第1 章 一元函數(shù)極限及連續(xù) ·········································································· 1
1.1 函數(shù)的概念 ····················································································· 1
1.1.1 集合與鄰域 ······························································································· 1
1.1.2 函數(shù)的概念 ······························································································· 2
1.1.3 函數(shù)的性質(zhì) ······························································································· 4
1.1.4 初等函數(shù)的概念 ························································································· 6
習(xí)題1.1 ································································································ 6
1.2 數(shù)列極限 ························································································ 7
1.2.1 數(shù)列的概念 ······························································································· 7
1.2.2 數(shù)列的極限 ······························································································· 8
1.2.3 數(shù)列極限的性質(zhì)及收斂準(zhǔn)則 ·········································································· 9
1.2.4 數(shù)列極限的四則運算法則 ··········································································· 14
習(xí)題1.2 ······························································································· 14
1.3 函數(shù)的極限 ···················································································· 15
1.3.1 函數(shù)極限的定義 ······················································································· 15
1.3.2 函數(shù)極限的性質(zhì) ······················································································· 19
習(xí)題1.3 ······························································································· 20
1.4 無窮小量、無窮大量 ········································································ 20
1.4.1 無窮小量 ································································································ 20
1.4.2 無窮大量 ································································································ 23
1.4.3 無窮大與無窮小的關(guān)系 ·············································································· 25
習(xí)題1.4 ······························································································· 25
1.5 函數(shù)極限的運算法則 ········································································ 26
1.5.1 函數(shù)極限的運算法則 ················································································· 26
1.5.2 復(fù)合函數(shù)的極限運算法則 ··········································································· 29
習(xí)題1.5 ······························································································· 30
1.6 兩個重要極限 ················································································· 30
1.6.1 重要極限 ················································································· 31
1.6.2 重要極限············································································· 32
習(xí)題1.6 ······························································································· 35
1.7 無窮小的比較 ················································································· 36
1.7.1 無窮小的比較 ·························································································· 36
1.7.2 關(guān)于等價無窮小量的性質(zhì)和定理 ·································································· 38
習(xí)題1.7 ······························································································· 40
1.8 函數(shù)的連續(xù)性 ················································································· 40
1.8.1 函數(shù)連續(xù)性的概念 ···················································································· 41
1.8.2 函數(shù)的間斷點 ·························································································· 44
1.8.3 連續(xù)函數(shù)的基本性質(zhì) ················································································· 45
1.8.4 初等函數(shù)的連續(xù)性 ···················································································· 46
習(xí)題1.8 ······························································································· 48
1.9 閉區(qū)間上連續(xù)函數(shù)的性質(zhì) ·································································· 49
1.9.1 最大值最小值定理 ···················································································· 49
1.9.2 零點定理 ································································································ 50
1.9.3 介值定理 ································································································ 51
習(xí)題1.9 ······························································································· 52
總習(xí)題一 ······························································································ 52
第2 章 導(dǎo)數(shù)與微分 ····················································································· 55
2.1 導(dǎo)數(shù)的概念 ···················································································· 55
2.1.1 導(dǎo)數(shù)的定義 ····························································································· 55
2.1.2 導(dǎo)數(shù)的幾何意義 ······················································································· 59
2.1.3 可導(dǎo)與連續(xù)的關(guān)系 ···················································································· 60
習(xí)題2.1 ······························································································· 61
2.2 函數(shù)的求導(dǎo)法則和基本求導(dǎo)公式 ························································· 62
2.2.1 函數(shù)求導(dǎo)的四則運算法則 ··········································································· 62
2.2.2 反函數(shù)求導(dǎo)法則 ······················································································· 63
2.2.3 復(fù)合函數(shù)求導(dǎo)法則 ···················································································· 64
習(xí)題2.2 ······························································································· 67
2.3 高階導(dǎo)數(shù) ······················································································· 69
習(xí)題2.3 ······························································································· 71
2.4 隱函數(shù)及參數(shù)方程確定的函數(shù)的導(dǎo)數(shù) ··················································· 72
2.4.1 隱函數(shù)的導(dǎo)數(shù) ·························································································· 72
2.4.2 對數(shù)求導(dǎo)法 ····························································································· 73
2.4.3 參數(shù)方程求導(dǎo)法則 ···················································································· 74
習(xí)題2.4 ······························································································· 77
2.5 函數(shù)的微分 ···················································································· 78
2.5.1 微分的概念 ····························································································· 78
2.5.2 可微與可導(dǎo)的關(guān)系 ···················································································· 79
2.5.3 微分運算法則和基本微分公式 ····································································· 79
2.5.4 微分的幾何意義 ······················································································· 81
2.5.5 微分在近似計算中的應(yīng)用 ··········································································· 81
習(xí)題2.5 ······························································································· 83
總習(xí)題二 ······························································································ 84
第3 章 微分中值定理與導(dǎo)數(shù)的應(yīng)用 ································································ 88
3.1 微分中值定理 ················································································· 88
習(xí)題3.1 ······························································································· 91
3.2 洛必達(dá)法則 ···················································································· 92
習(xí)題3.2 ······························································································· 96
3.3 函數(shù)的單調(diào)性、極值、最值 ······························································· 96
3.3.1 函數(shù)的單調(diào)性 ·························································································· 96
3.3.2 函數(shù)的極值 ····························································································· 99
3.3.3 函數(shù)的最大值與最小值 ············································································· 101
習(xí)題3.3 ····························································································· 102
3.4 曲線的凸性、拐點與漸近線 ····························································· 103
3.4.1 曲線的凸性和拐點 ··················································································· 103
3.4.2 曲線的漸近線 ························································································· 105
3.4.3 函數(shù)圖形的描繪 ······················································································ 106
習(xí)題3.4 ····························································································· 107
3.5 導(dǎo)數(shù)在經(jīng)濟(jì)中的應(yīng)用 ······································································ 108
3.5.1 邊際與邊際分析 ······················································································ 108
3.5.2 彈性與彈性分析 ······················································································ 109
習(xí)題3.5 ····························································································· 111
3.6 泰勒公式 ····················································································· 112
習(xí)題3.6 ····························································································· 117
總習(xí)題三 ···························································································· 117
第4 章 不定積分 ······················································································ 122
4.1 不定積分的概念與性質(zhì) ··································································· 122
4.1.1 原函數(shù)與不定積分 ··················································································· 122
4.1.2 基本積分表 ···························································································· 123
4.1.3 不定積分的性質(zhì) ······················································································ 124
4.1.4 不定積分的幾何意義 ················································································ 126
習(xí)題4.1 ····························································································· 127
4.2 換元積分法 ·················································································· 128
4.2.1 第一類換元積分法 ··················································································· 128
4.2.2 第二類換元積分法 ··················································································· 131
習(xí)題4.2 ····························································································· 135
4.3 分部積分法 ·················································································· 137
習(xí)題4.3 ····························································································· 140
4.4 有理函數(shù)的積分 ············································································ 141
4.4.1 有理函數(shù)及其積分 ··················································································· 141
4.4.2 可化為有理函數(shù)的積分舉例 ······································································· 146
習(xí)題4.4 ····························································································· 148
總習(xí)題四 ···························································································· 149
第5 章 定積分及其應(yīng)用 ············································································· 153
5.1 定積分的概念與性質(zhì) ····································································· 153
5.1.1 定積分的定義 ························································································· 153
5.1.2 定積分的幾何意義 ··················································································· 157
5.1.3 定積分的基本性質(zhì) ··················································································· 157
習(xí)題5.1 ····························································································· 160
5.2 微積分基本定理 ············································································ 160
5.2.1 積分上限函數(shù) ························································································· 161
5.2.2 牛頓-萊布尼茨公式 ·················································································· 162
習(xí)題5.2 ····························································································· 164
5.3 定積分的積分法 ············································································ 164
5.3.1 定積分的換元積分法 ················································································ 165
5.3.2 定積分的分部積分法 ················································································ 167
習(xí)題5.3 ····························································································· 169
5.4 廣義積分 ····················································································· 170
5.4.1 無窮區(qū)間上的廣義積分 ············································································· 170
5.4.2 無界函數(shù)的廣義積分 ················································································ 172
習(xí)題5.4 ····························································································· 174
5.5 定積分的應(yīng)用 ··············································································· 175
5.5.1 平面圖形的面積 ······················································································ 175
5.5.2 旋轉(zhuǎn)體的體積 ························································································· 179
5.5.3 定積分在經(jīng)濟(jì)學(xué)的應(yīng)用 ············································································· 182
習(xí)題5.5 ····························································································· 185
總習(xí)題五 ···························································································· 186
第6 章 多元函數(shù)微積分 ············································································· 189
6.1 空間解析幾何 ··············································································· 189
6.1.1 空間直角坐標(biāo)系 ······················································································ 189
6.1.2 空間兩點間的距離 ··················································································· 191
6.1.3 曲面及其方程 ························································································· 191
習(xí)題6.1 ····························································································· 195
6.2 多元函數(shù)的基本概念 ······································································ 196
6.2.1 平面點集 ······························································································· 196
6.2.2 平面點集的基本概念 ················································································ 197
6.2.3 二元函數(shù)的概念 ······················································································ 198
6.2.4 二元函數(shù)的極限 ······················································································ 199
6.2.5 二元函數(shù)的連續(xù)性 ··················································································· 200
習(xí)題6.2 ····························································································· 200
6.3 偏導(dǎo)數(shù) ························································································ 201
6.3.1 偏導(dǎo)數(shù)的概念 ························································································· 201
6.3.2 偏導(dǎo)數(shù)的計算 ························································································· 203
6.3.3 偏導(dǎo)數(shù)和連續(xù)的關(guān)系 ················································································ 203
習(xí)題6.3 ····························································································· 204
6.4 全微分 ························································································ 205
6.4.1 全微分的定義 ························································································· 205
6.4.2 可微性的應(yīng)用 ························································································· 208
習(xí)題6.4 ····························································································· 208
6.5 復(fù)合函數(shù)微分法與隱函數(shù)微分法 ······················································· 208
6.5.1 復(fù)合函數(shù)的求導(dǎo)法則 ················································································ 208
6.5.2 一階微分形式不變性 ················································································ 211
6.5.3 隱函數(shù)可微性定理 ··················································································· 211
習(xí)題6.5 ····························································································· 214
6.6 多元函數(shù)的極值及其求法 ································································ 215
6.6.1 二元函數(shù)的極值問題 ················································································ 215
6.6.2 條件極值與拉格朗日乘數(shù)法 ······································································· 216
習(xí)題6.6 ····························································································· 220
6.7 二重積分的概念與性質(zhì) ··································································· 221
6.7.1 二重積分的概念 ······················································································ 221
6.7.2 二重積分的性質(zhì) ······················································································ 223
習(xí)題6.7 ····························································································· 224
6.8 在直角坐標(biāo)系下的二重積分的計算 ···················································· 224
6.8.1 在直角坐標(biāo)系下二重積分的計算 ································································· 224
6.8.2 交換二次積分次序 ··················································································· 230
6.8.3 利用對稱性和奇偶性化簡二重積分的計算 ····················································· 232
習(xí)題6.8 ····························································································· 233
6.9 在極坐標(biāo)系下的二重積分的計算 ······················································· 234
習(xí)題6.9 ····························································································· 237
總習(xí)題六 ···························································································· 238
第7 章 常微分方程 ··················································································· 242
7.1 基本概念和例題 ············································································ 242
習(xí)題7.1 ····························································································· 244
7.2 一階可分離變量的微分方程 ····························································· 244
習(xí)題7.2 ····························································································· 245
7.3 一階線性微分方程 ········································································· 246
習(xí)題7.3 ····························································································· 248
7.4 一階微分方程的變量代換法 ····························································· 249
習(xí)題7.4 ····························································································· 251
7.5 可降階的微分方程 ········································································· 251
7.5.1 形如 y(n) ? f (x)的微分方程 ········································································ 251
7.5.2 形如 y? ? f (x, y?)的微分方程 ······································································ 252
7.5.3 形如 y? ? f (y, y?)的微分方程 ······································································ 252
習(xí)題7.5 ····························································································· 253
7.6 二階常系數(shù)線性微分方程 ································································ 253
7.6.1 二階常系數(shù)齊次線性微分方程的求解 ··························································· 253
7.6.2 二階常系數(shù)非齊次線性微分方程的求解 ························································ 256
習(xí)題7.6 ····························································································· 258
7.7 一般的二階線性微分方程 ································································ 259
習(xí)題7.7 ····························································································· 260
7.8 差分方程* ···················································································· 261
7.8.1 差分的概念與性質(zhì) ··················································································· 261
7.8.2 差分方程的概念 ······················································································ 262
7.8.3 一階常系數(shù)線性差分方程 ·········································································· 262
習(xí)題7.8 ····························································································· 264
總習(xí)題七 ···························································································· 264
第8 章 無窮級數(shù) ······················································································ 267
8.1 無窮級數(shù)的概念與性質(zhì) ··································································· 267
8.1.1 無窮級數(shù)的定義 ······················································································ 267
8.1.2 收斂級數(shù)的基本性質(zhì) ················································································ 270
習(xí)題8.1 ····························································································· 272
8.2 常數(shù)項級數(shù)的審斂法 ······································································ 273
8.2.1 正項級數(shù)及其審斂法 ················································································ 273
8.2.2 交錯級數(shù)及其審斂法 ················································································ 279
8.2.3 絕對收斂與條件收斂 ················································································ 281
習(xí)題8.2 ····························································································· 282
8.3 冪級數(shù) ························································································ 284
8.3.1 函數(shù)項級數(shù)的概念 ··················································································· 284
8.3.2 冪級數(shù)及其收斂性 ··················································································· 284
8.3.3 冪級數(shù)的運算 ························································································· 288
習(xí)題8.3 ····························································································· 290
8.4 函數(shù)展開成冪級數(shù) ········································································· 291
8.4.1 泰勒級數(shù) ······························································································· 291
8.4.2 函數(shù)展開成冪級數(shù) ··················································································· 293
8.4.3 函數(shù)的冪級數(shù)展開式的應(yīng)用 ······································································· 295
習(xí)題8.4 ····························································································· 298
總習(xí)題八 ···························································································· 298
主要參考文獻(xiàn) ···························································································· 302
習(xí)題參考答案 ···························································································· 303
