则定义域为?x|x?0或x???1?? 2?⒊在半径为R的半圆内内接一梯形，梯形的一个底边与半圆的直径重合，另一底边的两个端点在半圆上，试将梯形的面积表示成其高的函数． 解： D
A R O h E
B C
设梯形ABCD即为题中要求的梯形，设高为h，即OE=h，下底CD＝2R 直角三角形AOE中，利用勾股定理得
AE?OA2?OE2?R2?h2 则上底＝2AE?2R2?h2 h?2R?2R2?h2?hR?R2?h2 2sin3x⒋求lim．
x?0sin2xsin3xsin3x?3xsin3x3133解：lim?lim3x?lim3x?＝??
x?0sin2xx?0sin2xx?0sin2x2122?2x2x2xx2?1⒌求lim．
x??1sin(x?1)故S?????x2?1(x?1)(x?1)x?1?1?1?lim?lim???2 解：limx??1sin(x?1)x??1sin(x?1)x??1sin(x?1)1x?1tan3x⒍求lim．
x?0xtan3xsin3x1sin3x11?lim??lim??3?1??3?3 解：limx?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]?1x?1xe?1x?4xx?x解：lim( )?lim()?lim?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】形考作业2答案：
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
⒊设f(x)?e，则limx11e D. e 24 ⒋设f(x)?x(x?1)(x?2)?(x?99)，则f?(0)?（D ）．
C.
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)2lnx5． ⒉设f(ex)?e2x?5ex，则??xxdx1 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?)
4224 ⒌设y?x2x，则y??2x2x(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
222⑵y?cotx?xlnx y???cscx?x?2xlnx
xx3212xlnx?xx2⑶y? y??
ln2xlnxcosx?2xx(?sinx?2xln2)?3(coxs?2x)⑷y? y?? 34xx
1sinx(?2x)?(lnx?x2)cosxlnx?xx⑸y? y?? 2sinxsinx2
4⑹y?x?sinxlnx y??4x?3sinx?cosxlnx x
sinx?x23x(cosx?2x)?(sinx?x2)3xln3⑺y? y?? x32x3
ex1x??⑻y?etanx?lnx y??etan 2cosxxxx
⒉求下列函数的导数y?： ⑴y?e1?x2
y??e
1?x2x1?x2
⑵y?lncosx3
?sinx32y??3x??3x2tanx3 3cosx ⑶y?78xxx
?17y?x y??x8
8
⑷y?3x?x
1?2?111y??(x?x2)3(1?x2)
32
⑸y?cose
2xy???exsin(2ex)
⑹
y?cosex2x2
x2y???2xesine
⑺y?sinxcosnx
n
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?xexex?eex
y??xexxex(?elnx)?eexx
⒊在下列方程中，y?y(x)是由方程确定的函数，求y?： ⑴ycosx?e2y
y?cosx?ysinx?2e2yy?
ysinxy?? 2ycosx?2e
⑵y?cosylnx
1y??siny.y?lnx?cosy.
xcosyy??
x(1?sinylnx)
x2⑶2xsiny?
y