微積分原理（上）

目錄
第1 章 實數(shù)集與初等函數(shù)··················1
1.1 實數(shù)集····································1
1.1.1 集合及其運算························1
1.1.2 映射···································3
1.1.3 可數(shù)集································3
1.1.4 實數(shù)集的性質························5
1.1.5 戴德金原理···························8
1.1.6 確界原理·····························8
習題1.1····································.10
1.2 初等函數(shù)······························.11
1.2.1 函數(shù)的概念························.11
1.2.2 函數(shù)的一些特性··················.12
1.2.3 函數(shù)的運算························.13
1.2.4 基本初等函數(shù)·····················.14
1.2.5 反函數(shù)及其存在條件·············.18
1.2.6 反三角函數(shù)························.19
*1.2.7 雙曲函數(shù)和反雙曲函數(shù)··········.22
*1.2.8 雙曲函數(shù)與三角函數(shù)之間
的聯(lián)系·····························.24
習題1.2····································.24
第2 章 數(shù)列極限····························.27
2.1 數(shù)列極限的概念·····················.27
習題2.1····································.30
2.2 數(shù)列極限的性質·····················.31
習題2.2····································.35
2.3 幾類特殊的數(shù)列·····················.36
2.3.1 無窮大數(shù)列與無窮小數(shù)列·······.36
2.3.2 無窮大數(shù)列與無界數(shù)列··········.36
2.3.3 Stolz 定理·························.38
習題2.3 ···································.40
2.4 實數(shù)連續(xù)性定理····················.41
2.4.1 單調有界定理·····················.41
2.4.2 閉區(qū)間套定理·····················.43
2.4.3 Bolzano-Weierstrass 定理·········.44
2.4.4 柯西收斂準則·····················.45
*2.4.5 有限覆蓋定理·····················.47
*2.4.6 聚點定理··························.48
習題2.4 ···································.48
*2.5 上極限與下極限···················.50
習題2.5 ···································.54
第3 章 函數(shù)極限與連續(xù)··················.55
3.1 函數(shù)極限的概念····················.55
3.1.1 函數(shù)在一點的極限···············.55
3.1.2 函數(shù)在無窮遠處的極限··········.58
習題3.1 ···································.58
3.2 函數(shù)極限的性質及運算···········.59
3.2.1 函數(shù)極限的性質··················.59
3.2.2 函數(shù)極限的四則運算·············.60
3.2.3 復合函數(shù)的極限··················.62
習題3.2 ···································.62
3.3 函數(shù)極限的存在條件··············.63
3.3.1 函數(shù)極限與數(shù)列極限的關系·····.63
3.3.2 兩個重要極限·····················.65
3.3.3 無窮大量與無窮小量·············.67
3.3.4 等價無窮小量代換求極限········.69
習題3.3····································.71
3.4 函數(shù)的連續(xù)···························.73
3.4.1 函數(shù)連續(xù)的概念··················.73
3.4.2 間斷點及其分類··················.74
3.4.3 連續(xù)函數(shù)的局部性質·············.78
習題3.4····································.78
3.5 閉區(qū)間上連續(xù)函數(shù)的性質········.79
3.5.1 閉區(qū)間上連續(xù)函數(shù)的基本性質··.79
3.5.2 反函數(shù)的連續(xù)性··················.82
3.5.3 一致連續(xù)性························.83
習題3.5····································.86
第4 章 導數(shù)與微分·························.89
4.1 導數(shù)的概念···························.89
4.1.1 導數(shù)概念的引出··················.89
4.1.2 函數(shù)可導的條件與性質··········.91
習題4.1····································.92
4.2 求導法則······························.94
4.2.1 導數(shù)的四則運算法則·············.94
4.2.2 反函數(shù)求導法則··················.96
4.2.3 復合函數(shù)的導數(shù)——鏈式法則···.97
4.2.4 隱函數(shù)求導法則··················.98
4.2.5 參數(shù)方程求導法則················.99
習題4.2····································102
4.3 函數(shù)的微分···························103
4.3.1 可微的概念························103
4.3.2 可微與可導的關系················104
4.3.3 微分在函數(shù)近似計算中的應用··105
4.3.4 微分的運算法則··················106
習題4.3····································106
4.4 高階導數(shù)與高階微分··············106
4.4.1 高階導數(shù)·························.107
4.4.2 高階微分·························.109
4.4.3 復合函數(shù)的微分·················.109
習題4.4 ··································.110
第5 章 微分學基本定理及應用········.112
5.1 微分中值定理······················.112
5.1.1 極值的概念與費馬定理·········.112
5.1.2 微分中值定理····················.113
習題5.1 ··································.118
5.2 洛必達法則··························.120
5.2.1 0
0
型不定式極限·················.120
5.2.2 ∞
∞
型不定式極限················.123
5.2.3 其他類型不定式極限············.125
習題5.2 ··································.126
5.3 泰勒公式及應用···················.127
5.3.1 泰勒公式·························.128
5.3.2 基本初等函數(shù)的展開式·········.130
5.3.3 泰勒公式的應用·················.134
習題5.3 ··································.138
5.4 單調性與極值······················.140
5.4.1 函數(shù)的單調性····················.140
5.4.2 函數(shù)取極值的條件··············.142
習題5.4 ··································.145
5.5 函數(shù)的凸性與函數(shù)作圖··········.147
5.5.1 函數(shù)的凸性······················.147
5.5.2 曲線的漸近性····················.152
5.5.3 函數(shù)作圖·························.153
習題5.5 ··································.155
*5.6 方程求根的牛頓迭代公式·····.155
第6 章 不定積分···························.160
6.1 原函數(shù)與不定積分················.160
6.1.1 原函數(shù)與6