微積分與高考數(shù)學（第2版）

第１章　函數(shù)１ !!!!!!!!!!!!!!!!!!!!!!
１．１　函數(shù)的概念１ !!!!!!!!!!!!!!!!!!!!!!!
１．１．１　實數(shù)與數(shù)集２ !!!!!!!!!!!!!!!!!!!!
１．１．２　區(qū)間２ !!!!!!!!!!!!!!!!!!!!!!!
１．１．３　鄰域與去心鄰域２ !!!!!!!!!!!!!!!!!!
１．１．４　有界集３ !!!!!!!!!!!!!!!!!!!!!!
１．１．５　函數(shù)３ !!!!!!!!!!!!!!!!!!!!!!!
１．２　隱函數(shù)與反函數(shù)５ !!!!!!!!!!!!!!!!!!!!!
１．２．１　隱函數(shù)５ !!!!!!!!!!!!!!!!!!!!!!
１．２．２　反函數(shù)５ !!!!!!!!!!!!!!!!!!!!!!
１．２．３　反三角函數(shù)６ !!!!!!!!!!!!!!!!!!!!
習題１．２ ８ !!!!!!!!!!!!!!!!!!!!!!!!
１．３　參數(shù)方程與極坐標９ !!!!!!!!!!!!!!!!!!!!
１．３．１　參數(shù)方程表示的函數(shù)９ !!!!!!!!!!!!!!!!
１．３．２　極坐標９ !!!!!!!!!!!!!!!!!!!!!!
１．３．３　極坐標與直角坐標的關系１１ !!!!!!!!!!!!!
１．３．４　極坐標中常見的曲線１２ !!!!!!!!!!!!!!!
習題１．３ １４ !!!!!!!!!!!!!!!!!!!!!!!!
１．４　初等函數(shù)１４ !!!!!!!!!!!!!!!!!!!!!!!!
１．４．１　基本初等函數(shù)１４ !!!!!!!!!!!!!!!!!!
１．４．２　復合函數(shù)與初等函數(shù)１７ !!!!!!!!!!!!!!!
習題１．４ １８
第２章　極限２０ !!!!!!!!!!!!!!!!!!!!!!
２．１　數(shù)列的極限２０ !!!!!!!!!!!!!!!!!!!!!!!
２．１．１　數(shù)列與極限的定義２１ !!!!!!!!!!!!!!!!
２．１．２　極限的運算２３ !!!!!!!!!!!!!!!!!!!
習題２．１ ２５ !!!!!!!!!!!!!!!!!!!!!!!!
２．２　連續(xù)函數(shù)與函數(shù)的極限２６ !!!!!!!!!!!!!!!!!!
２．２．１　函數(shù)的極限與連續(xù)函數(shù)的定義２６ !!!!!!!!!!!
２．２．２　函數(shù)的單側極限２８ !!!!!!!!!!!!!!!!!
習題２．２ ２９ !!!!!!!!!!!!!!!!!!!!!!!!
２．３　極限的嚴格定義２９ !!!!!!!!!!!!!!!!!!!!!
２．３．１　數(shù)列極限的嚴格定義３０ !!!!!!!!!!!!!!!
２．３．２　一般函數(shù)極限的嚴格定義３１ !!!!!!!!!!!!!
２．３．３　極限的性質３２ !!!!!!!!!!!!!!!!!!!
２．４　連續(xù)函數(shù)的性質３４ !!!!!!!!!!!!!!!!!!!!!
２．４．１　連續(xù)的嚴格定義３４ !!!!!!!!!!!!!!!!!
２．４．２　閉區(qū)間內連續(xù)函數(shù)的性質３５ !!!!!!!!!!!!!
習題２．４ ３７ !!!!!!!!!!!!!!!!!!!!!!!!
２．５　自然常數(shù)ｅ ３８ !!!!!!!!!!!!!!!!!!!!!!!
２．５．１　銀行復利的問題３８ !!!!!!!!!!!!!!!!!
２．５．２　ｅ的應用４０ !!!!!!!!!!!!!!!!!!!!
習題２．５ ４１ !!!!!!!!!!!!!!!!!!!!!!!!
２．６　漸近線４２ !!!!!!!!!!!!!!!!!!!!!!!!!
習題２．６ ４４ !!!!!!!!!!!!!!!!!!!!!!!!
第３章　一元微分學４６ !!!!!!!!!!!!!!!!!!!
３．１　導數(shù)的概念４６ !!!!!!!!!!!!!!!!!!!!!!!
３．１．１　導數(shù)的例子４６ !!!!!!!!!!!!!!!!!!!
３．１．２　導數(shù)的定義４８ !!!!!!!!!!!!!!!!!!!
習題３．１ ５１ !!!!!!!!!!!!!!!!!!!!!!!!
３．２　導數(shù)的基本公式與四則運算求導法則５
習題３．２ ５５ !!!!!!!!!!!!!!!!!!!!!!!!
３．３　其他求導法則５６ !!!!!!!!!!!!!!!!!!!!!!
３．３．１　反函數(shù)與復合函數(shù)求導法則５６ !!!!!!!!!!!!
３．３．２　隱函數(shù)與參數(shù)方程求導法則５９ !!!!!!!!!!!!
習題３．３ ６２ !!!!!!!!!!!!!!!!!!!!!!!!
３．４　高階導數(shù)６６ !!!!!!!!!!!!!!!!!!!!!!!!
習題３．４ ７０ !!!!!!!!!!!!!!!!!!!!!!!!
３．５　微分７０ !!!!!!!!!!!!!!!!!!!!!!!!!!
習題３．５ ７２ !!!!!!!!!!!!!!!!!!!!!!!!
３．６　洛必達法則７２ !!!!!!!!!!!!!!!!!!!!!!!
３．６．１?。?
０和∞∞
型未定式７３ !!!!!!!!!!!!!!!!!
３．６．２　其他型未定式７４ !!!!!!!!!!!!!!!!!!
習題３．６ ７６ !!!!!!!!!!!!!!!!!!!!!!!!
３．７　極值的判定和最值７６ !!!!!!!!!!!!!!!!!!!!
習題３．７ ８０ !!!!!!!!!!!!!!!!!!!!!!!!
３．８　函數(shù)的凸向和作圖８１ !!!!!!!!!!!!!!!!!!!!
３．８．１　凸函數(shù)、曲線的凸向及拐點８１ !!!!!!!!!!!!
３．８．２　函數(shù)的分析作圖法８２ !!!!!!!!!!!!!!!!
習題３．８ ８４ !!!!!!!!!!!!!!!!!!!!!!!!
３．９　微分中值定理８５ !!!!!!!!!!!!!!!!!!!!!!
習題３．９ ８９ !!!!!!!!!!!!!!!!!!!!!!!!
３．１０　泰勒公式８９ !!!!!!!!!!!!!!!!!!!!!!!
習題３．１０ ９２ !!!!!!!!!!!!!!!!!!!!!!!
第４章　一元積分學９３ !!!!!!!!!!!!!!!!!!!
４．１　積分的概念與性質９３ !!!!!!!!!!!!!!!!!!!!
４．１．１　定積分的概念９３ !!!!!!!!!!!!!!!!!!
４．１．２　定積分的簡單性質９５ !!!!!!!!!!!!!!!!
習題４．１ ９６ !!!!!!!!!!!!!!!!!!!!!!!!
４．２　微積分基本定理９７
４．２．１　微積分基本定理第一部分９７ !!!!!!!!!!!!!
４．２．２　微積分基本定理第二部分９９ !!!!!!!!!!!!!
習題４．２ １００ !!!!!!!!!!!!!!!!!!!!!!!
第５章　多元函數(shù)微分學１０３ !!!!!!!!!!!!!!!!
５．１　多元函數(shù)的基本概念１０３ !!!!!!!!!!!!!!!!!!
５．１．１　預備知識１０３ !!!!!!!!!!!!!!!!!!!!
５．１．２　多元函數(shù)１０４ !!!!!!!!!!!!!!!!!!!!
習題５．１ １０５ !!!!!!!!!!!!!!!!!!!!!!!
５．２　多元函數(shù)的極限與連續(xù)１０６ !!!!!!!!!!!!!!!!!
習題５．２ １０８ !!!!!!!!!!!!!!!!!!!!!!!
５．３　偏導數(shù)與二階偏導數(shù)１０９ !!!!!!!!!!!!!!!!!!
５．３．１　偏導數(shù)１０９ !!!!!!!!!!!!!!!!!!!!!
５．３．２　二階偏導數(shù)１１１ !!!!!!!!!!!!!!!!!!!
習題５．３ １１２ !!!!!!!!!!!!!!!!!!!!!!!
５．４　全微分１１２ !!!!!!!!!!!!!!!!!!!!!!!!
習題５．４ １１６ !!!!!!!!!!!!!!!!!!!!!!!
５．５　復合函數(shù)求導法１１６ !!!!!!!!!!!!!!!!!!!!
習題５．５ １１９ !!!!!!!!!!!!!!!!!!!!!!!
５．６　隱函數(shù)求導法１２０ !!!!!!!!!!!!!!!!!!!!!
習題５．６ １２１ !!!!!!!!!!!!!!!!!!!!!!!
５．７　多元函數(shù)的一階泰勒公式１２１ !!!!!!!!!!!!!!!!
習題５．７ １２３ !!!!!!!!!!!!!!!!!!!!!!!
５．８　多元函數(shù)的極值１２３ !!!!!!!!!!!!!!!!!!!!
習題５．８ １２６ !!!!!!!!!!!!!!!!!!!!!!!
第６章　微積分在高中的應用１２７ !!!!!!!!!!!!!!
６．１　微積分的論證方法１２７ !!!!!!!!!!!!!!!!!!!
６．１．１　分析法１２７ !!!!!!!!!!!!!!!!!!!!!
６．１．２　綜合法１２９ !!!!!!!!!!!!!!!!!!!!!
６．１．３　構造法１３０
６．１．４　舉反例法１３１ !!!!!!!!!!!!!!!!!!!!
６．１．５　計算法１３２ !!!!!!!!!!!!!!!!!!!!!
６．１．６　反證法１３３ !!!!!!!!!!!!!!!!!!!!!
６．１．７　數(shù)學歸納法１３４ !!!!!!!!!!!!!!!!!!!
６．２　微積分在高考數(shù)學中的應用１３５ !!!!!!!!!!!!!!!
６．２．１　極坐標及反三角函數(shù)的應用１３５ !!!!!!!!!!!!
６．２．２　極限的應用１３６ !!!!!!!!!!!!!!!!!!!
６．２．３　中值定理的應用１４６ !!!!!!!!!!!!!!!!!
６．２．４　泰勒公式的應用１５０ !!!!!!!!!!!!!!!!!
６．２．５　微積分基本定理的應用１５７ !!!!!!!!!!!!!!
習題６（強基真題） １６０ !!!!!!!!!!!!!!!!!!!!!!
參考答案１６６ !!!!!!!!!!!!!!!!!!!!!!!
參考文獻１８１