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第二章一元函数微分学§2.1导数与微分一、主要内容㈠导数的概念1.导数:)(xfy在0x的某个邻域内有定义,xxfxxfxyxx)()(limlim000000)()(lim0xxxfxfxx00)(0xxxxdxdyxfy2.左导数:000)()(lim)(0xxxfxfxfxx右导数:000)()(lim)(0xxxfxfxfxx定理:)(xf在0x的左(或右)邻域上连续在其内可导,且极限存在;则:)(lim)(00xfxfxx(或:)(lim)(00xfxfxx)3.函数可导的必要条件:定理:)(xf在0x处可导)(xf在0x处连续4.函数可导的充要条件:定理:)(00xfyxx存在)()(00xfxf,且存在。5.导函数:),(xfy),(bax)(xf在),(ba内处处可导。y)(0xf)(xf6.导数的几何性质:y)(0xf是曲线)(xfy上点x00,yxM处切线的斜率。ox0x㈡求导法则1.基本求导公式:2.导数的四则运算:1ovuvu)(2ovuvuvu)(3o2vvuvuvu)0(v3.复合函数的导数:)]([),(),(xfyxuufydxdududydxdy,或)()]([})]([{xxfxf☆注意})]([{xf与)]([xf的区别:})]([{xf表示复合函数对自变量x求导;)]([xf表示复合函数对中间变量)(x求导。4.高阶导数:)(),(),()3(xfxfxf或)4,3,2(,])([)()1()(nxfxfnn函数的n阶导数等于其n-1导数的导数。㈢微分的概念1.微分:)(xf在x的某个邻域内有定义,)()(xoxxAy其中:)(xA与x无关,)(xo是比x较高阶的无穷小量,即:0)(lim0xxox则称)(xfy在x处可微,记作:xxAdy)(dxxAdy)()0(x2.导数与微分的等价关系:定理:)(xf在x处可微)(xf在x处可导,且:)()(xAxf3.微分形式不变性:duufdy)(不论u是自变量,还是中间变量,函数的微分dy都具有相同的形式。一、例题分析例1.设)(xf存在,且1)()2(lim000xxfxxfx,则)(0xf等于A.1,B.0,C.2,D.21.[]解:xxfxxfx)()2(lim0001)(22)()2(lim200002xfxxfxxfx∴21)(0xf(应选D)例2.设),()()(22xaxxf其中)(x在ax处连续;求)(af。解:axafxfafax)()(lim)(axaaaxaxax)()()()(lim2222)()(lim)())((limxaxaxxaxaxaxax)(2aa误解:)()()(2)(22xaxxxxf∴)(2)()()(2)(22aaaaaaaaf结果虽然相同,但步骤是错的。因为已知条件并没说)(x可导,所以)(x不一定存在。例3.设)(xf在1x处可导,且2)1(f,求:1)1()34(lim1xfxfx解:设)4(,3431txxt当1x时,1t1)4()1()(lim1)1()34(lim3111tftfxfxftx623)1(31)1()(lim31ftftft例4.设)(xf是可导的奇函数,且0)(0kxf,则)(0xf等于:A.k,B.k,C.k1,D.k1.[]解:)()(xfxf])([])([xfxf)()()(xfxxf)()(xfxf∴kxfxf)()(00(应选A)(结论:可导奇函数的导数是偶函数;可导偶函数的导数是奇函数。)例5.设1211)(2xxxxxf在1x处是否可导?解法一:22)1(1xxf2)1(lim)(lim211xxfxx2)2(lim)(lim11xxfxx∴)(xf在1x处连续121lim1)1()(lim)1(211xxxfxffxx2)1(lim11lim121xxxxx22lim122lim1)1()(lim)1(111xxxxxxfxff∴2)1()1()1(fff∴)(xf在1x处可导。解法二:22)1(1xxf2)1(lim)(lim211xxfxx2)2(lim)(lim11xxfxx∴)(xf在1x处连续当1x时,1212)(xxxxf∴22lim)(lim)1(11xxffxx22lim)(lim)1(11xxxff∴2)1()1()1(fff∴)(xf在1x处可导。例6.设001)(2xaexbxxfx求a,b的值,使)(xf处处可导。解:)(xf的定义域:),(x当0x时,bxxf1)(是初等函数,在)0,(内有定义,∴不论a和b为何值,)(xf在)0,(内连续;当0x时,xaexf2)(是初等函数,在),0(内有定义,∴不论a和b为何值,)(xf在),0(内连续;1)1()0(0xbxf1)1(lim)(lim00bxxfxxaaexfxxx200lim)(lim只有当1a时,)(xf在0x处连续;∴当1a时,)(xf处处连续;当0a时,可导可导020020)(221xexbxaexbxfxxabbxffxx00lim)(lim)0(22lim)(lim)0(200xxxexff只有当2b时,)(xf在0x处可导;∴当2,1ba,)(xf处处可导。例7.求下列函数的导数⑴)21ln(cosxy解:xvvuuy21lncosdxdvdvdududydxdy)21ln(sin21221sinxxvu⑵)arctan(tan2xy解:])n[arctan(ta2xy)(tan)(tan1tan2)(tan)(tan1122222xxxxxxxxxxx44222cossin2sin)(tan1sectan2⑶xxy2tan10解:)2tan(1010ln)10(2tan2tanxxyxxxx)2sec22(tan1010ln22tanxxxxx⑷222ryx(r为常数)解法一:22xry2222222)()(xrxrxry22xrx解法二:)()(222ryx022yyx22xrxyxy⑸)cos(xyy解法一:)()sin(])[cos(xyxyxyy)()sin(yxyxy∴)sin(1)sin(xyxxyyy解法二:设)cos(),(xyyyxF)sin(1),sin(xyxFxyyFyx)sin(1)sin(xyxxyyFFdxdyyx隐函数求导!⑹yxxylnln解法一:)ln()ln(yxxyyyxyxyyxlnln22lnlnlnlnxxxyyyxyxyyyxxy解法二:设yxxyyxFlnln),(yxxFyxyFyxln,ln22lnlnlnlnxxxyyyxyxyFFdxdyyxxyyx⑺3)2)(1(xxxy解:(对数法)3)2)(1(lnlnxxxy)]3ln()2ln()1[ln(21xxx})]3ln()2ln()1[ln({)(ln21xxxy)312111(211xxxyy∴3)2)(1()312111(21xxxxxxy⑻xxy解法一:(对数法)xxxyxlnlnln1lnln1xxxxyy∴)1(lnxxyx解法二:(指数法)xxxxeexyxlnln)ln()(lnlnxxeeyxxxx)1(lnxxx⑼xxxxycos)(sin2解法一:(对数法)设xxxyxycos21)(sin,22121,yyyyyyxxxyxln2ln2lnln1)2(ln21ln211xxxxxxyy∴)2(ln)2(ln2221xxxxxyxxxxysinlncosln2xxxxxyysincoscossinlnsin122∴)sinlnsincot(cos)(sincos2xxxxxyx21yyy)sinlnsincot(cos)(sin)1(lncos21xxxxxxxxx解法二:(指数法)xxxxeeysinlncosln2)sinln(cos)ln(2sinlncoslnxxexxexxxx)sinlnsincot(cos)(sin)1(lncos21xxxxxxxxx⑽yxxy解法一:xyyxlnlnxyyxyyxylnln∴22lnlnxxxyyyxyy解法二:设xyyxyxF),(yxyxyxxyxyyyxxyyyyxF)ln(lnln1yxyxyyxxyxyyxxxxyxxF)ln(lnln122lnln)ln()ln(xxxyyyxyxxxyFFdxdyyyxyxyyx例8.已知xxfsin)(,求)(xf。解:设2,txxt2sin)(ttf∴2sin)(xxf∴222cos2)(cos)(xxxxxf例9.求下列函数的二阶导数⑴)1ln(2xy解:212xxy222222)1(22)1(22)1(2xxxxxxy⑵0lnyxy解法一:01yyyxy02yyxyy∴xyyy1222)1()()1(2xyyxyyxyyyy2121)1()()1(222xyxyyxyyxyyxyy3223)1(])1([)1(2xyxyxyyyxyy343)1(23xyxyy解法二:01yyyxy02yyxyy∴xyyy120)(2yyxyy0)(22yyxyyxyyyyxyxyxyyxyyyxyyxyy131)(3211222343343)1(23)1()1(3xyxyyxyxyxyy例10.设xexy29,求:10,,)()10(nyyn。解:xexy2829xexy227289xexy2362789……xxeexy292999)9(2!921789xey210)10(210,22)(neyxnn结论:对于mxy,若mn,则0)(ny例11.设xxyln49,求)50(y。解:1)48(49xxy212)47()1(4849xxy313)46(21)1(474849x
本文标题:成考专升本高数(二)第二章笔记
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