ÊÔ¾íÒ»£º

µÃ ·Ö

Ò»¡¢µ¥ÏîÑ¡ÔñÌ⣨3·Ö¡Á5=15·Ö£©

1¡¢1¡¢ÏÂÁи÷ʽÕýÈ·µÄÊÇ£¨ £©

£¨A£©limA????n??n??n?1k??nAk; £¨B£©limn??An??n?1k??nAk;

£¨C£©limA????n??n??n?1k??nAk; £¨D£©limn??An??n?1k??nAk;

2¡¢ÉèPΪCantor¼¯£¬ÔòÏÂÁи÷ʽ²»³ÉÁ¢µÄÊÇ£¨ £© £¨A£©P? c (B) mP?0 (C) P'?P (D) P??P 3¡¢ÏÂÁÐ˵·¨²»ÕýÈ·µÄÊÇ£¨ £©

(A) ·²Íâ²à¶ÈΪÁãµÄ¼¯ºÏ¶¼¿É²â£¨B£©¿É²â¼¯µÄÈκÎ×Ó¼¯¶¼¿É²â (C) ¿ª¼¯ºÍ±Õ¼¯¶¼ÊDz¨À׶ú¼¯ £¨D£©²¨À׶ú¼¯¶¼¿É²â

4¡¢Éè?fn(x)?ÊÇEÉϵÄa.e.ÓÐÏ޵Ŀɲ⺯ÊýÁÐ,ÔòÏÂÃæ²»³ÉÁ¢µÄÊÇ( )£¨A£©Èôfn(x)?f(x), Ôòfn(x)?f(x) (B) sup?fn(x)?Êǿɲ⺯Êýn £¨C£©infn?fn(x)?Êǿɲ⺯Êý;£¨D£©Èôfn(x)?f(x),Ôòf(x)¿É²â

5¡¢Éèf(x)ÊÇ[a,b]ÉÏÓнç±ä²îº¯Êý£¬ÔòÏÂÃæ²»³ÉÁ¢µÄÊÇ£¨ £© (A) f(x)ÔÚ[a,b]ÉÏÓнç (B) f(x)ÔÚ[a,b]Éϼ¸ºõ´¦´¦´æÔÚµ¼Êý £¨C£©f'(x)ÔÚ[a,b]ÉÏL¿É»ý (D) ?baf'(x)dx?f(b)?f(a)

µÃ ·Ö

¶þ. Ìî¿ÕÌâ(3·Ö¡Á5=15·Ö)

1¡¢(CsA?CsB)?(A?(A?B))?_________

2¡¢ÉèEÊÇ?0,1?ÉÏÓÐÀíµãÈ«Ì壬ÔòE'=______,Eo=______,E=______.

3¡¢ÉèEÊÇRnÖе㼯£¬Èç¹û¶ÔÈÎÒ»µã¼¯T¶¼ÓÐ_________________________________£¬Ôò³ÆEÊÇL¿É²âµÄ

4¡¢f(x)¿É²âµÄ________Ìõ¼þÊÇËü¿ÉÒÔ±í³ÉÒ»Áмòµ¥º¯ÊýµÄ¼«ÏÞº¯Êý. £¨Ìî¡°³ä·Ö¡±£¬¡°±ØÒª¡±£¬¡°³äÒª¡±£©

5¡¢Éèf(x)Ϊ?a,b?ÉϵÄÓÐÏÞº¯Êý£¬Èç¹û¶ÔÓÚ?a,b?µÄÒ»Çзֻ®£¬Ê¹_____________________________________________________,Ôò³Æf(x)Ϊ

?a,b?ÉϵÄÓнç±ä²îº¯Êý¡£

Èý¡¢ÏÂÁÐÃüÌâÊÇ·ñ³ÉÁ¢?Èô³ÉÁ¢,ÔòÖ¤Ã÷Ö®;Èô²»³ÉÁ¢,Ôò¾Ù

µÃ ·Ö ·´Àý˵Ã÷.(5·Ö¡Á4=20·Ö)

1¡¢ÉèE?R1£¬ÈôEÊdzíÃܼ¯£¬ÔòCEÊÇÎÞ´¦³íÃܼ¯¡£

2¡¢ÈômE?0£¬ÔòEÒ»¶¨ÊÇ¿ÉÊý¼¯.

3¡¢Èô|f(x)|Êǿɲ⺯Êý£¬Ôòf(x)±ØÊǿɲ⺯Êý¡£

4£®Éèf(x)Ôڿɲ⼯EÉÏ¿É»ý·Ö£¬Èô?x?E,f(x)?0£¬Ôò?f(x)?0

E

µÃ ·Ö ËÄ¡¢½â´ðÌ⣨8·Ö¡Á2=16·Ö£©.

?x2,xΪÎÞÀíÊý1¡¢£¨8·Ö£©Éèf(x)?? £¬Ôòf(x)ÔÚ?0,1?ÉÏÊÇ·ñR?¿É»ý£¬ÊÇ·ñL??1,xΪÓÐÀíÊý¿É»ý£¬Èô¿É»ý£¬Çó³ö»ý·ÖÖµ¡£

2¡¢£¨8·Ö£©Çólim?n?0ln(x?n)?xecosxdx n

µÃ ·Ö Îå¡¢Ö¤Ã÷Ì⣨6·Ö¡Á4+10=34·Ö£©.

1¡¢£¨6·Ö£©Ö¤Ã÷?0,1?ÉϵÄÈ«ÌåÎÞÀíÊý×÷³ÉµÄ¼¯ÆäÊÆΪc.

2¡¢£¨6·Ö£©Éèf(x)ÊÇ???,???ÉϵÄʵֵÁ¬Ðøº¯Êý£¬Ôò¶ÔÓÚÈÎÒâ³£Êý

a,E?{x|f(x)?a}ÊDZռ¯¡£

3¡¢£¨6·Ö£©ÔÚ?a,b?ÉϵÄÈÎÒ»Óнç±ä²îº¯Êýf(x)¶¼¿ÉÒÔ±íʾΪÁ½¸öÔöº¯ÊýÖ®²î¡£

4¡¢£¨6·Ö£©ÉèmE??,f(x)ÔÚEÉÏ¿É»ý£¬en?E(|f|?n)£¬Ôòlimn?men?0.

n

5¡¢£¨10·Ö£©Éèf(x)ÊÇEÉÏa.e.ÓÐÏ޵ĺ¯Êý£¬Èô¶ÔÈÎÒâ??0£¬´æÔÚ±Õ×Ó¼¯F??E£¬Ê¹f(x)ÔÚF?ÉÏÁ¬Ðø£¬ÇÒm(E?F?)??£¬Ö¤Ã÷£ºf(x)ÊÇEÉϵĿɲ⺯Êý¡£(³½ò¶¨ÀíµÄÄ涨Àí)

ÊÔ¾íÒ» ´ð°¸£º

ÊÔ¾íÒ» £¨²Î¿¼´ð°¸¼°ÆÀ·Ö±ê×¼£©

Ò»¡¢1. C 2 D 3. B 4. A 5. D

¶þ¡¢1£®? 2¡¢?0,1?£» ? £» ?0,1? 3¡¢m*T?m*(T?E)?m*(T?CE)

?n?4¡¢³äÒª 5¡¢??|f(xi)?f(xi?1)|?³ÉÒ»ÓнçÊý¼¯¡£

?i?1?Èý¡¢1£®´íÎó¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö

ÀýÈ磺ÉèEÊÇ?0,1?ÉÏÓÐÀíµãÈ«Ì壬ÔòEºÍCE¶¼ÔÚ?0,1?ÖгíÃÜ ¡­¡­¡­¡­¡­¡­¡­¡­¡­..5·Ö

2£®´íÎó¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö ÀýÈ磺ÉèEÊÇCantor¼¯£¬ÔòmE?0£¬µ«E?c £¬ ¹ÊÆäΪ²»¿ÉÊý¼¯ ¡­¡­¡­¡­¡­¡­¡­¡­¡­.5·Ö 3£®´íÎó¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö

??x,x?E;ÀýÈ磺ÉèEÊÇ?a,b?ÉϵIJ»¿É²â¼¯£¬f(x)??

???x,x??a,b??E;Ôò|f(x)|ÊÇ?a,b?ÉϵĿɲ⺯Êý£¬µ«f(x)²»ÊÇ?a,b?ÉϵĿɲ⺯Êý¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..5·Ö

4£®´íÎó¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö

mE?0ʱ£¬¶ÔEÉÏÈÎÒâµÄʵº¯Êýf(x)¶¼ÓÐ?f(x)dx?0¡­5·Ö

EËÄ¡¢1£®f(x)ÔÚ?0,1?Éϲ»ÊÇR?¿É»ýµÄ£¬ÒòΪf(x)½öÔÚx?1´¦Á¬Ðø£¬¼´²»Á¬ÐøµãΪÕý²â¶È¼¯¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..3·Ö

ÒòΪf(x)ÊÇÓнç¿É²âº¯Êý£¬f(x)ÔÚ?0,1?ÉÏÊÇL?¿É»ýµÄ¡­6·Ö ÒòΪf(x)Óëxa.e.ÏàµÈ£¬½øÒ»²½£¬?2£®½â£ºÉèfn(x)?2?0,1?1f(x)dx??x2dx?¡­8·Ö

031ln(x?n)?xecosx£¬ÔòÒ×Öªµ±n??ʱ£¬fn(x)?0 n ¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..2·Ö

'ÓÖÒò??lnt??t???1?lntt2?0£¬(t?3)£¬ËùÒÔµ±n?3,x?0ʱ£¬

ln(x?n)n?n?xln(x?n)nx?n?n?xln3ln3n3?3(1?x)¡­¡­¡­¡­¡­¡­4·Ö ´Ó¶øʹµÃ|fln3n(x)|?3(1?x)e?x¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­6·Ö

µ«ÊDz»µÈʽÓұߵĺ¯Êý£¬ÔÚ?0,???ÉÏÊÇL¿É»ýµÄ£¬¹ÊÓÐ

lim??fn(x)dx???n00limnfn(x)dx?0¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­8·Ö

Îå¡¢1£®ÉèE?[0,1],A?E?Q,B?E\\(E?Q).

QBÊÇÎÞÏÞ¼¯£¬??¿ÉÊý×Ó¼¯M?B ¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö QAÊÇ¿ÉÊý¼¯£¬?A?M:M. ¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­.3·Ö QB?M?(B\\M),E?A?B?A?M?(B\\M),ÇÒ(A?M)?(B\\M)??,M?(B\\M)??,¡­¡­¡­¡­..5·Ö

?E:B,?B?c.¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­6·Ö 2£®?x?E?,Ôò´æÔÚEÖеĻ¥ÒìµãÁÐ{xn},ʹlimn??xn?x¡­¡­¡­.2·Ö

Qxn?E,?f(xn)?a¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­.3·Ö

Qf(x)ÔÚxµãÁ¬Ðø£¬?f(x)?limn??f(xn)?a

?x?E¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­5·Ö

?EÊDZռ¯.¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­.6·Ö

3. ¶Ô??1£¬???0£¬Ê¹¶ÔÈÎÒ⻥²»ÏཻµÄÓÐÏÞ¸ö(ai,bi)?(a,b) nnµ±?(bi?ai)??ʱ£¬ÓÐi?1?f(b)i?f(ai)?1¡­¡­¡­¡­¡­¡­2·Ö

i?1n½«[a,b]mµÈ·Ö£¬Ê¹

?xi?xi?1??£¬¶Ô?T:xi?1?z0?z1?L?zk?xi£¬i?1ÓÐ

?i?1kf(zi)?f(zi?1)?1£¬ËùÒÔ

f(x)ÔÚ[xi?1,xi]ÉÏÊÇÓнç±ä²îº¯

Êý¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­.5·Ö ËùÒÔ

V(f)?1,xi´Ó¶ø

V(f)?mb£¬Òò´Ë£¬f(x)ÊÇ[a,b]ÉϵÄÓнç±ä²îº¯

xi?1aÊý¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..6·Ö 4¡¢f(x)ÔÚEÉÏ¿É»ý?limn??mE(|f|?n)?mE(|f|???)?0¡­¡­2·Ö

¾Ý»ý·ÖµÄ¾ø¶ÔÁ¬ÐøÐÔ£¬

???0,???0,?e?E,me??£¬?e|f(x)|dx??¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­.4·Ö

¶ÔÉÏÊö??0,?k,?n?k,mE(|f|?n)??£¬´Ó¶øn?men??e|f(x)|dx??£¬nlimnn?men?0¡­¡­¡­¡­¡­¡­¡­6·Ö

5

£®

?n?N,´æÔÚ±Õ¼¯

F?F1n?E,m?En??2n,f(x)ÔÚ

FnÐø¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö ??Áî

F?UIF?n£¬Ôò

?x?F??k,x?n??kFn,?n?k,x?Fn?f(x)ÔÚFk?1n?kÐø¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­4·Ö ÓÖ¶ÔÈÎÒâk,m?E?F??m[E?(??F?n?kn)]?m[n??k(E?Fn)]

???m(E?F1n)?2k¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­.6·Ö n?k¹Êm(E?F)?0,f(x)ÔÚF?EÁ¬Ðø¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..8·Ö ÓÖm(E?F)?0,ËùÒÔf(x)ÊÇE?FÉϵĿɲ⺯Êý£¬´Ó¶øÊÇEÉ쵀 ¿É²âº¯Êý¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..10·Ö

ÊÔ¾í¶þ£º

ÓÐ

¼´

Á¬

Á¬

¡¶Êµ±äº¯Êý¡·ÊÔ¾í¶þ

רҵ________°à¼¶_______ÐÕÃû ѧºÅ ÌâºÅ Ò» ¶þ Èý ËÄ Îå ×Ü·Ö

µÃ·Ö ×¢ Òâ Ê Ïî

1¡¢±¾ÊÔ¾í¹²6Ò³¡£

2¡¢¿¼Éú´ðÌâʱ±ØÐë׼ȷÌîдרҵ¡¢°à¼¶¡¢Ñ§ºÅµÈÀ¸Ä¿£¬×Ö¼£ÒªÇå³þ¡¢¹¤Õû¡£

µÃ ·Ö Ò».µ¥ÏîÑ¡ÔñÌ⣨3·Ö¡Á5=15·Ö£©

1£®ÉèM,NÊÇÁ½¼¯ºÏ£¬Ôò M?(M?N)=£¨ £© (A) M (B) N (C) M?N (D) ?

2. ÏÂÁÐ˵·¨²»ÕýÈ·µÄÊÇ( )

(A) P0µÄÈÎÒ»ÁìÓòÄÚ¶¼ÓÐEÖÐÎÞÇî¶à¸öµã£¬ÔòP0ÊÇEµÄ¾Ûµã (B) P0µÄÈÎÒ»ÁìÓòÄÚÖÁÉÙÓÐÒ»¸öEÖÐÒìÓÚP0µÄµã£¬ÔòP0ÊÇEµÄ¾Ûµã (C) ´æÔÚEÖеãÁÐ?Pn?£¬Ê¹Pn?P0£¬ÔòP0ÊÇEµÄ¾Ûµã

(D) ÄÚµã±ØÊǾ۵ã

3. ÏÂÁжÏÑÔ( )ÊÇÕýÈ·µÄ¡£

£¨A£©ÈÎÒâ¸ö¿ª¼¯µÄ½»ÊÇ¿ª¼¯£»(B) ÈÎÒâ¸ö±Õ¼¯µÄ½»ÊDZռ¯£» (C) ÈÎÒâ¸ö±Õ¼¯µÄ²¢ÊDZռ¯£»(D) ÒÔÉ϶¼²»¶Ô£» 4. ÏÂÁжÏÑÔÖÐ( )ÊÇ´íÎóµÄ¡£

£¨A£©Áã²â¼¯Êǿɲ⼯£» £¨B£©¿ÉÊý¸öÁã²â¼¯µÄ²¢ÊÇÁã²â¼¯£» £¨C£©ÈÎÒâ¸öÁã²â¼¯µÄ²¢ÊÇÁã²â¼¯£»£¨D£©Áã²â¼¯µÄÈÎÒâ×Ó¼¯Êǿɲ⼯£» 5. Èôf(x)Êǿɲ⺯Êý£¬ÔòÏÂÁжÏÑÔ£¨ £©ÊÇÕýÈ·µÄ (A) f(x)ÔÚ?a,b?L?¿É»ý?|f(x)|ÔÚ?a,b?L?¿É»ý£»

(B) f(x)ÔÚ?a,b?R?¿É»ý?|f(x)|ÔÚ?a,b?R?¿É»ý (C) f(x)ÔÚ?a,b?L?¿É»ý?|f(x)|ÔÚ?a,b?R?¿É»ý; (D) f(x)ÔÚ?a,???R?¹ãÒå¿É»ý?f(x)ÔÚ?a,+??L?¿É»ý

µÃ ·Ö ¶þ. Ìî¿ÕÌâ(3·Ö¡Á5=15·Ö)

111¡¢ÉèAn?[,2?],n?1,2,L£¬ÔòlimAn?_________¡£

n??nn2¡¢ÉèPΪCantor¼¯£¬Ôò P? £¬mP?_____£¬P=________¡£

???3¡¢Éè?Si?ÊÇÒ»Áпɲ⼯£¬Ôòm??Si?______?mSi

?i?1?i?1?o4¡¢Â³½ò¶¨Àí£º______________________________________________________

_______________________________________________________________ 5¡¢ÉèF(x)Ϊ?a,b?ÉϵÄÓÐÏÞº¯Êý£¬Èç¹û_________________________________ _____________________________________________________________________________________________Ôò³ÆF(x)Ϊ?a,b?Éϵľø¶ÔÁ¬Ðøº¯Êý¡£

µÃ ·Ö Èý.ÏÂÁÐÃüÌâÊÇ·ñ³ÉÁ¢?Èô³ÉÁ¢,ÔòÖ¤Ã÷Ö®;Èô²»³ÉÁ¢,Ôò˵Ã÷Ô­Òò»ò¾Ù³ö·´Àý.(5·Ö¡Á4=20·Ö)

1¡¢ÓÉÓÚ?0,1???0,1???0,1?£¬¹Ê²»´æÔÚʹ?0,1?ºÍ?01 £¬?Ö®¼ä1?1¶ÔÓ¦µÄÓ³Éä¡£

2¡¢¿ÉÊý¸öÁã²â¶È¼¯Ö®ºÍ¼¯ÈÔΪÁã²â¶È¼¯¡£

3¡¢a.e.ÊÕÁ²µÄº¯ÊýÁбØÒÀ²â¶ÈÊÕÁ²¡£

4¡¢Á¬Ðøº¯ÊýÒ»¶¨ÊÇÓнç±ä²îº¯Êý¡£

µÃ ·Ö ËÄ.½â´ðÌâ(8·Ö¡Á2=16·Ö)

?x,xΪÎÞÀíÊý1¡¢Éèf(x)?? £¬Ôòf(x)ÔÚ?0,1?ÉÏÊÇ·ñR?¿É»ý£¬ÊÇ·ñL?¿É»ý£¬

?1,xΪÓÐÀíÊýÈô¿É»ý£¬Çó³ö»ý·ÖÖµ¡£

2¡¢Çó¼«ÏÞ lim?n??nxsin3nxdx. 2201?nx1

µÃ ·Ö Îå.Ö¤Ã÷Ìâ(6·Ö¡Á3+ 8?2 =34·Ö)

1.(6·Ö) 1¡¢Éèf(x)ÊÇ(??,??)ÉϵÄʵֵÁ¬Ðøº¯Êý£¬Ôò¶ÔÈÎÒâ³£Êý c£¬

E?{x|f(x)?c} ÊÇÒ»¿ª¼¯.

2.(6·Ö) Éè??0,?¿ª¼¯G?E,ʹm*(G?E)??£¬ÔòEÊǿɲ⼯¡£

3. £¨6·Ö£©ÔÚ?a,b?ÉϵÄÈÎÒ»Óнç±ä²îº¯Êýf(x)¶¼¿ÉÒÔ±íʾΪÁ½¸öÔöº¯ÊýÖ®²î¡£

4.£¨8·Ö£©É躯ÊýÁÐfn(x) (n?1,2,L)ÔÚÓн缯EÉÏ¡°»ù±¾ÉÏ¡±Ò»ÖÂÊÕÁ²ÓÚf(x)£¬Ö¤Ã÷£ºfn(x)a.e.ÊÕÁ²ÓÚf(x)¡£

5.£¨8·Ö£©Éèf(x)ÔÚE??a,b?ÉÏ¿É»ý£¬Ôò¶ÔÈκÎ??0£¬±Ø´æÔÚEÉϵÄÁ¬Ðøº¯Êý?(x)£¬Ê¹?|f(x)??(x)|dx??.

ab

ÊÔ¾í¶þ£¨²Î¿¼´ð°¸¼°ÆÀ·Ö±ê×¼£©

Ò»¡¢1£¬C 2, C 3, B 4, C 5, A ¶þ¡¢1£¬?0,2? 2£¬c £»0 £»? 3£¬ ?

4£¬Éèf(x)ÊÇEÉÏa.e.ÓÐÏ޵Ŀɲ⺯Êý£¬Ôò¶ÔÈÎÒâ??0£¬´æÔÚ±Õ×Ó¼¯E??E£¬Ê¹µÃf(x)ÔÚE?ÉÏÊÇÁ¬Ðøº¯Êý£¬ÇÒm(E\\E?)??¡£

5£¬¶ÔÈÎÒâ??0,???0£¬Ê¹¶Ô?a,b?Öл¥²»ÏཻµÄÈÎÒâÓÐÏÞ¸ö¿ªÇø¼ä

?ai,bi?,i?1,2,L,n,Ö»Òª??bi?ai???£¬¾ÍÓÐ?|F(bi)?F(ai)|??

i?1i?1nnÈý¡¢1£®´íÎó¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö

??(0)?r1??(1)?r?2¼Ç(0,1)ÖÐÓÐÀíÊýÈ«ÌåR?{r1,r2,L}?

??(rn)?rn?2,n?1,2L?1]ÖÐÎÞÀíÊý£¬??(x)?x,xΪ[0£¬1]0£¬1£©ÉϵÄ1?1Ó³Éä¡£¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­5·Ö ÏÔÈ»?ÊÇ[0£¬µ½£¨2£®ÕýÈ·¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö ÉèEiΪÁã²â¶È¼¯£¬ 0?m(UEi)??mE?0£¬ËùÒÔ£¬m(UEi)?0

***i?1i?1i?1???Òò´Ë£¬UEiÊÇÁã²â¶È¼¯¡£¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­5·Ö

i?1?3£®´íÎó¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö

?1,x?(0,n]ÀýÈ磺ȡE?(0,??),×÷º¯ÊýÁУºfn(x)??n?1,2,L

?0,x?(n,??)ÏÔÈ»fn(x)?1,µ±x?E¡£µ«µ±0???1ʱ£¬E[|fn?1|??]?(n,??) ÇÒm(n,??)???Õâ˵Ã÷fn(x)²»²â¶ÈÊÕÁ²µ½1.¡­¡­¡­¡­¡­¡­5·Ö 4£®´íÎó¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö

???xcos,0?x?1,ÀýÈ磺f(x)??ÏÔÈ»ÊÇ?0,1?µÄÁ¬Ðøº¯Êý¡£ 2x??0,x?0.Èç¹û¶Ô?0,1?È¡·Ö»®T:0?2nn1111??L???1£¬ÔòÈÝÒ×Ö¤Ã÷ 2n2n?13211|f(xi)?f(xi?1)|??£¬´Ó¶øµÃµ½V(f)??¡­¡­¡­¡­¡­¡­¡­5·Ö ?0i?1i?1iËÄ¡¢1£®f(x)ÔÚ?0,1?Éϲ»ÊÇR?¿É»ýµÄ£¬ÒòΪf(x)½öÔÚx?1´¦Á¬Ðø£¬ ¼´²»Á¬ÐøµãΪÕý²â¶È¼¯¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­3·Ö ÒòΪf(x)ÊÇÓнç¿É²âº¯Êý£¬ËùÒÔf(x)ÔÚ

?0,1?ÉÏÊÇL?¿É»ý

µÄ¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­. ¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­.6·Ö

11ÒòΪf(x)Óëxa.e.ÏàµÈ, ½øÒ»²½£¬?f(x)dx??xdx?¡­¡­8·Ö

0?0,1?2nxn??2É裬ÔòÒ×Öªµ±fn(x)?sin3nxdx221?nxfn(x)?0¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö

ʱ£¬

ÓÖ|fn(x)|?nx¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­4·Ö 221?nxµ«ÊDz»µÈʽÓұߵĺ¯Êý£¬ÔÚ?0,???ÉÏÊÇL¿É»ýµÄ¡­¡­¡­¡­¡­6·Ö ¹ÊÓÐlim?fn(x)dx??limfn(x)dx?0¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­8·Ö

n00n??Îå¡¢1£®?x?E,f(x)?c¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..1·Ö

Qf(x)ÔÚxµãÁ¬Ðø£¬?¶Ô??f(x)?c?0,?U(x,?),µ±y?U(x,?)ʱ£¬

ÓÐf(y)?f(x)??¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­3·Ö ??f(x)?c?f(y)?f(x)?f(x)?c?f(y)?c£¬?y?E¡­¡­5·Ö Òò´ËU(x,?)?E£¬´Ó¶øEΪ¿ª¼¯¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..6·Ö 2£®¶ÔÈκÎÕýÕûÊý

n£¬ÓÉÌõ¼þ´æÔÚ¿ª¼¯Gn?E,ʹ

m*(Gn?E)??1¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­1·Ö nÁîG?IGn£¬ÔòGÊǿɲ⼯¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­3·Ö

n?11¶ÔÒ»ÇÐÕýÕûÊýn³ÉÁ¢£¬Òò¶øm*(G?E)?0£¬¼´nÊÇÒ»Áã²â¶È¼¯£¬ËùÒÔÒ²¿ÉM?G?E²â.¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­5·Ö ÓÖÒòm*(G?E)?m*(Gn?E)?ÓÉE?G?(G?E)Öª£¬E¿É²â¡£¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­6·Ö 3¡¢Ò×Öªg(x)?V(f)ÊÇ?a,b?ÉϵÄÔöº¯Êý¡­¡­¡­¡­¡­¡­¡­¡­¡­2·Ö

axÁîh(x)?g(x)?f(x), Ôò¶ÔÓÚa?x1?x2?bÓÐ

h(x2)?h(x1)?g(x2)?g(x1)?[f(x2)?f(x1)]?V(f)?[f(x2)?f(x1)]?|f(x2)?f(x1)|?[f(x2)?f(x1)]?0x1x2

ËùÒÔh(x)ÊÇ?a,b?ÉϵÄÔöº¯Êý¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­4·Ö

Òò´Ëf(x)?g(x)?h(x)£¬ÆäÖÐg(x)Óëh(x)¾ùΪ?a,b?ÉϵÄÓÐÏÞÔöº¯Êý¡­¡­¡­¡­. ¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­.6·Ö

4¡¢ÒòΪfn(x)ÔÚEÉÏ¡°»ù±¾ÉÏ¡±Ò»ÖÂÊÕÁ²ÓÚf(x)£¬ËùÒÔ¶ÔÓÚÈÎÒâµÄk?Z?£¬´æÔڿɲâ

¼¯

Ek?E£¬

fn(x)ÔÚ

EkÉÏÒ»ÖÂÊÕÁ²ÓÚf(x)£¬ÇÒ

m(E\\Ek)?1¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­3·Ö kÁîE?UEk£¬Ôòfn(x)ÔÚE*ÉÏ´¦´¦ÊÕÁ²µ½f(x)¡­¡­¡­¡­¡­5·Ö

*k?1?m(E\\E)?m(E\\UEk)?m(E\\Ek)?*k?1?1£¬k=1,2L kËùÒÔm(E\\E*)?0¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­8·Ö 5¡¢Ö¤Ã÷£ºÉèen?E[|f|?n],ÓÉÓÚ

f(x)ÔÚEÉÏa.e.ÓÐÏÞ£¬¹Ê

men?0,(n??)¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­..2·Ö

ÓÉ»ý·ÖµÄ¾ø¶ÔÁ¬ÐøÐÔ£¬¶ÔÈκÎ???0,?N£¬Ê¹

N?meN??|f(x)|dx?eN?4¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­¡­4·Ö

ÁîBN?E\\eN£¬ÔÚBNÉÏÀûÓó½ò¶¨Àí£¬´æÔÚ±Õ¼¯FN?BNºÍÔÚR1ÉϵÄÁ¬Ðøº¯Êý?(x)ʹ£¨1£©

x?R14Nsup|?(x)|?sup|f(x)|?N¡­¡­¡­¡­¡­¡­¡­¡­6·Ö

x?FNm(BN\\FN)??;£¨2£©

x?FNʱ£¬f(x)??(x)£¬ÇÒ

ËùÒÔ

?ba|f(x)??(x)|dx??|f(x)??(x)|dx??|f(x)??(x)|dxeNBNeNeNBN\\FN??|f(x)|dx??|?(x)|dx???|f(x)??(x)|dx??

?4?N?meN?2N??4N??4??4??2¡­¡­¡­¡­¡­¡­¡­¡­...8·Ö