機(jī)電工業(yè)出版社高等數(shù)學(xué)第3章作業(yè)參考解答

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§2.1導(dǎo)數(shù)概念

一、填空題

1.設(shè)函數(shù)f(x)在點(diǎn)x0處可導(dǎo)且limx?0f(x0?2x)?f(x0?x)2?1,則f?(x0)?.

32x由于limx?0f(x0?2x)?f(x0?x)f(x0?2x)?f(x0)?f(x0)?f(x0?x)?limx?02x2x?f(x0?2x)?f(x0)f(x0?x)?f(x0)??f(x0?2x)?f(x0)1f(x0?x)?f(x0)??lim???lim??x?0??x?02x2x2x2?x?????f'(x0)?1'32f(x0)?f'(x0)?1?f'(x0)?2232.函數(shù)f(x)在點(diǎn)x0可導(dǎo)是連續(xù)的充分條件，函數(shù)f(x)在點(diǎn)x0處連續(xù)是可導(dǎo)的必要條件.

?3??xx11512?2?(x)?3.⑴()???2.⑵(x)??.⑶?.⑷.???36xx2x6xx?x?二、解：y??x2????2x,y?|x?3?2?3?6,y|x?3?9，則切線(xiàn)方程為：

y?9?6(x?3)即6x?y?9?0，

法線(xiàn)方程為：

1y?9??(x?3)即x?6y?57?0，

6

§2.2函數(shù)的求導(dǎo)法則

一、填空題

1.若曲線(xiàn)y?x?e上一點(diǎn)的切線(xiàn)與軸平行，該點(diǎn)=0。

xxxx2,則?f(0)???0，f?(0)?1。2．設(shè)f(x)?sinx?5解：f?(x)?cosx?2x,f?(0)?cos0?153.已知物體的運(yùn)動(dòng)方程為s?f(t),則物體的加速度a?f??(t).

2x?12x?12x?1?1?14.解：f?(x)?2e,f??(x)?4e,f???(x)?8e,f??(0)?4e,f???(0)?8e

二、

解：y??2?3x?2??3x?2???6?3x?2?２.解：y??2?arctanx??arctanx???３.解：y??2arctanx21?x111???lnlnxlnx?????????ln?lnx?ln?lnx?lnxxlnxln?lnx?４.解：y???1?sinx???sinx?21?sinx??cosx?2sinx?21?sinx2sinx?sinx三、

解：y??f?(x?1)(x?1)??12xf?(x?1)

２.解：y??四、

1??2f?x?f?(x)2??1?f(x)?1?f2(x)?1?f2(x)解：y??4x?11,y???4?2xxx22x2２.解：y??e?2xe?(1?2x2)ex,y???4xex?(1?2x2)2xex?(6x?4x3)ex

2222f?(x),五、解：y??f(x)y???f??(x)f(x)??f?(x)?f2(x)2

§2.3隱函數(shù)及其由參數(shù)方程所確定的函數(shù)的導(dǎo)數(shù)

??x?y?1?,?ln(x?y)一、解:?????222x?2yy?x2?y2?2x?1?y??y??2222x?y2y?x?y二、

x?2(3?x)41?ln(x?2)?4ln(3?x)?5ln(x?1)解：lny?ln(x?1)52y?145????y??y2(x?2)x?3x?1x?2(3?x)4(x?1)5?145????2(x?2)x?3x?1?

???143?544(3?x)?4x?2(3?x)?2x?2?(x?1)?5x?2(3?x)(x?1)?解法2：y???(x?1)10?143?4(3?x)?4x?2(3?x)(x?1)?5x?2(3?x)?2x?2????6(x?1)?(3?x)4?8(x?2)(3?x)3?(x?1)?10(x?2)(3?x)4???62x?2(x?1)(3?x)4(x?1)?8(x?2)(3?x)3(x?1)?10(x?2)(3?x)4?2x?2(x?1)6?(3?x)3?(3?x)(x?1)?8(x?2)(x?1)?10(x?2)(3?x)?2x?2(x?1)6

(3?x)3(x2?32x?73)?2x?2(x?1)6２.y??lnx?解法１：

xlny?xln?lnx?,y?1111???ln?lnx??x?ln?lnx???y??(lnx)x?ln?lnx??ylnxxlnxlnx???1?x?x?xlnlnx(xlnlnx)???lnx??lnlnx?解法２：y????lnx???e???lnx???yt'?a(cost?cost?tsint)?atsintdxcost???cott三、解：?'dysintx?a(?sint?sint?tcost)?atcost?td2x?cott???csc2t1d2x182????,????

3?1dy2atsintatsintatsin3tdy2t?3?3?aa4422四、解：x?y?e?y???0,1?y??y?ey?0?y??111?,?y??x?1y?01?ey1?e02在點(diǎn)（１，０）的切線(xiàn)方程為x?2y?1?02.

dy3t23tdy?3,x|t?2?5,y|t?2?8，則切線(xiàn)方程為y?8?3(x?5)即??，解：

dxt?2dx2t21y?3x?7，法線(xiàn)方程為y?8??(x?5)?x?3y?29?0

3

§2.5函數(shù)的微分

一、選擇題

1.若y?f(x)在x0處可微，則以下結(jié)論中不正確的是f(x)在x0處（B）A．連續(xù)，B.不一定連續(xù)，C.可導(dǎo)，D.極限存在２.函數(shù)y?f(x)在x處可導(dǎo)是函數(shù)y?f(x)在x處可微的（B）

A．必要條件，B.充分必要條件，C.充分條件，D.非充分非必要條件３.函數(shù)y?f(u)在u處可導(dǎo)，且u?sinx則dy?（B）

A．f?(sinx)dx，B.f?(sinx)cosxdx，C.?f(sinx)??d(sinx)，D.?f(sinx)??cosxdx二、填空:

1.d?ax?c??adx；2.d??cos?x?c??sin?dx；

???3.d?ln(1?x)?c??5.d2x?c1?1?dx；4.d??e?2x?c??e?2xdx；1?x?2??1????1?1?dx；6.d?tan3x?c??sec23xdx.x?3?1?x2?x三、（1）解:dy?

（2）

x21?x1?x2dx?1?1?x?322dx

解:dy?2tan(1?2x)sec(1?2x)4xdx?8xtan(1?2x)sec(1?2x)dx3.解：d(xy)?dex?y222222,ydx?xdy?ex?y(dx?dy)?ex?ydx?ex?ydy?

y?ex?yy?xyy(1?x)dy?x?ydx?dx?dx

e?xxy?xx(y?1)y(1?x)y?ex?ydxdx或dy?則dy?x?yx(y?1)e?x三、

解：f(x)?sinx,x0?30?0?6,?x??10???180

sin290?sin?6?cos?6(??180)?13??(?)?0.4849221802.解：3996?31000?4?1031?41000f(x)?31?x，1031?414?10[31?(?)]?10[1?0.0013]?9.987

23100010003(1?0)解法2：f(x)?3x，f?(x)?13x32，x0?1000,?x?996?1000??4

3996?31000?13100032(?4)?9.9867

其次章測(cè)試題

一、填空題1.由于y??e??arctanx?'??e?arctanx?arctanx??e?arctan?'x111?x2x??1e?arctan2x?1?x?x?dy??1e?arctanxdx

2x?1?x??x?1?t22．曲線(xiàn)?在t?2處的切線(xiàn)方程為y?3x?73?y?t???2?dsinx?C?cosxdx?3.?22???dy3t23tdy3?2??,??3解：

dx2t2dxt?22?x|t?2?5，y?8?3(x?5)?y?3x?7?y|?8?t?2二、選擇題

1.設(shè)f(x)在x?a的某鄰域內(nèi)有定義，則f(x)在x?a處可導(dǎo)的一個(gè)充分條件是（D）存在.由于limh?