µÚ¶þÕ ²¨º¯ÊýºÍѦ¶¨ÚÌ·½³Ìb ÏÂÔر¾ÎÄ

?2n(x)?Asinn?n?(x?a)?Aein?sinx,n?1,2,L (2.3-14) aaÇ¡ºÃÂú×ã±¾ÌâÖÐÊÆÚåÄÚ±¾Õ÷º¯ÊýËùÒªÇóµÄ·½³ÌºÍ±ß½çÌõ¼þ

??''(x)?k2?(x)?0,0?x?a (2.3-15) ???(0)??(a)?0Òò´Ë±¾ÌâµÄ½âΪ

?n(x)?Aein?sinn?x,0?x?a,n?1,2,L (2.3-16) a½«ÉÏÃæµÄ±¾Õ÷º¯Êý´úÈ붨̬Ѧ¶¨ÚÌ·½³Ì(2.3-2)ʽºó£¬Á¢¿ÌÄÜÁ¿±¾Õ÷ֵΪ

d2?n(x)?2h2n2h2 (2.3-17) En???222m?n(x)dx2ma¡¾ÎïÀíÌÖÂÛ¡¿

µÚÒ»½âÖÐÔڵõ½ÏµÊýB?0ºó£¬ÁíÒ»¸öϵÊýA?0£¬·ñÔò²¨º¯ÊýÔÚÈ«¿Õ¼äÖж¼µÈÓÚÁ㣬¶ÔÓ¦µÄÁ£×Ó²»´æÔÚ¡£Í¬Ñù£¬Á¿×ÓÊýn Ò²²»ÄܵÈÓÚÁ㣬Òò´ËÁ£×ӵĻù̬Ϊn = 1¡£

ÓÉÁ¿×Ó»¯Ìõ¼þ£¨1-2£©£¬µÃµ½

??pdq?2aÓɴ˽â³ö

2mE?nh,n?N (2.3-18)

h2n2?2h2n2En??,n?0,1,2,L (2.3-19)

8ma22ma2Á½ÕßÏà²î1¸öÄܼ¶¡£

´ËÍ⣬°´ÕÕÁ¿×ÓÁ¦Ñ§µÄ½á¹û£¬Á£×ӵĸÅÂÊ·Ö²¼Îª

?2wn(x)?|?n(x)|2?asin2nax,0?x?a,n?1,2,L (2.3-20)

µ«ÊÇ°´ÕÕ¾­µäÁ¦Ñ§£¬ÔÚ(x,x?dx)Çø¼äÄÚÕÒµ½Á£×ӵĸÅÂÊw(x)dxÓë¸ÃÁ£×ÓÔÚ´ËÇø¼äÄÚ¶ºÁôµÄʱ¼ä³ÉÕý±È£¬¼´w(x)dx?dt/T£¬ÆäÖÐTΪÁ£×ÓÔ˶¯µÄÖÜÆÚ¡£¿¼Âǵ½ÔÚÒ»¸öÖÜÆÚÖУ¬Á£×ÓÁ½´Î¾­¹ýͬһ¸öλÖã¬ÓÚÊǵõ½¸ÅÂÊÃܶÈΪ

w(x)?2dt2? (2.3-21) TdxTvÔÚ±¾ÌâÖУ¬T?2a/v£¬ÓÚÊǵõ½w(x)?1/a£¬ÎªÒ»¸öÓëλÖúÍÄÜÁ¿¶¼Î޹صij£Êý¡£ 2.4 Ö¤Ã÷¿í¶ÈΪ2aµÄһά¶Ô³ÆÎÞÏÞÉî·½ÊÆÚåÖб¾Õ÷º¯Êý?n(x)µÄ¹éÒ»»¯Òò×ÓÊÇA'?1/a¡£ ¡¾ÌâÒâ·ÖÎö¡¿

9

n???A'sin2a(x?a),|x|?aÒÑÖªÌõ¼þ£º±¾Õ÷º¯ÊýΪ?n(x)?? (2.4-1)

0,|x|?a??´ýÇóÎÊÌ⣺¹éÒ»»¯Òò×ÓA'£» Ï໥ÁªÏµ£º¹éÒ»»¯Ìõ¼þ

????|?(x)|2dx?1 (2.4-2)

¡¾Çó½â¹ý³Ì¡¿

½«±¾Õ÷º¯Êý(2.4-1)ʽ´úÈë¹éÒ»»¯Ìõ¼þ£¬µÃµ½

?????2?ndx??|A|si2na?2an?2a?x(2a)d?x?|A?|a

1Óɴ˵õ½¹éÒ»»¯Òò×ÓA??Óֽ⣺

1i?e ¡£ a ´ÓÎïÀíÉÏ¿´£¬Ò»Î¬ÎÞÏÞÉî·½ÊÆÚåÖÐÔ˶¯Á£×ÓµÄÄܼ¶ºÍ¹éÒ»»¯Òò×ÓÍêÈ«ÓÉÊÆÚåµÄ¿í¶È¾ö¶¨£¬ÓëÊÆÚåÔÚxÖáÉÏËù´¦µÄλÖÃÎ޹ء£ÓÉÉÏÌ⣬ÊÆÚå¿í¶ÈΪaʱ¹éÒ»»¯Òò×ÓΪ2/a£¬Òò´Ëµ±ÊÆÚå¿í¶ÈΪ2aʱ£¬¹éÒ»»¯Òò×ÓÓ¦¸ÃΪ2/(2a)?1/a¡£ ¡¾ÎïÀíÌÖÂÛ¡¿

Ò»°ãÀ´Ëµ£¬¹éÒ»»¯Òò×Ó¿ÉÒÔÊǸ´Êý£¬Æä·ù½Ç²¿·Ö³ÆΪÏàÒò×Ó¡£±¾ÌâÖйéÒ»»¯Òò×ÓµÄģΪa£¬ÏàÒò×ÓΪe£¬ÆäÖзù½Ç?¿ÉÒÔÈ¡ÈÎÒâʵÊý¡£ÏàÒò×ÓÖзù½ÇµÄÈ¡Öµ¼È²»Ó°ÏìϵͳµÄ¸ÅÂÊÃܶȺ͸ÅÂÊÁ÷Ãܶȣ¬Ò²²»Ó°ÏìÄÜÁ¿¡¢¶¯Á¿µÈÎïÀíÁ¿£¬Ã»ÓпÉÒÔ¹Û²ìµÄÎïÀíЧӦ¡£Òò´Ë£¬ÎªÁ˼òµ¥Æð¼û£¬Í¨³£ºöÂÔ¹éÒ»»¯Òò×ÓÖеÄÏàÒò×Ó£¬¼´°Ñ·ù½ÇȡΪÁã¡£ÕâÑù£¬±¾ÌâÖеĹéÒ»»¯Òò×Ó³ÉΪA'?1/a¡£

ÔÚ±¾ÌâÖУ¬ËùÓб¾Õ÷º¯ÊýµÄ¹éÒ»»¯Òò×Ó¶¼Ïàͬ£¬ÕâÊÇÒ»¸ö·Ç³£ÌØÊâµÄÇé¿ö¡£ÔÚÒ»°ãÇé¿öÏ£¬¹éÒ»»¯Òò×ÓËæÁ¿×ÓÊýnµÄ±ä»¯¶ø¸Ä±ä¡£ 2.5 ÇóһάгÕñ×Ó´¦ÔÚµÚÒ»¼¤·¢Ì¬Ê±¸ÅÂÊ×î´óµÄλÖᣠ¡¾ÌâÒâ·ÖÎö¡¿

ÒÑÖªÌõ¼þ£ºÒ»Î¬Ð³Õñ×ÓµÚÒ»¼¤·¢Ì¬µÄ²¨º¯Êý

1??x??(x)??2?xe2 (2.5-1)

2?22i?´ýÇóÎÊÌ⣺Çóxm£¬Ê¹µÃw(xm)?maxx?Rw(x)£» Ï໥ÁªÏµ£ºw(x)?|?(x)|¡£

2 10

¡¾Çó½â¹ý³Ì¡¿

Óɶ¨Ò壬¸ÅÂÊÃܶÈΪ

w(x)?|?(x)|?22?3??x2e??22x (2.5-2)

Âú×ãÊø¸¿Ì¬Ìõ¼þw(??)?0¡£Óɼ«ÖµÌõ¼þ

w'(x)?2?3??(2x?2?x)e23??2x2?0 (2.5-3)

x??1?/¡£ÈÝÒ×ÑéÖ¤£¬ÔÚפµã´¦ÓÐw\?0£¬µÃµ½¸ÅÂÊÃܶȺ¯ÊýµÄפµãΪx?0, w\?1/?)?0¡£ÓÚÊǵÃÖª£¬µ±x??1/?ʱ¸ÅÂÊÃܶȺ¯ÊýÈ¡¼«´óÖµ£¬ÔÚ±¾ÌâÖÐÒ²ÊÇ×î´óÖµ¡£

Óֽ⣺

ÔÚÎïÀíѧÖг£³£¿¼²ìÎïÀíÁ¿µÄÏà¶Ô±ä»¯ÂÊy'(x)/y(x)£¬ËüÍùÍù±È±ä»¯ÂÊy'(x)¸üÓÐÒâÒå¡£Ïà¶Ô±ä»¯ÂÊ¿ÉÒÔ±íʾ³É¶ÔÊýµ¼ÊýµÄÐÎʽ£¬¼´y'(x)/y(x)?[lny(x)]'£¬¼ÆËãºÜ·½±ã¡£´ÓÊýѧµÄ½Ç¶È¿´£¬¶ÔÊýº¯ÊýÊǵ¥µ÷Ôöº¯Êý£¬Äܹ»±£³Öº¯ÊýµÄÔö¼õÐÔ£¬²»¸Ä±ä¼«ÖµµãµÄÐÔÖʺÍλÖá£ÔÚ±¾ÌâÖÐÎÒÃÇÒ²¿ÉÒÔÓøÅÂÊÃܶȵĶÔÊýµ¼ÊýÀ´½øÐзÖÎö£¬¼´¼ÆËã

[lnw(x)]'?[ln2?3??2lnx??2x2]'?2?2?2x?0 (2.5-3) xÁ¢¿ÌµÃµ½x??1/?¡£ ¡¾ÎïÀíÌÖÂÛ¡¿

ÊýѧÉÏ¿ÉÒÔÖ¤Ã÷£¬ÔÚһάÊƳ¡ÖÐÔ˶¯µÄÊø¸¿Ì¬Á£×Ó£¬»ù̬ÔÚÇøÓòÄÚ²¿Ã»ÓÐÁãµã£¬µÚn¸ö¼¤·¢Ì¬ÓÐn¸öÁãµã£¨²»°üÀ¨±ß½çÉϵÄÁ½¸öÁãµã£©£¬Õâ¸ö½áÂÛ³ÆΪÁãµã¶¨Àí¡£¸ÅÂÊÃܶȺ¯ÊýΪ²¨º¯ÊýµÄÄ£·½£¬Òò´ËÒ²ÓÐn¸öÁãµã£¬¼ÓÉϱ߽çÉϵÄ2¸öÁãµã£¬Ò»¹²ÓÐn+2¸öÁãµã£¬¼´×îСֵ¡£Á¬Ðøº¯ÊýµÄÁ½¸öÏàÁÚ×îСֵ֮¼äÓÐÒ»¸ö¼«´óÖµ£¬Òò´ËµÚn¸öÄÜÁ¿±¾Õ÷̬µÄ¸ÅÂÊÃܶȾßÓÐn+1¸ö¼«´óÖµ¡£ÓÉÏÂÌâ¿ÉÖª£¬Ò»Î¬Êø¸¿Ì¬µÄ±¾Õ÷º¯Êý¾ßÓÐÈ·¶¨µÄÓî³Æ£¬¶ÔÓ¦µÄ¸ÅÂÊÃܶÈΪżº¯Êý£¬Æ伫´óÖµµÄ·Ö²¼¹ØÓÚyÖá¶Ô³Æ¡£ÔÚ±¾ÌâµÄÇé¿öÏ£¬n = 1£¬Òò´Ë¸ÅÂÊÃܶÈÓÐ2¸ö¼«´óÖµ£¬¶Ô³ÆµØ·Ö²¼ÔÚÔ­µãµÄÁ½²à¡£

2.6 ÔÚһάÊƳ¡ÖÐÔ˶¯µÄÁ£×Ó£¬ÊÆÄܶÔÔ­µã¶Ô³Æ£¬Ö¤Ã÷Á£×ӵĶ¨Ì¬²¨º¯Êý¾ßÓÐÈ·¶¨µÄÓî³Æ. ¡¾ÌâÒâ·ÖÎö¡¿

ÒÑÖªÌõ¼þ£ºÁ£×ÓÔÚһάÊƳ¡U(x)ÖÐÔ˶¯£¬U(?x)?U(x)¡£ ´ýÖ¤ÎÊÌ⣺±¾Õ÷º¯Êý¾ßÓÐÈ·¶¨µÄÓî³Æ£¬¼´?(?x)???(x) Ï໥ÁªÏµ£º±¾Õ÷º¯ÊýÂú×㶨̬Ѧ¶¨ÚÌ·½³Ì

11

h2d2?(x)?U(x)?(x)?E?(x) (2.6-1) ?2mdx2¡¾Çó½â¹ý³Ì¡¿

¶Ô¶¨Ì¬Ñ¦¶¨ÚÌ·½³Ì½øÐпռ䷴ÑÝ£¬¼´°Ñx»»³É?x£¬µÃµ½

h2d2?(?x)?U(x)?(?x)?E?(?x) (2.6-2) ?22mdx¼ÆËãÖÐÒѾ­ÀûÓÃÁËÊÆÄܵĶԳÆÐÔ¡£Óë(2.6-1)ʽ±È½Ïºó£¬¿ÉÒÔ¿´³ö?(?x)Ò²ÊǶ¨Ì¬Ñ¦¶¨ÚÌ·½³ÌµÄÒ»¸ö½â£¬¼´Ò²ÊDZ¾Õ÷º¯Êý£¬Óë?(x)¶ÔÓ¦ÓÚͬһ¸ö±¾Õ÷ÖµE¡£ Èç¹û?(?x)Óë?(x)ÏßÐÔÏà¹Ø£¬¼´

?(?x)?c?(x) (2.6-3) ÔÚÉÏʽÖÐÔÙ°Ñx»»³É?x£¬µÃµ½

?(x)?c?(?x)?c2?(x) (2.6-4)

2ÓÉÓÚ²¨º¯Êý²»ÄܺãµÈÓÚÁ㣬ÉÏʽ¸ø³öÁËÌõ¼þc?1£¬¼´c??1¡£´úÈëµ½(2.6-3)ʽºó£¬µÃµ½

( ) (2.6-5) ?(?x)???x Èç¹û?(?x)Óë?(x)ÏßÐÔÎ޹أ¬Ôò¿ÉÒÔÏßÐÔ×éºÏ³ÉÁ½¸öеĶÀÁ¢±¾Õ÷º¯Êý

?e(x)??(x)??(?x) (2.6-6)

?o(x)??(x)??(?x)ÈÝÒ×ÑéÖ¤

?e(?x)???e(x) (2.6-7)

?o(?x)???o(x)ÓÉ´Ë˵Ã÷ÁËÁ£×ӵĶ¨Ì¬²¨º¯Êý¾ßÓÐÈ·¶¨µÄÓî³Æ¡£ ¡¾ÎïÀíÌÖÂÛ¡¿

ÔÚһάÊø¸¿Ì¬µÄÇé¿öÏ£¬²¨º¯Êý²»´æÔÚ¼ò²¢ÏÖÏó£¬Õâʱ?(?x)Óë?(x)ÃèдÁËͬһ¸öÄÜÁ¿±¾Õ÷̬£¬ËüÃÇÖ®¼äÏßÐÔÏà¹Ø£¬¹Ê?(x)Ö»ÄÜÊÇÆ溯Êý»òżº¯Êý¡£ÓÉÓÚµÚn¸öÄܼ¶µÄ²¨º¯ÊýÓÐn¸öÁãµã£¬Òò´ËÔÚ¶Ô³ÆÊÆÚåÖУ¬ÓëżÊýÄܼ¶¶ÔÓ¦µÄ²¨º¯ÊýÊÇżº¯Êý£»ÓëÆæÊýÄܼ¶¶ÔÓ¦µÄ²¨º¯ÊýÊÇÆ溯Êý¡£

ÔÚһάɢÉä̬µÄÇé¿öÏ£¬²¨º¯Êý´æÔÚ¼ò²¢ÏÖÏó£¬Õâʱ?(?x)Óë?(x)¿ÉÒÔÃèдͬһ¸öÄÜ

12