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AB0=>28BAO OA=>, GB> DC=:F88 ?>;57=> :;0AA8D8F8@>20BL 8<5==> ?> MB><C ?@87=0:C. 8R4    fkgֳgֳ{ ?"6@ NNN?N K2>4 D>@<C; 48DD5@5=F8@>20=8O A ?><>ILN !! ?@>872>48BAO 2 A;54CNI5< ?>@O4:5. 1. !>AB02;O5< AB@C:BC@=CN AE5<C 4;O 8AE>4=>9 D>@<C;K. 2. ?@545;O5< :>MDD8F85=BK ?5@540G8 G0AB=KE A2O759 ('!) 4;O 2A5E AB@5;>: AE5<K. 3. !25@BK205< AE5<C, >?@545;OO 55 @57C;LB8@CNI89 :>MDD8F85=B ?5@540G8 ( )  MB> 8 5ABL 8A:><0O ?@>872>4=0O. 4Q,#8XH  0޽h ? 33___PPT10i.4]p=+D=' x= @B +X  0 og@ (  V   fDgֳgֳ{ ?"6@ NNN?N P 1) 8DD5@5=F8@>20=85 DC=:F88 >B DC=:F88: u=u(x), y=y(u) = y(u(x)); 2 B0:>9 70?8A8 y 5ABL 2=5H=OO, 0 u  2=CB@5==OO DC=:F88. -B>B A;CG09 C65 @0AA<0B@820;AO ?@8 2K2>45 ?@028;0 1, 8 ?>2B>@O5BAO 745AL 4;O A8AB5<0B870F88. = A2>48BAO : AE5<5 A ?>A;54>20B5;L=K<8 AB@5;:0<8:)(2% H_08X]8 q* ' *q 'e  # lDgֳgֳ{ ?"6@ NNN?Nq ' %x u y&&% % T qiq # E B   fgֳgֳD8c?"0@NNN?Nq qB   fgֳgֳD8c?"0@NNN?N qiqH   # lDgֳgֳ{ ?"6@ NNN?N3@ fk1,(2    H   # l,#Dgֳgֳ{ ?"6@ NNN?N*6  fk2,(2    8  7 7 [   # lp(Dgֳgֳ{ ?"6@ NNN?N  yx y&  B   # lgֳgֳD8c?"0@NNN?N  # l-Dgֳgֳ{ ?"6@ NNN?NS7 K=k1k2X(2       (     f[Dgֳgֳ{ ?"6@ NNN?N > ?@028;C 1, K = k1k2, 8;8 ?@8 48DD5@5=F8@>20=88 DC=:F88 >B DC=:F88 =5>1E>48<>  ?>A;54>20B5;L=KE AB@5;>: (745AL  ?@>872>4=K5 2=CB@5==59 8 2=5H=59 DC=:F89) ?5@5<=>68BL. !@02=8< A B@048F8>==K< 2K2>4><: dy/dx=dy/du du/dx = k2 k1. 07;8G85 2 B><, GB> ?@8 B@048F8>==>< A?>A>15 48DD5@5=F8@>20=85 =0G8=0NB "A=0@C68", ?>AB5?5==> ?@>=8:0O  2=CB@L 2K@065=8O, 0 2 !! >=> =0G8=05BAO :0: 1K  87=CB@8  ?@>AB> ?>B><C, GB> ?>;L7COAL AE5<>9, C4>1=55 42830BLAO 2 =0?@02;5=88 AB@5;>:  >B 2E>40 : 2KE>4C, 0 =5 =0>1>@>B. B MB>3> 87<5=O5BAO ?>@O4>: A;54>20=8O A><=>68B5;59, =>, @07C<55BAO, =5 @57C;LB0B. (#?><8=05< >1 MB>< @07;8G88 B>;L:> ?>B><C, GB> >=> <>65B A<CB8BL =0G8=0NI8E ?@8 ?5@5E>45 : 4@C3><C A?>A>1C). (2   W8  + l>8XH  0޽h ? 33___PPT10i.4]p=+D=' x= @B +c%  0 z$r$  ,`#(  \   fMDgֳgֳ{ ?"6@ NNN?N "2) @>872>4=0O >1@0B=>9 DC=:F88. 0 AE5<5 ?5@5E>4 : >1@0B=>9 DC=:F88 >7=0G05B 87<5=5=85 =0?@02;5=8O AB@5;:8, 55 2E>4 8 2KE>4 <5=ONBAO <5AB0<8:J 2 pJ   fEDgֳgֳ{ ?"6@ NNN?Nf vn0 3@0D8:5 MB> >7=0G05B >1<5= <5AB0<8 :>>@48=0B=KE >A59 A >4=>2@5<5==K< ?>2>@>B>< 8 75@:0;L=K< >B@065=85<. @8 MB>< C3;>2>9 :>MDD8F85=B 4;O ?@O<>9 DC=:F88 5ABL B0=35=A C3;0 =0:;>=0 :0A0B5;L=>9 : 3>@87>=B0;8, 0 4;O >1@0B=>9  :>B0=35=A B>3> 65 C3;0, 8;8 B0=35=A 4>?>;=8B5;L=>3> : =5<C C3;0. =K<8 A;>20<8, :>MDD8F85=BK ?5@540G8 >1@0B=>9 8 ?@O<>9 DC=:F89 5ABL 2708<=>->1@0B=K5 25;8G8=K: k2 = 1/k1, 8;8 dx/dy = 1/(dy/dx). (2( (((( (((((N8 qzSp   S  # l}Dgֳgֳ{ ?"6@ NNN?Nqp qx y&  B  # lgֳgֳD8c?"0@NNN?NdH  # lDgֳgֳ{ ?"6@ NNN?NF{ fk1,(2    S  # lDgֳgֳ{ ?"6@ NNN?NSo qx y&  B  B # lgֳgֳD8c?"0@NNN?N ` H   # ldDgֳgֳ{ ?"6@ NNN?N zf  fk2,(2    B    fgֳgֳD?"0@NNN?Nc ** B  # lgֳgֳD?"0@NNN?N * L 5 )#   5 B # lDgֳgֳ{ ?"6@ NNN?N@I Sa(2  @ eY (5B  ZZgֳgֳDjJ?"0@NNN?NeHB B ZgֳgֳD?"0@NNN?NffB B ZLgֳgֳD?"0@NNN?Ne*9  # lDgֳgֳ{ ?"6@ NNN?NY Wp/2-a(2  -  # lDgֳgֳ{ ?"6@ NNN?N J  Sy(2 -  # lDgֳgֳ{ ?"6@ NNN?N u*o  Sx(2 B   fgֳgֳD?"0@NNN?Nm))B  # lgֳgֳD?"0@NNN?N)B  TgֳgֳDjJ?"0@NNN?NHB  TgֳgֳD?"0@NNN?NHB   ZgֳgֳD?"0@NNN?N- ! # lDgֳgֳ{ ?"6@ NNN?Nf Sa(2  1 " # lLDgֳgֳ{ ?"6@ NNN?NRV Wp/2-a(2  - # # lDgֳgֳ{ ?"6@ NNN?NI, Sx(2 - $ # lhDgֳgֳ{ ?"6@ NNN?N)y Sy(2 * +  fDgֳgֳ{ ?"6@ NNN?N V?@O<0O DC=:F8OPf, ,  fDgֳgֳ{ ?"6@ NNN?N% U  X>1@0B=0ODC=:F8OPfH  0޽h ? 33___PPT10i.4]p=+D=' x= @B +  0   2(       fgֳgֳ{ ?"6@ NNN?N   3) @>872>4=0O M:A?>=5=BK (?>:070B5;L=>9 DC=:F88) y = ex . >A?>;L7C5<AO ?@028;>< 48DD5@5=F8@>20=8O >1@0B=>9 DC=:F88. (21,, ,,,@K   # lgֳgֳ{ ?"6@ NNN?N  qx y&  B   # lgֳgֳD8c?"0@NNN?N@   # lgֳgֳ{ ?"6@ NNN?N)f fk1,(2    K   # lgֳgֳ{ ?"6@ NNN?Nm  qx y&  B  @ # lgֳgֳD8c?"0@NNN?N` @   # l<gֳgֳ{ ?"6@ NNN?N(B fk2,(2    .    flgֳgֳ{ ?"6@ NNN?N  ZZ45AL MB> x=lny. ;O =55 dx/dy = 1/y (A<. 4.2, 3), ?>MB><C y2 = dy/dx = y = ex . @>872>4=0O M:A?>=5=F80;L=>9 DC=:F88 A >A=>20=85< e @02=0 A0<>9 DC=:F88. (2   $       433333D     g8 d   H F `    fgֳgֳ{ ?"6@ NNN?Nd%  x ex8     B    fgֳgֳD8c?"0@NNN?N*B    fhgֳgֳ{ ?"6@ NNN?N   fk1,(2    d    fgֳgֳ{ ?"6@ NNN?N7 z#  ln y y4  "  B  @  fgֳgֳD8c?"0@NNN?N :    fgֳgֳ{ ?"6@ NNN?Nvp fk2,(2    H   0޽h ? 33___PPT10i.4]p=+D=' x= @B +  0   `(   0 3 r0e0eA ? ? T   D 0 3 r0e0eA ? ?x    Dl  0A ? ?x   l  0A ? ?D  .8  d  Z̼D?"0@NNN?N  T4) 8DD5@5=F8@>20=85 AB5?5==>9 DC=:F88: *+'z  6A ? ? X8 jV jV  2 3 r0e0eA n? ?@ jV n DB   ZhD?"0@NNN?NL& x.@5>1@07C5< 55 : 284C: .02   TD?"0@NNN?N !>AB028< 4;O ?>A;54=53> 2K@065=8O AB@C:BC@=CN AE5<C 8 ?@>=C<5@C5< 2 =59 :>MDD8F85=BK ?5@540G8: _Z_   TD?"0@NNN?NpqR  4>?@545;8< :>MDD8F85=BK ?5@540G8 G0AB=KE A2O759, 8A?>;L7CO C65 8725AB=K5 ?@028;0 4;O ;>30@8D<0, ;8=59=>9 DC=:F88 8 M:A?>=5=BK:~Z~V  TxD?"0@NNN?NV '  ^>?@545;8< @57C;LB8@CNI89 :>MDD8F85=B ?5@540G8 : 0020 8 +vJ  +vJ   f8c?"0@NNN?NakWm  C xG *H\cI8c?"0@NNN?Nj m 2 3 r0e0eA ? ? -J  D  C xG0*H`IIu8c?"0@NNN?NjIm  ZXD?"0@NNN?NIv UyP$ 8  ZD?"0@NNN?N +K nk14(2 8  ZD?"0@NNN?NF f nk24(2 8  ZD?"0@NNN?N[{ nk34(2 z  6A0 ? ?: z  6A ? ?aG 2  # lgֳgֳ{ 8c?"6@`NNN?N9 VjiH  0޽h ??` 33___PPT10i.4]p=+D=' x= @B +Y  0 p h  #  (     T|D?"0@NNN?NE F @>:><<5=B8@C5< 2K2>4 D>@<C;K 4;O AB5?5==>9 DC=:F88. 1KG=> 55 2K2>4OB 2=0G0;5 4;O F5;KE ?>;>68B5;L=KE ?>:070B5;59, 8AE>4O 87 A2>9AB2 ?@>872545=8O. 0B5< 4>:07K20NB ?@8<5=8<>ABL : 4@>1=K< 8 >B@8F0B5;L=K< ?>:070B5;O<. "> 8 4@C3>5 B@51C5B ?@545;L=KE ?5@5E>4>2.  Z   T0D?"0@NNN?N `$45AL <K ?>;CG8;8 55 A@07C 4;O ;N1KE ?>:070B5;59 AB5?5=8, =8 @07C =5 ?@81530O : ?@545;L=K< ?5@5E>40<. -B> =5 7=0G8B, GB> >=8 2>>1I5 =5 =C6=K: : =8< ?@81530;8, ?>;CG0O 8A?>;L7C5<K5 745AL A@54AB20 - D>@<C;C 4;O ;>30@8D<0 8 ?@028;0 ?@5>1@07>20=89 !! 4;O DC=:F88 >1I53> 2840.  Z`  TD?"0@NNN?N0  h?@545;8< MB8< ?CB5< ?@>872>4=CN >1@0B=>9 25;8G8=K.  50258 L    + L z  6A ? ?L B  2  3 rgֳgֳ{ 1?"6@`NNN?N     T<N?"0@NNN?N  -B>B @57C;LB0B =0< C65 8725AB5= 87 ?. 4.2, 4, 8 B0< C:070=> =0 53> 871KB>G=>ABL.  SZS # T@N?"0@NNN?Nd >?CB=>5 70<5G0=85: =5 =C6=> ?CB0BL >1@0B=CN 25;8G8=C 8 >1@0B=CN DC=:F8N  MB> A>25@H5==> @07=K5 25I8. gZgH  0޽h ? 33___PPT10i.4]p=+D=' x= @B ++  0 B:p (     f Ngֳgֳ{ ?"6@ NNN?N &5) @>872>4=0O M:A?>=5=BK A ?@>872>;L=K< >A=>20=85<. K@078< >A=>20=85 AB5?5=8 G5@57 53> =0BC@0;L=K9 ;>30@8D<: y=ax=(eln a)x = exlna; ?>;CG05< DC=:F8N >B DC=:F88, 4;O :>B>@>9 A>AB028< AB@C:BC@=CN AE5<C: l(4;, ,,, , ,, ,,, , , ,,K%Dr    8 F %  F % U   f$Ngֳgֳ{ ?"6@ NNN?N %  yx y&  B   fgֳgֳD8c?"0@NNN?N     fH*Ngֳgֳ{ ?"6@ NNN?NF  K=k1k2X(2       (  8 q d   B  # lgֳgֳD8c?"0@NNN?N 8 B  # lgֳgֳD8c?"0@NNN?NM B    f2Ngֳgֳ{ ?"6@ NNN?Nz   fk1,(2    B    f7Ngֳgֳ{ ?"6@ NNN?NH  fk2,(2        fdMDD8F85=BK ?5@540G8 G0AB=KE A2O759 8 ?>;CG8< 2K@065=85 4;O 8A:><>9 ?@>872>4=>9: k1 = ln a; k2 = y; (2d  Dh    r  6A" ? ? ' l  0A ? ?f ) &F     fYNgֳgֳ{ ?"6@ NNN?Nwm  L8;8*(2 $2  # lgֳgֳ{ 8c?"6@`NNN?N H  0޽h ? 33___PPT10i.4]p=+D=' x= @B +  0  !(      fhwgֳgֳ{ ?"6@ NNN?N n6) @>872>4=0O :>A8=CA0. @5>1@07C5< :>A8=CA 2 A8=CA 4>?>;=8B5;L=>3> C3;0: y =cos x = sin(p/2 x), 8 A>AB028< AB@C:BC@=CN AE5<C 4;O 48DD5@5=F8@>20=8O DC=:F88 >B DC=:F88 :(21$$$$$$$$$J:R  M8    ]'lB  # lgֳgֳD8c?"0@NNN?N   B  # lgֳgֳD8c?"0@NNN?Nz    H  # lpNgֳgֳ{ ?"6@ NNN?NW(  fk1,(2    H  # luNgֳgֳ{ ?"6@ NNN?Nu  (  fk2,(2    J  # lzNgֳgֳ{ ?"6@ NNN?N6   hlx p/2  x sin(p/2  x) 7 $$$$$$$$$$$$$6    f8Ngֳgֳ{ ?"6@ NNN?Ns  "?@545;8< 4;O =55 :>MDD8F85=BK ?5@540G8 G0AB=KE A2O759 8 ?>;CG8< 2K@065=85 4;O 8A:><>9 ?@>872>4=>9: k1 =  1; k2 = cos(p/2  x) ; d$ $$,, $$ $$$$$$$$,$Xh     l  0A#  ? ?    l  0A  ? ? /*     faNgֳgֳ{ ?"6@ NNN?NIZP  >8;8(2 2  # lgֳgֳ{ 8c?"6@`NNN?NL w2  # lgֳgֳ{ 8c?"6@`NNN?N F F %   >wU   fNgֳgֳ{ ?"6@ NNN?N %  yx y&  B    fgֳgֳD8c?"0@NNN?N  !  f,Ngֳgֳ{ ?"6@ NNN?NF  K=k1k2X(2       (  H  0޽h ? 33___PPT10i.4]p=+D=' x= @B +  0  W(  r   fNgֳgֳ{ ?"6@ NNN?N 7) @>872>4=0O 0@:A8=CA0: y = arc sin x. >A?>;L7C5<AO ?@028;>< 48DD5@5=F8@>20=8O >1@0B=>9 DC=:F88. o$$$ $$@ SE   flNgֳgֳ{ ?"6@ NNN?NK  qx y&  B   fgֳgֳD8c?"0@NNN?N..:   fNgֳgֳ{ ?"6@ NNN?No0 fk1,(2    E   fNgֳgֳ{ ?"6@ NNN?NJv  qx y&  B @  fgֳgֳD8c?"0@NNN?N-i -:   fNgֳgֳ{ ?"6@ NNN?NK/ fk2,(2    `   # l|Ngֳgֳ{ ?"6@ NNN?Ns  x arc sin x4   B   # lgֳgֳD8c?"0@NNN?NMM@   # lNgֳgֳ{ ?"6@ NNN?N<9 fk1,(2    W   fNgֳgֳ{ ?"6@ NNN?NP@  sin y y4   B @  fgֳgֳD8c?"0@NNN?N< "<:   fNgֳgֳ{ ?"6@ NNN?N9> fk2,(2       fNgֳgֳ{ ?"6@ NNN?NCjp  :45AL MB> x=sin y. ;O =55 dx/dy =k2= cos y (A<. 4.2, 2), ?>MB><C  P 2   3l      l  0A% ? ? & 2  # lgֳgֳ{ 8c?"6@`NNN?N l  0A$ ? ? gs      fUgֳgֳ{ ?"6@ NNN?N j` [ L8;8*(2 $H  0޽h ? 33___PPT10i.4]p=+D=' x= @B +  0  (     fNgֳgֳ{ ?"6@ NNN?N 8) @>872>4=0O 0@::>A8=CA0: y = arc cos x. >A?>;L7C5<AO ?@028;>< 48DD5@5=F8@>20=8O >1@0B=>9 DC=:F88. q$$$ $$@<   HE   fUgֳgֳ{ ?"6@ NNN?NK  qx y&  B   fgֳgֳD8c?"0@NNN?N..:   f8Ugֳgֳ{ ?"6@ NNN?No0 fk1,(2    E   f4Ugֳgֳ{ ?"6@ NNN?NJv  qx y&  B @  fgֳgֳD8c?"0@NNN?N-i -:   fdUgֳgֳ{ ?"6@ NNN?NK/ fk2,(2    n    f$Ugֳgֳ{ ?"6@ NNN?Ns  x arc cos x4 (  B    fgֳgֳD8c?"0@NNN?NMM:    f*Ugֳgֳ{ ?"6@ NNN?N<9 fk1,(2    W    f/Ugֳgֳ{ ?"6@ NNN?NP@  cos y y4   B  @  fgֳgֳD8c?"0@NNN?N< "<:   f$5Ugֳgֳ{ ?"6@ NNN?N9> fk2,(2        f=Ugֳgֳ{ ?"6@ NNN?NCj  L45AL MB> x=cos y. ;O =55 dx/dy =k2=  sin y (A<. 4.3, 6), ?>MB><C  O 2          l  0A( ? ? & 2  # lgֳgֳ{ 8c?"6@`NNN?N l  0A& ? ? s      f\PUgֳgֳ{ ?"6@ NNN?N j` [ L8;8*(2 $H  0޽h ? 33___PPT10i.4]p=+D=' x= @B +A   0 X P ` 0(  0 0  fXUgֳgֳ{ ?"6@ NNN?N >(2$  0  f[Ugֳgֳ{ ?"6@ NNN?Nw 4.4. $C=:F88 A 42CE:0=0;L=>9 7028A8<>ABLN >B 0@3C<5=B0 (DC=:F88 >B 42CE DC=:F89 >4=>3> 0@3C<5=B0). =8 A>>B25BAB2CNB C65 8725AB=>9 =0< AE5<5 @8A. 1.3, ?>2B>@5==>9 =865 =0 @8A. 4.1, 4.2. K2>4 @O40 A;54CNI8E 87 =55 D>@<C; 48DD5@5=F8@>20=8O ;53:> C=8D8F8@C5BAO A ?><>ILN B01;8FK 4.1.  =59 @0AA<>B@5= @O4 DC=:F89, @07=>>1@07=KE ?> 284C, 8 >1J548=5==KE ;8HL >4=8< >1I8< ?@87=0:><  42CE:0=0;L=>ABLN 2;8O=8O. K@065=85 4;O ?>;=>9 ?@>872>4=>9 ( ) 4;O =8E 2A5E >48=0:>2> 8 8<55B 284: K=k11 " k12+ k21 " k22 . 07;8G0NBAO ;8HL 2K@065=8O 4;O 2E>4OI8E 2 MBC D>@<C;C :>MDD8F85=B>2 ?5@540G8 G0AB=KE A2O759 ('!). o(27  ,0               a SH 0 0޽h ? 33___PPT10i.4]p=+D=' x= @B +R' 0 i&a&0 %%$%(  $ $ T{U?"0@NNN?N >!E5<0 48DD5@5=F8@>20=8O M;5<5=B0@=KE DC=:F89 A 42CE:0=0;L=>9 7028A8<>ABLN >B 0@3C<5=B0. =0G5=8O '! A<. 2 H0?:5 B01;8FK 4.1.( f: F s  $ q  R $ C xGjHIj8c?"0@NNN?N) ?R $ 3 rG.=H<I.=8c?"0@NNN?N q R $ 3 rG.=HԒI.=8c?"0@NNN?N " $ TU?"0@NNN?Np# Yy P$ + $ T܄U?"0@NNN?Njw gk11,(2  +  $ THU?"0@NNN?N` $ gk12,(2  +  $ TU?"0@NNN?N  gk21,(2    $ TU?"0@NNN?Ns? WxP,   $ TdU?"0@NNN?N   WvP,   $ TU?"0@NNN?N   Uu, + $ TU?"0@NNN?N *  gk22,(2  R $B C xZGHI8c?"0@NNN?N) b $ C xGaHIB8c?"0@NNN?N 1  $ TU?"0@NNN?N g  Yy P$ [ $ TU?"0@NNN?N}(  k11 k12J (2  (  i $ T:070=K 420 MB0?0 A25@BK20=8O AE5<K %82%H $ 0޽h ? $ $$ $$$ $$$ $ $$$$$$$$$$$ 33___PPT10i.4]p=+D=' x= @B + 0 @ ( (  (h ( TU?"0@NNN?N4 p@8<5@ DC=:F88 A 42CE:0=0;L=>9 7028A8<>ABLN >B 0@3C<5=B0 98294F j9)  ( s R ( C xG,HI,8c?"0@NNN?N.ZkR ( 3 rGH)I8c?"0@NNN?NP c R ( 3 rGDHID8c?"0@NNN?N# ZA  ( TtU?"0@NNN?NAF \tg x P$ S ( TU?"0@NNN?Nt-  k11=cos x: (2 (  U  ( TU?"0@NNN?N  k12=1/cos x: (2  (  U  ( TZ?"0@NNN?N   k21= -sin x: (2  (    ( T4Z?"0@NNN?Nk WxP, 1  ( TPZ?"0@NNN?NP )  mcos x0P,, /  ( TZ?"0@NNN?Nj# J ksin x.,, z ( TZ?"0@NNN?N 9  k22= - sin x/cos2xX(2   (  R (B C xZGKHIK8c?"0@NNN?N.c 8  2 # ( 2 #& ( Z`Z?"0@NNN?N 5 2  \tg x P$ ! 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