高等数学(上)总复习

1复习《高等数学》（上）总基本要求基本内容；、认真看书和笔记掌握1、加强运算能力；3及典型题型。、通过总复习掌握综合4、成绩5dx9309.01.0）勤奋（聪明考题掌握基本题型；辅导系统、做十年、认真看《高等数学》22元函数微分学函数、极限、连续及一:一、基本题型、基本函数问题；1多种）；、极限的求法（102论；、连续性及间断点的讨3无穷小；、无穷小的比较及等价4性质及应用；、闭区间上连续函数的5几何意义；、导数、微分的定义及6及左右导数的应用；、左右极限、左右连续7微的关系；、极限、连续可导及可83参数方程求导；、复合函数、隐函数、、求导法则、基本公式9、高阶导数及公式10及其应用；、中值定理及泰勒公式11点；、最值应用、凹向及拐、函数的单调性、极值12种常用方法）；、不等式的证明（513、方程根的讨论。144二、典型例题的定义域、求xxysinln2161:解,sin00162xx由),(),[04得5))((,,,)(xffxxxxxf求、设11132:解1113)(),()(,)())((xfxfxfxfxff0x101xx或11013049xxxxxx,,,6.0),(0,)()(,32为连续的奇函数函数使函数及函数、求常数xxgxkexfxgkx:解,)(,)(00fxf必须为连续的奇函数要使1k有)()(xfxf又时当0x)()()()()(kexfxfxgx2)()(01122xeexx7、求极限4])()[(lim)(323231111xxxx])()][()()[(lim31313131311111xxxxxx])()()()(])())[(([lim3231313231313111111111xxxxxxxxxx348])(sin[lim)(cscxxxxexx23112220xexxxe230lncsclim)ln(sinlimxexxxxe21110)(limxxexxxe210xxexxe210lim19)(tanlim)(nnn243nnnn)tantan(lim2121nnnnnnnn112122222121221tantantantan])tantan[(lim4e10xxxxxxcosln)ln(arctancoslnlim)(214420)cosln()cosln(lim11112420xxxxxx11410xxxcoscoslim22411)sin(lim)(xxeeIxxx410125)sin(limxxeexxx410012)sin(limxxeeexxxx12434001)sin(limxxeexxx410012)sin(limxxeexxx41001211I12,])[(lim存在、如果极限xxxcx27645.及极限求c:解])[(limxxxcx2745存在])[(lim12711415ccccxxxxx012711415])[(limccccxxxx12711415ccccxxxx)(lim即51c13])[(limxxxcx2745从而])[(lim1271515xxxx])[(lim12711415ccccxxxxxxxx~)(110)(lim52751xxxx5714.lim,)()(nnnnknxkx求22117:解22211111)(nnnxn2212112nnxnnn)(222nnxlim15xxxxfIffxtan)cos(sinlim)(,)()(2021018求已知xxxxxxfxxfIxtancossincossin)()cossin(lim11111222012111))((f16.)(lim,)(,)(,)(存在证明连续存在且有时设nfxfxxfxn21019:证明,)(0xf单调增加)(xf单调增加)(nf,)(210xxf由dxxdxxfxx12110)(1110xfxf)()(1111xfxff)()()()(11f有界)()(11fnf.)(lim存在即nfn17)()ln(sinlim,,01530ccdtttxaxcbaxbx的值，使、确定常数,sin:00xaxx解,0c而(*))ln(lim0130dtttxbx,),(),()ln(恒正在定义域00113tt01003dtttbb)ln(,则若0b知由(*)18xxdtttxax0301)ln(sinlimxxxax)ln(coslim30120xxaxcoslim等价,,则上极限为若1a1a201xxcxcoslim2119、6012010111xxcebxxxxaxfx),ln()(,,)()(设.)(,,点可导在，使试确定0xxfcba:解)()()(00000fff由122ba,)(,点连续在时当0212xxfba)()(00ff由41c,)(,,,点可导在时当041212xxfcba200402172xxxxxxxf,sin,cos)(）（、的间断点及类型。求)(xf解：),(,,)(11220kkxxxxf的可能间断点为)(00f,22)(00f021;)(的第一类跳跃间断点为xfx0)(limxfx2,sinlim不存在422xx;)(的第二类间断点为xfx2)(limxfx1211xxxxcos)(lim2;)(的第一类可去间断点为xfx1)(lim)(xfkx122112xxxkxcos)(lim)()(1k不存在);()()(112kxfkx的第二类间断点为22.)(,,))((),()(fxxffxf使证明存在上连续，且在、设8证明：,)()(xxfxF作)())(())((xfxffxfF)(xfx)(xFxxfxxf)(,)(0若,)(0xxf若,)(),,(0000xxfx使存在,)())((000xFxfF则00200)()())((xFxFxfF.)(,)()(fFxxf即之间，使与在00023、计算下列各题9,)(lim,)()()(14121211110xxfffx存在且设.),()(处的切线方程在点求曲线211xfy:解xfxffx)()(lim)(1110xxfx2110)(limttfttx2110)(limttft412120)(lim2切线方程)(1221xy24dxdyxy求）（,)arcsin(sin23212242121112111123xxxxxdxdycossin)(sin)arcsin(sin25dxdyexfyxf求,)(ln)()(3)(')(ln)(ln')()(xfexfexxfdxdyxfxf1)(ln)(')(ln')(xfxfxfxexf126)(,xyexyey求）（4两边求导0yxyyeyxeyyy21)()()(xeyeyxeyyyyy3222)(xeyexyyeyyy27ttytxarctan)ln()(21533dxyd求,/)(/21211122ttttdtdxdtdtdxdy34223381211141441tttttdtdxttdtddxyd)(/)(ttdtdxtdtddxyd412222/)(28，、1141022xxy)(ny求11422xxy)(1111234134422xxxx1111111111nnnnnnxnxxnx)(!)()(,)(!)()()()(])()([!)()(111111123nnnnxxny29、11.)(,最大值求已知axxxfa11110:解axaxxaxaxxxaxxxf,,,)(11110111101111分成三个区间将),(,axx0),[),,(],,(aa0030axaxxaxaxxxaxxxf,)()(,)()(,)()()(2222221111011110111100)(,),(xf内在,)(单增xf上最大值在为],()()(0121110xfaaaf0)(,),(xfa内在,)(单减xf上最大值在为),[)()(axfaaaaf121113100)(,),(xfa令内在2ax唯一驻点,)(02af为极小值点2axxyoa2a上最大值在为),()(xfaa1232)()()()(lim)(为正整数且的某领域内连续，在点、设nxxxfxfxxfnxx2120000?)(处是否取得极值在问0xxxf:解02000nxxxxxfxf)()()(lim某领域使存在0x000nxxxfxf)()()(,)(为奇数当n1;)(点不取得极值在0xxf,)(为偶数当n2;)(点取得极小值在0xxf33.)(,,baaababea证明、设013:证明ababaaln)()ln(只要证明)(ln)()ln()(0xaxaxaaxf作01xaaxaaxfln)ln()(单减)(xf00)()(fbfababaaln)()ln(.)(baaaba即34)()()(])([,),(,)(),(2121211110014xftxtfxttxftbaxxxfba有任意实数及、证明对任意内、在:证法一0112121])([)()()(xttxfxftxtf只要证明,21xx不妨设:证明])([)()()(212111xttxfxftxtf)}(])([{]})([)(){(121212111xfxttxftxttxfxft;,等号成立时当21xx,时当21xx35))()(()()()(12112211xxttfxxtft21111xttxx)(22211xxttx)()]()()[)((12121ffxxtt))(())((12121fxxtt210:证法二2101xttxx)(记2000021))((!))(()()(xxfxxxfxfxf))(()()(000xxxfxfxf3