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

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2020/5/8,1,作業(yè),P50綜合題1.4.,P49習(xí)題2.411.13.14.,預(yù)習(xí)：P51—58,,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(jià)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(hào)性：,,,2020/5/8,6,,,（四）連續(xù)函數(shù)的運(yùn)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(guān)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(yàn)證基本初等函數(shù)：,,,[例],2020/5/8,9,（3）用連續(xù)函數(shù)四則運(yùn)算性質(zhì)證明基本初等函數(shù):,（2）用復(fù)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(yùn)用連續(xù)函數(shù)的四則運(yùn)算、復(fù)合運(yùn)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(wèn)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(diǎn)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(gòu)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(xiàng)式最多有三個(gè)實(shí)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(dǎo)數(shù)與微分,2020/5/8,29,一、引言,兩個(gè)典型背景示例,[例1]運(yùn)動(dòng)物體的瞬時(shí)速度,設(shè)汽車沿t軸作直線運(yùn)動(dòng),若己知其運(yùn)動(dòng)規(guī)律(路程與時(shí)間的函數(shù)關(guān)系)為求在時(shí)刻的瞬時(shí)速度.,,,,,,2020/5/8,30,[解],如果極限存在,這個(gè)極限值就是汽車的瞬時(shí)速度.,,,2020/5/8,31,[例2]曲線的切線斜率問(wèn)題,什麼是曲線的切線？,,,2020/5/8,32,,,,,,,2020/5/8,33,,,