答：当底连长为5米，高为2.5米时用料最省。 （三）计算题
cos1⒈
?xx2dx???cos1xd(1x)??sin1x?c
x⒉
?exdx?2?exdx?2ex?c
⒊
?1xlnxdx??1lnxd(lnx)?ln(lnx)?c
⒋?xsin2xdx??12?xd?cos2x???12xcos2x?12?cos2xdx??12xcos2x?14sin2x?c ?e3?lnx1xdx??e1(3?lnx)d(3?lnx)?1e⒌72(3?lnx)1?2 1⒍?1?2x111?2x1?21?2x13?210xe?2xdx??2ex?edx??e?e02?0240??4e?4 22e?e⒎1e2xxe?1ee1?11xlnxdx?2?1lnxdx?2lne2???1e2?1 12?1xdx?2?2???21??44elnxdx??1lnxe?ee⒏
?11x2x?12dx??11e???2e?1 1xx1
三、计算题（每小题11分，共44分） ⒈求下列函数的导数y?：
⑴
y?(xx?3)ex
解:y???xx?3??ex??xx?3??ex??3 ?(x2?3)ex?312x2ex
⑵
y?cotx?x2lnx
解：y???cotx????x2??lnx?x2?lnx????csc2x?x?2xlnx
y?x2⑶lnx
?x2??lnx?x2?解：
y???lnx?2xlnx?xln2x?ln2x
⑷y?cosx?2xx3
x3x3y???cosx?2??x??cosx?2??x??解：
?x(?sinx?2xln2)?3(cosx?2x)?x3?2x4
⑸y?lnx?x2sinx
2?sinx??lnx?x2??sinx??sinx(1解：
y???lnx?x?x?2x)?(lnx?x2)cosxsin2x?sin2x
6
⑹
y?x4?sinxlnx
解：
y???x4????sinx??lnx?sinx?lnx???4x3?sinxx?cosxlnx
⑺y?sinx?x23x
2x2xy???sinx?x??3??sinx?x?解：
?3x(cosx?2x)?(sinx?x2)3xln3???3?3x?232x ⑻
y?extanx?lnx
解：y???ex??tanx?ex?tanx????lnx???extanx?ex1cos2x?x ⒉求下列函数的导数y?：
⑴
y?ex
解：
y???ex????ex??1x?12?1x22xe
⑵y?lncosx
解：y??1cosx??sinx???sinxcos??tanx ⑶
y?xxx
解：y???7??1??x8????788x
??⑷y?sin2x
解：y??2sinx?sinx???2sinx?cosx?2sin2x
⑸y?sinx2
2解：
y??cosx?2x?2xcosx
cosex2⑹
y?
解：y???sinex2?ex2????2xex2sinex2
⑺
y?sinnxcosnx
解：
y???sinnx??cosnx?sinnx?cosnx???nsinn?1xcosxcosnx?nsinnxsin(nx)⑻y?5sinx
解：
y??5sinxln5?cosx?ln5cosx5sinx
⑼y?ecosx
??ecosx??sinx???sinxecosx解：y ⒊在下列方程中，y?y(x)是由方程确定的函数，求y?：
⑴
ycosx?e2y
解：
y?cosx?ysinx?2e2yy? y??ysinxcosx?2e2y
7
⑵
y?cosylnx
解：
y??siny.y?lnx?cosy.1cosx y??yx(1?sinylnx) x2⑶2xsiny?y
解：2xcosy.y??2siny?2yx?x2y?x22yx2xy?2ysinyy2 y?(2xcosy?y2)?y2?2siny y??2xy2cosy?x2
⑷y?x?lny
解：y??y?yy?1 y??y?1 ⑸lnx?ey?y2
解：1x?eyy??2yy? y??1x(2y?ey)
⑹
y2?1?exsiny
解：2yy??excosy.y??siny.ex y??exsiny2y?excosy
⑺ey?ex?y3
解：eyy??ex?3y2y? y??exey?3y2 ⑻y?5x?2y
x5?y?2yln2 y??5x解：y??5lnln51?2yln2
⒋求下列函数的微分dy：（注：dy?y?dx） ⑴y?cotx?cscx
解：y???csc2x?cscxcotx dy?(?1cos2x?cosxsin2x)dx ⑵y?lnxsinx
1sinx?lnxcosx1sinx?lnxcos解：y??xsin2x dy?xxsin2xdx ⑶y?sin2x
解：y??2sinxcosx dy?2sinxcosxdx
⑹y?tanex
3解：y??sec2ex?ex2x3 dy?sece?exdx?exsec2exdx
⒌求下列函数的二阶导数：
⑴
y?x
解：y??1x?12 y???1????1??x?321?3?2???4x222
⑵y?3x
解：y??3xln3 y???ln3?3x?ln3?ln23?3x ⑶y?lnx
解：y??1 y???12
x?x8