高等數(shù)學(xué)下復(fù)旦大學(xué)出版習(xí)題八

1、173習(xí)題八習(xí)題八1.判斷下列平面點(diǎn)集哪些是開集、閉集、區(qū)域、有界集、無界集？并分別指出它們的聚點(diǎn)集和邊界：(1)(xy)|x≠0(2)(xy)|1≤x2y24(3)(xy)|yx2(4)(xy)|(x1)2y2≤1∪(xy)|(x1)2y2≤1.解：(1)開集、無界集，聚點(diǎn)集：R2，邊界：(xy)|x=0.(2)既非開集又非閉集，有界集聚點(diǎn)集：(xy)|1≤x2y2≤4，邊界：(xy)|x2y2=1∪(xy)|x2y2=4.(3)開集

2、、區(qū)域、無界集，聚點(diǎn)集：(xy)|y≤x2，邊界：(xy)|y=x2.(4)閉集、有界集，聚點(diǎn)集即是其本身，邊界：(xy)|(x1)2y2=1∪(xy)|(x1)2y2=1.2.已知f(xy)=x2y2xytan試求.xy()ftxty解：222()()()tan().txftxtytxtytxtytfxyty?????3.已知試求()wuvfuvwuw???().fxyxyxy??解：f(xyxyxy)=(xy)xy(xy)xyxy=

3、(xy)xy(xy)2x.4.求下列各函數(shù)的定義域：2(1)ln(21)zyx???11(2)zxyxy????2224(3)ln(1)xyzxy????111(4)uxyz???(5)zxy??22(6)ln()1xzyxxy?????22(7)arccos.zuxy??解：2(1)()|210.Dxyyx????17533333333sin()0(2)00xyxyzxyxy????????????(3)2222222220(2)()

4、00xyxyzxyxyxy????????????解：(1)由于3333333322223333sin()sin()sin()0()xyxyxyxyyxxyxyxyxy?????????????又，且00lim()0xyyx????3333000sin()sinlimlim1xuyxyuxyu???????故.00lim0(00)xyzz????故函數(shù)在O(00)處連續(xù).(2)000sinlimlim1(00)0xuyuzzu?????

5、??故O(00)是z的間斷點(diǎn).(3)若P(xy)沿直線y=x趨于(0，0)點(diǎn)，則2222000limlim10xxyxxxzxx?????????若點(diǎn)P(xy)沿直線y=x趨于(0，0)點(diǎn)，則22222220000()limlimlim0()44xxxyxxxxzxxxx??????????????故不存在.故函數(shù)z在O(0，0)處不連續(xù).00limxyz??7.指出下列函數(shù)在向外間斷：(1)f(xy)=(2)f(xy)=233xyxy