??????(?????)-????????

admin

?????????(?????)-?????16

1. ???????????????????????????????????????

(1) ?D?{(x,y)|xy?0}. (2) ?D?{(x,y)|y?x2}. (3) ?D?{(x,y)|y?sin

1x

,x?0}.

(4) ?D?{(x,y)|x2?y2?1?y?0,1?x?2}.

2. ??????????????????{Pn}?E,Pn?P0??P0?E???. 3. f(x,y)?x2?y2?xytg

xy

,?f(tx,ty).

4??F(x,y)?lnxlny,??:?u?0,v?0,?

5. ??????????,????????,?????????:

(1)

f(x,y)?

6.??????

:

(1)

(x,y)?(0,0)

(2) f(x,y)?ln(y?x).

lim

xy

2

222

x?y

;(2)

1x?y

2

2

(x,y)?(0,0lim

22sin(x?y)x?y

2

22

2

(3)

(x,y)?(0,0)

lim(x?y)sin; (4)

(x,y)?(0,0)

lim.

7. ?????????0?0??????????? (1) f(x,y)?

y

2

2

2

x?y

(2) f(x,y)?(x?y)sin

1x

sin

1y

(3) f(x,y)?

xy

2

2

22

2

xy?(x?y)x?y

23

3

(4) f(x,y)?

x?y

1x

23

(5) f(x,y)?ysin

xy

32

(6) f(x,y)?

x?y

(7) f(x,y)?8.??:?(1)

e?e

xy

sinxy

(x,y)?(a,b)

limf(x,y)?A;(2)y?b?????,??

limf(x,y)??(y),

x?a

?limlimf(x,y)?A.

y?bx?a

9. ?????????:

(x,y)?(0,0)

lim

xyx?y

2

2

2

?0.

10. ???????????????????

?sinxy?,y?0

11.??????????: (1) f(x,y)??y.

??0, y?0

22

x?y?0

(2) f(x,y)?

x2?y2?0?

(3) f(x,y)?

?

0,x????

.

y,x????

x?22

,x?y?0?22p

,(p?0), 12???????????:f(x,y)??(x?y)

?0,x2?y2?0?

13. ?f(x,y)???????S?[a,b]?[c,d].?f?y?[c,d]?????,?x?[a,b](???y)?????,??f?S?????.

14?????D?R2??????f?D????????f????????f(D)???????16.8?,???????

15. ?f(x,y)???G?R??x??,?y????????:

|f(x,y?)?f(x,y??)|?L|y??y??|,

2

??(x,y?),(x,y??)?G,L???.??f?G?????.

16. ??????(x)?[a,b]???,?f(x,y)??(x),(x,y)?D?[a,b]?(??,??),??f?D?????????????

?????????(?????)-?????1

1.?a????,x????.??:

(1) a?x????; (2)?a?0?, ax????.

2. ?x?0,??:x?1

x?2,???????????.

3.?????????????????????

(1) S?{x|x2?2};

(2) S?{x|x?n!,n?N}.

(3) S?{x|x?(0,1)?????}.

(4) S?{x|1?1

2n,n?N?}.

4??A,B???????????

A?B?{z|z?x?y,x?A,y?B}.

????1?sup(A?B)?supA?supB; ?2?inf(A?B)?inf(A)?inf(B).

1

x5????????????x??6.

6??S?????. ??:infS???S???minS.

7. ???f(x)??2?x,x?0,?: x2,x?0

(1) f(?3),f(0),f(1);(2) f(?x)?f(0),f(??x)?f(0)(?x?0).

8. ???????????????????:

(1) y?(arcsinx), (2)

y?lg(1?

9. ??????,??y?

10?????f(x)?1

x222ax?bcx?d?????????. ?(0,1)??????.

11. ???f???[?a,a]?.??:(3) f???????????????.

12?????1? limn!

nnn???0.

?0. ?2?limsinn???n

(3) limn

ann???0(a?1).

?4???liman?a,?liman?k?a. n??n??

14.????liman?a,?lim|an|?|a|,???????? n??n??

15???1-2?????[0?1]??????f1(x)?f2(x)???????

16????

?2?x,x?0f(x)??x, ?2,x?0

??(1)f(?3),f(0),f(1);(2)f(?x)?f(0),f(??x)?f(0)(?x?0).

17???f(x)?1

x2?(0,1)???????

18??????????????????y?3x?1?(??,??)??????

19???????[?a,a]?????

?1?F(x)?f(x)?f(?x),x?[?a,a]?????

?2?G(x)?f(x)?f(?x),x?[?a,a]?????

?3?f???????????????

20. ????y?[x]??:

(1) ???????????,?5?????????1?.?????????y?

??????x???????(??????????30~50?);

(2) ??x????????y,??x?y???????.

?????????(?????)-?????9

1????????????,???????{?i},????????????????,????????:

(1) ?

110exdx, (2) ?badx(0?a?b). 2x2. ??????? ex?e?x

dx; (1)?02

?

(2)

(3)?30tan2xdx;

dx;

?09

(4)?0 4

(5)e

1

e1(lnx)2dx. x

3.????????: (1)lim1(1?23???n3); 4n??n

(2)lim?n??

?3?lim?111????. 222?(n?n)??(n?1)(n?2)1?2?n?1(sin?sin???sin?). n??nnnn

4?????f(x)?[a,b]????[?,?]?[a,b],?f(x)?[?,?]????

5??f(x),g(x)???[a,b]??????.??:???[a,b]??????f(x)?g(x),??f(x)?[a,b]????,g(x)?[a,b]???,??b

af(x)dx??g(x)dx. a

x??x??x?,x????b6.??:?f(x)???????,?supf(x)?inff(x)?sup|f(x?)?f(x??)|.

nb7.??:?f(x),g(x)??[a,b]???,?lim||T||?0?f(?i)g(?i)?xi??f(x)g(x)dx.

i?1a

8.???????: 1?

x2?10exdx?e. 2?:??f(x)?e?[0,1]???,?????????:

?,??e?e0?10exdx?e?,????[0,1].???f(x)?ex?[0,1]??222?2?e,?1??edx?e. 011x2

1sinx?x?dx?.(???,????1??dx?) 9???:1??00sinxx221

10

??????4e

e?6. 11??f(x)?[a,b]???,?f(x)????,??

12.?f(x)?g(x)??[a,b]???,??: ?ba(f(x))2dx?0.

M(x)?max{f(x),g(x)},m(x)?min{f(x),g(x)} x?[a,b]x?[a,b]

??[a,b]???.

13.??:?f(x),g(x)??[a,b]???,?lim

||T||?0?f(?i)g(?i)?xi??f(x)g(x)dx. i?1anb

14.?f(t)?????,u(x),v(x)??????,??????f?u?f?v,??:

dv(x)f(t)dt?f(v(x))v?(x)?f(u(x))u?(x). ?u(x)dx

15?????

(1)

(2)?a0x(a?0). ?11

(x2?x?1)3

20dx.

(3) ?0

a

01.

. (4) ?

?x?

(5)

20cos?d?. sin??cos?

a?pp16. ?f?(??,??)??p????????.???????a,??

17.?f?????.??:

???af(x)dx??f(x)dx. 0

(1)

(2) ?20f(sinx)dx??2f(cosx)dx. 0??

0xf(sinx)dx??2??

0f(sinx)dx.


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