高等数学加强版湘潭大学文科高等数学教学改革课题组课后答案.pdf

() 5.1 A 1 (1) axn n limkax kn n lim. (2) ) , 2 , 1( 0 nxn0limaxn n axn n lim. (1) axn n lim0NNNn axn. NnNkn ax kn ax kn n lim. (2) axn n lim0NNNnaxn. 0a aa ax a ax ax ax ax n n n n n 0a nn xx0 axn n lim. 2. (1) ; 2 1 34 12 lim n n n (2) 0)1(limnn n ; (3) 2 1 42 4 lim 2 2 nn nn n (4) 1lim 22 n an n . (1) 211551 4322(43)8 n nnnn a kn kn N NN NNnN x ax x a a n n n 0 1 ,NNn 2111 . 432 n nn 211 lim. 432 n n n (2) 11 1, 1 nn nnn 0 2 1 ,NNn 1 1.nn n lim(1)0. n nn (3) 4 1 Nn04n0 2 222 413434 2422 242 24 nnnn nnnnnn 2 331 , 2 24 n nnn 1 n 1 2 N 1 , 4max ,max 21 NNN 0 1 max4, NNn 2 1 42 4 2 2 nn nn 2 2 41 lim. 242 n nn nn (4) 0a0a 22 limlim11. nn na n lim(1)0.lim(1)0.lim(1)0.lim(1)0.lim(1)0.lim(1)0.lim(1)0.lim(1)0.lim(1)0.lim(1)0. 0404 4 4 413434413434413434413434413434 lim(1)0. 0 0 222222 413434413434 222222222 413434413434413434 nnnnnnnnnnnn 413434 2422 242 242422 242 24 222222222 2422 242 242422 242 24 maxmaxN Nmaxmax max4, 0 n a nann a n an 2 22 222 )( 1 1 22 n an n a 2 . 2 a n. 2 a NNn n a2 1 22 n an 1lim 22 n an n . 3.axn n limaxn n lim axn n lim0NNNn axn. Nn axn n xa lim. n n xa 1, n n xlim1. n n xlim n n x. 4 n x0lim n n y0lim nn n yx. n x0,M (1,2,). n xM n lim0, n n y0NNNn n n N N N NN NNnNn x xn n x , n y M . nnnn x yxyM M 0lim nn n yx. 5. (1) ; 0 sin lim x x x (2) ; 2 1 12 1 lim x x x (3) ; 1) 12(lim 1 x x (4) 4 lim2; x x (5) 2 2 1 12 lim; 213 x x xx (6) ( , )(1,2) lim(3)5. x y xy (1) 0 2 1 MxM sinsin1 0, xx xxx ; 0 sin lim x x x (2) 11322 , 2122 212121 x xxxx 0 1 2 1 , 2 MxM 112 , 21221 x xx 11 lim. 212 x x x (3) (21) 121,xx sinsin 0,0,0, x x xxxxxxxxxxxx 0,0,0,0,0, lim(3)5.lim(3)5.lim(3)5.lim(3)5.lim(3)5.lim(3)5.lim(3)5. 1 0,0, 1 xxxxxxxxx 0,0,0,0,0,0,0,0, sinsin limlim x x x 1132211322 2122 2121212122 212121 x x xxxxxxxx2122 2121212122 2121212122 2121212122 212121 0, 2 10x (21) 1212,xx 1 lim(21)1. x x (4) 44 2, 22 xx x x 01,204x 4 2. 2 x x 4 lim2. x x (5)01,x10.x 2 2 12121 2132133(21) xxx xxxx 1x1 0 xx110x20x1x. 211,x 11 3(21)3 xx x 3 2 12 1 2 2 xx x , 3 1x 31x. 1212112121 2132133(21)2132133(21) xxxxxx 2132133(21)2132133(21) 1212112121121211212112121 2132133(21)2132133(21)2132133(21)2132133(21)2132133(21)2132133(21) 121211212112121 2132133(21)2132133(21) xxxxxx 2132133(21) 121211212112121121211212112121 2132133(21)2132133(21)2132133(21)2132133(21) 3 , 1min10x 3 2 12 1 2 2 xx x . (6) (3)53(1)2312 ,xyxyxy 22 |1|(1)(2) ,xxy 22 |2|(1)(2) ,yxy 22 0(1)(2)xy (3)531234 .xyxy 1 4 22 0(1)(2)xy (3)5,xy ( , )(1,2) lim(3)5 x y xy 6. 2 yx 0 x x 0 xx 2 yx 22 2 000 2,yxxxxxx 2 0 00 limlim 20, xx yxxx 2 yx 0 x 7.sinyx, 0 x,x 0 xxxsinyx 000 sinsin2sincos, 22 xx yxxxx 0 |cos()| 1, 2 x x | |sin|, 22 xx lim(3)5 x 2 2 yxyxyxyxyx 2222222222 000000000000000 2222 2 2 222222 2, 22 yxxxxxxyxxxxxxyxxxxxx 000000000 limlim 2limlim 2yxxxyxxx 0000xxxxxx00000000 limlim 2limlim 2limlim 2limlim 2limlim 2limlim 2 0 0 x x 0 2 sincos21, 222 xxx yxx 0, 0 xx 0 sinsin,xxyx sinyx 0 x 0 x, sinyx, B 1limlimlim 212 axxax n n n n n n . ” lim n n xa0NNNn axn NnNn12Nn2 ax n 12 ax n2 limlim 212 axx n n n n . ”limlim 212 axx n n n n ,0 12 ,N NN 1 Nn ax n 12 2 Nn ax n2 ) 2 , 12 ( max 21 NNNNnaxn limaxn n . N N x x n2 2n x x n2 lim n n limlim 2 x x limlimlim 212 axxax n n n n n n . 2 PQ ,P Q ,P Q.xfy 0 xx 0 xxf 0 xfyxfz, 00, y xxy 00, y xyxf, 00, y xf. 3 n n xlim0sinlim 2 n x n n n . n n xlim)., 2 , 1(, 0nMxM n .sin 2 n M n x n x n n n ,0sin, 0 2 n M n x aNn M N n 0sinlim 2 n x n n n . 4 n xa. axn n lim0NNNnaxn. lim n n xa 0 0NN 0 nN 0 0n xa. 5.2 A 1. (1)lim 3n n n ; (2) 1 lim 1 n n ; (3) ) 1 2 1 1 1 (lim nnnn n (4) n nn 1 3 1 2 1 1lim (5) 1 2 11 (lim 222 nnnn n n (6) 2( 1) lim. 2 n n n n (1) sinsin 2 n n a . . NNNN sinsin 2 2 n n x x n n n n n MM NnNn n n NNNNNNnN 2 0 33 n nn n , 2 lim00, lim0, 3 n n nn , lim0. 3n n n (2) 11 1lim 11 n nn lim11, n 1 lim(1)1 n n , 1 lim 11. n n (3) 111 , 121 nn nnnnnnn lim1, n n nn , 1 1 lim n n n 111 lim()1. 12 n nnnn (4) n n n n 1 3 1 2 1 1 1 11lim n 1lim n n n 1 1 lim 11.lim 11.lim 11. 1 1 n n lim 11.lim 11.lim 11.lim 11. 111111111111 1212 nnnn111111 nnnnnnnnnnnnnn1212nnnnnnn1212121212 lim1,lim1,lim1, n n n n lim1, nnnn 1 1 3 1 2 1 1limn nn . (5) 22 22222 111 (, 2 nn n nnnnnnn 2 2 lim1; n n nn 2 2 lim1, n n n 222 111 lim ()1. 2 n n nnnn (6) 112( 1)3 , 2222 nnn n n 11 lim 22 n 31 lim 22 n n 2( 1)1 lim. 22 n n n n 2 (1) 1 lim sin x x x (2) 2 sec)1 (lim 1 x x x (3) 2 cot 2 0 lim 1 3tan x x x (4) 2 1 lim 3 x x x x (5) 2 2 lim 1 x x x x (6) 3 0 1tan1 sin lim x xx x ; (7) 31 limsinln 1sinln 1; x x xx (8) 1 1 lim 1 x x x (9) 21 lim 21 n x n n (10) 1 lim 1 x x x 1111 2222 limlim 3131 lim 2222 2( 1)12( 1)1 limlim n n n 2( 1)12( 1)12( 1)1 2 cot (11) 5 lim 1 x x x (12) 1 23 lim 21 x x x x . (1) 1 sin 1 lim sinlim1. 1 xx x x x x (2)1,xt . 2 2 sin 2 lim 2 2 sin lim 2 sec)1 (lim 001t t t tx x ttx (3) 2 2 3 1 cot 223 3tan 00 lim 1 3tanlim1 3tan. x x xx xxe (4) 4 3 21 4 4 141 limlim1. 333 x x xx xx e xxx (5) 2 2 11 limlimlimlim1. 11 111 11 xx x xx xxxx xxx xxx xx (6) 3 0 1tan1 sin lim x xx x 3 0 tansin lim 1tan1 sin x xx xxx 33 00 1tansin1sinsin .cos limlim 22cos xx xxxxx xxx 2 0 1sin1 cos11 lim. 2cos4 x xx xxx (7) 33 sin ln 1sinln 1 33 lim sinln 1limlimln 1 13 ln 1 xxx xx xx xx xx .3 3 33 limln 1limln 13. x x xx xx 1 lim sin ln 11, x x x limlim1.limlim1. xxxxxxxxxx xxx limlim1. xxx 111111111111111111111111111111111111111111111111 limlim1.limlim1. xxxx xxxxxx xxxx 111111 limlim1.limlim1.limlim1. 1111 limlim1.limlim1.limlim1.limlim1.limlim1. 1111111111111111 limlim1. 1111111111 limlim1.limlim1.limlim1. 11 1111 11 xxxxxx11 1111 1111 11111111111111 3 0 0 tansin limlim x x xxx 3 1tansin1sinsin .cos1tansin1sinsin .cos lim 1tansin1sinsin .cos 2 31 limsinln 1sinln 13 12. x x xx (8) 1 1 111 lim 1lim11 xx xx xxx 1 1 111 lim 1lim 11. x xx e xxe (9) 212 limlim 1 2121 nn xx n nn 11 22 1 2 1111 lim 111. 11 22 n x ee nn (10),txxt 1 lim 1 x x x 1 lim 1 t t t 11 lim. 1 1 t t e t ( 1) 11 lim 1lim 1 xx xx xx 1 1 1 lim 1. x x e x (11) 5 x u,xu, 55 5 5 5 5111 lim 1lim 1lim1lim 1 /5 x xuu xxxx e xxuu . (12) 232 1 2121 x xx , 21xu, 1 2 u x,x,u. 1 1 1 2222 2 2322 limlim 1lim 1 21 uu x xuu x ee xuu . 310 n x 4 1 )1 ( 1nn xx , 2 , 1n n x. 10 n x110 n x 1 1 (1), 4 nn xx ( 1)( 1)xxxxxxxxxx 1111111111 xxxxxx 11111111 lim 1lim 1lim 1lim 1 111111 lim 1lim 1lim 1lim 1lim 1lim 1lim 1lim 1 xxxxxxxx lim 1lim 1 xxxxxx 11 lim.lim. 11 t lim. e e lim. 1 1 t t t ( 1) lim 1lim 1lim 1lim 1 , 5 x x 5 51115111 5 lim 1lim 1lim1lim 1lim 1lim 1lim1lim 1 51115111 lim 1lim 1lim1lim 1lim 1lim 1lim1lim 1lim 1lim 1lim1lim 1lim 1lim 1lim1lim 1lim 1lim 1lim1lim 1lim 1lim 1lim1lim 1lim 1lim 1lim1lim 1 xxxx xxuuxxuu/5 xxxxxxxx /5 xxxxxxxx lim 1lim 1lim1lim 1 /5/5 xxxxxxxx /5 xxxxxxxx /5 lim 1lim 1lim1lim 1lim 1lim 1lim1lim 1 1 1 4(1) n n x x 0 )1 (4 )21 ( )1 (4 441 )1 (4 1 22 1 n n n nn n n nn x x x xx x x xx n x. 10 n x n x. n xAxn n lim 4 1 )1 ( 1nn xx 4 1 )1 (AA 0144 2 AA0) 12( 2 A 2 1 A 2 1 lim n n x. B 1) 1( 0lima a n n n . 0 n n a n x 0 n x n x. 1 11 limlim 1 ana n x x n n n n NNnN1 1 n n x x Nn n x. n xAxn n lim n nn n x na n a n na n a n x 111 1 1 n nn n n x na n xlim 1 limlim 1 A a A 1 n . limlim n x n n n 1 1 1 n n n x x x 1 (1)0,A a 0.A 0lim n n a n . 2 (1) 222 12 lim; 12 n n nnnnnnn (2) 222 lim. (1)(2)() n nnn nnnn (1) nnn n nnnnnnn n nn nn 22222 2 2 1 121 )2(2 ) 1( ) 1(2 ) 1( 1 21 22 nn nn nn n 2 1 )2(2 ) 1( lim 2 nn nn n2 1 ) 1(2 ) 1( lim 2 nn nn n 2 1 ) 2 2 1 1 (lim 222 nnn n nnnn n . (2) 222 )()2() 1( ) 1)()3)(2()2)(1( nn n n n n n nnnn n nn n nn n )(1()2)(1() 1(nnnn n nn n nn n 121 ) 12 1 1 1 ( ) 1)()3)(2()2)(1(n n n n nn n nnnn n nn n nn n n n 21 1 2 2 2 2 n n n n2 2 (2 2 ( ( 1 1 n n( ( n n 2 2 1 1 )2 )2 1 1 ( (limlim 2 n n n 2 1 2 1 1) 2 11 ( )(1()2)(1() 1(nn n nnnn n nn n nn n 2 1 )()2() 1( 121 222 nn n n n n n n n n n 2 1 ) 121 (lim n n n n n2 1 2 1 lim n 222 1 lim. (1)(2)()2 n nnn nnnn 3.10 1 x nn xx6 1 , 2 , 1n n x. 10 1 x4 2 x 21 xx. kn 1k 1kk xx 211 66 kkkk xxxx n x. 10 1 x nn xx6 1 0 n x , 2 , 1n n x n x.Axn n lim. 1 limlim6, nn nn xx AA6 06 2 AA 2A3A() 3lim n n x. 1 6 6 6 6 k k 6 6 6 6 k k 1 1 x6 6 k k 1 1 1 1 1 1 . . 10 1 x x , 2 , 1 4 x xxx x cba 1 03 lim. x xxx x x xxx x cbacba 1 0 1 0 ) 3 3 1 (lim) 3 (lim 3 3 3 3 0 3 lim1 3 xxx xxx abc x xxx abc x abc 1111 3 lim lim 3 0 3 0 xxx xxxabc abc xxx x x x ee 3 1(ln lnln ) ln3 3 . abc abc eeabc 5 n. ) 1( 1 n n n . 5.3 A (1) 22 4 sinzx yxyy; (2) ) 1, 0(xxxz y ; (3) 323 3zx yxyxy; (4) 22 1;zxyxy (5) ; xy zxe (6)(cossin ) x zeyxy; (1) 24sin z xyy x 2 4 cos2 z xxyy y 2 2 2 z y x 2 24cos z xy x y 2 24cos z xy y x 2 2 4 sin2 z xy y (2) 1y yx x z xx y z yln 2 2 2 (1) y z y yx x 2 2 2 (ln ) y z xx y . . 5.3 5.3 A A 22 zx yxyyzx yxyyzx yxyy; (2) ; (2) 323 zx yxyxyzx yxyxyzx yxyxyzx yxyxy; (4) 2 11 ln yy z yxxx x y 2 11 ln yy z yxxx y x (3) yyyx x z 322 33xxyyx y z 23 92 2 2 2 6xy x z 2 3 2 218 z xxy y 196 22 2 yyx yx z 196 22 2 yyx xy z (4) x xxxxyyx x z 1 22 002yxyx2; y yyyxyyx y z 1 22 xy2; 20222 2 2 x xxyxyx x z ; 11022 2 y yyyxyx yx z ; 11022 2 x xxxyxy xy z ; 20222 2 2 y yyxyxy x z . (5) xyxy z exye x 2xy z x e y 2 22 2 2 xyxyxyxyxy z yeyexy eyexy e x 2 3 2 xy z x e y 2 22 2 xyxyxyxyxy z xexex yexex ye x y 2 2 2 xyxy z xex ye x y (6)cos1 sin x z eyxy x ( sincos ) x z eyxy y 2 2 cos2 sin x z eyxy x 2 2 ( cossin ) x z eyxy y 10101010 y y y1; ; 10 010 0 x x xyxyxyxy exyeexye xyxyxyxyxyxy exye 2 sin1 cos x z eyxy x y 2 sin1 cos x z eyxy x y 2. 22 lnyxz0 2 2 2 2 y z x z )ln( 2 1 ln 2222 yxyxz 22 yx x x z 22 yx y y z 222 22 222 22 2 2 )()( 2)( yx xy yx xxyx x z 222 22 222 22 2 2 )()( 2)( yx yx yx yyyx y z 0 )()( 222 22 222 22 2 2 2 2 yx xy yx yx y z x z B 1. ),(yxfD 2 2 x f 2 2 y f 2 f x yxy f 2 D A xy f yx f 22 B),(yxfD C),(yxfDD 2. 1 , 23 y x ( )(0)n y= 12 ( 1)!( ) . 33 nn n 1 (23) ,yx 223 1 2(23) ,1 ( 2) 2 (23)yxyx ( )1 ( 1)! 2 (23) nnnn ynx ( )(0)n y= 12 ( 1)!( ) . 33 nn n )()2 222 ) ) 2 2 y y yx y y 0 )()2 2222) ) 2 2 y y x x B xy xy f xy y 3.),(vuf,fxg yyxg yg y 0,g y )( )( 2 2 vg vg vu f . ,uxg yvy),(vuf. ,uxg yvy),(vuf)( )( vg vg u )( 1 vgu f )( )( 2 2 vg vg vu f . 4.z=)32sin(yx x z y z xx z yy z xy z. x z=)32cos(2)32()32cos(yxyxyx x , y z=)32cos(3)32()32cos(yxyxyx y , xx z=)32sin(4)32cos(2yxyxx, yy z=)32sin(9)32cos(3yxyx y , xy z=)32sin(6)32cos(2yxyxy. 5.4 A 1. (1) sinzuvt,cos , t ue vt dt dz (2) xyyxzsin, 2 dx dz (3) 1 sin(),ln ,cos , xy ueyz xyt zt tdt du (4) ),(),(),( 2 tytx y x t x y tfu, f dt du (1) t z dt dv v z dt du u z dt dz sincos t v eutt 3 )32)3y yx3 3, 32 2sin(sin(2 2sin(sin()3, )32)3sin(3 32sin( cossincos tt t eettcossincos . t ettt (2) dz z dxz dy dtx dty dt zz dy xy dt 1cos 2cos2. 22 sin x xxx yx (3) )lnsin cos )(lncos(cos) ln1 )(lnsin(cos 2 1 2 1 tt t t ttt t t ttt dt du tt (4) 12 22 ( )( )( )( ) ( )( ) ( )( )( ) 2 ( )( )( ) duttttttttt ftft dtttt 2. (1) )( 2 xyxyfz (2) ),()(xyyxfyxz (3) )(),(yxy x y fz (4) 22 ( , (, )zf x yxy. (1) )( 2)(),( )( 2222232 xyfyxxyxf y z xyfxyxyyf x z (2) 1212 (,)()(),(,)()() zz f xy xyxyfyff xy xyxyfxf xy (3) 1212 2 1 ( )()(),( )(1() zyyzy ffxyffxy xxxyx x (4) 131232 2, 2. zz fxfyff xy 3. (1) 222 ),( zyx ezyxfuyxzsin 2 x u y u (2) yxvyxauvuez ax cos,sin),( x z y z (3) 22 (,), xy zf xye z x . z y (1) x z z f x f x u yxzexe zyxzyx sin222 222222 yxyx eyxx 2422 sin22 )sin21 (2 y z z f y f y u yxzeye zyxzyx cos22 2 222222 ( )()(),( )(1() zz (,)()(),(,)()()(,)()(),(,)()() 121212 (,)()(),(,)()() xyxy zzzzzz (,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()() zzzz (,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()() 121212 (,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()() 12121212 (,)()(),(,)()() 1212 xyxy ( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1() ( , (, )( , (, ). . )( 2 2 xy (,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()() xy (,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()()(,)()(),(,)()() xy zyzy ( )()(),( )(1()( )()(),( )(1()( )()(),( )(1() zyzy ( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1()( )()(),( )(1() 232 xy fxfyfffxfyff 232 2, 2.2, 2. 232 fxfyfffxfyfffxfyff2, 2.2, 2.2, 2.2, 2.2, 2.2, 2.2, 2.2, 2. 232 2, 2.2, 2.2, 2.2, 2.2, 2. xy fxfyff2, 2. 2 x2 e),),),) yxyx eyyxy 2422 sin4 )cossin(2 (2) ( , , )() ax zf x u veuv x v v f x u u f x f x z xexaevuae axaxax sincos)(2sin) 1( 2 ayxaeax .2) 1(1 axaxax eee y v v f y u u f y z (3) 22, . xy uxyve( , )uf u v 2 xy zfufvff xey xuxvxuv 12 2. xy xfyef ( 2 ) xy zfufvff yex yuyvyuv 12 2. xy yfxef 4 (1) ),( x y y z fxu k ku z u z y u y x u x (2) 222 1 zyx u0 2 2 2 2 2 2 z u y u x u (3) 2 () 2 y zxy x 22 3 0 2 zz xxyy xy (1) 12 2121 2 1 (,), k kkk uzyuzux kxfxyfxfff xy xyyxzy z u z y u y x u x 2121 (,) k kkk zyzyx z kx fx yfxfffku y xyxy . (2) 2 5 2 3 )( 3 , )( 222 2222 2 2 222 zyx xzyx x u zyx x x u , z z y y u u y y xy y y 2 2u u 1212 2. 12 12 2. yfxef 1212 2.2.2. 121212 2.2.2.2. 121212 kuku z z z 2 2 2 2 2 y y u x kxfxyfxfff kkkkkk121212kkkkkkkkk121212 uzyuzuxuzyuzux kkkkkk1212kkkkkk121212 kxfxyfxfffkxfxyfxfff uzyuzuxuzyuzux (,), kkkkkk1212 (,), 12 kxfxyfxfff 2 2 y u 2 5 2 5 )( 3 , )( 3 222 2222 2 2 222 2222 zyx zzyx z u zyx yzyx 0 2 2 2 2 2 2 z u y u x u . (3) 2 2 () 2 zy yxy xx () zy xxy yx 2 zz xxy xy 2 2 2 ()() 2 yy xyxyxyxxy xx 2 2222 3 ()() 22 y x yxyyx yxyy 22 3 0 2 zz xxyy xy 5),(yxzz0),(zyxzfdz. 0),(zyxzf 12 ()()0,f d xzf d yz 12 ()()0f zdxxdzfdydz 12 12 1 ()dzzf dxf dy xff . 6 (1) ( , ) ( , ), f x y uxf x y y 2u x y (2) f x y xfxyf y z),()( 1 yx z 2 (3) fxyyxfz,)(,)(, yx z 2 22 2 zz 22 xxyyxxyy xyxy 2222 xxyyxxyyxxyyxxyyxxyy xyxyxy 0),0),) )yzy f 3 2222 x yxyyx yxyyx yxyyx yxyyx yxyyx yxyy 2222 x yxyyx yxyy 0 ),(yf 1212 f d xzf d yzf d xzf d yz 1212 ()()0,()()0,()()0, 12121212 f d xzf d yz()()0,()()0,()()0,()()0, 1212 1212 f zdxxdzfdydzf zdxxdzfdydz 1212 ()()0()()0()()0 121212 f zdxxdzfdydzf zdxxdzfdydzf zdxxdzfdydz()()0()()0 (4) 0 0 0 , )( ),( 22 22 22 22 yx yx yx yxxy yxf 22 (0,0)(0,0) ,. ff x yy x (1) 1 ( , )( , )( , ), xx u fx yf x yxfx y xy 2 2 11 ( , )( , )( , )( , ) xxyyxy u fx yfx yfx yxfx y x yyy . (2)()( )( ) zyyy fxyff xxxx 2 2 ()( ) zyy xfxyf x yxx . (3) 12 ( ) z fxf x 2 111222 ( )( )( ) 1( ) z x fxyfy f x y . (4) 0 ( ,0)(0,0) (0,0)lim0. x x f xf f x 0y x yfyxf yf x x ), 0(),( lim), 0( 0 ,)0 )( ( 1 lim 22 22 0 y yx yxxy x x 0 (0, )(0,0) (0,0)lim1, xx xy y fyf f y , 0)0 , 0( y f 0x ( ,0) y fxx. 0 ( ,0) (0,0) (0,0)lim1. yy yx x fxf f x 7.)(),(min)( 21 xfxfxF2 , 0, x xfxxf 1 )(,)( 21 . 2 , 0),(xxF. x x ) )yfyf 0. ( ( ( 1 1 limlim 0 0 xyxy x x x x (0,0)lim(0,0)lim y fyf (0,0)lim(0,0)lim(0,0)lim )(xF ,01, ( ) 1 ,12. xx F x x x 10x ( )( )1 .F xx 21x 2 1 ) 1 ()( xx xF 1x 1 1)1 ( lim ) 1 ()1 ( lim) 1 ( 00 x x x FxF F xx . x x x FxF F xx 1 1 1 lim ) 1 ()1 ( lim) 1 ( 00 1 )1 ( lim )1 ( )1 (1 lim 00 xx x xx x xx . ) 1 () 1 (FF ) 1 (F 8.0 y yxexyf x , y F x yyxex 1,1, yy xy FeFxe x y Fdy dxF 11 11 yy yy ee xexe 9. y z z x ln,zf x y., y z x z ( , , )ln, xz F x y z zy x )1 ()1 )1) 1 (1x 1 (1 1 (1 ( 1 1. . x x x x 1 1 1 1 lim 0 1 ( limlim 0 0 1 (x ) 1 ( ) 1 (F) 1 ( ) 1 ( 0 0yxexyxex y y F x yyxexF x yyxex, y y F x yyxexF x yyxexF x yyxex, y y 2 11 , xy yz FF zzyy 22 1 . z xxz F zzz 2 1 , x z Fzz z xz xFxz z 2 2 1 . () y z F zzy xz yFy xz z 10. 22 10xyyf x0x 22 ,1F x yxy 2 ,2 ,0,10,0,120, xyxy Fx Fy FF y x F F dx dy y x 0 0x dx dy 33 22 222 2 1 )( yy xy y y x xy y yxy dx yd 1 0 2 2 x dx yd 11. 3 z ezxy,zf x y 2 2 x z , ,3, z F x y zezxy ,1, z xy Fy Fe , 11 x zz z Fzyy xFee 22 22223 () 1 . 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