《高等數學(xué)》針對高等學(xué)校本科生編寫(xiě)的數學(xué)教材，內容架構從學(xué)生的實(shí)際出發(fā)，全書(shū)共12章，分上、下兩冊出版。本書(shū)為上冊，共7章，主要內容包括函數、極限與連續，導數與微分，一元函數微分學(xué)的應用，不定積分，定積分，定積分的應用和微分方程，每一章除具體知識點(diǎn)講解外，還增設復習題板塊，書(shū)末還附有積分表及習題參考答案，可幫助學(xué)生及時(shí)鞏固應用理論知識。本教材覆蓋知識點(diǎn)全面、講解通俗易懂、試題類(lèi)型豐富，可更好地滿(mǎn)足教學(xué)需要。

黃文韜，男，教授，博士生導師，廣西師范大學(xué)黨委常委、副校長(cháng)，中國生物數學(xué)學(xué)會(huì )副理事長(cháng)，廣西數學(xué)會(huì )副理事長(cháng)。主要微分方程定性理論、計算機符號計算研究。近年來(lái)發(fā)表學(xué)術(shù)論文50余篇，其中SCI收錄30余篇，出版英文學(xué)術(shù)專(zhuān)著(zhù)2部，教材4部，主持完成國家自然科學(xué)基金1項，主持在研國家自然科學(xué)基金1項。

第1 章 函數、極限與連續 ············································································· 1

1.1 映射與函數 ························································································· 1

1.1.1 集合的概念 ··················································································· 1

1.1.2 區間與鄰域 ··················································································· 2

1.1.3 映射與函數的概念 ·········································································· 3

1.1.4 函數的幾種特性 ············································································· 6

1.1.5 反函數與復合函數、函數的運算 ························································ 9

1.1.6 基本初等函數、初等函數 ································································ 13

1.1.7 函數的參數表示和極坐標表示 ·························································· 18

習題1.1 ··································································································· 22

1.2 數列的極限 ························································································ 24

1.2.1 數列極限的定義 ············································································ 24

1.2.2 數列極限的性質(zhì) ············································································ 27

習題1.2 ··································································································· 29

1.3 函數的極限 ························································································ 30

1.3.1 自變量趨向無(wú)窮大時(shí)函數的極限 ······················································· 30

1.3.2 自變量趨于有限值時(shí)函數的極限 ······················································· 31

1.3.3 函數極限的性質(zhì) ············································································ 34

習題1.3 ··································································································· 37

1.4 無(wú)窮小量與無(wú)窮大量 ············································································ 38

1.4.1 無(wú)窮小量的概念 ············································································ 38

1.4.2 無(wú)窮小量的性質(zhì) ············································································ 39

1.4.3 無(wú)窮大量 ····················································································· 39

習題1.4 ··································································································· 42

1.5 極限的運算法則 ·················································································· 42

1.5.1 極限的四則運算法則 ······································································ 42

1.5.2 復合函數的極限運算法則 ································································ 45

習題1.5 ··································································································· 46

1.6 極限存在準則與兩個(gè)重要極限 ································································ 48

1.6.1 極限存在準則 ··············································································· 48

1.6.2 兩個(gè)重要極限 ··············································································· 50

習題1.6 ··································································································· 55

1.7 無(wú)窮小量的比較 ················································································· 56

1.7.1 無(wú)窮小量的比較 ············································································ 57

1.7.2 利用等價(jià)無(wú)窮小求極限 ··································································· 58

習題1.7 ··································································································· 60

1.8 函數的連續與間斷 ·············································································· 61

1.8.1 函數連續性的概念 ········································································· 61

1.8.2 連續函數的運算法則與初等函數的連續性 ··········································· 62

1.8.3 函數的間斷點(diǎn) ··············································································· 63

1.8.4 閉區間上連續函數的性質(zhì) ································································ 66

習題1.8 ··································································································· 69

拓展閱讀 ································································································· 70

總習題一 ································································································· 71

第2 章 導數與微分 ····················································································· 76

2.1 導數的概念 ······················································································· 76

2.1.1 引例 ··························································································· 76

2.1.2 導數的定義 ·················································································· 77

2.1.3 導數的幾何意義 ············································································ 80

2.1.4 函數可導性與連續性的關(guān)系 ····························································· 81

習題2.1 ··································································································· 82

2.2 導數的求導法則 ················································································· 83

2.2.1 導數的四則運算法則 ······································································ 83

2.2.2 反函數的求導法則 ········································································· 85

2.2.3 復合函數的求導法則 ······································································ 86

2.2.4 導數表(常數和基本初等函數的導數公式) ··········································· 87

習題2.2 ··································································································· 88

2.3 高階導數 ·························································································· 89

2.3.1 高階導數的概念 ············································································ 89

2.3.2 高階導數的計算 ············································································ 89

習題2.3 ··································································································· 92

2.4 隱函數和參數方程所確定的函數的導數及其相關(guān)變化率 ······························ 93

2.4.1 隱函數的導數 ··············································································· 93

2.4.2 參數方程所確定的函數的導數 ·························································· 95

2.4.3 相關(guān)變化率 ·················································································· 97

習題2.4 ··································································································· 98

2.5 函數的微分 ······················································································ 100

2.5.1 微分的概念 ················································································ 100

2.5.2 基本初等函數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算中的應用 ············································ 103

習題2.5 ································································································· 105

拓展閱讀 ······························································································· 107

總習題二 ······························································································· 108

第3 章 一元函數微分學(xué)的應用 ···································································· 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

習題3.1 ································································································· 119

3.2 洛必達(L' Hospital)法則 ····································································· 119

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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

習題3.2 ································································································· 124

3.3 泰勒公式 ························································································· 125

3.3.1 函數逼近 ··················································································· 125

3.3.2 泰勒中值定理 ············································································· 125

3.3.3 泰勒公式的應用 ·········································································· 128

習題3.3 ································································································· 129

3.4 函數的單調性、極值和最值 ································································· 130

3.4.1 函數的單調性 ············································································· 130

3.4.2 函數的極值 ················································································ 132

3.4.3 最大值與最小值問(wèn)題 ···································································· 135

習題3.4 ································································································· 136

3.5 曲線(xiàn)的凹凸性與拐點(diǎn) ·········································································· 137

3.5.1 曲線(xiàn)的凹凸性 ············································································· 138

3.5.2 曲線(xiàn)的拐點(diǎn) ················································································ 140

習題3.5 ································································································· 142

3.6 曲率 ······························································································· 142

3.6.1 弧微分 ······················································································· 142

3.6.2 曲率及其計算公式 ········································································ 143

習題3.6 ·································································································· 146

3.7 函數的性態(tài)和圖形 ············································································· 146

3.7.1 漸近線(xiàn) ······················································································· 147

3.7.2 函數圖形描繪 ·············································································· 148

習題3.7 ·································································································· 150

拓展閱讀 ································································································ 150

總習題三 ································································································ 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

習題4.1 ·································································································· 158

4.2 不定積分的換元積分法 ······································································· 159

4.2.1 第一類(lèi)換元法 ·············································································· 159

4.2.2 第二類(lèi)換元法 ·············································································· 162

習題4.2 ·································································································· 166

4.3 不定積分的分部積分法 ······································································· 167

習題4.3 ·································································································· 170

4.4 有理函數的積分 ················································································ 170

4.4.1 有理函數的積分 ··········································································· 170

4.4.2 三角有理式的積分 ········································································ 173

4.4.3 簡(jiǎn)單無(wú)理式的積分 ········································································ 174

習題4.4 ·································································································· 176

拓展閱讀 ································································································ 176

總習題四 ································································································ 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 定積分的近似計算 ········································································ 183

5.1.4 定積分的性質(zhì) ·············································································· 185

習題5.1 ································································································· 188

5.2 微積分基本公式 ················································································ 189

5.2.1 引例 ························································································· 190

5.2.2 積分上限的函數及其導數 ······························································ 190

5.2.3 牛頓-萊布尼茨公式 ······································································ 193

習題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

習題5.3 ································································································· 201

5.4 反常積分 ························································································· 202

5.4.1 無(wú)窮區間上的反常積分 ································································· 203

5.4.2 無(wú)界函數的反常積分 ···································································· 205

習題5.4 ································································································· 208

*

5.5 反常積分的審斂法和Γ函數 ······························································ 208

5.5.1 無(wú)窮區間上反常積分的判別法 ························································ 208

5.5.2 無(wú)界函數的反常積分的審斂法 ························································ 211

5.5.3 ? 函數 ······················································································ 213

*習題5.5 ································································································ 214

拓展閱讀 ······························································································· 215

總習題五 ······························································································· 216

第6 章 定積分的應用 ················································································ 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(xiàn)的弧長(cháng) ·········································································· 228

習題6.2 ································································································· 231

6.3 定積分在物理學(xué)上的應用 ···································································· 232

6.3.1 變力沿直線(xiàn)所做的功 ···································································· 232

6.3.2 水壓力 ······················································································ 233

6.3.3 引力 ························································································· 235

習題6.3 ································································································· 236

拓展閱讀 ······························································································· 237

總習題六 ······························································································· 239

第7 章 微分方程 ······················································································· 241

7.1 微分方程的基本概念 ·········································································· 241

習題7.1 ·································································································· 244

7.2 可分離變量的微分方程 ······································································· 244

習題7.2 ·································································································· 246

7.3 齊次方程 ························································································· 247

7.3.1 齊次方程及其解法 ········································································ 247

*

7.3.2 可化為齊次方程的微分方程 ··························································· 250

習題7.3 ·································································································· 252

7.4 一階線(xiàn)性微分方程 ············································································· 253

7.4.1 一階線(xiàn)性微分方程及常數變易法 ······················································ 253

7.4.2 伯努利方程 ················································································· 256

習題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 y和 y?的方程 y?? ? f (x) ··············································· 258

7.5.2 不含未知函數 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

習題7.5 ·································································································· 263

7.6 二階線(xiàn)性微分方程解的結構 ································································· 264

7.6.1 二階線(xiàn)性微分方程 ········································································ 264

7.6.2 線(xiàn)性微分方程解的結構 ·································································· 265

習題7.6 ·································································································· 267

7.7 二階常系數齊次線(xiàn)性微分方程 ······························································ 268

習題7.7 ·································································································· 272

7.8 二階常系數非齊次線(xiàn)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

習題7.8 ·································································································· 279

7.9 歐拉方程 ························································································· 280

習題7.9 ·································································································· 281

7.10 常系數線(xiàn)性微分方程組 ······································································ 281

習題7.10 ································································································ 283

拓展閱讀 ································································································ 283

總習題七 ································································································ 285

數學(xué)實(shí)驗(上) ···························································································· 287

Python 簡(jiǎn)介及基本操作 ············································································· 287

實(shí)驗1 割圓術(shù)、生長(cháng)模型 ········································································ 323

實(shí)驗2 陳酒出售的最佳時(shí)機問(wèn)題 ······························································· 327

實(shí)驗3 泰勒展開(kāi)與e 的計算 ····································································· 331

實(shí)驗4 氫燃料電池堆優(yōu)化設計 ·································································· 335

實(shí)驗5 智慧農場(chǎng)中根系層含水量估計與智能灌溉決策 ···································· 343

習題答案··································································································· 351

第1 章 函數、極限與連續 ······································································· 351

第2 章 導數與微分 ················································································ 357

第3 章 一元函數微分學(xué)的應用 ································································· 364

第4 章 不定積分 ··················································································· 369

第5 章 定積分 ······················································································ 374

第6 章 定積分的應用 ············································································· 378

第7 章 微分方程 ··················································································· 380

主要參考文獻 ···························································································· 387

附錄 積分表 ···························································································· 389

前言

《高等數學(xué)》作為理工科學(xué)生的必修基礎課程之一，歷來(lái)是培養學(xué)生數學(xué)素養、邏輯思維能力和科學(xué)研究能力的重要途徑. 隨著(zhù)科技的飛速進(jìn)步和社會(huì )的發(fā)展，這門(mén)課程的教學(xué)內容與方法也在不斷地變化和拓展. 本書(shū)分為上、下兩冊，上冊?xún)热莅ㄒ辉⒎e分學(xué)與常微分方程，下冊則涵蓋空間解析幾何、多元微積分學(xué)與級數等內容. 本書(shū)既可作為普通高等學(xué)校理工科專(zhuān)業(yè)的本科教材，也可作為教學(xué)參考書(shū).

在當今高等數學(xué)的教材市場(chǎng)中，已經(jīng)存在諸多質(zhì)量較高的教材. 然而，隨著(zhù)新一輪科技革命和產(chǎn)業(yè)變革的推進(jìn)，傳統的教學(xué)內容和方式也面臨著(zhù)更新與革新的需求. 在這種時(shí)代背景下，我們編寫(xiě)本書(shū)，旨在為數學(xué)教育和人才培養提供一種新的選擇. 具體來(lái)說(shuō)，以下幾個(gè)方面的考慮促使了本書(shū)的誕生.

第一，在全球科技迅猛發(fā)展的今天，數學(xué)作為科學(xué)的基礎工具，在推動(dòng)技術(shù)創(chuàng )新、產(chǎn)業(yè)升級、社會(huì )進(jìn)步方面發(fā)揮著(zhù)不可或缺的作用. 國家提出以“新”提“質(zhì)”，以“數”賦能新質(zhì)生產(chǎn)力的戰略目標，強調數學(xué)教育要注重創(chuàng )新性、實(shí)用性和社會(huì )應用，培養具備跨學(xué)科視野和實(shí)踐能力的高素質(zhì)人才. 本書(shū)編寫(xiě)之初，即立足于數字技術(shù)日新月異的發(fā)展趨勢，力求將現代數學(xué)方法與實(shí)際產(chǎn)業(yè)需求結合，為學(xué)生提供具有前瞻性的數學(xué)教育內容，為他們成為適應新技術(shù)時(shí)代的創(chuàng )新型人才打下基礎.

第二，課程思政已成為教育體系中的重要組成部分，數學(xué)作為一門(mén)基礎學(xué)科，也不應置身事外. 本書(shū)在內容設計上，從方法論和辯證統一的角度，努力深挖數學(xué)與思政教育的結合點(diǎn). 我們不僅關(guān)注學(xué)生數學(xué)知識的掌握，還注重培養學(xué)生正確的世界觀(guān)、人生觀(guān)和價(jià)值觀(guān). 通過(guò)將思想政治教育有機融入數學(xué)教學(xué)，我們旨在啟發(fā)學(xué)生樹(shù)立正確的人生理想、社會(huì )責任感與使命擔當，激發(fā)他們?yōu)閲液蜕鐣?huì )做貢獻的意識. 同時(shí)，本書(shū)通過(guò)對數學(xué)思想和方法的剖析，提升學(xué)生的批判性思維和創(chuàng )新意識，致力培養他們成為有理想、有信仰、有擔當的新時(shí)代青年.

第三，信息技術(shù)的迅猛發(fā)展，尤其是計算機與互聯(lián)網(wǎng)的普及以及人工智能的發(fā)展，帶來(lái)了教育模式的深刻變革. 在這種背景下，本書(shū)在內容編排上力求既扎根基礎，又面向應用，兼顧數學(xué)思想和方法的傳授. 同時(shí)，注重數學(xué)的幾何背景與實(shí)際意義，以確保學(xué)生不僅掌握數學(xué)方法，更能理解其背后的思想. 為了滿(mǎn)足新時(shí)代教育的需求，進(jìn)一步提高學(xué)生解決實(shí)際問(wèn)題和動(dòng)手操作的能力，本書(shū)特別增加了Python 數學(xué)實(shí)驗的內容，注重信息技術(shù)與數學(xué)的結合，支持學(xué)生利用Python 等工具進(jìn)行數學(xué)實(shí)驗，探索數學(xué)在現實(shí)世界中的應用. 通過(guò)實(shí)踐性強的數學(xué)實(shí)驗，學(xué)生不僅能更好地掌握數學(xué)的基本概念，還能親身體驗數學(xué)在現實(shí)問(wèn)題中的應用，增強動(dòng)手能力和創(chuàng )新思維. 人工智能的發(fā)展，特別是DeepSeek、豆包、Kimi等大模型的應用，為數學(xué)教育注入了全新活力. 這些大模型具備強大的數據分析、智能解答以及模擬演示能力，能夠為學(xué)生提供多元化的學(xué)習輔助. 但我們建議，在高等數學(xué)基礎知識的學(xué)習階段，學(xué)生應避免過(guò)度依賴(lài)大模型，以防造成思維惰性，削弱創(chuàng )新能力. 在獨立完成解題、建模、分析等任務(wù)后，學(xué)生可利用大模型進(jìn)行驗證和比較，將其作為一種高效的輔助工具.

第四，當今社會(huì )對高素質(zhì)應用型人才的需求日益增加，數學(xué)作為基礎學(xué)科，其應用價(jià)值也愈加突出. 本書(shū)在注重理論知識講解的同時(shí)，也關(guān)注數學(xué)在實(shí)際工程、技術(shù)及科學(xué)研究中的應用. 我們不僅強調數學(xué)的普遍性與抽象性，還在書(shū)中加入了大量與實(shí)際問(wèn)題相關(guān)的案例和習題，力求讓學(xué)生在學(xué)好數學(xué)基礎的同時(shí)，初步具備利用數學(xué)作為工具解決實(shí)際問(wèn)題，特別是工程問(wèn)題、科學(xué)研究問(wèn)題的能力. 通過(guò)對數學(xué)方法的應用和實(shí)踐，學(xué)生將能夠更好地應對未來(lái)職業(yè)生涯中的挑戰，尤其是在科技創(chuàng )新、工程技術(shù)、數據分析等領(lǐng)域中發(fā)揮重要作用.

總之，《高等數學(xué)》作為一門(mén)經(jīng)典的基礎課程，不僅關(guān)乎學(xué)科知識本身的傳授、創(chuàng )新，更承載著(zhù)國家對未來(lái)人才的培養使命. 希望本書(shū)能夠幫助學(xué)生在數學(xué)的海洋中遨游，激發(fā)他們的探索精神，培養他們的創(chuàng )新能力和批判性思維，為未來(lái)的科學(xué)技術(shù)發(fā)展和社會(huì )進(jìn)步貢獻力量.

本書(shū)的編寫(xiě)工作安排如下：第1~2章由李紹剛和劉期懷老師負責，第3章由蔣利華老師負責，第4章由林昕茜老師負責，第5 章由楊龍老師負責，第6 章由熊峰和黃逸飛老師負責，第7 章由李光云老師負責，第8章由齊恩鳳老師負責，第9 章由陳翠玲和吳果林老師負責，第10 章由李玉山老師負責，第11 章由徐勐戩老師負責，第12 章由黃良力和郭述鋒老師負責，數學(xué)實(shí)驗由黃文韜、何東平和黃婷老師負責. 最后由黃文韜和劉期懷老師進(jìn)行統稿. 本教材編寫(xiě)工作得到了廣西師范大學(xué)數學(xué)與統計學(xué)院、桂林電子科技大學(xué)數學(xué)與計算科學(xué)學(xué)院和桂林航天工業(yè)學(xué)院理學(xué)院領(lǐng)導和老師們的大力支持和幫助，在此一并表示誠摯的感謝.

限于編者水平，書(shū)中難免存在疏漏之處，懇請廣大讀者批評指正.

編 者

2025年6月

針對高等學(xué)校本科生編寫(xiě)的數學(xué)教材，覆蓋知識點(diǎn)全面、講解通俗易懂、試題類(lèi)型豐富，可更好地滿(mǎn)足學(xué)習高等數學(xué)的需要。

集合是數學(xué)中的一個(gè)最基本的概念，已滲透到數學(xué)的各個(gè)分支，成為現代數學(xué)的基礎和語(yǔ)言. 一般地，具有某種特定性質(zhì)的事物的總體稱(chēng)為集合. 組成這個(gè)集合的對象稱(chēng)為該集合的元素.

集合的表示方法主要有列舉法和描述法. 列舉法是將集合的元素一一列舉出來(lái)，寫(xiě)在一個(gè)花括號內.

我們學(xué)習過(guò)的許多數學(xué)運算都是成對互逆出現的，如加法與減法、乘法與除法. 在微分學(xué)部分，我們學(xué)習了如何求一個(gè)函數的導函數，在這一章我們將討論的它的逆運算，即尋找一個(gè)可導函數，使它的導函數等于已知函數.

微分方程作為一門(mén)重要的數學(xué)學(xué)科分支，其產(chǎn)生與發(fā)展緊密伴隨著(zhù)人類(lèi)生產(chǎn)實(shí)踐的需求. 歷經(jīng)三百余年的歷程，它不僅在數學(xué)領(lǐng)域占據著(zhù)關(guān)鍵地位，還廣泛滲透到物理學(xué)、化學(xué)、腦科學(xué)、生物學(xué)、經(jīng)濟學(xué)等眾多學(xué)科，對推動(dòng)科學(xué)技術(shù)的進(jìn)步發(fā)揮了不可替代的作用. 如今，在21 世紀，微分方程正以全新的科學(xué)思維模式，展現出蓬勃的生機與活力.