高等数学(北大版)习题答案 第二章

习题2.1201.,.,.()2(0)(1),;(2),?(3)lim,?xlOxxmxxxlxxmmxmx设一物质细杆的长为其质量在横截面的分布上可以看作均匀的现取杆的左端点为坐标原点杆所在直线为轴设从左端点到细杆上任一点之间那一段的质量为给自变量一个增量求的相应增量求比值问它的物理意义是什么求极限问它的物理意义是什么2222222000(1)2()22(2)22(2).2(2)(2)2(2).(3)limlim2(2)4.limxxxmxxxxxxxxxxxmxxxmxxxxxxxxmmxxxxxx是到那段细杆的平均线密度.是细杆在点的线密度.解333032233222000002.,:(1);(2)2,0;(3)sin5.()(1)lim(33)limlim(33)3.2()2(2)lim2lim(2limxxxxxxyaxypxpyxaxxaxyxxxxxxxxaaxxxxaxxpxxpxxxxypxxxp根据定义求下列函数的导函数解0000000)()2lim()()212lim.25(2)52cossinsin5()sin522(3)limlim55(2)552cossinsin5(2)2222lim5limcoslim5522xxxxxxxxxxxxxpxxxxxxxxppxxxxxxxxxxyxxxxxxxxx5cos5.2xx00223.()(,()):(1)2,(0,1);(2)2,(3,11).(1)2ln2,(0)ln2,1ln2(-0),(ln2)1.(2)2,(3)6,:116(3).4.2(0)(,)(0,0)xxyfxMxfxyMyxByyyxyxyxyyxypxpMxyxy求下列曲线在指定点处的切线方程切线方程切线方程试求抛物线上任一点处的切线斜率解,0,.2pFx,并证明:从抛物线的焦点发射光线时其反射线一定平行于轴200022222222,,().22(),.,2222,.222,.pppypxyMPMNYyXxyypxpyxNXyXxXxxyppppFNxFMxyxpxpppxpxxxFNFNMFMNMPQxPMQFNMFMN过点的切线方程：切线与轴交点(,0),故过作平行于轴则证2005.2341,.224,1,6,4112564(1),42.:6(1),.444yxxyxyxxykyxyxyxyx曲线上哪一点的切线与直线平行并求曲线在该点的切线和法线方程切线方程:法线方程解323226.,,;(),,,(1)():(2)();(3)().()lim()lim,lim()limrRrRrRrRrgrGMrrRRgrRMGGMrRrgrrgrgrrGMrGMrRgrgrRRGMgrr离地球中心处的重力加速度是的函数其表达式为其中是地球的半径是地球的质量是引力常数.问是否为的连续函数作的草图是否是的可导函数明显地时连续.解,2lim(),()rRGMgrgrrRR在连续.(2)33(3)()2(),()(),().rRgrGMGMgRgRgRgrrRRR时可导.在不可导227.(),:(1,3)(),(0)3,(2)1.3(),()2.34111113,,3(),()3.2222PxyPxPPabcPxaxbxcPxaxbbabbacabPxxx求二次函数已知点在曲线上且解3222222222228.:(1)87,241.(2)(53)(62),5(62)12(53)903610.(3)(1)(1)tan(1)tan,(2)tan(1)sec.9(92)(56)5(9)51254(4),56(56)yxxyxyxxyxxxxxyxxxxxyxxxxxxxxxxxxyyxx求下列函数的导函数22.(56)122(5)1(1),.11(1)xxyxyxxx23322222226(6)(1),.1(1)1(21)(1)1(7),.(8)10,1010ln1010(1ln10).sincossin(9)cos,cossin.(10)sin,sincos(sxxxxxxxxxxxxxxyxyxxxxxeexxxxyyeeeyxyxxxxxxyxxyxxxxxyexyexexeincos).xx00000001001100009.:()()()(),()0().()()(1)(2).()()(),()0()()()()()()(()()())()(),(mkkkkkPxPxxxgxgxxPxmxPxkxPxkkPxxxgxgxPxkxxgxxxgxxxkgxxxgxxxhxhx定义若多项式可表为则称是的重根今若已知是的重根,证明是的重根证00)()0,()(1)kgxxPxk由定义是的重根.000000010.()(,),()(),().()(0),(0)0.()(0)()(0)()(0)(0)limlimlim(0),(0)0.()()11.(),lim22xxxxfxaafxfxfxfxfffxffxffxffffxxxfxxfxxfxxfx若在中有定义且满足则称为偶函数设是偶函数,且存在试证明设在处可导证明证=000000000000000000000().()()()()()()1limlim22()()()()1lim2()()()()11limlim[()22xxxxxxfxxfxxfxxfxfxxfxxxxfxxfxfxxfxxxfxxfxfxxfxfxxx证002()]().12.,(0/2)()((),()):.fxfxyxttPtxtytOPxtt一质点沿曲线运动且已知时刻时质点所在位置满足直线与轴的夹角恰为求时刻时质点的位置速度及加速度.222222422222()()()tan,()tan,()()(tan,tan),()(sec,2tansec),()(2sectan,2sec4tansec)2sec(sec,2tan).ytxtxttyttxtxtttvttttvttttttttt位置解1/1/1/1/1/000013.,0()10,00.1111(0)limlim1,(0)limlim0.1114.()||(),()()0.().()limxxxxxxxxxxxxfxexxxxeeffxexefxxaxxxaafxxafa求函数在的左右导数设其中在处连续且证明在不可导-+解证()()()()(),()lim()().axaaxxxaxaaafaxaxa+-fy=x2习题2.22222222222222221.,:sin(1)(cos)sin,.(cos)sin.2111(2)[ln(1)],.[ln(1)](1).111(3)112,.111121121xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx下列各题的计算是否正确指出错误并加以改正错错错332222222()2223.111(4)ln|2sin|(14sin)cos,.2sin1ln|2sin|(14sincos).2sin2.(())()|.()1.(1)(),(0),(),(sin);(2)(),(sin);(3)ugxxxxxxxxxxxxxxxxxfgxfufxxfxffxfxddfxfxdxdx错记现设求求2222223(())(())?.(1)()2,(0)0,()2,(sin)2sin.(2)()()224.(sin)(sin)(sin)2sincossin2.(3)(())(()),(())(())().fgxfgxfxxffxxfxxdfxfxxxxxdxdfxfxxxxxdxfgxfgxfgxfgxgx与是否相同指出两者的关系与不同解222233312232323.2236(1),.111(2)sec,(cos)(cos)(cos)(cos)(sin)tansec.(3)sin3cos5,3cos35sin5.(4)sincos3,3sincoscos33sinsin33sinxxyyxxxyxyxxxxxxxyxxyxxyxxyxxxxx求下列函数的导函数:2(coscos3sinsin3)3sincos4.xxxxxxx22222222222232222222241sin2sincoscos(1sin)(sin)2(5),coscossin2cos2(1sin)(sin).cos1(6)tantan,tansecsec13tansectantan(sec1)tan.(7)sin,saxaxxxxxxxxyyxxxxxxxxyxxxyxxxxxxxxxyebxyae524222422222incos(sincos).(8)cos1,5cos1(sin1)15cos1sin1.111(9)lntan,sec24224tan2411112tancos2sin24242axaxbxbebxeabxbbxxyxyxxxxxxxxxyyxxxx222cos42411sec.cossin()211()()1(10)ln(0,),.22()xxxxxaxaxaxayaxayaxaaxaxaxa2222222222222224.:111(1)arcsin(0),.111111(2)arctan(0),.1(3)arccos(||1),2arccos.11111(4)arctan,.111(5)ar22xyayaaaxxaxyayaaaaaxxaxyxxxyxxxyyxxxxxayax求下列函数的导函数csin(0),xaa22222222222222222222222222222222222222222121122211.2(6)ln(0)221112221.2222(7)arcsin,1xxayaxaaxxaxaaxaxa