习题1—1解答 1． 设f(x,y)?xy?x11x1，求f(?x,?y),f(,),f(xy,), yxyyf(x,y)11xy1xyyxxy22解f(?x,?y)?xy?xy；f(,)??;f(xy,)?x?y;1f(x,y)?yxy2?x
2． 设f(x,y)?lnxlny，证明：f(xy,uv)?f(x,u)?f(x,v)?f(y,u)?f(y,v)
f(xy,uv)?ln(xy)?ln(uv)?(lnx?lny)(lnu?lnv)?lnx?lnu?lnx?lnv?lny?lnu?lny?lnv?f(x,u)?f(x,v)?f(y,u)?f(y,v)
3． 求下列函数的定义域，并画出定义域的图形： （1）f(x,y)?1?x2?2y?1;
2（2）f(x,y)?4x?y2ln(1?x?y)xa222;
（3）f(x,y)?1??yb22?zc22;
（4）f(x,y,z)?x?2y?2z2.
1?x?y?z解（1）D?{(x,y)x?1,y?1
?
y 1 -1 O -1 1 x （2）D?(x,y)0?x2?y2?1,y2?4x
?? y 1 -1 O -1 1 x 1
222??xyz（3）D??(x,y)2?2?2?1? abc??z c -a
-b a x O b y （4）D?(x,y,z)x?0,y?0,z?0,x2?y2?z2?1
4．求下列各极限： （1）lim1?xyx?y22??z １ O x 1?00?1?1
１ y １ x?0y?1=
（2）limln(x?ex?y2?2y)2x?1y?0?ln(1?e)1?0(2?0?ln2
（3）limxy?4xyx?0y?0?limxy?4)(2?xy(2?xy?4)x?0y?0xy?4)??14
（4）limsin(xy)yx?2y?0?limsin(xy)xyx?2y?0?x?2
5．证明下列极限不存在： （1）limx?0y?0x?yx?y; （2）limxy22222x?0y?0xy?(x?y)
（1）证明 如果动点P(x,y)沿y?2x趋向(0,0) 则limx?yx?yx?2xx?2x??3；
x?0y?2x?0?limx?0如果动点P(x,y)沿x?2y趋向(0,0)，则limx?yx?yy?0x?2y?0?lim3yyy?0?3
2
所以极限不存在。
（2）证明： 如果动点P(x,y)沿y?x趋向(0,0)
xy22222则limx?0y?x?0xy?(x?y)?limxx44x?0?1；
如果动点P(x,y)沿y?2x趋向(0,0)，则lim所以极限不存在。
6．指出下列函数的间断点： （1）f(x,y)?y?2xy?2x2xy22222x?0y?2x?0xy?(x?y)?lim4x442x?04x?x?0
； （2）z?lnx?y。
解 （1）为使函数表达式有意义，需y?2x?0，所以在y?2x?0处，函数间断。 （2）为使函数表达式有意义，需x?y，所以在x?y处，函数间断。 习题1—2 1．（1）z?xy1y?yx
?z?x??yx2；
?z?y?1x?xy2.
(2)
?z?x?ycos(xy)?2ycos(xy)sin(xy)?y[cos(xy)?sin(2xy)]
?z?y?xcos(xy)?2xcos(xy)sin(xy)?x[cos(xy)?sin(2xy)]
(3)
?z?x?y(1?xy)y?1y?y(1?xy)2y?1,
1?zz?yx1?xy lnz=yln，两边同时对y求偏导得?ln(1?xy)?y,
?z?y?z[ln1(?xy)?xy1?xy]?(1?xy)[ln1(?xy)?yxy1?xy];
(4)
?z?x1??x?2yxx3y2?x?2yx(x?y)33,
3
1?z?y?xx?2yx2?13x?y
;?u(5)?x?yzyxz?1,?u?y?1zyxzlnx,?u?z??yz2yxzlnx;
(6)
?u?x?z(x?y)z?12z1?(x?y)z(x?y),
?u?y?u?zz?12z??1?(x?y)z,
?(x?y)ln(x?y)1?(x?y)2z;
2.(1)
zx?y,zy?x,zxx?0,zxy?1,zyy?;
(2) zx?asin2(ax?by),zy?bsin2(ax?by),
zxx?2acos2(ax?by),zxy?2abcos2(ax?by),zyy?2bcos2(ax?by).
22 3 fx?y?2xz,fy?2xy?z,fz?2yz?x,fxx?2z,fxz?2x,fyz?2z,
fxx(0,0,1)?2,fxz(1,0,2)?2,fyz(0,?1,0)?0.
2224
zx??2sin2(x?t2),zt?sin2(x?t2yt2),zxt?2cos2(x?t2)?0.
t2),ztt??cos2(x?t2)
2ztt?zxt??2cos2(x?)?2cos2(x?5.(1) zx??12yx2yex, zy?1xex,dz??yx2yexdx?
1xyexdy;
(2) z?ln(x2?y),zx?2xx?y22,zy?yx?y22,dz? xx?y22dx?yx?y22dy;
(3)zx2yx???2 , zy?2y2x?y1?()1?x?y1?ydx?xdyxx?2dz? ,; 222y2x?yx?y()x 4
(4) ux?yzxyz?1,uy?zxyz?1yzlnx,uz?yxyzyzlnx, lnxdz.
du?yzxdx?zxlnxdy?yxyz6. 设对角线为z,则z?22x?y,zx?xx?y22,zy?yx?y22, dz?xdx?ydyx?y22
当x?6,y?8,?x?0.05,?y??0.1时,?z?dz?6?0.05?8?(?0.1)6?822 =-0.05(m).
7. 设两腰分别为x、y,斜边为z,则z?zx?xx?y22x?y,
22，zy?yx?y22, dz?xdx?ydyx?y22,
设x、y、z的绝对误差分别为?x、?y、?z, 当x?7,y?24,?x??x?0.1,?y???z?dz?7?0.1?24?0.17?2422y?0.1时, z?7?2422?25
=0.124,z的绝对误差?z?0.124
z的相对误差
?zz?0.12425?0.496%.
8. 设内半径为r，内高为h，容积为V，则
V??rh,Vr?2?rh,Vh??r,dV?2?rhdr??rdh,
222当r?4,h?20,?r?0.1,?h?0.1时,
?V?dV?2?3.14?4?20?0.1?3.14?4?0.1?55.264(cm).
23习题1—3
yx? )21.
dudx??fdx?xdx??fdy?ydx??fdz?zdx?1?(zxyzax41?(22zxyz?xyzxyz2?ae)2ax?1?(?2a(ax?1)
)2=
y[z?axz?2axy(ax?1)]z?xy??f?????x?222=
(ax?1)e(1?ax)22ax(ax?1)?xe.
342.
?z?x?f?????x=
?1??2?x1?x?y22?arcsin??4x4x?y=
5