單擊此處編輯母版標(biāo)題樣式,,單擊此處編輯母版文本樣式,,第二級(jí),,第三級(jí),,第四級(jí),,第五級(jí),,*,*,P59,習(xí)題3.1,作 業(yè),預(yù)習(xí)P60 —67. P70 —78,8. 9 (3)(6). 11(2)(6). 12. 13.,11/26/2024,1,,第五講 導(dǎo)數(shù)與微分（一）,二、導(dǎo)數(shù)定義與性質(zhì),五、基本導(dǎo)數(shù)（微分）公式,一、引言,三、函數(shù)的微分,四、可導(dǎo)、可微與連續(xù)的關(guān)系,11/26/2024,2,,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(chē),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.,11/26/2024,3,,[解],如果極限,存在, 這個(gè)極限值就是質(zhì)點(diǎn)的,,瞬時(shí)速度.,11/26/2024,4,,[例2] 曲線的切線斜率問(wèn)題,什麼是曲線的切線？,11/26/2024,5,,11/26/2024,6,,11/26/2024,7,,二、導(dǎo)數(shù)定義與性質(zhì),1. 導(dǎo)數(shù)定義：,11/26/2024,8,,[注意1],導(dǎo)數(shù)的等價(jià)定義：,11/26/2024,9,,[注意2] 導(dǎo)數(shù)的意義：,物理意義,幾何意義,,導(dǎo)數(shù)是函數(shù)在一點(diǎn)的變化率,11/26/2024,10,,例:線密度問(wèn)題,11/26/2024,11,,左導(dǎo)數(shù),右導(dǎo)數(shù),2. 單側(cè)導(dǎo)數(shù)定義：,定理：,11/26/2024,12,,3. 導(dǎo)函數(shù)定義：,11/26/2024,13,,三、函數(shù)的微分,,,導(dǎo)數(shù)是從函數(shù)對(duì)自變量變化的速度來(lái),,研究;而微分則是直接研究函數(shù)的增量，,,這有許多方便之處。,（一）函數(shù)的微分的定義,11/26/2024,14,,11/26/2024,15,,四、可導(dǎo)、可微與連續(xù)的關(guān)系,定理1: 函數(shù)可微與可導(dǎo)是等價(jià)的,11/26/2024,16,,[證] (1),11/26/2024,17,,[證] (2),11/26/2024,18,,定理2：,[證],[注意] 可導(dǎo)必連續(xù), 連續(xù)不一定可導(dǎo)！,11/26/2024,19,,[解],11/26/2024,20,,尖點(diǎn),11/26/2024,21,,[解],有鉛垂切線,11/26/2024,22,,[解],振蕩,不存在,！,11/26/2024,23,,11/26/2024,24,,微分的幾何意義,微分三角形,11/26/2024,25,,11/26/2024,26,,五、基本導(dǎo)數(shù)（微分）公式,11/26/2024,27,,11/26/2024,28,,微分基本公式,11/26/2024,29,,5. 利用定義求導(dǎo)的例子,[解],11/26/2024,30,,[解],11/26/2024,31,,[解],11/26/2024,32,,[解],11/26/2024,33,,問(wèn)題：如何求其他函數(shù)的導(dǎo)數(shù)？,基本導(dǎo)數(shù)公式,導(dǎo)數(shù)運(yùn)算法則,其他基本初等函數(shù),初等函數(shù),? ?,四則,復(fù)合,反函數(shù),隱函數(shù),參數(shù)方程,對(duì)數(shù)微分法,11/26/2024,34,,