北京邮电大学双语高等数学教学组 2011年第一版
Advanced Mathematics School of Science, BUPT Oct. 2011 1.1
Part A
1. (1) {}1,2,3,4,5,6,7,8BA∪=, {}8AB∩=, {}\1,3,5,7AB=, {}\2,4,6BA=.
(2) {}all parallelogramsBA∪=, {}all rectanglesBA∩=,{}\all parallelograms except rectanglesAB=, \BA=∅.
(3) {}1,2,3,AB∪=, {}2,4,6,AB∩=, {}\1,3,5,AB=, \BA=∅.
2. .{}5
15,9c
i
iA
==∩.
3. {}13BxA∪=<≤ AB∩=∅. 5. (1) 1
,
2⎛⎤
−∞⎜⎥
⎝⎦. (2) ()(),,αβγ∪+∞. (3) 2
2,2
33kkππ
ππ⎡⎤
++
⎢⎥
⎣⎦. (4)
()0,+∞.(5)
()4,2−−.(6)
(]3,2−−.
(7) ()(]1,22,4∪. (8)
2
,1
2⎡⎤
−⎢⎥
⎣⎦. (9) ()0,+∞. (10) [)0,2. (11) ()1,+∞.
6. (1) []1,0−.(2)[]0,1.(3) ()2,21,kkk+∈⎡⎤⎣⎦Zππ
.(4) 1
1,0
2
1
,
2aaxaa⎧
≤≤−<≤
⎪⎪
⎨
∅>⎪
⎪⎩.
7. (1) No. (2) No.(3) No. (4)Yes.(5) No. (6) Yes.(7) No. (8)Yes.(9) No. (10) Yes.
8. (1) Yes.(2) Yes.(3) Yes.
11. ()53,1
3,12
352xx
fxxx
xx−<⎧
⎪
=−≤<⎨
⎪
−≥⎩.
12. (1)3yu=, sinuv=, vw= and 12wx=−.(2)arccosyu=, 2
2x
u−
=.(3)1
y
u=, 1uv=+,arctanwv=, 2wx=.
(4)10yu=, 12ux=+.(5)2yu=,arcsinuv=,2vx=.(6)()ln1yu=+, 1uv=+, 2vx=.(7)2uy=, 3uv=, sinvx=.
13. ()()()3sin2sin2,,fxxxx=−∈−∞+∞φ, ()()()()3sin2,,fxxxx=−∈−∞+∞φ,
()()()3579,233,ffxxxxxxx=−+−+∈−∞+∞.
14. ()()()()1,01/,||1
0,01,||1
1,0,||1xex
xfgxgx
xexfx−<<⎧⎧
⎪⎪
= ==⎨⎨
⎪⎪
>>=
⎩⎩ 北京邮电大学双语高等数学教学组 2011年第一版
Advanced Mathematics School of Science, BUPT Oct. 2011 15. (
)()
()0,1
1,10
1,01x
fxhxx
hxx⎧>
⎪
=+−≤<⎨
⎪
−−≤≤⎩ ()
()20,1
1,11
3
1,1
3x
gxxx
xx⎧
⎪<−
⎪⎪
=−−≤≤⎨
⎪
⎪−>
⎪⎩. 17. (1) ()21,01xyy=−−≤≤.(2) 12yxe−=−.(3) ()1
arcsin,22
32y
xy=−≤≤.(4) ()
()
31
log,01y
xy
y−
=<<. (5) 21
1
1
2y
y
x⎛⎞+
−⎜⎟
−⎝⎠
=[)1,1y∈−.(6)
2,
,12
log,1
2y
xyy
yyy−⎧
⎪
=≤≤⎨
⎪
<<+∞∞<
⎩<
.
Part B
1. (1) ()()()0,0fgxx==, ()()()
22,12gfxxx=−≤≤.
(2) ()()()arcsin1,04
2x
fgxx⎛⎞
=−≤≤⎜⎟
⎝⎠, ()(
)()
21
arcsin1,01
2
1
arcsin1,12
22xx
gfx
x
x⎧
−≤≤
⎪
⎪
=⎨
⎛⎞
⎪−<≤⎜⎟
⎪⎝⎠⎩.
2. ()()23122
,0
1fx
x
xxx−+
−∞<<
−+=.
3. (
)(]
[]11,1,0
1,1,2xx
fx
xx−⎧−+∈−⎪
=⎨
−∈⎪⎩
6. ()1fxx=+.
8. ()22fxx=−. 2
2
2111
24fxxx
xxx⎛⎞⎛⎞
−=−−=+−⎜⎟⎜⎟
⎝⎠⎝⎠.
1.2
Part A
1. (1) No. (2) Yes. (3) Yes,.(4) No.
2. (1) wrong.(2) wrong(3) wrong..
5. wrong
6. wrong 9. (1)1
2. (2)1
3. (3)2. (4)2. (5)
1
3.
(6) 1
e. 北京邮电大学双语高等数学教学组 2011年第一版
Advanced Mathematics School of Science, BUPT Oct. 2011 12. 1
110
1
1100,
,
,limll
lll
mm
mmmnlm
anananaa
lm
bbnbnbnb
lm−
−
−→∞
−⎧<
⎪
++++⎪
==⎨
++++⎪
⎪∞>⎩
.
13. (1) convergent. (2) divergent. (3) convergent.
1.3
Part A
1. (1)>0ε∀, 0X∃>, ()fxAε−< holds for all xX>. (2)0ε∀>, 0∃>δ, ()fxAε−< holds for all
0xxδ−<.
(3)0M∀<, 0∃>δ, ()fxM< holds for all 2xδ−<.
2. (1) wrong. (2) right. (2) right. (2) wrong. (2)wrong.
3. (1) wrong. (2) right. (2) wrong.
5. (1) No. (2) No. (3) No. (4) No. (5) No. (6) No. (7) Yes. (8) No.
6. (1) 3− (2) 1 (3) 1 (4) 3 (5)334
2 (6)cosx (7)sinx− (8)0
7. (1) 12 (2) 1
4 (3) 1− (4) 2
π (5)6e− (6)2e− (7)π (8)2e
8. (1) 1a=− 2b=− (2) 1
2a=
Part B
1. (1) 4
3 (2) 2e− (3)1/2e− (4) 2
eπ
3. some 0ε>, there exists a 0δ>, such that ()fxAε−> holds for all ()
0,xUxδ∈.
4. 2
eπ
1.4
Part A
1. 0ε∀>, there exists a 0δ>, such that ()xαε< holds for all ()
0,xUxδ∈.
2. 0M∀>, there exists a0X>, such that ()fxM> holds for all xX>.
3. (1)wrong. (2) wrong. (3)wrong. (4) wrong. (5) wrong. 北京邮电大学双语高等数学教学组 2011年第一版
Advanced Mathematics School of Science, BUPT Oct. 2011 4. (1)wrong. (2) wrong.
5. (1) x (2) x (3) x(4)4/3x
7. (1) 2 (2) 1
3 (3)1
2−
Part B
1. (2)
1yx=− 2. (1)1
1
2ba± ⎧
⎪
⎨
=±=
⎪
⎩ (2) 3
16a=, 1
2b= 2c=
1.5
Part A
2. Wrong
5. (1) 2x= is a removable discontinuous point or discontinuous point of the first type.
2x=− is an essential discontinuous point or discontinuous point of the second type.
(2) 1x= is an essential discontinuous point.
(3) 0x= is a jump discontinuous point or discontinuous point of first type.
(4) 1x=± are both jump discontinuous point or discontinuous point of the first type.
6. (1) 0x= is an essential discontinuous point or discontinuous point of second type.
(2) 1x= is an essential discontinuous point or discontinuous point of second type.
(3) 0x= is a continuous point.
(4) 0x= is a jump discontinuous or a discontinuous of the first type.
(5) 1x=−21xk=+, (k
+∈N)are essential discontinuous points or discontinuous points of second type.0x= is a jump
discontinuous or a discontinuous point of the first kind..1x= is a continuous point. 7. (1)
2π
(2) 1 (3) 2− (4) 1
2e−
(5) 0
9. (1) 0a= (2) 3a= (3) 2a= 3
2b=−