班级姓名学号1第二章导数与微分1..1,102fxxf试按定义求设200200(1)(1)10(1)10'(1)limlim1020limlim(1020)20xxxxfxfxfxxxxxx2.下列各题中均假定0xf存在,按导数定义观察下列极限,指出此极限表示什么,并将答案填在括号内。⑴xxfxxfx000lim(0'()fx);⑵xxfx0lim('(0)f),其中存在;且0,00ff⑶hhxfhxfh000lim(02'()fx).3.求下列函数的导数:⑴yxy,4则34x⑵yxy,32则1323x⑶yxy,1则3212x⑷yxxy,53则115165x4.求曲线.21,3cos程处的切线方程和法线方上点xy3'sin,'()32yxy所以切线方程为13()223yx化简得332(1)03xy班级姓名学号2法线方程为12()233yx化简得323(3)0xy5.讨论函数0001sin2xxxxy在0x处的连续性和可导性.20(0)01limsin0(0)()xfxfx因为有界量乘以无穷小所以函数在0x处连续因为20001sin(0)(0)1limlimlimsin0xxxxfxfxxxxx所以函数在0x处可导.6.已知是否存在?又及求0,00,002fffxxxxxf2'00(0)(0)(0)limlim0hhfhfhfhh'00(0)(0)(0)limlim1hhfhfhfhh''(0)(0)ff'(0)f不存在7..,00sinxfxxxxxf求已知当0x时,'()(sin)'cosfxxx;当0x时,'()()'1fxx;班级姓名学号3当0x时'00(0)(0)(0)limlim1hhfhfhfhh++'00(0)(0)sin(0)limlim1hhfhfhfhh-'(0)1f综上,cos,0'()1,0xxfxx8.求下列函数的导数:(1);54323xxxy(2);1227445xxxy2222222232242222csccot(1)2csc2'(1)2(1)csccot4csc(1)23(3)(3ln)(2ln)(2)'(3ln)(94)ln32(3ln)xxxxxyxxxxxxxxxxxxxxxyxxxxxxxxxx2'364yxx652'20282yxxx(3);3253xxexy(4);1sectan2xxy2'152ln23xxyxe2'2secsectanyxxx(5);log3lg2ln2xxxy(6);7432xxy班级姓名学号4123'ln10ln2yxxx'422yx(7);lnxxy(8);cosln2xxxy21ln'xxxyx221'2lncoscoslnsinyxxxxxxxxx21lnxx22lncoscoslnsinxxxxxxxx(9);1csc22xxy2222csccot(1)2csc2'(1)xxxxxyx2222(1)csccot4csc(1)xxxxxx(10).ln3ln223xxxxy2232223(3)(3ln)(2ln)(2)'(3ln)xxxxxxxxyxx4222(94)ln32(3ln)xxxxxxxx9.已知.,cos21sin4dd求因为1sincossin2dd所以42212222422284dd班级姓名学号510..1轴交点处的切线方程与写出曲线xxxy令0y,得11xx或因为2'1yx,所以11'2,'2xxyy曲线在(1,0)处的切线方程为2(1)yx,即220xy;曲线在(1,0)处的切线方程为2(1)yx,即220xy。11.求下列函数的导数:(1)可分解为:函数452xy4,25yuux其导数y38(25)x(2)函数可分解为:23xey2,3uyeuxy其导数236xxe(3)可分解为:函数22xay22,yuuaxy其导数22xax(4)可分解为:函数xeyarctanarctan,xyuuey其导数21xxee12.写出下列函数的导数(只需写出结果):(1)yxy,cos343sin(43)x(2)yxy,ln21221xx班级姓名学号6(3)yxy,sin22sincosxx(4)yxy,arctan2421xx(5)yxy,tan2222sec()xx(6),logyxxya12221(1)lnxxxa(7)yxy,coslntanx(8)yxy,arcsin2121xx13.求下列函数的导数(要有解题步骤):(1);2arcsin2xy(2);arctanxey(3);lnlnlnxy(4).cossinnxxyn14.设:dxdyyxf的导数可导,求下列函数(1);2xfy(2).cossin22xfxfy22'()dyxfxdx22'(sin)2sincos'(cos)2cossindyfxxxfxxxdx22sin2['(sin)'()]xfxfcoxx班级姓名学号715.求下列函数的导数:(1)22sinsinxxy(2)xy1cosln(3)xey1sin2(4)xxy16.求下列函数的二阶导数:(1)xxyln221'4yxx21''4yx(2)teytsin'sincos(cossin)tttyetetett''(cossin)(sincos)2costttyettettet(3)1ln2xxy2222221212111'1111xxxxyxxxxx班级姓名学号83223221''(1)22(1)xyxxx17.若:22dxydxf阶导数存在,求下列函数的二(1)2xfy(2)xfyln2222'()()''()()['()][()]dyfxdxfxdyfxfxfxdxfx18.求下列函数的n阶导数的一般表达式:(1)xy2sin(2)xxyln19.求下列函数所指定阶的导数:(1),cosxeyx求.4y(2),2sin2xxy求.50y20.求下列方程所确定的隐函数:dxdyy的导数(1)0333axyyx(2)yxey1方程两边关于x求导得:方程两边关于x求导得:2233330dydyxyayaxdxdxyydydyexedxdx所以22223333dyayxayxdxyaxyax所以1yydyedxxe222222222'()2['()2''()]2'()4''()dyxfxdxdyfxxxfxdxfxxfx班级姓名学号921..42,42323232程处的切线方程和法线方在点求曲线aaayx方程两边关于x求导得:113322033dyxydx所以13313dyxydxxy从而切线斜率122(,)441aadykdx,法线斜率2111kk所以切线方程为22()44yaxa,即202xya;法线方程为2244yaxa,即0xy。22..122dxydyxeyy的二阶导数所确定的隐函数求由方程23.用对数求导法求下列函数的导数:dxdy(1)55225xxy(2)54132xxxy班级姓名学号1024.求由参数方程tytx2cossin,所确定的曲线在4t处的切线方程和法线方程.25.:22dxyd的函数的二阶导数求下列参数方程所确定(1)tteyex23(2)tftftytfx,.存在且不为零设tf22233tttdydyedtedxdxedt'()''()'()''()dydyfttftftdttdxdxftdt223244339tttedyedxe221''()dydxft26.注水入深m8上顶直径m8的正圆锥形容器中,其速率为min43m.当水深为m5时,其表面上升的速率为多少?27.求下列函数的微分:⑴;2sinxxy⑵;1ln2xysin2sindyxdxxdx2[ln(1)]dydxsin22cos2xdxxxdx2ln(1)ln(1)xdx班级姓名学号11(sin22cos2)xxxdx2ln(1)(1)1xdxx2ln(1)1xdxx⑶;3cosxeyx⑷.1arcsin2xycos(3)(3)xxdyxdeedcoxx2arcsin1dydxcos(3)sin(3)xxxedxexdx22111(1)dxx(sin(3)cos(3))xexxdx21xdxxx28.将适当的函数填入下列括号内,使等式成立:⑴22;dxCdx⑵2332dxCxdx;⑶sincosdxCxdx⑷cossin;xdCxdx⑸1ln(1);1dxCdxx⑹22;2xxedCedx⑺12;dxCdxx⑻2tan3sec33xdCxdx29.计算三角函数值29cos的近似值。因为cos29cos(301)所以cos29cos30sin30180310.874762218030.计算根式665的近似值。班级姓名学号12因为1166665(65)(641)所以1156661111(65)64(64)222.0052663219231.当x较小时,证明下列近似公式:(利用()(0)'(0)fxffx)(1);是角的弧度值xxxtan(2).1lnxx(tan)'secxx1(ln(1))'1xx0tan0xx0ln(1)0xx0(tan)'1xx0[ln(1)]'1xx所以tanxx所以ln(1)xx