新編微積分（上）

目 錄
第 1 章 實數(shù)集、函數(shù)及其應用 ................................................ (1)
1.1 實數(shù)集................................................................ (1)
1.1.1 實數(shù)集及其性質 ................................................ (1)
1.1.2 絕對值與不等式 ................................................ (1)
1.1.3 區(qū)間與鄰域 ..................................................... (2)
1.1.4 確界原理 ....................................................... (3)
習題 1.1 ............................................................... (4)
1.2 函數(shù)................................................................. (5)
1.2.1 函數(shù)的概念 ..................................................... (5)
1.2.2 函數(shù)的某些特性 ................................................ (9)
習題 1.2 .............................................................. (11)
1.2.3 應用事例與探究課題 ........................................... (13)
第 2 章 極限及其應用 .................................................... (16)
2.1 數(shù)列極限及其應用 ..................................................... (16)
2.1.1 數(shù)列極限的概念 ............................................... (16)
2.1.2 收斂數(shù)列的性質 ............................................... (19)
2.1.3 數(shù)列收斂性的判別 ............................................. (22)
習題 2.1 .............................................................. (27)
2.1.4 應用事例與探究課題 ........................................... (29)
2.2 函數(shù)極限及其應用 ..................................................... (32)
2.2.1 函數(shù)極限的概念 ............................................... (32)
2.2.2 函數(shù)極限的性質 ............................................... (35)
2.2.3 函數(shù)極限存在的判別 ........................................... (38)
2.2.4 無窮小與無窮大 ............................................... (42)
習題 2.2 .............................................................. (44)
2.2.5 應用事例與探究課題 ........................................... (46)
第 3 章 連續(xù)性及其應用 .................................................. (48)
3.1 函數(shù)的連續(xù)性及其應用................................................. (48)
3.1.1 函數(shù)連續(xù)的概念 ............................................... (48)
3.1.2 連續(xù)函數(shù)的基本性質與初等函數(shù)的連續(xù)性 ...................... (51)
3.1.3 閉區(qū)間上連續(xù)函數(shù)的性質 ...................................... (52)
習題 3.1 .............................................................. (58)
3.1.4 應用事例與探究課題 ........................................... (61)
3.2 實數(shù)的連續(xù)性及其應用................................................. (63)
3.2.1 閉區(qū)間套定理 ................................................. (63)
3.2.2 聚點定理 ...................................................... (64)
3.2.3 有限覆蓋定理 ................................................. (66)
習題 3.2 .............................................................. (67)
第 4 章 一元微分學及其應用 ............................................... (68)
4.1 導數(shù)及其應用.......................................................... (68)
4.1.1 導數(shù)的定義 .................................................... (68)
習題 4.1 .............................................................. (72)
4.1.2 求導法則 ...................................................... (73)
習題 4.2 .............................................................. (77)
4.1.3 隱函數(shù)與參數(shù)方程所確定的導數(shù) ............................... (79)
習題 4.3............................................................... (81)
4.1.4 高階導數(shù) ...................................................... (81)
習題 4.4 .............................................................. (83)
4.1.5 應用事例與探究課題 ........................................... (84)
4.2 微分................................................................ (86)
4.2.1 微分的定義 .................................................... (86)
4.2.2 微分的運算法則 ............................................... (88)
4.2.3 高階微分 ...................................................... (89)
習題 4.5 .............................................................. (90)
4.3 微分學基本定理及其應用 .............................................. (90)
4.3.1 中值定理 ...................................................... (90)
習題 4.6 .............................................................. (95)
4.3.2 待定式極限 .................................................... (96)
習題 4.7 .............................................................. (99)
4.3.3 泰勒公式 ..................................................... (100)
習題 4.8 ............................................................. (104)
4.3.4 函數(shù)的單調性與極值 ......................................... (105)
習題 4.9 ............................................................. (108)
4.3.5 函數(shù)的凸性與拐點 ........................................... (110)
習題 4.10 ............................................................ (113)
4.3.6 曲線的漸近線與函數(shù)的圖像 ................................... (113)
習題 4.11 ............................................................ (116)
4.3.7 應用事例與探究課題 ......................................... (116)
第 5 章 一元積分學及其應用 .............................................. (120)
5.1 不定積分及其應用.................................................... (120)
5.1.1 不定積分的概念 .............................................. (120)
習題 5.1 ............................................................. (122)
5.1.2 換元積分法與分部積分法 ..................................... (123)
習題 5.2 ............................................................. (127)
5.1.3 有理函數(shù)與可化為有理函數(shù)的不定積分 ....................... (128)
習題 5.3 ............................................................. (133)
5.1.4 應用事例與探究課題 ......................................... (134)
5.2 定積分及其應用 ...................................................... (136)
5.2.1 定積分的概念與可積條件 ..................................... (136)
習題 5.4 ............................................................. (142)
5.2.2 定積分的性質 ................................................ (143)
習題 5.5 ............................................................. (147)
5.2.3 微積分學基本定理 ............................................ (148)
習題 5.6 ............................................................. (153)
5.2.4 應用事例與探究課題 ......................................... (155)
5.3 定積分的應用 ........................................................ (157)
5.3.1 微元法 ....................................................... (157)
5.3.2 平面圖形的面積 .............................................. (158)
5.3.3 利用平行截面面積求體積 ..................................... (161)
5.3.4 平面曲線的弧長 .............................................. (163)
5.3.5 旋轉曲面的面積 .............................................. (165)
習題 5.7 ............................................................. (166)
5.3.6 應用事例與探究課題 ......................................... (167)
5.4 反常積分及其應用.................................................... (168)
5.4.1 無窮積分 ..................................................... (168)
習題 5.8 ............................................................. (172)
5.4.2 瑕積分 ....................................................... (173)
習題 5.9 ............................................................. (178)
5.4.3 應用事例與探究課題 ......................................... (179)
第 6 章 常微分方程、常差分方程及其應用 ................................... (182)
6.1 常微分方程及其應用.................................................. (182)
6.1.1 基本概念 ..................................................... (182)
6.1.2 初等積分法 ................................................... (183)
習題 6.1 ............................................................. (191)
6.1.3 線性微分方程組 .............................................. (193)
習題 6.2 ............................................................. (204)
6.1.4 高階線性微分方程 ............................................ (205)
習題 6.3 ............................................................. (213)
6.1.5 應用事例與探究課題 ......................................... (213)
6.2 常差分方程及其應用.................................................. (214)
6.2.1 基本概念 ..................................................... (214)
6.2.2 線性常差分方程 .............................................. (215)
習題 6.4 ............................................................. (219)
參考文獻 ............................................................（220）