1x1lim(3)x0
x
limx0
(1x1)(1x1)
x(1x1)
limx0
xx(1x1)
2
limx0
11x
52x42x
3
11132
(4)
x3x5lim2x
3x2x4
sin3x
31
xlim
x2
3
x
sin3x
5x
lim
x
(5)
sin5xx03xsin5x53sin3x5x3limlimx053xx0sin5x5
0
lim
limx2
(6)
x
2
4
sin(x2)
limx2
(x2)sin(x2)
(x2)
4
x2limlim(x2)x2x2
sin(x2)
xsin
2.设函数
14
1
f(x)
xa,sinxx
b,xxx
00,0
问:(1)当
a,b为何值时,f(x)在x
0处有极限存在?0处连续.
(2)当
a,b为何值时,f(x)在x
3.计算下列函数的导数或微分:
(1)
yx
2
2
x
log2x
2
2
,求
y
y
(x2
)(2x
)(log2x)(22
)
2x2x
ln2
1xln2
(2)
y
axbcx
d
,求
y
y
(axb)(cxd)(axb)(cxd)
(cxd)2
a(cxd)c(axb)
adbc
(cxd)
2
(cxd)
2
(3)
y
1,求
3x
5
y
1
3
y(3x5)
2
,y
13
2
(3x5)
2
(3x5)
32
(3x5)
2
yxxe
x
,求
y
y
(x)(xex
)
12x
(xe
x
x(ex
))
1(x1)e
x
2x
(5)
y
eax
sinbx,求dy
y(eax
)sinbx
eax
(sinbx)eax(ax)sinbxeax
cosbx(bx)
aeax
sinbxbeax
cosbxeax
(asinbx
bcosbx)
dyeax(asinbxbcosbx)dx
4)
(1
(6)
y
1
ex
x
xx,求dy
32
yex,y
1
1e()
x
x
1
3x2
12
1x2ex
1
3x2
12
dy(y
1x2ex
e
3
x)dx2
x2
,求
12
(7)
cosx
dy
x2
ysinx(x)e(x)
2
12x
sinx2xe
x2
dy(
12x
n
sinx2xe
x2
)dx
(8)
y
sinxnsin
n1
sinnx,求y
cosnx(nx)
nsin
n1
y
(9)
x(sinx)xcosxncosnx
y
ln(x
1xx
111
x
1x
1x2),求y
(x(1
1
3
y
x
2
1x
x)
x
x
2
2
1111
,求
x
2
1
(121
x)x
2
2
2
)
x
2
(10)
y2
cot
1x
2
2xx
y
cot
1y2x
ln2(cot1
11
x)
(x2
x6
2)
2
cot
1x
ln2(csc
2
1
11
32
15
x)(x)(2
x6
x6
)
ln2
cot
1x
x
2
2csc21
13
5216x2x6
x
4.下列各方程中
y
是
x的隐函数,试求y
或
dy
(1)
x
2
y
2
xy3x1,求dy2x2yyyxy
30
(2y
x)y
y2x3
y
y2x3dy
y2x3
2yx2yx
dx
(2)
sin(xy)e
xy
4x,求ycos(xy)(1
y)e
xy
(yxy)
4
y
4cos(xy)ye
xy
cos(xy)xe
xy
5.求下列函数的二阶导数:
(1)
y
ln(1x2
),求y2x
2
y
,y
2(1x2
)
2x2x2(1x)1
x
2
(1
x2)
2
(1x2)
2
(2)
y
1x,求
y
及
y(1)
x
1
1
3
1
5
3
yx2
x2
,y
12
x
2
12
x
2
,y
34
x
2
14
x
2
,y(1)1
1.计算下列不定积分
(1)
3e
x
x
dxx)x
2
3edx
x
dx1x
1ln312x
3e
3
x
c
43
3
(2)
(1
(x)dx
2
2xx
2
25
5
x2
c
(3)
x
2
x
(4)
4dx21
dx
2
(x1
2)dx1
12
x
2
2xc12
ln|12x|c
322
12xx2
212x12
2
d(12x)
2
2
(5)
xdxxd(2x)
13
(2x)c
(6)
sinx
x
dx
2sinxdx
x
2xd(cos)
2x2
4sin1)xxln(x
1dxcx2
c
2cosx
x2
C
x
2cosdx
2
(7)
xsindx
2
x
2xcos
2xcos
ln(x
1)dx
(8)
xln(x1)x
xd(ln(x1))xln(x
1)
(11)
1xx1)dx
xln(xxln(x
2.计算下列定积分
1)1)(x1)ln(xc