习题1—1解答 1． 设f(x,y)?xy?x11x1，求f(?x,?y),f(,),f(xy,), yxyyf(x,y)解f(?x,?y)?xy?111yx1yx?;f(xy,)?x2?y2;?2；f(,)?
xyxyxyf(x,y)xy?xy2． 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?x?（2）f(x,y)?2
y2?1;
;
4x?y2ln(1?x2?y2)x2y2z2（3）f(x,y)?1?2?2?2;
abc（4）f(x,y,z)?x?y?z1?x?y?z222.
解（1）D?{(x,y)x?1,y?1? （2）D?(x,y)0?x?y?1,y?4x?222?
??x2y2z2（3）D??(x,y)2?2?2?1?
abc??（4）D?(x,y,z)x?0,y?0,z?0,x?y?z?1
?222?
4．求下列各极限：
1
（1）limx?0y?11?xy1?0=?1 22x?y0?1ln(x?ey)x?y22（2）limx?1y?0?ln(1?e0)1?0?ln2
（3）lim2?xy?4(2?xy?4)(2?xy?4)1?lim??
x?0x?0xy4xy(2?xy?4)y?0y?0（4）limsin(xy)sin(xy)?lim?x?2
x?2x?2yxyy?0y?05．证明下列极限不存在：
x2y2x?y（1）lim ; （2）lim22x?0xy?(x?y)2x?0x?yy?0y?0（1）证明 如果动点P(x,y)沿y?2x趋向(0,0) 则limx?yx?2x?lim??3；
x?0x?0x?2xx?yy?2x?0x?y3y?lim?3
y?0y?0x?yyx?2y?0如果动点P(x,y)沿x?2y趋向(0,0)，则lim所以极限不存在。
（2）证明： 如果动点P(x,y)沿y?x趋向(0,0)
x2y2x4?lim4?1； 则lim222x?0x?0xxy?(x?y)y?x?0x2y24x4?lim4?0 如果动点P(x,y)沿y?2x趋向(0,0)，则lim2222x?0x?0xy?(x?y)4x?xy?2x?0所以极限不存在。
6．指出下列函数的间断点：
y2?2x（1）f(x,y)?； （2）z?lnx?y。
y?2x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?xy? yx2
?z1y?z1x??2；??2. ?xyx?yxy (2)
?z?ycos(xy)?2ycos(xy)sin(xy)?y[cos(xy)?sin(2xy)] ?x?z?xcos(xy)?2xcos(xy)sin(xy)?x[cos(xy)?sin(2xy)] ?y(3)
?z?y(1?xy)y?1y?y2(1?xy)y?1, ?x1?zx?ln(1?xy)?y, z?y1?xy lnz=yln(1+xy)，两边同时对y求偏导得
xyxy?z?z[ln1(?xy)?]?(1?xy)y[ln1(?xy)?]; ?y1?xy1?xy12y2?z11?3x3??;x?2y?zx3y??(4), ?yx?y x?2y?xx(x3?y)x?2xxy?uyz?1?u1z?u?x,?xlnx,??2xzlnx(5)?x; z?yz?zzyyyz(x?y)z?1?uz(x?y)z?1(?y)?u?u(x?y)zlnx(6), , ; ?????x1?(x?y)2z?y?z1?(x?y)2z1?(x?y)2z2.(1)
zx?y,zy?x,zxx?0,zxy?1,zyy?0;
(2) zx?asin2(ax?by),zy?bsin2(ax?by),
zxx?2a2cos2(ax?by),zxy?2abcos2(ax?by),zyy?2b2cos2(ax?by).
222 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.
4
ttttzx??2sin2(x?),zt?sin2(x?),zxt?2cos2(x?),ztt??cos2(x?)
2222tt2ztt?zxt??2cos2(x?)?2cos2(x?)?0.
22 3
yy115.(1) zx??2ex, zy?ex,dz??2exdx? exdy;
xxxx (2) z?yyyyyyxx1z?dz? dx?dy; ,,ln(x2?y2),zx?2y2x2?y2x2?y2x2?y2x?y2y12?ydx?xdyyxxxdz???2z?? (3)zx? , ,; yy2y2x2?y2x2?y2x?y21?()1?()xx?yz?1,uy?zxyzlnx,uz?yxyzlnx, (4) ux?yzxdu?yzxyz?1dx?zxyzlnxdy?yxyzlnxdz.
6. 设对角线为z,则z?x2?y2,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?x2?y2,
, dz?zx?xx?y22，zy?yx?y22xdx?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??y?0.1时, z?72?242?25
?z?dz?7?0.1?24?0.17?2422 =0.124,z的绝对误差?z?0.124
z的相对误差
?z0.124??0.496%. z258. 设内半径为r，内高为h，容积为V，则
V??r2h,Vr?2?rh,Vh??r2,dV?2?rhdr??r2dh,
当r?4,h?20,?r?0.1,?h?0.1时,
?V?dV?2?3.14?4?20?0.1?3.14?42?0.1?55.264(cm3).
习题1—3
4
xyyx?2du?fdx?fdy?fdzzzz????? ?aeax?1.?2a(ax?1)
xyxyxydx?xdx?ydx?zdx1?()21?()21?()2zzzy[z?axz?2axy(ax?1)](ax?1)eax(1?a2x2)==. 422axz2?x2y2(ax?1)?xe??z?f???f????2.=?x???x???x1??24x3arcsin1?x2?y2x?y444x3?arcsin??4=422x?y1?x?y?x?xln(x4?y4)(1?x?y)(x?y)2222
??z?f???f????=?y???y???y1??24y3arcsin1?x2?y2x?y3. (1)
444y3?arcsin??4=422x?y1?x?y?y?yln(x4?y4)(1?x?y)(x?y)2222.
?u?uxyxy=2xf1?yef2, =?2yf1?xef2.
?y?xx1y?u?u1?u=?f1, =?2?f1?f2,=?2?f2.
?yzy?xy?zz(2)
(3)
?u?u?u=f1?yf2?yzf3,=xf2?xzf3,=xyf3.
?y?z?x?u?u?u=2xf1?yf2?f3=2yf1?xf2?f3,=f3.
?y?x?z?z?z?xf1?f2, ?yf1,?y?x(4)
4 .(1)
?f1?2z?y?y?f11?y??y2f11, 2?x?x?2z??yf1??f1?y?f1?f1?y(f11?x?f12)?f1?xyf11?yf12, ??x?y?y?y?f1?f2?2z?2???xf?f?x??x(f?x?f)?f?x?f?xf11?2xf12?f2212111221222?y?y?y?y 5