2011-2012(1)高数a试题及答案(实验)

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1、2011-2012学年第一学期期末考试试题（A卷）课程名称高等数学总分得分一、单项选择题（共15分每小题3分）1.下列函数在定义域内（）是奇函数2x−x（A）f()xxx=()ln；（B）fx()=+ee；sinx⎛⎞ax−（C）fx()=；（D）fx()=ln⎜⎟（>0)a.x⎝⎠ax+2sin()x−12.极限lim=（）x→1x−11（A）1；（B）0；（C）2；（D）.2x3.x=π是函数fx()=的()．tanx（A）可去间断点；（B）跳跃间断点；（C）无穷间断点；（D）振荡间断点.−−xx4、若f()x的一个原函数

2、为Fx()，则∫e(fe)dx为（）x−x（A）F(e)+C；（B）−FC(e)+；−x−xF(e)（C）F(e)+C；（D）+C.x5、在MATLAB中，求函数sin(2)x在[0,1]上的定积分的命令是（）（A）int(sin(2),,0,1)xx；（B）int(sin2,,0,1)xx；（C）int(sin[2∗x],0,1)；（D）int(sin(2∗xx),,0,1).1、D；2、C；3、C；4、B；5、D.得分二、填空题（共15分每小题3分）21、函数f()xx=−+1lnx的定义域为.2⎧⎪x=+1t2、曲线⎨在

3、t=2处的切线方程为.3⎪⎩yt=第1页共7页−arcsinx3、设函数y=e，则dy=d(arcsinx)．124、定积分∫xsinxdx=.−1x∫f()tdt05、若f()x为连续函数，且已知ff()00=,0′()=2，则lim的值为.2x→0x−arcsinx1、(0,1]；2、yx=37−；3、−e；4、0；5、1.三、求解下列各题（共5小题，每小题8分，共40分）得分x⎛⎞x1．求极限lim⎜⎟.x→∞⎝⎠1+xx⎛⎞x+−11解：法一：原式=lim⎜⎟······························

4、··········································3分x→0⎝⎠1+x−x−+⋅(1x)⎛⎞11+x=−lim1⎜⎟···············································································6分x→0⎝⎠1+x1=····································································································

5、·············8分e−x⎛⎞x+1法二：原式=lim⎜⎟······················································································3分x→0⎝⎠xx⋅−(1)⎛⎞1=+lim1⎜⎟··························································································6分x→0⎝⎠x1=·····················

6、····························································································8分e得分2ydydy2.设函数yfx=()由方程xy+=ee确定，求,.2dxdxx=0x=0ddyyy解：等式两边求导得yx++=e0·························································3分ddxxdyy所以=−，··································

7、·················································4分ydexx+第2页共7页d1y因为x=0，y=1，所以=−····························································5分dexx=0yyddyy2(e)xy+−(1e+⋅)dyddxx所以=−22yd(xx+e)2dy2所以x=0，=−e·································································

8、···········8分2dx得分2x3.证明：当x>0时，ln(1+>−xx).22x证明：设fx()ln(1)=+−+xx，·····································································2分22