高等數(shù)學導數(shù)與微分習題課件

1、高等數(shù)學導數(shù)與微分習題1. )()()(aFxfaxxf處可導且在,)(,)(.01aFaxxF處連續(xù)在函數(shù)例)(|)()(, )()()()(xFaxxgxFaxxf21討論.處的連續(xù)性與可導性在ax , )()()()(.xFaxxf1解,)(,點連續(xù)在連續(xù)axFax .)(點連續(xù)在axfaxafxfafax)()(lim)(axxFaxax0)()(lim. )()(limaFxFax高等數(shù)學導數(shù)與微分習題2, )(|)()(xFaxxg2,)(,|點連續(xù)在連續(xù)axFax .)(點連續(xù)在axgaxagxgagax)()(lim)(0axxFaxax00)()(lim, )()(limaF

2、xFax0. )()(aFag,)(0aF, )()(, )()(agagaFaF.)(處不可導在axxg高等數(shù)學導數(shù)與微分習題3)(.xf研究函數(shù)例2011xxx00 x.處的連續(xù)性與可導性在0 xxxxfxx1100lim)(lim.解110 xxxxlim110 xxxlim,0,)()(0000ff.)(處連續(xù)在0 xxf( 0,)()(000f.)(,處不可導在所以0 xxf0000 xfxffx)()(lim)(xxxx110limxxxx20/lim)(. 不存在,(0)/(211xx 高等數(shù)學導數(shù)與微分習題4,)(,)(.03xfxf可導函數(shù)例,sin)()(:在交點處相切和曲

3、線證明xxfyxfy21.切線即曲線在交點處有公共, ),(.00yx設兩曲線的交點坐標為證000 xxfxfsin)()(,sin10 x, )(00 xfy , )(xfy 1對于曲線, )(),(0100 xfkyx處切線的斜率為在,sin)(xxfy 2對于曲線處切線的斜率為在),(00yx02xxxxfxxfkcos)(sin)(0000 xxfxxfcos)(sin)(0100 xxcos,sin. )(0 xf .,證畢所以21kk 高等數(shù)學導數(shù)與微分習題5,)(.為周期的可導函數(shù)是以例Txf4.)(為周期仍以證明Txf ,)()(.成立對證xxfTxf0)( :是常數(shù)兩端同時求

4、導得T.)()(證畢0 xfTxf.,.偶函數(shù)的導數(shù)是奇函數(shù)奇函數(shù)的導數(shù)是偶函數(shù)例5,)(.為奇函數(shù)設證xf,)()(成立對xxfxf:求導得等式兩端同時關于 x)()()(xfxf1)()(xfxf.)(為偶函數(shù)xf .同理可證后半段高等數(shù)學導數(shù)與微分習題6.lnlog:.xyx對函數(shù)求導例6,ln)ln(ln:.xxy 換底解xxyln)ln(lnuuuuxlnln211uuuulnxln.lnlnlnxxx21高等數(shù)學導數(shù)與微分習題7.:.xxxxxy對函數(shù)求導例7,.xxz 解)ln()(ln xxzzzzxxxxxx1 ln)ln(,ln x 1;ln xxxxx1,xxxz )ln

5、()(ln xxzzzzxxxxxxxxxxx1ln)ln(,lnln11xxxxxx.lnln11xxxxxxxxxxxx.lnlnln111xxxxxxxxxxxyx高等數(shù)學導數(shù)與微分習題8xxxxxy.又解xxexxlnlnxxexxxxlnlnxxxxx1lnxxxxxxxxx1ln)(xxxln1.lnln11xxxxxxxxxxxxxxeelnln高等數(shù)學導數(shù)與微分習題9.cos)sin()(xxxxa112,sin,)sin)(.xxuxxxaxdyd28已知例.udyd求udxdxdydudyd.解)(xuxdyd1.tanarc)(yyyy2211.)(,tanarc)(.x

6、xxx19 求已知例,tanarc)(, )(.yyyxxy 則記解1)()(yx 112111yyytanarc高等數(shù)學導數(shù)與微分習題10.,.xdydyxxy求例10,lnln.yxxy解,lnlnyyxyxyxy11xyxyxyylnln.lnlnxyxxyyxy22,lnln.yxxy又解:兩端同時微分得xydln)(xxdy,ln)(yydxyxd,ln)(ln)(ydxyxdyxxdyxydyx22,ln)(ln)(22yyyxxdxxyxyd.lnlnxyxxyyxyxdyd22高等數(shù)學導數(shù)與微分習題11.,sin.2202111xdydyyx求例,cos.021ydyydxd解

7、ydyydxdcos22.cos yxdyd222222020)cos()sin(yyyxdydyyycos)cos(sin22222.)cos(sin324yy高等數(shù)學導數(shù)與微分習題12)sgn() |ln(|ln|xxxxxx11 .|)(|xxxxy 2, |)(uuuuyxu 則記2xdududydyx)ln(lnuuuuu11 )sgn(xx1, |ln)(xy 1)(. 1解)sgn(|xx1,.012x設例:y求xx |1xln.?)sgn()sgn(xxx1uyx?)(0 x)(0 x高等數(shù)學導數(shù)與微分習題13.|arccos.的導數(shù)求例xy113,|.時當解1x|xxy111

8、12221|xxxx221xxxx)sgn(|122xxx.112xx高等數(shù)學導數(shù)與微分習題14.14例xxfxxfx )()(lim0hxfhxfhxh)()(lim0 hxfhxfh)()(lim0. )(xf . )()()(limxfxxxfxfx 0 xxxfxxfx )()(lim0 xxxfxfxfxxfx )()()()(lim0 xxxfxfxxfxxfxx )()(lim)()(lim00. )(xf 2高等數(shù)學導數(shù)與微分習題15)(.xf15例12xx1xbxa.)(,處可導在使的值選取1xxfba( 1,)(.處應連續(xù)在解1xxf,)()(1101ffbabxafx)(

9、lim)(0101.1ba2112xxf)(1111xfxffx)()(lim)(111xbxaxlim1111xaxax)(lim11xaxaxlim,a.,12ba,(1高等數(shù)學導數(shù)與微分習題16)(.xf16例01xx)(ln0 xx )0,)ln()()(010000 xxff,lim)(0000 xfx.)(處連續(xù)在0 xxf010 xxf)ln()(011xx10000 xfxffx)()(lim)(000 xxxlim110 )(f)(xf011xx01x01x)(xf011xx01x. )(xf 類似地可續(xù)求),0. )(xf求高等數(shù)學導數(shù)與微分習題17. )()(, )()(

10、 )()(11002117fxfxxxxxf及求設例)(ln)()(xfxfxf解)ln()ln()ln(ln)(ln10021xxxxxf100121111xxxx11002111xxxxxfx)()( )(lim)()()(lim10021xxxx!99)(ln)()(xfxfxf?)( nf高等數(shù)學導數(shù)與微分習題18.,lnarctanyxxxy求設例11114112118222,.21xu設解,lnarctan114121uuuy則1111411212uuuyu)(411u,4221xx 21xux,21xx.)(23121xxxyx高等數(shù)學導數(shù)與微分習題190245219txdydt

11、ttyttx求設例,.,.不能用公式求導不存在時當解tdydtdxdt0tttttxytx 245200)(limlim)sgn()sgn(limtttt 2450)sgn(ttt 0.00txdyd所以高等數(shù)學導數(shù)與微分習題20.,),()(220020 xdydyxxyxfyyx求所確定由方程設函數(shù)例,lnln:.xyyx11兩邊取對數(shù)得解,lnlnxxyy即,ln)ln(11xyy,lnln11yxy211111)ln()ln()ln( yyyxyxy.)(ln)ln()ln(322111yyxxxyy高等數(shù)學導數(shù)與微分習題21. )(,)()(xfxxxxf求設例2212200 xxx

12、xf)()(22xx)(22xx)(22xx.)abs()abs()(.連續(xù)解2xxxxf )(022200 xxxxf)(顯然xx432xx432xx432,時當0 x0000 xfxffx)()(lim)(,)(lim)(0002020 xxxfx.)(00 f,)(lim000220 xxxx高等數(shù)學導數(shù)與微分習題22,時當2x22202xfxffx)()(lim)(22202xfxffx)()(lim)(,)(lim420220 xxxx2200 xxxxf)()(22xx)(22xx)(22xx )(02)()(22ff.)(不可導在2xxf,)(lim4202202xxxx2200 xxxxf)(xx432xx432xx432因此高等數(shù)學導數(shù)與微分習題23.,)(sincosyxxyx求設例22xyyy)ln(.解)sinlncosln(xxxyxxxxxxxxsincossinlnsin)(sincos21.,)(nyxxy求設例114232213441142222xxxxy.解1111234xx,)(!)()(11111nnnxnx,)(!)()(11111nnnxnx.)()(!)()(111111123nnnnxxny高等數(shù)學導數(shù)與微分習題24:.導出試從例yydxd124.)()(.;)(.5233322321