基本介紹
- 書名：微積分複習指導與深化訓練
- 作者：劉強,賈尚暉
- 出版時間：2016-03
出版信息,內容簡介,目錄,
出版信息
微積分複習指導與深化訓練
作 譯 者：劉強,賈尚暉
出版時間：2016-03 千 字 數：460
版 次：01-01 頁 數：284
開 本：16開
裝 幀：
I S B N ：9787121282515
內容簡介
本書依據全國高校數學基礎課程授課的內容組織編寫，按微積分（經管類）課程等知識點分專題進行講授，對所涉及的知識點和考點進行分類整合，精選了典型例題和拓展習題進行講解或解答，化解難點。本書編寫特色在於知識的高度綜合性和交叉性，在一定高度上進行數學思想的糅合。知識點可以前後穿插，以訓練學生的數學思維能力，鍛鍊學生的獨立思考的能力，提高學生的解題水平。
目錄
第1 章
函式 ··································································································································· 1
1.1 知識要點 ··························································································································· 1
1.1.1 函式、鄰域的概念 ····································································································· 1
1.1.2 函式的基本特性 ········································································································· 1
1.1.3 反函式與複合函式 ····································································································· 2
1.1.4 基本初等函式與初等函式 ··························································································· 3
1.1.5 一些常用公式 ············································································································ 3
1.2 典型例題分析 ··················································································································· 4
1.2.1 題型一：函式定義域的求解 ························································································ 4
1.2.2 題型二：函式表達式的求解 ························································································ 4
1.2.3 題型三：反函式的求解 ······························································································· 5
1.2.4 題型四：複合函式的求解 ··························································································· 6
1.2.5 題型五：函式的四種基本特性 ····················································································· 7
1.3 深化訓練 ··························································································································· 8
1.4 深化訓練詳解 ··················································································································· 9
1.5 綜合提高訓練 ················································································································· 10
第2 章極限與連續 ····················································································································· 12
2.1 知識要點 ························································································································· 12
2.1.1 極限的概念 ············································································································· 12
2.1.2 無窮小量與無窮大量 ································································································ 12
2.1.3 極限的性質與運算法則 ····························································································· 14
2.1.4 極限存在準則與兩個重要極限 ··················································································· 14
2.1.5 函式的連續性 ·········································································································· 15
2.1.6 函式的間斷點 ·········································································································· 15
2.1.7 連續函式的性質 ······································································································· 16
2.1.8 閉區間上的連續函式的性質 ······················································································ 16
2.1.9 一些重要的結論 ······································································································· 17
2.2 典型例題分析 ················································································································· 17
2.2.1 題型一：極限的概念與性質問題 ··············································································· 17
2.2.2 題型二：利用極限的四則運算法則求極限 ·································································· 18
2.2.3 題型三：利用單側極限的性質求極限 ········································································· 19
2.2.4 題型四：利用兩個重要極限求極限 ············································································ 20
2.2.5 題型五：利用等價無窮小量替換求極限 ····································································· 21
2.2.6 題型六：利用極限存在準則求極限 ············································································ 22
2.2.7 題型七：函式的連續性問題 ······················································································ 23
2.2.8 題型八：連續函式的等式證明問題 ············································································ 24
2.3 深化訓練 ························································································································· 25
2.4 深化訓練詳解 ················································································································· 27
2.5 綜合提高訓練 ················································································································· 31
第3 章導數與微分 ····················································································································· 35
3.1 知識要點 ························································································································· 35
3.1.1 導數的概念 ············································································································· 35
3.1.2 導數的幾何意義 ······································································································· 35
3.1.3 基本導數公式 ·········································································································· 35
3.1.4 導數的四則運算法則 ································································································ 36
3.1.5 常用求導法則 ·········································································································· 36
3.1.6 高階導數 ················································································································· 37
3.1.7 微分的概念與性質 ··································································································· 38
3.1.8 導數在經濟學中的套用 ····························································································· 39
3.2 典型例題分析 ················································································································· 40
3.2.1 題型一：導數與微分的定義問題 ··············································································· 40
3.2.2 題型二：分段函式的求導問題 ··················································································· 42
3.2.3 題型三：導數的幾何意義 ························································································· 43
3.2.4 題型四：導函式的幾何特性問題 ··············································································· 44
3.2.5 題型五：利用可導性求參數值（域） ········································································· 44
3.2.6 題型六：高階導數問題 ····························································································· 45
3.2.7 題型七：反函式、複合函式的求導問題 ····································································· 46
3.2.8 題型八：隱函式的求導問題 ······················································································ 47
3.2.9 題型九：導函式的連續性問題 ··················································································· 48
3.2.10 題型十：導數在經濟學中的套用 ·············································································· 49
3.3 深化訓練 ························································································································· 49
3.4 深化訓練詳解 ················································································································· 52
3.5 綜合提高訓練 ················································································································· 56
第4 章中值定理與導數的套用 ·································································································· 57
4.1 知識要點 ························································································································· 57
4.1.1 中值定理 ················································································································· 57
4.1.2 洛必達法則 ············································································································· 57
4.1.3 函式的單調區間 ······································································································· 58
4.1.4 函式的極值與最值 ··································································································· 58
4.1.5 函式的凹凸區間與拐點 ····························································································· 58
4.1.6 曲線的漸近線 ·········································································································· 59
4.1.7 函式作圖 ················································································································· 59
4.1.8 一些常用的麥克勞林公式 ························································································· 59
4.2 典型例題分析 ················································································································· 60
4.2.1 題型一：利用中值定理證明等式問題 ········································································· 60
4.2.2 題型二：利用中值定理證明不等式問題 ····································································· 62
4.2.3 題型三：利用洛必達法則求極限 ··············································································· 63
4.2.4 題型四：關於函式的單調性與極值問題 ····································································· 64
4.2.5 題型五：函式的凹凸性與拐點問題 ············································································ 64
4.2.6 題型六：顯式不等式的證明問題 ··············································································· 66
4.2.7 題型七：函式的零點（方程的根）問題 ····································································· 68
4.2.8 題型八：漸近線問題 ································································································ 68
4.2.9 題型九：泰勒公式的套用 ························································································· 69
4.2.10 題型十：套用題 ····································································································· 70
4.3 深化訓練 ························································································································· 71
4.4 深化訓練詳解 ················································································································· 73
4.5 綜合提高訓練 ················································································································· 80
第5 章不定積分 ························································································································· 83
5.1 知識要點 ························································································································· 83
5.1.1 不定積分的概念與幾何意義 ······················································································ 83
5.1.2 不定積分的性質 ······································································································· 83
5.1.3 換元積分法 ············································································································· 83
5.1.4 分部積分法 ············································································································· 85
5.1.5 有理函式的積分法 ··································································································· 85
5.1.6 三角函式有理式的積分法 ························································································· 86
5.1.7 簡單無理函式的積分法 ····························································································· 86
5.1.8 常用積分公式表 ······································································································· 86
5.2 典型例題分析 ················································································································· 87
5.2.1 題型一：不定積分的定義與性質問題 ········································································· 87
5.2.2 題型二：求解分段函式的不定積分 ············································································ 88
5.2.3 題型三：直接積分法求解不定積分 ············································································ 89
5.2.4 題型四：利用換元積分法求解不定積分 ····································································· 90
5.2.5 題型五：利用分部積分法求解不定積分 ····································································· 91
5.2.6 題型六：求解三角函式有理式的不定積分 ·································································· 93
5.2.7 題型七：求解有理函式的不定積分 ············································································ 95
5.2.8 題型八：求解簡單無理函式的不定積分 ····································································· 97
5.3 深化訓練 ························································································································· 98
5.4 深化訓練詳解 ················································································································· 99
5.5 綜合提高訓練 ··············································································································· 107
第6 章定積分 ···························································································································· 115
6.1 知識要點 ························································································································ 115
6.1.1定積分的定義 ········································································································· 115 ·VII·
6.1.2 定積分的幾何意義與物理意義 ·················································································· 115
6.1.3 定積分的基本性質 ·································································································· 116
6.1.4 變上限積分函式 ······································································································ 117
6.1.5 定積分的計算 ········································································································· 117
6.1.6 廣義積分 ················································································································ 117
6.1.7 定積分的幾何套用 ·································································································· 118
6.1.8 定積分的經濟套用 ·································································································· 119
6.1.9 幾個重要的結論 ····································································································· 120
6.2 典型例題分析 ··············································································································· 120
6.2.1 題型一：利用幾何意義計算定積分 ·········································································· 120
6.2.2 題型二：有關定積分的性質問題 ············································································· 121
6.2.3 題型三：利用定積分的定義求解極限 ······································································· 122
6.2.4 題型四：變限積分問題 ··························································································· 123
6.2.5 題型五：利用換元法、分部積分法求解定積分 ························································· 126
6.2.6 題型六：利用奇偶性、周期性計算定積分 ································································ 127
6.2.7 題型七：分段函式的積分問題 ················································································· 128
6.2.8 題型八：某些不易求出原函式的積分計算問題 ························································· 129
6.2.9 題型九：定積分相關的證明問題 ············································································· 130
6.2.10 題型十：廣義積分問題 ························································································· 131
6.2.11 題型十一：積分的套用問題··················································································· 132
6.3 深化訓練 ······················································································································· 134
6.4 深化訓練詳解 ··············································································································· 137
6.5 綜合提高訓練 ··············································································································· 147
第7 章多元函式微積分學 ······································································································· 152
7.1 知識要點 ······················································································································· 152
7.1.1 二元函式的定義 ····································································································· 152
7.1.2 二元函式的極限與連續 ··························································································· 152
7.1.3 偏導數 ·················································································································· 152
7.1.4 全微分 ·················································································································· 153
7.1.5 高階偏導數 ··········································································································· 154
7.1.6 多元函式的求導法則 ······························································································ 155
7.1.7 二元函式的極值 ····································································································· 155
7.1.8 二重積分的概念與性質 ··························································································· 156
7.1.9 利用直角坐標系計算二重積分 ················································································· 158
7.1.10 利用極坐標計算二重積分 ······················································································ 158
7.1.11 利用對稱性求解二重積分 ······················································································ 159
7.2 典型例題分析 ··············································································································· 160
7.2.1 題型一：多元函式的概念問題 ················································································· 160
7.2.2 題型二：求解多元函式的極限 ················································································· 161
7.2.3 題型三：求解多元函式的偏導數 ············································································· 162
7.2.4 題型四：計算多元函式的全微分 ············································································· 164
7.2.5 題型五：抽象複合函式的偏導數的求解 ··································································· 166
7.2.6 題型六：隱函式的求導問題 ···················································································· 167
7.2.7 題型七：求多元函式的極值和最值 ·········································································· 169
7.2.8 題型八：二重積分的計算 ······················································································· 171
7.2.9 題型九：實際套用題 ······························································································ 175
7.3 深化訓練 ······················································································································· 176
7.4 深化訓練詳解 ··············································································································· 179
7.5 綜合提高訓練 ··············································································································· 190
第8 章無窮級數 ······················································································································· 196
8.1 知識要點 ······················································································································· 196
8.1.1 無窮級數的概念 ····································································································· 196
8.1.2 無窮級數的性質 ····································································································· 196
8.1.3 常見級數的斂散性 ································································································· 197
8.1.4 正項級數斂散性的判別法 ······················································································· 197
8.1.5 任意項級數的斂散性 ······························································································ 198
8.1.6 函式項級數的概念 ································································································· 198
8.1.7 冪級數的概念 ········································································································ 199
8.1.8 冪級數的和函式的性質 ··························································································· 199
8.1.9 函式的冪級數展開 ································································································· 200
8.1.10 常見的麥克勞林公式 ···························································································· 200
8.2 典型例題分析 ··············································································································· 200
8.2.1 題型一：利用定義與性質判斷級數的斂散性 ····························································· 200
8.2.2 題型二：判斷正項級數的斂散性 ············································································· 202
8.2.3 題型三：判斷任意項級數的斂散性 ·········································································· 203
8.2.4 題型四：函式項級數收斂域的求解 ·········································································· 205
8.2.5 題型五：討論冪級數的收斂半徑及收斂域 ································································ 205
8.2.6 題型六：求冪級數的和函式 ···················································································· 208
8.2.7 題型七：函式展開成冪級數問題 ·············································································· 211
8.2.8 題型八：無窮級數的套用問題 ················································································· 213
8.3 深化訓練 ······················································································································· 213
8.4 深化訓練詳解 ··············································································································· 215
8.5 綜合提高訓練 ··············································································································· 224
第9 章常微分方程 ··················································································································· 229
9.1 知識要點 ······················································································································· 229
9.1.1 微分方程的概念 ····································································································· 229
9.1.2 一階微分方程及解法 ······························································································ 229
9.1.3二階線性微分方程 ································································································· 231 ·IX·
9.2 典型例題分析 ··············································································································· 232
9.2.1 題型一：分離變數法求解微分方程 ·········································································· 232
9.2.2 題型二：求解齊次微分方程 ···················································································· 233
9.2.3 題型三：求解一階線性微分方程 ············································································· 235
9.2.4 題型四：求解伯努利方程 ······················································································· 236
9.2.5 題型五：求解二階線性微分方程 ············································································· 237
9.2.6 題型六：套用題 ····································································································· 238
9.3 深化訓練 ······················································································································· 240
9.4 深化訓練詳解 ··············································································································· 242
9.5 綜合提高訓練 ··············································································································· 247
2013 年考研數學三高等數學考題 ······························································································ 251
2014 年考研數學三高等數學考題 ······························································································ 256
2015 年考研數學三高等數學考題 ······························································································ 262
2016 年考研數學三高等數學考題 ······························································································ 268
