高等数学题库第02章(导数与微分)(4)

1010?2?tanx???2?sinx?3. limx?0sinx? . 10?210

2??x?1?t4.曲线?,在t?2处的切线方程为 . 3x?y?7?0 3??y?t5.已知曲线f?x??xn在点(1,1)处的切线与x轴的交点为??n,0?,

1则limf??n?? . n??e6.设函数y?y?x?由方程ex?ydyysin?xy??ex?y? . x?y ?cos?xy??0确定,则dxe?xsin?xy??x?t?ln?1?t??6t?5??t?1? d2y?7.设函数y?y?x?由参数方程?所确定,则 . 32tdx2?y?t?t3?dy?3x?2?'28.已知y?f? ? . ?,f(x)?arctanx,则

4dxx?0?3x?2?1?x??1?2?n!

9.设f?x??,则f?n??x?? .

1?x?1?x?n?1n三．求下列函数的二阶导数

1.y?x3lnx x?6lnx?5?

2sin2x1?2.y?cos2x?lnx ??2cos2x?lnx??2cos2xx??x? ?3.y?ln1?x ??2?21?x2?1?x??22?

4.y?e?x?sinx ?2e?xcosx 5.设y?y?x?由方程e?xy?e所确定,求y.

y''y?x?e?''y2?yey???2?x?ey?? ??6.设y?y?x?由方程y?1?xe所确定,求y?0? y?0??y''2e2y?1?xe?y3?2e2

x?0y?0d2y7.设f?x?存在,求下列函数y的二阶导数2.

dx''(1)y?fx2 f''x2?(2x)2?2f'(x2)

????(2)y?f(lnx)

1fx2?''?lnx??f'(lnx)?

f''?x?f'?x?'(3)y?arctan?f?x?? ???x? ??2fx?f2221?f?x?1?f?x???四．计算题

?x?2?1?cos???1.设曲线参数方程为?,求曲线在??处的切线方程.

4?y?4sin? y?2x?4?42

‘?d2yd2y1?x?f?t?2.设?,其中二阶可导,求. ??ft?22‘’’dxdxf?t???y?tf?t??f?t?3.计算1.05的近似值. 1.05?1.025

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