
xxf #. 1!#" xy
$. xy "
%. xy #"
2 !"#$%&"'%( )%*"(+%( 300
"(,%&"'% (-+."/+#
- 011234 567 8494:;:2<
!" #$%#&'" ())*+,- '+./#0#+ ,1#23'/4+* /"#- 5#/26)*- 3*+ ,78,&# 2'7 4%," 4+# /.+' 5'"+. 9*/,&' /’ #73:+, %;$&- +# 3*+ 34/+," 5#" 3*+ 5#3#-95,6#0#+ /, <=9* >,;/,3$"54- "?".3*3,- 2'7 #2'$$4'7+ #2’ #73.+ 3'+ '$"9/.. (39" :3#+ >+;93.- ' 3$.2'- 5#3#95,7:- ,1#23'/4+;+ 93'+ 56-5)' 5#" 3"- 5;+"54- 3'/4- (4)),"@*, 2#$#<'):, 72,$<'):). A2&9*-, /, 2$'917>: 9, 5"+*/#3"54- /,8.?'7-, ' B$%"/:?*- ,&%, ,2"+':9," /48'?' 5#3#95,7:- 3*- ,1#23'/4+*- /"#- 5#/26)*- 2'7 ,&+#" 9:/,$# >+;93: ;- “4)"5# 3'7 B$%"/:?*”. C ,2./,+* ,D4)"D* 93' 0:3*/# #73. 4>"+, 93"- #$%4- 3'7 17'7 #"E+#, .-3#+ =$%"9, * 9793*/#3"5: ,1#$/'>: #)>,<$"5E+ /,8.?;+ 93* >,;/,3$&#. F' ,2./,+' 2#$=?,">/# ?,&%+," 3'+ 3$.2' /, 3'+ '2'&' * G)>,<$# ,1#$-/.0,3#" 93'+ 2$'9?"'$"9/. 3*- ,1#23'/4+*- /"#- 2#$#<'):-.
(93; 2)( xxfy !! * ,D&9;9*
/"#- 2#$#<'):- /, 5'$71: 3*+ #$%: 3;+ #D.+;+ 5#" ),( 00 yxM
4+# 9*/,&' 3*-, 93' '2'&' 0*3,&3#" +# 5#3#95,7#93,& /"# ,1#23'/4+* !. C 5#3#95,7: #73: /2'$,& +# >&+," #+ 2$'9?"'$&9'7/, 4+# =))' %#$#53*$"93"5. 9*/,&' 3*- !, .-2;- 2.%. 3' 9*/,&' " 93' '2'&' 34/+," 3'+ =D'+# 3;+ 3,3/*/4+;+. H,;$'6/, 4+# =))' 9*/,&' 3*- 2#$#<'):-, 3' ),( 11 yxN , 2')6 >,"3'+"5.
3'7 #, 343'"' E93, hxx "! 01 (3' h 8,;$,&3#" ,?E /"# #2,"$',)=%"93*
/,3#<'): 3'7 0x ). I3*+ 2,$&23;9* #73: 3# '$8'>E+"# 3$&>;+# #$" 5#" %&" /2'$'6+ +# 8,;$*8'6+ 5#3= 2$'94>>"9* ./'"# 5#" =$# 8# "9%6," 5#-
3= 2$'94>>"9* * #+#)'>&# "$#$
N #$!
#. B+ 849'7/, s"$ ! , 3.3, ?"#?'%"5=
8# "9%6,":
s
hs
y
y "!
0
1 : %&'
()* "!
s
hyy 101 :
s
hyyy 001 !+ :
s
y
h
yy 001 !+
. (1)
F' 2$E3' /4)'- #73:- 3*- 5#3= 2$'94>>"9* "9.3*3#- >$=1,3#":
hxh
xhhxx
h
xhx
h
xfhxf
h
xfxf"!
+""!
+"!
+"!
+0
20
20
20
20
200001 2
2)()()()()(
5#" 439" * (1) >&+,3#" s
yhx 0
02 !" . B+ 3E$# 849'7/,, .2;- '" /#8*/#3"-
5'& 3'7 17'7 #"E+#, 0!h <$&95'7/, #2. 3*+ 3,),73#&# .3" s
yx 0
02 ! :
y
x1=x0+h
x &(x1)
P(x0)
! %
#
T O
y0
y1
2 !"#$%&"'%( )%*"(+%( 301
0
0
2x
ys ! . !"#$%&'"()* +',-." (' /012%' 2-)34* ),( 00 yxM , -$'/5,'$%&'6-
12 )-. (0" (2+26()%) (' 147'* sTP ! -'6 1)* 5%"2, )18/#* (' /012%' !.
9 26:2%) "! 2%"), 0 &0(';12"0 23)-('18"0 (0* -)$)<'+4*. 9 -$'0=';-12"0 5,)5,7)/%) 4()" 8")* )-. ('6* 5$.1'6* -'6 '54=0/)" ,/('$,7>, /(0" 8""',) (0* -)$)=?='6.
',-.-/0 1,2,3435670 @(' 52;(2$' 1,/. ('6 17'6 ),?"), ', 1):01)(,7'% 2%A)" 7)('$:?/2, ") 12()/A01)(%/'6" .+0 (0 1)7$'/72+4 5,)5,7)/%) -)$)=?=,/0* /2 23)$-1'=4 '$,/18"#" 7)"."#" 7), (;-#", 12 (0 <'4:2,) 7)(>++0+) 2-,+2=18-"#" /61<.+#". B$#('-.$', -$'* )6(4" (0" 7)(2;:6"/0 6-4$C)" ', I. Newton 7), ' G. Leibniz. O Leibniz /61<.+,&2 (0" )-2,$'2+>A,/(0 12()-<'+4 1,)* -'/.(0()* x 12 dx (5,)3'$,7. ('6 x). 8(/,, -.A. =,) (0 /6">$(0-
/0 2xy ! ('6 -$'0=';12"'6 -)$)52%=1)('*, 0 )"(%/(',A0 12()<'+4 ('6 y
(5,)3'$,7. ('6 y) 4()":
2222222 )(2)(2)()( dxxdxxdxxdxxxdxxxddy "!#""!#"!! .
B)$)+2%-'"()* (0" -'+; 1,7$4 (/6=7$,".12"0 12 (,* >++2*) -'/.(0() 2)(dx -$'876-(2 0 xdxdy 2! (25? 0 -)$>=#='* x2 '"'1>&'"()" “5,)3'-
$,7.* /6"(2+2/(4*”) 7), (2+,7> 0 xdx
dy2! , 8")* /61<'+,/1.* -'6 5,)(0-
$2%(), 18A$, /412$), A#$%* .1#* ") 8A2, ".01) -0+%7'6. D2 ('" ($.-' )6(. ' Leibniz )-852,C2 (' 1677 ('" 7)".") =,) ('" 6-'+'=,/1. (0* 12-()<'+4* ('6 =,"'18"'6 5;' 12()<+0(?" x 7), y, -'6 )-'(2+2% 1,) “-$#(.-='"0” 1'$34 ('6 /012$,"'; 7)".") (0* -)$)=?='6 2".* =,"'18"'6 /6-")$(4/2#"
xydyydxxxyd #""! ))(()(
xydxdyydxxdyxy #"""!
dxdyydxxdy ""! .
B)$)+2%-'"()* 7), 25? (0" -'+; 1,7$4 -'/.(0() dxdy , -)%$"'612 (0
/A8/0
ydxxdyxyd "!)( .
D2 (0" 2,/)=#=4 7), 7):,8$#/0 )6(?" (#" 7)"."#" 7), /61<'+,/1?", 0 8""',) (0* -)$)=?='6 2C2+%A:072 /’ 8") 2C),$2(,7> )-'(2+2/1)(,7. 2$-=)+2%' 7), 5,2;$6"2 /2 12=>+' <):1. (,* 23)$1'=8* (0* 1):01)(,74* )-">+6/0*. B)$>++0+) .1#*, ', )/>32,2* -'6 2-,/01>")12 )-'(2+';/)" 1,) 5,)$74 -$.7+0/0 =,) ('6* 1):01)(,7';* -'6 )"(,12(?-,&)" 12 7$,(,-7. -"2;1) () :218+,) (0* 2-,/(410* ('6*. E -$?('* )6/(0$.* '$,/1.* )6(4* (0* 8""',)*, -'6 /(0$%&2(), /(0" 8""',) ('6 '$%'6, 5.:072 =,) -$?-(0 3'$> (' 1823 )-. ('" A.L. Cauchy:
2 !"#$%&"'%( )%*"(+%( 302
“!"#$ % &'$()"%&% )(xfy ! *#)#+,$-. &'$-/01 &’ ,$# 2.(&"%+# "%1 +-"#-
34%"01 x 5#. 267-8 &’ #'"0 "% +-"#34%"0 +.# ".+0 *6' #$05-. &’ #'"9 "6 2.(&"%+#, "9"- 5(7- #*-.)6-4(/.&"% #:;%&% "%1 +-"#34%"01 *#)(<-. +.# #*-.)6-4(/.&"% #:;%&% "%1 &'$()"%&%1. ='$-*>1, #$ "-7-8 ix !" , "9"- 6. 2'6 9)6. "6' *%4856' 2.#?6)>$
i
xfixf
x
y )()( #$!
""
7# -8$#. #*-.)6-4(/.&"-1 *6&9"%"-1. @44( -$> #'"68 6. 2'6 9)6. 7# *)6-&-<<8A6'$ -*’ (*-.)6$ 5#. "#'"9/)6$# "6 9).6 +%2,$, "6 *%4856 +*6)-8 $# &'<548$-. *)61 5(*6.6 (446 9).6, 7-".59 0 #)$%".59, @'"9 "6 9).6, 9"#$ '*()/-. ,/-. +.# 6).&+,$% ".+0 <.# 5(7- &'<5-5).+,$6 x, #44( +-"#3(44-"#. +#A8 +- "6 x.
B +6)?0 "%1 $,#1 &'$()"%&%1 *6' 7# -5?)(A-. "6 9).6 "6' 49<6'
i
xfixf )()( #$
7# -;#)"("#. #*9 "% +6)?0 "%1 26&+,$%1 &'$()"%&%1 )(xfy ! .
C.# $# ;-/D)8&6'+- #'"0$ "%$ -;()"%&%, 28$6'+- &"% $,# &'$()"%&% "6 9-
$6+# ,-./01023 456/.7848 5#. "% &'+3648A6'+-, +- "% 3607-.# -$91 "9-
$6', y % 0 )(xf % ”.
!" #$"%&'(# #)%*+ %,+ ,'-./*, , Cauchy )0,1*2-." %-3 0#'#242,)3 %5+ 6#.-74+ .)+#'%8."5+ 7#- #09:"-;" %,)3 7#+*+"3 %&3 0#'#242-.&3. <.=. 2-# %,+ -:-#(%"'# .&/#+%-7* 7#+*+# %&3 0#'#242,) /-#3 .>+?"%&3 .)+@'-%&.&3, 9:5." %&+ #7*1,)?& #0*:"-;&:
“E&"D z +.# 2-:"-)% &'$()"%&% "6' x, &'$2-9+-$% +- "%$ *)>"% )(xfy !
+,&D "6' ":*6' )(yFz ! . B z 0 )]([ xfF -8$#. #'"0 *6' 6$6+(A-"#. &'$()-
"%&% +.#1 &'$()"%&%1 "%1 +-"#34%"01 x 5#. #$ 6. #*-.)6-4(/.&"-1 5#. "#'-"9/)6$-1 #';0&-.1 "D$ yx, 5#. z &'+364.&"6:$ +- x" , y" , z" #$"8&"6.-/#, "9"- 7# -8$#.
x
y
y
yFyyF
x
yFyyF
x
z
""
""
""
""
&#$
!#$
!)()()()(
. (1)
@*9 #'"0$, *-)$>$"#1 &"# 9).#, ,/6'+-
)()]([)( xfxfFyyFz %&%!%&%!% ”(*)
(*) A+# #:>+#%, .&/"(, #)%83 %&3 #0*:"-;&3, 0,) #$,'@ %&+ -.*%&%# (1), "(+#- *%- 2-# /-7'93, /& /&:"+-793 %-/93 %,) x" , /0,'"( +# -.=>"- 0)()( !#$! xfxxfy "" .
!"!#"$%&'(#!) "!*()+!)
3.1 ,!%()'! !"!#"$%&+,
,-./01 2345-6727
!"##$% &"'$% ()*+ ,'-.* ,/'"0(1-2"+)/1 ,'"3#45/)/, ,"0 * #6(* )"0% /,/1)78 ,"'78/ /+)8()'"&* )*% ,/'/9:91(*%. ;$)"1/ ,'"3#45/)/ 78+/1 91/ ,/'-<7195/
)/ ,/'/=-)>:
— ? 76'7(* )*% @$(*% )(tS 7+A% =1+*)"6 )* B'"+1=4 ()1954 t, /+ 78+/1 9+>()4 *
)/B6)*)- )"0 )(t! ,"0, A,>% 9+>'82"057, 78+/1 * ,/'-9>9"% )*% (0+-')*(*% @$(*% )(tSx ! .
— ? 76'7(* )*% )/B6)*)/% )(t! 7+A% =1+*)"6 )* B'"+1=4 ()1954 t, /+ 78+/1 9+>-
()4 * 7,1)-B0+(4 )"0 )(t" ,"0, A,>% 9+>'82"057, 78+/1 * ,/'-9>9"% )*% (0+-'-
)*(*% )(t!! ! .
— ? 76'7(* )"0 ,#*@0(5"6 )(tN 51/% ="1+>+8/% 3/=)*'1<8>+ )* B'"+1=4 ()19-54 t, /+ 78+/1 9+>()A% " '0@5A% /6.*(*% )(tN " )"0 ,#*@0(5"6.
;" ="1+A B/'/=)*'1()1=A )>+ ,'"3#*5-)>+ /0):+ 78+/1 A)1, <8+7)/1 51/ (0+-'-
)*(* f =/1 2*)78)/1 +/ 3'7@78 51/ -##* (0+-')*(* F 91/ )*+ ","8/ +/ 1(B671 )()( xfxF !" (7 $+/ <1-()*5/ #. C<*9"65/()7 $)(1 ()"+ ,/'/=-)> "'1(5A.
!"#$%!$
D()> f 51/ (0+-')*(* "'1(5$+* (7 $+/ <1-()*5/ #. &'()*+ ,-./'01,1 4
23'/45-,3 016 f ,05 8(1) "+"5-27)/1 =-@7 (0+-')*(* F ,"0 78+/1 ,/'/9>-
98(15* ()" # =/1 1(B671
)()( xfxF !" , 91/ =-@7 #x# .
(1) E,"<71=+67)/1 A)1 =-@7 (0+7B4% (0+-')*(* (7 <1-()*5/ # $B71 ,/'-9"0(/ ()" <1-()*5/ /0-
)A.
3
3 O!"#!$%&'(#") !"*()+") 304
!"# $#%&'(")*#, + ,-.&%/+,+ 3)( xxF ! (0.#" *"# $#%&)1-,# /+2 23)( xxf ! ,/1
", #314 23 3)( xx !# . 5#%#/+%14*( 6/" 7#" 68(2 1" ,-.#%/9,("2 /+2 *1%392
cxFcxxG $!$! )()( 3 , 6$1- %c ", (0.#" $#%&)1-,(2 /+2 f ,/1 ", #314
23 3)( xcx !#$ . !(."7& ",:4(" /1 $#%#7&/; <(=%+*#:
!"#$%&'
>,/; f *"# ,-.&%/+,+ 1%",*?.+ ,( ?.# '"&,/+*# !. @. F (0.#" *"# $#%&)1--
,# /+2 f ,/1 !, /6/(
& 68(2 1" ,-.#%/9,("2 /+2 *1%392
cxFxG $! )()( , %c ",
(0.#" $#%&)1-,(2 /+2 f ,/1 ! 7#"
& 7&<( &88+ $#%&)1-,# G /+2 f ,/1 ! $#0%.(" /+ *1%39
cxFxG $! )()( , %c ".
,-"./(0$
& A&<( ,-.&%/+,+ /+2 *1%392 cxFxG $! )()( , 6$1- %c "', (0.#" *"# $#%&)1--
,# /+2 f ,/1 !, #314
)()())(()( xfxFcxFxG !#!#$!# , )"# 7&<( !x% .
& >,/; G (0.#" *"# &88+ $#%&)1-,# /+2 f ,/1 !. B6/( )"# 7&<( !x% ",:41-. )()( xfxF !# 7#" )()( xfxG !# , 1$6/(
)()( xFxG #!# , )"# 7&<( !x% .
C%#, ,4*3;.# *( /1 $6%",*# /+2 § 2.6, -$&%:(" ,/#<(%& c /?/1"#, =,/(
cxFxG $! )()( , )"# 7&<( !x% . !
,123456 6768792:;<
B1 ,4.181 68;. /;. $#%#)1-,=. *"#2 ,-.&%/+,+2 f ,’ ?.# '"&,/+*# ! 1.1-
*&D(/#" ()*+,-. ./.0/1*23( -45 f ,-. ., ,-*E180D(/#" ( dxxf )( 7#" '"#E&-
D(/#" “181789%;*# (3 /1- x ./( x”. F+8#'9,
( $! cxFdxxf )()( , %c ",
6$1- F *"# $#%&)1-,# /+2 f ,/1 !.
!"# $#%&'(")*#,
( $! cxxdx +*,-. , #314 xx ,-.)+*( !# .
3 O!"#!$%&'(#") !"*()+") 305
!"# $%& $'#"% "%( %')*$+,- $% .#'/*$% %0%,01'23. "'%,4"$-/ #$/:
5/. ,67- *(&6'$+*+ f, ".'.828)*/3+ *- 9&. :/6*$+3. !, /*;4-/
! "#$ cxfdxxf )()( , %c &
< :/.:/,.*). -4'-*+= $%( .#'/*$%( %0%,0+'>3.$%= -)&./ .&$)*$'%?+ "%'-).
$+= ".'.8>8/*+= ,./ 098-$./ !"!#"$%&'(. < *$.7-'6 c 098-$./ ')*+,%- !"!-
#"$%&'(.. !"# $%& ")&.,. $2& ".'.8>82& @.*/,>& *(&.'$1*-2& @')*,%(3- $%& ".'.,6-
$2 ")&.,. .#'/*$2& %0%,0+'236$2&.
A/ $4"%/ $%( ")&.,. .($%4 /*;4%(& *- ,67- /0-')(1* *$% %"%)% %/ ".'.*$6*-/= $%( x "%( -3?.&)B%&$./ 9;%(& &#+3..
2345657 58937:;4 8<86<=9;>5:;4
1. ! dx0 c# 6. ! xdx+3 cx "'# *(&
2. ! dx1 cx "# 7. dxx! 2*(&
1cx "# -?
3. ! dxx
1cx "# ||ln 8. ! dx
x2+31
cx "'# *?
4. ! dxx" c"x"
""
#"
1
1
, 1'(" 9. ! dxe x ce x "#
5. ! xdx*(& cx "# +3 10. ! dx" x c"
" x
"#ln
C(&9"-/. $%( %'/*3%4 $%( .#'/*$%( %0%,0+'>3.$%= ,./ $2& ,.&#&2& ".'.8>-
8/*+= -)&./ %/ -D1= :4% /:/#$+$-=:
!& %/ *(&.'$1*-/= f ,./ g 9;%(& ".'68%(*. *’ 9&. :/6*$+3. !, $#$-
) ! !# dxxf#dxxf# )()( , *!%#
) ! ! !"#" dxxgdxxfdxxgxf )()())()((
C43?2&. 3- $%(= ".'."6&2 $4"%(= 9;%(3- 8/. ".'6:-/83.:
! ! "## cx
dxxdxx3
4443
22
! ! !'#' dxexdxdxex xx 2+33)2+33(
! !'# dxexdx x2+33
cex x "''# 2*(&3
3 O!"#!$%&'(#") !"*()+") 306
! ! !"#"
dxx
dxx
xdx
x
x 1313
!!"
"# dxxdxx 2
1
2
1
3
cxxcxx
$"#$"# 2
1
2
32
1
2
3
22
2
1
2
33 .
,-.%+"*,)
1. N! "#$%$& '()*#+,', f +-+./!, 0'+$ , 1#!2/34 +,5 6!#*'+!', )!
7/-#8$+!/ !69 +. ',:$&. 3)(2,A 3!/ )! /'8;$/ 12)( "#% xxf , 1/! 3*%$ &x ''''.
!/)$
!"#$%& 12)( "#% xxf , '()*+# %$,%)($-.:
! ! "#% dxxdxxf )12()(
22
1)( cxxcxf $"#$ , &21 , cc '(
((((((((((((((((((((((((((((((((((((((((((((((((((( 122)( ccxxxf "$"# , &21 , cc '
cxxxf $"# 2)( , &c '.
/$, 0, %$'1(#2,$ 3 f ,"4 2) 53+#6) )3,2(A "1'"#$ -,$ ,1-#6 3)2( #f &, $5)%7-
0,+,, 322 2 #$" c , %38,%& 1#c . !")+'09:, 1)( 2 $"# xxxf .
2. < $&'6#!=, )( xE , !69 +,) 60>,', x :.)*7?) $)95 6#.@9)+.5 100)(0 )))))))) x
:/!5 "/.:,8!)&!5, :$+!"*>>$+!/ :$ #(%:9 xxE """"####%%%% 100)( ('$ 8/>/*7$5 7#!8-
:-5 !)* :.)*7! 6#.@9)+.5), $)0 . #(%:95 :$+!".>45 +.( 39'+.(5 6!#!1?145 $&)!/ '+!%$#95 3!/ /'.;+!/ :$ 2 ('$ 8/>/*7$5 7#!8:-5 !)* :.)*7! 6#.@9)+.5).
A! "#$%$& +. 3-#7.5 +,5 "/.:,8!)&!5 !69 +,) 6!#!1?14 100 :.)*7?) 6#.@9-
)+.5, (6.%-+.)+!5 9+/ +. 3-#7.5 $&)!/ :,7-) 9+!) , "/.:,8!)&! 7$) 6!#*1$/ 6#.@9)+!.
!/)$
;0 )(xP #60,$ 2) -'1%): -,$ )(xK #60,$ 2) -452): ",1,<9<&: <$, x +)0.%#: "1)=402):, 242#
)()()( xKxExP "# ,
3 O!"#!$%&'(#") !"*()+") 307
!"#$%
xxxKxExP !"!!"#!#"# 982100)()()( .
&'()*+
xxP !"# 98)( ,
!"#$%
$$ !"# dxxdxxP )98()(
,)- ./)
cx
xxP %%%%!!!!""""2
98)(2
, &c '.
01$)2 ' 3-!4'5)26) *%2 ")/.7%- "/!8#2$), $! ,9/*!: %62)- 4'*92, *'()*+ -;5<%- 0)0( "P ,!"#$% 0"c . ="!492>:,
298)(
2xxxP !!!!"""" .
?/), $! ,9/*!: )"# 100 4!2.*%: "/!8#2$!: %62)-
2
10010098)100(
2
!("P 480050009800 "!" (;% 5-(-.*%: */)549:).
,)#$)-()
!" #$!%!&
1. @) A"!(!76;%$% $) !(!,('/B4)$)
i) $ %% dxxxx );A2'4( 3 ii) dxx
xx$
%% 12
iii) $ dxxx3 iv) $ %%
dxx
x
2
83
v) $ )*+
,-. %! dxx
xe x 2;A23
vi) $ ))*
+,,-
.! dx
xx 22 '41
;A21
vii) dxx
x$ %
%2
3.
3 O!"#!$%&'(#") !"*()+") 308
2. N! "#$%&$ &' ()*+#&'(' f, ,$ -$.%/ /#0(,/1 &/ .0+(&',! ),0( !" ,
20! &'* /-/%! 0(31$0
xxf
1)( #$ 4!0 1)9( #f .
3. 5! "#$%&$ &' ()*+#&'(' f, 20! &'* /-/%! 0(31$0 3)( #$$ xf , 6)1( #$f
4!0 4)0( #f .
4. 5! "#$%&$ &' ()*+#&'(' f , 20! &'* /-/%! 0(31$0 212)( 2 "#$$ xxf 4!0 ' 2#!6047 &'8 -!#+(&!(' (&/ (',$%/ &'8 )1,1(A 93$0 4:%(' 3.
5. ; -:'<)(,=8 )(tN , ($ $4!&/,,1#0!, ,0!8 4/0*>*%!8 "!4&'#0.%>*, !)-
?+*$&!0 ,$ #)<,= 20/
20
1)( tetN #$ !*+ :$-&=. 5! "#$%&$ &'* !1?'('
&/) -:'<)(,/1 (&! -#@&! 60 :$-&+.
6. A0! "0/,'3!*%! 93$0 .0!-0(&@($0 =&0 20! $"./,!.0!%! -!#!2>27 x $-?!#&',+&>* 93$0 /#0!4= 4=(&/8 xx 52 " (30:0+.$8 .#!3,98 !*+ ,/*+.!
-#/B=*&/8). 5! "#$%&$ &' ()*+#&'(' 4=(&/)8 &'8 $"./,!.0!%!8 -!#!-
2>278, !* $%*!0 2*>(&= =&0 &! (&!<$#+ $"./,!.0!%! 9?/.! &'8 "0/,'-
3!*%!8, =&!* .$* -!#+2$0 4!*9*! $?+#&',!, $%*!0 100 (30:0+.$8 .#!3-,98).
7. A0! *9! 2$@&#'(' $?@#)?'8 -$&#$:!%/) 93$0 #)<,= +*&:'('8 -/) .%-
*$&!0 !-= &/* &1-/ 2
4
31020)( tttR %"#$ , =-/) )(tR $%*!0 / !#0<,=8,
($ 30:0+.$8, &>* "!#$:0@* -/) !*&:7<'4!* (&/)8 t -#@&/)8 ,7*$8 :$0&/)#2%!8 &'8. 5! "#$%&$ -=(! "!#9:0! <! 93/)* !*&:'<$% &/)8 8
-#@&/)8 ,7*$8 :$0&/)#2%!8 &'8.
!" #$%&%'
1. C <$#,/4#!(%! ! $*=8 (@,!&/8, -/) -$#0"+::$&!0 !-= 9*! D)4&04=
)2#=, $:!&&@*$&!0 ,$ #)<,= kte"# %% , =-/) #, " $%*!0 <$&0498 (&!<$#98 4!0 t / 3#=*/8. C !#3047 <$#,/4#!(%! )0(T &/) (@,!&/8 $%*!0 #T "0 ,
=-/) 0! ' <$#,/4#!(%! &/) )2#/1 ' /-/%! ,$ 4!&+::':/ ,'3+*',!
.0!&'#$%&!0 (&!<$#7. 5! "#$%&$ &' <$#,/4#!(%! &/) (@,!&/8 &' 3#/-
*047 (&02,7 t.
2. E*!8 "0/,73!*/8, / /-/%/8 $-$*.1$0 x 30:0+.$8 .#!3,98 (&' "$:&%>('
&'8 -!#!2>278 &/) $#2/(&!(%/) &/), !*!,9*$0 *! 93$0 49#./8 )(xP .
30:0+.$8 .#!3,98 !-= !)&7 &'* $-9*.)('. A0! !*+:)(' &'8 -!#!2>-
278 9.$0?$ =&0 / #)<,=8 ,$&!"/:78 &/) 49#./)8 )(xP , -/) /6$%:$&!0
3 O!"#!$%&'(#") !"*()+") 309
!"#$ %&'$()!# *)"+, (,$%"*- *&. "/$ "0&/ 2000/8,5)( xexP !"# . 1*
23%,"% "/ !)$/4-5. 5'3(/6 &/) /7%,4%"*- !% *08#!# "#6 %&'$()!#6 *&. 4000000 !% 6000000 (3*9:'6.
3. ;&. "#$ &<4#!# %$.6 $'/) &3/=.$"/6 :-*6 %"*-3%,*6 (-*&-!"<>#5% ."- / 3)>:.6 :%"*2/4+6 "/) 5.!"/)6 )(tK (,$%"*- *&. "/$ "0&/
ttK 6,0800)( !"# (!% 9-4-?(%6 (3*9:'6 "#$ #:'3*), %$< / 3)>:.6 :%-"*2/4+6 "#6 %,!&3*8#6 )(tE !"/ "'4/6 "@$ t #:%3<$ (,$%"*- *&. "/$ "0&/ ttE 3,01000)( $"# (!% 9-4-?(%6 (3*9:'6 "#$ #:'3*). 1* 23%,"% "/
!)$/4-5. 5'3(/6 "#6 %"*-3%,*6 *&. "#$ "3,"# '@6 5*- "#$ '5"# #:'3*
&*3*A@A+6.
4. B!"@ gf , (0/ !)$*3"+!%-6 :% )0()0( gf " , 1)1()1( $" gf 5*- )()( xgxf ##"## A-* 5?>% %x &. 1* *&/(%,8%"% ."-:
i) xxgxf $" )()( , A-* 5?>% %x &.
ii) A$ # !)$?3"#!# g '9%- (0/ 3,C%6 !", :% !" '' 0 , "."% # !)$?3-
"#!# f '9%- :-* "/)4?9-!"/$, 3,C* !"/ ),( !" .
3.2 ME,"-"( "!"#!$%&)$)
D &,$*5*6 "@$ *.3-!"@$ /4/54#3@:?"@$, &/) (<!*:% &*3*&?$@, (%$ %,$*- *3-
5%".6 A-* $* )&/4/A,!/):% "/ /4/54+3@:* :,*6 /&/-*!(+&/"% !)$?3"#!#6, .-
&@6 &.9. "* /4/54#3<:*"* dxxx( $12 2 5*- ( dxxe x . E% "'"/-%6 &%3-&"<!%-6 /
)&/4/A-!:.6 A,$%"*- *&4/0!"%3/6 :% "# 2/+>%-* "@$ &*3*5?"@ :%>.(@$ /4/-
54+3@!#6.
+./0102 03043567892 4:;< =:6<>0?;@2
F :'>/(/6 *)"+ %573?C%"*- :% "/$ "0&/:
( ( #!"# dxxgxfxgxfdxxgxf )()()()()()(
&/) %,$*- !)$'&%-* "/) 5*$.$* &*3*A<A-!#6 "/) A-$/:'$/) (0/ &*3*A@A,!-:@$ !)$*3"+!%@$ gf , !% '$* (-?!"#:* #.
G3?A:*"-, A-* 5?>% #x% , '9/):%
)()()()())()(( xgxfxgxfxgxf #$#"# ,
/&."%
)()())()(()()( xgxfxgxfxgxf #!#"# .
3 O!"#!$%&'(#") !"*()+") 310
!"#$%&'(
! ! ! "#"$" dxxgxfdxxgxfdxxgxf )()())()(()()(
), *+#,-&.$.,
! ! "#%$" dxxgxfcxgxfdxxgxf )()()()()()( . (1)
!"/*,) 0# #1#21)3'$. 0#4 ,/-0/3#4 $%1#4( 05( (1) "/3*%6/* $*. +0.7/38 #1#-
21)3'+5(, 0# c $"#3/9 &. ".3.1/*:7/9, #";0/ %6#4$/ 0#& ".3."8&' 0-"#. !
< ".3."8&' 0-"#( 635+*$#"#*/90.* =*. 0#& 4"#1#=*+$; #1#2153'$80'& $/ 05& "3#>";7/+5 ;0* 0# #1#21)3'$. 0#4 ?@ $%1#4( 4"#1#=9A/0.* /42#1;0/3..
B*. ".38,/*=$., .( 4"#1#=9+#4$/ 0# #1#21)3'$. ! dxxe x . C6#4$/:
! ! ! %#$#$"$ cexedxexedxexdxxe xxxxxx )( .
D&, 0E3., ,#2*$8+#4$/ &. 4"#1#=9+#4$/ 0# ".3."8&' #1#21)3'$., .118A#-
&0.( 0#4( 3;1#4( 0'& x 2.* xe , ?39+2#4$/
! ! !#$"
&&'
())*
+$ dxe
xe
xdxe
xdxxe xxxx
222
222
.
F# 0/1/40.9#, ;$'(, #1#21)3'$. /9&.* "*# +-&7/0# ."; 0# .36*2;.
,-.%+"*,)
1. N! "#$%$&'()$*+ )! $%$,%-./0!)!
i) ! dxex x2 ii) ! xdxx-02
iii) ! % xdxx 1)ln(4 3 iv) ! xdxe x-02 .
!/)$
i) C6#4$/
! ! ! !#$"#$"$ dxxeexdxexexdxexdxex xxxxxx 2)()( 22222
! !%#$"#$ dxexeexdxexex xxxxx 22)(2 22
cexeex xxx %%#$ 222 .
G/ 0#& 9,*# 03;"# 4"#1#=9A#4$/ #1#2153E$.0. 05( $#3:)(
3 O!"#!$%&'(#") !"*()+") 311
! dxexP x!)(
!"#$ )(xP "#%$&'$(# )#$ x *+, *!"! .
ii) -.#$(/
! ! ! ##$%#$%&$% cxxxxdxxxdxxxxdxx 20(4
121$'
2
121$'
2
121$'
2
1)21$'(
2
120( .
2/ )#' 34,# )5!"# $"#%#637#$(/ #%#*%05&(+)+ )08 (#59:8
! dxx!xP )(0()( , ! dxx!xP )(1$')(
!"#$ )(xP "#%$&'$(# )#$ x *+, *!"! .
iii) -.#$(/
! ! ! #$#%&#%# dxx
xxxxxxdxxxxdxx1
)(ln)(ln)(ln)14( 4443
! #$$#%#$#% cxx
xxxdxxxxx4
ln)()1(ln)(4
434 .
2/ )#' 34,# )5!"# $"#%#637#$(/ #%#*%05&(+)+ )08 (#59:8
! dxx!xP )ln()( ,
!"#$ )(xP "#%$&'$(# )#$ x *+, *!"! .
iv) ;<)#$(/ !% dxxeI x )2(0( , #"!)/ <.#$(/
! !$%&% dxxexedxxeI xxx )2(1$'2)2(0()2(0()(
! &$% dxxexe xx )2(1$')(2)2(0(
!$$% xdxexexe xxx 20(4)2(1$'2)2(0(
Ixexe xx 4)2(1$'2)2(0( $$% .
="#(<'>8,
1)2(1$'2)2(0(5 cxexeI xx #$% ,
#"!)/
cxexeI xx #$% )2(1$'5
2)2(0(
5
1.
2/ )#' 34,# )5!"# $"#%#637#$(/ #%#*%05&(+)+ )08 (#59:8
3 O!"#!$%&'(#") !"*()+") 312
! dxx!e x" )(!" , ! dxx!e x" )(#$%
&'($ *, !"!" .
2. ! "#$%&'()* )(tP , 200 ######## t , (+,* ")#$*, "-& ".-/0&12 ,") '&34562&'$
10 0-+6-78796, ,&:;627,+ (2 .&%() ('2 ;7-(, ,6; /7-*) "-& <=627,+ ,") 7-6 7>"- /10)( ttetP $$$$%%%% , 200 ######## t , )"-& t 2=6,+ - ,.+%()* 796 2756 (27; 7$ '&345-
62&'$. ?, @.2%2= - "#$%&'()* )(tP 7$* ")#$* t 4.)6+, (27; 7$ '&34562&'$,
,6 369.=A-&(2 )7+ - "#$%&'()* 87,6 10000 0;7-+0-+ 0,7; 7$ '7+3(8 7$* '&34562&'$*.
!,)$
)*($"+
! !$% dttedttP 10/1)(
! %$ tdte t )(10 10/
!&'$ dtete tt 10/10/ 1010
cete tt (&$ 10/10/ 10010 ,
('&,+
cetetP tt (&$ 10/10/ 10010)( , -./ 01'(.( !"c .
2,/% 0$t , ( '3!4$#"&5 +6%/. 10000. 7$%+'85:
101001000010001010000)0( 00 $)$(&')$ cceeP .
9:;/, ( '3!4$#"&5 ,!5 '&3!5, t *;&%./ "+,1 ,! #$-*8%+$#!, +6%/.
1010010010)( 10/10/ (&$ tt etetP .
"-./-01234 56 789:/79;39734
<+ ,! "=4(>( /$,? $'(3(-6@($"+ (3(03!;8"/,/ '($ =*($% ? "'(;(A% %/ '1-
;($% ,! "(;B? ! % dxxgxgf )())(( . C "=4(>(5 (3(03?;D#!5 "+ /%,.0/,1#,/#!
+0B;1@+,/. "+ ,(% /0&3($4( ,A'(:
! !$% duufdxxgxgf )()())(( ,
&'($ )(xgu $ 0/. dxxgdu )(%$
E '/;/'1%D ,A'(5 *;!#."('(.+6,/. "+ ,!% ';(F'&4+#! &,. ,( (3(03?;D"/
! duuf )( ,($ >+$,=;($ "=3($5 $'(3(-6@+,/. +$0(3&,+;/.
3 O!"#!$%&'(#") !"*()+") 313
! "#$%&'() *+, *-#+, ",*+- .*)/01&*"' .*+ 234.*$ 5"3$3" #"/"262'.)7 .-38&-*)7 .,39/*).)7. :/92;"*', "3 F &03"' ;'" #"/92+,." *)7 f, *$*&
)()( ufuF !" , (1)
+#$*&
))(())(( xgfxgF !"
5"' 9/"
# # ""!" dxxgxgFdxxgxgf )())(()())((
# "! dxxgF )))((( ("<+- )())(())((( xgxgFxgF ""!" )
cxgF $! ))((
cuF $! )( , ($#+, )(xgu !
#! duuf )( (=$24 *)7 (1)) !
>'" #"/9%&'2;", "7 ,#+=+20.+,;& *+ +=+5=?/4;" dxxx# $12 2 . @A*+,;&
12 $! xu 5"' xdxdxxdu 2)1( 2 !"$! , +#$*& *+ +=+5=?/4;" 2/9<&*"':
!$# dxxx 12 2 duu#
#! duu 2/1
cu $! 2/3
3
2
cx $$! 2/32 )1(3
2
cx $$! 32 )1(3
2.
,-.%+"*E)
1. N! "#$%$&'()$*+ ,! $%$-%./01!,!
i) # $dx
e
ex
x
2)(1 ii) # xdx23 .
3 O!"#!$%&'(#") !"*()+") 314
!,)$
i) !"#$%&' xeu !"1 , $()#' dxedxedu xx "#!" )1( . *($&"+,-,
$ $ ""! 22)1( u
dudx
e
ex
x
$ !!
%"!%"% ce
cu
duux1
112
ii) ./$%&' $ "xdx'0 $ dxx
x
1%+2&
. *($&"+,-, 3+ 4"1$%&' xu 1%+" , $()#'
xdxdxxdu 2&)1%+( %"#" , "/$%&':
$ "xdx'0 $ !%"!%"% cxcuduu
|1%+|ln||ln1
.
2. !" #$%&%'()*%+, -" %&%.&/012"-"
i) $ ! dx-
x )6
/2(2 ii) $ %dx
x21
1 iii) $ % dxxx 992 1)( .
!,)$
i) !"#$%&' 6
2!
xu !" , $()#' dxdx!
xdu 2)6
2( "#!" .
*($&"+,-,
$ "!2
1)
62(2& dx
!x $$ "&! ududx
!x 2&
2
12)
622&(
c!
xcu !!%"!%" )6
2(1%+2
11%+2
1.
ii) !"#$%&' xu 21%" , $()#' dxdxxdu 2)21( %"#%" .
*($&"+,-,
$ $ !%"%"%
cuduu
dxx
||ln2
11
2
1
21
1cx !%%" |21|ln
2
1.
iii) !"#$%&' 12 %" xu , $()#' xdxdu 2" . 563
$ "% dxxx 992 )1( $ !%"!" cxcu
duu 1002100
99 )1(200
1
1002
1
2
1.
3. !" #$%&%'()*%+, -" %&%.&/012"-"
i) $ !%!
dxxx
x
65
122
ii) $ !%!%
dxxx
xx
65
732
2
.
3 O!"#!$%&'(#") !"*()+") 315
!,)$
i) ! "#$%&'("( 65
12)(
2 !"!
#xx
xxf )*+, -+./0 0&,"102 '0 $ }3,2{" 34, 5&%6+'4,
)3)(2(
12)(
""!
#xx
xxf .
7$48('021+ -&4514',3029 4&,:1029 !, " )'",, ;"'+ $4 ,"*2+,
32)3)(2(
12
"!
"#
""!
x
B
x
A
xx
x , 5,4 3%:+ }3,2{"%!x .
<+ 4-4=0,6> -4&0$014"';$ )*0#1+ '+=,3%:
123)2( !!#"! BAxBA , 5,4 3%:+ }3,2{"%!x .
! '+=+#'4/4 ,"?'('4 ,"*2+, 5,4 3%:+ }3,2{"%!x , 4$ 34, 1?$0 4$
&'
&()
#!!
#"!
0123
02
BA
BA >, ,"0.2$414,
&'
&()
#
"#
7
5
B
A.
@-01)$A9,
* * * * "!
""
#+,-
./0
"!
""
#!"
!dx
xdx
xdx
xxxx
x
3
7
2
5
3
7
2
5
65
122
cxx !"!""# |3|ln7|2|ln5 .
<+ '0$ /.,0 '&?-0 +&548?14"'+ 5,4 '0$ #-0=05,"1? 0=03=(&A1%'A$ '(9 10&6>9
* !!!
dx#x$x%
&x'2
, 1+ 042 1" %#$
ii) 7$ +3'+=)"0#1+ '( .,4/&+"( '0# -0=#A$210# 732 !" xx 1+ '0 -0=#;$#10
652 !" xx , B&/"30#1+ ?',
65
121
65
7322
2
!"!
!#!"!"
xx
x
xx
xx.
@-01)$A9,
* #!"!"
dxxx
xx
65
732
2
* dx1 * !"!
! dxxx
x
65
122
cxxx !"!""# |3|ln7|2|ln5 (=?5A '0# (i)).
M+ '0$ /.,0 '&?-0 #-0=05/80#1+ 0=03=(&;14'4 '(9 10&6>9
* !!dx#x$x%
xP2
)(,
3 O!"#!$%&'(#") !"*()+") 316
!"#$ )(xP "#%$&'$(# )#$ x *+,(#- (./+%-).0#$ 1 23#$ )#$ 2 4+5 042 !" !"# .
,)#$)-()
!" #$!%!&
1. 6+ $"#%#/23.). )+ #%#4%70&(+)+
i) # " dxex x2 ii) # $" dxexx x22 )123( iii) # xdxx ln3
iv) # xdxx 27(2 2 v) # xdxx 23$'4 vi) # xdxln ,
vii) # 2
ln
x
xdx viii) # xdxe x 23$' ix) # xdxe x7(
2. N+ $"#%#/23.). )+ #%#4%70&(+)+
i) # xdx37( ii) # "$" dxxxx )2()7164( 32
iii) # $$
dxxx
x42 )6(
3 iv) # $
dxx
x
3
2
2
v) dxxx# $1 .
3. N+ $"#%#/23.). )+ #%#4%70&(+)+
i) # dxee xx7( ii) # $dx
e
ex
x
1 iii) # dx
xx ln
1
iv) # $$dx
ee
exx
x
)1ln()1( v) dx
x
x#%&'
()*
2
17(.
B" #$!%!&
1. 6+ $"#%#/23.). )+ #%#4%70&(+)+
i) # $dx
x
x23$'1
27( ii) # + dxxx )3$'ln(.8 iii) # + dxex x7(3$' .
3 O!"#!$%&'(#") !"*()+") 317
2. !" #$%&%'()*+* +" %&%,&-./0"+"
i) ! "#
dxxx
x43
3 11 ii) ! #
dxx
x
12 iii) ! # dxxx )1ln( 2 .
3. N" #$%&%'()*+* +" %&%,&-./0"+"
i) ! dxxx 22 ln ii) dtt! 2)(ln iii) ! dxee xx)#12 .
4. N" #$%&%'()*+* +" %&%,&-./0"+"
i) ! xdx*2 ,"3 ! dxx
x2)#1
ii) ! dxx
x2-0
)#1 ,"3 ! #
dxx
x2-0)#11
iii) ! xdx3-0 ,"3 ! xdx3)#1 .
5. 4* +- 5%67*3" +81 +9$81
2
2)#11-0 2 !! $% ,"3
2
2)#11)#1 2 !! #%
1" #$%&%'()*+* +" %&%,&-./0"+":
i) ! xdx2-0 ii) ! xdx2)#1 iii) ! xdxx 22 )#1-0 .
6. M* +- 5%67*3" +81 +9$81
)(-0)(-0)#1-02 "!"!"! ##$% ,
)()#1)()#1)#1)#12 "!"!"! ##$%
)()#1)()#1-0-02 "!"!"! #$$%
1" #$%&%'()*+* +" %&%,&-./0"+":
i) ! xdxx 2)#1-0 ii) xdxx 5)#13)#1 iii) ! xdxx 4-02-0
7. !" #$%&%'()*+* +" %&%,&-./0"+"
i) ! #$$
dxxx
x
23
322
ii) ! #$#
dxxx
x
23
232
iii) ! ##$
dxxx
xx
23
22
3
iv) ! $dx
x 1
22
.
3 O!"#!$%&'(#") !"*()+") 318
3.3 ,(-."%(#/) /0()&)/()
*12345
!"# $%#&'#()*+# ,*-./01# *230)* 4"1, 4"0+ '+5%26#7)* "& 87+.%"&8& 9:8&; )(tSy ! *+4; ,1+&"#(, )$#%#()* +0 <%#()* "&+ "0=("&"0 ,01 "&+ *$1".=7+8&
"#7 ,1+&"#(. >#//:; -#%:;, 4)5;, *2+01 '+58"? & "0=("&"0 )(t!! ! ? & *$1".-
=7+8& )(t"" ! "#7 ,1+&"#( ,01 6&"*2"01 & 9:8& "#7. @10 $0%.3*1')0:
— A+ :+0 ,1+&"4 ,1+*2"01 *797'%.))5; )* 8"09*%? "0=("&"0 c, '10 +0 $%#831#-
%28#7)* "& 9:8& "#7 )(tSy ! , 0%,*2 +0 /(8#7)* 5; $%#; y "&+ *A2858&
cy !" . (1)
— A+ 8* :+0 8B)0 ).60; m 08,*2"01 3(+0)& )(tFF ! , "4"* "# 8B)0 ,1+*2"01 )* *$1".=7+8& )(t"" ! & #$#20, 8()-5+0 )* "# 2# +4)# "&; )&=0+1,?;, 32+*"01 0$4
"#+ "($# "mF ! ?, 18#3(+0)0, ymF ""! , 4$#7 )(tSy ! & 87+.%"&8& 9:8&; "#7
8B)0"#;. C$#):+5;, '10 +0 $%#831#%28#7)* "& 9:8& )(tSy ! "#7 8B)0"#;, 0%-
,*2 +0 /(8#7)* "&+ *A2858&
Fym !""# . (2)
CA18B8*1; 4$5; #1 (1) ,01 (2) /:'#+"01 !"#$%&"'() *+",-,*"). @*+1,.,
./012.1
3"#$%&"'4 *+5,6,7 /:'*"01 ,.9* *A2858& $#7 $*%1:=*1 "& )*"0</&"? x, )10
.'+58"& 87+.%"&8& )(xfy ! ,01 ,.$#1*; 0$4 "1; $0%0'B'#7; "&; ...,, yy """ .
@10 $0%.3*1')0, #1 *A18B8*1;
xy 2!" , yy 2!" , 0!$"" yy
*2+01 310-#%1,:; *A18B8*1;.
D )*'0/("*%& 0$4 "1; ".A*1; "5+ $0%0'B'5+ $#7 *)-0+26#+"01 8"&+ *A2858&
#+#).6*"01 89+7 "&; 310-#%1,?; *A2858&;. E"81 #1 *A18B8*1; xy 2!" ,01 yy 2!" *2+01 310-#%1,:; *A18B8*1; $%B"&; ".A*5;, *+B & 0!$"" yy *2+01 3*7-
":%0; ".A*5;. F.9* 87+.%"&8& )(xfy ! $#7 *$0/&9*(*1 "& 310-#%1,? *A2858& /:'*"01 :;,7
"&; *A2858&;. @10 $0%.3*1')0, & 87+.%"&8& 2xy ! *2+01 )10 /(8& "&; 310-#%1,?; *A2858&;
xy 2!" , 0-#( xxy 2)( 2 !"!" .
G# 8(+#/# 4/5+ "5+ /(8*5+ )10; 310-#%1,?; *A2858&; /:'*"01 <*="'4 :;,7 "&; *A2858&;.
3 O!"#!$%&'(#") !"*()+") 319
!"# $#%&'(")*#, + )(,"-. /01+ 2+3 (45161+3 xy 2!" (5,#" + cxy #! 2 , $c %.
789,& :+2&*( (-(5,+ 2+ /01+ )(xfy ! 2+3 '"#;<%"-.3 (45161+3 $<8 "-#,<$<"(5 *"# !"#$%& '()*&%+ )( 00 xfy ! . !"# ,# =%<0*( 2+ /01+ #82., =%51-<8*( $%>2#
2+ )(,"-. /01+ 2+3 (45161+3 -#" *( 2+ =<.?("# 2+3 #%9"-.3 18,?.-+3 $%<1'"<-
%5:<8*( 2+ :+2<0*(,+ /01+.
!"# $#%&'(")*#, + /01+ )(xfy ! 2+3 '"#;<%"-.3 (45161+3 xy 2!" , $<8 "-#,<-
$<"(5 2+, #%9"-. 18,?.-+ 2)1( !f , (5,#" + 18,&%2+1+ 12 #! xy , #;<0 #$@ 2+
)(,"-. /01+ cxy #! 2 , )"# 1!x -#" 2!y (5,#" 1!c .
72+ 18,A9("# ?# #19</+?<0*( *@,< *( '8< ("'"-A3 *<%;A3 '"#;<%"->, (4"1>1(-6, $%>2+3 2&4(63:
& B"3 (4"1>1("3 *( 96%":@*(,(3 *(2#=/+2A3 -#"
& T"3 )%#**"-A3 '"#;<%"-A3 (4"1>1("3 $%>2+3 2&4(63.
,-./01-234 56-7875-4 95 :;1-<=95>54 95?.@AB?34 C9(" #$<'("92(5 $("%#*#2"-&, @2" < %8?*@3 *(2#=</.3, 63 $%<3 2< 9%@,<, 2<8
$/+?81*<0 )(tPy ! *"#3 -<",6,5#3, + <$<5# '(, ($+%(&:(2#" #$@ (462(%"-<03 $#%&)<,2(3, (5,#" #,&/<)<3 2<8 $/+?81*<0. D+/#'., "190("
)()( tP!tP !" ,
@$<8 ! ?(2"-. 12#?(%&.
E, < #%9"-@3 $/+?81*@3 2+3 -<",6,5#3 (5,#" 0P , '+/#'. 0)0( PP ! , )"# ,# =%<0-
*( 2<, $/+?81*@ )(tP 012(%# #$@ 9%@,< t, ?# /01<8*( 2+, $#%#$&,6 '"#;<%"-.
(45161+.
F$("'. 0)( '! tPy , + (45161+ )%&;(2#"
!tP
tP!
")(
)(,
<$@2( </<-/+%>,<,2#3 -#" 2# '8< *A/+ 2+3, A9<8*( '"#'<9"-&:
( "dt
tP
tP
)(
)((! dt!
1)(ln ct!tP #! .
1)( ct!etP #! ,
t!cetP !)( , *( 1cec ! .
F$("'. 0)0( PP ! , (5,#" 0Pc ! , <$@2( .t
0 ePtP !)( .
3 O!"#!$%&'(#") !"*()+") 320
! "#$#"%&' ()#*+$),- ./01'12 345.6#) ()#*+$),- ./01'12 7. !"#$%&'()(* '(+,-./+0*. 8.&),%,
O123453
6$,78#$9: (;<="=/ '( !"#$%&'()(* '(+,-./+0* 345.6#) ,%9. ./01'12 62: 7+$*-:
)()( x!yy" !"# (1),
;"+< )(xfy ! 2 %5&'162 1<&%$6212, )(y" 1<&%$6212 6+< y ,#) )(x! 1<&%$-
6212 6+< x.
8)# &# 3=1+<7. 62& ./01'12 #<6- +3+,32$>&+<7. ,#) 6# (=+ 7432 62: ': "$+: x.
?@+<7.
$$ !" dxx!dxyy" )()( .
A".)(- )(xfy ! , .0&#) dxydxxfdy "!"! )( , +";6. 4@+<7.
$ $! dxx,dyy- )()( . (2)
B& )(yA .0&#) 7)# "#$%5+<1# )(y" ,#) )(xB 7)# "#$%5+<1# 62: )(x! , 6;6. 2
(2) 5$%*.6#)
cxByA %! )()( , &c '. (3)
B"; 62& 6.3.<6#0# ./01'12 "$+1()+$0C+<7. 62 5.&),- 3=12 62: ()#*+$),-: ./0-1'12:.
3>5?25
! )1;626# (2) 7#: .")6$4".) &# 5$%*+<7. 62 ()#*+$),- ./01'12 (1) 162& “%6<-
"2” 7+$*- 62:
dxx!dyy" )()( !
,#) &# +3+,32$>&+<7. 6# 7432 62:, 6+ 7.& "$>6+ 743+: 62: ': "$+: y, 6+ (. (.=6.$+ 743+: 62: ': "$+: x.
./0%+"*H
N, .@A8B) 8$ C$,78#$90* (;$=D=($*
i) 02 !"( yxy , 0)x 9,$ 0*y
ii) 22xyy !" , 0)y
iii) 0!"yy-x , 0*y .
!1)$
3 O!"#!$%&'(#") !"*()+") 321
i) H !"#$%$& '()*!+,- .-,./0-1)
x
yy
21!"
xdx
dy
y
21! ,
dxx
dyy
21! .
# #! dxx
dyy
21
1||ln2ln cxy $! 12ln cx $! , %1c &
2lnln 211
2
cxeeey xccx !'!! $ , 0(c .
2(,, 2cxy ! , 34/5 *!%x , 0(c (60. 1).
ii) 7 !"#$%$& '()*!+,- .-,./0-1)
22xydx
dy!
xdxy
dy2
2! .
# !2y
dy# xdx2
cxy
$!) 21
2(, cx
y$
)!2
1, 34/5 %c &* (60. 2).
iii) 7 !"#$%$& '()*!+,- .-,./0-1)
0!$dx
dyyx
xdxydy )!
# #)! xdxydy
1
22
22c
xy$)!
cyx !$ 22 , 0(c
2(,, 2xcy )! , 0(c (60. 3).
y>0
O
y
x
c !
1
2
c=1
c=2 1
c=0
c=1 O
x
y 2
3 2 1 O
y
x
c=9
c=4
c=1
3
3 O!"#!$%&'(#") !"*()+") 322
*,-../012 3/-45,/012 67/8986/2 :,9;<2 ;=76>2 !"# $% &'()*+ ,-./012'34 #$) ($2 5)"67-#
*8*6.37 )(984) 2 *7-#-7: $2' Kicrhhoff. ;%-
675+,
)()()(
tVtIRdt
tdIL !"#" . (1)
<)7 -7 "/2(5)2/0(2'34 $%- =-$7(%, )(tI , $2'
/4837$2: "2' 5)7//=4) $2 *8*6.37, 40-7) 7->-
,*% -7 68(2'34 $% 5)7?2/)*+ 4@0(.(% (1). A 4@0(.(% 7'$+ 6=,4$7) ,/733)*+
5)7?2/)*+ 4@0(.(% "/B$%: $>@4.:. <4-)*>:
!"#$%!$
&'())*+, -*(./'*+, 0123435 6'7859 8:1049 6=,4$7) *>C4 4@0(.(% $%: 32/?+:
)()( x!yx"y !#$ ,
#"2' )(xfy ! 40-7) % >,-.($% ('->/$%(% *7) )(x" , )(x! ('-7/$+(4): $2' x.
<)7 $%- 4"06'(% $%: 4@0(.(%: 7'$+:: — !-71%$2834 3)7 "7/>,2'(7 )(xA $%: ('->/$%(%: )(x" *7) ="4)$7
— D2667"67()>12'34 $7 3=6% $%: 4@0(.(%: 34 )( xAe .
E$(), =92'34 5)7529)*>
)()()( )()( xAxAxA ex!yex"ey !"#$
)()()( )()( xAxAxA ex!yexAey !"$#$
)()()( )()( xAxAxA ex!yeey !"#$ $
)()( )()( xAxA ex!ye !$
% %!$ dxex!dxye xAxA )()( )()(
cxBye xA #! )()( ,
#"2' )(xB 3)7 "7/>,2'(7 $%: )()( xAex! .
?@A%+"*H
1. ;( <=>02 5 -*(./'*+, 0123435
22 !#$ yy .
R
4
I
L
3 O!"#!$%&'(#") !"*()+") 323
!,)$
!"#$%& '$( "()*+,-.( /01 2)( !x! #23($ 0 xxA 2)( ! , ",44("4(.$*5,-'# 6($ /(
%-, '740 /01 #82.9.01 '# xe 2 . :/.$, 7;,-'# %$(%,;$6*
xxx eyeey 222 22 !"#
)()( 22 #!# xx eye
ceye xx "! 22
xcey 21 $"! , %c &.
-)#$).()
!" #$!%!&
1. N( 4<.#/# /$1 %$(=,)$671 #8$.>.#$1:
i) 24xyy $!# , 0'y ii) xyy !# , 0'y
iii) 21
!#yxy
iv) xey y.-3$!# .
2. ?( 4<.#/# /$1 %$(=,)$671 #8$.>.#$1:
i) 32 !"# yy ii) xeyy $!"# 2
iii) xyy 2!"# iv) xxyy !"# 2 .
3. N( @)#2/# /0 4<.0 /01 %$(=,)$6&1 #82.9.01 222 yxy !# , 0(y , /01 ,",2(1 0 +)(=$6& "()*./(.0 %$7);#/($ ("A /, .0'#2, )3,0( $A .
4. ?( @)#2/# /0 4<.0 /01 %$(=,)$6&1 #82.9.01 yy 32$!# ",- $6(3,",$-#2 /0 .-3B&60 3/2)0( !y .
5. ?( 4<.#/# /$1 %$(=,)$671 #8$.>.#$1:
i) x
yx
y22 .-3
1
.-31
!"# , (3 3)0( $!y
ii) xyyx ln)1( !"#" , (3 10)1( !y .
'" #$!%!&
3 O!"#!$%&'(#") !"*()+") 324
1. ! "#$%&' $() '*+,$-.,(/ -+/0%$(1 ! &+ "#% '*+,$-.,2 ,/,*30% .,%-
#(4(.+5 $'# +65&3&' tIdt
dI '0!" . 7# 0)0( !I , #% 8-+5$+ $'# "#$%&'
)(tI .
2. 9% 8-+5$+ $' */&' $'1 :.%;(-.,<1 +65&3&'1 xy eyye 22
!# , ' (4(5%
:."-=+$%. %42 $( &'0+5( )2,2(A .
3. 9% */&+$+ $' :.%;(-.,< +65&3&' xyx
y !$#1
.
4. ! ,*5&' $'1 +;%4$(0"#'1 0.%1 >-%00<1 )(C 0+ +65&3&' )(xyy ! ,
0%y &$( &'0+5( ),( yxM +5#%. 5&' 0+ xy . 9% 8-+5$+ $'# +65&3&' $'1 )(C , %# +5#%. >#3&$2 2$. :."-=+$%. %42 $( &'0+5( )1,0(A .
5. ?&$3 &"#$ ,, ' &$%@+-"1, 0+ 0%% "$ .
i) 9% */&+$+ $'# +65&3&' t"e#y$y $!"# .
ii) 7# )(tyy ! +5#%. 0.% */&' $'1 +65&3&'1, #% %4(:+56+$+ 2$. .&=/+. 0)(lim !
"()ty
t.
6. ?=+. %4(:+.=$+5 4+.-%0%$.,A 2$. ( -)@021 0+$%8(*<1 $'1 @+-0(,-%&5-%1 % +#21 &B0%$(1, 2$%# %)$2 8-+@+5 &+ 4+-.8A**(# &$%@+-<1 @+-0(-
,-%&5%1 & 0+ T% % , +5#%.
)( &%kdt
%d$$! , 0%k .
9% 8-+5$+ $' @+-0(,-%&5% )(t% , %# 0)0( %% ! .
7. C 4*'@)&021 )(tPP ! 0.%1 =B-%1 0+$%#%&$+/+. 0+ &$%@+-2 -)@02
0%m . D5#+$%. 2$. ( -)@021 %/6'&'1 $() 4*'@)&0(/ ', %# :+# )4<-=+ ' 0+$%#A&$+)&', @% <$%# %#A*(>(1 $() '.
i) 9% :.,%.(*(><&+$+ 2$. ( 4*'@)&021 ' .,%#(4(.+5 $'# +65&3&'
mkPP $!# , 0%k &$%@+-A.
ii) 9% 8-+5$+ $' &)#A-$'&' )(tPP ! , %# 0)0( PP !
iii) 9% %4(:+56+$+ 2$.:
— A# 0kPm * , $2$+ ( 4*'@)&021 %)6A#+$%..
— A# 0kPm % , $2$+ ( 4*'@)&021 0+.B#+$%..
3 O!"#!$%&'(#") !"*()+") 325
— A! 0kPm ! , "#"$ % &'()*+,#- &./.,0!$1 +".)$/#-.
8. 2+"3 )(tyy ! "% 45%- 6.1 )(tVV ! % #76%- "%* !$/%4 ,1.- 8$9.,$-
!:- "( ;/%!16: +"17,: t. < 8$9.,$!: .8$1=>$1 .&# ,1. 6*6'16: %&:
$,?.8%4 ! &%* ?/@+6$".1 +"%! &*),0!. "(-. A4,B3!. ,$ "% !#,% "%*
Torricelli % /*),#- ,$".?%':- "%* #76%* "%* !$/%4 $@!.1
gy!dt
dV2"! , 10!g m/s2.
i) C! ( 8$9.,$!: $@!.1 6*'1!8/16: ,$
45%- 3,6m, .6"@!. 1m 6.1 ( .6"@!. "(-
%&:- $@!.1 0,1m, !. .&%8$@9$"$ #"1 "% y
16.!%&%1$@ "(! $9@+3+(
yy50
5"!#
ii) D. ?/$@"$ "% 45%- )(ty , .! $@!.1 7!3-
+"# #"1 "( ;/%!16: +"17,: 0!t ( 8$-
9.,$!: :".! 7$,="(.
iii) E#+%- ;/#!%- ). ;/$1.+"$@ 71. !. .8$1=+$1 "$'$@3- ( 8$9.,$!:;
(F@!$".1 #"1 % #76%- "%* 6*'@!8/%* $@!.1 "r#V 2! ).
9. 2!.- ?(,."%8#"(- .&%-
"$'$@".1 .&# ,1. ,&.".-
/@. 6.1 0!.! &*6!3":,
$!G ( 6./81= &.@>$1 "%
/#'% "(- .!"@+".+(-,
#&3- B.@!$".1 +"% +;:-
,.. HI".! % 81.6#&"(- S
?/@+6$".1 +"( )0+( $, %
&*6!3":- B%/"@>$".1 $-
!G, #".! ?/@+6$".1 +"(
)0+( Q, % &*6!3":- $6-
B%/"@>$".1 6.1 &/%6.'$@
('$6"/16# $/0)1+,. +"(! 6./81=. J."= "( 81=/6$1. .*": +"(! 6./81=
$B./,#>$".1 ('$6"/$7$/"16: 84!.,( % &%* 16.!%&%1$@ "(! $9@+3+(
ERCdt
dE 1"! , 21 ttt $$ .
#&%* CR, +".)$/0-. D. ?/$@"$ "(! )(tE , .! 01 )( EtE ! .
"
V(t) y(t)
R
&!'()*
Q
P S C
E0
3 O!"#!$%&'(#") !"*()+") 326
10. !"#$%&' #( )*& +'&,&' )*- .irchhoff /0' )* +"+1%#' )*- 20-31'&*" 456#')*7 045"(0
)(tERIdt
dIL !" .
i) A& !R 12! , HL 4! ,
VE 60! ,
') &' 89(:)( );& <&)'4; )(tI )*- 9("#')*7, t sec #()= )* +1(:40#* )*- +-+1>#')*7.
8) &' 89(:)( )* )(lim tIt "#$
. ?: 4-#3(9':&()(;
ii) A& 4)* +"+1%#' '&): /0' #3')'9:' 3*- 2:&(0 4)'@(96 ;1(+)9(/(9)0+6
2"&'#; " 59;40#*3*064*-#( #0' /(&&6)90' 3*- 2:&(0 ttE 3;#60)( ! , &'
89(:)( );& <&)'4; )(tI .
3.4 "%()+,-" "!"#!$%&+.
,/01234 51610789:7; <=67>
A4)% ,)0 @<1*-#( &' 89*"#( )* (#8'2,& )*- 5%9:*- ! 3*- 3(90+1(:()'0 '3, );
/9'$0+6 3'9=4)'4; );7 4-&=9);4;7 2)( xxf ! , )*& =B*&' )%& x +'0 )07 (-@(:(7 0!x +'0 1!x (C'9'8*10+, 5%9:* !5. 5).
1
!
1 O x
y
y=x2 5
2v
1v
vv%1
1
1 O x
y 6
. . .
y=x2
D0' #<@*2*7 &' 39*4(//:4*-#( )* E;)*"#(&* (#8'2,& (:&'0 ; (B67:
F%9:E*-#( )* 20=4);#' ]1,0[ 4( # 04*#6+; -3*20'4)6#')', #6+*-7 #
x$ 1! , #(
=+9' )' 4;#(:':
3 O!"#!$%&'(#") !"*()+") 327
00 !x , !
x1
1 ! , !
x2
2 ! , ………….…., !!
x!1
1
"!" , 1!!
!!
x! .
# !"#$%&'()*$+ &% ),-)./01% $+ 234+15 &% *6)71%4&8$%&% %*&3 9%1 :;# � +<3"14&# &1$8  f 4+ 9%-=0% %6> %*&3. (!". 6). ?1% 6,)4=..14# &)* +$2%7):
6)* (#&3$+ +'0%1 &) 3-,)14$%, !" , &@0 +$2%7/0 &@0 6%,%630@ ),-).@0'@0. A#<%78, &):
!!
!f
!!f
!!f
!f"!
1112111)0( $
%&
'() "
**$%&
'()*$
%&
'()*! +
,,-
.
//0
1$%&
'() "
**$%&
'()*$
%&
'()*!
222
2 1210
1
!!
!!!+
])1(21[1 222
3"***! !
!+
2
2
3 6
132
6
)12()1(1
!!!!!!
!*"
!"2"
! .
# B0, &/,%, 4"#$%&'4)*$+ &% ),-)./01% $+ 234+15 &% 6%,%630@ *6)71%4&8$%&% 9%1 :;# � $=.14&# &1$8  f 4+ 9%-=0% %6’
%*&3 (!". 7), &>&+ &) 3-,)14$%
!!!
f!!
f!!
f#!11211
$%&
'()**$
%&
'()*$
%&
'()! +
&@0 +$2%7/0 &@0 ),-).@0'@0 %*&/0 +'0%1 $1% %9>$# 6,)4=..14# &)* (#&):$+0)* +$-
2%7):. C'0%1 >$@5,
!!!
f!!
f!!
f#!112112$%&
'()**$
%&
'()*$
%&
'()! +
,,-
.
//0
1$%&
'()**$
%&
'()*$
%&
'()!
222211
!!
!!!+
)21(1 222
3!
!***! +
2
2
3 6
132
6
)12)(1(1
!!!!!!
!**
!**
! .
v
v"1
2v
1v
1
1 O x
y 7
. . .
y=x2
3 O!"#!$%&'(#") !"*()+") 328
!" #$%"&'()", *'+,, ('-./*) ! -0123(%.4 '(%.5& %+) "# 3.4 "E . 6$7./8 429&-
(4 "" !!# !! , ":*%(
""""!!#
"#"#!! limlim .
;:(4/8 3
1limlim $$
"#"# """"!# , <9"='(
3
1$! .
% >), %?0., 29$'.%12"='( %. "0@"A?)4. '( -B2(4, %. :.0.:B)+ =:"/4.2%8'.%.
],[ 1 $$ xx & , "$ ...,,2,1$ 3.4 &C$ %$) %4'8 %$, 2=)B0%$2$, 2( ":"4"/8:"%( ()/4B'(2" 2$-
'(1" $% , "$ ...,,3...,,2,1$ , 3.@()*, /4.2%8-
'.%",, (D9. 8), %*%( %" B@0"42'.
)(1
)(1
)(1
21 "" %f"
%f"
%f"
S '''$ (
%+) ('-./?) %+) "0@"A+)1+) .=%?) (1).4 '4. .3*'$ :0"2<AA42$ %"= #$%"&'()"= ('-
-./"&. ;:(4/8 )()()( 1 $$$ xf%fxf !!& A4.
"$ ...,,2,1$ , @. (1).4
)(1
)(1
)(1
1 $$$ xf"
%f"
xf"
!!& ,
":*%( @. 429&(4
""" !S# !! .
;1).4 *'+,, !E# """"$$
'"#"#limlim . E0. @. 429&(4 !S""
$"#
lim .
",-./01 2/34567
F2%+ f '4. 2=)(98, 2=)B0%$2$ 2( <).
/4B2%$'. ],[ &' , '( 0)( )xf A4. 3B@( ],[ &'x* 3.4 ( %" 9+01" :"= "01#(%.4
.:* %$ A0.G438 :.0B2%.2$ %$, f, %") B5"). %+) x 3.4 %4, (=@(1(, 'x $ ,
&x $ .
H4. ). "012"='( %" ('-./*) %"= 9+01"=
( (D9. 9) (0A.#*'.2%( *:+, 2%" :0"$-
A"&'()" :.0B/(4A'.. 6$7./8:
1
1 %v
f (%k)
%k %2 %1 O x
y 8
. . . . . .
y=x2
)x& a
v$ &
x"-1 x2 ... x1 x"=& .=x0 %" %k
(
%2 %1 O
x
y=f (x) y
f(%1) f(%2) f(%k)
f(%")
xk ... xk-1
9
3 O!"#!$%&'(#") !"*()+") 329
! !"#$%&'() *& +,-.*/(0 ],[ !" .) # ,.&(12/ '3&+,0.*1(0*0, (12&'4
#"!
x$ "# , () *0 ./()$0 !xxxx" # #$$$$# ...210 .
! 5) 2-6) '3&+,-.*/(0 ],[ 1 %% xx " )3,789&'() 0'60$#)*0 8:0 ./()$& %& 20, .;/-
(0*$%&'() *0 &9<:,0 3&' 8;&': =-./ x$ 20, >?/ *0 )( %&f . @& -6#&,.(0
*": )(=0+<: *": &9":$": 0'*<: )$:0,
x$&f&fx$&fx$&fx$&fS ### )]()([)()()( 121 %%#%%%# && .
! Y3&7&9$%&'() *& ##S
%'(lim .
A3&+),2:>)*0, B*, *& ##S
'(lim '3-#;), .*& ) 20, )$:0, 0:)C-#*/*& 03B */: )3,7&-
91 *": ./()$": %& . @& B#,& 0'*B &:&(-%)*0, !"#$%&' *&' )3,38+&' ;"#$&' '
20, .'(=&7$%)*0, () )(&, . D$:0, E0:)#B B*, 0)( *'( .
$ -../01 2/3 /4056-./3 /7/8794:612/;
F.*" (,0 .':-#*/./ f ) * # + , - . .*& ],[ !" . G) *0 ./()$0
!xxxx" # #$$$$# ...210 ;"#$%&'() *& +,-.*/(0 ],[ !" .) # ,.&(12/ '3&+,0-
.*1(0*0 (12&'4 #"!
x$ "# .
5*/ .':8;),0 )3,789&'() 0'60$#)*0 8:0 ],[ 1 %%% xx& "+ , 9,0 2-6) }...,,2,1{ #%+ ,
20, .;/(0*$%&'() *& -6#&,.(0
x$&fx$&fx$&fx$&fS #%# )()()()( 21 %%%%%# &&
*& &3&$& .'(=&7$%)*0,, .>:*&(0, "4 )C14:
,#
##
%%# x$&fS
1
)( (1) .
(1) @& -6#&,.(0 0'*B &:&(-%)*0, 8:0 -6#&,.(0 RIEMANN.
xv-1 &v
y=f(x)
&k
&2 &1
x
x2 x1 xv=! a=x0 O
y10
3 O!"#!$%&'(#") !"*()+") 330
A!"#$%&'($)*% +)%,
“," +-%" )". */-"012*)"3 !S , #45*#6 )" !"
#$%
&'()*
,
--,.x/f
1
)(lim (1) .!7-8$% 1)" +
&*% $0'*% *'$97-)4)" *!+ )4' $!%5":6 );' $'#%72$1;' 142$0;' "# ”.
," !*-*!7'; +-%" (1) "'"27<$)*% !"#$%&'! !(!)(*"+%, )43 1.'$8"(3 1.'7--
)4143 f *!+ )" $ 1)" %, 1.2="50<$)*% 2$ ,0
1dxxf )( &*% #%*=7<$)*% “"5"&56-;-
2* )43 f *!+ )" $ 1)" %”. >45*#6,
, ' !!"
#$$%
&(
()*
$
%
!
""!
x&#fdxxf1
)(lim)(
," 1(2="5" , "?$05$)*% 1)"' Leibniz &*% "'"27<$)*% $-%.!(! "5"&56-;143. @.)+ $0'*% $!%26&.'14 )". *-8%&"( :-722*)"3 S )43 5A943 Summa (7/-"%12*).
B% *-%/2"0 $ &*% % "'"27<"')*% /"#, )43 "5"&56-;143. C A''"%* “+-%*” $#D #$'
A8$% )4' 0#%* A''"%* )". "-0". )". 2". &$?*5*0".. E)4' A&?-*14 ,%
$dxxf )( )"
:-722* x $0'*% 2%* 2$)*=54)6 &*% 2!"-$0 '* *')%&*)*1)*/$0 2$ "!"%"#6!")$ 75-
5" :-722*. F)1%, :%* !*-7#$%:2*, "% $&?-71$%3 ,%
$dxxf )( , ,
%
$dttf )( 1.2="50-
<".' )" 0#%" "-%12A'" "5"&56-;2* &*% $0'*% !-*:2*)%&+3 *-%/2+3, 1$ *')0/$14
2$ )" , dxxf )( !". $0'*% A'* 1('"5" 1.'*-)61$;'.
G0'*%, +2;3, 8-61%2" '* $!$&)$0'".2$ )"' !*-*!7'; "-%12+ &*% :%* )%3 !$-%-!)D1$%3 !". $0'*% %$ - 6 %$ ( , ;3 $963:
. , ,/(%
$
$
%dxxfdxxf )()(
. , ($
$dxxf 0)(
@!+ )".3 "-%12"(3 )". $2=*#"( &*% )". "-%-12A'". "5"&54-D2*)"3 !-"&(!)$% +)%:
@' 0)( 0xf :%* &7/$ ],[ %$x1 , )+)$ )" "5"-
&56-;2* ,%
$dxxf )( #0'$% )" $2=*#+' )('E
)". 8;-0". ' !". !$-%&5$0$)*% *!+ )4 :-*?%-&6 !*-71)*14 )43 f )"' 79"'* xx2 &*% )%3 $./$0$3 $x ( &*% %x ( (E8. 11). >45*#6,
, (0
1&Edxxf )()( .
G!"2A';3,
@' 0)( 0xf , )+)$ , 0%
$dxxf 0)( .
% $
'
O x
y=f (x)
y 11
3 O!"#!$%&'(#") !"*()+") 331
,-.%+"*H
N! !"#$%&'(%) *+& ! "#/
00/cdxc )( , ,&! #"#&#$-"#+% $c %%%%.
.1"2,(3$
i) !" !" # , #$#% ! "#"##"
""!c""ccdx )()(0 .
ii) !" !" & , #$#%, %&%'() * cxf #)( %+",' -."%/)0 -#1 ],[ !" , 2/1.3%
! !#!
"
!
"dxxfcdx )(
]))()()([(lim 21 x#$fx#$fx#$f %%'''#
()*
)]()()([lim 21 %%$f$f$f
%"!
'''"
#()
*
+,
-./
0 '''"
#()
)(lim ccc%"!
%*
)(lim "!cc%%"!
%"#+
,
-./
0 1"
#()
iii) !" !" 2 , #$#%
!! "#""#"#"
!
!
""!c!"ccdxcdx )()( .
./0120
!" 02c , #$#% #1 !!
"cdx %45678%' #1 %39,($"
%"$0 16:1;<"+1. 3% 97-* "! " 4,' =>10 c
(?/. 12).
(4567879: 7;< ;=5>?@A;< ;B;CB8=D?07;:
@% #* 91):%', #1. 16'-31= #1. 16'-32"1. 1A14A*6B3,#10 ,&1(%'4"=1"#,' #,
&,6,47#< :%<6)3,#,.
! " O x
y=c
y12
3 O!"#!$%&'(#") !"*()+") 332
!"#$%&' 1(
!"#$ gf , ! " # $ % $ & ' "%&'(#)"*+, "#- ],[ () .'+ !*+, ". /0#* +"12-%&
# $ $%(
)
(
)dxxf+dxxf+ )()(
# $ $ $&%&(
)
(
)
(
)dxxgdxxfdxxgxf )()()]()([
.'+ 3*&+.4
# $ $ $&%&(
)
(
)
(
)dxxg*dxxf+dxxg*xf+ )()()]()([
!"#$%&' 2(
5& 6 f *7&'+ ! " # $ % , ' "* 8+4"#69' - .'+ -.() !,, , #0#* +"12*+
$$$ &%(
.
.
)
(
)dxxfdxxfdxxf )()()(
:+' ;'(48*+39', '& 3)(3
0$ %dxxf .'+ 7)(4
0$ %dxxf , #0#*
473)()()()()(3
0
4
0
0
3
4
0
4
3%&'%&'%&% $ $$ $$ dxxfdxxfdxxfdxxfdxxf .
)%&"*#)%
5& 0)( (xf .'+ (.) )) (<1. 13), 6 ;'(';4&$
+8+0#6#' 86=>&*+ 0#+: )()()( 21 /0/0/0 &%
'?-2
$%.
)dxxf/0 )()( 1 , $%
(
.dxxf/0 )()( 2
.'+
$%(
)dxxf/0 )()( .
!"#$%&' 3(
!"#$ f 9+' ! " # $ % , ' "%&4(#6"6 "* @&' 8+4"#69' ],[ () . 5& 0)( (xf 3+'
.4A* ],[ ()x! .'+ 6 "%&4(#6"6 f 8*& *7&'+ ;'&#-2 968@& "#- 8+4"#69' '%#0,
#0#*
$ *(
)dxxf 0)( .
( . )
/2 /1
O x
y=f (x)
y13
3 O!"#!$%&'(#") !"*()+") 333
,)#$)-()
!" #$!%!&
1. A! ! "4
19)( dxxf , ! "
4
311)( dxxf "#$ ! "
8
113)( dxxf , !# %&'()' )# *+*-
"+,&-.#)#:
i) !3
4)( dxxf ii) !
8
4)( dxxf
iii) !3
1)( dxxf iv) !
8
3)( dxxf .
2. /# #0*1'(2')' 3)$
! !"e
edt
ttdt
1
1 1lnln .
3. /# 40*+*5(6')' )* ! 7)6$, -6)'
! ! "#
$#$
!
!
dxx
dxx
x
1
1
22
2
31
5
1
4.
4. 8! ! "3
15)( dxxf "#$ ! $"
3
12)( dxxg !# 40*+*5(6')' )# *+*"+,&-.#)#:
i) ! $3
1))(6)(2( dxxgxf ii) ! $
1
3))()(2( dxxgxf .
3.5 H )./,%'$)$ !"x
0dttfF(x) )(
9 40*+*5$6.3: '!3: *+*"+,&-.#)*: !"
#dxxf )( "#)'4;'(#! #03 )*! *&$6.3 '(!#$
64!<;=: .(# 1>6"*+, "#$ 0*+> "*0$#6)$"< 1$#1$"#6(#. ?),! 0#&@5&#A* #4)< ;#
#!#B,)<6*4.' )&30* 40*+*5$6.*> *+*"+,&=.@)=! C=&(: ), C&<6, )*4 *&$-6.*>. ?’ #4)3 ;# .#: %*,;<6'$ )* 5!=6)3, =: $ % & % ' ( ) * % + $ % ) , - & # . / 0 / ' / ! ' - , 1 . ( ! / 2 ' / 3 ( 4 & / 2. D #031'$2, )*4 ;'=&<.#)*: #4)*>
6),&(B')#$ 6)* '03.'!* ;'-&,.#, )* *0*(* .#: '2#6A#+(B'$ ),! >0#&2, 0#&@-
5*46#: .$#: 64!'C*>: 64!@&),6,: f 6' 7!# 1$@6),.# 5.
3 O!"#!$%&'(#") !"*()+") 334
!"#$%&'
!" f #$"%& '&% ()"#*+, ()"-./0(0 (# 1"% 2&-(/0'% ! 3%& " #$"%& 1"% (0'#$4
/4) !, /5/# 0 ()"-./0(0
!"x
"dttfxF )()( , !x# ,
#$"%& '&% 6%.-74)(% /0, f (/4 !. 809%2+ &(*:#&:
)()( xfdttfx
a"
$%&'(
)* ! , 7&% 3-;# !x# .
<&% 6%.-2#&7'%
xtdtx
2
0
2 0'0' "$%&'(
)* ! 3%& xtdt
x
lnln0
"$%&'(
)* ! .
()*+,A
+ =646/&3- /4 ()'61.%('% /4) 6%.%6-">
;#>.+'%/4, 6.43:6/#& (?*. 14) >, #@+,:
!,
"-,hx
xdttfxFhxF )()()(
"='A%25" /4) *>.$4) #.
hxf ./ )( , 7&% '&3.- 00h .
B.%, 7&% '&3.- 00h #$"%&
)()()(
xfh
xFhxF/
-,,
465/#
)()()(
lim)(0
xfh
xFhxFxF
h"
-,"$
1
+ !65 /4 6%.%6-"> ;#C.0'% 3%& /4 ;#C.0'% 6%.%7C7&(0, (:";#/0, ()"-./0-
(0, 6.43:6/#& 5/&:
)())(()( xgxgfdttfg(x)
,$."
$%&'(
)* ! ,
'# /0" 6.4D65;#(0 5/& /% *.0(&'464&4:'#"% (:'A49% 1*4)" "50'%.
<&% 6%.-2#&7'%,
xxxxxxtdtx
ln93)ln3()()(lnln 2233
0
3
""$."$
%%&
'(()
*!
!"#$%&' (!-.-/012-3 4-156.7 89: 9/9;/65<80;9= /9>0?.9=)
E(/> f '&% ()"#*+, ()"-./0(0 (’ 1"% 2&-(/0'% ],[ $" . !" G #$"%& '&% 6%.--
74)(% /0, f (/4 ],[ $" , /5/#
! -"$
""G$Gdttf )()()(
$ x " O
x
F(x) f (x)
y=f (x)
y14
3 O!"#!$%&'(#") !"*()+") 335
,-"./(0$
!"#$%&' #( )* +,*-.*"#(&* /(0,-#', - 12&3,)-1- !"x
!dttfxF )()( (4&'5 #5'
+',3.*21' )-6 f 1)* ],[ "! . 7+(589 :'5 - G (4&'5 #5' +',3.*21' )-6 f 1)*
],[ "! , /' 2+3,;(5 #c $ )<)*5*, 01)(
cxFxG %" )()( . (1)
=+> )-& (1), .5' !x " , <;*2#( ! "%"%"!
!ccdttfc!F!G )()()( , *+>)( )(!Gc " .
7+*#<&%6,
)()()( !GxFxG %" ,
*+>)(, .5' "x " , <;*2#(
! %"%""
!!Gdttf!G"F"G )()()()()(
:'5 3,'
! &""
!!G"Gdttf )()()( . !
?*@@<6 $*,<6, .5' &' '+@*+*591*2#( )56 (:$,31(56 #'6, 12#A*@4B*2#( )- 85'-
$*,3 )()( !G"G & #( "!xG )]([ , *+>)( - 51>)-)' )*2 +','+3&% /(%,9#')*6 .,3-
$()'5 "
!
"
!
"! dxxfxGdxxf! !"" ][ )()]([)( .
C5' +',38(5.#',
! "&"'(
)*+
,"
3
1
3
1
2
42
1
2
9
2
xxdx
! "%&"&"# # #xxdx0
0 212&012&]12&[-#
! "&""e
e exdxx1
1 11lnln][ln1
.
/1,%+"*H
1. !"#$%&' ( )*#+,%()(
dttxFx
! &"2
2 1)(
3 O!"#!$%&'(#") !"*()+") 336
i) !" #$%&%' () *%+') )$,-.)/ (01 F.
ii) !" .%2%(0&%' 31 *$)1 (0 .)4)()4'" 5", (" "5$6("(" 0 F.
!,)$
i) ! "#$%&'("( 1)( 2 !" ttf )*+, -+./0 0&,"102 '0 "2$030
),1[]1,( #$%!!# .
4,5 $5 0&/6+'5, ( F, -&)-+, '5 %7&5 2, x '0# 03073(&815'09 $5 5$:70#$ "'0
/.,0 .,%"'(15 '0# -+./0# 0&,"102 '(9 f. ;&5, -&)-+, ),1[ #$&x , 0-<'+ '0 -+./0
0&,"102 '(9 F +/$5, '0 "2$030 ),1[ #$ .
ii) 4,5 ),1[ #$&x )*0#1+:
11)( 2
2
2 !"'
(()
*++,
-!"' . xdttxF
x
.
=-+,.: ( F +/$5, "#$+*:9 "'0 ),1[ #$ 75, ,"*2+, 0)( /' xF >,5 7%?+ ),1( #$&x , (
"#$%&'("( F +/$5, >$("/@9 52A0#"5 "'0 ),1[ #$ , 0-<'+ -5&0#",%6+, +3%*,"'0 '0
0)1( "F .
+-./0/1 /2/32456789
0 B '2-09 '(9 03073:&@"(9 75'% -5&%>0$'+9 >,5 '0 0&,"1)$0 03073:&@15
-5/&$+, '( 10&C:
. . '!"'!
"
!
"!" dxxgxfxgxfdxxgxf )()()]()([)()( ,
<-0# gf '', +/$5, "#$+*+/9 "#$5&':"+,9 "'0 ],[ !" .
4,5 -5&%.+,>15, 59 #-030>/"0#1+ '0 03073:&@15 ."2/
0"#$
#xdxx$ . D*0#1+:
. . '!"'"2/
0
2/
0
2/
0(1)((1)(1( ][
# ## xdxxxxdxxxI
.!"2/
0
2/
0(1(1 ][
## xdxxx
2/
0
2/
0][][ "#$(1 ## xxx $"
2
21
2
!"!"##
.
0 B '2-09 03073:&@"(9 1+ 5335>: 1+'5E3(':9 >,5 '0 0&,"1)$0 03073:&@15
-5/&$+, '( 10&C:
3 O!"#!$%&'(#") !"*()+") 337
! !"#!
"
u
uduufdxxgxgf
2
1)()())(( ,
!"#$ gf #, %&'() *$'%+%&, *$'(-./*%),, )(xgu " , dxxgdu )(#" 0() )(1 "gu " , )(2 !gu " .
1)( "(-23%)45(, (, $"#6#4&*#$5% .# #6#06/-75( !"e
dxx
xI
1
ln.
8+#$5%:
! #"e
dxxxI1
)(lnln
A' 9:*#$5% xu ln" , .!.% dxxdu )(ln #" , 01ln1 ""u 0() 1ln2 "" eu . ;"#5:'7,,
! "$%
&'(
)""
1
0
1
0
2
2
1
2
uuduI .
,-.%+"*H
N! "#$%$&'()$*+ ,! $%$-%./01!,!
i) dxx
xx! *+2
1
2 1 ii) dx
x
x! +4
1
1 iii) dxx!*
5
1|2-| .
!/)$
i) <;+#$5% ! ! ! !*+"*+2
1
2
1
2
1
2
1
22 11dx
xdx
x
xdx
x
xdx
x
xx
! ! !*+"2
1
2
1
2
1
11 dx
xdxxdx
2ln2
52ln1
2
12ln
22
1
2
1
2
1
2
][][ *"*+*"*+$%&
'()" xx
x.
ii) 8+#$5% ! ! !+"+4
1
4
1
4
1
11dx
xdx
x
xdx
x
x! ! *+"
4
1
4
1
2/12/1 dxxdxx
3
202
3
2
2/12/3
4
1
4
1
3
4
1
2/14
1
2/3
][ "+$%&
'()"$
%
&'(
)+$
%
&'(
)" xx
xx.
3 O!"#!$%&'(#") !"*()+") 338
iii) !"#$%& !"#
$%&%
'%2,2
2,2|2|
xx
xxx , '()*+#
( ( (% %%
')'*+
,-.
/%)*
+
,-.
/%'%)%'%
5
1
2
1
5
2
5
2
22
1
2
92
9
2
92
222)2()2(|2| x
xxxdxxdxxdxx .
,)#$)-()
!" #$!%!&
1. N, *")-)./0#1# 1, )-)2-345+,1,:
i) ( )%2
0
2 )123( dxxx ii) ( )e
dxx
x
13
1
iii) ( %2/
0)3+20*6(
!dxxx iv) ( 0
12
345 )
2
1
21
dxx
x .
2. N, ,")%#/7#1# 81$
( ( ')
)))2
1
1
222
3
2
3
52
5
7dx
x
xdx
x
xx.
3. 9, ,")%#/7#1# 81$
( %'6"
##f"fdxxfxf 22 ))(())(()()(2 .
4. :6 3 .4,;$2& ",4<01,03 13= 0*6<41303= f %$'4(#1,$ ,"8 1, 03+#/,
)0,0(A 2,$ )1,1(B , 6, >4#/1# 136 1$+& 1)* )-)2-345+,1)= ( 61
0)( dxxf ,
#;80)6 3 f 6 #/6,$ 0*6#(&= 01) ]1,0[ .
5. 9, >4#/1# 1$= ",4,.5.)*= 1?6 0*6,41&0#?6
i) ( %'x
dttxF0*6
1
21)( ii) ('1 0*6
)(x
$d$$
xF
6. i) 9, >4#/1# 136 ",4<.?.) 13= 0*6<41303= )1ln()( 2 ))' xxxf
3 O!"#!$%&'(#") !"*()+") 339
ii) !" "#$%&'(&)& *)+ ! "#"
1
02
)21ln(1
1dx
x.
!" #$%&%'
1. ,- ! "#x
xxdtttg0
64)( .+" /01& $x %, -" 23&')& )$ )1(g .
2. !" "#$%&'(&)& *)+ 4 56-03)454 dtexfx
x
t!!"
#1
256-)( &'-"+ 5)"1&37.
3. ,- !&
#2
0)(
x
tdt
e
txf , -" #3$5%+$3'5&)& )" %+"5)78")" 8$-$)$-'"9 /"+
)" )$#+/0 "/3*)")" )49 f.
4. A- !#x
dttxfxF0
)()( , -" 23&')& )4- )(xF ' .
5. !" "#$%&'(&)& *)+ 4 56-03)454 ! ! ""
"#
x x
dtt
dtt
xF1
/1
122 1
1
1
1)( &'-"+
5)"1&37 5)$ ),0( (" /"+ -" 23&')& )$- ):#$ )49.
6. !" 23&')& )$ !"
)"
h
hdtt
h
2
2
2
05
1lim .
7. !" 6#$;$.'5&)& )" $;$/;43<8")"
i) ! &
6
42 4
dxx
x ii) ! "&"
2/
0
)]56-(4848)56-(48[
!
dxxxxxx .
8. N" 6#$;$.'5&)& )" $;$/;43<8")"
i) ! &&2
0
2 )|1|( dxxx ii) !&!
!dxxf )( , "-
*+,
-.--&
#!xx
x!xxf
0,480,
)(
iii) ! "&3
0
2 |23| dxxx .
9. N" 6#$;$.'5&)& )" $;$/;43<8")"
i) !2
1
lne
dxx
x ii) ! &
1
0dxxe x
iii) ! "1
0
2 )9ln( dxxx iv) !2/
0256-
!x xdxe .
3 O!"#!$%&'(#") !"*()+") 340
10. !" !"2/
0
2#$!
xdxxI , !"2/
0
2%&"!
xdxxJ , "' &()*)+,%-.- .' )*)/*#01-
$'.'
JI # , JI $ , ", J.
11. 2%.3 $4' %&"50.#%# f $- f %% %&"-67 /'4 +4' .#" )(),' 4%68-4
! "%%#!
xdxxfxf0
2#$)()( )( .
!" 1)( "!f , $- .# 9)7:-4' .#; )*)/*703%#; /'.5 ('05+)".-;, "'
&()*)+,%-.- .) )0(f .
12. 2%.3 )4 %&"'0.7%-4; gf , , $- f %% , g %% %&"-6-,; %.) ],[ #$ . !" 0)()( "" $g$f /'4 )()( #g#f %"% , "' '()<-,=-.- >.4
! $%"%%$%%"#
$#g#f#gdxxgxfxgxf" ))()()(()()()()( )( .
3.6 ,-&%$+. +-)$) '(+$) '"/
"!"#!$%&'(#"/ !"*()+"/
?- .# 9)7:-4' .)& :-$-*41<)&; :-307$'.); .)& )*)/*#03.4/)8 *)+4%$)8 $()-
0)8$-, .10', "' '()<-,=)&$- .) ('0'/5.3 :-10#$' ()& -,"'4 +"3%.> 3; % & ' ( ) * $ + , - ) . / 0 * 1 . 2 3 4 5 3 ) ( 6 7 0 5 4 8 9 4 : 0 - * 4 8.
!"#$%&'
!" $4' %&"50.#%# f -,"'4 %&"-67; %.) ],[ #$ , .>.- &(506-4 @"', .)&*564%.)", ),( #$; & .@.)4), 1%.-
! $"#
$$#;fdxxf ))(()(
.0"1-(2$
A-30)8$- .# %&"50.#%# !"x
$dttfxF )()( . B %&"50.#%# '&.7 -,"'4 ('0'+3+,%4-
$# %.) ],[ #$ /'4 4%68-4 )()( xfxF "% . C()$@"3;, %8$D3"' $- .) :-10#$' $@%#; .4$7; .)& <4'D)04/)8 *)+4%$)8 &(506-4 ),( #$; & .@.)4), 1%.-
$#$F#F;F
$$
"% )()()( . (1)
C,"'4 >$3;,
)()( ;f;F "% , !"#
$dttf#F )()( /'4 ! ""
$
$dttf$F 0)()( .
3 O!"#!$%&'(#") !"*()+") 341
3 O!"#!$%&'(#") !"*()+") 341
!"#$%&'(, ) *+,-)-. (1) /0123-.* !"
dttf#f
"
!
!"# )(
)( 4, *+#56&.$.,
# !""
!!"#fdttf ))(()( . !
!"#$%#
7 .0*8$,( !"
dxxf#f
"
!
!"# )(
)( 9%/3-.* &'() *+&, -)( +:&10-)+)( f +-# ],[ "! ;.*
+:$<#9=>3-.* $3 _f .
?3'$3-0*;1, ) $%+) -*$4 _f $*.( $) .0&)-*;4( +:-
&10-)+)( f +-# 5*1+-)$. ],[ "! ".0*+-1&3* -# 6@#( -#: #08#/'&=#: "#: %A3* <1+) -# ],[ "! ;.* 3$<.-
5,& =+# $3 -# 3$<.5,& -#: A'0=#: & "#: "30*;93=3--.* .", -) /0.2*;4 ".01+-.+) -)( f, -#& 1B#&. xx$
;.* -*( 3:83=3( !x " ;.* "x " (CA. 15).
,-.%+"*$
-(*. ) (/012*)() xxf ")( . 34 526768 9)(0,%/ '*(+ 9(*6 _f/f ")( .
!0)$
DA#:$3 29
18
9
3
2
9
9
0
2/39
0 ""&'(
)*+
""# xdxx_
f .
!"#$%&'(, .0;3= &. <0383= )9,0(%# %-+*, E+-3 2)( "#f . DA#:$3
422)( ","," ###f .
F0., -# >)-#6$3&# # 3=&.* -# 4.
.)#$),()
:; #<:=:!
1. N. <03=-3 -) $%+) -*$4 _f -)( +:&3A#6( +:&10-)+)( f +-# 5*1+-)$.
]1,0[ , .& 5=&3-.* ,-* # "!1
00)1)(( dxxf .
"
f
# ! O x
y=f (x)
y15
3 O!"#!$%&'(#") !"*()+") 342
2. !" # f $%"&' ()"$*+, (-. ],[ !" , # (-&/$01 2&' ! "#!
"dx#xf 0))(( , "&
&3.4$%5$-$ 6-' # 78(# -'7+ -#, f (-. ],[ !" $%"&' #.
3. 9& :0$/$% # 78(# -'7+ -#, 7$-&:;#-+, x (-. 4'1(-#7& ],[ !" .
!" #$%&%'
1. <%"."-&' .' ()"&0-+($', 2)( xxf " 2&' 2
1)(
xxg " , .0'(78"$, (’ 8"&
4'1(-#7& ],[ !" , 0$" . 9& )3.;.=%($-$ -', _f , g 2&' "& &3.4$%5$-$
6-' 1$% g_f .
2. > -&*?-#-& $ -.) &%7&-., (’
8"& &==$%. &2-%"&, R 2&' 7+-
2.), & , ($ &36(-&(# r &36 -." 2$"-0'26 15."& -.) &==$%.) $%-
"&' )(4
)( 22 rRn
Pr$ #"
&, 63.) %
# 4'&@.01 3'8($A, 7$-&5? -A" 120A" BA, -.) &==$%.) 2&' n -. '5B-
4$, -.) &%7&-., ((-&/$01).
&) 9& :0$%-$ -# 78(# -&*?-#-& -.) &%7&-.,, 6-&" ],0[ Rr' .
:) 9& :0$%-$ -# 78='(-# -&*?-#-& 2&' "& -# ()=20%"$-$ 7$ -# 78(# -&-
*?-#-&
3. C(-A f 7'& 3&0&=A=%('7# (-. ]1,0[ ()"10-#(#, 7$ ! "1
0)1()( fdxxf .
9& &3.4$%5$-$ 6-' # fC 8*$' 7'&, -.);1*'(-.", .0'D6"-'& $@&3-.78"#.
3.7 E+,-."/ 01(10."2 3&%("2
( E-#" 3&01=0&@. 4.4 $%4&7$ 6-', &" 7'& ()"10-
-#(# f $%"&' ()"$*+, ($ 8"& 4'1(-#7& ],[ !" 2&' 0)( )xf ='& 21/$ ],[ !"x' , -6-$ -. $7:&46" -.)
*A0%.) & 3.) .0%D$-&' &36 -# =0&@'2+ 3&01(-&-
(# -#, f , -', $)/$%$, "x " , !x " 2&' -." 15."&
xx* (E*. 16) $%"&'
B A
R u(r) r
l
!
&
" O x
y=f (x)
y16
!"!
"dxxf&E )()(
3 O!"#!$%&'(#") !"*()+") 343
!"# $#%&'(")*#, +, (*-#'./ +,0 12%3,0 $,0 $(-%"45(3(+#" #$. +6 )%#7"48 $#%&9+#96 +6:
xxf !)( , +,/ &;,/# xx" 4#" +": (0<(3(: 0!x ,
1!x (=1. 17) (3/#" 39, *(
# # !$%&
'()!!
1
0
1
0
1
0
2/32/1
3
2
3
2xdxxdxx .
* >9+2, +?%#, '0, 90/#%+89(": f 4#" g, 90/(1(3: 9+, '"&9+6*# ],[ !" *( 0)()( ++ xgxf )"# 4&<( ],[ !"x, 4#" # +, 12%3, $,0 $(%"45(3(+#" #$. +": )%#-
7"4@: $#%#9+&9(": +2/ gf , 4#" +": (0<(3(: "x ! 4#" !x ! (=1. 18#).
#
(") O x
y=g(x)
y=f (x) y
#1
(!) O x
y=f (x) y
#2
($) O x
y=g(x)
y18
A#%#+6%,B*( .+"
# # # -!-!-!!
"
!
"
!
"dxxgxfdxxgdxxf#%#%#% ))()(()()()()()( 21 .
C$,*@/2:,
!"# $#%&'(")*#, +, (*-#'./ +,0 12%3,0 #
$,0 $(%"45(3(+#" #$. +": )%#7"4@: $#%#-
9+&9(": +2/ 90/#%+89(2/ 2)( .-! xxf 4#" 2)( xxg ! (=1. 19) (3/#" 39, *(:
#- -!1
2)]()([)( dxxgxf#E
#- -.-!1
2
2 )2( dxxx
2
9
32
2
1
2
32
!$%&
'() -.-!
-
xx
x.
* D +B$,: (1) -%@<64( *( +6/ $%,E$.<(96 .+":
1
1 O x
y x!
y17
(1) # -!!
"dxxgxf#E ))()(()(
x
y=x2
#
-2 1
y=-x+2
O
y19
3 O!"#!$%&'(#") !"*()+") 344
(i) )()( xgxf ! !"# $%&' ],[ !"x" $#"
(ii) (" gf , ')*#" +, #-*,."$/0 1.( ],[ !" .
2# #3(4')5(6+', .7-#, 8." ( .93(0 (1) "1:9'" $#" :;-)0 .,* 638&'1, (ii). <-%!-+#.", '3'"4= (" 16*#-.=1'"0 gf , ')*#" 16*':')0 1.( ],[ !" , &# 63%-:'" #-"&+80 "c # ./.("(0 71.' 0)()( !$!$ cxgcxf , !"# $%&' ],[ !"x" . >)*#" ?#*'-8 8." .(
:;-)( # (@:. 20#) /:'" .( )4"( '+A#48* +' .( :;-)( # % (@:. 20A).
! "
(")
#
O x
y
y=g (x)
y=f (x)
! "
(!)
#
O x
y
y=f (x)+c
y=g (x)+c
20
>3(+/*;0, 19+?;*# +' .(* .93( (1), /:(6+':
& & '($'$(%(!
"
!
"dxxgxfdxcxgcxf#$#$ ))()(()])(())([()()( .
B-#,
& '(!
"dxxgxf#E ))()(()(
) C' ., A(=&'"# .(6 3-(,!(9+'*(6 .93(6 +3(-(9+' *# 63(D(!)1(6+' .( '+A#-
48* .(6 :;-)(6 # 3(6 3'-"$D')'.#" #38 .(* %5(*# xx% , ., !-#?"$= 3#-%1.#1, +"#0 16-
*%-.,1,0 g, +' 0)( *xg !"# $%&' ],[ !"x"
$#" ."0 '6&')'0 "x ( $#" !x ( (@:. 21).
<-%!+#.", '3'"4= ( %5(*#0 xx% ')*#" , !-#-
?"$= 3#-%1.#1, .,0 16*%-.,1,0 0)( (xf ,
/:(6+'
& '(!
"dxxgxf#E ))()(()(
& &'('(!
"
!
"dxxgdxxg )()]([ .
>3(+/*;0, #* !"# +"# 16*%-.,1, g "1:9'" 0)( *xg !"# $%&' ],[ !"x" , .8.'
&'(!
"dxxg#E )()(
!
#
" O
x
y=g (x)
y21
3 O!"#!$%&'(#") !"*()+") 345
!"# $#%&'(")*#, +, (*-#'./ +,0 12%3,0 ! $,0
$(%"45(3(+#" #$. +6 )%#7"48 $#%&9+#96 +6: 1)( 2 !" xxg 4#" +,/ &;,/# xx# (<1. 22) (3/#"
39, *(
$ $! !!"!!"
1
1
1
1
22 )1()1()( dxxdxx!E
3
4
3
1
1
3
"%&
'()
*!"
!
xx .
+ =+#/ 6 '"#7,%& )()( xgxf ! '(/ '"#+6%(3 9+#>(%. $%.96*, 9+, ],[ "# ,
.$2: 9+, <18*# 23, +.+( +, (*-#'./ +,0 12%3,0 ! $,0 $(%"45(3(+#" #$.
+": )%#7"4?: $#%#9+&9(": +2/ gf ,
4#" +": (0>(3(: #x " 4#" "x " (3/#" 39, *( +, &>%,"9*# +2/ (*-#'@/ +2/ 12%32/ 21 ,!! 4#" 3! . A65#'8,
)()()()( 321 !$!$!$!$ ,,"
$ !"%
#dxxgxf ))()(( $ $ !,!,
&
%
"
&dxxgxfdxxfxg ))()(())()((
$ $ $ !,!,!"%
#
&
%
"
&dxxgxfdxxgxfdxxgxf |)()(||)()(||)()(|
$ !""
#dxxgxf |)()(|
B$,*?/2:,
$ !""
#dxxgxf!E |)()(|)(
!"# $#%&'(")*#, #: 0$,5,)39,0*( +, (*-#'./ +,0 12%3,0 ! $,0 $(%"45(3(+#" #$. +": )%#7"4?: $#%#9+&9(": +2/ 90/#%+89(2/ 3)( xxf " ,
xxg ")( 4#" +": (0>(3(: 2!"x , 1"x . (<1. 24).
C%1"4& -%394,0*( +": %3D(: 4#" +, $%.96*, +6: '"#7,%&: )()( xgxf ! 9+, '"&9+6*# ]1,2[! .
B$("'8
)1)(1()1()()( 23 ,!"!"!"! xxxxxxxxgxf ,
?1,0*( +,/ #4.5,0>, $3/#4#:
O
y
y=x2+1
x 1
-1
-1 !
22
y=g (x) y=f (x)
!3
O
!2
!1
y
x & " # %
23
y=x
y=x3
' -2 -1
1 x
y24
3 O!"#!$%&'(#") !"*()+") 346
!"#$%&'&("), (*+", ,-./0 ('& -"+"-%&1 -2&"3", 45',#6
! ! !!" "
"
""#"#"$"$
1
2
0
1
1
0
1
2))()(())()(())()((|)()(|)( dxxfxgdxxgxfdxxfxgxgxf!E
! ! !"
" ""#"#"$
1
2
0
1
1
0
333 )()()( dxxxdxxxdxxx
4
11
422442
1
0
420
1
241
2
42
$%&
'()
*"#%
&
'()
*"#%
&
'()
*"$
"
"
"
xxxxxx.
!"#$%#
78#91&" #6 (" -"+"-%&1 (' !"
#dxxf )( 62-
&": 2;' #6 (' %<+':;#" (1& 6#$"=*& (1& 51+21& -', $+2;3'&(": -%&1 "-. ('& %>'&"
xx+ #62'& (' %<+':;#" (1& 6#$"=*& (1& 51+21& -', $+2;3'&(": 3%(1 "-. ('& %>'&"
xx+ (75. 25)
,-.%+"*,)
1. N& '()*)+ ,- ).'&/01 ,-2 34(+-2 & 5-2 5)(678)+),&6 &50 ,69 :(&;67<9 5&(&=,>=)69 ,41 =21&(,?=)41 xxf @.)( $ , xxg =21)( $ 7&6 ,69 )2*)+)9 0$x
7&6 /x 2$ .
!0)$
?+5:3% $+2;3',#6 (:) +2@6) 3": ('
-+.;0#' (0) =:"9'+%) )()( xgxf "
;(' =:%;(0#" ]2,0[ $ . 7(' =:%;(0#"
",(. 45',#6
xxxgxf ;,&0#)()( $,$
169 $, x
4
$x $, A
4
5$x $
B-'#4&1), C:" (' -+.;0#' (0) =:"9'+%) xxxgxf ;,&0#)()( "$" 45',#6 ('& "3.-
D',<' -2&"3":
x 2" 1" 0 1
)()( xgxf " " 0 + 0 " 0
x " +
"
+ " a
y
%
25
- %
y=;,&x
y=0#x
-/4 5-/4
2- x
y26
3 O!"#!$%&'(#") !"*()+") 347
x 0
4
!
4
5!
!2
)()( xgxf !
! 0 + 0 !
!"#$%&'&("), (*+", ,-./0 ('& -1&"2" ",(.&, 34',#5
" !#!
dxxgxf"#2
0
|)()(|)(
" " " $!$!$$!#4/
0
4/5
4/
2
4/5)6,&0#()6,&0#()6,&0#(
! !
!
!
!dxxxdxxxdxxx
!!
!!
! xxxxxx 24/5
4/54/
4/
0][][][ 0#6,&0#6,&0#6,& $$!!$$#
24212212 #$$$!# .
2. !" #$%&%' () %*#"+,- (). /0$'). 1). 1%$234%'%("2 "1, (5 6$"7238 1"$9-
:(":5 (5; xxf ln)( # , ()- 9<)-" (0- x 3"2 (5- %7"1()*=-5 (5; fC :() :5-
*%') 1),(eA .
!,)$
7 5816960 (0) 5:"-('#3&0) (0) fC 6(' 60-
#51' )1,(eA 51&";
))((1: exefy$ !%#! . (1)
<-5;=> x
xxf1
)(ln)( #%#% , 34',#5 e
ef1
)( #% .
<-'#3&9), 0 (1) ?+%:5(";:
xe
yexe
y1
)(1
1 #&!#! .
@' A0('B#5&' 5#$"=.& # 51&"; 16' #5 (' 5#$"-
=.& (', (+;?*&', #51'& (' 5#$"=.& (', 49+1', -', '+1A5("; "-. (0 fC ('& %8'-
&" xx% 2"; (;) 5,C515) 1#x 2"; ex # , =0D"=>
" " " ''(
)**+
,!!-.
/012#!#
e e e
e
e
dxx
xxxx
exdxxdx
eE
0 1 1
1
0
2 1]ln[
2
1ln
1
2
2ln
2
1
110
2
][][ !#$!-
.
/01
2#
exxx
x
e
ee
e
.
3. !" .1)4)62:(%' () %*#"+,- - (). 3.3423)> +':3). 222 .yx #$ .
y=lnx
$
e
1
1 % x
y27
3 O!"#!$%&'(#") !"*()+") 348
!,)$
!" #$%&'&(%" 1C )*+,% -.,/%&0 1,.234,3# 4#5 36+2.4#3#5
],[,)( 22 !!xx!xf !"!# ,
,/"' -%, 0$y )*+,%
%#& 222 !yx 22 x!y !# .
7+ 1E )*+,% 4" )$8,9:+ 4"6 #$%&6&(*"6, 4:4)
12EE # . ;1)%90 0)( 'xf -%, &2<) ],[ !!x !" ,
=>"6$)
(! !#!
!
dxx!" 221 . (1)
;1)%90 !x! ))! , =>"6$) 11 ))!!x
. ;1"$=+?5, 612.>)% *+,
-./!"
2,
2
##$ 4=4"%",
@34)
$!x #$# . (1)
A43%, =>"6$) $!x #$# , *+,
-./!"
2,
2
##$ , "1:4) $d$!dx 36+# . ;1%1(="+, -%, !x !#
)*+,% 2
#$ !# &,% -%, !x # )*+,% 2
#$ # . ;1"$=+?5,
(( !!
0!#0!#2/
2/
22
2/
2/
2221 36+#$136+#$
#
#
#
#
$d$$!$d$!$!!"
(( !!
#0#2/
2/
22
2/
2/
22 36+36+36+#
#
#
#
$d$!$d$$! (;1)%90 036+ $$ )
24
2#$22
236+1 22/
2/
2/
2/
22#!$$!$d$!
#
#
#
#
#*+
,-.
/ &#&
# (! !
.
B., 212 #!"" ## .
C) 4"+ *9%" 4.:1" ,1"9)%&+'"6$) :4% 4" )$8,9:+ 4#5 =(()%D#5 12
2
2
2
#&%y
&x
)*+,%
*3" $) #&% .
!
!!
!!
! y ! x# !2 2
' x
(c2)
(c1) E1
y28
3 O!"#!$%&'(#") !"*()+") 349
,)#$)-()
!" #$!%!&
1. N! "#$%$&'()*) *$ )+,!-./ *$" 012'$" #$" #)234%)')*!3 !#. *5 &2!-
6347 #!28(*!(5 *59 ("/82*5(59 32)( 2 !"# xxxf , *39 )":)')9 0#x ,
2#x 4!3 *$" 8;$/! *1/ x.
2. N! "#$%$&'()*) *$ )+,!-./ *$" 012'$" #$" #)234%)')*!3 !#. *5 &2!-
6347 #!28(*!(5 *59 ("/82*5(59 f, *$/ 8;$/! *1/ x 4!3 *39 )":)')9 #$" -'/$/*!3 48:) 6$28:
i) 3)( xxf # , 0#x , 27#x ii) x
xf2("/
1)( # , 0#x ,
3
!x # .
3. N! "#$%$&'()*) *$ )+,!-./ *$" 012'$" #$" #)234%)')*!3 !#. *5 &2!-
6347 #!28(*!(5 *59 ("/82*5(59 xxxf 3)( 2 "# 4!3 *$/ 8;$/! *1/ x.
4. N! "#$%$&'()*) *$ )+,!-./ *$" 012'$" #$" #)234%)')*!3 !#. *39 &2!-
634<9 #!2!(*8()39 *1/ ("/!2*7()1/ 3)( xxf # 4!3 22)( xxxg "# .
5. =! "#$%$&'()*) *$ )+,!-./ *$" 012'$" #$" #)234%)')*!3 !#. *5 &2!-
6347 #!28(*!(5 *59 ("/82*5(59 24)( xxf "# 4!3 *5/ )":)'!
02 #"" yx .
'" #$!%!&
1. >(*1 5 ("/82*5(5 23)( xxf #
i) =! ,2)'*) *5/ );'(1(5 *59 )6!#*$+</59 *59 fC (*$ (5+)'$ *59 )3,1(A .
ii) =! "#$%$&'()*) *$ )+,!-./ *$" 012'$" #$" #)234%)')*!3 !#. *5
&2!6347 #!28(*!(5 *59 f, *5/ )6!#*$+</5 *59 (*$ " 4!3 *$/ 8;$-
/! *1/ x.
2. N! "#$%$&'()*) *$ )+,!-./ *$" 012'$" #$" #)234%)')*!3 !#. *5 &2!-
6347 #!28(*!(5 *59 $%
$&'
(
)!"#
1,2
1,3)(
2
xx
xxxf , *39 )":)')9 1"#x , 2#x
4!3 *$/ 8;$/! *1/ x.
3 O!"#!$%&'(#") !"*()+") 350
3. N! "#$%&$ &' $("!)*+ &', -.#%', /', /$#012$%$&!0 !/* &3 4#!5016
/!#78&!83 &39 8,+7#&3839 !"
!#$
%&'
('&')
2,52
2,34)(
2
xx
xxxxf 1!0 &'+ 7:'+!
&.+ x.
4. ;! ,/'2'4%8$&$ &' $("!)*+ &', -.#%', /', /$#012$%$&!0 !/* &09 4#!-
501<9 /!#!8&78$09 &.+ 8,+!#&68$.+
1)( ') xxf 1!0 3
1)(
&)
xxg .
5. i) N! ,/'2'4%8$&$ &' $("!)*+, )(!E , &', -.#%', /', /$#012$%$&!0 !/*
&09 4#!501<9 /!#!8&78$09 &.+ 8,+!#&68$.+ x
exf ))( , xxg ln)( ) ,
&'+ 7:'+! &.+ x 1!0 &3+ $,=$%! !x ) , e! * .
ii) N! "#$%&$ &' *#0' )(lim !"! &+,
.
6. N! ,/'2'4%8$&$ &' $("!)*+ &',
4#!(('810!8(<+', -.#%', &',
)0/2!+'> 8-6(!&'9.
7. ;! ,/'2'4%8$&$ &' $("!)*+ &',
4#!(('810!8(<+', -.#%', &', )0/2!-
+'> 8-6(!&'9.
3 1 # x
y=x y=3x
y=3
y
A
3 # x
y=x2'1
y
y=x2'2x+2
3 O!"#!$%&'(#") !"*()+") 351
8. !"#$%&' ( )*#+,%()(
xxf (-)( !
i) .& /,$"%$ %'0 $1')2)$'0 %3# $4&5%6-7#3# %(0
fC )%& )(-$"& )0,0(O
8&' )0,(!A .
ii) .& /,$"%$ %6 $-/&9:# %6* ;3,"6* 56* 5$,'8<$"$%&' &5: %( =,&4'8> 5&,+)%&)( %(0 f 8&' %'0 $4&5%:-$#$0 )%& )(-$"& " 8&' #.
9. i) .& *56<6=")$%$ %6 $-/&9:# %6* ;3,"6* 56* 5$,'8<$"$%&' &5: %(
=,&4'8> 5,+)%&)( %(0 )*#+,%()(0 xxf !)( , %(# $4&5%:-$#> %(0 )%6 )(-$"6 )1,1( 8&' %6# +16#& %3# x.
ii) .& /,$"%$ %(# $*?$"& $x ! , ( 656"& ;3,"@$' %6 ;3,"6 &*%: )$ 9A6
')$-/&9'8+ ;3,"&.
10. .& *56<6=")$%$ %6 $-/&9:# %6* ;3,"6* 56* 5$,'8<$"$%&' &5: %'0 =,&-
4'870 5&,&)%+)$'0 %3# )*#&,%>)$3# xxf ln)( ! , x
xg1
ln)( ! 8&' %(#
$*?$"& 2ln!y .
11. i) N& /,$"%$ )*#+,%()( f %(0 656"&0 ( =,&4'8> 5&,+)%&)( 9'7,;$-%&' &5: %6 )(-$"6 )2,0(A 8&' ( 8<")( %(0 )%6 )(-$"6 ))(,( xfxM
$"#&' 32 "x .
ii) B6'6 $"#&' %6 $-/&9:# %6* ;3,"6* 56* 6,"@6*# ( fC 8&' 6 +16#&0 %3# x.
12. C)%3 ( )*#+,%()( )3)(1()( ""! xxxf .
i) .& /,$"%$ %'0 $1')2)$'0 %3# $4&5%6-7#3# %(0 =,&4'8>0 5&,+)%&-
)(0 %(0 f )%& )(-$"& %#, 56* ( fC %7-#$' %6# +16#& %3# x.
ii) D# & $"#&' %6 )(-$"6 %6->0 %3# $4&5%6-7#3#, #& &569$"1$%$ :%' (
fC ;3,"@$' %6 %,"=3#6 #%& )$ 9A6 ;3,"& 56* 6 <:=60 %3# $-/&-
92# %6*0 $"#&' 1
2.
x
y
A(5,0) O(0,0)
3 O!"#!$%&'(#") !"*()+") 352
*,-(#,) .)#$),()
!" #$%&%'
1. i) !" #$%&'()*)'+&,-, -%. ".-'/"-0&-"&% x!u !" 1'" ." "*)2,34,--, 5-'
# "!
dxxxf0
)%(( #!
dxxf!
0)%((
2
ii) !" 6*)7)13&,-, -) )7)/7+$8(" # $
!
dxx
xx
02%(3
%(.
2. i) !" 6*)7)13&,-, -) )7)/7+$8(" # !
2/1
02 1
1dx
x
ii) !" 6*)7)13&,-, -) )7)/7+$8(" #2/
3/ %(1
!
!
dxx
.
3. !" 6*)7)13&,-, -) )7)/7+$8("
# $$du
uu )2)(1(
1
/"' &-% &6.9#,'" -" )7)/7%$:("-":
i) dxxx
x# $$ )2%()(1%((
&6. ii) dx
ee
exx
x
# $$ )2)(1(.
4. ;. # $"
$1
02
12
1dt
t
tI
"
" , !%" ,
i) !" 6*)7)13&,-, -) 0<$)'&(" 1$$ "" #I , !%"
ii) !" 6*)7)13&,-, -" )7)/7%$:("-" 0I , 1I , 2I .
5. ;. % &6.0$-%&% f ,3."' &6.,#+= &-) &, ." "*)2,34,-, 5-'
# # # ''(
)**+
,"!
x x u
dudttfduuxuf0 0 0
)())(( .
6. >3.,-"' % &6.0$-%&% #"x
dttfxF1
)()( , 5*)6 duutft
# !"1
2 1)( .
3 O!"#!$%&'(#") !"*()+") 353
i) !" #$%&'% '( )%*&( ($+,-(. '/0 ,10"$'2,%/0 f 3"+ F.
ii) !" ")(*%&4%'% 5'+ 6 F %&0"+ 706,&/8 ".4(1," 3"+ 31$'2.
7. 9&0(0'"+ '" (:(3:6$;-"'"
!"x
t tdtexF0
2,10)( 3"+ !"x
t tdtexG0
26-)( , !#x .
i) !" 1)(:(7&,%'% '" (:(3:6$;-"'"
)()( xGxF $ 3"+ )()( xGxF %
3"+ ,'6 ,10<=%+" '" (:(3:6$;-"'" )(xF 3"+ )(xG .
ii) !" 1)(:(7&,%'% '" (:(3:6$;-"'"
!"!
!t tdteI
22,10 3"+ !"
!
!t tdteJ
226- .
8. >( =/$&( )(1 )%$+3:%&%'"+ ")5 '6 7$"?+32 )"$@,'",6 '68 ,10@$'6,68 1)( 2 $" xxf 3"+ '60 %1A%&" 5"y =/$&B%'"+ ")5 '60 %1A%&" 12 $" "y ,
0&" , ,% *.( +,%-#"*+3@ =/$&". !" #$%&'% '60 '+-2 '(1 ".
9. i) !" #$%A%& '( %-#"*50 )(#$ '(1 =/$&(1 )(1 )%$+3:%&%'"+ ")5 '6
7$"?+32 )"$@,'",6 '68 ,10@$'6,68 2
1)(
xxf " , '(0 @4(0" '/0 x
3"+ '+8 %1A%&%8 1"x , #x " , 0&# .
ii) !" #$%A(.0 (+ '+-<8 '(1 # <',+, ;,'% 2
1)( "#$ .
iii) N" #$%A(.0 '" )(lim0
#E#'
3"+ )(lim #E# $('
.
10. C,'/ gf , *.( ,10"$'2,%+8 ,10%=%&8 ,'( ],[ %" . !" ")(*%&4%'% 5'+:
i) D0 )()( xgxf ) 7+" 3@A% ],[ %"x# , '5'% ! !)%
"
%
"dxxgdxxf )()( .
ii) D0 m 6 %:@=+,'6 3"+ & 6 -<7+,'6 '+-2 '68 f ,'( ],[ %" , '5'%
! %**%%
""%Mdxxf"%m )()()(
iii) E% '6 #(2A%+" '68 "0+,5'6'"8 xx &%? 7+" 3@A% +,-
./0#
2,0!
x , 0"
")(*%&4%'% 5'+ 6 ,10@$'6,6 x
xxf
6-)( " , +
,-
./0#
2,0!
x %&0"+ 706,&/8
?A&0(1," 3"+ ,'6 ,10<=%+" 0" ")(*%&4%'% 5'+:
3 O!"#!$%&'(#") !"*()+") 354
!) !x
x
!3"#
2
33!! $%! &'() "#
$%&'(
3,
6
!!x &!%
*) ) !!3/
6/ 2
1"#4
3!
!
dxx
x.
iv) N! !+,-)./)0) 10% " 234'50"2" 2
)( xexf *+ ).4!% $4"2.67 8(.4,3-
2! 20, ),0[ ,- &!% 20" 2349:)%!, #) 0" *,;()%! 0"7 !4%210"0!7
xe x -.1 $%! &'() !(x , 4! !+,-)./)0) 10%:
!) 1122 !!* *xex $%! &'() ]1,0[(x &!%
*) ) !! *1
01
3
2dxe x .
,%&'$),() #-'-."$)$)
!. "# $%&#'() %*+ ,(- *%.%$),/ *#.(*,01#(- 2% $3$401#,# ,5 6.)''% -,
%2 5 (173.(1'+- #82%( %49&:- $%( ,5 6.)''% /, %2 5 (173.(1'+- #82%( ;#3<:- <($%(545602,%- 1367.+2/- ,92 %*)2,91: 1%-.
1. <2:=)% ) ) )-+-/
0
/
0
/
0dxxgdxxfdxxgxf )()())()(( > ?
2. <2:=)% ) ) )1+1"
#
"
#
"
#dxxgdxxfdxxgxf )()()()( > ?
3. >4 /0 + , 010) ) +/
00)( dxxf . > ?
4. >4 ) +/
00)( dxxf , 010) &!0’ !4'$&" (! ).4!% 0)( +xf $%!
&'() ],[ /0(x .
5. >4 0)( .xf $%! &'() ],[ /0(x , 010) ) ./
00)( dxxf . > ?
> ?
3 O!"#!$%&'(#") !"*()+") 355
6. !" ! "#
$0)( dxxf , #$#% &'#’ '"()&* +' %,"'-
0)( "xf )-' &(+% ],[ #$%x .
7. ! !& &''('
$
$
$
$dxxxdxx )1()1( 244 , )-' &(+% 0)$ . ! .
8. ! !*&4/
0
4/
0
2 /0"ln2)*11ln(+ +
xdxdxx . ! .
9. ! ! ,&* dxxfxxxfdxxf )()()( . ! .
10. ! *e
xdx1
ln !1 1ln
e
dtt
. ! .
11. A" ! *&1
00)1)(( dxxf #$#% 1*f . ! .
12. !" ! *#
$0)( dxxf , #$#% 0)( *-f )-' &(23-3 ),( #$- % . ! .
13. !" ! *#
$0)( dxxf &'- * f 4%" %,"'- 2'"#35 1*46" /#3
],[ #$ , #$#% * f 2',7"%- 403, #308(9-/#3", %#%7$/*1%: #-16:.
14. ;3 383&8<7=1' dxxx!& &1
1
3 )( 2'7-/#("%- #3 %1>'4$" #30
9=7,30 230 2%7-&8%,%#'- '2$ #* )7'?-&< 2'7(/#'/* #*: /0"(7#*/*: xxxf &* 3)( &'- #3" (@3"' #=" x.
!!. "# $%&#'() %*+ ,(- *%.%$),/ *#.(*,01#(- 2% $3$401#,# ,5 1/1,6 %*)-
2,515
1. !" x!xf *1)( *, &'- 0)0( *f , #$#% #3 )1(f -/35#'- 1%
!) !1
& , A) !1
, B) !2&
, C) !2
.
2. To 383&8<7=1' ! &dx
x4
1 /#3 ),4( .' %,"'- ,/3 1%
! .
! .
! .
3 O!"#!$%&'(#") !"*()+") 356
!) cx !" )4ln( , ") cx !"" )4ln( ,
#) cx !" )4ln( , $) cx !"" )4ln( .
3. %& &'&(')*+,- # $%&
'() " dx
xx
21
./& ),0( *! 012-3 1.& ,0
!) cx
x
!$%&
'() "
3
13
, ") $%&
'() !$%&
'() "
2
11
12
xxx ,
#) cx
!"
3
)ln1( 3
, $) cxx
x!"" 2
1
3
3
,
4) cx
xx
!$%&
'() !
$%&
'() "
2
3
11
3
1
.
4. %& &'&(')*+,- dxx |1| 21
1
"#" 012-3 1.& ,0
!) 3
4, ") 0, #)
3
4" , $)
3
2, 4)
3
5.
5. %& &'&(')*+,- # xdxln 012-3 1.& ,0
!) cx!
1, ") c
x!
2
ln 2
, #) )1(ln "xx , $) cx!
3
ln 3
.
6. 5./+ gf , 67& 8-*-9+91.3,0: .72-*/).03: ,0 .720;01: 8-*-9<9&7: ./& ],[ !" . !2 )()( xgxf + 93- (=>0 ],[ !"x, , /?/0 (-/’ -2=9(@ >-
3.;A03:
!) )()( xgxf -+- , ],[ !"x, , ") # #+!
"
!
"dxxgdxxf )()(
#) ## + dxxgdxxf )()( , ],[ !"x, ,
$) # #+"
!
"
!dxxgdxxf )()( .
7. %& 0,B-6?2 /&7 9*-,,&.(3--
.,C2&7 ;+*1&7 /&7 638'-2&A
.;),-/&: 012-3 1.& ,0
x
5 -3 O
y
3 O!"#!$%&'(#") !"*()+") 357
!) !"5
3)( dxxf , ") !
"3
5)( dxxf .
#) ! !""
0
3
5
0)()( dxxfdxxf , $) ! !
"#
3
0
5
0)()( dxxfdxxf .
8. A% )()( xgxf $%$ &'( )*+, ]1,1["&x )(' 2)0()0( #% gf , -.-, &'( )*+, ]1,1["&x '/01,':
!) 2)()( "% xgxf , ") !" %"1
14))()(( dxxgxf .
#) )()( xgxf ' , ]1,1["&x $) 2' fC , gC 3045% )4'%. /67,84 /-4 ]1,1[" .
9. 9/-: 6 /5%*;-6/6 !%x
dttfxF1
)()(
.<45 f 6 /5%*;-6/6 -45 ='<>(%41
/0?7(-4@. A.-, 6 )1(F $ ,8%(' 8/6 7,
!) 0, ") 1, #) 2, $) 2
1.
10. 9/-: 6 /5%*;-6/6 f -45 ='<>(-
%41 /0?7(-4@. A%
2)( 1 %!E , 1)( 2 %!" )(' 3)( 3 %!"
-.-, -4 !#
$dxxf )( ,8%(' 8/4 7,
!) 6, ") 4" , #) 4,
$) 0, B) 2.
11. 9/-: 6 /5%*;-6/6 !%x
dttfxF0
)()( ,
.<45 f 6 /5%*;-6/6 -45 ='<>(%41
/0?7(-4@. A.-,
!) 2)( xxF % ,
") ()*
'+'
%x
xxxF
1,2
10,2)( ,
#) ()*
'+'
%xx
xxxF
1,2
10,)(
2
, $) ()*
'"+'
%xx
xxxF
1,12
10,)(
2
.
!!!.
O 3
Cf
2
1 x
y
O
y
x
$ % &
#
!3 !1
!2
2
1 O
y
x
3 O!"#!$%&'(#") !"*()+") 358
1. !"#" $%& '$ %$($)*'+ ,-./$'$ $0'#%(",+%121# '3 4($5#). %$(*,'$-
,3 /#$6 72,36 '36 8#$5"(#).6 19:,+,36 yyx !" , /1 0, #yx .
O x
y
O x
y
O x
y
(;) (<) (=)
O x
y
O x
y
(>) (?)
2. !"#$ $%& '$ %$($)*'+ "7")73(@/$'$ 1:0$# )$7@6 "(#,/A0$;
;) $ %
1
0 1
1dx
x, <) $
2/
03/!
xdx , =) $!
xdx015
>) $1
0ln xdx , ?) $ %
2
0
21 dxx , >) $ &
1
0 1
1dx
x.
3. B$ 10'"%:,1'1 '" 7*C"6 ,'#6 %$($)*'+ %(*91#6:
$ $ $"'()
*+,%-!"! dx
xx
xxdx
xxdx
x
111)(
1
$ '()
*+,%%! dx
xx
2
11
$&! dxx
11 .
D($ $ $&! dxx
dxx
11
1, "%&'1 10 ! !
4. B$ 10'"%:,1'1 '" 7*C"6 ,'#6 %$($)*'+ %(*91#6
3 O!"#!$%&'(#") !"*()+") 359
duu
u
dxx
I ! !" "#$%
&'(")
*+
*+
1
12
1
1
2
2
1
11
1
1
1
!" "+*
"+1
121
1Idu
u. (!"#$%&
ux
1+ '()*& )
12
duu
dx "+ .
+,-$ II "+ '()*& 0+I . ,.*), )%/0, &12$3 4*'(', $5'6
!" ,*
+1
12
01
1dx
xI ,
&(&378 01
12,
* x 93$ :4;& ]1,1["-x .
5. !&/-'6%& *< #.24-*<#<
!+x
dttfxF0
)()( ,
)('. f < #.24-*<#< *'. 73(=$-
2'6 #>8%$*'0. ?$ #.%(=<-@#&*& *$ ($-$:4*/ :&24.
+)0(F , +)2(F , +)3(F , +)4(F , +)6(F .
6 4 3 2
4
2
O
y
x
Cf
3 O!"#!$%&'(#") !"*()+") 360
()'"%(#" )$+,(&+-
-./012 3456.7838 - -9.037: :;:1;2.<=>
! "#$%&'($( )(*")+" $,& -.%/0(*1$,& -%# )#&23%&$"4 *5 $%& #-%0%64-)*' 5*/"27& 8"4 %4 424"+$5.59 2#)8%0+59 -%# -".%#)41:%#&, %2;6()"& $%#9 *"<(*"$48%=9 "-' $(& ".>"4'$($" )$(& 5-4&'()( 65&487& *5<'2,&
*3$.()(9 5*/"27&, 424"+$5." 5-4?"&547& -%# -5.4805+%&$"4 "-' 8"*-=059. @"<%.4)$48; )$% :;$(*" "#$' #-;.A5 ( )#*/%0; $,& ".>"+,& B00;&,& 8"4 424"+$5." $%# C.>4*;2(. D4 42359 $%# C.>4*;2( -1&, )$% -.'/0(*"
$%# 5*/"2%= #-;.A"& ( "?5$(.+" $(9 2(*4%#.6+"9 $%# )=6>.%&%# %0%80(-
.,$48%= 0%64)*%=. E"."8$(.4)$48' -".12546*" "-%$505+ % $.'-%9 #-%0%-
64)*%= $%# 5*/"2%= *4"9 5-4?1&54"9 -%# -5.4805+5$"4 "-' 3&" $*;*" -"-
."/%0;9 8"4 3&" 5#<=6."**% $*;*" (-"."/%048' >,.+%).
F)$, 3&" -"."/%048'
>,.+% *5 /1)( !" 8"4 8%.#?; # ($% )(*5+% $(9 -"."/%0;9 -%# 3>54 $(
*364)$( "-')$")( "-'
$( /1)(). D C.>4*;2(9, ?3.&%&$"9 $49 >%.239 #!
8"4 #", 2(*4%#.65+ 2#%
&3" -"."/%0481 >,.+"
*5 /1)549 #!, #" 8"4 8%.#?39 1O , 2O "&$+)$%4>". G$( )#&3>54", >.()4*%-%47&$"9 65,*5$.4839 424'$($59 $(9 -"."/%0;9, "-%2548&=54 '$4 64" $" 5*/"21 $,& $.47& $.467-
&,& OAB , AOO1 8"4 BOO2 4)>=54 ( )>3)(
)]()[(4)( 21 BOOAOOOAB !" . (1)
G#&5>+:%&$"9 $(& +24" 24"248")+" )$" &3" -"."/%0481 >,.+", /.+)854 '$4
)]()[(4)( 14131 OOOAOOAOO !" (2)
8"4 )]()[(4)( 26252 BOOOOOBOO !" (3)
H5 $%& $.'-% "#$', $% 5*/"2'& $ $%# -"."/%048%= >,.+%# *-%.5+ &" -.%-
)5664)$5+ (“5A"&$0(<5+”) "-' 3&" 1<.%4)*" 5*/"27& 56656."**3&,& $.4-67&,& ,9 5A;9:
!!!!!" )]()[()]()[()( 1411413 OOOOAOOOOAOOOABE
#!!! )]()[( 2625 BOOOOO
#!!!!" )(4
1)(
4
1)(
4
1)( 21 BOOAOOOABOAB
O5
O2
O6
B A
O3
O1
O4 O
3 O!"#!$%&'(#") !"*()+") 361
!""""# )]()[(4
1)(
4
1)( 21 BOOAOOOABOAB
!"$%&
'()""# )(
4
1)(
4
1)(
2
OABOABOAB
!"#$ %&'() *('%+,, "+,-%).() /)( .0 12+0)34( .#' (("%&+#') ,+#' 4)($ *2&'053($ /%#4%.+)-6$ "+0,705 4% "+8.0 ,+0 .0 )(*+,- # -() 9,/0
4
1#. . :0 12+0)34( (5., 7&'%.() 364%+( (", .0 /'#3., .;"0
)(3
4
4
11
)(
1*+,
*+,.
-#
/#
/.
:0 %4<(7,' 90)",' .05 "(+(<09)-0; =#+&05 %&'() &30 4% .( 3
4 .05 %4<(-
70; .05 .+)/8'05 "05 0+&>05' .( 1-+( .?$ <13?$ -() ? -0+5*6 .?$ "(+(-
<096$(*)
!"#$ %&' "()*+,-'&' '.()&/� .'1 23'"&)-40#0 4&%1 .'1 %&) "(5*+6-' &)7 2-*'8)9, )1 1842$ � '(:';#0 <++,0#0 =0>(1%'0 "'('"4(' 2?4+1?6 -2&/ &60 '0/"&7?6 &6$ @+=2*('$ .'1 &60 23'(-)=, &6$ %2 =2#-2&(1./ "()-*+,-'&'. A&6 81/(.21' &)7 17)7 '1>0' 81'"1%&>B6.2 5&1 ) 7")+)=1%-5$ � 2-*'8>0 -")(2; 0' =;021 -2 -1' 81'81.'%;' '0&;%&()36 "()$ '7&,0 &6$ "'('=>=1%6$.
",-./012 232435,6/7 - $ 0112-7 829 :/;7<2=
@(+(-.?+)3.)-, "(+17%)/4( .?$ 'A($ 4%2,705 ('.)4%.8")3?$ "+0<9?41-
.#' 5"090/)340; %4<(78' -(.1 .0' 170 ()8'( ("0.%9%& 0 .+,"0$ 4% .0' 0"0&0 0 J. Wallis ('(-195B% .0 1655 4)( 'A( ('(95.)-6 A-*+(3? /)( .0
%4<(7,' .05 -;-905 -() .0' (+)24, ".
(*)
C 7)(.;"#3? 3.0 A+/0 .05 D+=)467? “:%.+(/#')34,$ 0+20/#'&05 -8'05 .046$” %&'(): “"(' .414( "%+)%-=,4%'0' 5", %52%&($ -() 0+20/#'&05 -8'05 .041$ %"&.+).0' %3.) .05 .+)/8'05 .05 <13)' A=0'.0$ .(' (5.1' -() ;B0$ &30' .# .414(.)”. E D+=)467?$ 3.?' "+(/4(.)-,.?.( %+/13.?-% 9&/0 7)(*0+%.)-1 ("0*%;/0'.($ .?' A''0)( .05 ("%&+05, =+?3)40"0&?3% "%"%+(34A'0 "9620$ ,+#' .05 "(+("1'# (2+0&34(.0$ -() A7%)F% ,.)
.0 >?.0;4%'0 %4<(7, )30;.() 4% )(3
4 *+, ("0-9%&0'.($ (4% ("(/#/6 3% 1.0"0) .)$ "%+)".83%)$ '( %&'()
4%/(9;.%+0 6 4)-+,.%+0 (", (5.,.
3 O!"#!$%&'(#") !"*()+") 362
! Wallis "#$%&'# ()*
&+,-.-/,0 1,*+(2%03
R2!"# , 4$%,'# 2&) *-25)* 203 !" '# 5'*
2+6+*2* +6-037 # -*, *89 -:"# '&+#50 1,*5%#-'&7 .;<'# +,* -:"#2&
(=/. '46+*). >8<7 #5)*, ?)<'29 *89 2&) @3-/#5-1#,* ?#<+#2%5*, -:"# +,*
*89 *32(7 2,7 -:"#2#7 #5-)*, +('& *):/0?& 2<) 1.0 2+&+:2<) '2* 0805* 4<%5A#, 2& 1,:+#2%0 $".
B.4., ?,* 2&) -:"#2& %&, 803 #5)*, .;07 8%07 2&) 3802#5)03'* 203 0%"0-
?<)503 2%,?$)03 &$", ,'4.#, 222 ))(( $$$%#"%%& '!'(!)! RRR
1&/. 22 $%& '! R . >+0,* 8%0-.82#, 22 4$*+ '! R , 22 9$,- '! R
-.0.-. CD0. 380/9?,'# +# 20) 2%980 *329 9/#7 2,7 (8#8#%*'+()03 8/6-
"037) -:"#2#7 803 “#E*)2/0.)” 20 2#2*%2&+9%,0 !'", 0 Wallis 8%*?+*20-
805&'# +,* “+#2:=*'& '20 :8#,%0” +# 20) #E67 '3//0?,'+9:
“! ()*+, -+. #/0+123#-+, )(45 #.-65 -45 7#/8-45 90+, -+ :/0+;23# -45 3<*12-45 -;365 -+., (=>(. -45 #7-1545) <15#; 1=;+, 3< -+ ()*+ -+. -<-#0->-3+01+. (-+ +9+1+ “<?#5-(+@5” #.-8, +; 7:/<-<,) 90+, -+ -<-0:*45+ 3< 9(<.0: ->5 #7-15# (=>(. -+ -<-0:*45+ !'A", -+ +9+1+ “<?#5-(+@5” +; #-7-15<,-90+<7-:2<;, -45 7#/8-45)”.
F,*238<+()0 '# '3+=0/,-6 ?/$''*, 20 '3+8(%*'+* *329 203 Wallis ?5)#-2*,
4
4
1
)(2#2%:?<)0)(,02#2*%2&+9%40
2
2222222 9
R
R9
!'A"!'"
RRR
#R#RR!!!
(((('('('
..
(1)
C329 20 +5?+* G#<+#2%5*7, H/?#=%*7 -*, “8%<29?0)03” *8#,%0'2,-0. /0-
?,'+0., ,'013)*+#5 03',*'2,-: +# 2& '.?4%0)& '4('&
/ !'1
0
2
41
0dxx .
B%:?+*2,, *) "#<%6'03+# 1!R (1&/. 20 +0)*1,*50 -.-/0 122 !( yx ) -*, 1,*,%('03+# 2&) *-25)* (1&/. 20 1,:'2&+* ]1,0[ '# 5 5'* 2+6+*2* +6-037
20 -*"()* v
1, 292# 20 8%$20 +(/07 2&7 8%0&?0.+#)&7 ,'92&2*7 (1) ?5)#2*,
11
2
3
44
5
6789
:;< '
'((789
:;<'(7
89
:;<'(7
89
:;<'
22221
12
11
10
11
==
====. .
B A R O
B C
D E % #
A
&
M
3 O!"#!$%&'(#") !"*()+") 363
!"#$ $%&' $(&' )* +,-%. (*/*01#& .23*4 #, 0*#5#./, 1)/,46%* #7' 6"-
31/#767' 21)( xxf !" ((," (/,0-(#.4 *($ #73 .826&67 #," 0-09,"), &' (/,' #73 (/,7:,-%.37 +4*%;/467 #," +4*6#<%*#,' ]1,0[ , 0*4 #, $/4$ #,"
$#*3 #$%& , .23*4 #,
' !1
0
21 dxx .
= ;33,4* #," ,9,097/5%*#,', $(&' 0*4 ,4 199.' ).%.945+.4' ;33,4.' #7' *319"67', ;:43*3 *3#40.2%.3, 6"6#7%*#40<' 0/4#40<' 0*4 ,/26#70*3 %. 9,-
:40< *"6#7/$#7#* 6#7 +41/0.4* #," 19," *453*. = ;33,4* #," ,9,097/5-
%*#,' (," >/764%,(,4,-%. 6<%./* 6#, 6>,9.2,, 6#7/2?.#*4 6#,3 .($%.3,
,/46%$ #," 6"%@$9," '(
)dxxf )( (," ;+&6. , A. Riemann #, 1854:
“!"#$%&'" '() )*%+%,-.) /('01 121 ...,,, !&xxx 2%, 3$.4*%1/)( )15'"4) 4/)
) *)( 3 *)/5 4"($5 '"67-%,8 *)( 4,'3%+.9%,'" :5$(1 4,1/%'.)8 /% )!1x '"
1* , /% 12 xx ! '" ...,2* /% 1!! &( x '" &* *)( /) 61;4() -"/(*5 *+54')/) '"
i+ . <=/" > /('; /%, )-$%.4')/%8
)()()()( 133232212111 &&&& *+**+**+**+)* ########" !xfxfxffS ,
-) "?)$/5/)( )2= />1 "*+%6; /#1 @()4/>'5/#1 i* *)( /#1 2%4%/;/#1 i+ .
A1 7:"( />1 (@(=/>/), )1"?)$/;/#8 />8 "*+%6;8 /#1 i* *)( i+ , 1) /".1"(
2$%8 71) 4/)-"$= =$(% ! *)-08 =+) /) i* 6.1%1/)( )2"($%"+5:(4/), /=/" >
/('; ),/; %1%'59"/)( '(
)dxxf )( . A1 @"1 7:"( ),/; />1 (@(=/>/), /=/" /%
'(
)dxxf )( @"1 7:"( *)171) 1=>')”.
I.
1. !"#$%& ! ! !"#"$
%
$
%
$
%dxxgdxxfdxxgxf )()())()(( .
2. !"#$%& ! ! !&#"$
%
$
%
$
%dxxgdxxfdxxgxf )()()()(
3. '( $% # , )*)% ! #$
%0)( dxxf .
4. '( ! #$
%0)( dxxf , )*)% +,),(-.+/ 0, %1(,& 0)( #xf .&, +-0%
],[ $%'x .
5. '( 0)( (xf .&, +-0% ],[ $%'x , )*)% ! ($
%0)( dxxf .
6. '( ! ($
%0)( dxxf , )*)% +,),(-.+/ 0, %1(,& 0)( (xf .&, +-0%
],[ $%'x .
7. ! !) )""*"
%
%
%
%dxxxdxx )1()1( 244 , .&, +-0% 0+% .
8. ! !#)4/
0
4/
0
2 "2(ln2)/31ln(, ,
xdxdxx .
9. ! ! -)#$
%
$
%dxxfxxxfdxxf )()()( .
10. ! #e
xdx1
ln !1 1ln
e
dtt
.
11. A( ! #)1
00)1)(( dxxf )*)% 1#f .
12. '( ! #$
%0)( dxxf , )*)% 0)( #.f .&, +-45&5 ),( $%. ' .
13. '( ! #$
%0)( dxxf +,& / f 6%( %1(,& 4,()5$ 3/67( ")5 ],[ $% , )*)%
/ f 4,18(%& 625, )529-#&")5(, %)%8*"/3%: )&37:.
14. ;5 595+9<8=3, dxxx!) )1
1
3 )( 4,8&")-(%& )5 %3>,6*( )52 #=8152
452 4%8&+9%1%),& ,4* )/ .8,?&+< 4,8-"),"/ )/: "2(-8)/"/: xxxf )# 3)( +,& )5( -@5(, )=( x.
!!!. 3. !" #$%&'()#%# %& *+,&- )%.- '"/"0+%1 '/+2#.-:
! ! !"#$%
&'()*+"+ dx
xx
xxdx
xxdx
x
111)(
1
! #$%
&'())+ dx
xx
2
11
!,+ dxx
11 .
3/" ! !,+ dxx
dxx
11
1, &'4%# 10 + !
4. !" #$%&'()#%# %& *+,&- )%.- '"/"0+%1 '/+2#.-
duu
u
dxx
I ! !) )#$%
&'()*
,+
,+
1
12
1
1
2
2
1
11
1
1
1
! )+,
)+ Iduu 21
1.
(56)"7#u
x1
+ &'4%# )1
2du
udx )+ .
89/" II )+ &'4%# 0+I . 9:%4, 471-, #($". +%&'&, ";&<
!) -,
+1
12
01
1dx
xI , #'#.=> 0
1
12-
, x ?." 0+,# ]1,1[).x .