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(zhì)························5
1.1.5 戴德金原理···························8
1.1.6 確界原理·····························8
習(xí)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(xí)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(xí)題2.1····································.30
2.2 數(shù)列極限的性質(zhì)·····················.31
習(xí)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(xí)題2.3 ···································.40
2.4 實數(shù)連續(xù)性定理····················.41
2.4.1 單調(diào)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(xí)題2.4 ···································.48
*2.5 上極限與下極限···················.50
習(xí)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(xí)題3.1 ···································.58
3.2 函數(shù)極限的性質(zhì)及運算···········.59
3.2.1 函數(shù)極限的性質(zhì)··················.59
3.2.2 函數(shù)極限的四則運算·············.60
3.2.3 復(fù)合函數(shù)的極限··················.62
習(xí)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(guān)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(xí)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(zhì)·············.78
習(xí)題3.4····································.78
3.5 閉區(qū)間上連續(xù)函數(shù)的性質(zhì)········.79
3.5.1 閉區(qū)間上連續(xù)函數(shù)的基本性質(zhì)··.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(xí)題3.5····································.86
第4 章 導(dǎo)數(shù)與微分·························.89
4.1 導(dǎo)數(shù)的概念···························.89
4.1.1 導(dǎo)數(shù)概念的引出··················.89
4.1.2 函數(shù)可導(dǎo)的條件與性質(zhì)··········.91
習(xí)題4.1····································.92
4.2 求導(dǎo)法則······························.94
4.2.1 導(dǎo)數(shù)的四則運算法則·············.94
4.2.2 反函數(shù)求導(dǎo)法則··················.96
4.2.3 復(fù)合函數(shù)的導(dǎo)數(shù)——鏈式法則···.97
4.2.4 隱函數(shù)求導(dǎo)法則··················.98
4.2.5 參數(shù)方程求導(dǎo)法則················.99
習(xí)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(dǎo)的關(guān)系················104
4.3.3 微分在函數(shù)近似計算中的應(yīng)用··105
4.3.4 微分的運算法則··················106
習(xí)題4.3····································106
4.4 高階導(dǎo)數(shù)與高階微分··············106
4.4.1 高階導(dǎo)數(shù)·························.107
4.4.2 高階微分·························.109
4.4.3 復(fù)合函數(shù)的微分·················.109
習(xí)題4.4 ··································.110
第5 章 微分學(xué)基本定理及應(yīng)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(xí)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(xí)題5.2 ··································.126
5.3 泰勒公式及應(yīng)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(yīng)用·················.134
習(xí)題5.3 ··································.138
5.4 單調(diào)性與極值······················.140
5.4.1 函數(shù)的單調(diào)性····················.140
5.4.2 函數(shù)取極值的條件··············.142
習(xí)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(xí)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