医用高等数学习题解答（第1,2,3,6章） - 41 -
?x?u2u2x??2ulnv, zv??, u?y??2, v?y??2, z?????3. zuy?zu?uy?zv?vy?2ulnv????y2???v?(?2) vy???x??2??y?xx2x2?11??????2ln(3x?2y)?2? ??2?ln(3x?2y)???yy3x?2yy?y3x?2y?x1x1y?1yyyy?1yy??????z??z??yx??xlnx，?2x?2z 4. z??(x)?yx, z?(x)?xlnxxyxxyyylnxylnx5. fx?(x,y)?e?x(?x)?xsin(x?2y)?e?xcos(x?2y)(x?2y)?x??e?xsin(x?2y)?e?xcos(x?2y)
2??????e?x?cos(x?2y)?sin(x?2y)?, fx?(0,)?e0?cos()?sin()???1
422????fy?(x,y)?e?xcos(x?2y)(x?2y)?y?2e?xcos(x?2y), fy?(0,)?2e0cos()?0
42?????x???x?x?x另解：fx?(x,)??esin(x?)???e?xcosx???ecosx?esinx??e(cosx?sinx),
4?2?fx?(0,)??e0?cos0?sin0???1
4???fy?(0,y)?(e0sin(0?2y))?y?(sin2y)??2cos2y,fy?(0,)?2cos()?0 426. y?1,x?0?y?x,x?0，如图。 xyy?xysinx3?xsiny?9ysinx 7. ??xsiny3?xsiny?9??x?y??xo?x?y?xx???x?0?sinxy? ??3?xsiny?9??6???xsiny?y?0???y??x1111111??111??8. z?y2x2，z?y?(x2y2)?y?x2(y2)?y?x2y2 x?(xy)x?y(x)x?2212121212四、解答题题解 1. 求定义域
?x?1?1?x2?0??(1)?2, D??(x,y)|x|?1,|y|?1?
y?1?y?1?0?(2)D?(x,y,z)x?0,y?0,z?0 (3)见选择题4
2.不连续区域D?(x,y)1?x2?y2?0
????医用高等数学习题解答（第1,2,3,6章） - 42 -
3.求极限
2?xy?42?xy?4?11(1) ?????(x,y)?(0,0)?
xy42?xy?42?xy?4(2) sin(xy)xsin(xy)??x??0?(x,y)?(0,0)? yxxyx2?y21(3) 4????(x,y)?(0,0)?
x?y4x2?y2(4) x?y?0?(x,y)?(?,?)?
x2?y24.求偏导数
2x2x?(1) z?x?2esiny, zy?ecosy y?1y?(2) z??yx, z?xlnx xy(3) z?x?111, z?y??
x?lnyx?lnyy11?xy2(4) z?x??y2x, z?y?x1?xy2
yy?1yy1?yzzz??(5) u???x, u?xlnx?, u?xlnx? xyzzzz21yy2, z??y2(1?xy)y?1 (6)两端取对数lnz?yln(1?xy), z?x?yx?zz1?xy1?xy?1xxy?xy?y?????? z??ln(1?xy)?y, z?zln(1?xy)? ?(1?xy)ln(1?xy)?yy????1?xy?z1?xy1?xy???v?1v?????y?uvlnu?0?vyuv?1 也可设z?u, u?1?xy, v?y, z?x?zu?ux?zv?vx?vuv?1????z??x?uvlnu?1?uv(y?zu?uy?zv?vy?vuxv?lnu) u5.求给定点的偏导数
????3???z(?,)?2cos(2??)?3, z(?,)?cos(2??)?(1) z??2cos(2x?y), z?cos(2x?y)， xyxy6666222???(2) z?x?3x?2y, zy?3y?2x?2y，zx(2,?1)?12?2?14，zy(2,?1)?3?4?2??3
6.证：z?ln(x?y)，z?x?11?, z?y?x?y2x11?
x?y2y医用高等数学习题解答（第1,2,3,6章） - 43 -
?x?z?x?y?zy?x?7.证：z?11111??y????x?y2xx?y2y2x?y???2x?y??1 22?xy?x?y , z?e?tdt??e?tdt??e?tdt??ae?tdt??ae?tdt，z?x?ey??e2a2x2x2y2ya?x?y?x?z??(?z)?e?e?exy222?y2
8.求二阶导数
6(x2?y2)?6x?2x6(y2?x2)6x6y6(x2?y2)?6y?2y???2(1) z?, z?y?2，z?，z? xx?yy?x?22222222(x2?y2)2(x?y)(x?y)x?yx?y0?(x2?y2)?6x?2y?12xy0?(x2?y2)?6y?2x?12xy6(x2?y2)?????2z??z??，， xyyx22222222222222(x?y)(x?y)(x?y)(x?y)(x?y)?(2) z?x?cos(x?2y), zy?2cos(x?2y)
???????z?xx??sin(x?2y)，zyy??4sin(x?2y)，zxy??2sin(x?2y)，zyx??2sin(x?2y)
??(3) u?x?y?z, uy?x?z, uz?x?y
?????????????????u?xx?0, uyy?0, uzz?0, uxy?1, uxz?1，uyx?1, uyz?1, uzx?1, uzy?1
(4) z?x?e?1?z??exyy?xy?1xy?，z??, z??e?yxx?e2yyx?2?ey?xy?x?xy11??2?2ey，z?yy?eyy?xy?x?xyx2?4?ey?xy?2x?3?ey?xy?x22x???y4?y3?? ???xy1?2?ey?xy?1x?x????，?z?eyx2?y2y3?y???1??ey?xy1?2?ey?xy?1x????y2y3?? ??9.验证函数z?x?1?x?1???y?????21y?2, z?y?2yx?y1?x?1???y?????2?x?x??1, xv??1, yu??1, yv???1 ?，xu222yx?y??z????zux?xu?zy?yu?y?xy?x?????, z?z?x?z?y?， vxvyvx2?y2x2?y2??zv??zuy?xy?x2y2(u?v)u?v???? 2222222222x?yx?yx?y(u?v)?(u?v)u?v10.求多元复合函数的导数
(1)见（三、填空题1） (2)见（三、填空题3）
dz?z?z?yyx(1?x)exyxxx?????e?, z?y?, y?(3)z? x?x?e，22dx?x?y?x1?(xy)21?(xy)21?x2e2x1?(xy)1?(xy)(4)可设z?f(u,v), u?x1?x??fu?(u,v), zv??fv?(u,v), u???, v?x2,zu?, u?, v?xyx?2x, vy?0
yyy2医用高等数学习题解答（第1,2,3,6章） - 44 -
?z?z?u?z?v1?z?z?u?z?v?x????fu?(u,v)?2xfv?(u,v), ????2fu?(u,v) ?x?u?x?v?xy?y?u?y?v?yy??eusinv, zv??eucosv, u???, v?(5)zux?y, uy?x, vx?1y?1
?z?z?u?z?v????eusinv?y?eucosv?1?exy?ysin(x?y)?cos(x?y)? ?x?u?x?v?x?z?z?u?z?v????eusinv?x?eucosv?1?exy?xsin(x?y)?cos(x?y)? ?y?u?y?v?y11.求函数的全微分
22??(1) z?x?2x?y，zy?2xy?cosy，dz?z?xdx?zydy?(2x?y)dx?(2xy?cosy)dy
(2) z?x?1?x?1???y????2?1?y1y1?xy22，z?x?x?x?1???y????2??1?y2?xy21?xy22
dz?1x2y1?2ydx?y2?xx21?2ydy??x??dx?dy?? 2?yx??y1?2y1yexy(ex?ey)?exyexexy(y?1)ex?yeyxexy(ex?ey)?exyeyexy(x?1)ey?xex(3) z???，z? x?y?xy2xy2(e?e)(e?e)(ex?ey)2(ex?ey)2exyxyyxdz?x[(y?1)e?ye]dx?[(x?1)e?xe]dy y2(e?e)??????2222?(4) z?x?3x?2y, zy?3y?2x，dz??3x?2y?dx??3y?2x?dy
?R2?a2?z222?12.设内接长方体长,宽,高分别是x,y,z且满足：?2?1??1?
R??x???y???2??2??长方形体积为：V?xyz。F?xyz??(x?zaR14212y?z2?a2) 4yx?Fx??yz??x/2?0???1212??xy?F?xz??y/2?0??x?y?2z，代入 x?y?z2?a2?0 ?y?zy44?F??xy?2?z?0???z??y4z?z?2aa，x?y? 332另解：V?xya?1212x?y 44医用高等数学习题解答（第1,2,3,6章） - 45 -
?1??x112Vx??y?a2?x2?y2?x?44121222a?x?y?44???1???y?, V??x?a2?1x2?1y2?y2y??44121222a?x?y??44????? ????Vx??02aa 令? 解之得 x?y?, z??V?033?y13.求函数的极值
?z??x??1x?2x?y?1?0(1) z?x?xy?y?x?y?1,? ???z?2y?x?1?0?y?1?y22???z??A?z?xx?2xx?2????2??C?z?yy??2?B2?AC??3, 故 f极小值(?1,1)?0 ?z?yy?z???1?B?z???1xy?xy???z?xx??2y?a??z??ay?2xy?y?0?(2) z?axy?x2y?xy2,?x, ?z? ?x?y?yy??2x2?3???zy?ax?x?2xy?0?xy?a?2x?2y?z?22a???A?z??xx?3?aaa32a3a2?2???,当a?0时是极大值,当a?0时是极小值。 , f(,)???C?z?yy?B?AC??332739?a?B?z????xy?3?112由x?y?1得y? 14.求条件极值z?x(1?x)?x?x, z??1?2x?0?x?, 22111z????2由定理9知有极大值, 故 z极大值(,)?
22415.求条件极值F(x,y)?x?y???22?xy???1? ?ab????F?2x??0?x?ab2ax?2??2?ax?by?0???a?b? ???Fy?2y??0??2b?bx?ay?ab?y?ab???x?y?1a2?b2???ab?????A?z?xx?2, C?zyy?2, B?zxy?0, B?AC??4, 故 z极小值?z?15?14x?32y?8xy?2x2?10y316.根据条件?
?x?y?1.52a2b2?2 2a?b