清華微積分(高等數(shù)學)課件第四講連續(xù)函數(shù)的性質(zhì).ppt

2020/5/8,1,作業(yè),P50綜合題1.4.,P49習題2.411.13.14.,預習：P5158,2020/5/8,2,連續(xù)函數(shù)的性質(zhì),第四講,一、連續(xù)函數(shù)的基本性質(zhì),二、初等函數(shù)的連續(xù)性,三、閉區(qū)間上連續(xù)函數(shù)的性質(zhì),2020/5/8,3,一、函數(shù)連續(xù)性的基本性質(zhì),（一）連續(xù)性定義的等價形式：,2020/5/8,4,（二）連續(xù)函數(shù)的有界性：,2020/5/8,5,（三）連續(xù)函數(shù)的保號性：,2020/5/8,6,（四）連續(xù)函數(shù)的運算性質(zhì)：,2020/5/8,7,（六）初等函數(shù)的連續(xù)性,初等函數(shù)在其定義區(qū)間上是連續(xù)的。,（五）關于反函數(shù)的連續(xù)性,結(jié)論：,2020/5/8,8,1.基本初等函數(shù)的連續(xù)性,（1）由連續(xù)定義可驗證基本初等函數(shù)：,例,2020/5/8,9,（3）用連續(xù)函數(shù)四則運算性質(zhì)證明基本初等函數(shù):,（2）用復合函數(shù)及反函數(shù)的連續(xù)性證明基本初等函數(shù):,2020/5/8,10,2.初等函數(shù)的連續(xù)性,由基本初等函數(shù)的連續(xù)性,運用連續(xù)函數(shù)的四則運算、復合運算就推出所有初等函數(shù)在其定義區(qū)間上處處連續(xù).,2020/5/8,11,解,非初等函數(shù)連續(xù)性問題舉例,2020/5/8,12,2020/5/8,13,解,2020/5/8,14,2020/5/8,15,1.有界性定理：,2.最大最小值定理：,三、閉區(qū)間上連續(xù)函數(shù)的性質(zhì),2020/5/8,16,3.零點定理：,2020/5/8,17,4.介值定理：,推論：,2020/5/8,18,f(x),g(x),2020/5/8,19,證,構造輔助函數(shù),介值定理的證明,2020/5/8,20,則有,且,2020/5/8,21,解,試算,根據(jù)代數(shù)基本定理三次多項式最多有三個實根,2020/5/8,22,證,2020/5/8,23,證,2020/5/8,24,2020/5/8,25,證,2020/5/8,26,證,2020/5/8,27,矛盾！,2020/5/8,28,導數(shù)與微分,2020/5/8,29,一、引言,兩個典型背景示例,例1運動物體的瞬時速度,設汽車沿t軸作直線運動,若己知其運動規(guī)律(路程與時間的函數(shù)關系)為求在時刻的瞬時速度.,2020/5/8,30,解,如果極限存在,這個極限值就是汽車的瞬時速度.,2020/5/8,31,例2曲線的切線斜率問題,什麼是曲線的切線？,2020/5/8,32,2020/5/8,33,