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北京邮电大学高等数学第一册答案

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

3. some 0ε>, there exists a 0δ>, such that ()fxAε−> holds for all ()

0,xUxδ∈.

4. 2

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)

(2) 1 (3) 2− (4) 1

2e−

(5) 0

9. (1) 0a= (2) 3a= (3) 2a= 3

2b=−

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