第 1 章 函數(shù)、極限及連續(xù)················································································1
1.1 函數(shù)·······························································································.1
1.1.1 函數(shù)的定義 ············································································.1
1.1.2 函數(shù)的性質(zhì) ············································································.3
1.1.3 復(fù)合函數(shù) ···············································································.4
1.1.4 基本初等函數(shù) ·········································································.5
1.1.5 初等函數(shù) ···············································································.5
1.2 極限的概念······················································································.5
1.2.1 數(shù)列的極限 ············································································.5
1.2.2 函數(shù)的極限 ············································································.7
1.3 極限的計算······················································································.9
1.3.1 極限的四則運算······································································.9
1.3.2 無窮大與無窮小·····································································.12
1.3.3 兩個重要極限 ········································································.16
1.4 函數(shù)連續(xù)的定義···············································································.18
1.4.1 函數(shù)的連續(xù)性 ········································································.18
1.4.2 函數(shù)的間斷點 ········································································.20
1.4.3 初等函數(shù)的連續(xù)性··································································.22
1.4.4 閉區(qū)間上連續(xù)函數(shù)的性質(zhì)·························································.23
1.5 知識點匯總·····················································································.24
1.6 本章真題歸類··················································································.25
1.7 本章檢測························································································.28
第 2 章 一元函數(shù)微分學················································································.30
2.1 導數(shù)的概念·····················································································.31
2.1.1 導數(shù)的定義 ···········································································.31
2.1.2 導數(shù)的幾何意義·····································································.33
2.1.3 可導與連續(xù)的關(guān)系··································································.34
2.2 導數(shù)的計算·····················································································.35
2.2.1 導數(shù)的四則運算·····································································.35
2.2.2 復(fù)合函數(shù)和反函數(shù)的導數(shù)·························································.37
2.2.3 隱函數(shù)的導數(shù) ········································································.38
2.2.4 對數(shù)求導法 ···········································································.39
2.2.5 高階求導 ··············································································.41
2.2.6 參數(shù)方程確定的函數(shù)的導數(shù)······················································.42
2.3 函數(shù)的微分·····················································································.44
2.3.1 微分的定義及運算··································································.44
2.3.2 微分在近似計算中的應(yīng)用·························································.46
2.4 微分中值定理及其應(yīng)用······································································.47
2.4.1 微分中值定理 ········································································.47
2.4.2 洛必達法則 ···········································································.49
2.5 導數(shù)的應(yīng)用·····················································································.52
2.5.1 函數(shù)的單調(diào)性 ········································································.52
2.5.2 函數(shù)的極值及其求法·······························································.55
2.5.3 函數(shù)的最值及其求法·······························································.57
2.5.4 函數(shù)圖象的凹凸性與漸近線······················································.58
2.5.5 曲線的漸近線 ········································································.59
2.6 知識點匯總·····················································································.62
2.7 本章真題歸類··················································································.63
2.8 本章檢測························································································.71
第 3 章 一元函數(shù)積分學················································································.72
3.1 不定積分基礎(chǔ)知識············································································.72
3.1.1 不定積分的概念·····································································.72
3.1.2 不定積分的幾何意義·······························································.73
3.1.3 不定積分的性質(zhì)·····································································.73
3.1.4 基本積分公式 ········································································.74
3.2 不定積分的計算方法·········································································.77
3.2.1 第一類換元法（湊微分法）······················································.77
3.2.2 第二類換元法 ········································································.78
3.2.3 分部積分法 ···········································································.81
3.3 定積分的基本概念············································································.82
3.3.1 定積分的概念 ········································································.82
3.3.2 定積分的幾何意義··································································.83
3.3.3 定積分的性質(zhì) ········································································.84
3.4 微積分基本定理和定積分的計算··························································.86
3.4.1 變上限積分函數(shù)·····································································.86
3.4.2 微積分基本定理·····································································.87
3.4.3 定積分的換元積分法·······························································.89
3.4.4 分部積分法 ···········································································.90
3.4.5 無窮區(qū)間上的反常積分····························································.92
3.5 定積分的應(yīng)用··················································································.93
3.5.1 直角坐標系下平面圖形面積的計算·············································.93
3.5.2 旋轉(zhuǎn)體體積的計算方法····························································.95
3.5.3 曲線弧長的計算方法·······························································.96
3.6 知識點匯總·····················································································.98
3.7 本章真題歸類··················································································.99
3.8 本章檢測······················································································.107
第 4 章 微分方程·························································································109
4.1 微分方程的概念·············································································.109
4.2 一階微分方程··················································································110
4.2.1 可分離變量的微分方程····························································111
4.2.2 齊次微分方程 ·······································································112
4.2.3 一階線性微分方程··································································114
4.2.4 可降階的高階微分方程···························································117
4.3 二階常系數(shù)線性微分方程···································································119
4.3.1 二階常系數(shù)齊次線性微分方程···················································119
4.3.2 二階常系數(shù)非齊次線性微分方程·············································.121
4.4 知識點匯總···················································································.123
4.5 本章真題歸類················································································.124
4.6 本章檢測······················································································.127
第 5 章 多元函數(shù)微積分················································································128
5.1 多元函數(shù)的基本概念·······································································.129
5.2 多元函數(shù)的偏導數(shù)與全微分······························································.131
5.2.1 多元函數(shù)的一階偏導數(shù)··························································.131
5.2.2 多元函數(shù)的高階偏導數(shù)··························································.133
5.2.3 多元函數(shù)全微分···································································.135
5.2.4 多元復(fù)合函數(shù)求導································································.136
5.2.5 隱函數(shù)的偏導數(shù)···································································.139
5.3 二重積分······················································································.141
5.3.1 二重積分的概念和性質(zhì)··························································.141
5.3.2 二重積分的計算···································································.145
5.4 知識點匯總···················································································.152
5.5 本章真題歸類················································································.153
5.6 本章檢測······················································································.159
第 6 章 常數(shù)項級數(shù)······················································································161
6.1 常數(shù)項級數(shù)的概念與性質(zhì)·································································.161
6.1.1 數(shù)項級數(shù)的定義與斂散性·······················································.161
6.1.2 常數(shù)項級數(shù)的基本性質(zhì)··························································.165
6.2 正項級數(shù)的審斂法··········································································.166
6.2.1 比較審斂法 ·········································································.166
6.2.2 比值審斂法 ·········································································.170
6.2.3 根值審斂法 ·········································································.171
6.3 任意項級數(shù)的審斂法·······································································.172
6.3.1 交錯級數(shù)與萊布尼茨判別法····················································.172
6.3.2 絕對收斂與條件收斂·····························································.173
6.4 知識點匯總···················································································.174
6.5 本章真題歸類················································································.175
6.6 本章檢測······················································································.180
第 7 章 新增考點預(yù)測···················································································182
7.1 曲率····························································································.182
7.1.1 弧微分 ···············································································.182
7.1.2 曲率及其計算公式································································.183
7.1.3 曲率圓與曲率半徑································································.184
7.2 泰勒公式······················································································.184
7.2.1 泰勒公式 ············································································.184
7.2.1 麥克勞林公式 ······································································.186
7.3 函數(shù)圖形的繪制·············································································.187
7.4 定積分的物理學應(yīng)用·······································································.189
7.4.1 變力沿直線所做的功·····························································.189
7.4.2 水壓力 ···············································································.190
7.4.3 引力 ··················································································.191
7.5 定積分的幾何應(yīng)用··········································································.192
7.5.1 平面圖形的面積···································································.192
7.5.2 體積 ··················································································.193
7.5 三重積分······················································································.197
7.5.1 三重積分的定義···································································.197
7.5.2 直角坐標系下三重積分的計算·················································.198