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µÚÒ»Õ Ð÷ ÂÛ
1. Éèx>0,xµÄÏà¶ÔÎó²îΪ¦Ä,ÇólnxµÄÎó²î.
2. ÉèxµÄÏà¶ÔÎó²îΪ2£¥,ÇóxµÄÏà¶ÔÎó²î.
3. ÏÂÁи÷Êý¶¼ÊǾ¹ýËÄÉáÎåÈëµÃµ½µÄ½üËÆÊý,¼´Îó²îÏÞ²»³¬¹ý×îºóһλµÄ°ë¸öµ¥Î»,ÊÔÖ¸³öËüÃÇÊǼ¸Î»
ÓÐЧÊý×Ö: 4. ÀûÓù«Ê½(3.3)ÇóÏÂÁи÷½üËÆÖµµÄÎó²îÏÞ:
******nx1?1.1021,x2?0.031,x3?385.6,x4?56.430,x5?7?1.0.
***********(i)x1?x2?x4,(ii)x1x2x3,(iii)x2/x4,ÆäÖÐx1,x2,x3,x4¾ùΪµÚ3ÌâËù¸øµÄÊý.
5. ¼ÆËãÇòÌå»ýҪʹÏà¶ÔÎó²îÏÞΪ1£¥,ÎʶÈÁ¿°ë¾¶RʱÔÊÐíµÄÏà¶ÔÎó²îÏÞÊǶàÉÙ? 6. ÉèY0?28,°´µÝÍÆ¹«Ê½
Yn?Yn?1?1100783 ( n=1,2,¡)
Y¼ÆËãµ½Y100.ÈôÈ¡783¡Ö27.982(ÎåλÓÐЧÊý×Ö),ÊÔÎʼÆËã100½«Óжà´óÎó²î?
27. Çó·½³Ìx?56x?1?0µÄÁ½¸ö¸ù,ʹËüÖÁÉÙ¾ßÓÐËÄλÓÐЧÊý×Ö(783¡Ö27.982).
8.
?µ±N³ä·Ö´óʱ,ÔõÑùÇó
S?1gt2??N11?x2dx?
29. Õý·½Ðεı߳¤´óԼΪ100©M,Ó¦ÔõÑù²âÁ¿²ÅÄÜʹÆäÃæ»ýÎó²î²»³¬¹ý1©M?
210. Éè¼Ù¶¨gÊÇ׼ȷµÄ,¶ø¶ÔtµÄ²âÁ¿ÓСÀ0.1ÃëµÄÎó²î,Ö¤Ã÷µ±tÔö¼ÓʱSµÄ¾ø¶ÔÎó²îÔö¼Ó,
¶øÏà¶ÔÎó²îÈ´¼õС.
n?111. ÐòÁÐnÂú×ãµÝÍÆ¹ØÏµnʱÎó²îÓжà´ó?Õâ¸ö¼ÆËã¹ý³ÌÎȶ¨Âð?
{y}y?10y?1(n=1,2,¡),Èôy0?2?1.41(ÈýλÓÐЧÊý×Ö),¼ÆËãµ½y1012. ¼ÆËãf?(2?1),È¡
62?1.4,ÀûÓÃÏÂÁеÈʽ¼ÆËã,ÄÄÒ»¸öµÃµ½µÄ½á¹û×îºÃ?
1(2?1)6,(3?22),31(3?22)3,99?702.
13. f(x)?ln(x?¼Û¹«Ê½
x?1),Çóf(30)µÄÖµ.Èô¿ªÆ½·½ÓÃÁùλº¯Êý±í,ÎÊÇó¶ÔÊýʱÎó²îÓжà´ó?Èô¸ÄÓÃÁíÒ»µÈ
ln(x?x?1)??ln(x?1022 ¼ÆËã,Çó¶ÔÊýʱÎó²îÓжà´ó?
x?1)
214. ÊÔÓÃÏûÔª·¨½â·½³Ì×é15. ÒÑÖªÈý½ÇÐÎÃæ»ý
s??x1?10x2?10x1?x2?2.10;12absinc,¼Ù¶¨Ö»ÓÃÈýλÊý¼ÆËã,Îʽá¹ûÊÇ·ñ¿É¿¿?
?0?c?2,ÇÒ²âÁ¿a ,b ,c µÄÎó²î·Ö±ðΪ?a,?b,?c.Ö¤
ÆäÖÐcΪ»¡¶È,
?ss?aaÃ÷Ãæ»ýµÄÎó²î?sÂú×ã
???bb??cc.
µÚ¶þÕ ²åÖµ·¨
1. ¸ù¾Ý(2.2)¶¨ÒåµÄ·¶µÂÃÉÐÐÁÐʽ,Áî
1Vn(x)?Vn(x0,x1,?,xn?1,x)??11x0?xn?1xx022????x0nn?xn?1x2?xn?1xn
Ö¤Ã÷Vn(x)ÊÇn´Î¶àÏîʽ,ËüµÄ¸ùÊÇx0,?,xn?1,ÇÒ
Vn(x)?Vn?1(x0,x1,?,xn?1)(x?x0)?(x?xn?1).
2. µ±x= 1 , -1 , 2 ʱ, f(x)= 0 , -3 , 4 ,Çóf(x)µÄ¶þ´Î²åÖµ¶àÏîʽ.
3. ¸ø³öf(x)=ln x µÄÊýÖµ±íÓÃÏßÐÔ²åÖµ¼°¶þ´Î²åÖµ¼ÆËãln 0.54 µÄ½üËÆÖµ. x lnx 0.4 -0.916291 0.5 -0.693147 0.6 -0.510826 0.7 -0.357765 0.8 -0.223144
4. ¸ø³öcos x,0¡ã¡Üx ¡Ü90¡ãµÄº¯Êý±í,²½³¤h =1¡ä=(1/60)¡ã,Èôº¯Êý±í¾ßÓÐ5λÓÐЧÊý×Ö,Ñо¿ÓÃÏßÐÔ
²åÖµÇócos x ½üËÆÖµÊ±µÄ×ÜÎó²î½ç. 5. Éè6. Éè
xk?x0?khxj,k=0,1,2,3,Çó
x0?x?x3maxl2(x).
Ϊ»¥Òì½Úµã(j=0,1,¡,n),ÇóÖ¤:
ni) ii) 7. Éè
?xj?0nj?0kjlj(x)?x(k?0,1,?,n);k
j?(x?x)lj(x)???k?1,2,?,n).k
2f(x)?C?a,b?ÇÒf(a)?f(b)?0,ÇóÖ¤
xmaxa?x?bf(x)?18(b?a)2maxa?x?bf?(x).
x8. ÔÚ?4?x?4Éϸø³öf(x)?eµÄµÈ¾à½Úµãº¯Êý±í,ÈôÓöþ´Î²åÖµÇóeµÄ½üËÆÖµ,Ҫʹ½Ø¶ÏÎó²î²»³¬¹ý
10?6,ÎÊʹÓú¯Êý±íµÄ²½³¤hӦȡ¶àÉÙ?
n449. Èôyn?2,Çó?yn¼°?yn.
10. Èç¹ûf(x)ÊÇm´Î¶àÏîʽ,¼Ç?f(x)?f(x?h)?f(x),Ö¤Ã÷f(x)µÄk½×²î·Ö?f(x)(0?k?m)ÊÇ
m?k´Î¶àÏîʽ,²¢ÇÒ?m?lkf(x)?0(lΪÕýÕûÊý).
11. Ö¤Ã÷?(fkgk)?fk?gk?gk?1?fk.
n?1n?112. Ö¤Ã÷k?0n?1?fk?gk?fngn?f0g0??gk?1?fk.k?0
13. Ö¤Ã÷
??j?02yj??yn??y0.
n?114. Èôf(x)?a0?a1x???an?1x?anxÓÐn¸ö²»Í¬Êµ¸ùx1,x2,?,xn,Ö¤Ã÷
nn?j?1xjkf?(xj)??0,0?k?n?2;an,k?n?1.?115. Ö¤Ã÷n½×¾ù²îÓÐÏÂÁÐÐÔÖÊ: i)
Fx,x,?,xn??cf?x0,x1,?,xn?ÈôF(x)?cf(x),Ôò?01;
Fx,x,?,xn??f?x0,x1,?,xn??g?x0,x1,?,xn?ii) ÈôF(x)?f(x)?g(x),Ôò?01.
01774??20,21,?,28?f?2,2,?,2ff(x)?x?x?3x?1?¼°??. 16. ,Çó?17. Ö¤Ã÷Á½µãÈý´Î°£¶ûÃ×ÌØ²åÖµÓàÏîÊÇ
R3(x)?f(4)(?)(x?xk)(x?xk?1)/4!,??(xk,xk?1)22 ²¢ÓÉ´ËÇó³ö·Ö¶ÎÈý´Î°£¶ûÃ×ÌØ²åÖµµÄÎó²îÏÞ.
18. ÇóÒ»¸ö´ÎÊý²»¸ßÓÚ4´ÎµÄ¶àÏîʽP(x),ʹËüÂú×ãP(0)?P(?k?1)²¢ÓÉ´ËÇó³ö·Ö¶ÎÈý´Î°£¶ûÃ×ÌØ²åÖµ
µÄÎó²îÏÞ. 19. ÊÔÇó³öÒ»¸ö×î¸ß´ÎÊý²»¸ßÓÚ4´ÎµÄº¯Êý¶àÏîʽP(x),ÒÔ±ãʹËüÄܹ»Âú×ãÒÔϱ߽çÌõ¼þ
P(0)?P?(0)?0,P(1)?P?(1)?1,P(2)?1.
20. Éèf(x)?C?a,b?,°Ñ?a,b?·ÖΪnµÈ·Ö,ÊÔ¹¹ÔìÒ»¸ǫ̈½×ÐεÄÁã´Î·Ö¶Î²åÖµº¯Êý?n(x)²¢Ö¤Ã÷µ±
n??ʱ,?n(x)ÔÚ?a,b?ÉÏÒ»ÖÂÊÕÁ²µ½f(x).
21. Éèf(x)?1/(1?x),ÔÚ?5?x?5ÉÏÈ¡n?10,°´µÈ¾à½ÚµãÇó·Ö¶ÎÏßÐÔ²åÖµº¯ÊýIh(x),¼ÆËã¸÷½Úµã¼ä
Öе㴦µÄIh(x)Óëf(x)µÄÖµ,²¢¹À¼ÆÎó²î.
22. Çóf(x)?xÔÚ?a,b?ÉϵķֶÎÏßÐÔ²åÖµº¯ÊýIh(x),²¢¹À¼ÆÎó²î.
2423. Çóf(x)?xÔÚ?a,b?Éϵķֶΰ£¶ûÃ×ÌØ²åÖµ,²¢¹À¼ÆÎó²î. 24. ¸ø¶¨Êý¾Ý±íÈçÏÂ: 0.25 0.30 0.39 0.45 xj 20.53 0.7280 yj0.5000 0.5477 0.6245 0.6708 ÊÔÇóÈý´ÎÑùÌõ²åÖµS(x)²¢Âú×ãÌõ¼þ i) ii) 25. Èô
bS?(0.25)?1.0000,S?(0.53)?0.6868; S?(0.25)?S?(0.53)?0.
f(x)?C22?a,b?,S(x)ÊÇÈý´ÎÑùÌõº¯Êý,Ö¤Ã÷
bai)
??af?(x)?dx???S?(x)?dx?2??abf?(x)?S?(x)?dx?2?S?(x)?f?(x)?S?(x)?dxa2b;
ii) Èôf(xi)?S(xi)(i?0,1,?,n),ʽÖÐxiΪ²åÖµ½Úµã,ÇÒa?x0?x1???xn?b,Ôò
?baS?(x)?f?(x)?S?(x)?dx?S?(b)?f?(b)?S?(b)??S?(a)?f?(a)?S?(a)?.
26. ±à³ö¼ÆËãÈý´ÎÑùÌõº¯ÊýS(x)ϵÊý¼°ÆäÔÚ²åÖµ½ÚµãÖеãµÄÖµµÄ³ÌÐò¿òͼ(S(x)¿ÉÓÃ(8.7)ʽµÄ±í´ïʽ).
µÚÈýÕ º¯Êý±Æ½üÓë¼ÆËã
1. (a)ÀûÓÃÇø¼ä±ä»»ÍƳöÇø¼äΪ?(b)¶Ôf(x)?sinxÔÚ?²¿·ÖºÍÎó²î×ö±È½Ï. 2. ÇóÖ¤:
a,b?µÄ²®¶÷˹̹¶àÏîʽ.
0,?/2?ÉÏÇó1´ÎºÍÈý´Î²®¶÷˹̹¶àÏîʽ²¢»³öͼÐÎ,²¢ÓëÏàÓ¦µÄÂí¿ËÀÍÁÖ¼¶Êý
(a)µ±m?f(x)?Mʱ,m?Bn(f,x)?M. (b)µ±f(x)?xʱ,Bn(f,x)?x. 0,2??3. ÔÚ´ÎÊý²»³¬¹ý6µÄ¶àÏîʽÖÐ,Çóf(x)?sin4xÔÚ?µÄ×î¼ÑÒ»Ö±ƽü¶àÏîʽ. a,b?4. ¼ÙÉèf(x)ÔÚ?ÉÏÁ¬Ðø,Çóf(x)µÄÁã´Î×î¼ÑÒ»Ö±ƽü¶àÏîʽ.
5. ѡȡ³£Êýa,ʹ
maxx?ax0?x?13´ïµ½¼«Ð¡,ÓÖÎÊÕâ¸ö½âÊÇ·ñΨһ?
0,?/2?6. Çóf(x)?sinxÔÚ?ÉϵÄ×î¼ÑÒ»´Î±Æ½ü¶àÏîʽ,²¢¹À¼ÆÎó²î. 0,17. Çóf(x)?eÔÚ??ÉϵÄ×î¼ÑÒ»´Î±Æ½ü¶àÏîʽ.
x?1,1?8. ÈçºÎѡȡr,ʹp(x)?x?rÔÚ?ÉÏÓëÁãÆ«²î×îС?rÊÇ·ñΨһ?
20,19. Éèf(x)?x?3x?1,ÔÚ??ÉÏÇóÈý´Î×î¼Ñ±Æ½ü¶àÏîʽ.
4310. Áî
Tn(x)?Tn(2x?1),x??0,1?***,ÇóT0(x),T1(x),T2(x),T3(x).
11. ÊÔÖ¤?T(x)?*nÊÇÔÚ?0,1?ÉÏ´øÈ¨
??1x?x2µÄÕý½»¶àÏîʽ.
12. ÔÚ??1,1?ÉÏÀûÓòåÖµ¼«Ð¡»¯Çó1f(x)?tgxµÄÈý´Î½üËÆ×î¼Ñ±Æ½ü¶àÏîʽ.
?1x13. Éèf(x)?eÔÚ??1,1?ÉϵIJåÖµ¼«Ð¡»¯½üËÆ×î¼Ñ±Æ½ü¶àÏîʽΪLn(x),Èôf?Ln?Óнç,Ö¤Ã÷¶ÔÈκÎ
n?1,´æÔÚ³£Êý?n¡¢?n,ʹ
?nTn?1(x)?f(x)?Ln(x)??nTn?1(x)(?1?x?1).
14. ÉèÔÚ??1,1?ÉÏ
Îó²î.
?(x)?1?12x?18x?2324x?315384x?41653840x5,ÊÔ½«?(x)½µµÍµ½3´Î¶àÏîʽ²¢¹À¼Æ
15. ÔÚ??1,1?ÉÏÀûÓÃÃݼ¶ÊýÏîÊýÇóf(x)?sinxµÄ3´Î±Æ½ü¶àÏîʽ,ʹÎó²î²»³¬¹ý0.005.
16. f(x)ÊÇ??a,a?ÉϵÄÁ¬ÐøÆæ(ż)º¯Êý,Ö¤Ã÷²»¹ÜnÊÇÆæÊý»òżÊý,f(x)µÄ×î¼Ñ±Æ½ü¶àÏîʽFn(x)?Hn*Ò²ÊÇÆæ(ż)º¯Êý.
??ax?b?sinx?17. Çóa¡¢bʹ?022dxΪ×îС.²¢Óë1Ìâ¼°6ÌâµÄÒ»´Î±Æ½ü¶àÏîʽÎó²î×÷±È½Ï.
18. f(x)¡¢g(x)?C1?a,b?,¶¨Òå
(a)(f,g)??baf?(x)g?(x)dx;(b)(f,g)?x6?baf?(x)g?(x)dx?f(a)g(a); ÎÊËüÃÇÊÇ·ñ¹¹³ÉÄÚ»ý? 19. ÓÃÐíÍß×Ȳ»µÈʽ(4.5)¹À¼Æ
?101?xdx1?1µÄÉϽç,²¢Óûý·ÖÖÐÖµ¶¨Àí¹À¼ÆÍ¬Ò»»ý·ÖµÄÉÏϽç,²¢±È½ÏÆä½á¹û.
221?120. Ñ¡Ôña,ʹÏÂÁлý·ÖÈ¡µÃ×îСֵ:?21. Éè¿Õ¼ä
µÄ×î¼Ñƽ·½±Æ½ü,²¢±È½ÏÆä½á¹û. 22.
f(x)?xun(x)?(x?ax)dx,?100x?axdx12.
2???span?1,x?,?2?span?x,x101?,·Ö±ðÔÚ?4x?C?0,1?¡¢?2ÉÏÇó³öÒ»¸öÔªËØ,ʹµÃÆäΪ
??span?1,x?1,1?ÔÚ?ÉÏ,ÇóÔÚ12,x?ÉϵÄ×î¼Ñƽ·½±Æ½ü.
sin?(n?1)arccosx?1?x223. ÊǵڶþÀàÇбÈÑ©·ò¶àÏîʽ,Ö¤Ã÷ËüÓеÝÍÆ¹ØÏµ
un?1?x??2xun?x??un?1?x?.
?1,1?24. ½«ÔÚ?Éϰ´ÀÕÈõ¶àÏîʽ¼°ÇбÈÑ©·ò¶àÏîʽչ¿ª,ÇóÈý´Î×î¼Ñƽ·½±Æ½ü¶àÏîʽ²¢
»³öÎó²îͼÐÎ,ÔÙ¼ÆËã¾ù·½Îó²î.
2f(x)?sin1x?1,1?25. °Ñf(x)?arccosxÔÚ?ÉÏÕ¹³ÉÇбÈÑ©·ò¼¶Êý.
26. ÓÃ×îС¶þ³Ë·¨ÇóÒ»¸öÐÎÈçy?a?bxµÄ¾Ñ鹫ʽ,ʹËüÓëÏÂÁÐÊý¾ÝÄâºÏ,²¢Çó¾ù·½Îó²î.
xi yi 219 19.0 25 32.3 31 49.0 38 73.3 44 97.8 27. ¹Û²âÎïÌåµÄÖ±ÏßÔ˶¯,µÃ³öÒÔÏÂÊý¾Ý: ʱ¼ät(Ãë) ¾àÀës(Ã×) 0 0 0.9 10 1.9 30 3.0 50 3.9 80 5.0 110 ÇóÔ˶¯·½³Ì. 28. ÔÚij»¯Ñ§·´Ó¦Àï,¸ù¾ÝʵÑéËùµÃ·Ö½âÎïµÄŨ¶ÈÓëʱ¼ä¹ØÏµÈçÏÂ: ʱ¼ä 0 5 10 15 20 25 30 35 40 Ũ¶È 0 1.27 2.16 2.86 3.44 3.87 4.15 4.37 4.51 45 4.58 50 4.62 55 4.64 ÓÃ×îС¶þ³ËÄâºÏÇóy?f(t).
29. ±à³öÓÃÕý½»¶àÏîʽ×ö×îС¶þ³ËÄâºÏµÄ³ÌÐò¿òͼ. 30. ±à³ö¸Ä½øFFTËã·¨µÄ³ÌÐò¿òͼ.
31. ÏÖ¸ø³öÒ»ÕżÇ¼?xk???4,3,2,1,0,1,2,3?,ÊÔÓøĽøFFTËã·¨Çó³öÐòÁÐ?xk?µÄÀëɢƵÆ×
?Ck?(k?0,1?,,7 )µÚËÄÕ ÊýÖµ»ý·ÖÓëÊýֵ΢·Ö
1. È·¶¨ÏÂÁÐÇó»ý¹«Ê½ÖеĴý¶¨²ÎÊý,ʹÆä´úÊý¾«¶È¾¡Á¿¸ß,²¢Ö¸Ã÷Ëù¹¹Ôì³öµÄÇó»ý¹«Ê½Ëù¾ßÓеĴúÊý¾«
¶È:
(1)??h(2)??2h(3)??1(4)?h012hhf(x)dx?A?1f(?h)?A0f(0)?A1f(h)f(x)dx?A?1f(?h)?A0f(0)?A1f(h); ; ;
f?(h)?1f(x)dx??f(?1)?2f(x1)?3f(x2)?/3f(x)dx?h?f(0)?f(h)?/1?ah2?f?(0)?.
2. ·Ö±ðÓÃÌÝÐι«Ê½ºÍÐÁÆÕɹ«Ê½¼ÆËãÏÂÁлý·Ö:
?(1)
10x4?x2dx,n?8?; (2)
10(1?ex?x)2dx,n?10; .
(3)?1; (4)?03. Ö±½ÓÑéÖ¤¿ÂÌØË¹¹«Ê½(2.4)¾ßÓÐ5´Î´úÊý¾«¶È.
xdx,n?49?6?sin?dx,n?624. ÓÃÐÁÆÕɹ«Ê½Çó»ý·Ö?0²¢¼ÆËãÎó²î. 5. ÍÆµ¼ÏÂÁÐÈýÖÖ¾ØÐÎÇó»ý¹«Ê½:
1edx?x?(1)
(2)
babaf(x)dx?(b?a)f(a)?f(x)dx?(b?a)f(b)?a?b2)?f?(?)2f?(?)2f?(?)24(b?a)(b?a)2;
2?;
3?(3)
baf(x)dx?(b?a)f((b?a).
b6. Ö¤Ã÷ÌÝÐι«Ê½(2.9)ºÍÐÁÆÕɹ«Ê½(2.11)µ±n??ʱÊÕÁ²µ½»ý·Ö?a7. Óø´»¯ÌÝÐι«Ê½Çó»ý·Ö?a¼ÆÉáÈëÎó²î)?
bf(x)dx.
f(x)dx,ÎÊÒª½«»ý·ÖÇø¼ä?a,b?·Ö³É¶àÉٵȷÖ,²ÅÄܱ£Ö¤Îó²î²»³¬¹ý?(Éè²»
28. ÓÃÁú±´¸ñ·½·¨¼ÆËã»ý·Ö??10edx??x,ÒªÇóÎó²î²»³¬¹ý10.
S?a?20?59. ÎÀÐǹìµÀÊÇÒ»¸öÍÖÔ²,ÍÖÔ²Öܳ¤µÄ¼ÆË㹫ʽÊÇ
c221?()sin?d?a,ÕâÀïaÊÇÍÖÔ²µÄ°ë³¤Öá,cÊǵØÇòÖÐÐÄÓë¹ìµÀÖÐÐÄ(ÍÖÔ²ÖÐÐÄ)µÄ¾àÀë,¼ÇhΪ½üµØµã¾àÀë,HΪԶµØµã¾àÀë,R?6371¹«ÀïΪµØÇò°ë¾¶,Ôòa?(2R?H?h)/2,c?(H?h)/2.ÎÒ¹úµÚÒ»¿ÅÈËÔìÎÀÐǽüµØµã¾àÀëh?439¹«Àï,Ô¶µØµã¾àÀëH?2384¹«Àï,ÊÔÇóÎÀÐǹìµÀµÄÖܳ¤. 10. Ö¤Ã÷µÈʽ
ËÆÖµ.
nsin?n????323!n??545!n??ÊÔÒÀ¾Ýnsin(?/n)(n?3,6,12)µÄÖµ,ÓÃÍâÍÆËã·¨Çó?µÄ½ü