第1 章 函數(shù)、極限與連續(xù) ············································································· 1
1.1 映射與函數(shù) ························································································· 1
1.1.1 集合的概念 ··················································································· 1
1.1.2 區(qū)間與鄰域 ··················································································· 2
1.1.3 映射與函數(shù)的概念 ·········································································· 3
1.1.4 函數(shù)的幾種特性 ············································································· 6
1.1.5 反函數(shù)與復(fù)合函數(shù)、函數(shù)的運(yùn)算 ························································ 9
1.1.6 基本初等函數(shù)、初等函數(shù) ································································ 13
1.1.7 函數(shù)的參數(shù)表示和極坐標(biāo)表示 ·························································· 18
習(xí)題1.1 ··································································································· 22
1.2 數(shù)列的極限 ························································································ 24
1.2.1 數(shù)列極限的定義 ············································································ 24
1.2.2 數(shù)列極限的性質(zhì) ············································································ 27
習(xí)題1.2 ··································································································· 29
1.3 函數(shù)的極限 ························································································ 30
1.3.1 自變量趨向無窮大時(shí)函數(shù)的極限 ······················································· 30
1.3.2 自變量趨于有限值時(shí)函數(shù)的極限 ······················································· 31
1.3.3 函數(shù)極限的性質(zhì) ············································································ 34
習(xí)題1.3 ··································································································· 37
1.4 無窮小量與無窮大量 ············································································ 38
1.4.1 無窮小量的概念 ············································································ 38
1.4.2 無窮小量的性質(zhì) ············································································ 39
1.4.3 無窮大量 ····················································································· 39
習(xí)題1.4 ··································································································· 42
1.5 極限的運(yùn)算法則 ·················································································· 42
1.5.1 極限的四則運(yùn)算法則 ······································································ 42
1.5.2 復(fù)合函數(shù)的極限運(yùn)算法則 ································································ 45
習(xí)題1.5 ··································································································· 46
1.6 極限存在準(zhǔn)則與兩個(gè)重要極限 ································································ 48
1.6.1 極限存在準(zhǔn)則 ··············································································· 48
1.6.2 兩個(gè)重要極限 ··············································································· 50
習(xí)題1.6 ··································································································· 55
1.7 無窮小量的比較 ················································································· 56
1.7.1 無窮小量的比較 ············································································ 57
1.7.2 利用等價(jià)無窮小求極限 ··································································· 58
習(xí)題1.7 ··································································································· 60
1.8 函數(shù)的連續(xù)與間斷 ·············································································· 61
1.8.1 函數(shù)連續(xù)性的概念 ········································································· 61
1.8.2 連續(xù)函數(shù)的運(yùn)算法則與初等函數(shù)的連續(xù)性 ··········································· 62
1.8.3 函數(shù)的間斷點(diǎn) ··············································································· 63
1.8.4 閉區(qū)間上連續(xù)函數(shù)的性質(zhì) ································································ 66
習(xí)題1.8 ··································································································· 69
拓展閱讀 ································································································· 70
總習(xí)題一 ································································································· 71
第2 章 導(dǎo)數(shù)與微分 ····················································································· 76
2.1 導(dǎo)數(shù)的概念 ······················································································· 76
2.1.1 引例 ··························································································· 76
2.1.2 導(dǎo)數(shù)的定義 ·················································································· 77
2.1.3 導(dǎo)數(shù)的幾何意義 ············································································ 80
2.1.4 函數(shù)可導(dǎo)性與連續(xù)性的關(guān)系 ····························································· 81
習(xí)題2.1 ··································································································· 82
2.2 導(dǎo)數(shù)的求導(dǎo)法則 ················································································· 83
2.2.1 導(dǎo)數(shù)的四則運(yùn)算法則 ······································································ 83
2.2.2 反函數(shù)的求導(dǎo)法則 ········································································· 85
2.2.3 復(fù)合函數(shù)的求導(dǎo)法則 ······································································ 86
2.2.4 導(dǎo)數(shù)表(常數(shù)和基本初等函數(shù)的導(dǎo)數(shù)公式) ··········································· 87
習(xí)題2.2 ··································································································· 88
2.3 高階導(dǎo)數(shù) ·························································································· 89
2.3.1 高階導(dǎo)數(shù)的概念 ············································································ 89
2.3.2 高階導(dǎo)數(shù)的計(jì)算 ············································································ 89
習(xí)題2.3 ··································································································· 92
2.4 隱函數(shù)和參數(shù)方程所確定的函數(shù)的導(dǎo)數(shù)及其相關(guān)變化率 ······························ 93
2.4.1 隱函數(shù)的導(dǎo)數(shù) ··············································································· 93
2.4.2 參數(shù)方程所確定的函數(shù)的導(dǎo)數(shù) ·························································· 95
2.4.3 相關(guān)變化率 ·················································································· 97
習(xí)題2.4 ··································································································· 98
2.5 函數(shù)的微分 ······················································································ 100
2.5.1 微分的概念 ················································································ 100
2.5.2 基本初等函數(shù)的微分公式和微分法則 ··············································· 101
2.5.3 微分的幾何意義及在近似計(jì)算中的應(yīng)用 ············································ 103
習(xí)題2.5 ································································································· 105
拓展閱讀 ······························································································· 107
總習(xí)題二 ······························································································· 108
第3 章 一元函數(shù)微分學(xué)的應(yīng)用 ···································································· 113
3.1 微分中值定理 ··················································································· 113
3.1.1 羅爾中值定理 ············································································· 113
3.1.2 拉格朗日中值定理 ······································································· 115
3.1.3 柯西中值定理 ············································································· 117
習(xí)題3.1 ································································································· 119
3.2 洛必達(dá)(L' Hospital)法則 ····································································· 119
3.2.1 “
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”型未定式 ·········································································· 120
3.2.2 “
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”型未定式 ·········································································· 121
3.2.3 其他型的未定式 ·········································································· 123
習(xí)題3.2 ································································································· 124
3.3 泰勒公式 ························································································· 125
3.3.1 函數(shù)逼近 ··················································································· 125
3.3.2 泰勒中值定理 ············································································· 125
3.3.3 泰勒公式的應(yīng)用 ·········································································· 128
習(xí)題3.3 ································································································· 129
3.4 函數(shù)的單調(diào)性、極值和最值 ································································· 130
3.4.1 函數(shù)的單調(diào)性 ············································································· 130
3.4.2 函數(shù)的極值 ················································································ 132
3.4.3 最大值與最小值問題 ···································································· 135
習(xí)題3.4 ································································································· 136
3.5 曲線的凹凸性與拐點(diǎn) ·········································································· 137
3.5.1 曲線的凹凸性 ············································································· 138
3.5.2 曲線的拐點(diǎn) ················································································ 140
習(xí)題3.5 ································································································· 142
3.6 曲率 ······························································································· 142
3.6.1 弧微分 ······················································································· 142
3.6.2 曲率及其計(jì)算公式 ········································································ 143
習(xí)題3.6 ·································································································· 146
3.7 函數(shù)的性態(tài)和圖形 ············································································· 146
3.7.1 漸近線 ······················································································· 147
3.7.2 函數(shù)圖形描繪 ·············································································· 148
習(xí)題3.7 ·································································································· 150
拓展閱讀 ································································································ 150
總習(xí)題三 ································································································ 151
第4 章 不定積分 ······················································································· 154
4.1 不定積分的概念與性質(zhì) ······································································· 154
4.1.1 原函數(shù)與不定積分的概念 ······························································· 154
4.1.2 基本積分表 ················································································· 155
4.1.3 不定積分的性質(zhì) ··········································································· 156
習(xí)題4.1 ·································································································· 158
4.2 不定積分的換元積分法 ······································································· 159
4.2.1 第一類換元法 ·············································································· 159
4.2.2 第二類換元法 ·············································································· 162
習(xí)題4.2 ·································································································· 166
4.3 不定積分的分部積分法 ······································································· 167
習(xí)題4.3 ·································································································· 170
4.4 有理函數(shù)的積分 ················································································ 170
4.4.1 有理函數(shù)的積分 ··········································································· 170
4.4.2 三角有理式的積分 ········································································ 173
4.4.3 簡(jiǎn)單無理式的積分 ········································································ 174
習(xí)題4.4 ·································································································· 176
拓展閱讀 ································································································ 176
總習(xí)題四 ································································································ 177
第5 章 定積分 ·························································································· 179
5.1 定積分的概念和性質(zhì) ·········································································· 179
5.1.1 引例 ·························································································· 179
5.1.2 定積分的定義 ·············································································· 181
5.1.3 定積分的近似計(jì)算 ········································································ 183
5.1.4 定積分的性質(zhì) ·············································································· 185
習(xí)題5.1 ································································································· 188
5.2 微積分基本公式 ················································································ 189
5.2.1 引例 ························································································· 190
5.2.2 積分上限的函數(shù)及其導(dǎo)數(shù) ······························································ 190
5.2.3 牛頓-萊布尼茨公式 ······································································ 193
習(xí)題5.2 ································································································· 194
5.3 定積分的換元法和分部積分法 ······························································ 196
5.3.1 定積分的換元法 ·········································································· 196
5.3.2 定積分的分部積分法 ···································································· 200
習(xí)題5.3 ································································································· 201
5.4 反常積分 ························································································· 202
5.4.1 無窮區(qū)間上的反常積分 ································································· 203
5.4.2 無界函數(shù)的反常積分 ···································································· 205
習(xí)題5.4 ································································································· 208
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5.5 反常積分的審斂法和Γ函數(shù) ······························································ 208
5.5.1 無窮區(qū)間上反常積分的判別法 ························································ 208
5.5.2 無界函數(shù)的反常積分的審斂法 ························································ 211
5.5.3 ? 函數(shù) ······················································································ 213
*習(xí)題5.5 ································································································ 214
拓展閱讀 ······························································································· 215
總習(xí)題五 ······························································································· 216
第6 章 定積分的應(yīng)用 ················································································ 220
6.1 定積分的元素法 ················································································ 220
6.2 定積分在幾何學(xué)上的應(yīng)用 ···································································· 222
6.2.1 平面圖形的面積 ·········································································· 222
6.2.2 體積 ························································································· 225
6.2.3 平面曲線的弧長(zhǎng) ·········································································· 228
習(xí)題6.2 ································································································· 231
6.3 定積分在物理學(xué)上的應(yīng)用 ···································································· 232
6.3.1 變力沿直線所做的功 ···································································· 232
6.3.2 水壓力 ······················································································ 233
6.3.3 引力 ························································································· 235
習(xí)題6.3 ································································································· 236
拓展閱讀 ······························································································· 237
總習(xí)題六 ······························································································· 239
第7 章 微分方程 ······················································································· 241
7.1 微分方程的基本概念 ·········································································· 241
習(xí)題7.1 ·································································································· 244
7.2 可分離變量的微分方程 ······································································· 244
習(xí)題7.2 ·································································································· 246
7.3 齊次方程 ························································································· 247
7.3.1 齊次方程及其解法 ········································································ 247
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7.3.2 可化為齊次方程的微分方程 ··························································· 250
習(xí)題7.3 ·································································································· 252
7.4 一階線性微分方程 ············································································· 253
7.4.1 一階線性微分方程及常數(shù)變易法 ······················································ 253
7.4.2 伯努利方程 ················································································· 256
習(xí)題7.4 ·································································································· 257
7.5 可降階的二階微分方程 ······································································· 258
7.5.1 不含未知函數(shù) y和 y?的方程 y?? ? f (x) ··············································· 258
7.5.2 不含未知函數(shù) y的方程 y?? ? f (x, y?) ·················································· 259
7.5.3 不含自變量x的方程 y?? ? f ( y, y?) ····················································· 262
習(xí)題7.5 ·································································································· 263
7.6 二階線性微分方程解的結(jié)構(gòu) ································································· 264
7.6.1 二階線性微分方程 ········································································ 264
7.6.2 線性微分方程解的結(jié)構(gòu) ·································································· 265
習(xí)題7.6 ·································································································· 267
7.7 二階常系數(shù)齊次線性微分方程 ······························································ 268
習(xí)題7.7 ·································································································· 272
7.8 二階常系數(shù)非齊次線性微分方程 ··························································· 272
7.8.1 ( ) e ( ) x
m fx P x ?
? 型 ········································································· 273
7.8.2 1( ) e ( )cos x
m f x P x x ? ? ? 或 2( ) e ( )sin x
m f x P x x ? ? ? 型 ································ 276
習(xí)題7.8 ·································································································· 279
7.9 歐拉方程 ························································································· 280
習(xí)題7.9 ·································································································· 281
7.10 常系數(shù)線性微分方程組 ······································································ 281
習(xí)題7.10 ································································································ 283
拓展閱讀 ································································································ 283
總習(xí)題七 ································································································ 285
數(shù)學(xué)實(shí)驗(yàn)(上) ···························································································· 287
Python 簡(jiǎn)介及基本操作 ············································································· 287
實(shí)驗(yàn)1 割圓術(shù)、生長(zhǎng)模型 ········································································ 323
實(shí)驗(yàn)2 陳酒出售的最佳時(shí)機(jī)問題 ······························································· 327
實(shí)驗(yàn)3 泰勒展開與e 的計(jì)算 ····································································· 331
實(shí)驗(yàn)4 氫燃料電池堆優(yōu)化設(shè)計(jì) ·································································· 335
實(shí)驗(yàn)5 智慧農(nóng)場(chǎng)中根系層含水量估計(jì)與智能灌溉決策 ···································· 343
習(xí)題答案··································································································· 351
第1 章 函數(shù)、極限與連續(xù) ······································································· 351
第2 章 導(dǎo)數(shù)與微分 ················································································ 357
第3 章 一元函數(shù)微分學(xué)的應(yīng)用 ································································· 364
第4 章 不定積分 ··················································································· 369
第5 章 定積分 ······················································································ 374
第6 章 定積分的應(yīng)用 ············································································· 378
第7 章 微分方程 ··················································································· 380
主要參考文獻(xiàn) ···························································································· 387
附錄 積分表 ···························································································· 389
