解：limtan3xsin3x1sin3x11?lim?lim??3?1??3?3
x?0x?0xxcos3xx?03xcos3x11?x2?1⒎求lim．
x?0sinx1?x2?1(1?x2?1)(1?x2?1)x2?lim?lim解：lim2x?0x?0x?0sinx(1?x?1)sinx(1?x2?1)sinx ?limx?0
x(1?x2?1)sinxx?0?0
?1?1??1⒏求lim(x??x?1x)． x?3111(1?)x[(1?)?x]?1?1x?1xex?4x?lim?x解：lim( )?lim(x)?lim??ex3x??x?3x??x??33xx??e11?(1?)[(1?)3]3xxx3x2?6x?8⒐求lim2．
x?4x?5x?4x2?6x?8?x?4??x?2??limx?2?4?2?2
解：lim2?limx?4x?5x?4x?4?x?4??x?1?x?4x?14?131?⒑设函数
?(x?2)2,x?1?f(x)??x,?1?x?1
?x?1,x??1?讨论f(x)的连续性，并写出其连续区间． 解：分别对分段点x??1,x?1处讨论连续性 （1）
x??1?x??1?limf?x??limx??1x??1?x??1?limf?x??lim?x?1???1?1?0x??1?x??1?
所以limf?x??limf?x?，即f?x?在x??1处不连续 （2）
x?1?x?1?limf?x??lim?x?2???1?2??1x?1?x?1?22limf?x??limx?1f?1??1
所以limf?x??limf?x??f?1?即f?x?在x?1处连续
x?1?x?1?由（1）（2）得f?x?在除点x??1外均连续 故f?x?的连续区间为???,?1???1,???
《高等数学基础》第二次作业
第3章 导数与微分
（一）单项选择题
f(x)f(x)?（C ）． 存在，则limx?0x?0xx A. f(0) B. f?(0) C. f?(x) D. 0cvx
f(x0?2h)?f(x0)?（D ）． ⒉设f(x)在x0可导，则limh?02h A. ?2f?(x0) B. f?(x0) C. 2f?(x0) D. ?f?(x0)
⒈设f(0)?0且极限limf(1??x)?f(1)?（A ）．
?x?0?x A. e B. 2e
11 C. e D. e
24 ⒋设f(x)?x(x?1)(x?2)?(x?99)，则f?(0)?（D ）．
⒊设f(x)?e，则limx A. 99 B. ?99 C. 99! D. ?99! ⒌下列结论中正确的是（ C ）．
A. 若f(x)在点x0有极限，则在点x0可导． B. 若f(x)在点x0连续，则在点x0可导． C. 若f(x)在点x0可导，则在点x0有极限． D. 若f(x)在点x0有极限，则在点x0连续．
（二）填空题
1?2xsin,x?0? ⒈设函数f(x)??，则f?(0)? 0 ． x?x?0?0,df(lnx)2lnxx2xx5． ⒉设f(e)?e?5e，则??dxxx1 ⒊曲线f(x)?x?1在(1,2)处的切线斜率是k?
2π22? ⒋曲线f(x)?sinx在(,1)处的切线方程是y?x?(1?)
42242x2x ⒌设y?x，则y??2x(1?lnx)
1 ⒍设y?xlnx，则y???
x（三）计算题
⒈求下列函数的导数y?：
3x⑴y?(xx?3)e y??(x?3)e?x2e
2xx321⑵y?cotx?x2lnx y???csc2x?x?2xlnx
2xlnx?xx2⑶y? y?? 2lnxlnxcosx?2xx(?sinx?2xln2)?3(coxs?2x)⑷y? y?? 3xx4
1sinx(?2x)?(lnx?x2)cosx2lnx?xx⑸y? y?? sinxsin2x
⑹y?x4?sinxlnx y??4x?
3sinx?cosxlnx xsinx?x23x(cosx?2x)?(sinx?x2)3xln3⑺y? y?? x32x3
ex1x??⑻y?etanx?lnx y??etan
co2sxxxx
⒉求下列函数的导数y?： ⑴y?e1?x2
y??e
1?x2x1?x2
⑵y?lncosx
3?sinx32y??3x??3x2tanx3 3cosx ⑶y?xxx
7y?x y??x8
8
⑷y?3x?78?1x
1?2?111y??(x?x2)3(1?x2)
32
⑸y?cos2ex
y???exsin(2ex)
⑹
y?cosex2x2
x2y???2xesine
⑺y?sinnxcosnx
y??nsinn?1xcosxcosnx?nsinnxsin(nx)
⑻
y?5sinx2
y??2xln5cosx5 ⑼
2sinx2
y?esin2x
y??sin2xe
⑽
sin2x
y?x?ex2x2x2
y??x(x?2xlnx)?2xe ⑾
x2
y?xxex?eex
y??xeexxex(?elnx)?eexx
⒊在下列方程中，是由方程确定的函数，求：
⑴ycosx?e2y
y?cosx?ysinx?2e2yy?
y??ysinxcosx?2e2y
⑵y?cosylnx
y??siny.y?lnx?cosy.1x
y??cosyx(1?sinylnx)
⑶2xsiny?x2y
2xcosy.y??2siny?2yx?x2y?y2y??2xy?2ysiny2xy2cosy?x2
⑷y?x?lny
y??y?y?1 y??yy?1
⑸lnx?ey?y2 1x?eyy??2yy? y??1x(2y?ey)
⑹y2?1?exsiny
2yy??excosy.y??siny.ex
y??exsiny2y?excosy
y?(2xcosy?x22yxy2)?y2?2siny
2018年电大高等数学基础形成性考核册及复习题考试题资料附答案
