2010年湖南师范大学高等数学试题
一， 计算题
1已知![](https://wkrtcs.bdimg.com/rtcs/image?w=132&md5sum=24560a60fee11ca3a97d5937185ace1e&sign=2c9e234a84&rtcs_flag=1&rtcs_ver=3.2&l=webapp&bucketNum=26&ipr={"t":"img","w":"132","h":"45","dataType":"gif","c":"word/media/image1.png"})
，求
2讨论函数
![](https://wkrtcs.bdimg.com/rtcs/image?w=148&md5sum=24560a60fee11ca3a97d5937185ace1e&sign=2c9e234a84&rtcs_flag=1&rtcs_ver=3.2&l=webapp&bucketNum=26&ipr={"t":"img","w":"148","h":"80","dataType":"gif","c":"word/media/image3.png"})
在![](https://wkrtcs.bdimg.com/rtcs/image?w=37&md5sum=24560a60fee11ca3a97d5937185ace1e&sign=2c9e234a84&rtcs_flag=1&rtcs_ver=3.2&l=webapp&bucketNum=26&ipr={"t":"img","w":"37","h":"19","dataType":"gif","c":"word/media/image4.png"})
点的可微性
3设![](https://wkrtcs.bdimg.com/rtcs/image?w=85&md5sum=24560a60fee11ca3a97d5937185ace1e&sign=2c9e234a84&rtcs_flag=1&rtcs_ver=3.2&l=webapp&bucketNum=26&ipr={"t":"img","w":"85","h":"24","dataType":"gif","c":"word/media/image5.png"})
，求极限
4，设函数![](https://wkrtcs.bdimg.com/rtcs/image?w=73&md5sum=24560a60fee11ca3a97d5937185ace1e&sign=2c9e234a84&rtcs_flag=1&rtcs_ver=3.2&l=webapp&bucketNum=26&ipr={"t":"img","w":"73","h":"27","dataType":"gif","c":"word/media/image7.png"})
，由方程确定，其中具有一阶连续的导数，求
5计算![](https://wkrtcs.bdimg.com/rtcs/image?w=108&md5sum=24560a60fee11ca3a97d5937185ace1e&sign=2c9e234a84&rtcs_flag=1&rtcs_ver=3.2&l=webapp&bucketNum=26&ipr={"t":"img","w":"108","h":"52","dataType":"gif","c":"word/media/image11.png"})
6，将函数![](https://wkrtcs.bdimg.com/rtcs/image?w=120&md5sum=24560a60fee11ca3a97d5937185ace1e&sign=2c9e234a84&rtcs_flag=1&rtcs_ver=3.2&l=webapp&bucketNum=26&ipr={"t":"img","w":"120","h":"27","dataType":"gif","c":"word/media/image12.png"})
展为的幂级数，并指出其收敛域
二， 应用题
1， 在椭球面![](https://wkrtcs.bdimg.com/rtcs/image?w=108&md5sum=24560a60fee11ca3a97d5937185ace1e&sign=2c9e234a84&rtcs_flag=1&rtcs_ver=3.2&l=webapp&bucketNum=26&ipr={"t":"img","w":"108","h":"24","dataType":"gif","c":"word/media/image14.png"})
上求距离平面的最近点和最远点的距离，最远点和最远距离
2， 一质量为![](https://wkrtcs.bdimg.com/rtcs/image?w=17&md5sum=24560a60fee11ca3a97d5937185ace1e&sign=2c9e234a84&rtcs_flag=1&rtcs_ver=3.2&l=webapp&bucketNum=26&ipr={"t":"img","w":"17","h":"15","dataType":"gif","c":"word/media/image16.png"})
的船以速度行驶，时动力被关闭，假设水的阻力正比于，其中是瞬时速度，为常数，求速度与经过距离的函数关系
三， 证明题
设函数![](https://wkrtcs.bdimg.com/rtcs/image?w=39&md5sum=24560a60fee11ca3a97d5937185ace1e&sign=2c9e234a84&rtcs_flag=1&rtcs_ver=3.2&l=webapp&bucketNum=26&ipr={"t":"img","w":"39","h":"27","dataType":"gif","c":"word/media/image22.png"})
对一切实数适合且，，证明：对一切自然数，有
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