微积分第七章习题解答
第七章习题解答
1.求下列函数的定义域。
221z,1,x,y,, 22,,,,解:D:x,yx,y,1
2222z,ln4,x,y,x,1,,,, 22,,,,解:D:x,yx,y,4,x,1
223z,sinx,y,,,, 22,,,,,,解:D:x,y2k,,x,y,2k,1,,k,0,,1,,2,?
2222224z,ln16,x,yx,y,41,x,y,,,,,,,, 22,,,,解:D:x,y4,x,y,16
115z,,,,
x,yx,y
,,,,解:D:x,y,x,y,x,x,0
,,6z,x,y 2,,,,解:D:x,y0,y,x,x,0
22arcsin3,x,y,,7z,,,2x,y
222,,,,解:D:x,y2,x,y,4,x,y
2228z,arcsinx,y,lnln10,x,4y,,,,,, 2222,,,,,,,,解:D:x,yx,y,1,10,x,4y,1,x,y,1,x,y,1,x,4y,9
2.求下列函数的极限。
122xy1lim,sin,,,,22x,0xy,,,y,0 112222,,解:limx,ysinx,y,ulimusin,022x,0u,0u,,xy,y,0
xy,2lim,,22x,,xy,,,y,,0
xyxy,11解:lim,lim,lim,lim,lim,0,0,022222222x,,x,,x,,x,,x,,,,,,,,xyxyxyyx,,,y,,0y,,0y,,0y,,0y,,0xy,,xy
sinxy3lim,,x,0xy,2 sinxysinxysinu解:lim,lim,yxy,ulim,y,1,2,2x,0x,0u,0xxyuy,2y,2y,2
2xy4lim,,22x,0,xyy,0 22xyxyy解:lim,lim,,02222x,x,002,,xyxyy,0y,0
3求下列函数的一阶偏导数。
xy1zsincos,,,yx
,z1xyyxy解:,coscos,sinsin 2,xyyxyxx
,zxxy1xy,,coscos,sinsin2yyxyx,yx2z,lnx,lny,,,,
,z1解:, ,xx,lny
,z11,,,yx,lnyy
x3z,arcsin,,y
y,z11 解:,,,222,xyyx,y,,x,,1,,,y,,
,xy,,,z1x,,,,,,2,,2222,yyyx,y,,,,x,,1,,,y,,
,,,,4z,arccosx,y
,z1 解:,,2,x,,1,x,y
,z1,,2,y,,1,x,y
4求下列函数在定点的偏导数。
33,,,,1z,x,y,3xy求z,zxy,,1,2,,1,2
,z,z22,,解:,3x,3y?,3x,3y,,3,,1,2,x,x,,1,2 ,z,z22,,,3y,3x?,3y,3x,9,,1,2,y,y,,1,2
22xyx,y,,,,2z,z,z求,,xy221,1,,1,1,,x,y
2222224545,z,,yx,y,xy,2xx,y,xyx,y,2xxy,y,zxy,y,,,,,,,,?,,0解:1,1,,222222222,x,x,,x,yx,y,,x,y,,1,1,,
2222225454,,,,,,,,,zxx,y,xy,2yx,y,xyx,y,2yx,xy,zx,xy,,?,,0,,1,1222222222,y,y,,,,,,x,yx,yx,y,,1,1
22x,y,,,,3z,e求z,zxy,,0,1,,0,1
2222,z,zx,yx,y,,解:,e,2x?,2xe,0,,0,1,x,x,,0,1
2222,z,zx,yx,y,,,e,2y?,2ye,2e,,0,1,y,y,,0,1
,,,,4z,lnxy求z,zxy,,,1,,1,,1,1
,1,1zz解:,?,,,1,,,,1,1,xx,xx,,,0,11
,z1,z1,?,,1,,1,1,yy,yy,,1,1
5求下列函数的二阶偏导数。
22,,,,1lnz,x,y
22222,,,,22222,zx,zx,y,x,xy,x解:,,,222222222,xx,y,x,,,,x,yx,y
22222 ,,,,22222,zy,zx,y,y,yx,y,,,222222222,yx,y,y,,,,x,yx,y22224,z,x,y,x,y,z,,,222222,x,y,y,x,,,,x,yx,y
2z,xlnx,y,,,,
2,zx,zx,y,xx,y12,,解:,ln,,x,y,,,,222,xx,yx,y,xx,y,,x,y,,
2 ,zx,z,x,,22,yx,y,y,,x,y
22,z1,x,y,z,,,,2222,x,yx,y,y,x,,x,y,,x,y
xz,y,xye3cossin,,,,
,zxxx解:,sinye,cosy,xsinye,siny,cosy,xsinye,,,,,,,x
2,zxxxx,y,e,y,y,xye,y,y,xye,z,y,esinsin,,,,cossin2sincossin2sin2,x
2,z,zxx,,,,,,siny,xcosye,,cosy,xsinye,,z2,y,y
22,z,zx,,,cosy,siny,xcosye,,x,y,y,x
22y,xz,4,,22x,y
22222,xx,y,y,xx,xy224,z,,,,,,解:222222,xx,yx,y,,,,
22222,222222,yx,y,xy,x,y,xyx,y442243,,,,,,,z,,2432222,xx,yx,y,,,,
22222y,,,,x,y,y,xyxy224,z,,222222,y,,,,x,yx,y
22222222222,,,,,,xx,y,xy,x,y,yxx,y442243,z,,2432222,y,,,,x,yx,y
2222,222222,,,,,,,8xyx,y,4xy,2x,y,2y8xyy,x8xyz,z,z,,,,433222222,x,y,y,x,,,,,,x,yx,yx,y
6求下列函数的全微分。
yz,x,,1
,z,zy,1y dz,dx,dy,yxdx,xlmxdy解:,x,y
22x,y2z,e,,
222222,z,zx,yx,yx,y ,,解:dz,dx,dy,2xedx,2yedy,2exdx,ydy,x,y
223z,x,xy,siny,,
,z,z2 ,,,,解:dz,dx,dy,2x,ydx,2xy,cosydy,x,y
22z,x,y,,4ln
,z,zy1x11,,dz,dx,dy,dx,dy,xdx,ydy解:2222222222,x,yx,yx,yx,yx,yx,y
x,,5z,arcsiny
yxy,z,z111,xdzdxdydxdydxdy解:,,,,,,22222222xyy,,yyyxyyx,,,,,,xx,,,,1,1,,,,,yy,,,,
7求下列复合函数的偏导数。
dzv,,1zu,u,sinxv,cosx求dx
z,z,dzdudvcosx,1cosx,1v,1v2,,,,,,解:,,vu,cosxu,lnu,sinx,cosxsinx,sinxlnsinxdxu,dxv,dx
y222,,,,,,求2zulnvuvxyz,z,,xyx
22,,zz,,,,yy2yzuvu,,22解:,,,,,,,,,2ulnv2x2ln,,xy,,2322 ,,,,,xuxvxv,,,xxxxy,,
32,,zz,,,y2yzuv1u,,22,,,,,2ulnv,,2y,,2lnx,y,,,2222,y,u,y,v,yxv,,,xxxy,,
yuv22,,,,,,3zeulnxyvarctanz,z,,,,求xyx
,,,,zz,,,,yyuzuv12x1xv,,uvuvuv,,,,,,,,,veuee解:,,,,2222222222,,,,,xuxvx,,xxyxy,,y,,xy2xy,,,,,1,,x,,
,,,xvyuuv,,,e,,22,xy,,
,,,,zz,,,zuv12y11yvxu,,uvuvuv,,,,,,,,,veuee,,,,222222222,,,,,yuyvyx,,xyxy,,y,,xy2xy,,,,,1,,x,,
,,,yvxuuv,,,e22,,,xy,,
8求下列隐函数的偏导数。
22xy1sinxyxye,,,,,,
22xy解:令Fxysinxyxye,,,,,,,
FF,, 2xyx2xycosxyy2xyey.cosxyx2xyex,,,,,,,,,,,,,,xy,,
F,2xy,,dycosxyy2xyey.,,,,yx,?,,,,,,2xyF,,,dxxcosxyx2xyex,,,,
y,
xy,,2y,x
xy,,解:令Fxy,y,x
,F,F,,xy1x1y,ylny,yx.,xy,xlnx,x,y
,F,xy1,xy1dyylny,yxylny,yx ,x?,,,,,,,x1yyx1,Fdxxy,xlnxxlnx,xy
,y
,xy1xxy,,,,yxylny,xyxyylny,yx,,,yx1yyx,,,,xyxlnx,yxyxxlnx,xy
xyxy2?,yx,,,,,,ylny,yylnx,yylnx,1,,,,,yx2,,,,,,xlnx,xxlny,xxlny,1
3332,,,,,,z,z,z9.设函数,,由方程x,y,z,a确定,求。 z,fx,yxxyyxy
3333,,解:令Fx,y,z,x,y,z,a
,F,F,F222,3x,3y,3z,x,y,z
,F,F2222,z3x,z3y,yyx,x?,,,,,,,,,,,,2222,F,F,x,y3zz3zz
,z,z
2,z,,,z,x222,,,2xz,2x2xz,x,2z,2432,z,,,x2x,xz,,,xz,,,,,,,,22352,x,xzz,,z
2,z,z,,,y22,,x,2z,,,,2x2222,z,x,y2xy,,z,,,,,,,,2352,x,y,yzz,,z
2,,,z,z,y222,,,2yz,y,2z,2yz,2y,,2432,y,,,z,y2y,yz,,z,,,,,,,,22352 ,y,yzz,,z
10(求下列函数的极值。
331fx,y,x,y,9xy,27,,,,
解:
,f,2,3x,9y,0,x,3,,x,,,,,fy,32,,.,3y,9x,0,,y,
222,f,f,f,6x,,9,6y22,x,y,x,y
222,f,f,f,,在3,3处,A,,6x,18B,,,9C,,6y,1822,,,,3,33,3,x,y,x,y,,,,,,3,33,33,3
2?B,AC,81,18,18,,243,0,且A,18,0
33,,,,,,?在3,3处fx,y,x,y,9xy,27有极小值,极小值为f3,3,0
33(2)fx,y,x,3xy,y,,
解:
,f,2,3x,3y,0,x,1,,x,,,,,fy,12,,.,3y,3x,0,,y,
222,f,f,f,6x,,3,6y22,x,y,x,y
222,f,f,f,,在1,1处,A,,6x,6B,,,3C,,6y,622,,,,1,11,1,x,y,x,y,,,,,,1,11,11,1
2?B,AC,9,6,6,,27,0,且A,6,0
33,,,,,,?在1,1处fx,y,x,3xy,y有极小值,极小值为f1,1,,1
33(3)fx,y,x,3xy,y,,
解:
,f,2,3x,3y,0,x,,1,,x,,,,,fy,,12,,.,3y,3x,0,,y,
222,f,f,f,6x,3,6y22,x,y,x,y
222,f,f,f,,在,1,,1处,A,,6x,,6B,,3C,,6y,,622,,,,,,,,1,11,1,x,y,x,y,,,,,,,,,,,,,1,11,11,1
2,,,,?B,AC,9,,6,,6,,27,0,且A,,6,0
33,,,,,,?在,1,,1处fx,y,x,3xy,y有极大值,极大值为f,1,,1,111(求下列函数的条件极值。
(1)z,xy,x,y,2
,,,,,,解:令:Fx,y,,xy,x,y,2
,,f,,y,,0,,xx,1,,,f,,,.,x,,0,y,1,,,y,,,,,1 ,,,f,x,y,2,0,,,,
,,点1,1是唯一驻点,在x,y,2上任取一点,有y,2,x
22,,,,?z,xy,x2,x,2x,x,,x,1,1,1
,,,,,,?在1,1处,fx,y,xy有极大值,极大值为f1,1,1
,,,,(2)z,xy,1,x,1y,1,1x,0,y,0.
,,,,,,,,,,解:令:Fx,y,,xy,1,x,1y,1,1
,,f,,,,y,y,1,0,,xx,2x,0,,,12,f,,,,,,.,x,x,1,0,y,2y,0,,,12,y,,,,,,,2,0,,12,,f,,,,,x,1y,1,1,0,,,,
,,?x,0,y,0?点0,0舍去
1x,,,,,,点2,2是唯一驻点,在x,1y,1,1上任取一点,有y,1,,x,1x,1
22,,xxx,2,,?z,xy,1,x,1,3,,4,3,,3,z2,2x,1x,1x,1
,,,,,,?在2,2处,fx,y,xy,1有极小值,极小值为f2,2,3
22212(求函数在区域的最值。 z,xyx,y,1
解:
,z,2,y,,0,x,0,,x,,,,,zy,0,,.,2xy,0,,y,
22,,点是唯一驻点x,y,0,01,
2,,,,,,,,当时可正可负,故不是极值点,xy,z,xyxy,,0,0,,0,0
22最大值,最小值只能在边界上取得?x,y,1
2222?x,y,1,y,1,x
223,,z,xy,x,x,x,xx,11
32,z,,x,,x,,1303
,,,,,,z,,x,z,,,z,,6230,23033x,x,,33
3323分别是最大值点,最小值点?x,,x,,z,xy,x,x33
,,,,在由知z,z,,1,01,00,
33222在上取得最大值在上取得最小值z,xyx,y,x,x,,1,33
,,3623,,?z,,,max,,339,,
,,3623,,z,,,,,min,,339,,
Rx,y13(设销售收入(单位:万元)与花费在两种广告宣传的费用(单位:万元)
22,,Rx,y,15,14x,32y,8xy,2x,10y之间的关系为
(1) 在广告费用不限的情况下,求使总利润最大的广告策略; (2) 若提供的广告费用为1.5万元,求使总利润最大的广告策略;
22,,,,解:1设销售净利润为:L,R,C,15,14x,32y,8xy,2x,10y,x,y
22,15,13x,31y,8xy,2x,10y,L,,13,8y,4x,0,x,0.75,,x,0,,,,Ly,1.250,,.,31,8x,20y,0,,y,
222,L,L,L,,?又在x,y处:,,4,,,8,,,200022,x,y,x,y
2?B,AC,64,80,,16,0A,,4,0
22,,点x,y是L,15,13x,31y,8xy,2x,10y唯一驻点,00
,,,,?点x,y,0.75,1.25是L的最大值点,最大值为:00
,,,,L0.75,1.25,39.25万元
,,,,2条件为:x,y,1.5
,,,,,,令Fx,y,,L,x,y,1.5
,F,,,13,8y,4x,,0,,xx,0,,,F,,,,31,8x,20x,,0y,1.5,,,x,,,,,1 ,,F,,x,y,1.5,0,,,,
,,点0,1.5是唯一驻点,由问题实际含义知:
,,,,当x,0,y,1.5时,销售净利润最大,为:L0,1.5,39万元
14. 欲造一正方形体的盒子,所用材料其料底部的价格为顶与侧面价格的两倍,若此盒容积
为324立方米,各边长为多少时,其造价最低。
解:设盒子的长,宽,高分别为:x,y,z
324由题意知:x,y,z,324,z,xy
324324648648造价函数为:L,3xy,2xz,2yz,3xy,2x,2y,3xy,,xyxyyx
648,,L,3y,,0x2,x,6,x ,,,648y,6,,,L,3x,,0y2,y,
,,6,6是L唯一驻点,
324由实际意义知:当x,6,y,6,z,,9时,造价最省。6,6
15(工厂生产两种产品,其产量分别为,总成本函数为 Q,Q12
22,, CQ,Q,2Q,2QQ,Q,37.5121122
两种产品的价格函数分别为:
,,PQ,Q,70,2Q,3Q11212
,,PQ,Q,110,3Q,5Q21212
为使利润最大,试确定两种产品的产量及最大利润。 解:设收入函数为:RQ,Q,pQ,pQ,,121122
22销售净利润为:L,R,C,Q70,2Q,3Q,Q110,3Q,5Q,2Q,2QQ,Q,37.5,,,,,,1122121122
22,70Q,110Q,4Q,6Q,,4QQ,37.5121212
,L,,70,8Q,4Q,012,,QQ,5,,11,,,,LQ,7.5,2,.,110,12Q,4Q,021,,Q,2
22,,点5,7.5是L,70Q,110Q,4Q,6Q,,4QQ,37.5唯一驻点,121212
由实际意义知:当Q,5,Q,7.5时,利润最大,12
,,最大利润为:L5,7.5,550
16.在经济学中Cobb-Douglas生产函数模型为:
,,,1,, ,,fx,y,cxy
式中x代表劳动力的数量,y代表资本的数量(确切地说是y个单位资本),c与,,,0,,,1是系数,由各工厂的具体情况而定,函数值表示生产量。现某个制造商的生产函数为:
31
44 ,,fx,y,100xy
每个劳动力与每个单位资本的成本分别为150元和250元,该制造商的总预算是50000元,问他该如何分配这笔钱用于雇佣劳动力与资本,以使生产量最大。
1344,,解:因为生产函数为:fx,y,100xy,,,条件为:x,y,150x,250y,50000,0
13,,,44,,,,,,L,fx,y,x,y,100xy,150x,250y,50000
,33,,L,44,25xy,150,0,,x,
,,11x,250,,L,44,.,75xy,250,0,,,y,50,y,,
,,L,150x,250y,50000,0,,,,,
,,点250,50是唯一驻点,
由实际意义知:当x,250,y,50时,生产量最大,2
,,最大生产量为:f250,50,16719(单位)
17. 某工厂生产一种产品同时在两个市场销售,售价分别为,销售量分别为,p,p,q,q1212需求函数分别为总成本函数为,,,厂q,24,0.2p,q,10,0.05pC,35,40q,q112212家如何确定两个市场的售价,才能获得最大利润,最大利润为多少,
22,,解:设收入函数为:RQ,Q,pQ,pQ,24p,0.2p,10p,0.05p1211221122
22,,,,,,销售净利润为:L,R,C,24p,0.2p,10p,0.05p,35,40Q,Q112212
22,32p,0.2p,12p,0.05p,13951122
,L,,32,0.4p,01,,pp,80,,11,,,,Lp,120,2,.,12,0.1p,02,,p,2
22,,点80,120是L,32p,0.2p,12p,0.05p,1395唯一驻点,1122
由实际意义知:当p,80,p,120时,利润最大,12
,,最大利润为:L80,120,605
,1.50.10.318.已知某产品的需求量为,其中Q为需求量,p为价格,x为广告Q,200000pxy
费用,y为推销费用,如果生产企业的可变成本为每件产品25元,固定成本(不含广告和推销费用)为7000元,求最佳经营时的价格,以及广告费和推销费。
,1.50.10.3,,解:设成本函数为:CQ,7000,25Q,7000,25,200000pxy
,1.50.10.3,0.50.10.3,,收入函数为:Rp,pQ,p,200000pxy,200000pxy
,0.50.10.3,1.50.10.3,,销售净利润为:L,R,C,200000pxy,7000,25,200000pxy
,0.50.10.3,1,,,200000pxy1,25p,70000
,L,,1.5,10.10.3,,,200000p1.5,25p,0.5xy,0,,p,p,75,,L,,,0.5,0.90.3,1,,.,0.1,200000pxy1,25p,0,x,355.554,,,x,,y,1066.662,,L,,0.50.1,0.7,1,,,0.3,200000pxy1,25p,0,,y,
,,点75,355.552,1066.662是L唯一驻点,
由实际意义知:当p,75,x,355.554,y,1066.662时,利润最大,
,,,,最大利润为:L75,355.554,1066.662,2053333与
19.为测定刀具的磨损速度,按每隔一小时测量一次厚度的方式得到若表一的实测数据。时间t(小时) 0 1 2 3 4 5 6 7 刀具厚度y(毫米)27.0 26.8 26.5 26.3 26.1 25.7 25.3 24.8
根据这组实测数据建立变量y与t之间的经验公式,,。 y,ft
解: 设所求经验公式为: y,at,b
为用最小二乘法计算,由已知对数据,得下表
i 2tyty t iiiii
1 0 27.0 0 0
2 1 26.8 1 26.8
3 2 26.5 4 53.0
4 3 26.3 9 78.9
5 4 26.1 16 104.4
6 5 25.7 25 128.5
7 6 25.3 36 151.8
8 7 24.8 49 173.6
9
28 215.7 140 717 8
,i,1
带入公式得:
nnn1,28,215.7ty,t,y717,,,,iiii,n8i,1i,1i,1,a,,,,0.30352nn28,28,1,,2140, t,t,,,,,ii8ni,1i,1,,,
nn,11byat27.125,,,,,,,iinni,1i,1,
得经验公式为: y,27.125,0.3035t
20.某地区通过抽样调查,收集到人均收入(单位:千元)和平均每百户拥有洗衣机的台数的统计资料如图表二
人均收入x 单位:千元) 1.5 1.8 2.4 3.0 3.5 3.9 4.4 4.8 5.0
洗衣机的台数y 4.8 5.7 7.0 8.3 10.9 12.4 13.1 13.6 15.3
试根据这组实测数据建立变量y与x之间的经验公式,,. y,fx
解: 设所求经验公式为: y,ax,b
为用最小二乘法计算,由已知对数据,得下表
i 2xy x y x iiiii
1 1.5 4.8 2.25 7.2
2 1.8 5.7 3.24 10.26
3 2.4 7.0 5.76 16.8
4 3.0 8.3 9 24.9
5 3.5 10.9 12.25 38.15
6 3.9 12.4 15.21 48.36
7 4.4 13.1 19.36 57.64
8 4.8 13.6 23.04 65.28
9 5.0 15.3 25 76.5
30.3 91.1 115.11 345.09 9
,i,1
带入公式得:
nnn1,30.3,90.1xy,x,y345.09,,,,iiii,n9i,1i,1i,1,a,,,2.9302nn30.3,30.3,1,,2115.11, x,x,,,,,ii9ni,1i,1,,,
nn,11byax0.2569,,,,,,,iinni,1i,1,
得经验公式为: y,2.930x,0.2569
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