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Abstract

Gọi X bị chặn trên trong R(bị chặn dưới) khi và chỉ khi tồn tại ít nhất một cận trên (cận dưới) của X trong R.

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References (114)

1. TÀI LIỆU THAM KHẢO
2. G. M. FICHTENGÔN, Giáo trình phép tính vi tích phân, Tập 1,2,3. Nauka, Moskva,1969. (tiếng Nga)
3. G. M. FICHTENGÔN, Cơ sở giải tích toán học, Tập 1,2,3. NXB Đại học và Trung học chuyên nghiệp, Hà nội, 1977.
4. K. MAURIN, PWN, Warszawa, 1976. .
5. c Czes
6. R. A. ADAMS, Calculus-a complete, Addison,Wesley, New York,Don Mills, 1991.
7. NGUYỄN ĐÌNH TRÍ (chủ biên), Toán học cao cấp ,Tập 1,2,3. NXB Đại học và Giáo dục chuyên nghiệp, Hà nội, 1990.
8. JEAN-MARIE MONIER, Giáo trình toán, Tập 1,2,3,4. NXB Giáo dục, Hà nội, 1999 (dịch từ tiếng Pháp, DUNOD, Paris,1999)
9. Môc lôc Ch-¬ng I: Giíi h¹n cña d•y sè .................................................................................... 3
10. Sè thùc .................................................................................................................................... 3
11. C¸c tÝnh chÊt c¬ b ¶n cña tËp sè thùc..................................................................................... 3
12. TËp sè thùc më réng ............................................................................................................. 6
13. C¸c kho ¶ng sè thùc............................................................................................................... 7
14. Gi¸ trÞ tuyÖt ®èi cña sè thùc.................................................................................................. 7
15. 1.5. Kho ¶ng c¸ch th«ng th-êng trong R...................................................................................... 8
16. Sè phøc ................................................................................................................................... 9
17. 2.1. §Þnh nghÜa vµ c¸c d¹ng sè phøc ........................................................................................... 9
18. C¸c phÐp to¸n trªn tËp C.......................................................................................................10
19. 2.3. ¸p dông sè phøc vµo l-îng gi¸c ..........................................................................................17
20. D•y sè thùc.............................................................................................................................19
21. sè thùc..................................................................................19
22. TÝnh chÊt cña d•y héi tô .......................................................................................................20
23. TÝnh ®¬n ®iÖu cña d•y sè......................................................................................................26
24. D•y con.................................................................................................................................31
25. Ch-¬ng II: Hμm sè mét biÕn sè....................................................................................34
26. 1. C¸c kh¸i niÖm c¬ b ¶n vÒ hµm sè ..........................................................................................34
27. C¸c ®Þnh nghÜa c¬ b ¶n ..........................................................................................................34
28. C¸c hµm sè th«ng dông ........................................................................................................37
29. Hµm sè s¬ cÊp.......................................................................................................................47
30. 2. Giíi h¹n cña hµm sè ..............................................................................................................47
31. Kh¸i niÖm vÒ giíi h¹n...........................................................................................................47
32. 2.2. TÝnh chÊt cña hµm cã giíi h¹n..............................................................................................49
33. C¸c giíi h¹n ®¸ng nhí ..........................................................................................................56
34. 3. §¹i l-îng v« cïng bÐ (VCB) vµ ®¹i l-îng v« cïng lín (VCL)...........................................58
35. 3.1. §¹i l-îng VCB .....................................................................................................................58
36. 3.2. §¹i l-îng VCL .....................................................................................................................60
37. Sù liªn tôc cña hµm sè...........................................................................................................62
38. C¸c kh¸i niÖm c¬ b ¶n ...........................................................................................................62
39. 4.2. C¸c phÐp to¸n ®¹i sè cña hµm liªn tôc..................................................................................63
40. 4.3. TÝnh chÊt cña hµm sè liªn tôc trªn mét ®o¹n........................................................................65
41. TÝnh liªn tôc ®Òu...................................................................................................................66
42. Ch-¬ng III: PhÐp tÝnh vi ph©n hμm sè mét biÕn sè..........................................68
43. 1. §¹o hµm .................................................................................................................................68
44. 1.1. §¹o hµm t¹i mét ®iÓm ..........................................................................................................68
45. 1.2. C¸c phÐp tÝnh ®¹i sè cña c¸c hµm kh ¶ vi t¹i mét ®iÓm.........................................................71
46. 1.3. §¹o hµm trªn mét kho ¶ng (¸nh x¹ ®¹o hµm) .......................................................................74
47. 1.4. §¹o hµm cña c¸c hµm sè th«ng th-êng................................................................................75
48. Vi ph©n cña hµm sè ...............................................................................................................81
49. 2.1. §Þnh nghÜa vi ph©n t¹i mét ®iÓm ..........................................................................................81
50. Vi ph©n trªn mét kho ¶ng ......................................................................................................82
51. 3. §¹o hµm vµ vi ph©n cÊp cao .................................................................................................84
52. 3.1. §¹o hµm cÊp cao ..................................................................................................................84
53. Vi ph©n cÊp cao ....................................................................................................................85
54. Líp cña mét hµm ..................................................................................................................86
55. C¸c ®Þnh lÝ vÒ gi¸ trÞ trung b×nh...........................................................................................91
56. 4.1. §Þnh lÝ PhÐc ma (Fermat) .....................................................................................................91
57. 4.2. §Þnh lÝ R«n (Rolle)...............................................................................................................92
58. 4.3. §Þnh lÝ sè gia h¹n. (®Þnh lÝ Lag¬r¨ng (Lagrange))................................................................93
59. 4.4. §Þnh lÝ sè gia h÷u h¹n suy réng ( §Þnh lÝ C«si (Cauchy)).....................................................95
60. 5. øng dông c¸c ®Þnh lÝ vÒ gi¸ trÞ trung b×nh ..........................................................................98
61. C«ng thøc Taylo (Taylor), c«ng thøc Macl«ranh (McLaurin)..............................................98
62. 5.2. Qui t¾c L«pitan (L' Hospital)............................................................................................. 102
63. Sù biÕn thiªn cña hµm sè ................................................................................................... 105
64. TÝnh ®¬n ®iÖu cña hµm kh ¶ vi ........................................................................................... 105
65. 6.2. §iÒu kiÖn hµm sè ®¹t cùc trÞ.............................................................................................. 107
66. 7. Bµi to¸n t×m gi¸ trÞ lín nhÊt, gi¸ trÞ bÐ nhÊt .................................................................... 109
67. Hµm liªn tôc trªn ®o¹n kÝn [a,b]........................................................................................ 109
68. 7.2. Hµm liªn tôc trªn kho ¶ng më, kho ¶ng v« h¹n................................................................... 110
69. Hµm låi ................................................................................................................................ 110
70. Kh¸i niÖm vÒ hµm låi, hµm lâm vµ ®iÓm uèn.................................................................... 110
71. 8.2. §iÒu kiÖn hµm låi .............................................................................................................. 113
72. TiÖm cËn cña ®-êng cong .................................................................................................. 115
73. Kh¸i niÖm chung vÒ tiÖm cËn ............................................................................................ 115
74. 9.2. Ph©n lo¹i vµ c¸ch t×m tiÖm cËn .......................................................................................... 116
75. 10. Bµi to¸n kh ¶o s¸t hµm sè .................................................................................................... 117
76. Ch-¬ng IV: PhÐp tÝnh tÝch ph©n.............................................................................. 122
77. 1. Kh¸i niÖm vÒ tÝch ph©n x¸c ®Þnh....................................................................................... 122
78. 1.1. §Þnh nghÜa tÝch ph©n x¸c ®Þnh........................................................................................... 122
79. 1.2. §iÒu kiÖn tån t¹i ................................................................................................................ 123
80. Líp c¸c hµm kh ¶ tÝch......................................................................................................... 125
81. C¸c tÝnh chÊt cña tÝch ph©n x¸c ®Þnh ................................................................................. 126
82. C«ng thøc Niut¬n-LÐpnÝt (Newbnitz)................................................................................ 129
83. 2. Hai ph-¬ng ph¸p c¬ b ¶n tÝnh tÝch ph©n x¸c ®Þnh ........................................................... 135
84. PhÐp ®aei biÕn ..................................................................................................................... 135
85. PhÐp tÝch ph©n tõng phÇn................................................................................................... 135
86. 3. Ph-¬ng ph¸p tÝnh tÝch ph©n bÊt ®Þnh.............................................................................. 141
87. B ¶ng c¸c nguyªn hµm th«ng dông..................................................................................... 141
88. 3.2. Hai ph-¬ng ph¸p c¬ b ¶n tÝnh tÝch ph©n bÊt ®Þnh ............................................................... 142
89. C¸ch tÝnh tÝch ph©n bÊt ®Þnh cña c¸c hµm sè h÷u tØ .......................................................... 145
90. TÝnh nguyªn hµm c¸c ph©n thøc h÷u tØ ®èi víi mét sè hµm th«ng dông ........................... 147
91. Mét sè øng dông cña tÝch ph©n x¸c ®Þnh.......................................................................... 152
92. TÝnh ®iÖn tÝch h×nh ph¼ng.................................................................................................. 153
93. 4.2. TÝnh ®é dµi ®-êng cong ph¼ng .......................................................................................... 155
94. TÝnh thÓ tÝch vËt thÓ ........................................................................................................... 157
95. TÝnh diÖn tÝch mAEt trßn xoay ............................................................................................. 159
96. 5. TÝch ph©n suy réng............................................................................................................. 161
97. 5.1. TÝch ph©n suy réng víi cËn v« h¹n .................................................................................... 161
98. 5.2. TÝch ph©n suy réng víi hµm d-íi dÊu tÝch ph©n cã cùc ®iÓm ........................................... 167
99. Chuçi sè ............................................................................................................................... 173
100. C¸c kh¸i niÖm chung ......................................................................................................... 173
101. Chuçi sè d-¬ng.................................................................................................................. 176
102. Chuçi ®an dÊu.................................................................................................................... 183
103. Chuçi cã sè h¹ng mang dÊu bÊt k×..................................................................................... 185
104. 2. Chuçi hµm ........................................................................................................................... 187
105. C¸c kh¸i niÖm chung vÒ chuçi hµm................................................................................... 187
106. 2.2. Sù héi tô ®Òu cña chuçi hµm.............................................................................................. 188
107. 3. Chuçi luü thõa .................................................................................................................... 194
108. C¸c kh¸i niÖm chung vÒ chuçi luü thõa............................................................................. 194
109. 3.2. Khai triÓn mét hµm sè thµnh chuçi luü thõa ..................................................................... 201
110. Chuçi Phuriª (Fourier) ...................................................................................................... 209
111. C¸c kh¸i niÖm chung ......................................................................................................... 209
112. 4.2. §iÒu kiÖn ®ñ ®Ó hµm sè khai triÓn thµnh chuçi Fourier .................................................... 213
113. Khai triÓn thµnh chuçi Fourier cña mét hµm sè bÊt kú ..................................................... 217
114. Tμi liÖu tham kh ¶o ........................................................................................................ 224

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