高等数学下 复旦大学出版 习题八

习题八

1. 判断下列平面点集哪些是开集、闭集、区域、有界集、无界集？并分别指出它们的聚点集和边界： (1) {(x,y)|x≠0};

(2) {(x,y)|1≤x2+y2<4}; (3) {(x,y)|y

(4) {(x,y)|(x-1)2+y2≤1}∪{(x,y)|(x+1)2+y2≤1}.

解：(1)开集、无界集，聚点集：R2，边界：{(x,y)|x=0}. (2)既非开集又非闭集，有界集, 聚点集：{(x,y)|1≤x2+y2≤4}，

边界：{(x,y)|x2+y2=1}∪{(x,y)| x2+y2=4}. (3)开集、区域、无界集， 聚点集：{(x,y)|y≤x2}， 边界：{(x,y)| y=x2}.

(4)闭集、有界集，聚点集即是其本身，

边界：{(x,y)|(x-1)2+y2=1}∪{(x,y)|(x+1)2+y2=1}. 2. 已知f(x,y)=x2+y2-xytan

xy,试求f(tx,ty). 解：f(tx,ty)?(tx)2?(ty)2?tx?tytantxty?t2f(x,y). 3. 已知f(u,v,w)?uw?wu?v,试求f(x?y,x?y,xy). 解：f(x+y, x-y, xy) =(x+y)xy+(xy)x+y+x-y =(x+y)xy+(xy)2x. 4. 求下列各函数的定义域：

(1)z?ln(y2?2x?1);

(2)z?1x?y?1x?y; 4x?y2(3)z?ln(1?x2?y2); (4)u?1x?1y?1z; (5)z?x?y; (6)z?ln(y?x)?x1?x2?y2;(7)u?arccoszx2?y2.

解：(1)D?{(x,y)|y2?2x?1?0}.

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(2)D?{(x,y)|x?y?0,x?y?0}.

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

(4)D?{(x,y,z)|x?0,y?0,z?0}.

(5)D?{(x,y)|x?0,y?0,x2?y}. (6)D?{(x,y)|y?x?0,x?0,x2?y2?1}. (7)D?{(x,y,z)|x2?y2?0,x2?y2?z2?0}.5. 求下列各极限：

(1)limln(x?ey)xy??10x2?y2; (3)lim2?xy?4x;

y??00xy(5)limsinxyx;y??00x

解：(1)原式=ln(1?e0)12?02?ln2.

(2)原式=+∞. (3)原式=lim4?xy?4xy??00xy(2?xy?4)??14.

(4)原式=limxy(xy?1?1)xy??00xy?1?1?2.

(5)原式=limsinxyx?y?1?0?0.

y??00xy1(6)原式=lim2(x2?y2)2x2?y2x22x2??lim(x2?y2)?0. y??0(x?y)ey20xy??002e6. 判断下列函数在原点O(0，0)处是否连续：

?sin(x3?y3)(1)z???x2?y2,x2?y2?0,

??0,x2?y2?0;

(2)lim1xy??00x2?y2;

(4)limxyxy??00xy?1?1;

(6)lim1?cos(x2?y2)x?0(x2?y2)ex2?y2.y?0

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?sin(x3?y3),x3?y3?0,?33(2)z??x?y

?0,x3?y3?0;??x2y222,x?y?0,?222(3) (2)z??xy?(x?y)

22?0,x?y?0;?sin(x3?y3)x3?y3sin(x3?y3)sin(x3?y3)解：(1)由于0? ?2?(x?y)?2223333x?yx?yx?yx?ysin(x3?y3)sinu?lim?1, 又lim(x?y)?0，且lim33x?0x?0u?0x?yuy?0y?0故limz?0?z(0,0).

x?0y?0故函数在O(0,0)处连续. (2)limz?limx?0y?0sinu?1?z(0,0)?0

u?0u故O(0,0)是z的间断点.

(3)若P(x,y) 沿直线y=x趋于(0，0)点，则

x2?x2limz?lim22?1, x?0x?0x?x?0y?x?0若点P(x,y) 沿直线y=-x趋于(0，0)点，则

x2(?x)2x2limz?lim2?lim2?0 x?0x?0x?(?x)2?4x2x?0x?4y??x?0故limz不存在.故函数z在O(0，0)处不连续.

x?0y?07. 指出下列函数在向外间断：

x?y2(1) f(x,y)=3;

x?y3y2?2x(2) f(x,y)=2;

y?2x?x?x2?2ey,(4)f(x,y)=?y??0,2(3) f(x,y)=ln(1－x2－y2);

y?0,y?0.

解：(1)因为当y=-x时，函数无定义，所以函数在直线y=-x上的所有点处间断，而在其余点处均连续.

(2)因为当y2=2x时，函数无定义，所以函数在抛物线y2=2x上的所有点处间断.而在其余各点处均连续.

(3)因为当x2+y2=1时，函数无定义，所以函数在圆周x2+y2=1上所有点处间断.而在其余各点

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处均连续.

(4)因为点P(x,y)沿直线y=x趋于O(0,0)时.

x?0limf(x,y)?limx?0x?1e??. 2xy?x?0故(0，0)是函数的间断点，而在其余各点处均连续. 8. 求下列函数的偏导数：

(1)z=x2

y+xy2;

(2)s=u2?v2uv;

(3)z=xlnx2?y2;

(4)z=lntanxy; (5)z=(1+xy)y; (6)u=zxy;

y(7)u=arctan(x-y)z

; (8)u?xz.

解：(1)

?z1?z2?x?2xy?xy2, ?y?x2?y3. (2)s?uv?vu ?s?u?1v?v?su1u2, ?v??v2?u. (3)?z111x2?x?lnx2?y2?x?x2?y2?2x2?y2?2x?2ln(x2?y2)?x2?y2,?z?y?x?1x2?y2?12x2?y2?2y?xyx2?y2. (4)

?z?1?sec2x122x?x?y?ycscy, tanxyy

?z1xx2x?y??sec2?(?y??2x2)y2cscy. tanxyy(5)两边取对数得lnz?yln(1?xy)

故

?z2?(1?xy)y??yln(1?xy)??yyx?(1?xy)?1?xy?y2(1?xy)y?1?x. 176