历年试题及答案
2、求lim
(1 x) (1 2x)x 0sinx
1
x
1x12x
.
1x
12x
解: lim
00
(1 x) (1 2x)x 0sinx
1
2x
lim
(1 x) (1 2x)x 0x
11
1 ln(1 x) 1ln(1 2x) x2x
lim (1 x) (1 2x) x(2x 1) 2x2 2x 0x(x 1)x 11
2x (2x 1)ln(1 2x) x (x 1)ln(1 x)2x
lim (1 x)x (1 2x) 22x 0x(x 1)2x(2x 1) 00
1
x (x 1)ln(1 x) 2x (2x 1)ln(1 2x) 2x
lim(1 x) lim(1 2x) x 0 22x 0x(x 1)2x(2x 1)
1
x
elim
x 0
x (x 1)ln(1 x)2x (2x 1)ln(1 2x)
elim 22x 0x2x
ln(1 x) 2ln(1 2x)
elim
x 0x 02x4xee e .
22 elim
2007(x p)
3、求p的值,使 (x p)edx 0.
ab
2
解:
b
a
(x p)
2007
e
(x p)2
dx
t x p
b p
a p
t2007etdt
2
被积函数是奇函数, 要积分为零, 当且仅当积分区间对称,即: a p b p, 解得: p 4、计算
a b
. 2
a
dx e
b
max{b2x2,a2y2}
dy,(a 0,b 0).
dy emax{bx
D
22
22
解:
a
dx emax{bx
22
b
22
,a2y2},a2y2}
d , 其中D如右图
emax{bx
D1
22
,a2y2}
d
emax{bx
D2
22
,a2y2}
d eayd ebxd
D1
D2
bxa0
dy
b
ayb0
e
a2y2
dx dx ebxdy
a
22
aba2y2bab2x2
yedy xedx 00ba